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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 17947 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 35154 | . 2 ⊢ {〈〈𝐴, 𝐴〉, 𝐴〉} ∈ GrpOp |
3 | opex 5349 | . . . . 5 ⊢ 〈𝐴, 𝐴〉 ∈ V | |
4 | 3 | rnsnop 6076 | . . . 4 ⊢ ran {〈〈𝐴, 𝐴〉, 𝐴〉} = {𝐴} |
5 | 4 | eqcomi 2830 | . . 3 ⊢ {𝐴} = ran {〈〈𝐴, 𝐴〉, 𝐴〉} |
6 | eqid 2821 | . . 3 ⊢ (GId‘{〈〈𝐴, 𝐴〉, 𝐴〉}) = (GId‘{〈〈𝐴, 𝐴〉, 𝐴〉}) | |
7 | 5, 6 | grpoidcl 28285 | . 2 ⊢ ({〈〈𝐴, 𝐴〉, 𝐴〉} ∈ GrpOp → (GId‘{〈〈𝐴, 𝐴〉, 𝐴〉}) ∈ {𝐴}) |
8 | elsni 4578 | . 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 1533 ∈ wcel 2110 Vcvv 3495 {csn 4561 〈cop 4567 ran crn 5551 ‘cfv 6350 GrpOpcgr 28260 GIdcgi 28261 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-rep 5183 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5322 ax-un 7455 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3497 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4833 df-iun 4914 df-br 5060 df-opab 5122 df-mpt 5140 df-id 5455 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-rn 5561 df-res 5562 df-ima 5563 df-iota 6309 df-fun 6352 df-fn 6353 df-f 6354 df-f1 6355 df-fo 6356 df-f1o 6357 df-fv 6358 df-riota 7108 df-ov 7153 df-grpo 28264 df-gid 28265 |
This theorem is referenced by: zrdivrng 35225 |
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