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Theorem ipfval 20495
Description: The inner product operation as a function. (Contributed by Mario Carneiro, 14-Aug-2015.)
Hypotheses
Ref Expression
ipffval.1 𝑉 = (Base‘𝑊)
ipffval.2 , = (·𝑖𝑊)
ipffval.3 · = (·if𝑊)
Assertion
Ref Expression
ipfval ((𝑋𝑉𝑌𝑉) → (𝑋 · 𝑌) = (𝑋 , 𝑌))

Proof of Theorem ipfval
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 oveq12 6985 . 2 ((𝑥 = 𝑋𝑦 = 𝑌) → (𝑥 , 𝑦) = (𝑋 , 𝑌))
2 ipffval.1 . . 3 𝑉 = (Base‘𝑊)
3 ipffval.2 . . 3 , = (·𝑖𝑊)
4 ipffval.3 . . 3 · = (·if𝑊)
52, 3, 4ipffval 20494 . 2 · = (𝑥𝑉, 𝑦𝑉 ↦ (𝑥 , 𝑦))
6 ovex 7008 . 2 (𝑋 , 𝑌) ∈ V
71, 5, 6ovmpoa 7121 1 ((𝑋𝑉𝑌𝑉) → (𝑋 · 𝑌) = (𝑋 , 𝑌))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 387   = wceq 1507  wcel 2050  cfv 6188  (class class class)co 6976  Basecbs 16339  ·𝑖cip 16426  ·ifcipf 20471
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-13 2301  ax-ext 2751  ax-sep 5060  ax-nul 5067  ax-pow 5119  ax-pr 5186  ax-un 7279
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-mo 2547  df-eu 2584  df-clab 2760  df-cleq 2772  df-clel 2847  df-nfc 2919  df-ne 2969  df-ral 3094  df-rex 3095  df-rab 3098  df-v 3418  df-sbc 3683  df-csb 3788  df-dif 3833  df-un 3835  df-in 3837  df-ss 3844  df-nul 4180  df-if 4351  df-pw 4424  df-sn 4442  df-pr 4444  df-op 4448  df-uni 4713  df-iun 4794  df-br 4930  df-opab 4992  df-mpt 5009  df-id 5312  df-xp 5413  df-rel 5414  df-cnv 5415  df-co 5416  df-dm 5417  df-rn 5418  df-res 5419  df-ima 5420  df-iota 6152  df-fun 6190  df-fn 6191  df-f 6192  df-fv 6196  df-ov 6979  df-oprab 6980  df-mpo 6981  df-1st 7501  df-2nd 7502  df-ipf 20473
This theorem is referenced by:  ipcn  23552  cnmpt1ip  23553  cnmpt2ip  23554
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