To gyronaut, my blessing for your thinking.
Vegetation already combines water and carbon dioxide in a process called photosynthesis to produce carbohydrates, in the presence of chlorophyll in several steps, as in the following:
nCO
2 + nH
2O + solar energy -> (CH
2O)
n + nO
2
when n=6 we have glucose.
As for fuel (hydrocarbon) synthesis, this involves the addition of hydrogen to CO, or to CO
2, with water as a by-product.
Hydrogen can be obtained from water electrolysis, from the carbon-steam reaction, the water-gas shift reaction, and from acids reacting with metals; otherwise, from hydrocarbons by catalytic reforming, by partial oxidation, or by thermal decomposition, the last three options should be discarded as far as hydrocarbons are meant to be produced, not destroyed.
CO, as obtained from decomposition of CO
2, needs a very high activation energy of about 460 kJ/mol. However, the carbon-steam reaction mentioned above provides CO as a co-product to hydrogen by:
C + H
2O -> CO + H
2
The fuel-making reaction would be then represented by:
m CO + (2
m+1) H
2 -> C
mH
2m+2 +
m H
2O
CO + hydrogen can also render monohydric alcohols and glycols.
From what you said, one doesn't need CO if one has CO
2.
As for the reaction of CO
2 with hydrogen, one may obtain, for example, methane (the main component of natural gas):
CO
2 + 4H
2 -> CH
4 +2 H
2O.
In short, by splitting water with electricity as you suggest (not so cheap nor simple) you get hydrogen (and oxygen as a useful by-product). Since carbon dioxide is available, you should think of combining it with hydrogen, and voila, you'd have a fuel, with pure water as a by-product.
As a matter of fact this kind of (catalysed) reaction is being actively studied many years now, for the potential production of fuels.