| Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
| Mirrors > Home > MPE Home > Th. List > mndcl | Structured version Visualization version GIF version | ||
| Description: Closure of the operation of a monoid. (Contributed by NM, 14-Aug-2011.) (Revised by Mario Carneiro, 6-Jan-2015.) (Proof shortened by AV, 8-Feb-2020.) |
| Ref | Expression |
|---|---|
| mndcl.b | ⊢ 𝐵 = (Base‘𝐺) |
| mndcl.p | ⊢ + = (+g‘𝐺) |
| Ref | Expression |
|---|---|
| mndcl | ⊢ ((𝐺 ∈ Mnd ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 + 𝑌) ∈ 𝐵) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | mndmgm 18754 | . 2 ⊢ (𝐺 ∈ Mnd → 𝐺 ∈ Mgm) | |
| 2 | mndcl.b | . . 3 ⊢ 𝐵 = (Base‘𝐺) | |
| 3 | mndcl.p | . . 3 ⊢ + = (+g‘𝐺) | |
| 4 | 2, 3 | mgmcl 18656 | . 2 ⊢ ((𝐺 ∈ Mgm ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 + 𝑌) ∈ 𝐵) |
| 5 | 1, 4 | syl3an1 1164 | 1 ⊢ ((𝐺 ∈ Mnd ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 + 𝑌) ∈ 𝐵) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ w3a 1087 = wceq 1540 ∈ wcel 2108 ‘cfv 6561 (class class class)co 7431 Basecbs 17247 +gcplusg 17297 Mgmcmgm 18651 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-ext 2708 ax-nul 5306 |
| 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-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ne 2941 df-ral 3062 df-rex 3071 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-iota 6514 df-fv 6569 df-ov 7434 df-mgm 18653 df-sgrp 18732 df-mnd 18748 |
| This theorem is referenced by: mnd4g 18761 mndpropd 18772 issubmnd 18774 prdsplusgcl 18781 imasmnd 18788 xpsmnd0 18791 idmhm 18808 mhmf1o 18809 mndvcl 18810 mhmvlin 18814 issubmd 18819 0mhm 18832 mhmco 18836 mhmeql 18839 submacs 18840 mndind 18841 prdspjmhm 18842 pwsdiagmhm 18844 pwsco1mhm 18845 pwsco2mhm 18846 gsumwmhm 18858 grpcl 18959 mhmmnd 19082 mulgnn0cl 19108 cntzsubm 19356 oppgmnd 19373 lsmssv 19661 frgp0 19778 frgpadd 19781 mulgnn0di 19843 mulgmhm 19845 gsumval3eu 19922 gsumval3 19925 gsumzcl2 19928 gsumzaddlem 19939 gsumzmhm 19955 gsummptfzcl 19987 srgcl 20190 srgacl 20202 srgbinomlem 20227 srgbinom 20228 ringcl 20247 ringpropd 20285 c0mhm 20460 mat2pmatghm 22736 pm2mpghm 22822 cpmadugsumlemF 22882 tsmsadd 24155 mndcld 33027 cmn246135 33038 cmn145236 33039 omndadd2d 33085 omndadd2rd 33086 slmdacl 33215 slmdvacl 33218 gsumncl 34555 primrootsunit1 42098 aks6d1c1 42117 aks6d1c5lem0 42136 aks6d1c5lem3 42138 aks6d1c5lem2 42139 aks6d1c5 42140 aks6d1c6lem1 42171 ofaddmndmap 48259 lincsum 48346 mndtccatid 49184 |
| Copyright terms: Public domain | W3C validator |