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Mirrors > Home > MPE Home > Th. List > Mathboxes > gidsn | Structured version Visualization version GIF version |
Description: Obsolete as of 23-Jan-2020. Use mnd1id 18664 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 36688 | . 2 ⊢ {〈〈𝐴, 𝐴〉, 𝐴〉} ∈ GrpOp |
3 | opex 5463 | . . . . 5 ⊢ 〈𝐴, 𝐴〉 ∈ V | |
4 | 3 | rnsnop 6220 | . . . 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 29745 | . 2 ⊢ ({〈〈𝐴, 𝐴〉, 𝐴〉} ∈ GrpOp → (GId‘{〈〈𝐴, 𝐴〉, 𝐴〉}) ∈ {𝐴}) |
8 | elsni 4644 | . 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 4627 〈cop 4633 ran crn 5676 ‘cfv 6540 GrpOpcgr 29720 GIdcgi 29721 |
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 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7720 |
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 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-id 5573 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-riota 7360 df-ov 7407 df-grpo 29724 df-gid 29725 |
This theorem is referenced by: zrdivrng 36759 |
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