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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 6915 | . . . . 5 ⊢ (Base‘𝑅) ∈ V | |
6 | 2, 5 | eqeltrdi 2837 | . . . 4 ⊢ (𝜑 → 𝑉 ∈ V) |
7 | erex 8755 | . . . 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 17532 | . 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 17538 | . 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 17519 | 1 ⊢ (𝜑 → ∙ :((𝑉 / ∼ ) × (𝑉 / ∼ ))⟶(𝑉 / ∼ )) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 = wceq 1533 ∈ wcel 2098 Vcvv 3473 {csn 4632 〈cop 4638 ∪ ciun 5000 class class class wbr 5152 ↦ cmpt 5235 × cxp 5680 ⟶wf 6549 ‘cfv 6553 (class class class)co 7426 Er wer 8728 [cec 8729 / cqs 8730 Basecbs 17187 /s cqus 17494 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2699 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7746 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-id 5580 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-fo 6559 df-fv 6561 df-ov 7429 df-er 8731 df-ec 8733 df-qs 8737 |
This theorem is referenced by: qusaddf 17543 qusmulf 17545 |
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