Theorem ipfeq 21635

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	⊢ , = (·𝑖‘𝑊)
ipffval.3	⊢ · = (·if‘𝑊)

Assertion

Ref	Expression
ipfeq	⊢ ( , Fn (𝑉 × 𝑉) → · = , )

Proof of Theorem ipfeq

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ipffval.1	. . 3 ⊢ 𝑉 = (Base‘𝑊)
2		ipffval.2	. . 3 ⊢ , = (·𝑖‘𝑊)
3		ipffval.3	. . 3 ⊢ · = (·if‘𝑊)
4	1, 2, 3	ipffval 21633	. 2 ⊢ · = (𝑥 ∈ 𝑉, 𝑦 ∈ 𝑉 ↦ (𝑥 , 𝑦))
5		fnov 7547	. . 3 ⊢ ( , Fn (𝑉 × 𝑉) ↔ , = (𝑥 ∈ 𝑉, 𝑦 ∈ 𝑉 ↦ (𝑥 , 𝑦)))
6	5	biimpi 216	. 2 ⊢ ( , Fn (𝑉 × 𝑉) → , = (𝑥 ∈ 𝑉, 𝑦 ∈ 𝑉 ↦ (𝑥 , 𝑦)))
7	4, 6	eqtr4id 2788	1 ⊢ ( , Fn (𝑉 × 𝑉) → · = , )

Syntax hints:   → wi 4   = wceq 1539   × cxp 5665   Fn wfn 6537  ‘cfv 6542  (class class class)co 7414   ∈ cmpo 7416  Basecbs 17230  ·_𝑖cip 17282  ·_ifcipf 21610

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2706  ax-sep 5278  ax-nul 5288  ax-pow 5347  ax-pr 5414  ax-un 7738

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2808  df-nfc 2884  df-ne 2932  df-ral 3051  df-rex 3060  df-rab 3421  df-v 3466  df-sbc 3773  df-csb 3882  df-dif 3936  df-un 3938  df-in 3940  df-ss 3950  df-nul 4316  df-if 4508  df-pw 4584  df-sn 4609  df-pr 4611  df-op 4615  df-uni 4890  df-iun 4975  df-br 5126  df-opab 5188  df-mpt 5208  df-id 5560  df-xp 5673  df-rel 5674  df-cnv 5675  df-co 5676  df-dm 5677  df-rn 5678  df-res 5679  df-ima 5680  df-iota 6495  df-fun 6544  df-fn 6545  df-f 6546  df-fv 6550  df-ov 7417  df-oprab 7418  df-mpo 7419  df-1st 7997  df-2nd 7998  df-ipf 21612

This theorem is referenced by: (None)