Theorem rngoisohom 35262

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 2824	. . . 4 ⊢ (1st ‘𝑅) = (1st ‘𝑅)
2		eqid 2824	. . . 4 ⊢ ran (1st ‘𝑅) = ran (1st ‘𝑅)
3		eqid 2824	. . . 4 ⊢ (1st ‘𝑆) = (1st ‘𝑆)
4		eqid 2824	. . . 4 ⊢ ran (1st ‘𝑆) = ran (1st ‘𝑆)
5	1, 2, 3, 4	isrngoiso 35260	. . 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 501	. 2 ⊢ (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) ∧ 𝐹 ∈ (𝑅 RngIso 𝑆)) → 𝐹 ∈ (𝑅 RngHom 𝑆))
7	6	3impa 1106	1 ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngIso 𝑆)) → 𝐹 ∈ (𝑅 RngHom 𝑆))

Syntax hints:   → wi 4   ∧ wa 398   ∧ w3a 1083   ∈ wcel 2113  ran crn 5559  –1-1-onto→wf1o 6357  ‘cfv 6358  (class class class)co 7159  1^st c1st 7690  RingOpscrngo 35176   RngHom crnghom 35242   RngIso crngiso 35243

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 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2160  ax-12 2176  ax-ext 2796  ax-sep 5206  ax-nul 5213  ax-pr 5333

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1539  df-ex 1780  df-nf 1784  df-sb 2069  df-mo 2621  df-eu 2653  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2966  df-ral 3146  df-rex 3147  df-rab 3150  df-v 3499  df-sbc 3776  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-nul 4295  df-if 4471  df-sn 4571  df-pr 4573  df-op 4577  df-uni 4842  df-br 5070  df-opab 5132  df-id 5463  df-xp 5564  df-rel 5565  df-cnv 5566  df-co 5567  df-dm 5568  df-rn 5569  df-iota 6317  df-fun 6360  df-fn 6361  df-f 6362  df-f1 6363  df-fo 6364  df-f1o 6365  df-fv 6366  df-ov 7162  df-oprab 7163  df-mpo 7164  df-rngoiso 35258

This theorem is referenced by:  rngoisoco  35264