Theorem lmimgim 21003

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 21002	. . 3 ⊢ (𝐹 ∈ (𝑅 LMIso 𝑆) → 𝐹 ∈ (𝑅 LMHom 𝑆))
2		lmghm 20969	. . 3 ⊢ (𝐹 ∈ (𝑅 LMHom 𝑆) → 𝐹 ∈ (𝑅 GrpHom 𝑆))
3	1, 2	syl 17	. 2 ⊢ (𝐹 ∈ (𝑅 LMIso 𝑆) → 𝐹 ∈ (𝑅 GrpHom 𝑆))
4		eqid 2733	. . 3 ⊢ (Base‘𝑅) = (Base‘𝑅)
5		eqid 2733	. . 3 ⊢ (Base‘𝑆) = (Base‘𝑆)
6	4, 5	lmimf1o 21001	. 2 ⊢ (𝐹 ∈ (𝑅 LMIso 𝑆) → 𝐹:(Base‘𝑅)–1-1-onto→(Base‘𝑆))
7	4, 5	isgim 19178	. 2 ⊢ (𝐹 ∈ (𝑅 GrpIso 𝑆) ↔ (𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ 𝐹:(Base‘𝑅)–1-1-onto→(Base‘𝑆)))
8	3, 6, 7	sylanbrc 583	1 ⊢ (𝐹 ∈ (𝑅 LMIso 𝑆) → 𝐹 ∈ (𝑅 GrpIso 𝑆))

Syntax hints:   → wi 4   ∈ wcel 2113  –1-1-onto→wf1o 6487  ‘cfv 6488  (class class class)co 7354  Basecbs 17124   GrpHom cghm 19128   GrpIso cgim 19173   LMHom clmhm 20957   LMIso clmim 20958

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705  ax-sep 5238  ax-nul 5248  ax-pow 5307  ax-pr 5374  ax-un 7676

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2882  df-ne 2930  df-ral 3049  df-rex 3058  df-rab 3397  df-v 3439  df-sbc 3738  df-csb 3847  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4283  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4861  df-iun 4945  df-br 5096  df-opab 5158  df-mpt 5177  df-id 5516  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-rn 5632  df-res 5633  df-ima 5634  df-iota 6444  df-fun 6490  df-fn 6491  df-f 6492  df-f1 6493  df-fo 6494  df-f1o 6495  df-fv 6496  df-ov 7357  df-oprab 7358  df-mpo 7359  df-1st 7929  df-2nd 7930  df-map 8760  df-ghm 19129  df-gim 19175  df-lmhm 20960  df-lmim 20961

This theorem is referenced by: (None)