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Theorem ipfeq 19814
 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 6666 . . 3 ( , Fn (𝑉 × 𝑉) ↔ , = (𝑥𝑉, 𝑦𝑉 ↦ (𝑥 , 𝑦)))
21biimpi 205 . 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 19812 . 2 · = (𝑥𝑉, 𝑦𝑉 ↦ (𝑥 , 𝑦))
72, 6syl6reqr 2663 1 ( , Fn (𝑉 × 𝑉) → · = , )
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1475   × cxp 5036   Fn wfn 5799  ‘cfv 5804  (class class class)co 6549   ↦ cmpt2 6551  Basecbs 15695  ·𝑖cip 15773  ·ifcipf 19789 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-1st 7059  df-2nd 7060  df-ipf 19791 This theorem is referenced by: (None)
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