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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 18763 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 37594 | . 2 ⊢ {〈〈𝐴, 𝐴〉, 𝐴〉} ∈ GrpOp |
3 | opex 5461 | . . . . 5 ⊢ 〈𝐴, 𝐴〉 ∈ V | |
4 | 3 | rnsnop 6226 | . . . 4 ⊢ ran {〈〈𝐴, 𝐴〉, 𝐴〉} = {𝐴} |
5 | 4 | eqcomi 2735 | . . 3 ⊢ {𝐴} = ran {〈〈𝐴, 𝐴〉, 𝐴〉} |
6 | eqid 2726 | . . 3 ⊢ (GId‘{〈〈𝐴, 𝐴〉, 𝐴〉}) = (GId‘{〈〈𝐴, 𝐴〉, 𝐴〉}) | |
7 | 5, 6 | grpoidcl 30442 | . 2 ⊢ ({〈〈𝐴, 𝐴〉, 𝐴〉} ∈ GrpOp → (GId‘{〈〈𝐴, 𝐴〉, 𝐴〉}) ∈ {𝐴}) |
8 | elsni 4641 | . 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 1534 ∈ wcel 2099 Vcvv 3463 {csn 4624 〈cop 4630 ran crn 5674 ‘cfv 6544 GrpOpcgr 30417 GIdcgi 30418 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-rep 5281 ax-sep 5295 ax-nul 5302 ax-pow 5360 ax-pr 5424 ax-un 7736 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-ral 3052 df-rex 3061 df-reu 3366 df-rab 3421 df-v 3465 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4324 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4907 df-iun 4996 df-br 5145 df-opab 5207 df-mpt 5228 df-id 5571 df-xp 5679 df-rel 5680 df-cnv 5681 df-co 5682 df-dm 5683 df-rn 5684 df-res 5685 df-ima 5686 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 7370 df-ov 7417 df-grpo 30421 df-gid 30422 |
This theorem is referenced by: zrdivrng 37665 |
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