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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 2821 | . . 3 ⊢ (comp‘𝐶) = (comp‘𝐶) | |
6 | 1, 2, 3, 4, 5 | coaval 17327 | . 2 ⊢ (𝜑 → (𝐺 · 𝐹) = 〈𝑋, 𝑍, ((2nd ‘𝐺)(〈𝑋, 𝑌〉(comp‘𝐶)𝑍)(2nd ‘𝐹))〉) |
7 | eqid 2821 | . . 3 ⊢ (Base‘𝐶) = (Base‘𝐶) | |
8 | 2 | homarcl 17287 | . . . 4 ⊢ (𝐹 ∈ (𝑋𝐻𝑌) → 𝐶 ∈ Cat) |
9 | 3, 8 | syl 17 | . . 3 ⊢ (𝜑 → 𝐶 ∈ Cat) |
10 | eqid 2821 | . . 3 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶) | |
11 | 2, 7 | homarcl2 17294 | . . . . 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 17294 | . . . . 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 17298 | . . . . 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 17298 | . . . . 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 16955 | . . 3 ⊢ (𝜑 → ((2nd ‘𝐺)(〈𝑋, 𝑌〉(comp‘𝐶)𝑍)(2nd ‘𝐹)) ∈ (𝑋(Hom ‘𝐶)𝑍)) |
23 | 2, 7, 9, 10, 13, 16, 22 | elhomai2 17293 | . 2 ⊢ (𝜑 → 〈𝑋, 𝑍, ((2nd ‘𝐺)(〈𝑋, 𝑌〉(comp‘𝐶)𝑍)(2nd ‘𝐹))〉 ∈ (𝑋𝐻𝑍)) |
24 | 6, 23 | eqeltrd 2913 | 1 ⊢ (𝜑 → (𝐺 · 𝐹) ∈ (𝑋𝐻𝑍)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1533 ∈ wcel 2110 〈cop 4572 〈cotp 4574 ‘cfv 6354 (class class class)co 7155 2nd c2nd 7687 Basecbs 16482 Hom chom 16575 compcco 16576 Catccat 16934 Homachoma 17282 compaccoa 17313 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5189 ax-sep 5202 ax-nul 5209 ax-pow 5265 ax-pr 5329 ax-un 7460 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4567 df-pr 4569 df-op 4573 df-ot 4575 df-uni 4838 df-iun 4920 df-br 5066 df-opab 5128 df-mpt 5146 df-id 5459 df-xp 5560 df-rel 5561 df-cnv 5562 df-co 5563 df-dm 5564 df-rn 5565 df-res 5566 df-ima 5567 df-iota 6313 df-fun 6356 df-fn 6357 df-f 6358 df-f1 6359 df-fo 6360 df-f1o 6361 df-fv 6362 df-ov 7158 df-oprab 7159 df-mpo 7160 df-1st 7688 df-2nd 7689 df-cat 16938 df-doma 17283 df-coda 17284 df-homa 17285 df-arw 17286 df-coa 17315 |
This theorem is referenced by: coapm 17330 arwass 17333 |
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