gicref - Metamath Proof Explorer

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Theorem gicref 19182

Description: Isomorphism is reflexive. (Contributed by Mario Carneiro, 21-Apr-2016.)

**Assertion**
Ref	Expression
gicref	⊢ (𝑅 ∈ Grp → 𝑅 ≃_𝑔 𝑅)

**Proof of Theorem gicref**
Step	Hyp	Ref	Expression
1		eqid 2731	. . . 4 ⊢ (Base‘𝑅) = (Base‘𝑅)
2	1	idghm 19141	. . 3 ⊢ (𝑅 ∈ Grp → ( I ↾ (Base‘𝑅)) ∈ (𝑅 GrpHom 𝑅))
3		cnvresid 6560	. . . 4 ⊢ ^◡( I ↾ (Base‘𝑅)) = ( I ↾ (Base‘𝑅))
4	3, 2	eqeltrid 2835	. . 3 ⊢ (𝑅 ∈ Grp → ^◡( I ↾ (Base‘𝑅)) ∈ (𝑅 GrpHom 𝑅))
5		isgim2 19175	. . 3 ⊢ (( I ↾ (Base‘𝑅)) ∈ (𝑅 GrpIso 𝑅) ↔ (( I ↾ (Base‘𝑅)) ∈ (𝑅 GrpHom 𝑅) ∧ ^◡( I ↾ (Base‘𝑅)) ∈ (𝑅 GrpHom 𝑅)))
6	2, 4, 5	sylanbrc 583	. 2 ⊢ (𝑅 ∈ Grp → ( I ↾ (Base‘𝑅)) ∈ (𝑅 GrpIso 𝑅))
7		brgici 19181	. 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 2111 class class class wbr 5091 I cid 5510 ^◡ccnv 5615 ↾ cres 5618 ‘cfv 6481 (class class class)co 7346 Basecbs 17117 Grpcgrp 18843 GrpHom cghm 19122 GrpIso cgim 19167 ≃_𝑔 cgic 19168

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2113 ax-9 2121 ax-10 2144 ax-11 2160 ax-12 2180 ax-ext 2703 ax-sep 5234 ax-nul 5244 ax-pow 5303 ax-pr 5370 ax-un 7668

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2535 df-eu 2564 df-clab 2710 df-cleq 2723 df-clel 2806 df-nfc 2881 df-ne 2929 df-ral 3048 df-rex 3057 df-rab 3396 df-v 3438 df-sbc 3742 df-csb 3851 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-nul 4284 df-if 4476 df-pw 4552 df-sn 4577 df-pr 4579 df-op 4583 df-uni 4860 df-iun 4943 df-br 5092 df-opab 5154 df-mpt 5173 df-id 5511 df-xp 5622 df-rel 5623 df-cnv 5624 df-co 5625 df-dm 5626 df-rn 5627 df-res 5628 df-ima 5629 df-suc 6312 df-iota 6437 df-fun 6483 df-fn 6484 df-f 6485 df-f1 6486 df-fo 6487 df-f1o 6488 df-fv 6489 df-ov 7349 df-oprab 7350 df-mpo 7351 df-1st 7921 df-2nd 7922 df-1o 8385 df-map 8752 df-mgm 18545 df-sgrp 18624 df-mnd 18640 df-grp 18846 df-ghm 19123 df-gim 19169 df-gic 19170

This theorem is referenced by: gicer 19187

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