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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 15992 | . . . 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 790 | . . . . . . . 8 ⊢ ((𝜑 ∧ (𝑟 = 𝑅 ∧ 𝑒 = ∼ )) → 𝑟 = 𝑅) | |
5 | 4 | fveq2d 6107 | . . . . . . 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 2647 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑟 = 𝑅 ∧ 𝑒 = ∼ )) → (Base‘𝑟) = 𝑉) |
9 | eceq2 7671 | . . . . . . 7 ⊢ (𝑒 = ∼ → [𝑥]𝑒 = [𝑥] ∼ ) | |
10 | 9 | ad2antll 761 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑟 = 𝑅 ∧ 𝑒 = ∼ )) → [𝑥]𝑒 = [𝑥] ∼ ) |
11 | 8, 10 | mpteq12dv 4663 | . . . . 5 ⊢ ((𝜑 ∧ (𝑟 = 𝑅 ∧ 𝑒 = ∼ )) → (𝑥 ∈ (Base‘𝑟) ↦ [𝑥]𝑒) = (𝑥 ∈ 𝑉 ↦ [𝑥] ∼ )) |
12 | qusval.f | . . . . 5 ⊢ 𝐹 = (𝑥 ∈ 𝑉 ↦ [𝑥] ∼ ) | |
13 | 11, 12 | syl6eqr 2662 | . . . 4 ⊢ ((𝜑 ∧ (𝑟 = 𝑅 ∧ 𝑒 = ∼ )) → (𝑥 ∈ (Base‘𝑟) ↦ [𝑥]𝑒) = 𝐹) |
14 | 13, 4 | oveq12d 6567 | . . 3 ⊢ ((𝜑 ∧ (𝑟 = 𝑅 ∧ 𝑒 = ∼ )) → ((𝑥 ∈ (Base‘𝑟) ↦ [𝑥]𝑒) “s 𝑟) = (𝐹 “s 𝑅)) |
15 | qusval.r | . . . 4 ⊢ (𝜑 → 𝑅 ∈ 𝑍) | |
16 | elex 3185 | . . . 4 ⊢ (𝑅 ∈ 𝑍 → 𝑅 ∈ V) | |
17 | 15, 16 | syl 17 | . . 3 ⊢ (𝜑 → 𝑅 ∈ V) |
18 | qusval.e | . . . 4 ⊢ (𝜑 → ∼ ∈ 𝑊) | |
19 | elex 3185 | . . . 4 ⊢ ( ∼ ∈ 𝑊 → ∼ ∈ V) | |
20 | 18, 19 | syl 17 | . . 3 ⊢ (𝜑 → ∼ ∈ V) |
21 | ovex 6577 | . . . 4 ⊢ (𝐹 “s 𝑅) ∈ V | |
22 | 21 | a1i 11 | . . 3 ⊢ (𝜑 → (𝐹 “s 𝑅) ∈ V) |
23 | 3, 14, 17, 20, 22 | ovmpt2d 6686 | . 2 ⊢ (𝜑 → (𝑅 /s ∼ ) = (𝐹 “s 𝑅)) |
24 | 1, 23 | eqtrd 2644 | 1 ⊢ (𝜑 → 𝑈 = (𝐹 “s 𝑅)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1475 ∈ wcel 1977 Vcvv 3173 ↦ cmpt 4643 ‘cfv 5804 (class class class)co 6549 ↦ cmpt2 6551 [cec 7627 Basecbs 15695 “s cimas 15987 /s cqus 15988 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fv 5812 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-ec 7631 df-qus 15992 |
This theorem is referenced by: qusin 16027 qusbas 16028 quss 16029 qusaddval 16036 qusaddf 16037 qusmulval 16038 qusmulf 16039 qusgrp2 17356 qusring2 18443 qustps 21335 qustgpopn 21733 qustgplem 21734 qustgphaus 21736 |
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