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Mirrors > Home > MPE Home > Th. List > Mathboxes > crngocom | Structured version Visualization version GIF version |
Description: The multiplication operation of a commutative ring is commutative. (Contributed by Jeff Madsen, 8-Jun-2010.) |
Ref | Expression |
---|---|
crngocom.1 | ⊢ 𝐺 = (1st ‘𝑅) |
crngocom.2 | ⊢ 𝐻 = (2nd ‘𝑅) |
crngocom.3 | ⊢ 𝑋 = ran 𝐺 |
Ref | Expression |
---|---|
crngocom | ⊢ ((𝑅 ∈ CRingOps ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝐻𝐵) = (𝐵𝐻𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | crngocom.1 | . . . . 5 ⊢ 𝐺 = (1st ‘𝑅) | |
2 | crngocom.2 | . . . . 5 ⊢ 𝐻 = (2nd ‘𝑅) | |
3 | crngocom.3 | . . . . 5 ⊢ 𝑋 = ran 𝐺 | |
4 | 1, 2, 3 | iscrngo2 35277 | . . . 4 ⊢ (𝑅 ∈ CRingOps ↔ (𝑅 ∈ RingOps ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝐻𝑦) = (𝑦𝐻𝑥))) |
5 | 4 | simprbi 499 | . . 3 ⊢ (𝑅 ∈ CRingOps → ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝐻𝑦) = (𝑦𝐻𝑥)) |
6 | oveq1 7165 | . . . . 5 ⊢ (𝑥 = 𝐴 → (𝑥𝐻𝑦) = (𝐴𝐻𝑦)) | |
7 | oveq2 7166 | . . . . 5 ⊢ (𝑥 = 𝐴 → (𝑦𝐻𝑥) = (𝑦𝐻𝐴)) | |
8 | 6, 7 | eqeq12d 2839 | . . . 4 ⊢ (𝑥 = 𝐴 → ((𝑥𝐻𝑦) = (𝑦𝐻𝑥) ↔ (𝐴𝐻𝑦) = (𝑦𝐻𝐴))) |
9 | oveq2 7166 | . . . . 5 ⊢ (𝑦 = 𝐵 → (𝐴𝐻𝑦) = (𝐴𝐻𝐵)) | |
10 | oveq1 7165 | . . . . 5 ⊢ (𝑦 = 𝐵 → (𝑦𝐻𝐴) = (𝐵𝐻𝐴)) | |
11 | 9, 10 | eqeq12d 2839 | . . . 4 ⊢ (𝑦 = 𝐵 → ((𝐴𝐻𝑦) = (𝑦𝐻𝐴) ↔ (𝐴𝐻𝐵) = (𝐵𝐻𝐴))) |
12 | 8, 11 | rspc2v 3635 | . . 3 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝐻𝑦) = (𝑦𝐻𝑥) → (𝐴𝐻𝐵) = (𝐵𝐻𝐴))) |
13 | 5, 12 | mpan9 509 | . 2 ⊢ ((𝑅 ∈ CRingOps ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → (𝐴𝐻𝐵) = (𝐵𝐻𝐴)) |
14 | 13 | 3impb 1111 | 1 ⊢ ((𝑅 ∈ CRingOps ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝐻𝐵) = (𝐵𝐻𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∧ w3a 1083 = wceq 1537 ∈ wcel 2114 ∀wral 3140 ran crn 5558 ‘cfv 6357 (class class class)co 7158 1st c1st 7689 2nd c2nd 7690 RingOpscrngo 35174 CRingOpsccring 35273 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-sbc 3775 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-br 5069 df-opab 5131 df-mpt 5149 df-id 5462 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-iota 6316 df-fun 6359 df-fv 6365 df-ov 7161 df-1st 7691 df-2nd 7692 df-rngo 35175 df-com2 35270 df-crngo 35274 |
This theorem is referenced by: crngm23 35282 crngohomfo 35286 isidlc 35295 dmncan2 35357 |
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