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| Mirrors > Home > MPE Home > Th. List > qusval | Structured version Visualization version GIF version | ||
| Description: Value of a quotient structure. (Contributed by Mario Carneiro, 23-Feb-2015.) | 
| Ref | Expression | 
|---|---|
| qusval.u | ⊢ (𝜑 → 𝑈 = (𝑅 /s ∼ )) | 
| qusval.v | ⊢ (𝜑 → 𝑉 = (Base‘𝑅)) | 
| qusval.f | ⊢ 𝐹 = (𝑥 ∈ 𝑉 ↦ [𝑥] ∼ ) | 
| qusval.e | ⊢ (𝜑 → ∼ ∈ 𝑊) | 
| qusval.r | ⊢ (𝜑 → 𝑅 ∈ 𝑍) | 
| Ref | Expression | 
|---|---|
| qusval | ⊢ (𝜑 → 𝑈 = (𝐹 “s 𝑅)) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | qusval.u | . 2 ⊢ (𝜑 → 𝑈 = (𝑅 /s ∼ )) | |
| 2 | df-qus 17555 | . . . 4 ⊢ /s = (𝑟 ∈ V, 𝑒 ∈ V ↦ ((𝑥 ∈ (Base‘𝑟) ↦ [𝑥]𝑒) “s 𝑟)) | |
| 3 | 2 | a1i 11 | . . 3 ⊢ (𝜑 → /s = (𝑟 ∈ V, 𝑒 ∈ V ↦ ((𝑥 ∈ (Base‘𝑟) ↦ [𝑥]𝑒) “s 𝑟))) | 
| 4 | simprl 770 | . . . . . . . 8 ⊢ ((𝜑 ∧ (𝑟 = 𝑅 ∧ 𝑒 = ∼ )) → 𝑟 = 𝑅) | |
| 5 | 4 | fveq2d 6909 | . . . . . . 7 ⊢ ((𝜑 ∧ (𝑟 = 𝑅 ∧ 𝑒 = ∼ )) → (Base‘𝑟) = (Base‘𝑅)) | 
| 6 | qusval.v | . . . . . . . 8 ⊢ (𝜑 → 𝑉 = (Base‘𝑅)) | |
| 7 | 6 | adantr 480 | . . . . . . 7 ⊢ ((𝜑 ∧ (𝑟 = 𝑅 ∧ 𝑒 = ∼ )) → 𝑉 = (Base‘𝑅)) | 
| 8 | 5, 7 | eqtr4d 2779 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑟 = 𝑅 ∧ 𝑒 = ∼ )) → (Base‘𝑟) = 𝑉) | 
| 9 | eceq2 8787 | . . . . . . 7 ⊢ (𝑒 = ∼ → [𝑥]𝑒 = [𝑥] ∼ ) | |
| 10 | 9 | ad2antll 729 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑟 = 𝑅 ∧ 𝑒 = ∼ )) → [𝑥]𝑒 = [𝑥] ∼ ) | 
| 11 | 8, 10 | mpteq12dv 5232 | . . . . 5 ⊢ ((𝜑 ∧ (𝑟 = 𝑅 ∧ 𝑒 = ∼ )) → (𝑥 ∈ (Base‘𝑟) ↦ [𝑥]𝑒) = (𝑥 ∈ 𝑉 ↦ [𝑥] ∼ )) | 
| 12 | qusval.f | . . . . 5 ⊢ 𝐹 = (𝑥 ∈ 𝑉 ↦ [𝑥] ∼ ) | |
| 13 | 11, 12 | eqtr4di 2794 | . . . 4 ⊢ ((𝜑 ∧ (𝑟 = 𝑅 ∧ 𝑒 = ∼ )) → (𝑥 ∈ (Base‘𝑟) ↦ [𝑥]𝑒) = 𝐹) | 
| 14 | 13, 4 | oveq12d 7450 | . . 3 ⊢ ((𝜑 ∧ (𝑟 = 𝑅 ∧ 𝑒 = ∼ )) → ((𝑥 ∈ (Base‘𝑟) ↦ [𝑥]𝑒) “s 𝑟) = (𝐹 “s 𝑅)) | 
| 15 | qusval.r | . . . 4 ⊢ (𝜑 → 𝑅 ∈ 𝑍) | |
| 16 | 15 | elexd 3503 | . . 3 ⊢ (𝜑 → 𝑅 ∈ V) | 
| 17 | qusval.e | . . . 4 ⊢ (𝜑 → ∼ ∈ 𝑊) | |
| 18 | 17 | elexd 3503 | . . 3 ⊢ (𝜑 → ∼ ∈ V) | 
| 19 | ovexd 7467 | . . 3 ⊢ (𝜑 → (𝐹 “s 𝑅) ∈ V) | |
| 20 | 3, 14, 16, 18, 19 | ovmpod 7586 | . 2 ⊢ (𝜑 → (𝑅 /s ∼ ) = (𝐹 “s 𝑅)) | 
| 21 | 1, 20 | eqtrd 2776 | 1 ⊢ (𝜑 → 𝑈 = (𝐹 “s 𝑅)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1539 ∈ wcel 2107 Vcvv 3479 ↦ cmpt 5224 ‘cfv 6560 (class class class)co 7432 ∈ cmpo 7434 [cec 8744 Basecbs 17248 “s cimas 17550 /s cqus 17551 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pr 5431 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-ral 3061 df-rex 3070 df-rab 3436 df-v 3481 df-sbc 3788 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-nul 4333 df-if 4525 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-br 5143 df-opab 5205 df-mpt 5225 df-id 5577 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-iota 6513 df-fun 6562 df-fv 6568 df-ov 7435 df-oprab 7436 df-mpo 7437 df-ec 8748 df-qus 17555 | 
| This theorem is referenced by: qusin 17590 qusbas 17591 quss 17592 qusaddval 17599 qusaddf 17600 qusmulval 17601 qusmulf 17602 qusgrp2 19077 qusrng 20178 qusring2 20332 qustps 23731 qustgpopn 24129 qustgplem 24130 qustgphaus 24132 qusvsval 33381 quslmod 33387 quslmhm 33388 | 
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