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Theorem ipfeq 21077
Description: If the inner product operation is already a function, the functionalization of it is equal to the original operation. (Contributed by Mario Carneiro, 14-Aug-2015.)
Hypotheses
Ref Expression
ipffval.1 ๐‘‰ = (Baseโ€˜๐‘Š)
ipffval.2 , = (ยท๐‘–โ€˜๐‘Š)
ipffval.3 ยท = (ยทifโ€˜๐‘Š)
Assertion
Ref Expression
ipfeq ( , Fn (๐‘‰ ร— ๐‘‰) โ†’ ยท = , )

Proof of Theorem ipfeq
Dummy variables ๐‘ฅ ๐‘ฆ are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ipffval.1 . . 3 ๐‘‰ = (Baseโ€˜๐‘Š)
2 ipffval.2 . . 3 , = (ยท๐‘–โ€˜๐‘Š)
3 ipffval.3 . . 3 ยท = (ยทifโ€˜๐‘Š)
41, 2, 3ipffval 21075 . 2 ยท = (๐‘ฅ โˆˆ ๐‘‰, ๐‘ฆ โˆˆ ๐‘‰ โ†ฆ (๐‘ฅ , ๐‘ฆ))
5 fnov 7491 . . 3 ( , Fn (๐‘‰ ร— ๐‘‰) โ†” , = (๐‘ฅ โˆˆ ๐‘‰, ๐‘ฆ โˆˆ ๐‘‰ โ†ฆ (๐‘ฅ , ๐‘ฆ)))
65biimpi 215 . 2 ( , Fn (๐‘‰ ร— ๐‘‰) โ†’ , = (๐‘ฅ โˆˆ ๐‘‰, ๐‘ฆ โˆˆ ๐‘‰ โ†ฆ (๐‘ฅ , ๐‘ฆ)))
74, 6eqtr4id 2792 1 ( , Fn (๐‘‰ ร— ๐‘‰) โ†’ ยท = , )
Colors of variables: wff setvar class
Syntax hints:   โ†’ wi 4   = wceq 1542   ร— cxp 5635   Fn wfn 6495  โ€˜cfv 6500  (class class class)co 7361   โˆˆ cmpo 7363  Basecbs 17091  ยท๐‘–cip 17146  ยทifcipf 21052
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5260  ax-nul 5267  ax-pow 5324  ax-pr 5388  ax-un 7676
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3449  df-sbc 3744  df-csb 3860  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-pw 4566  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-iun 4960  df-br 5110  df-opab 5172  df-mpt 5193  df-id 5535  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-rn 5648  df-res 5649  df-ima 5650  df-iota 6452  df-fun 6502  df-fn 6503  df-f 6504  df-fv 6508  df-ov 7364  df-oprab 7365  df-mpo 7366  df-1st 7925  df-2nd 7926  df-ipf 21054
This theorem is referenced by: (None)
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