Theorem rngoisohom 33779

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 RngIso 𝑆)) → 𝐹 ∈ (𝑅 RngHom 𝑆))

Proof of Theorem rngoisohom

Step	Hyp	Ref	Expression
1		eqid 2622	. . . 4 ⊢ (1st ‘𝑅) = (1st ‘𝑅)
2		eqid 2622	. . . 4 ⊢ ran (1st ‘𝑅) = ran (1st ‘𝑅)
3		eqid 2622	. . . 4 ⊢ (1st ‘𝑆) = (1st ‘𝑆)
4		eqid 2622	. . . 4 ⊢ ran (1st ‘𝑆) = ran (1st ‘𝑆)
5	1, 2, 3, 4	isrngoiso 33777	. . 3 ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) → (𝐹 ∈ (𝑅 RngIso 𝑆) ↔ (𝐹 ∈ (𝑅 RngHom 𝑆) ∧ 𝐹:ran (1st ‘𝑅)–1-1-onto→ran (1st ‘𝑆))))
6	5	simprbda 653	. 2 ⊢ (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) ∧ 𝐹 ∈ (𝑅 RngIso 𝑆)) → 𝐹 ∈ (𝑅 RngHom 𝑆))
7	6	3impa 1259	1 ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngIso 𝑆)) → 𝐹 ∈ (𝑅 RngHom 𝑆))

Syntax hints:   → wi 4   ∧ wa 384   ∧ w3a 1037   ∈ wcel 1990  ran crn 5115  –1-1-onto→wf1o 5887  ‘cfv 5888  (class class class)co 6650  1^st c1st 7166  RingOpscrngo 33693   RngHom crnghom 33759   RngIso crngiso 33760

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-rngoiso 33775

This theorem is referenced by:  rngoisoco  33781