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Theorem ipfeq 20789
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 fnov 7272 . . 3 ( , Fn (𝑉 × 𝑉) ↔ , = (𝑥𝑉, 𝑦𝑉 ↦ (𝑥 , 𝑦)))
21biimpi 219 . 2 ( , Fn (𝑉 × 𝑉) → , = (𝑥𝑉, 𝑦𝑉 ↦ (𝑥 , 𝑦)))
3 ipffval.1 . . 3 𝑉 = (Base‘𝑊)
4 ipffval.2 . . 3 , = (·𝑖𝑊)
5 ipffval.3 . . 3 · = (·if𝑊)
63, 4, 5ipffval 20787 . 2 · = (𝑥𝑉, 𝑦𝑉 ↦ (𝑥 , 𝑦))
72, 6syl6reqr 2878 1 ( , Fn (𝑉 × 𝑉) → · = , )
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1538   × cxp 5541   Fn wfn 6339  cfv 6344  (class class class)co 7146  cmpo 7148  Basecbs 16481  ·𝑖cip 16568  ·ifcipf 20764
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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5190  ax-nul 5197  ax-pow 5254  ax-pr 5318  ax-un 7452
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-ral 3138  df-rex 3139  df-rab 3142  df-v 3482  df-sbc 3759  df-csb 3867  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-pw 4524  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4826  df-iun 4908  df-br 5054  df-opab 5116  df-mpt 5134  df-id 5448  df-xp 5549  df-rel 5550  df-cnv 5551  df-co 5552  df-dm 5553  df-rn 5554  df-res 5555  df-ima 5556  df-iota 6303  df-fun 6346  df-fn 6347  df-f 6348  df-fv 6352  df-ov 7149  df-oprab 7150  df-mpo 7151  df-1st 7681  df-2nd 7682  df-ipf 20766
This theorem is referenced by: (None)
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