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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 2724 | . . . 4 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
2 | 1 | idghm 19152 | . . 3 ⊢ (𝑅 ∈ Grp → ( I ↾ (Base‘𝑅)) ∈ (𝑅 GrpHom 𝑅)) |
3 | cnvresid 6618 | . . . 4 ⊢ ◡( I ↾ (Base‘𝑅)) = ( I ↾ (Base‘𝑅)) | |
4 | 3, 2 | eqeltrid 2829 | . . 3 ⊢ (𝑅 ∈ Grp → ◡( I ↾ (Base‘𝑅)) ∈ (𝑅 GrpHom 𝑅)) |
5 | isgim2 19186 | . . 3 ⊢ (( I ↾ (Base‘𝑅)) ∈ (𝑅 GrpIso 𝑅) ↔ (( I ↾ (Base‘𝑅)) ∈ (𝑅 GrpHom 𝑅) ∧ ◡( I ↾ (Base‘𝑅)) ∈ (𝑅 GrpHom 𝑅))) | |
6 | 2, 4, 5 | sylanbrc 582 | . 2 ⊢ (𝑅 ∈ Grp → ( I ↾ (Base‘𝑅)) ∈ (𝑅 GrpIso 𝑅)) |
7 | brgici 19192 | . 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 2098 class class class wbr 5139 I cid 5564 ◡ccnv 5666 ↾ cres 5669 ‘cfv 6534 (class class class)co 7402 Basecbs 17149 Grpcgrp 18859 GrpHom cghm 19134 GrpIso cgim 19178 ≃𝑔 cgic 19179 |
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 2695 ax-rep 5276 ax-sep 5290 ax-nul 5297 ax-pow 5354 ax-pr 5418 ax-un 7719 |
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 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-ral 3054 df-rex 3063 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-nul 4316 df-if 4522 df-pw 4597 df-sn 4622 df-pr 4624 df-op 4628 df-uni 4901 df-iun 4990 df-br 5140 df-opab 5202 df-mpt 5223 df-id 5565 df-xp 5673 df-rel 5674 df-cnv 5675 df-co 5676 df-dm 5677 df-rn 5678 df-res 5679 df-ima 5680 df-suc 6361 df-iota 6486 df-fun 6536 df-fn 6537 df-f 6538 df-f1 6539 df-fo 6540 df-f1o 6541 df-fv 6542 df-ov 7405 df-oprab 7406 df-mpo 7407 df-1st 7969 df-2nd 7970 df-1o 8462 df-mgm 18569 df-sgrp 18648 df-mnd 18664 df-grp 18862 df-ghm 19135 df-gim 19180 df-gic 19181 |
This theorem is referenced by: gicer 19198 |
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