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 19245

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 2740	. . . 4 ⊢ (Base‘𝑅) = (Base‘𝑅)
2	1	idghm 19204	. . 3 ⊢ (𝑅 ∈ Grp → ( I ↾ (Base‘𝑅)) ∈ (𝑅 GrpHom 𝑅))
3		cnvresid 6571	. . . 4 ⊢ ^◡( I ↾ (Base‘𝑅)) = ( I ↾ (Base‘𝑅))
4	3, 2	eqeltrid 2844	. . 3 ⊢ (𝑅 ∈ Grp → ^◡( I ↾ (Base‘𝑅)) ∈ (𝑅 GrpHom 𝑅))
5		isgim2 19238	. . 3 ⊢ (( I ↾ (Base‘𝑅)) ∈ (𝑅 GrpIso 𝑅) ↔ (( I ↾ (Base‘𝑅)) ∈ (𝑅 GrpHom 𝑅) ∧ ^◡( I ↾ (Base‘𝑅)) ∈ (𝑅 GrpHom 𝑅)))
6	2, 4, 5	sylanbrc 589	. 2 ⊢ (𝑅 ∈ Grp → ( I ↾ (Base‘𝑅)) ∈ (𝑅 GrpIso 𝑅))
7		brgici 19244	. 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 2119 class class class wbr 5079 I cid 5519 ^◡ccnv 5624 ↾ cres 5627 ‘cfv 6492 (class class class)co 7363 Basecbs 17177 Grpcgrp 18907 GrpHom cghm 19185 GrpIso cgim 19230 ≃_𝑔 cgic 19231

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1974 ax-7 2015 ax-8 2121 ax-9 2129 ax-10 2152 ax-11 2168 ax-12 2189 ax-ext 2712 ax-sep 5225 ax-nul 5235 ax-pow 5301 ax-pr 5369 ax-un 7685

This theorem depends on definitions: df-bi 208 df-an 397 df-or 854 df-3an 1094 df-tru 1550 df-fal 1560 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2543 df-eu 2573 df-clab 2719 df-cleq 2732 df-clel 2815 df-nfc 2889 df-ne 2936 df-ral 3055 df-rex 3065 df-rab 3393 df-v 3434 df-sbc 3731 df-csb 3839 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-nul 4269 df-if 4462 df-pw 4538 df-sn 4563 df-pr 4565 df-op 4569 df-uni 4846 df-iun 4930 df-br 5080 df-opab 5142 df-mpt 5161 df-id 5520 df-xp 5631 df-rel 5632 df-cnv 5633 df-co 5634 df-dm 5635 df-rn 5636 df-res 5637 df-ima 5638 df-suc 6323 df-iota 6448 df-fun 6494 df-fn 6495 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499 df-fv 6500 df-ov 7366 df-oprab 7367 df-mpo 7368 df-1st 7938 df-2nd 7939 df-1o 8402 df-map 8772 df-mgm 18606 df-sgrp 18685 df-mnd 18701 df-grp 18910 df-ghm 19186 df-gim 19232 df-gic 19233

This theorem is referenced by: gicer 19250

W3C validator