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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 16718 | . . 3 ⊢ (𝜑 → (𝐺 · 𝐹) = 〈𝑋, 𝑍, ((2nd ‘𝐺)(〈𝑋, 𝑌〉 ∙ 𝑍)(2nd ‘𝐹))〉) |
7 | 6 | fveq2d 6195 | . 2 ⊢ (𝜑 → (2nd ‘(𝐺 · 𝐹)) = (2nd ‘〈𝑋, 𝑍, ((2nd ‘𝐺)(〈𝑋, 𝑌〉 ∙ 𝑍)(2nd ‘𝐹))〉)) |
8 | ovex 6678 | . . 3 ⊢ ((2nd ‘𝐺)(〈𝑋, 𝑌〉 ∙ 𝑍)(2nd ‘𝐹)) ∈ V | |
9 | ot3rdg 7184 | . . 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 2672 | 1 ⊢ (𝜑 → (2nd ‘(𝐺 · 𝐹)) = ((2nd ‘𝐺)(〈𝑋, 𝑌〉 ∙ 𝑍)(2nd ‘𝐹))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1483 ∈ wcel 1990 Vcvv 3200 〈cop 4183 〈cotp 4185 ‘cfv 5888 (class class class)co 6650 2nd c2nd 7167 compcco 15953 Homachoma 16673 compaccoa 16704 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-reu 2919 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-ot 4186 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-1st 7168 df-2nd 7169 df-doma 16674 df-coda 16675 df-homa 16676 df-arw 16677 df-coa 16706 |
This theorem is referenced by: arwass 16724 |
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