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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 17556 | . . . 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 771 | . . . . . . . 8 ⊢ ((𝜑 ∧ (𝑟 = 𝑅 ∧ 𝑒 = ∼ )) → 𝑟 = 𝑅) | |
5 | 4 | fveq2d 6911 | . . . . . . 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 2778 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑟 = 𝑅 ∧ 𝑒 = ∼ )) → (Base‘𝑟) = 𝑉) |
9 | eceq2 8785 | . . . . . . 7 ⊢ (𝑒 = ∼ → [𝑥]𝑒 = [𝑥] ∼ ) | |
10 | 9 | ad2antll 729 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑟 = 𝑅 ∧ 𝑒 = ∼ )) → [𝑥]𝑒 = [𝑥] ∼ ) |
11 | 8, 10 | mpteq12dv 5239 | . . . . 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 2793 | . . . 4 ⊢ ((𝜑 ∧ (𝑟 = 𝑅 ∧ 𝑒 = ∼ )) → (𝑥 ∈ (Base‘𝑟) ↦ [𝑥]𝑒) = 𝐹) |
14 | 13, 4 | oveq12d 7449 | . . 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 3502 | . . 3 ⊢ (𝜑 → 𝑅 ∈ V) |
17 | qusval.e | . . . 4 ⊢ (𝜑 → ∼ ∈ 𝑊) | |
18 | 17 | elexd 3502 | . . 3 ⊢ (𝜑 → ∼ ∈ V) |
19 | ovexd 7466 | . . 3 ⊢ (𝜑 → (𝐹 “s 𝑅) ∈ V) | |
20 | 3, 14, 16, 18, 19 | ovmpod 7585 | . 2 ⊢ (𝜑 → (𝑅 /s ∼ ) = (𝐹 “s 𝑅)) |
21 | 1, 20 | eqtrd 2775 | 1 ⊢ (𝜑 → 𝑈 = (𝐹 “s 𝑅)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1537 ∈ wcel 2106 Vcvv 3478 ↦ cmpt 5231 ‘cfv 6563 (class class class)co 7431 ∈ cmpo 7433 [cec 8742 Basecbs 17245 “s cimas 17551 /s cqus 17552 |
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 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-sbc 3792 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-iota 6516 df-fun 6565 df-fv 6571 df-ov 7434 df-oprab 7435 df-mpo 7436 df-ec 8746 df-qus 17556 |
This theorem is referenced by: qusin 17591 qusbas 17592 quss 17593 qusaddval 17600 qusaddf 17601 qusmulval 17602 qusmulf 17603 qusgrp2 19089 qusrng 20198 qusring2 20348 qustps 23746 qustgpopn 24144 qustgplem 24145 qustgphaus 24147 qusvsval 33360 quslmod 33366 quslmhm 33367 |
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