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Mirrors > Home > MPE Home > Th. List > gicref | Structured version Visualization version GIF version |
Description: Isomorphism is reflexive. (Contributed by Mario Carneiro, 21-Apr-2016.) |
Ref | Expression |
---|---|
gicref | ⊢ (𝑅 ∈ Grp → 𝑅 ≃𝑔 𝑅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2821 | . . . 4 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
2 | 1 | idghm 18367 | . . 3 ⊢ (𝑅 ∈ Grp → ( I ↾ (Base‘𝑅)) ∈ (𝑅 GrpHom 𝑅)) |
3 | cnvresid 6427 | . . . 4 ⊢ ◡( I ↾ (Base‘𝑅)) = ( I ↾ (Base‘𝑅)) | |
4 | 3, 2 | eqeltrid 2917 | . . 3 ⊢ (𝑅 ∈ Grp → ◡( I ↾ (Base‘𝑅)) ∈ (𝑅 GrpHom 𝑅)) |
5 | isgim2 18399 | . . 3 ⊢ (( I ↾ (Base‘𝑅)) ∈ (𝑅 GrpIso 𝑅) ↔ (( I ↾ (Base‘𝑅)) ∈ (𝑅 GrpHom 𝑅) ∧ ◡( I ↾ (Base‘𝑅)) ∈ (𝑅 GrpHom 𝑅))) | |
6 | 2, 4, 5 | sylanbrc 585 | . 2 ⊢ (𝑅 ∈ Grp → ( I ↾ (Base‘𝑅)) ∈ (𝑅 GrpIso 𝑅)) |
7 | brgici 18404 | . 2 ⊢ (( I ↾ (Base‘𝑅)) ∈ (𝑅 GrpIso 𝑅) → 𝑅 ≃𝑔 𝑅) | |
8 | 6, 7 | syl 17 | 1 ⊢ (𝑅 ∈ Grp → 𝑅 ≃𝑔 𝑅) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2110 class class class wbr 5058 I cid 5453 ◡ccnv 5548 ↾ cres 5551 ‘cfv 6349 (class class class)co 7150 Basecbs 16477 Grpcgrp 18097 GrpHom cghm 18349 GrpIso cgim 18391 ≃𝑔 cgic 18392 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5182 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-iun 4913 df-br 5059 df-opab 5121 df-mpt 5139 df-id 5454 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-suc 6191 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-ov 7153 df-oprab 7154 df-mpo 7155 df-1st 7683 df-2nd 7684 df-1o 8096 df-mgm 17846 df-sgrp 17895 df-mnd 17906 df-grp 18100 df-ghm 18350 df-gim 18393 df-gic 18394 |
This theorem is referenced by: gicer 18410 |
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