Theorem ipffn 21577

Description: The inner product operation is a function. (Contributed by Mario Carneiro, 20-Sep-2015.)

Hypotheses

Ref	Expression
ipffn.1	⊢ 𝑉 = (Base‘𝑊)
ipffn.2	⊢ , = (·if‘𝑊)

Assertion

Ref	Expression
ipffn	⊢ , Fn (𝑉 × 𝑉)

Proof of Theorem ipffn

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

Step	Hyp	Ref	Expression
1		ipffn.1	. . 3 ⊢ 𝑉 = (Base‘𝑊)
2		eqid 2728	. . 3 ⊢ (·𝑖‘𝑊) = (·𝑖‘𝑊)
3		ipffn.2	. . 3 ⊢ , = (·if‘𝑊)
4	1, 2, 3	ipffval 21574	. 2 ⊢ , = (𝑥 ∈ 𝑉, 𝑦 ∈ 𝑉 ↦ (𝑥(·𝑖‘𝑊)𝑦))
5		ovex 7448	. 2 ⊢ (𝑥(·𝑖‘𝑊)𝑦) ∈ V
6	4, 5	fnmpoi 8069	1 ⊢ , Fn (𝑉 × 𝑉)

Syntax hints:   = wceq 1534   × cxp 5671   Fn wfn 6538  ‘cfv 6543  (class class class)co 7415  Basecbs 17174  ·_𝑖cip 17232  ·_ifcipf 21551

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 2699  ax-sep 5294  ax-nul 5301  ax-pow 5360  ax-pr 5424  ax-un 7735

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2937  df-ral 3058  df-rex 3067  df-rab 3429  df-v 3472  df-sbc 3776  df-csb 3891  df-dif 3948  df-un 3950  df-in 3952  df-ss 3962  df-nul 4320  df-if 4526  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4905  df-iun 4994  df-br 5144  df-opab 5206  df-mpt 5227  df-id 5571  df-xp 5679  df-rel 5680  df-cnv 5681  df-co 5682  df-dm 5683  df-rn 5684  df-res 5685  df-ima 5686  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-fv 6551  df-ov 7418  df-oprab 7419  df-mpo 7420  df-1st 7988  df-2nd 7989  df-ipf 21553

This theorem is referenced by: (None)