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Mirrors > Home > MPE Home > Th. List > comffval2 | Structured version Visualization version GIF version |
Description: Value of the functionalized composition operation. (Contributed by Mario Carneiro, 4-Jan-2017.) |
Ref | Expression |
---|---|
comfffval2.o | โข ๐ = (compfโ๐ถ) |
comfffval2.b | โข ๐ต = (Baseโ๐ถ) |
comfffval2.h | โข ๐ป = (Homf โ๐ถ) |
comfffval2.x | โข ยท = (compโ๐ถ) |
comffval2.x | โข (๐ โ ๐ โ ๐ต) |
comffval2.y | โข (๐ โ ๐ โ ๐ต) |
comffval2.z | โข (๐ โ ๐ โ ๐ต) |
Ref | Expression |
---|---|
comffval2 | โข (๐ โ (โจ๐, ๐โฉ๐๐) = (๐ โ (๐๐ป๐), ๐ โ (๐๐ป๐) โฆ (๐(โจ๐, ๐โฉ ยท ๐)๐))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | comfffval2.o | . . 3 โข ๐ = (compfโ๐ถ) | |
2 | comfffval2.b | . . 3 โข ๐ต = (Baseโ๐ถ) | |
3 | eqid 2725 | . . 3 โข (Hom โ๐ถ) = (Hom โ๐ถ) | |
4 | comfffval2.x | . . 3 โข ยท = (compโ๐ถ) | |
5 | comffval2.x | . . 3 โข (๐ โ ๐ โ ๐ต) | |
6 | comffval2.y | . . 3 โข (๐ โ ๐ โ ๐ต) | |
7 | comffval2.z | . . 3 โข (๐ โ ๐ โ ๐ต) | |
8 | 1, 2, 3, 4, 5, 6, 7 | comffval 17678 | . 2 โข (๐ โ (โจ๐, ๐โฉ๐๐) = (๐ โ (๐(Hom โ๐ถ)๐), ๐ โ (๐(Hom โ๐ถ)๐) โฆ (๐(โจ๐, ๐โฉ ยท ๐)๐))) |
9 | comfffval2.h | . . . 4 โข ๐ป = (Homf โ๐ถ) | |
10 | 9, 2, 3, 6, 7 | homfval 17671 | . . 3 โข (๐ โ (๐๐ป๐) = (๐(Hom โ๐ถ)๐)) |
11 | 9, 2, 3, 5, 6 | homfval 17671 | . . 3 โข (๐ โ (๐๐ป๐) = (๐(Hom โ๐ถ)๐)) |
12 | eqidd 2726 | . . 3 โข (๐ โ (๐(โจ๐, ๐โฉ ยท ๐)๐) = (๐(โจ๐, ๐โฉ ยท ๐)๐)) | |
13 | 10, 11, 12 | mpoeq123dv 7492 | . 2 โข (๐ โ (๐ โ (๐๐ป๐), ๐ โ (๐๐ป๐) โฆ (๐(โจ๐, ๐โฉ ยท ๐)๐)) = (๐ โ (๐(Hom โ๐ถ)๐), ๐ โ (๐(Hom โ๐ถ)๐) โฆ (๐(โจ๐, ๐โฉ ยท ๐)๐))) |
14 | 8, 13 | eqtr4d 2768 | 1 โข (๐ โ (โจ๐, ๐โฉ๐๐) = (๐ โ (๐๐ป๐), ๐ โ (๐๐ป๐) โฆ (๐(โจ๐, ๐โฉ ยท ๐)๐))) |
Colors of variables: wff setvar class |
Syntax hints: โ wi 4 = wceq 1533 โ wcel 2098 โจcop 4630 โcfv 6543 (class class class)co 7416 โ cmpo 7418 Basecbs 17179 Hom chom 17243 compcco 17244 Homf chomf 17645 compfccomf 17646 |
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 2696 ax-rep 5280 ax-sep 5294 ax-nul 5301 ax-pow 5359 ax-pr 5423 ax-un 7738 |
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 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-ral 3052 df-rex 3061 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3769 df-csb 3885 df-dif 3942 df-un 3944 df-in 3946 df-ss 3956 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5227 df-id 5570 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-ov 7419 df-oprab 7420 df-mpo 7421 df-1st 7991 df-2nd 7992 df-homf 17649 df-comf 17650 |
This theorem is referenced by: (None) |
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