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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 6810 | . . 3 ⊢ (𝜑 → (〈𝑋, 𝑌〉(compf‘𝐶)𝑍) = (〈𝑋, 𝑌〉(compf‘𝐷)𝑍)) |
3 | 2 | oveqd 6810 | . 2 ⊢ (𝜑 → (𝐺(〈𝑋, 𝑌〉(compf‘𝐶)𝑍)𝐹) = (𝐺(〈𝑋, 𝑌〉(compf‘𝐷)𝑍)𝐹)) |
4 | eqid 2771 | . . 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 16567 | . 2 ⊢ (𝜑 → (𝐺(〈𝑋, 𝑌〉(compf‘𝐶)𝑍)𝐹) = (𝐺(〈𝑋, 𝑌〉 · 𝑍)𝐹)) |
14 | eqid 2771 | . . 3 ⊢ (compf‘𝐷) = (compf‘𝐷) | |
15 | eqid 2771 | . . 3 ⊢ (Base‘𝐷) = (Base‘𝐷) | |
16 | eqid 2771 | . . 3 ⊢ (Hom ‘𝐷) = (Hom ‘𝐷) | |
17 | comfeqval.2 | . . 3 ⊢ ∙ = (comp‘𝐷) | |
18 | comfeqval.3 | . . . . . 6 ⊢ (𝜑 → (Homf ‘𝐶) = (Homf ‘𝐷)) | |
19 | 18 | homfeqbas 16563 | . . . . 5 ⊢ (𝜑 → (Base‘𝐶) = (Base‘𝐷)) |
20 | 5, 19 | syl5eq 2817 | . . . 4 ⊢ (𝜑 → 𝐵 = (Base‘𝐷)) |
21 | 8, 20 | eleqtrd 2852 | . . 3 ⊢ (𝜑 → 𝑋 ∈ (Base‘𝐷)) |
22 | 9, 20 | eleqtrd 2852 | . . 3 ⊢ (𝜑 → 𝑌 ∈ (Base‘𝐷)) |
23 | 10, 20 | eleqtrd 2852 | . . 3 ⊢ (𝜑 → 𝑍 ∈ (Base‘𝐷)) |
24 | 5, 6, 16, 18, 8, 9 | homfeqval 16564 | . . . 4 ⊢ (𝜑 → (𝑋𝐻𝑌) = (𝑋(Hom ‘𝐷)𝑌)) |
25 | 11, 24 | eleqtrd 2852 | . . 3 ⊢ (𝜑 → 𝐹 ∈ (𝑋(Hom ‘𝐷)𝑌)) |
26 | 5, 6, 16, 18, 9, 10 | homfeqval 16564 | . . . 4 ⊢ (𝜑 → (𝑌𝐻𝑍) = (𝑌(Hom ‘𝐷)𝑍)) |
27 | 12, 26 | eleqtrd 2852 | . . 3 ⊢ (𝜑 → 𝐺 ∈ (𝑌(Hom ‘𝐷)𝑍)) |
28 | 14, 15, 16, 17, 21, 22, 23, 25, 27 | comfval 16567 | . 2 ⊢ (𝜑 → (𝐺(〈𝑋, 𝑌〉(compf‘𝐷)𝑍)𝐹) = (𝐺(〈𝑋, 𝑌〉 ∙ 𝑍)𝐹)) |
29 | 3, 13, 28 | 3eqtr3d 2813 | 1 ⊢ (𝜑 → (𝐺(〈𝑋, 𝑌〉 · 𝑍)𝐹) = (𝐺(〈𝑋, 𝑌〉 ∙ 𝑍)𝐹)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1631 ∈ wcel 2145 〈cop 4322 ‘cfv 6031 (class class class)co 6793 Basecbs 16064 Hom chom 16160 compcco 16161 Homf chomf 16534 compfccomf 16535 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-8 2147 ax-9 2154 ax-10 2174 ax-11 2190 ax-12 2203 ax-13 2408 ax-ext 2751 ax-rep 4904 ax-sep 4915 ax-nul 4923 ax-pow 4974 ax-pr 5034 ax-un 7096 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 837 df-3an 1073 df-tru 1634 df-ex 1853 df-nf 1858 df-sb 2050 df-eu 2622 df-mo 2623 df-clab 2758 df-cleq 2764 df-clel 2767 df-nfc 2902 df-ne 2944 df-ral 3066 df-rex 3067 df-reu 3068 df-rab 3070 df-v 3353 df-sbc 3588 df-csb 3683 df-dif 3726 df-un 3728 df-in 3730 df-ss 3737 df-nul 4064 df-if 4226 df-pw 4299 df-sn 4317 df-pr 4319 df-op 4323 df-uni 4575 df-iun 4656 df-br 4787 df-opab 4847 df-mpt 4864 df-id 5157 df-xp 5255 df-rel 5256 df-cnv 5257 df-co 5258 df-dm 5259 df-rn 5260 df-res 5261 df-ima 5262 df-iota 5994 df-fun 6033 df-fn 6034 df-f 6035 df-f1 6036 df-fo 6037 df-f1o 6038 df-fv 6039 df-ov 6796 df-oprab 6797 df-mpt2 6798 df-1st 7315 df-2nd 7316 df-homf 16538 df-comf 16539 |
This theorem is referenced by: catpropd 16576 cidpropd 16577 oppccomfpropd 16594 monpropd 16604 funcpropd 16767 natpropd 16843 fucpropd 16844 xpcpropd 17056 hofpropd 17115 |
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