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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 7167 | . . 3 ⊢ (𝜑 → (〈𝑋, 𝑌〉(compf‘𝐶)𝑍) = (〈𝑋, 𝑌〉(compf‘𝐷)𝑍)) |
3 | 2 | oveqd 7167 | . 2 ⊢ (𝜑 → (𝐺(〈𝑋, 𝑌〉(compf‘𝐶)𝑍)𝐹) = (𝐺(〈𝑋, 𝑌〉(compf‘𝐷)𝑍)𝐹)) |
4 | eqid 2821 | . . 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 16964 | . 2 ⊢ (𝜑 → (𝐺(〈𝑋, 𝑌〉(compf‘𝐶)𝑍)𝐹) = (𝐺(〈𝑋, 𝑌〉 · 𝑍)𝐹)) |
14 | eqid 2821 | . . 3 ⊢ (compf‘𝐷) = (compf‘𝐷) | |
15 | eqid 2821 | . . 3 ⊢ (Base‘𝐷) = (Base‘𝐷) | |
16 | eqid 2821 | . . 3 ⊢ (Hom ‘𝐷) = (Hom ‘𝐷) | |
17 | comfeqval.2 | . . 3 ⊢ ∙ = (comp‘𝐷) | |
18 | comfeqval.3 | . . . . . 6 ⊢ (𝜑 → (Homf ‘𝐶) = (Homf ‘𝐷)) | |
19 | 18 | homfeqbas 16960 | . . . . 5 ⊢ (𝜑 → (Base‘𝐶) = (Base‘𝐷)) |
20 | 5, 19 | syl5eq 2868 | . . . 4 ⊢ (𝜑 → 𝐵 = (Base‘𝐷)) |
21 | 8, 20 | eleqtrd 2915 | . . 3 ⊢ (𝜑 → 𝑋 ∈ (Base‘𝐷)) |
22 | 9, 20 | eleqtrd 2915 | . . 3 ⊢ (𝜑 → 𝑌 ∈ (Base‘𝐷)) |
23 | 10, 20 | eleqtrd 2915 | . . 3 ⊢ (𝜑 → 𝑍 ∈ (Base‘𝐷)) |
24 | 5, 6, 16, 18, 8, 9 | homfeqval 16961 | . . . 4 ⊢ (𝜑 → (𝑋𝐻𝑌) = (𝑋(Hom ‘𝐷)𝑌)) |
25 | 11, 24 | eleqtrd 2915 | . . 3 ⊢ (𝜑 → 𝐹 ∈ (𝑋(Hom ‘𝐷)𝑌)) |
26 | 5, 6, 16, 18, 9, 10 | homfeqval 16961 | . . . 4 ⊢ (𝜑 → (𝑌𝐻𝑍) = (𝑌(Hom ‘𝐷)𝑍)) |
27 | 12, 26 | eleqtrd 2915 | . . 3 ⊢ (𝜑 → 𝐺 ∈ (𝑌(Hom ‘𝐷)𝑍)) |
28 | 14, 15, 16, 17, 21, 22, 23, 25, 27 | comfval 16964 | . 2 ⊢ (𝜑 → (𝐺(〈𝑋, 𝑌〉(compf‘𝐷)𝑍)𝐹) = (𝐺(〈𝑋, 𝑌〉 ∙ 𝑍)𝐹)) |
29 | 3, 13, 28 | 3eqtr3d 2864 | 1 ⊢ (𝜑 → (𝐺(〈𝑋, 𝑌〉 · 𝑍)𝐹) = (𝐺(〈𝑋, 𝑌〉 ∙ 𝑍)𝐹)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2110 〈cop 4566 ‘cfv 6349 (class class class)co 7150 Basecbs 16477 Hom chom 16570 compcco 16571 Homf chomf 16931 compfccomf 16932 |
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 5182 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 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 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 4561 df-pr 4563 df-op 4567 df-uni 4832 df-iun 4913 df-br 5059 df-opab 5121 df-mpt 5139 df-id 5454 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-ov 7153 df-oprab 7154 df-mpo 7155 df-1st 7683 df-2nd 7684 df-homf 16935 df-comf 16936 |
This theorem is referenced by: catpropd 16973 cidpropd 16974 oppccomfpropd 16991 monpropd 17001 funcpropd 17164 natpropd 17240 fucpropd 17241 xpcpropd 17452 hofpropd 17511 |
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