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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 2823 | . . 3 ⊢ (comp‘𝐶) = (comp‘𝐶) | |
6 | 1, 2, 3, 4, 5 | coaval 17330 | . 2 ⊢ (𝜑 → (𝐺 · 𝐹) = 〈𝑋, 𝑍, ((2nd ‘𝐺)(〈𝑋, 𝑌〉(comp‘𝐶)𝑍)(2nd ‘𝐹))〉) |
7 | eqid 2823 | . . 3 ⊢ (Base‘𝐶) = (Base‘𝐶) | |
8 | 2 | homarcl 17290 | . . . 4 ⊢ (𝐹 ∈ (𝑋𝐻𝑌) → 𝐶 ∈ Cat) |
9 | 3, 8 | syl 17 | . . 3 ⊢ (𝜑 → 𝐶 ∈ Cat) |
10 | eqid 2823 | . . 3 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶) | |
11 | 2, 7 | homarcl2 17297 | . . . . 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 497 | . . 3 ⊢ (𝜑 → 𝑋 ∈ (Base‘𝐶)) |
14 | 2, 7 | homarcl2 17297 | . . . . 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 498 | . . 3 ⊢ (𝜑 → 𝑍 ∈ (Base‘𝐶)) |
17 | 12 | simprd 498 | . . . 4 ⊢ (𝜑 → 𝑌 ∈ (Base‘𝐶)) |
18 | 2, 10 | homahom 17301 | . . . . 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 17301 | . . . . 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 16958 | . . 3 ⊢ (𝜑 → ((2nd ‘𝐺)(〈𝑋, 𝑌〉(comp‘𝐶)𝑍)(2nd ‘𝐹)) ∈ (𝑋(Hom ‘𝐶)𝑍)) |
23 | 2, 7, 9, 10, 13, 16, 22 | elhomai2 17296 | . 2 ⊢ (𝜑 → 〈𝑋, 𝑍, ((2nd ‘𝐺)(〈𝑋, 𝑌〉(comp‘𝐶)𝑍)(2nd ‘𝐹))〉 ∈ (𝑋𝐻𝑍)) |
24 | 6, 23 | eqeltrd 2915 | 1 ⊢ (𝜑 → (𝐺 · 𝐹) ∈ (𝑋𝐻𝑍)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 〈cop 4575 〈cotp 4577 ‘cfv 6357 (class class class)co 7158 2nd c2nd 7690 Basecbs 16485 Hom chom 16578 compcco 16579 Catccat 16937 Homachoma 17285 compaccoa 17316 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-ral 3145 df-rex 3146 df-reu 3147 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-op 4576 df-ot 4578 df-uni 4841 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-id 5462 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-ov 7161 df-oprab 7162 df-mpo 7163 df-1st 7691 df-2nd 7692 df-cat 16941 df-doma 17286 df-coda 17287 df-homa 17288 df-arw 17289 df-coa 17318 |
This theorem is referenced by: coapm 17333 arwass 17336 |
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