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Theorem lmimlmhm 20902
Description: An isomorphism of modules is a homomorphism. (Contributed by Stefan O'Rear, 21-Jan-2015.)
Assertion
Ref Expression
lmimlmhm (𝐹 ∈ (𝑅 LMIso 𝑆) → 𝐹 ∈ (𝑅 LMHom 𝑆))

Proof of Theorem lmimlmhm
StepHypRef Expression
1 eqid 2724 . . 3 (Base‘𝑅) = (Base‘𝑅)
2 eqid 2724 . . 3 (Base‘𝑆) = (Base‘𝑆)
31, 2islmim 20900 . 2 (𝐹 ∈ (𝑅 LMIso 𝑆) ↔ (𝐹 ∈ (𝑅 LMHom 𝑆) ∧ 𝐹:(Base‘𝑅)–1-1-onto→(Base‘𝑆)))
43simplbi 497 1 (𝐹 ∈ (𝑅 LMIso 𝑆) → 𝐹 ∈ (𝑅 LMHom 𝑆))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2098  1-1-ontowf1o 6532  cfv 6533  (class class class)co 7401  Basecbs 17143   LMHom clmhm 20857   LMIso clmim 20858
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 2163  ax-ext 2695  ax-sep 5289  ax-nul 5296  ax-pr 5417
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rex 3063  df-rab 3425  df-v 3468  df-sbc 3770  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-nul 4315  df-if 4521  df-sn 4621  df-pr 4623  df-op 4627  df-uni 4900  df-br 5139  df-opab 5201  df-id 5564  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-iota 6485  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-ov 7404  df-oprab 7405  df-mpo 7406  df-lmhm 20860  df-lmim 20861
This theorem is referenced by:  lmimgim  20903  lmiclcl  20908  lmicrcl  20909  lmimlbs  21699  lmimdim  33167  lnmlmic  42319
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