gicref - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > gicref		Structured version Visualization version GIF version

Theorem gicref 19204

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 2729	. . . 4 ⊢ (Base‘𝑅) = (Base‘𝑅)
2	1	idghm 19163	. . 3 ⊢ (𝑅 ∈ Grp → ( I ↾ (Base‘𝑅)) ∈ (𝑅 GrpHom 𝑅))
3		cnvresid 6595	. . . 4 ⊢ ^◡( I ↾ (Base‘𝑅)) = ( I ↾ (Base‘𝑅))
4	3, 2	eqeltrid 2832	. . 3 ⊢ (𝑅 ∈ Grp → ^◡( I ↾ (Base‘𝑅)) ∈ (𝑅 GrpHom 𝑅))
5		isgim2 19197	. . 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 19203	. 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 2109 class class class wbr 5107 I cid 5532 ^◡ccnv 5637 ↾ cres 5640 ‘cfv 6511 (class class class)co 7387 Basecbs 17179 Grpcgrp 18865 GrpHom cghm 19144 GrpIso cgim 19189 ≃_𝑔 cgic 19190

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-sep 5251 ax-nul 5261 ax-pow 5320 ax-pr 5387 ax-un 7711

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-ral 3045 df-rex 3054 df-rab 3406 df-v 3449 df-sbc 3754 df-csb 3863 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-nul 4297 df-if 4489 df-pw 4565 df-sn 4590 df-pr 4592 df-op 4596 df-uni 4872 df-iun 4957 df-br 5108 df-opab 5170 df-mpt 5189 df-id 5533 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-suc 6338 df-iota 6464 df-fun 6513 df-fn 6514 df-f 6515 df-f1 6516 df-fo 6517 df-f1o 6518 df-fv 6519 df-ov 7390 df-oprab 7391 df-mpo 7392 df-1st 7968 df-2nd 7969 df-1o 8434 df-map 8801 df-mgm 18567 df-sgrp 18646 df-mnd 18662 df-grp 18868 df-ghm 19145 df-gim 19191 df-gic 19192

This theorem is referenced by: gicer 19209

W3C validator