rngoisohom - Mathbox for Jeff Madsen

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Theorem rngoisohom 38292

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 2737	. . . 4 ⊢ (1^st ‘𝑅) = (1^st ‘𝑅)
2		eqid 2737	. . . 4 ⊢ ran (1^st ‘𝑅) = ran (1^st ‘𝑅)
3		eqid 2737	. . . 4 ⊢ (1^st ‘𝑆) = (1^st ‘𝑆)
4		eqid 2737	. . . 4 ⊢ ran (1^st ‘𝑆) = ran (1^st ‘𝑆)
5	1, 2, 3, 4	isrngoiso 38290	. . 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 1110	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 1087 ∈ wcel 2114 ran crn 5623 –1-1-onto→wf1o 6489 ‘cfv 6490 (class class class)co 7358 1^st c1st 7931 RingOpscrngo 38206 RingOpsHom crngohom 38272 RingOpsIso crngoiso 38273

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-sep 5231 ax-nul 5241 ax-pr 5368

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-ral 3053 df-rex 3063 df-rab 3391 df-v 3432 df-sbc 3730 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-nul 4275 df-if 4468 df-pw 4544 df-sn 4569 df-pr 4571 df-op 4575 df-uni 4852 df-br 5087 df-opab 5149 df-id 5517 df-xp 5628 df-rel 5629 df-cnv 5630 df-co 5631 df-dm 5632 df-rn 5633 df-iota 6446 df-fun 6492 df-fn 6493 df-f 6494 df-f1 6495 df-fo 6496 df-f1o 6497 df-fv 6498 df-ov 7361 df-oprab 7362 df-mpo 7363 df-rngoiso 38288

This theorem is referenced by: rngoisoco 38294

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