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Mirrors > Home > MPE Home > Th. List > coahom | Structured version Visualization version GIF version |
Description: The composition of two composable arrows is an arrow. (Contributed by Mario Carneiro, 11-Jan-2017.) |
Ref | Expression |
---|---|
homdmcoa.o | ⊢ · = (compa‘𝐶) |
homdmcoa.h | ⊢ 𝐻 = (Homa‘𝐶) |
homdmcoa.f | ⊢ (𝜑 → 𝐹 ∈ (𝑋𝐻𝑌)) |
homdmcoa.g | ⊢ (𝜑 → 𝐺 ∈ (𝑌𝐻𝑍)) |
Ref | Expression |
---|---|
coahom | ⊢ (𝜑 → (𝐺 · 𝐹) ∈ (𝑋𝐻𝑍)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | homdmcoa.o | . . 3 ⊢ · = (compa‘𝐶) | |
2 | homdmcoa.h | . . 3 ⊢ 𝐻 = (Homa‘𝐶) | |
3 | homdmcoa.f | . . 3 ⊢ (𝜑 → 𝐹 ∈ (𝑋𝐻𝑌)) | |
4 | homdmcoa.g | . . 3 ⊢ (𝜑 → 𝐺 ∈ (𝑌𝐻𝑍)) | |
5 | eqid 2760 | . . 3 ⊢ (comp‘𝐶) = (comp‘𝐶) | |
6 | 1, 2, 3, 4, 5 | coaval 16919 | . 2 ⊢ (𝜑 → (𝐺 · 𝐹) = 〈𝑋, 𝑍, ((2nd ‘𝐺)(〈𝑋, 𝑌〉(comp‘𝐶)𝑍)(2nd ‘𝐹))〉) |
7 | eqid 2760 | . . 3 ⊢ (Base‘𝐶) = (Base‘𝐶) | |
8 | 2 | homarcl 16879 | . . . 4 ⊢ (𝐹 ∈ (𝑋𝐻𝑌) → 𝐶 ∈ Cat) |
9 | 3, 8 | syl 17 | . . 3 ⊢ (𝜑 → 𝐶 ∈ Cat) |
10 | eqid 2760 | . . 3 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶) | |
11 | 2, 7 | homarcl2 16886 | . . . . 5 ⊢ (𝐹 ∈ (𝑋𝐻𝑌) → (𝑋 ∈ (Base‘𝐶) ∧ 𝑌 ∈ (Base‘𝐶))) |
12 | 3, 11 | syl 17 | . . . 4 ⊢ (𝜑 → (𝑋 ∈ (Base‘𝐶) ∧ 𝑌 ∈ (Base‘𝐶))) |
13 | 12 | simpld 477 | . . 3 ⊢ (𝜑 → 𝑋 ∈ (Base‘𝐶)) |
14 | 2, 7 | homarcl2 16886 | . . . . 5 ⊢ (𝐺 ∈ (𝑌𝐻𝑍) → (𝑌 ∈ (Base‘𝐶) ∧ 𝑍 ∈ (Base‘𝐶))) |
15 | 4, 14 | syl 17 | . . . 4 ⊢ (𝜑 → (𝑌 ∈ (Base‘𝐶) ∧ 𝑍 ∈ (Base‘𝐶))) |
16 | 15 | simprd 482 | . . 3 ⊢ (𝜑 → 𝑍 ∈ (Base‘𝐶)) |
17 | 12 | simprd 482 | . . . 4 ⊢ (𝜑 → 𝑌 ∈ (Base‘𝐶)) |
18 | 2, 10 | homahom 16890 | . . . . 5 ⊢ (𝐹 ∈ (𝑋𝐻𝑌) → (2nd ‘𝐹) ∈ (𝑋(Hom ‘𝐶)𝑌)) |
19 | 3, 18 | syl 17 | . . . 4 ⊢ (𝜑 → (2nd ‘𝐹) ∈ (𝑋(Hom ‘𝐶)𝑌)) |
20 | 2, 10 | homahom 16890 | . . . . 5 ⊢ (𝐺 ∈ (𝑌𝐻𝑍) → (2nd ‘𝐺) ∈ (𝑌(Hom ‘𝐶)𝑍)) |
21 | 4, 20 | syl 17 | . . . 4 ⊢ (𝜑 → (2nd ‘𝐺) ∈ (𝑌(Hom ‘𝐶)𝑍)) |
22 | 7, 10, 5, 9, 13, 17, 16, 19, 21 | catcocl 16547 | . . 3 ⊢ (𝜑 → ((2nd ‘𝐺)(〈𝑋, 𝑌〉(comp‘𝐶)𝑍)(2nd ‘𝐹)) ∈ (𝑋(Hom ‘𝐶)𝑍)) |
23 | 2, 7, 9, 10, 13, 16, 22 | elhomai2 16885 | . 2 ⊢ (𝜑 → 〈𝑋, 𝑍, ((2nd ‘𝐺)(〈𝑋, 𝑌〉(comp‘𝐶)𝑍)(2nd ‘𝐹))〉 ∈ (𝑋𝐻𝑍)) |
24 | 6, 23 | eqeltrd 2839 | 1 ⊢ (𝜑 → (𝐺 · 𝐹) ∈ (𝑋𝐻𝑍)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1632 ∈ wcel 2139 〈cop 4327 〈cotp 4329 ‘cfv 6049 (class class class)co 6813 2nd c2nd 7332 Basecbs 16059 Hom chom 16154 compcco 16155 Catccat 16526 Homachoma 16874 compaccoa 16905 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1871 ax-4 1886 ax-5 1988 ax-6 2054 ax-7 2090 ax-8 2141 ax-9 2148 ax-10 2168 ax-11 2183 ax-12 2196 ax-13 2391 ax-ext 2740 ax-rep 4923 ax-sep 4933 ax-nul 4941 ax-pow 4992 ax-pr 5055 ax-un 7114 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3an 1074 df-tru 1635 df-ex 1854 df-nf 1859 df-sb 2047 df-eu 2611 df-mo 2612 df-clab 2747 df-cleq 2753 df-clel 2756 df-nfc 2891 df-ne 2933 df-ral 3055 df-rex 3056 df-reu 3057 df-rab 3059 df-v 3342 df-sbc 3577 df-csb 3675 df-dif 3718 df-un 3720 df-in 3722 df-ss 3729 df-nul 4059 df-if 4231 df-pw 4304 df-sn 4322 df-pr 4324 df-op 4328 df-ot 4330 df-uni 4589 df-iun 4674 df-br 4805 df-opab 4865 df-mpt 4882 df-id 5174 df-xp 5272 df-rel 5273 df-cnv 5274 df-co 5275 df-dm 5276 df-rn 5277 df-res 5278 df-ima 5279 df-iota 6012 df-fun 6051 df-fn 6052 df-f 6053 df-f1 6054 df-fo 6055 df-f1o 6056 df-fv 6057 df-ov 6816 df-oprab 6817 df-mpt2 6818 df-1st 7333 df-2nd 7334 df-cat 16530 df-doma 16875 df-coda 16876 df-homa 16877 df-arw 16878 df-coa 16907 |
This theorem is referenced by: coapm 16922 arwass 16925 |
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