Theorem ipffn 21865

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 2760	. . 3 ⊢ (·𝑖‘𝑊) = (·𝑖‘𝑊)
3		ipffn.2	. . 3 ⊢ , = (·if‘𝑊)
4	1, 2, 3	ipffval 21862	. 2 ⊢ , = (𝑥 ∈ 𝑉, 𝑦 ∈ 𝑉 ↦ (𝑥(·𝑖‘𝑊)𝑦))
5		ovex 7447	. 2 ⊢ (𝑥(·𝑖‘𝑊)𝑦) ∈ V
6	4, 5	fnmpoi 8068	1 ⊢ , Fn (𝑉 × 𝑉)

This proof depends on syntax axioms:   = wceq 1570   × cxp 5653   Fn wfn 6528  ‘cfv 6533  (class class class)co 7414  Basecbs 17302  ·_𝑖cip 17348  ·_ifcipf 21839

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-sep 5251  ax-nul 5263  ax-pow 5330  ax-pr 5398  ax-un 7737

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-iota 6489  df-fun 6535  df-fn 6536  df-f 6537  df-fv 6541  df-ov 7417  df-oprab 7418  df-mpo 7419  df-1st 7987  df-2nd 7988  df-ipf 21841

This theorem is used by: (None)