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Theorem rngoisohom 35376
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
StepHypRef Expression
1 eqid 2822 . . . 4 (1st𝑅) = (1st𝑅)
2 eqid 2822 . . . 4 ran (1st𝑅) = ran (1st𝑅)
3 eqid 2822 . . . 4 (1st𝑆) = (1st𝑆)
4 eqid 2822 . . . 4 ran (1st𝑆) = ran (1st𝑆)
51, 2, 3, 4isrngoiso 35374 . . 3 ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) → (𝐹 ∈ (𝑅 RngIso 𝑆) ↔ (𝐹 ∈ (𝑅 RngHom 𝑆) ∧ 𝐹:ran (1st𝑅)–1-1-onto→ran (1st𝑆))))
65simprbda 502 . 2 (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) ∧ 𝐹 ∈ (𝑅 RngIso 𝑆)) → 𝐹 ∈ (𝑅 RngHom 𝑆))
763impa 1107 1 ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngIso 𝑆)) → 𝐹 ∈ (𝑅 RngHom 𝑆))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399  w3a 1084  wcel 2114  ran crn 5533  1-1-ontowf1o 6333  cfv 6334  (class class class)co 7140  1st c1st 7673  RingOpscrngo 35290   RngHom crnghom 35356   RngIso crngiso 35357
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 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-sep 5179  ax-nul 5186  ax-pr 5307
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ral 3135  df-rex 3136  df-rab 3139  df-v 3471  df-sbc 3748  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4266  df-if 4440  df-sn 4540  df-pr 4542  df-op 4546  df-uni 4814  df-br 5043  df-opab 5105  df-id 5437  df-xp 5538  df-rel 5539  df-cnv 5540  df-co 5541  df-dm 5542  df-rn 5543  df-iota 6293  df-fun 6336  df-fn 6337  df-f 6338  df-f1 6339  df-fo 6340  df-f1o 6341  df-fv 6342  df-ov 7143  df-oprab 7144  df-mpo 7145  df-rngoiso 35372
This theorem is referenced by:  rngoisoco  35378
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