Theorem ipfeq 20788

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 7276	. . 3 ⊢ ( , Fn (𝑉 × 𝑉) ↔ , = (𝑥 ∈ 𝑉, 𝑦 ∈ 𝑉 ↦ (𝑥 , 𝑦)))
2	1	biimpi 218	. 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 20786	. 2 ⊢ · = (𝑥 ∈ 𝑉, 𝑦 ∈ 𝑉 ↦ (𝑥 , 𝑦))
7	2, 6	syl6reqr 2875	1 ⊢ ( , Fn (𝑉 × 𝑉) → · = , )

Syntax hints:   → wi 4   = wceq 1533   × cxp 5547   Fn wfn 6344  ‘cfv 6349  (class class class)co 7150   ∈ cmpo 7152  Basecbs 16477  ·_𝑖cip 16564  ·_ifcipf 20763

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-sep 5195  ax-nul 5202  ax-pow 5258  ax-pr 5321  ax-un 7455

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3496  df-sbc 3772  df-csb 3883  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4467  df-pw 4540  df-sn 4561  df-pr 4563  df-op 4567  df-uni 4832  df-iun 4913  df-br 5059  df-opab 5121  df-mpt 5139  df-id 5454  df-xp 5555  df-rel 5556  df-cnv 5557  df-co 5558  df-dm 5559  df-rn 5560  df-res 5561  df-ima 5562  df-iota 6308  df-fun 6351  df-fn 6352  df-f 6353  df-fv 6357  df-ov 7153  df-oprab 7154  df-mpo 7155  df-1st 7683  df-2nd 7684  df-ipf 20765

This theorem is referenced by: (None)