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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 17501 | . 2 ⊢ (𝐺 ∈ Mnd → 𝐺 ∈ Mgm) | |
2 | mndcl.b | . . 3 ⊢ 𝐵 = (Base‘𝐺) | |
3 | mndcl.p | . . 3 ⊢ + = (+g‘𝐺) | |
4 | 2, 3 | mgmcl 17446 | . 2 ⊢ ((𝐺 ∈ Mgm ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 + 𝑌) ∈ 𝐵) |
5 | 1, 4 | syl3an1 1167 | 1 ⊢ ((𝐺 ∈ Mnd ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 + 𝑌) ∈ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1072 = wceq 1632 ∈ wcel 2139 ‘cfv 6049 (class class class)co 6813 Basecbs 16059 +gcplusg 16143 Mgmcmgm 17441 Mndcmnd 17495 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1871 ax-4 1886 ax-5 1988 ax-6 2054 ax-7 2090 ax-9 2148 ax-10 2168 ax-11 2183 ax-12 2196 ax-13 2391 ax-ext 2740 ax-nul 4941 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3an 1074 df-tru 1635 df-ex 1854 df-nf 1859 df-sb 2047 df-eu 2611 df-clab 2747 df-cleq 2753 df-clel 2756 df-nfc 2891 df-ral 3055 df-rex 3056 df-rab 3059 df-v 3342 df-sbc 3577 df-dif 3718 df-un 3720 df-in 3722 df-ss 3729 df-nul 4059 df-if 4231 df-sn 4322 df-pr 4324 df-op 4328 df-uni 4589 df-br 4805 df-iota 6012 df-fv 6057 df-ov 6816 df-mgm 17443 df-sgrp 17485 df-mnd 17496 |
This theorem is referenced by: mnd4g 17508 mndpropd 17517 issubmnd 17519 prdsplusgcl 17522 imasmnd 17529 idmhm 17545 mhmf1o 17546 issubmd 17550 0mhm 17559 mhmco 17563 mhmeql 17565 submacs 17566 mrcmndind 17567 prdspjmhm 17568 pwsdiagmhm 17570 pwsco1mhm 17571 pwsco2mhm 17572 gsumccat 17579 gsumwmhm 17583 grpcl 17631 mhmmnd 17738 mulgnnclOLD 17758 mulgnn0cl 17759 mulgnndirOLD 17771 cntzsubm 17968 oppgmnd 17984 lsmssv 18258 frgp0 18373 frgpadd 18376 mulgnn0di 18431 mulgmhm 18433 gsumval3eu 18505 gsumval3 18508 gsumzcl2 18511 gsumzaddlem 18521 gsumzmhm 18537 gsummptfzcl 18568 srgcl 18712 srgacl 18724 srgbinomlem 18744 srgbinom 18745 ringcl 18761 ringpropd 18782 mndvcl 20399 mhmvlin 20405 mat2pmatghm 20737 pm2mpghm 20823 cpmadugsumlemF 20883 tsmsadd 22151 omndadd2d 30017 omndadd2rd 30018 slmdacl 30071 slmdvacl 30074 gsumncl 30923 c0mhm 42420 ofaddmndmap 42632 lincsum 42728 |
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