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Mirrors > Home > ILE Home > Th. List > mndidcl | 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 2177 | . 2 ⊢ (+g‘𝐺) = (+g‘𝐺) | |
4 | 1, 3 | mndid 12780 | . 2 ⊢ (𝐺 ∈ Mnd → ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ((𝑥(+g‘𝐺)𝑦) = 𝑦 ∧ (𝑦(+g‘𝐺)𝑥) = 𝑦)) |
5 | 1, 2, 3, 4 | mgmidcl 12751 | 1 ⊢ (𝐺 ∈ Mnd → 0 ∈ 𝐵) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1353 ∈ wcel 2148 ‘cfv 5216 Basecbs 12456 +gcplusg 12530 0gc0g 12695 Mndcmnd 12771 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-sep 4121 ax-pow 4174 ax-pr 4209 ax-un 4433 ax-cnex 7901 ax-resscn 7902 ax-1re 7904 ax-addrcl 7907 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ral 2460 df-rex 2461 df-reu 2462 df-rmo 2463 df-rab 2464 df-v 2739 df-sbc 2963 df-csb 3058 df-un 3133 df-in 3135 df-ss 3142 df-pw 3577 df-sn 3598 df-pr 3599 df-op 3601 df-uni 3810 df-int 3845 df-br 4004 df-opab 4065 df-mpt 4066 df-id 4293 df-xp 4632 df-rel 4633 df-cnv 4634 df-co 4635 df-dm 4636 df-rn 4637 df-res 4638 df-iota 5178 df-fun 5218 df-fn 5219 df-fv 5224 df-riota 5830 df-ov 5877 df-inn 8918 df-2 8976 df-ndx 12459 df-slot 12460 df-base 12462 df-plusg 12543 df-0g 12697 df-mgm 12729 df-sgrp 12762 df-mnd 12772 |
This theorem is referenced by: mndbn0 12786 hashfinmndnn 12787 mndpfo 12793 idmhm 12814 mhmf1o 12815 issubmd 12819 submid 12822 0subm 12825 0mhm 12827 mhmco 12828 mhmeql 12830 dfgrp2 12856 grpidcl 12858 mhmid 12933 mhmmnd 12934 mulgnn0cl 12953 mulgnn0z 12963 srgidcl 13112 srg0cl 13113 ringidcl 13156 |
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