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Theorem ipfeq 21039
Description: If the inner product operation is already a function, the functionalization of it is equal to the original operation. (Contributed by Mario Carneiro, 14-Aug-2015.)
Hypotheses
Ref Expression
ipffval.1 𝑉 = (Base‘𝑊)
ipffval.2 , = (·𝑖𝑊)
ipffval.3 · = (·if𝑊)
Assertion
Ref Expression
ipfeq ( , Fn (𝑉 × 𝑉) → · = , )

Proof of Theorem ipfeq
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ipffval.1 . . 3 𝑉 = (Base‘𝑊)
2 ipffval.2 . . 3 , = (·𝑖𝑊)
3 ipffval.3 . . 3 · = (·if𝑊)
41, 2, 3ipffval 21037 . 2 · = (𝑥𝑉, 𝑦𝑉 ↦ (𝑥 , 𝑦))
5 fnov 7483 . . 3 ( , Fn (𝑉 × 𝑉) ↔ , = (𝑥𝑉, 𝑦𝑉 ↦ (𝑥 , 𝑦)))
65biimpi 215 . 2 ( , Fn (𝑉 × 𝑉) → , = (𝑥𝑉, 𝑦𝑉 ↦ (𝑥 , 𝑦)))
74, 6eqtr4id 2795 1 ( , Fn (𝑉 × 𝑉) → · = , )
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1541   × cxp 5629   Fn wfn 6488  cfv 6493  (class class class)co 7353  cmpo 7355  Basecbs 17075  ·𝑖cip 17130  ·ifcipf 21014
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-sep 5254  ax-nul 5261  ax-pow 5318  ax-pr 5382  ax-un 7668
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2887  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3406  df-v 3445  df-sbc 3738  df-csb 3854  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4281  df-if 4485  df-pw 4560  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4864  df-iun 4954  df-br 5104  df-opab 5166  df-mpt 5187  df-id 5529  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6445  df-fun 6495  df-fn 6496  df-f 6497  df-fv 6501  df-ov 7356  df-oprab 7357  df-mpo 7358  df-1st 7917  df-2nd 7918  df-ipf 21016
This theorem is referenced by: (None)
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