Theorem ipfval 21645

Description: The inner product operation as a function. (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
ipfval	⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → (𝑋 · 𝑌) = (𝑋 , 𝑌))

Proof of Theorem ipfval

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

Step	Hyp	Ref	Expression
1		oveq12 7433	. 2 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → (𝑥 , 𝑦) = (𝑋 , 𝑌))
2		ipffval.1	. . 3 ⊢ 𝑉 = (Base‘𝑊)
3		ipffval.2	. . 3 ⊢ , = (·𝑖‘𝑊)
4		ipffval.3	. . 3 ⊢ · = (·if‘𝑊)
5	2, 3, 4	ipffval 21644	. 2 ⊢ · = (𝑥 ∈ 𝑉, 𝑦 ∈ 𝑉 ↦ (𝑥 , 𝑦))
6		ovex 7457	. 2 ⊢ (𝑋 , 𝑌) ∈ V
7	1, 5, 6	ovmpoa 7581	1 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → (𝑋 · 𝑌) = (𝑋 , 𝑌))

Syntax hints:   → wi 4   ∧ wa 394   = wceq 1534   ∈ wcel 2099  ‘cfv 6554  (class class class)co 7424  Basecbs 17213  ·_𝑖cip 17271  ·_ifcipf 21621

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-sep 5304  ax-nul 5311  ax-pow 5369  ax-pr 5433  ax-un 7746

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3464  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4326  df-if 4534  df-pw 4609  df-sn 4634  df-pr 4636  df-op 4640  df-uni 4914  df-iun 5003  df-br 5154  df-opab 5216  df-mpt 5237  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-iota 6506  df-fun 6556  df-fn 6557  df-f 6558  df-fv 6562  df-ov 7427  df-oprab 7428  df-mpo 7429  df-1st 8003  df-2nd 8004  df-ipf 21623

This theorem is referenced by:  ipcn  25265  cnmpt1ip  25266  cnmpt2ip  25267