Theorem rngoisohom 37510

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 2725	. . . 4 ⊢ (1st ‘𝑅) = (1st ‘𝑅)
2		eqid 2725	. . . 4 ⊢ ran (1st ‘𝑅) = ran (1st ‘𝑅)
3		eqid 2725	. . . 4 ⊢ (1st ‘𝑆) = (1st ‘𝑆)
4		eqid 2725	. . . 4 ⊢ ran (1st ‘𝑆) = ran (1st ‘𝑆)
5	1, 2, 3, 4	isrngoiso 37508	. . 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 497	. 2 ⊢ (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) ∧ 𝐹 ∈ (𝑅 RingOpsIso 𝑆)) → 𝐹 ∈ (𝑅 RingOpsHom 𝑆))
7	6	3impa 1107	1 ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RingOpsIso 𝑆)) → 𝐹 ∈ (𝑅 RingOpsHom 𝑆))

Syntax hints:   → wi 4   ∧ wa 394   ∧ w3a 1084   ∈ wcel 2098  ran crn 5673  –1-1-onto→wf1o 6542  ‘cfv 6543  (class class class)co 7416  1^st c1st 7989  RingOpscrngo 37424   RingOpsHom crngohom 37490   RingOpsIso crngoiso 37491

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5294  ax-nul 5301  ax-pr 5423

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2931  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3465  df-sbc 3769  df-dif 3942  df-un 3944  df-in 3946  df-ss 3956  df-nul 4319  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-br 5144  df-opab 5206  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-ov 7419  df-oprab 7420  df-mpo 7421  df-rngoiso 37506

This theorem is referenced by:  rngoisoco  37512