rngoisohom - Mathbox for Jeff Madsen

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Theorem rngoisohom 37968

Description: A ring isomorphism is a ring homomorphism. (Contributed by Jeff Madsen, 16-Jun-2011.)

**Assertion**
Ref	Expression
rngoisohom	⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RingOpsIso 𝑆)) → 𝐹 ∈ (𝑅 RingOpsHom 𝑆))

**Proof of Theorem rngoisohom**
Step	Hyp	Ref	Expression
1		eqid 2729	. . . 4 ⊢ (1^st ‘𝑅) = (1^st ‘𝑅)
2		eqid 2729	. . . 4 ⊢ ran (1^st ‘𝑅) = ran (1^st ‘𝑅)
3		eqid 2729	. . . 4 ⊢ (1^st ‘𝑆) = (1^st ‘𝑆)
4		eqid 2729	. . . 4 ⊢ ran (1^st ‘𝑆) = ran (1^st ‘𝑆)
5	1, 2, 3, 4	isrngoiso 37966	. . 3 ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) → (𝐹 ∈ (𝑅 RingOpsIso 𝑆) ↔ (𝐹 ∈ (𝑅 RingOpsHom 𝑆) ∧ 𝐹:ran (1^st ‘𝑅)–1-1-onto→ran (1^st ‘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 2109 ran crn 5632 –1-1-onto→wf1o 6498 ‘cfv 6499 (class class class)co 7369 1^st c1st 7945 RingOpscrngo 37882 RingOpsHom crngohom 37948 RingOpsIso crngoiso 37949

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 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-sep 5246 ax-nul 5256 ax-pr 5382

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-ral 3045 df-rex 3054 df-rab 3403 df-v 3446 df-sbc 3751 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4293 df-if 4485 df-pw 4561 df-sn 4586 df-pr 4588 df-op 4592 df-uni 4868 df-br 5103 df-opab 5165 df-id 5526 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-iota 6452 df-fun 6501 df-fn 6502 df-f 6503 df-f1 6504 df-fo 6505 df-f1o 6506 df-fv 6507 df-ov 7372 df-oprab 7373 df-mpo 7374 df-rngoiso 37964

This theorem is referenced by: rngoisoco 37970

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