![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 2737 | . . 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 17539 | . 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 17532 | . . 3 โข (๐ โ (๐๐ป๐) = (๐(Hom โ๐ถ)๐)) |
11 | 9, 2, 3, 5, 6 | homfval 17532 | . . 3 โข (๐ โ (๐๐ป๐) = (๐(Hom โ๐ถ)๐)) |
12 | eqidd 2738 | . . 3 โข (๐ โ (๐(โจ๐, ๐โฉ ยท ๐)๐) = (๐(โจ๐, ๐โฉ ยท ๐)๐)) | |
13 | 10, 11, 12 | mpoeq123dv 7426 | . 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 2780 | 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 1541 โ wcel 2106 โจcop 4590 โcfv 6493 (class class class)co 7351 โ cmpo 7353 Basecbs 17043 Hom chom 17104 compcco 17105 Homf chomf 17506 compfccomf 17507 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2708 ax-rep 5240 ax-sep 5254 ax-nul 5261 ax-pow 5318 ax-pr 5382 ax-un 7664 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2887 df-ne 2942 df-ral 3063 df-rex 3072 df-reu 3352 df-rab 3406 df-v 3445 df-sbc 3738 df-csb 3854 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-nul 4281 df-if 4485 df-pw 4560 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4864 df-iun 4954 df-br 5104 df-opab 5166 df-mpt 5187 df-id 5529 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-iota 6445 df-fun 6495 df-fn 6496 df-f 6497 df-f1 6498 df-fo 6499 df-f1o 6500 df-fv 6501 df-ov 7354 df-oprab 7355 df-mpo 7356 df-1st 7913 df-2nd 7914 df-homf 17510 df-comf 17511 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |