Theorem ipfval 20934

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 7325	. 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 20933	. 2 ⊢ · = (𝑥 ∈ 𝑉, 𝑦 ∈ 𝑉 ↦ (𝑥 , 𝑦))
6		ovex 7349	. 2 ⊢ (𝑋 , 𝑌) ∈ V
7	1, 5, 6	ovmpoa 7469	1 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → (𝑋 · 𝑌) = (𝑋 , 𝑌))

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1540   ∈ wcel 2105  ‘cfv 6465  (class class class)co 7316  Basecbs 16986  ·_𝑖cip 17041  ·_ifcipf 20910

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2707  ax-sep 5237  ax-nul 5244  ax-pow 5302  ax-pr 5366  ax-un 7629

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3404  df-v 3442  df-sbc 3726  df-csb 3842  df-dif 3899  df-un 3901  df-in 3903  df-ss 3913  df-nul 4267  df-if 4471  df-pw 4546  df-sn 4571  df-pr 4573  df-op 4577  df-uni 4850  df-iun 4938  df-br 5087  df-opab 5149  df-mpt 5170  df-id 5506  df-xp 5613  df-rel 5614  df-cnv 5615  df-co 5616  df-dm 5617  df-rn 5618  df-res 5619  df-ima 5620  df-iota 6417  df-fun 6467  df-fn 6468  df-f 6469  df-fv 6473  df-ov 7319  df-oprab 7320  df-mpo 7321  df-1st 7877  df-2nd 7878  df-ipf 20912

This theorem is referenced by:  ipcn  24490  cnmpt1ip  24491  cnmpt2ip  24492