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Theorem ipffn 19918
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 , = (·if𝑊)
Assertion
Ref Expression
ipffn , Fn (𝑉 × 𝑉)

Proof of Theorem ipffn
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ipffn.1 . . 3 𝑉 = (Base‘𝑊)
2 eqid 2621 . . 3 (·𝑖𝑊) = (·𝑖𝑊)
3 ipffn.2 . . 3 , = (·if𝑊)
41, 2, 3ipffval 19915 . 2 , = (𝑥𝑉, 𝑦𝑉 ↦ (𝑥(·𝑖𝑊)𝑦))
5 ovex 6635 . 2 (𝑥(·𝑖𝑊)𝑦) ∈ V
64, 5fnmpt2i 7187 1 , Fn (𝑉 × 𝑉)
Colors of variables: wff setvar class
Syntax hints:   = wceq 1480   × cxp 5074   Fn wfn 5844  cfv 5849  (class class class)co 6607  Basecbs 15784  ·𝑖cip 15870  ·ifcipf 19892
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4743  ax-nul 4751  ax-pow 4805  ax-pr 4869  ax-un 6905
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3419  df-csb 3516  df-dif 3559  df-un 3561  df-in 3563  df-ss 3570  df-nul 3894  df-if 4061  df-pw 4134  df-sn 4151  df-pr 4153  df-op 4157  df-uni 4405  df-iun 4489  df-br 4616  df-opab 4676  df-mpt 4677  df-id 4991  df-xp 5082  df-rel 5083  df-cnv 5084  df-co 5085  df-dm 5086  df-rn 5087  df-res 5088  df-ima 5089  df-iota 5812  df-fun 5851  df-fn 5852  df-f 5853  df-fv 5857  df-ov 6610  df-oprab 6611  df-mpt2 6612  df-1st 7116  df-2nd 7117  df-ipf 19894
This theorem is referenced by: (None)
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