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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 17686 | . 2 โข (๐ โ (โจ๐, ๐โฉ๐๐) = (๐ โ (๐๐ป๐), ๐ โ (๐๐ป๐) โฆ (๐(โจ๐, ๐โฉ ยท ๐)๐))) |
9 | oveq12 7435 | . . 3 โข ((๐ = ๐บ โง ๐ = ๐น) โ (๐(โจ๐, ๐โฉ ยท ๐)๐) = (๐บ(โจ๐, ๐โฉ ยท ๐)๐น)) | |
10 | 9 | adantl 480 | . 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 7461 | . 2 โข (๐ โ (๐บ(โจ๐, ๐โฉ ยท ๐)๐น) โ V) | |
14 | 8, 10, 11, 12, 13 | ovmpod 7579 | 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 394 = wceq 1533 โ wcel 2098 Vcvv 3473 โจcop 4638 โcfv 6553 (class class class)co 7426 Basecbs 17187 Hom chom 17251 compcco 17252 compfccomf 17654 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2699 ax-rep 5289 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7746 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-id 5580 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-ov 7429 df-oprab 7430 df-mpo 7431 df-1st 7999 df-2nd 8000 df-comf 17658 |
This theorem is referenced by: comfval2 17690 comfeqval 17695 |
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