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| Mirrors > Home > MPE Home > Th. List > Mathboxes > rngoiso1o | Structured version Visualization version GIF version | ||
| Description: A ring isomorphism is a bijection. (Contributed by Jeff Madsen, 16-Jun-2011.) |
| Ref | Expression |
|---|---|
| rngisoval.1 | ⊢ 𝐺 = (1st ‘𝑅) |
| rngisoval.2 | ⊢ 𝑋 = ran 𝐺 |
| rngisoval.3 | ⊢ 𝐽 = (1st ‘𝑆) |
| rngisoval.4 | ⊢ 𝑌 = ran 𝐽 |
| Ref | Expression |
|---|---|
| rngoiso1o | ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RingOpsIso 𝑆)) → 𝐹:𝑋–1-1-onto→𝑌) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | rngisoval.1 | . . . 4 ⊢ 𝐺 = (1st ‘𝑅) | |
| 2 | rngisoval.2 | . . . 4 ⊢ 𝑋 = ran 𝐺 | |
| 3 | rngisoval.3 | . . . 4 ⊢ 𝐽 = (1st ‘𝑆) | |
| 4 | rngisoval.4 | . . . 4 ⊢ 𝑌 = ran 𝐽 | |
| 5 | 1, 2, 3, 4 | isrngoiso 37985 | . . 3 ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) → (𝐹 ∈ (𝑅 RingOpsIso 𝑆) ↔ (𝐹 ∈ (𝑅 RingOpsHom 𝑆) ∧ 𝐹:𝑋–1-1-onto→𝑌))) |
| 6 | 5 | simplbda 499 | . 2 ⊢ (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) ∧ 𝐹 ∈ (𝑅 RingOpsIso 𝑆)) → 𝐹:𝑋–1-1-onto→𝑌) |
| 7 | 6 | 3impa 1110 | 1 ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RingOpsIso 𝑆)) → 𝐹:𝑋–1-1-onto→𝑌) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1087 = wceq 1540 ∈ wcel 2108 ran crn 5686 –1-1-onto→wf1o 6560 ‘cfv 6561 (class class class)co 7431 1st c1st 8012 RingOpscrngo 37901 RingOpsHom crngohom 37967 RingOpsIso crngoiso 37968 |
| 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 |
| 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-sbc 3789 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 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-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-ov 7434 df-oprab 7435 df-mpo 7436 df-rngoiso 37983 |
| This theorem is referenced by: rngoisoco 37989 |
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