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Mirrors > Home > MPE Home > Th. List > giclcl | Structured version Visualization version GIF version |
Description: Isomorphism implies the left side is a group. (Contributed by Stefan O'Rear, 25-Jan-2015.) |
Ref | Expression |
---|---|
giclcl | ⊢ (𝑅 ≃𝑔 𝑆 → 𝑅 ∈ Grp) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | brgic 18193 | . . 3 ⊢ (𝑅 ≃𝑔 𝑆 ↔ (𝑅 GrpIso 𝑆) ≠ ∅) | |
2 | n0 4191 | . . 3 ⊢ ((𝑅 GrpIso 𝑆) ≠ ∅ ↔ ∃𝑓 𝑓 ∈ (𝑅 GrpIso 𝑆)) | |
3 | 1, 2 | bitri 267 | . 2 ⊢ (𝑅 ≃𝑔 𝑆 ↔ ∃𝑓 𝑓 ∈ (𝑅 GrpIso 𝑆)) |
4 | gimghm 18188 | . . . 4 ⊢ (𝑓 ∈ (𝑅 GrpIso 𝑆) → 𝑓 ∈ (𝑅 GrpHom 𝑆)) | |
5 | ghmgrp1 18144 | . . . 4 ⊢ (𝑓 ∈ (𝑅 GrpHom 𝑆) → 𝑅 ∈ Grp) | |
6 | 4, 5 | syl 17 | . . 3 ⊢ (𝑓 ∈ (𝑅 GrpIso 𝑆) → 𝑅 ∈ Grp) |
7 | 6 | exlimiv 1890 | . 2 ⊢ (∃𝑓 𝑓 ∈ (𝑅 GrpIso 𝑆) → 𝑅 ∈ Grp) |
8 | 3, 7 | sylbi 209 | 1 ⊢ (𝑅 ≃𝑔 𝑆 → 𝑅 ∈ Grp) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∃wex 1743 ∈ wcel 2051 ≠ wne 2962 ∅c0 4173 class class class wbr 4926 (class class class)co 6975 Grpcgrp 17904 GrpHom cghm 18139 GrpIso cgim 18181 ≃𝑔 cgic 18182 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1759 ax-4 1773 ax-5 1870 ax-6 1929 ax-7 1966 ax-8 2053 ax-9 2060 ax-10 2080 ax-11 2094 ax-12 2107 ax-13 2302 ax-ext 2745 ax-rep 5046 ax-sep 5057 ax-nul 5064 ax-pow 5116 ax-pr 5183 ax-un 7278 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 835 df-3an 1071 df-tru 1511 df-ex 1744 df-nf 1748 df-sb 2017 df-mo 2548 df-eu 2585 df-clab 2754 df-cleq 2766 df-clel 2841 df-nfc 2913 df-ne 2963 df-ral 3088 df-rex 3089 df-reu 3090 df-rab 3092 df-v 3412 df-sbc 3677 df-csb 3782 df-dif 3827 df-un 3829 df-in 3831 df-ss 3838 df-nul 4174 df-if 4346 df-pw 4419 df-sn 4437 df-pr 4439 df-op 4443 df-uni 4710 df-iun 4791 df-br 4927 df-opab 4989 df-mpt 5006 df-id 5309 df-xp 5410 df-rel 5411 df-cnv 5412 df-co 5413 df-dm 5414 df-rn 5415 df-res 5416 df-ima 5417 df-suc 6033 df-iota 6150 df-fun 6188 df-fn 6189 df-f 6190 df-f1 6191 df-fo 6192 df-f1o 6193 df-fv 6194 df-ov 6978 df-oprab 6979 df-mpo 6980 df-1st 7500 df-2nd 7501 df-1o 7904 df-ghm 18140 df-gim 18183 df-gic 18184 |
This theorem is referenced by: gicer 18200 |
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