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Mirrors > Home > MPE Home > Th. List > qusaddvallem | Structured version Visualization version GIF version |
Description: Value of an operation defined on a quotient structure. (Contributed by Mario Carneiro, 24-Feb-2015.) |
Ref | Expression |
---|---|
qusaddf.u | ⊢ (𝜑 → 𝑈 = (𝑅 /s ∼ )) |
qusaddf.v | ⊢ (𝜑 → 𝑉 = (Base‘𝑅)) |
qusaddf.r | ⊢ (𝜑 → ∼ Er 𝑉) |
qusaddf.z | ⊢ (𝜑 → 𝑅 ∈ 𝑍) |
qusaddf.e | ⊢ (𝜑 → ((𝑎 ∼ 𝑝 ∧ 𝑏 ∼ 𝑞) → (𝑎 · 𝑏) ∼ (𝑝 · 𝑞))) |
qusaddf.c | ⊢ ((𝜑 ∧ (𝑝 ∈ 𝑉 ∧ 𝑞 ∈ 𝑉)) → (𝑝 · 𝑞) ∈ 𝑉) |
qusaddflem.f | ⊢ 𝐹 = (𝑥 ∈ 𝑉 ↦ [𝑥] ∼ ) |
qusaddflem.g | ⊢ (𝜑 → ∙ = ∪ 𝑝 ∈ 𝑉 ∪ 𝑞 ∈ 𝑉 {〈〈(𝐹‘𝑝), (𝐹‘𝑞)〉, (𝐹‘(𝑝 · 𝑞))〉}) |
Ref | Expression |
---|---|
qusaddvallem | ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → ([𝑋] ∼ ∙ [𝑌] ∼ ) = [(𝑋 · 𝑌)] ∼ ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | qusaddf.u | . . . 4 ⊢ (𝜑 → 𝑈 = (𝑅 /s ∼ )) | |
2 | qusaddf.v | . . . 4 ⊢ (𝜑 → 𝑉 = (Base‘𝑅)) | |
3 | qusaddflem.f | . . . 4 ⊢ 𝐹 = (𝑥 ∈ 𝑉 ↦ [𝑥] ∼ ) | |
4 | qusaddf.r | . . . . 5 ⊢ (𝜑 → ∼ Er 𝑉) | |
5 | fvex 6677 | . . . . . 6 ⊢ (Base‘𝑅) ∈ V | |
6 | 2, 5 | eqeltrdi 2921 | . . . . 5 ⊢ (𝜑 → 𝑉 ∈ V) |
7 | erex 8307 | . . . . 5 ⊢ ( ∼ Er 𝑉 → (𝑉 ∈ V → ∼ ∈ V)) | |
8 | 4, 6, 7 | sylc 65 | . . . 4 ⊢ (𝜑 → ∼ ∈ V) |
9 | qusaddf.z | . . . 4 ⊢ (𝜑 → 𝑅 ∈ 𝑍) | |
10 | 1, 2, 3, 8, 9 | quslem 16810 | . . 3 ⊢ (𝜑 → 𝐹:𝑉–onto→(𝑉 / ∼ )) |
11 | qusaddf.c | . . . 4 ⊢ ((𝜑 ∧ (𝑝 ∈ 𝑉 ∧ 𝑞 ∈ 𝑉)) → (𝑝 · 𝑞) ∈ 𝑉) | |
12 | qusaddf.e | . . . 4 ⊢ (𝜑 → ((𝑎 ∼ 𝑝 ∧ 𝑏 ∼ 𝑞) → (𝑎 · 𝑏) ∼ (𝑝 · 𝑞))) | |
13 | 4, 6, 3, 11, 12 | ercpbl 16816 | . . 3 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) ∧ (𝑝 ∈ 𝑉 ∧ 𝑞 ∈ 𝑉)) → (((𝐹‘𝑎) = (𝐹‘𝑝) ∧ (𝐹‘𝑏) = (𝐹‘𝑞)) → (𝐹‘(𝑎 · 𝑏)) = (𝐹‘(𝑝 · 𝑞)))) |
14 | qusaddflem.g | . . 3 ⊢ (𝜑 → ∙ = ∪ 𝑝 ∈ 𝑉 ∪ 𝑞 ∈ 𝑉 {〈〈(𝐹‘𝑝), (𝐹‘𝑞)〉, (𝐹‘(𝑝 · 𝑞))〉}) | |
15 | 10, 13, 14 | imasaddvallem 16796 | . 2 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → ((𝐹‘𝑋) ∙ (𝐹‘𝑌)) = (𝐹‘(𝑋 · 𝑌))) |
16 | 4 | 3ad2ant1 1129 | . . . 4 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → ∼ Er 𝑉) |
17 | 6 | 3ad2ant1 1129 | . . . 4 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → 𝑉 ∈ V) |
18 | 16, 17, 3 | divsfval 16814 | . . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → (𝐹‘𝑋) = [𝑋] ∼ ) |
19 | 16, 17, 3 | divsfval 16814 | . . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → (𝐹‘𝑌) = [𝑌] ∼ ) |
20 | 18, 19 | oveq12d 7168 | . 2 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → ((𝐹‘𝑋) ∙ (𝐹‘𝑌)) = ([𝑋] ∼ ∙ [𝑌] ∼ )) |
21 | 16, 17, 3 | divsfval 16814 | . 2 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → (𝐹‘(𝑋 · 𝑌)) = [(𝑋 · 𝑌)] ∼ ) |
22 | 15, 20, 21 | 3eqtr3d 2864 | 1 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → ([𝑋] ∼ ∙ [𝑌] ∼ ) = [(𝑋 · 𝑌)] ∼ ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∧ w3a 1083 = wceq 1533 ∈ wcel 2110 Vcvv 3494 {csn 4560 〈cop 4566 ∪ ciun 4911 class class class wbr 5058 ↦ cmpt 5138 ‘cfv 6349 (class class class)co 7150 Er wer 8280 [cec 8281 / cqs 8282 Basecbs 16477 /s cqus 16772 |
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-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-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-fo 6355 df-fv 6357 df-ov 7153 df-er 8283 df-ec 8285 df-qs 8289 |
This theorem is referenced by: qusaddval 16820 qusmulval 16822 |
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