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| Mirrors > Home > MPE Home > Th. List > mndidcl | Structured version Visualization version GIF version | ||
| Description: The identity element of a monoid belongs to the monoid. (Contributed by NM, 27-Aug-2011.) (Revised by Mario Carneiro, 27-Dec-2014.) |
| Ref | Expression |
|---|---|
| mndidcl.b | ⊢ 𝐵 = (Base‘𝐺) |
| mndidcl.o | ⊢ 0 = (0g‘𝐺) |
| Ref | Expression |
|---|---|
| mndidcl | ⊢ (𝐺 ∈ Mnd → 0 ∈ 𝐵) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | mndidcl.b | . 2 ⊢ 𝐵 = (Base‘𝐺) | |
| 2 | mndidcl.o | . 2 ⊢ 0 = (0g‘𝐺) | |
| 3 | eqid 2737 | . 2 ⊢ (+g‘𝐺) = (+g‘𝐺) | |
| 4 | 1, 3 | mndid 18757 | . 2 ⊢ (𝐺 ∈ Mnd → ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ((𝑥(+g‘𝐺)𝑦) = 𝑦 ∧ (𝑦(+g‘𝐺)𝑥) = 𝑦)) |
| 5 | 1, 2, 3, 4 | mgmidcl 18679 | 1 ⊢ (𝐺 ∈ Mnd → 0 ∈ 𝐵) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2108 ‘cfv 6561 Basecbs 17247 +gcplusg 17297 0gc0g 17484 Mndcmnd 18747 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-iota 6514 df-fun 6563 df-fv 6569 df-riota 7388 df-ov 7434 df-0g 17486 df-mgm 18653 df-sgrp 18732 df-mnd 18748 |
| This theorem is referenced by: mndbn0 18763 hashfinmndnn 18764 mndpfo 18770 mndpsuppss 18778 prdsidlem 18782 imasmnd 18788 xpsmnd0 18791 idmhm 18808 mhmf1o 18809 mndvlid 18812 mndvrid 18813 issubmd 18819 submid 18823 0subm 18830 0mhm 18832 mhmco 18836 mhmeql 18839 submacs 18840 mndind 18841 prdspjmhm 18842 pwsdiagmhm 18844 pwsco1mhm 18845 pwsco2mhm 18846 gsumvallem2 18847 dfgrp2 18980 grpidcl 18983 mhmid 19081 mhmmnd 19082 mulgnn0cl 19108 mulgnn0z 19119 cntzsubm 19356 oppgmnd 19373 gex1 19609 mulgnn0di 19843 mulgmhm 19845 subcmn 19855 gsumval3 19925 gsumzcl2 19928 gsumzaddlem 19939 gsumzsplit 19945 gsumzmhm 19955 gsummpt1n0 19983 simpgnideld 20119 srgidcl 20196 srg0cl 20197 ringidcl 20262 gsummgp0 20315 c0mgm 20459 c0mhm 20460 c0snmgmhm 20462 c0snmhm 20463 pwssplit1 21058 rngqiprngimf1 21310 dsmm0cl 21760 dsmmacl 21761 mhmcompl 22384 mdet0 22612 mndifsplit 22642 gsummatr01lem3 22663 pmatcollpw3fi1lem1 22792 tmdmulg 24100 tmdgsum 24103 tsms0 24150 tsmssplit 24160 tsmsxp 24163 mndlactfo 33032 mndractfo 33034 mndlactf1o 33035 mndractf1o 33036 gsumwun 33068 cntzsnid 33072 submomnd 33087 omndmul2 33089 omndmul3 33090 omndmul 33091 ogrpinv0le 33092 gsumle 33101 slmd0vcl 33227 ply1degltdimlem 33673 lvecendof1f1o 33684 sibf0 34336 sitmcl 34353 primrootsunit1 42098 primrootscoprmpow 42100 primrootscoprbij 42103 evl1gprodd 42118 ringexp0nn 42135 aks6d1c5lem2 42139 pwssplit4 43101 mgpsumz 48278 lco0 48344 mndtccatid 49184 |
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