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Mirrors > Home > MPE Home > Th. List > qusaddflem | Structured version Visualization version GIF version |
Description: The operation of a quotient structure is a function. (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 |
---|---|
qusaddflem | ⊢ (𝜑 → ∙ :((𝑉 / ∼ ) × (𝑉 / ∼ ))⟶(𝑉 / ∼ )) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | qusaddf.u | . . 3 ⊢ (𝜑 → 𝑈 = (𝑅 /s ∼ )) | |
2 | qusaddf.v | . . 3 ⊢ (𝜑 → 𝑉 = (Base‘𝑅)) | |
3 | qusaddflem.f | . . 3 ⊢ 𝐹 = (𝑥 ∈ 𝑉 ↦ [𝑥] ∼ ) | |
4 | qusaddf.r | . . . 4 ⊢ (𝜑 → ∼ Er 𝑉) | |
5 | fvex 6676 | . . . . 5 ⊢ (Base‘𝑅) ∈ V | |
6 | 2, 5 | syl6eqel 2918 | . . . 4 ⊢ (𝜑 → 𝑉 ∈ V) |
7 | erex 8302 | . . . 4 ⊢ ( ∼ Er 𝑉 → (𝑉 ∈ V → ∼ ∈ V)) | |
8 | 4, 6, 7 | sylc 65 | . . 3 ⊢ (𝜑 → ∼ ∈ V) |
9 | qusaddf.z | . . 3 ⊢ (𝜑 → 𝑅 ∈ 𝑍) | |
10 | 1, 2, 3, 8, 9 | quslem 16804 | . 2 ⊢ (𝜑 → 𝐹:𝑉–onto→(𝑉 / ∼ )) |
11 | qusaddf.c | . . 3 ⊢ ((𝜑 ∧ (𝑝 ∈ 𝑉 ∧ 𝑞 ∈ 𝑉)) → (𝑝 · 𝑞) ∈ 𝑉) | |
12 | qusaddf.e | . . 3 ⊢ (𝜑 → ((𝑎 ∼ 𝑝 ∧ 𝑏 ∼ 𝑞) → (𝑎 · 𝑏) ∼ (𝑝 · 𝑞))) | |
13 | 4, 6, 3, 11, 12 | ercpbl 16810 | . 2 ⊢ ((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 | . 2 ⊢ (𝜑 → ∙ = ∪ 𝑝 ∈ 𝑉 ∪ 𝑞 ∈ 𝑉 {〈〈(𝐹‘𝑝), (𝐹‘𝑞)〉, (𝐹‘(𝑝 · 𝑞))〉}) | |
15 | 10, 13, 14, 11 | imasaddflem 16791 | 1 ⊢ (𝜑 → ∙ :((𝑉 / ∼ ) × (𝑉 / ∼ ))⟶(𝑉 / ∼ )) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1528 ∈ wcel 2105 Vcvv 3492 {csn 4557 〈cop 4563 ∪ ciun 4910 class class class wbr 5057 ↦ cmpt 5137 × cxp 5546 ⟶wf 6344 ‘cfv 6348 (class class class)co 7145 Er wer 8275 [cec 8276 / cqs 8277 Basecbs 16471 /s cqus 16766 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-fo 6354 df-fv 6356 df-ov 7148 df-er 8278 df-ec 8280 df-qs 8284 |
This theorem is referenced by: qusaddf 16815 qusmulf 16817 |
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