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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 2825 | . . . 4 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
2 | 1 | idghm 18026 | . . 3 ⊢ (𝑅 ∈ Grp → ( I ↾ (Base‘𝑅)) ∈ (𝑅 GrpHom 𝑅)) |
3 | cnvresid 6201 | . . . 4 ⊢ ◡( I ↾ (Base‘𝑅)) = ( I ↾ (Base‘𝑅)) | |
4 | 3, 2 | syl5eqel 2910 | . . 3 ⊢ (𝑅 ∈ Grp → ◡( I ↾ (Base‘𝑅)) ∈ (𝑅 GrpHom 𝑅)) |
5 | isgim2 18058 | . . 3 ⊢ (( I ↾ (Base‘𝑅)) ∈ (𝑅 GrpIso 𝑅) ↔ (( I ↾ (Base‘𝑅)) ∈ (𝑅 GrpHom 𝑅) ∧ ◡( I ↾ (Base‘𝑅)) ∈ (𝑅 GrpHom 𝑅))) | |
6 | 2, 4, 5 | sylanbrc 580 | . 2 ⊢ (𝑅 ∈ Grp → ( I ↾ (Base‘𝑅)) ∈ (𝑅 GrpIso 𝑅)) |
7 | brgici 18063 | . 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 2166 class class class wbr 4873 I cid 5249 ◡ccnv 5341 ↾ cres 5344 ‘cfv 6123 (class class class)co 6905 Basecbs 16222 Grpcgrp 17776 GrpHom cghm 18008 GrpIso cgim 18050 ≃𝑔 cgic 18051 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-8 2168 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2391 ax-ext 2803 ax-rep 4994 ax-sep 5005 ax-nul 5013 ax-pow 5065 ax-pr 5127 ax-un 7209 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3an 1115 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-ral 3122 df-rex 3123 df-reu 3124 df-rab 3126 df-v 3416 df-sbc 3663 df-csb 3758 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-nul 4145 df-if 4307 df-pw 4380 df-sn 4398 df-pr 4400 df-op 4404 df-uni 4659 df-iun 4742 df-br 4874 df-opab 4936 df-mpt 4953 df-id 5250 df-xp 5348 df-rel 5349 df-cnv 5350 df-co 5351 df-dm 5352 df-rn 5353 df-res 5354 df-ima 5355 df-suc 5969 df-iota 6086 df-fun 6125 df-fn 6126 df-f 6127 df-f1 6128 df-fo 6129 df-f1o 6130 df-fv 6131 df-ov 6908 df-oprab 6909 df-mpt2 6910 df-1st 7428 df-2nd 7429 df-1o 7826 df-mgm 17595 df-sgrp 17637 df-mnd 17648 df-grp 17779 df-ghm 18009 df-gim 18052 df-gic 18053 |
This theorem is referenced by: gicer 18069 |
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