![]() |
Mathbox for Jeff Madsen |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > gidsn | Structured version Visualization version GIF version |
Description: Obsolete as of 23-Jan-2020. Use mnd1id 18668 instead. The identity element of the trivial group. (Contributed by FL, 21-Jun-2010.) (Proof shortened by Mario Carneiro, 15-Dec-2013.) (New usage is discouraged.) |
Ref | Expression |
---|---|
ablsn.1 | ⊢ 𝐴 ∈ V |
Ref | Expression |
---|---|
gidsn | ⊢ (GId‘{⟨⟨𝐴, 𝐴⟩, 𝐴⟩}) = 𝐴 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ablsn.1 | . . 3 ⊢ 𝐴 ∈ V | |
2 | 1 | grposnOLD 36750 | . 2 ⊢ {⟨⟨𝐴, 𝐴⟩, 𝐴⟩} ∈ GrpOp |
3 | opex 5465 | . . . . 5 ⊢ ⟨𝐴, 𝐴⟩ ∈ V | |
4 | 3 | rnsnop 6224 | . . . 4 ⊢ ran {⟨⟨𝐴, 𝐴⟩, 𝐴⟩} = {𝐴} |
5 | 4 | eqcomi 2742 | . . 3 ⊢ {𝐴} = ran {⟨⟨𝐴, 𝐴⟩, 𝐴⟩} |
6 | eqid 2733 | . . 3 ⊢ (GId‘{⟨⟨𝐴, 𝐴⟩, 𝐴⟩}) = (GId‘{⟨⟨𝐴, 𝐴⟩, 𝐴⟩}) | |
7 | 5, 6 | grpoidcl 29767 | . 2 ⊢ ({⟨⟨𝐴, 𝐴⟩, 𝐴⟩} ∈ GrpOp → (GId‘{⟨⟨𝐴, 𝐴⟩, 𝐴⟩}) ∈ {𝐴}) |
8 | elsni 4646 | . 2 ⊢ ((GId‘{⟨⟨𝐴, 𝐴⟩, 𝐴⟩}) ∈ {𝐴} → (GId‘{⟨⟨𝐴, 𝐴⟩, 𝐴⟩}) = 𝐴) | |
9 | 2, 7, 8 | mp2b 10 | 1 ⊢ (GId‘{⟨⟨𝐴, 𝐴⟩, 𝐴⟩}) = 𝐴 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1542 ∈ wcel 2107 Vcvv 3475 {csn 4629 ⟨cop 4635 ran crn 5678 ‘cfv 6544 GrpOpcgr 29742 GIdcgi 29743 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-ral 3063 df-rex 3072 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-iun 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-id 5575 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-riota 7365 df-ov 7412 df-grpo 29746 df-gid 29747 |
This theorem is referenced by: zrdivrng 36821 |
Copyright terms: Public domain | W3C validator |