Theorem lmimgim 21373

Description: An isomorphism of modules is an isomorphism of groups. (Contributed by Stefan O'Rear, 21-Jan-2015.) (Revised by Mario Carneiro, 6-May-2015.)

Assertion

Ref	Expression
lmimgim	⊢ (𝐹 ∈ (𝑅 LMIso 𝑆) → 𝐹 ∈ (𝑅 GrpIso 𝑆))

Proof of Theorem lmimgim

Step	Hyp	Ref	Expression
1		lmimlmhm 21372	. . 3 ⊢ (𝐹 ∈ (𝑅 LMIso 𝑆) → 𝐹 ∈ (𝑅 LMHom 𝑆))
2		lmghm 21339	. . 3 ⊢ (𝐹 ∈ (𝑅 LMHom 𝑆) → 𝐹 ∈ (𝑅 GrpHom 𝑆))
3	1, 2	syl 18	. 2 ⊢ (𝐹 ∈ (𝑅 LMIso 𝑆) → 𝐹 ∈ (𝑅 GrpHom 𝑆))
4		eqid 2763	. . 3 ⊢ (Base‘𝑅) = (Base‘𝑅)
5		eqid 2763	. . 3 ⊢ (Base‘𝑆) = (Base‘𝑆)
6	4, 5	lmimf1o 21371	. 2 ⊢ (𝐹 ∈ (𝑅 LMIso 𝑆) → 𝐹:(Base‘𝑅)⟶1-1onto(Base‘𝑆))
7	4, 5	isgim 19508	. 2 ⊢ (𝐹 ∈ (𝑅 GrpIso 𝑆) ↔ (𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ 𝐹:(Base‘𝑅)⟶1-1onto(Base‘𝑆)))
8	3, 6, 7	sylanbrc 595	1 ⊢ (𝐹 ∈ (𝑅 LMIso 𝑆) → 𝐹 ∈ (𝑅 GrpIso 𝑆))

This proof depends on syntax axioms:   → wi 4   ∈ wcel 2147  ⟶1-1ontowf1o 6560  ‘cfv 6561  (class class class)co 7446  Basecbs 17419   GrpHom cghm 19459   GrpIso cgim 19503   LMHom clmhm 21327   LMIso clmim 21328

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5253  ax-nul 5264  ax-pow 5332  ax-pr 5396  ax-un 7778

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-iun 4956  df-br 5107  df-opab 5172  df-mpt 5191  df-id 5551  df-xp 5662  df-rel 5663  df-cnv 5664  df-co 5665  df-dm 5666  df-rn 5667  df-res 5668  df-ima 5669  df-iota 6515  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7449  df-oprab 7450  df-mpo 7451  df-1st 8028  df-2nd 8029  df-map 8880  df-ghm 19460  df-gim 19505  df-lmhm 21330  df-lmim 21331

This theorem is used by: (None)