Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 6566 | . . 3 ⊢ (𝜑 → (〈𝑋, 𝑌〉(compf‘𝐶)𝑍) = (〈𝑋, 𝑌〉(compf‘𝐷)𝑍)) |
3 | 2 | oveqd 6566 | . 2 ⊢ (𝜑 → (𝐺(〈𝑋, 𝑌〉(compf‘𝐶)𝑍)𝐹) = (𝐺(〈𝑋, 𝑌〉(compf‘𝐷)𝑍)𝐹)) |
4 | eqid 2610 | . . 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 16183 | . 2 ⊢ (𝜑 → (𝐺(〈𝑋, 𝑌〉(compf‘𝐶)𝑍)𝐹) = (𝐺(〈𝑋, 𝑌〉 · 𝑍)𝐹)) |
14 | eqid 2610 | . . 3 ⊢ (compf‘𝐷) = (compf‘𝐷) | |
15 | eqid 2610 | . . 3 ⊢ (Base‘𝐷) = (Base‘𝐷) | |
16 | eqid 2610 | . . 3 ⊢ (Hom ‘𝐷) = (Hom ‘𝐷) | |
17 | comfeqval.2 | . . 3 ⊢ ∙ = (comp‘𝐷) | |
18 | comfeqval.3 | . . . . . 6 ⊢ (𝜑 → (Homf ‘𝐶) = (Homf ‘𝐷)) | |
19 | 18 | homfeqbas 16179 | . . . . 5 ⊢ (𝜑 → (Base‘𝐶) = (Base‘𝐷)) |
20 | 5, 19 | syl5eq 2656 | . . . 4 ⊢ (𝜑 → 𝐵 = (Base‘𝐷)) |
21 | 8, 20 | eleqtrd 2690 | . . 3 ⊢ (𝜑 → 𝑋 ∈ (Base‘𝐷)) |
22 | 9, 20 | eleqtrd 2690 | . . 3 ⊢ (𝜑 → 𝑌 ∈ (Base‘𝐷)) |
23 | 10, 20 | eleqtrd 2690 | . . 3 ⊢ (𝜑 → 𝑍 ∈ (Base‘𝐷)) |
24 | 5, 6, 16, 18, 8, 9 | homfeqval 16180 | . . . 4 ⊢ (𝜑 → (𝑋𝐻𝑌) = (𝑋(Hom ‘𝐷)𝑌)) |
25 | 11, 24 | eleqtrd 2690 | . . 3 ⊢ (𝜑 → 𝐹 ∈ (𝑋(Hom ‘𝐷)𝑌)) |
26 | 5, 6, 16, 18, 9, 10 | homfeqval 16180 | . . . 4 ⊢ (𝜑 → (𝑌𝐻𝑍) = (𝑌(Hom ‘𝐷)𝑍)) |
27 | 12, 26 | eleqtrd 2690 | . . 3 ⊢ (𝜑 → 𝐺 ∈ (𝑌(Hom ‘𝐷)𝑍)) |
28 | 14, 15, 16, 17, 21, 22, 23, 25, 27 | comfval 16183 | . 2 ⊢ (𝜑 → (𝐺(〈𝑋, 𝑌〉(compf‘𝐷)𝑍)𝐹) = (𝐺(〈𝑋, 𝑌〉 ∙ 𝑍)𝐹)) |
29 | 3, 13, 28 | 3eqtr3d 2652 | 1 ⊢ (𝜑 → (𝐺(〈𝑋, 𝑌〉 · 𝑍)𝐹) = (𝐺(〈𝑋, 𝑌〉 ∙ 𝑍)𝐹)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1475 ∈ wcel 1977 〈cop 4131 ‘cfv 5804 (class class class)co 6549 Basecbs 15695 Hom chom 15779 compcco 15780 Homf chomf 16150 compfccomf 16151 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-1st 7059 df-2nd 7060 df-homf 16154 df-comf 16155 |
This theorem is referenced by: catpropd 16192 cidpropd 16193 oppccomfpropd 16210 monpropd 16220 funcpropd 16383 natpropd 16459 fucpropd 16460 xpcpropd 16671 hofpropd 16730 |
Copyright terms: Public domain | W3C validator |