Theorem ipfeq 20316

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		fnov 7000	. . 3 ⊢ ( , Fn (𝑉 × 𝑉) ↔ , = (𝑥 ∈ 𝑉, 𝑦 ∈ 𝑉 ↦ (𝑥 , 𝑦)))
2	1	biimpi 208	. 2 ⊢ ( , Fn (𝑉 × 𝑉) → , = (𝑥 ∈ 𝑉, 𝑦 ∈ 𝑉 ↦ (𝑥 , 𝑦)))
3		ipffval.1	. . 3 ⊢ 𝑉 = (Base‘𝑊)
4		ipffval.2	. . 3 ⊢ , = (·𝑖‘𝑊)
5		ipffval.3	. . 3 ⊢ · = (·if‘𝑊)
6	3, 4, 5	ipffval 20314	. 2 ⊢ · = (𝑥 ∈ 𝑉, 𝑦 ∈ 𝑉 ↦ (𝑥 , 𝑦))
7	2, 6	syl6reqr 2850	1 ⊢ ( , Fn (𝑉 × 𝑉) → · = , )

Syntax hints:   → wi 4   = wceq 1653   × cxp 5308   Fn wfn 6094  ‘cfv 6099  (class class class)co 6876   ↦ cmpt2 6878  Basecbs 16181  ·_𝑖cip 16269  ·_ifcipf 20291

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2354  ax-ext 2775  ax-sep 4973  ax-nul 4981  ax-pow 5033  ax-pr 5095  ax-un 7181

This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2590  df-eu 2607  df-clab 2784  df-cleq 2790  df-clel 2793  df-nfc 2928  df-ne 2970  df-ral 3092  df-rex 3093  df-rab 3096  df-v 3385  df-sbc 3632  df-csb 3727  df-dif 3770  df-un 3772  df-in 3774  df-ss 3781  df-nul 4114  df-if 4276  df-pw 4349  df-sn 4367  df-pr 4369  df-op 4373  df-uni 4627  df-iun 4710  df-br 4842  df-opab 4904  df-mpt 4921  df-id 5218  df-xp 5316  df-rel 5317  df-cnv 5318  df-co 5319  df-dm 5320  df-rn 5321  df-res 5322  df-ima 5323  df-iota 6062  df-fun 6101  df-fn 6102  df-f 6103  df-fv 6107  df-ov 6879  df-oprab 6880  df-mpt2 6881  df-1st 7399  df-2nd 7400  df-ipf 20293

This theorem is referenced by: (None)