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Mirrors > Home > MPE Home > Th. List > gicrcl | Structured version Visualization version GIF version |
Description: Isomorphism implies the right side is a group. (Contributed by Mario Carneiro, 6-May-2015.) |
Ref | Expression |
---|---|
gicrcl | ⊢ (𝑅 ≃𝑔 𝑆 → 𝑆 ∈ Grp) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | brgic 18412 | . . 3 ⊢ (𝑅 ≃𝑔 𝑆 ↔ (𝑅 GrpIso 𝑆) ≠ ∅) | |
2 | n0 4313 | . . 3 ⊢ ((𝑅 GrpIso 𝑆) ≠ ∅ ↔ ∃𝑓 𝑓 ∈ (𝑅 GrpIso 𝑆)) | |
3 | 1, 2 | bitri 277 | . 2 ⊢ (𝑅 ≃𝑔 𝑆 ↔ ∃𝑓 𝑓 ∈ (𝑅 GrpIso 𝑆)) |
4 | gimghm 18407 | . . . 4 ⊢ (𝑓 ∈ (𝑅 GrpIso 𝑆) → 𝑓 ∈ (𝑅 GrpHom 𝑆)) | |
5 | ghmgrp2 18364 | . . . 4 ⊢ (𝑓 ∈ (𝑅 GrpHom 𝑆) → 𝑆 ∈ Grp) | |
6 | 4, 5 | syl 17 | . . 3 ⊢ (𝑓 ∈ (𝑅 GrpIso 𝑆) → 𝑆 ∈ Grp) |
7 | 6 | exlimiv 1930 | . 2 ⊢ (∃𝑓 𝑓 ∈ (𝑅 GrpIso 𝑆) → 𝑆 ∈ Grp) |
8 | 3, 7 | sylbi 219 | 1 ⊢ (𝑅 ≃𝑔 𝑆 → 𝑆 ∈ Grp) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∃wex 1779 ∈ wcel 2113 ≠ wne 3019 ∅c0 4294 class class class wbr 5069 (class class class)co 7159 Grpcgrp 18106 GrpHom cghm 18358 GrpIso cgim 18400 ≃𝑔 cgic 18401 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-rep 5193 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 ax-un 7464 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-ral 3146 df-rex 3147 df-reu 3148 df-rab 3150 df-v 3499 df-sbc 3776 df-csb 3887 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-nul 4295 df-if 4471 df-pw 4544 df-sn 4571 df-pr 4573 df-op 4577 df-uni 4842 df-iun 4924 df-br 5070 df-opab 5132 df-mpt 5150 df-id 5463 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-res 5570 df-ima 5571 df-suc 6200 df-iota 6317 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-ov 7162 df-oprab 7163 df-mpo 7164 df-1st 7692 df-2nd 7693 df-1o 8105 df-ghm 18359 df-gim 18402 df-gic 18403 |
This theorem is referenced by: (None) |
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