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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 2818 | . 2 ⊢ (+g‘𝐺) = (+g‘𝐺) | |
4 | 1, 3 | mndid 17909 | . 2 ⊢ (𝐺 ∈ Mnd → ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ((𝑥(+g‘𝐺)𝑦) = 𝑦 ∧ (𝑦(+g‘𝐺)𝑥) = 𝑦)) |
5 | 1, 2, 3, 4 | mgmidcl 17864 | 1 ⊢ (𝐺 ∈ Mnd → 0 ∈ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1528 ∈ wcel 2105 ‘cfv 6348 Basecbs 16471 +gcplusg 16553 0gc0g 16701 Mndcmnd 17899 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-ral 3140 df-rex 3141 df-reu 3142 df-rmo 3143 df-rab 3144 df-v 3494 df-sbc 3770 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-iota 6307 df-fun 6350 df-fv 6356 df-riota 7103 df-ov 7148 df-0g 16703 df-mgm 17840 df-sgrp 17889 df-mnd 17900 |
This theorem is referenced by: mndbn0 17915 hashfinmndnn 17916 mndpfo 17922 prdsidlem 17931 imasmnd 17937 idmhm 17953 mhmf1o 17954 issubmd 17959 submid 17963 0subm 17970 0mhm 17972 mhmco 17976 mhmeql 17978 submacs 17979 mndind 17980 prdspjmhm 17981 pwsdiagmhm 17983 pwsco1mhm 17984 pwsco2mhm 17985 gsumvallem2 17986 dfgrp2 18066 grpidcl 18069 mhmid 18158 mhmmnd 18159 mulgnn0cl 18182 mulgnn0z 18192 cntzsubm 18404 oppgmnd 18420 gex1 18645 mulgnn0di 18875 mulgmhm 18877 subcmn 18886 gsumval3 18956 gsumzcl2 18959 gsumzaddlem 18970 gsumzsplit 18976 gsumzmhm 18986 gsummpt1n0 19014 simpgnideld 19150 srgidcl 19197 srg0cl 19198 ringidcl 19247 gsummgp0 19287 pwssplit1 19760 dsmm0cl 20812 dsmmacl 20813 mndvlid 20932 mndvrid 20933 mdet0 21143 mndifsplit 21173 gsummatr01lem3 21194 pmatcollpw3fi1lem1 21322 tmdmulg 22628 tmdgsum 22631 tsms0 22677 tsmssplit 22687 tsmsxp 22690 cntzsnid 30623 submomnd 30638 omndmul2 30640 omndmul3 30641 omndmul 30642 ogrpinv0le 30643 gsumle 30652 slmd0vcl 30776 sibf0 31491 sitmcl 31508 pwssplit4 39567 c0mgm 44108 c0mhm 44109 c0snmgmhm 44113 c0snmhm 44114 mgpsumz 44338 mndpsuppss 44347 lco0 44410 |
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