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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 18729 | . 2 ⊢ (𝐺 ∈ Mnd → 𝐺 ∈ Mgm) | |
2 | mndcl.b | . . 3 ⊢ 𝐵 = (Base‘𝐺) | |
3 | mndcl.p | . . 3 ⊢ + = (+g‘𝐺) | |
4 | 2, 3 | mgmcl 18631 | . 2 ⊢ ((𝐺 ∈ Mgm ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 + 𝑌) ∈ 𝐵) |
5 | 1, 4 | syl3an1 1160 | 1 ⊢ ((𝐺 ∈ Mnd ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 + 𝑌) ∈ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1084 = wceq 1534 ∈ wcel 2099 ‘cfv 6546 (class class class)co 7416 Basecbs 17208 +gcplusg 17261 Mgmcmgm 18626 Mndcmnd 18722 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-ext 2697 ax-nul 5303 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-sb 2061 df-clab 2704 df-cleq 2718 df-clel 2803 df-ne 2931 df-ral 3052 df-rex 3061 df-rab 3420 df-v 3464 df-sbc 3776 df-dif 3949 df-un 3951 df-ss 3963 df-nul 4323 df-if 4524 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4906 df-br 5146 df-iota 6498 df-fv 6554 df-ov 7419 df-mgm 18628 df-sgrp 18707 df-mnd 18723 |
This theorem is referenced by: mnd4g 18736 mndpropd 18747 issubmnd 18749 prdsplusgcl 18753 imasmnd 18760 xpsmnd0 18763 idmhm 18780 mhmf1o 18781 mndvcl 18782 mhmvlin 18786 issubmd 18791 0mhm 18804 mhmco 18808 mhmeql 18811 submacs 18812 mndind 18813 prdspjmhm 18814 pwsdiagmhm 18816 pwsco1mhm 18817 pwsco2mhm 18818 gsumwmhm 18830 grpcl 18931 mhmmnd 19054 mulgnn0cl 19080 cntzsubm 19328 oppgmnd 19347 lsmssv 19637 frgp0 19754 frgpadd 19757 mulgnn0di 19819 mulgmhm 19821 gsumval3eu 19898 gsumval3 19901 gsumzcl2 19904 gsumzaddlem 19915 gsumzmhm 19931 gsummptfzcl 19963 srgcl 20172 srgacl 20184 srgbinomlem 20209 srgbinom 20210 ringcl 20229 ringpropd 20263 c0mhm 20438 mat2pmatghm 22720 pm2mpghm 22806 cpmadugsumlemF 22866 tsmsadd 24139 cmn246135 32909 cmn145236 32910 omndadd2d 32947 omndadd2rd 32948 slmdacl 33077 slmdvacl 33080 gsumncl 34399 primrootsunit1 41809 aks6d1c1 41828 aks6d1c5lem0 41847 aks6d1c5lem3 41849 aks6d1c5lem2 41850 aks6d1c5 41851 aks6d1c6lem1 41882 ofaddmndmap 47758 lincsum 47848 mndtccatid 48450 |
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