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Mirrors > Home > MPE Home > Th. List > ipfval | Structured version Visualization version GIF version |
Description: The inner product operation as a function. (Contributed by Mario Carneiro, 14-Aug-2015.) |
Ref | Expression |
---|---|
ipffval.1 | โข ๐ = (Baseโ๐) |
ipffval.2 | โข , = (ยท๐โ๐) |
ipffval.3 | โข ยท = (ยทifโ๐) |
Ref | Expression |
---|---|
ipfval | โข ((๐ โ ๐ โง ๐ โ ๐) โ (๐ ยท ๐) = (๐ , ๐)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | oveq12 7370 | . 2 โข ((๐ฅ = ๐ โง ๐ฆ = ๐) โ (๐ฅ , ๐ฆ) = (๐ , ๐)) | |
2 | ipffval.1 | . . 3 โข ๐ = (Baseโ๐) | |
3 | ipffval.2 | . . 3 โข , = (ยท๐โ๐) | |
4 | ipffval.3 | . . 3 โข ยท = (ยทifโ๐) | |
5 | 2, 3, 4 | ipffval 21075 | . 2 โข ยท = (๐ฅ โ ๐, ๐ฆ โ ๐ โฆ (๐ฅ , ๐ฆ)) |
6 | ovex 7394 | . 2 โข (๐ , ๐) โ V | |
7 | 1, 5, 6 | ovmpoa 7514 | 1 โข ((๐ โ ๐ โง ๐ โ ๐) โ (๐ ยท ๐) = (๐ , ๐)) |
Colors of variables: wff setvar class |
Syntax hints: โ wi 4 โง wa 397 = wceq 1542 โ wcel 2107 โcfv 6500 (class class class)co 7361 Basecbs 17091 ยท๐cip 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: ipcn 24633 cnmpt1ip 24634 cnmpt2ip 24635 |
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