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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 2735 | . . . 4 ⊢ (1st ‘𝑅) = (1st ‘𝑅) | |
2 | eqid 2735 | . . . 4 ⊢ ran (1st ‘𝑅) = ran (1st ‘𝑅) | |
3 | eqid 2735 | . . . 4 ⊢ (1st ‘𝑆) = (1st ‘𝑆) | |
4 | eqid 2735 | . . . 4 ⊢ ran (1st ‘𝑆) = ran (1st ‘𝑆) | |
5 | 1, 2, 3, 4 | isrngoiso 37965 | . . 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 1109 | 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 1086 ∈ wcel 2106 ran crn 5690 –1-1-onto→wf1o 6562 ‘cfv 6563 (class class class)co 7431 1st c1st 8011 RingOpscrngo 37881 RingOpsHom crngohom 37947 RingOpsIso crngoiso 37948 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-sbc 3792 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-ov 7434 df-oprab 7435 df-mpo 7436 df-rngoiso 37963 |
This theorem is referenced by: rngoisoco 37969 |
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