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Mirrors > Home > MPE Home > Th. List > lmimgim | Structured version Visualization version GIF version |
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.) |
Ref | Expression |
---|---|
lmimgim | ⊢ (𝐹 ∈ (𝑅 LMIso 𝑆) → 𝐹 ∈ (𝑅 GrpIso 𝑆)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lmimlmhm 20909 | . . 3 ⊢ (𝐹 ∈ (𝑅 LMIso 𝑆) → 𝐹 ∈ (𝑅 LMHom 𝑆)) | |
2 | lmghm 20876 | . . 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 2726 | . . 3 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
5 | eqid 2726 | . . 3 ⊢ (Base‘𝑆) = (Base‘𝑆) | |
6 | 4, 5 | lmimf1o 20908 | . 2 ⊢ (𝐹 ∈ (𝑅 LMIso 𝑆) → 𝐹:(Base‘𝑅)–1-1-onto→(Base‘𝑆)) |
7 | 4, 5 | isgim 19184 | . 2 ⊢ (𝐹 ∈ (𝑅 GrpIso 𝑆) ↔ (𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ 𝐹:(Base‘𝑅)–1-1-onto→(Base‘𝑆))) |
8 | 3, 6, 7 | sylanbrc 582 | 1 ⊢ (𝐹 ∈ (𝑅 LMIso 𝑆) → 𝐹 ∈ (𝑅 GrpIso 𝑆)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2098 –1-1-onto→wf1o 6535 ‘cfv 6536 (class class class)co 7404 Basecbs 17150 GrpHom cghm 19135 GrpIso cgim 19179 LMHom clmhm 20864 LMIso clmim 20865 |
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 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7721 |
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 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-ral 3056 df-rex 3065 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-id 5567 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-iota 6488 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-ov 7407 df-oprab 7408 df-mpo 7409 df-ghm 19136 df-gim 19181 df-lmhm 20867 df-lmim 20868 |
This theorem is referenced by: (None) |
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