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Mirrors > Home > ILE Home > Th. List > mndcl | 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 12712 | . 2 ⊢ (𝐺 ∈ Mnd → 𝐺 ∈ Mgm) | |
2 | mndcl.b | . . 3 ⊢ 𝐵 = (Base‘𝐺) | |
3 | mndcl.p | . . 3 ⊢ + = (+g‘𝐺) | |
4 | 2, 3 | mgmcl 12667 | . 2 ⊢ ((𝐺 ∈ Mgm ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 + 𝑌) ∈ 𝐵) |
5 | 1, 4 | syl3an1 1271 | 1 ⊢ ((𝐺 ∈ Mnd ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 + 𝑌) ∈ 𝐵) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ w3a 978 = wceq 1353 ∈ wcel 2148 ‘cfv 5212 (class class class)co 5869 Basecbs 12442 +gcplusg 12515 Mgmcmgm 12662 Mndcmnd 12706 |
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 4118 ax-pow 4171 ax-pr 4206 ax-un 4430 ax-cnex 7890 ax-resscn 7891 ax-1re 7893 ax-addrcl 7896 |
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-rab 2464 df-v 2739 df-sbc 2963 df-un 3133 df-in 3135 df-ss 3142 df-pw 3576 df-sn 3597 df-pr 3598 df-op 3600 df-uni 3808 df-int 3843 df-br 4001 df-opab 4062 df-mpt 4063 df-id 4290 df-xp 4629 df-rel 4630 df-cnv 4631 df-co 4632 df-dm 4633 df-rn 4634 df-res 4635 df-iota 5174 df-fun 5214 df-fn 5215 df-fv 5220 df-ov 5872 df-inn 8906 df-2 8964 df-ndx 12445 df-slot 12446 df-base 12448 df-plusg 12528 df-mgm 12664 df-sgrp 12697 df-mnd 12707 |
This theorem is referenced by: mnd4g 12719 mndpropd 12730 issubmnd 12732 idmhm 12747 mhmf1o 12748 issubmd 12752 submid 12755 0mhm 12760 mhmco 12761 mhmeql 12763 grpcl 12772 mhmmnd 12866 mulgnn0cl 12885 mulgnn0z 12895 srgcl 12976 srgacl 12988 ringcl 13019 ringpropd 13040 |
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