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Mathbox for Jeff Madsen |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > rngoisohom | Structured version Visualization version GIF version |
Description: A ring isomorphism is a ring homomorphism. (Contributed by Jeff Madsen, 16-Jun-2011.) |
Ref | Expression |
---|---|
rngoisohom | ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RingOpsIso 𝑆)) → 𝐹 ∈ (𝑅 RingOpsHom 𝑆)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2728 | . . . 4 ⊢ (1st ‘𝑅) = (1st ‘𝑅) | |
2 | eqid 2728 | . . . 4 ⊢ ran (1st ‘𝑅) = ran (1st ‘𝑅) | |
3 | eqid 2728 | . . . 4 ⊢ (1st ‘𝑆) = (1st ‘𝑆) | |
4 | eqid 2728 | . . . 4 ⊢ ran (1st ‘𝑆) = ran (1st ‘𝑆) | |
5 | 1, 2, 3, 4 | isrngoiso 37451 | . . 3 ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) → (𝐹 ∈ (𝑅 RingOpsIso 𝑆) ↔ (𝐹 ∈ (𝑅 RingOpsHom 𝑆) ∧ 𝐹:ran (1st ‘𝑅)–1-1-onto→ran (1st ‘𝑆)))) |
6 | 5 | simprbda 498 | . 2 ⊢ (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) ∧ 𝐹 ∈ (𝑅 RingOpsIso 𝑆)) → 𝐹 ∈ (𝑅 RingOpsHom 𝑆)) |
7 | 6 | 3impa 1108 | 1 ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RingOpsIso 𝑆)) → 𝐹 ∈ (𝑅 RingOpsHom 𝑆)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1085 ∈ wcel 2099 ran crn 5679 –1-1-onto→wf1o 6547 ‘cfv 6548 (class class class)co 7420 1st c1st 7991 RingOpscrngo 37367 RingOpsHom crngohom 37433 RingOpsIso crngoiso 37434 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-sep 5299 ax-nul 5306 ax-pr 5429 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-rab 3430 df-v 3473 df-sbc 3777 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4324 df-if 4530 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-br 5149 df-opab 5211 df-id 5576 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-ov 7423 df-oprab 7424 df-mpo 7425 df-rngoiso 37449 |
This theorem is referenced by: rngoisoco 37455 |
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