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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 2739 | . . 3 ⊢ (comp‘𝐶) = (comp‘𝐶) | |
6 | 1, 2, 3, 4, 5 | coaval 17443 | . 2 ⊢ (𝜑 → (𝐺 · 𝐹) = 〈𝑋, 𝑍, ((2nd ‘𝐺)(〈𝑋, 𝑌〉(comp‘𝐶)𝑍)(2nd ‘𝐹))〉) |
7 | eqid 2739 | . . 3 ⊢ (Base‘𝐶) = (Base‘𝐶) | |
8 | 2 | homarcl 17403 | . . . 4 ⊢ (𝐹 ∈ (𝑋𝐻𝑌) → 𝐶 ∈ Cat) |
9 | 3, 8 | syl 17 | . . 3 ⊢ (𝜑 → 𝐶 ∈ Cat) |
10 | eqid 2739 | . . 3 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶) | |
11 | 2, 7 | homarcl2 17410 | . . . . 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 498 | . . 3 ⊢ (𝜑 → 𝑋 ∈ (Base‘𝐶)) |
14 | 2, 7 | homarcl2 17410 | . . . . 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 499 | . . 3 ⊢ (𝜑 → 𝑍 ∈ (Base‘𝐶)) |
17 | 12 | simprd 499 | . . . 4 ⊢ (𝜑 → 𝑌 ∈ (Base‘𝐶)) |
18 | 2, 10 | homahom 17414 | . . . . 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 17414 | . . . . 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 17062 | . . 3 ⊢ (𝜑 → ((2nd ‘𝐺)(〈𝑋, 𝑌〉(comp‘𝐶)𝑍)(2nd ‘𝐹)) ∈ (𝑋(Hom ‘𝐶)𝑍)) |
23 | 2, 7, 9, 10, 13, 16, 22 | elhomai2 17409 | . 2 ⊢ (𝜑 → 〈𝑋, 𝑍, ((2nd ‘𝐺)(〈𝑋, 𝑌〉(comp‘𝐶)𝑍)(2nd ‘𝐹))〉 ∈ (𝑋𝐻𝑍)) |
24 | 6, 23 | eqeltrd 2834 | 1 ⊢ (𝜑 → (𝐺 · 𝐹) ∈ (𝑋𝐻𝑍)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1542 ∈ wcel 2114 〈cop 4523 〈cotp 4525 ‘cfv 6340 (class class class)co 7173 2nd c2nd 7716 Basecbs 16589 Hom chom 16682 compcco 16683 Catccat 17041 Homachoma 17398 compaccoa 17429 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2711 ax-rep 5155 ax-sep 5168 ax-nul 5175 ax-pow 5233 ax-pr 5297 ax-un 7482 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2541 df-eu 2571 df-clab 2718 df-cleq 2731 df-clel 2812 df-nfc 2882 df-ne 2936 df-ral 3059 df-rex 3060 df-reu 3061 df-rab 3063 df-v 3401 df-sbc 3682 df-csb 3792 df-dif 3847 df-un 3849 df-in 3851 df-ss 3861 df-nul 4213 df-if 4416 df-pw 4491 df-sn 4518 df-pr 4520 df-op 4524 df-ot 4526 df-uni 4798 df-iun 4884 df-br 5032 df-opab 5094 df-mpt 5112 df-id 5430 df-xp 5532 df-rel 5533 df-cnv 5534 df-co 5535 df-dm 5536 df-rn 5537 df-res 5538 df-ima 5539 df-iota 6298 df-fun 6342 df-fn 6343 df-f 6344 df-f1 6345 df-fo 6346 df-f1o 6347 df-fv 6348 df-ov 7176 df-oprab 7177 df-mpo 7178 df-1st 7717 df-2nd 7718 df-cat 17045 df-doma 17399 df-coda 17400 df-homa 17401 df-arw 17402 df-coa 17431 |
This theorem is referenced by: coapm 17446 arwass 17449 |
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