Theorem ipfeq 21675

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 21673	. 2 ⊢ · = (𝑥 ∈ 𝑉, 𝑦 ∈ 𝑉 ↦ (𝑥 , 𝑦))
5		fnov 7516	. . 3 ⊢ ( , Fn (𝑉 × 𝑉) ↔ , = (𝑥 ∈ 𝑉, 𝑦 ∈ 𝑉 ↦ (𝑥 , 𝑦)))
6	5	biimpi 218	. 2 ⊢ ( , Fn (𝑉 × 𝑉) → , = (𝑥 ∈ 𝑉, 𝑦 ∈ 𝑉 ↦ (𝑥 , 𝑦)))
7	4, 6	eqtr4id 2810	1 ⊢ ( , Fn (𝑉 × 𝑉) → · = , )

Syntax hints:   → wi 4   = wceq 1554   × cxp 5638   Fn wfn 6505  ‘cfv 6510  (class class class)co 7385   ∈ cmpo 7387  Basecbs 17221  ·_𝑖cip 17267  ·_ifcipf 21650

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1809  ax-4 1823  ax-5 1924  ax-6 1981  ax-7 2022  ax-8 2138  ax-9 2146  ax-10 2169  ax-11 2185  ax-12 2206  ax-ext 2728  ax-sep 5240  ax-nul 5250  ax-pow 5316  ax-pr 5384  ax-un 7707

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 857  df-3an 1097  df-tru 1557  df-fal 1567  df-ex 1794  df-nf 1798  df-sb 2085  df-mo 2560  df-eu 2590  df-clab 2735  df-cleq 2748  df-clel 2831  df-nfc 2905  df-ne 2952  df-ral 3071  df-rex 3081  df-rab 3409  df-v 3450  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4281  df-if 4475  df-pw 4551  df-sn 4577  df-pr 4579  df-op 4583  df-uni 4860  df-iun 4945  df-br 5095  df-opab 5157  df-mpt 5176  df-id 5535  df-xp 5646  df-rel 5647  df-cnv 5648  df-co 5649  df-dm 5650  df-rn 5651  df-res 5652  df-ima 5653  df-iota 6466  df-fun 6512  df-fn 6513  df-f 6514  df-fv 6518  df-ov 7388  df-oprab 7389  df-mpo 7390  df-1st 7959  df-2nd 7960  df-ipf 21652

This theorem is referenced by: (None)