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Mirrors > Home > MPE Home > Th. List > comfeqval | Structured version Visualization version GIF version |
Description: Equality of two compositions. (Contributed by Mario Carneiro, 4-Jan-2017.) |
Ref | Expression |
---|---|
comfeqval.b | ⊢ 𝐵 = (Base‘𝐶) |
comfeqval.h | ⊢ 𝐻 = (Hom ‘𝐶) |
comfeqval.1 | ⊢ · = (comp‘𝐶) |
comfeqval.2 | ⊢ ∙ = (comp‘𝐷) |
comfeqval.3 | ⊢ (𝜑 → (Homf ‘𝐶) = (Homf ‘𝐷)) |
comfeqval.4 | ⊢ (𝜑 → (compf‘𝐶) = (compf‘𝐷)) |
comfeqval.x | ⊢ (𝜑 → 𝑋 ∈ 𝐵) |
comfeqval.y | ⊢ (𝜑 → 𝑌 ∈ 𝐵) |
comfeqval.z | ⊢ (𝜑 → 𝑍 ∈ 𝐵) |
comfeqval.f | ⊢ (𝜑 → 𝐹 ∈ (𝑋𝐻𝑌)) |
comfeqval.g | ⊢ (𝜑 → 𝐺 ∈ (𝑌𝐻𝑍)) |
Ref | Expression |
---|---|
comfeqval | ⊢ (𝜑 → (𝐺(〈𝑋, 𝑌〉 · 𝑍)𝐹) = (𝐺(〈𝑋, 𝑌〉 ∙ 𝑍)𝐹)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | comfeqval.4 | . . . 4 ⊢ (𝜑 → (compf‘𝐶) = (compf‘𝐷)) | |
2 | 1 | oveqd 7152 | . . 3 ⊢ (𝜑 → (〈𝑋, 𝑌〉(compf‘𝐶)𝑍) = (〈𝑋, 𝑌〉(compf‘𝐷)𝑍)) |
3 | 2 | oveqd 7152 | . 2 ⊢ (𝜑 → (𝐺(〈𝑋, 𝑌〉(compf‘𝐶)𝑍)𝐹) = (𝐺(〈𝑋, 𝑌〉(compf‘𝐷)𝑍)𝐹)) |
4 | eqid 2798 | . . 3 ⊢ (compf‘𝐶) = (compf‘𝐶) | |
5 | comfeqval.b | . . 3 ⊢ 𝐵 = (Base‘𝐶) | |
6 | comfeqval.h | . . 3 ⊢ 𝐻 = (Hom ‘𝐶) | |
7 | comfeqval.1 | . . 3 ⊢ · = (comp‘𝐶) | |
8 | comfeqval.x | . . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐵) | |
9 | comfeqval.y | . . 3 ⊢ (𝜑 → 𝑌 ∈ 𝐵) | |
10 | comfeqval.z | . . 3 ⊢ (𝜑 → 𝑍 ∈ 𝐵) | |
11 | comfeqval.f | . . 3 ⊢ (𝜑 → 𝐹 ∈ (𝑋𝐻𝑌)) | |
12 | comfeqval.g | . . 3 ⊢ (𝜑 → 𝐺 ∈ (𝑌𝐻𝑍)) | |
13 | 4, 5, 6, 7, 8, 9, 10, 11, 12 | comfval 16962 | . 2 ⊢ (𝜑 → (𝐺(〈𝑋, 𝑌〉(compf‘𝐶)𝑍)𝐹) = (𝐺(〈𝑋, 𝑌〉 · 𝑍)𝐹)) |
14 | eqid 2798 | . . 3 ⊢ (compf‘𝐷) = (compf‘𝐷) | |
15 | eqid 2798 | . . 3 ⊢ (Base‘𝐷) = (Base‘𝐷) | |
16 | eqid 2798 | . . 3 ⊢ (Hom ‘𝐷) = (Hom ‘𝐷) | |
17 | comfeqval.2 | . . 3 ⊢ ∙ = (comp‘𝐷) | |
18 | comfeqval.3 | . . . . . 6 ⊢ (𝜑 → (Homf ‘𝐶) = (Homf ‘𝐷)) | |
19 | 18 | homfeqbas 16958 | . . . . 5 ⊢ (𝜑 → (Base‘𝐶) = (Base‘𝐷)) |
20 | 5, 19 | syl5eq 2845 | . . . 4 ⊢ (𝜑 → 𝐵 = (Base‘𝐷)) |
21 | 8, 20 | eleqtrd 2892 | . . 3 ⊢ (𝜑 → 𝑋 ∈ (Base‘𝐷)) |
22 | 9, 20 | eleqtrd 2892 | . . 3 ⊢ (𝜑 → 𝑌 ∈ (Base‘𝐷)) |
23 | 10, 20 | eleqtrd 2892 | . . 3 ⊢ (𝜑 → 𝑍 ∈ (Base‘𝐷)) |
24 | 5, 6, 16, 18, 8, 9 | homfeqval 16959 | . . . 4 ⊢ (𝜑 → (𝑋𝐻𝑌) = (𝑋(Hom ‘𝐷)𝑌)) |
25 | 11, 24 | eleqtrd 2892 | . . 3 ⊢ (𝜑 → 𝐹 ∈ (𝑋(Hom ‘𝐷)𝑌)) |
26 | 5, 6, 16, 18, 9, 10 | homfeqval 16959 | . . . 4 ⊢ (𝜑 → (𝑌𝐻𝑍) = (𝑌(Hom ‘𝐷)𝑍)) |
27 | 12, 26 | eleqtrd 2892 | . . 3 ⊢ (𝜑 → 𝐺 ∈ (𝑌(Hom ‘𝐷)𝑍)) |
28 | 14, 15, 16, 17, 21, 22, 23, 25, 27 | comfval 16962 | . 2 ⊢ (𝜑 → (𝐺(〈𝑋, 𝑌〉(compf‘𝐷)𝑍)𝐹) = (𝐺(〈𝑋, 𝑌〉 ∙ 𝑍)𝐹)) |
29 | 3, 13, 28 | 3eqtr3d 2841 | 1 ⊢ (𝜑 → (𝐺(〈𝑋, 𝑌〉 · 𝑍)𝐹) = (𝐺(〈𝑋, 𝑌〉 ∙ 𝑍)𝐹)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1538 ∈ wcel 2111 〈cop 4531 ‘cfv 6324 (class class class)co 7135 Basecbs 16475 Hom chom 16568 compcco 16569 Homf chomf 16929 compfccomf 16930 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-reu 3113 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-ov 7138 df-oprab 7139 df-mpo 7140 df-1st 7671 df-2nd 7672 df-homf 16933 df-comf 16934 |
This theorem is referenced by: catpropd 16971 cidpropd 16972 oppccomfpropd 16989 monpropd 16999 funcpropd 17162 natpropd 17238 fucpropd 17239 xpcpropd 17450 hofpropd 17509 |
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