gicref - Metamath Proof Explorer

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Theorem gicref 18802

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 2738	. . . 4 ⊢ (Base‘𝑅) = (Base‘𝑅)
2	1	idghm 18764	. . 3 ⊢ (𝑅 ∈ Grp → ( I ↾ (Base‘𝑅)) ∈ (𝑅 GrpHom 𝑅))
3		cnvresid 6497	. . . 4 ⊢ ^◡( I ↾ (Base‘𝑅)) = ( I ↾ (Base‘𝑅))
4	3, 2	eqeltrid 2843	. . 3 ⊢ (𝑅 ∈ Grp → ^◡( I ↾ (Base‘𝑅)) ∈ (𝑅 GrpHom 𝑅))
5		isgim2 18796	. . 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 18801	. 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 2108 class class class wbr 5070 I cid 5479 ^◡ccnv 5579 ↾ cres 5582 ‘cfv 6418 (class class class)co 7255 Basecbs 16840 Grpcgrp 18492 GrpHom cghm 18746 GrpIso cgim 18788 ≃_𝑔 cgic 18789

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1799 ax-4 1813 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2156 ax-12 2173 ax-ext 2709 ax-rep 5205 ax-sep 5218 ax-nul 5225 ax-pow 5283 ax-pr 5347 ax-un 7566

This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1784 df-nf 1788 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2888 df-ne 2943 df-ral 3068 df-rex 3069 df-reu 3070 df-rab 3072 df-v 3424 df-sbc 3712 df-csb 3829 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-nul 4254 df-if 4457 df-pw 4532 df-sn 4559 df-pr 4561 df-op 4565 df-uni 4837 df-iun 4923 df-br 5071 df-opab 5133 df-mpt 5154 df-id 5480 df-xp 5586 df-rel 5587 df-cnv 5588 df-co 5589 df-dm 5590 df-rn 5591 df-res 5592 df-ima 5593 df-suc 6257 df-iota 6376 df-fun 6420 df-fn 6421 df-f 6422 df-f1 6423 df-fo 6424 df-f1o 6425 df-fv 6426 df-ov 7258 df-oprab 7259 df-mpo 7260 df-1st 7804 df-2nd 7805 df-1o 8267 df-mgm 18241 df-sgrp 18290 df-mnd 18301 df-grp 18495 df-ghm 18747 df-gim 18790 df-gic 18791

This theorem is referenced by: gicer 18807

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