| Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
| 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 2737 | . . 3 ⊢ (comp‘𝐶) = (comp‘𝐶) | |
| 6 | 1, 2, 3, 4, 5 | coaval 18113 | . 2 ⊢ (𝜑 → (𝐺 · 𝐹) = 〈𝑋, 𝑍, ((2nd ‘𝐺)(〈𝑋, 𝑌〉(comp‘𝐶)𝑍)(2nd ‘𝐹))〉) |
| 7 | eqid 2737 | . . 3 ⊢ (Base‘𝐶) = (Base‘𝐶) | |
| 8 | 2 | homarcl 18073 | . . . 4 ⊢ (𝐹 ∈ (𝑋𝐻𝑌) → 𝐶 ∈ Cat) |
| 9 | 3, 8 | syl 17 | . . 3 ⊢ (𝜑 → 𝐶 ∈ Cat) |
| 10 | eqid 2737 | . . 3 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶) | |
| 11 | 2, 7 | homarcl2 18080 | . . . . 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 494 | . . 3 ⊢ (𝜑 → 𝑋 ∈ (Base‘𝐶)) |
| 14 | 2, 7 | homarcl2 18080 | . . . . 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 495 | . . 3 ⊢ (𝜑 → 𝑍 ∈ (Base‘𝐶)) |
| 17 | 12 | simprd 495 | . . . 4 ⊢ (𝜑 → 𝑌 ∈ (Base‘𝐶)) |
| 18 | 2, 10 | homahom 18084 | . . . . 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 18084 | . . . . 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 17728 | . . 3 ⊢ (𝜑 → ((2nd ‘𝐺)(〈𝑋, 𝑌〉(comp‘𝐶)𝑍)(2nd ‘𝐹)) ∈ (𝑋(Hom ‘𝐶)𝑍)) |
| 23 | 2, 7, 9, 10, 13, 16, 22 | elhomai2 18079 | . 2 ⊢ (𝜑 → 〈𝑋, 𝑍, ((2nd ‘𝐺)(〈𝑋, 𝑌〉(comp‘𝐶)𝑍)(2nd ‘𝐹))〉 ∈ (𝑋𝐻𝑍)) |
| 24 | 6, 23 | eqeltrd 2841 | 1 ⊢ (𝜑 → (𝐺 · 𝐹) ∈ (𝑋𝐻𝑍)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 〈cop 4632 〈cotp 4634 ‘cfv 6561 (class class class)co 7431 2nd c2nd 8013 Basecbs 17247 Hom chom 17308 compcco 17309 Catccat 17707 Homachoma 18068 compaccoa 18099 |
| 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 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-ot 4635 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-ov 7434 df-oprab 7435 df-mpo 7436 df-1st 8014 df-2nd 8015 df-cat 17711 df-doma 18069 df-coda 18070 df-homa 18071 df-arw 18072 df-coa 18101 |
| This theorem is referenced by: coapm 18116 arwass 18119 |
| Copyright terms: Public domain | W3C validator |