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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 6919 | . . . . . 6 ⊢ (Base‘𝑅) ∈ V | |
| 6 | 2, 5 | eqeltrdi 2849 | . . . . 5 ⊢ (𝜑 → 𝑉 ∈ V) | 
| 7 | erex 8769 | . . . . 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 17588 | . . 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 17594 | . . 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 17574 | . 2 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → ((𝐹‘𝑋) ∙ (𝐹‘𝑌)) = (𝐹‘(𝑋 · 𝑌))) | 
| 16 | 4 | 3ad2ant1 1134 | . . . 4 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → ∼ Er 𝑉) | 
| 17 | 6 | 3ad2ant1 1134 | . . . 4 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → 𝑉 ∈ V) | 
| 18 | 16, 17, 3 | divsfval 17592 | . . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → (𝐹‘𝑋) = [𝑋] ∼ ) | 
| 19 | 16, 17, 3 | divsfval 17592 | . . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → (𝐹‘𝑌) = [𝑌] ∼ ) | 
| 20 | 18, 19 | oveq12d 7449 | . 2 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → ((𝐹‘𝑋) ∙ (𝐹‘𝑌)) = ([𝑋] ∼ ∙ [𝑌] ∼ )) | 
| 21 | 16, 17, 3 | divsfval 17592 | . 2 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → (𝐹‘(𝑋 · 𝑌)) = [(𝑋 · 𝑌)] ∼ ) | 
| 22 | 15, 20, 21 | 3eqtr3d 2785 | 1 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → ([𝑋] ∼ ∙ [𝑌] ∼ ) = [(𝑋 · 𝑌)] ∼ ) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1087 = wceq 1540 ∈ wcel 2108 Vcvv 3480 {csn 4626 〈cop 4632 ∪ ciun 4991 class class class wbr 5143 ↦ cmpt 5225 ‘cfv 6561 (class class class)co 7431 Er wer 8742 [cec 8743 / cqs 8744 Basecbs 17247 /s cqus 17550 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-fo 6567 df-fv 6569 df-ov 7434 df-er 8745 df-ec 8747 df-qs 8751 | 
| This theorem is referenced by: qusaddval 17598 qusmulval 17600 | 
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