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Mirrors > Home > MPE Home > Th. List > coa2 | Structured version Visualization version GIF version |
Description: The morphism part of arrow composition. (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)) |
coaval.x | ⊢ ∙ = (comp‘𝐶) |
Ref | Expression |
---|---|
coa2 | ⊢ (𝜑 → (2nd ‘(𝐺 · 𝐹)) = ((2nd ‘𝐺)(〈𝑋, 𝑌〉 ∙ 𝑍)(2nd ‘𝐹))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | homdmcoa.o | . . . 4 ⊢ · = (compa‘𝐶) | |
2 | homdmcoa.h | . . . 4 ⊢ 𝐻 = (Homa‘𝐶) | |
3 | homdmcoa.f | . . . 4 ⊢ (𝜑 → 𝐹 ∈ (𝑋𝐻𝑌)) | |
4 | homdmcoa.g | . . . 4 ⊢ (𝜑 → 𝐺 ∈ (𝑌𝐻𝑍)) | |
5 | coaval.x | . . . 4 ⊢ ∙ = (comp‘𝐶) | |
6 | 1, 2, 3, 4, 5 | coaval 17322 | . . 3 ⊢ (𝜑 → (𝐺 · 𝐹) = 〈𝑋, 𝑍, ((2nd ‘𝐺)(〈𝑋, 𝑌〉 ∙ 𝑍)(2nd ‘𝐹))〉) |
7 | 6 | fveq2d 6669 | . 2 ⊢ (𝜑 → (2nd ‘(𝐺 · 𝐹)) = (2nd ‘〈𝑋, 𝑍, ((2nd ‘𝐺)(〈𝑋, 𝑌〉 ∙ 𝑍)(2nd ‘𝐹))〉)) |
8 | ovex 7183 | . . 3 ⊢ ((2nd ‘𝐺)(〈𝑋, 𝑌〉 ∙ 𝑍)(2nd ‘𝐹)) ∈ V | |
9 | ot3rdg 7699 | . . 3 ⊢ (((2nd ‘𝐺)(〈𝑋, 𝑌〉 ∙ 𝑍)(2nd ‘𝐹)) ∈ V → (2nd ‘〈𝑋, 𝑍, ((2nd ‘𝐺)(〈𝑋, 𝑌〉 ∙ 𝑍)(2nd ‘𝐹))〉) = ((2nd ‘𝐺)(〈𝑋, 𝑌〉 ∙ 𝑍)(2nd ‘𝐹))) | |
10 | 8, 9 | ax-mp 5 | . 2 ⊢ (2nd ‘〈𝑋, 𝑍, ((2nd ‘𝐺)(〈𝑋, 𝑌〉 ∙ 𝑍)(2nd ‘𝐹))〉) = ((2nd ‘𝐺)(〈𝑋, 𝑌〉 ∙ 𝑍)(2nd ‘𝐹)) |
11 | 7, 10 | syl6eq 2872 | 1 ⊢ (𝜑 → (2nd ‘(𝐺 · 𝐹)) = ((2nd ‘𝐺)(〈𝑋, 𝑌〉 ∙ 𝑍)(2nd ‘𝐹))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2110 Vcvv 3495 〈cop 4567 〈cotp 4569 ‘cfv 6350 (class class class)co 7150 2nd c2nd 7682 compcco 16571 Homachoma 17277 compaccoa 17308 |
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 2156 ax-12 2172 ax-ext 2793 ax-rep 5183 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5322 ax-un 7455 |
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 3497 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4562 df-pr 4564 df-op 4568 df-ot 4570 df-uni 4833 df-iun 4914 df-br 5060 df-opab 5122 df-mpt 5140 df-id 5455 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-rn 5561 df-res 5562 df-ima 5563 df-iota 6309 df-fun 6352 df-fn 6353 df-f 6354 df-f1 6355 df-fo 6356 df-f1o 6357 df-fv 6358 df-ov 7153 df-oprab 7154 df-mpo 7155 df-1st 7683 df-2nd 7684 df-doma 17278 df-coda 17279 df-homa 17280 df-arw 17281 df-coa 17310 |
This theorem is referenced by: arwass 17328 |
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