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Mirrors > Home > MPE Home > Th. List > comfval | Structured version Visualization version GIF version |
Description: Value of the functionalized composition operation. (Contributed by Mario Carneiro, 4-Jan-2017.) |
Ref | Expression |
---|---|
comfffval.o | โข ๐ = (compfโ๐ถ) |
comfffval.b | โข ๐ต = (Baseโ๐ถ) |
comfffval.h | โข ๐ป = (Hom โ๐ถ) |
comfffval.x | โข ยท = (compโ๐ถ) |
comffval.x | โข (๐ โ ๐ โ ๐ต) |
comffval.y | โข (๐ โ ๐ โ ๐ต) |
comffval.z | โข (๐ โ ๐ โ ๐ต) |
comfval.f | โข (๐ โ ๐น โ (๐๐ป๐)) |
comfval.g | โข (๐ โ ๐บ โ (๐๐ป๐)) |
Ref | Expression |
---|---|
comfval | โข (๐ โ (๐บ(โจ๐, ๐โฉ๐๐)๐น) = (๐บ(โจ๐, ๐โฉ ยท ๐)๐น)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | comfffval.o | . . 3 โข ๐ = (compfโ๐ถ) | |
2 | comfffval.b | . . 3 โข ๐ต = (Baseโ๐ถ) | |
3 | comfffval.h | . . 3 โข ๐ป = (Hom โ๐ถ) | |
4 | comfffval.x | . . 3 โข ยท = (compโ๐ถ) | |
5 | comffval.x | . . 3 โข (๐ โ ๐ โ ๐ต) | |
6 | comffval.y | . . 3 โข (๐ โ ๐ โ ๐ต) | |
7 | comffval.z | . . 3 โข (๐ โ ๐ โ ๐ต) | |
8 | 1, 2, 3, 4, 5, 6, 7 | comffval 17453 | . 2 โข (๐ โ (โจ๐, ๐โฉ๐๐) = (๐ โ (๐๐ป๐), ๐ โ (๐๐ป๐) โฆ (๐(โจ๐, ๐โฉ ยท ๐)๐))) |
9 | oveq12 7316 | . . 3 โข ((๐ = ๐บ โง ๐ = ๐น) โ (๐(โจ๐, ๐โฉ ยท ๐)๐) = (๐บ(โจ๐, ๐โฉ ยท ๐)๐น)) | |
10 | 9 | adantl 483 | . 2 โข ((๐ โง (๐ = ๐บ โง ๐ = ๐น)) โ (๐(โจ๐, ๐โฉ ยท ๐)๐) = (๐บ(โจ๐, ๐โฉ ยท ๐)๐น)) |
11 | comfval.g | . 2 โข (๐ โ ๐บ โ (๐๐ป๐)) | |
12 | comfval.f | . 2 โข (๐ โ ๐น โ (๐๐ป๐)) | |
13 | ovexd 7342 | . 2 โข (๐ โ (๐บ(โจ๐, ๐โฉ ยท ๐)๐น) โ V) | |
14 | 8, 10, 11, 12, 13 | ovmpod 7457 | 1 โข (๐ โ (๐บ(โจ๐, ๐โฉ๐๐)๐น) = (๐บ(โจ๐, ๐โฉ ยท ๐)๐น)) |
Colors of variables: wff setvar class |
Syntax hints: โ wi 4 โง wa 397 = wceq 1539 โ wcel 2104 Vcvv 3437 โจcop 4571 โcfv 6458 (class class class)co 7307 Basecbs 16957 Hom chom 17018 compcco 17019 compfccomf 17421 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2707 ax-rep 5218 ax-sep 5232 ax-nul 5239 ax-pow 5297 ax-pr 5361 ax-un 7620 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 846 df-3an 1089 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2887 df-ne 2942 df-ral 3063 df-rex 3072 df-reu 3286 df-rab 3287 df-v 3439 df-sbc 3722 df-csb 3838 df-dif 3895 df-un 3897 df-in 3899 df-ss 3909 df-nul 4263 df-if 4466 df-pw 4541 df-sn 4566 df-pr 4568 df-op 4572 df-uni 4845 df-iun 4933 df-br 5082 df-opab 5144 df-mpt 5165 df-id 5500 df-xp 5606 df-rel 5607 df-cnv 5608 df-co 5609 df-dm 5610 df-rn 5611 df-res 5612 df-ima 5613 df-iota 6410 df-fun 6460 df-fn 6461 df-f 6462 df-f1 6463 df-fo 6464 df-f1o 6465 df-fv 6466 df-ov 7310 df-oprab 7311 df-mpo 7312 df-1st 7863 df-2nd 7864 df-comf 17425 |
This theorem is referenced by: comfval2 17457 comfeqval 17462 |
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