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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 38004 | . . . 4 ⊢ (𝑅 ∈ CRingOps ↔ (𝑅 ∈ RingOps ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝐻𝑦) = (𝑦𝐻𝑥))) |
| 5 | 4 | simprbi 496 | . . 3 ⊢ (𝑅 ∈ CRingOps → ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝐻𝑦) = (𝑦𝐻𝑥)) |
| 6 | oveq1 7438 | . . . . 5 ⊢ (𝑥 = 𝐴 → (𝑥𝐻𝑦) = (𝐴𝐻𝑦)) | |
| 7 | oveq2 7439 | . . . . 5 ⊢ (𝑥 = 𝐴 → (𝑦𝐻𝑥) = (𝑦𝐻𝐴)) | |
| 8 | 6, 7 | eqeq12d 2753 | . . . 4 ⊢ (𝑥 = 𝐴 → ((𝑥𝐻𝑦) = (𝑦𝐻𝑥) ↔ (𝐴𝐻𝑦) = (𝑦𝐻𝐴))) |
| 9 | oveq2 7439 | . . . . 5 ⊢ (𝑦 = 𝐵 → (𝐴𝐻𝑦) = (𝐴𝐻𝐵)) | |
| 10 | oveq1 7438 | . . . . 5 ⊢ (𝑦 = 𝐵 → (𝑦𝐻𝐴) = (𝐵𝐻𝐴)) | |
| 11 | 9, 10 | eqeq12d 2753 | . . . 4 ⊢ (𝑦 = 𝐵 → ((𝐴𝐻𝑦) = (𝑦𝐻𝐴) ↔ (𝐴𝐻𝐵) = (𝐵𝐻𝐴))) |
| 12 | 8, 11 | rspc2v 3633 | . . 3 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝐻𝑦) = (𝑦𝐻𝑥) → (𝐴𝐻𝐵) = (𝐵𝐻𝐴))) |
| 13 | 5, 12 | mpan9 506 | . 2 ⊢ ((𝑅 ∈ CRingOps ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → (𝐴𝐻𝐵) = (𝐵𝐻𝐴)) |
| 14 | 13 | 3impb 1115 | 1 ⊢ ((𝑅 ∈ CRingOps ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝐻𝐵) = (𝐵𝐻𝐴)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1087 = wceq 1540 ∈ wcel 2108 ∀wral 3061 ran crn 5686 ‘cfv 6561 (class class class)co 7431 1st c1st 8012 2nd c2nd 8013 RingOpscrngo 37901 CRingOpsccring 38000 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 ax-un 7755 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-iota 6514 df-fun 6563 df-fv 6569 df-ov 7434 df-1st 8014 df-2nd 8015 df-rngo 37902 df-com2 37997 df-crngo 38001 |
| This theorem is referenced by: crngm23 38009 crngohomfo 38013 isidlc 38022 dmncan2 38084 |
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