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Mirrors > Home > ILE Home > Th. List > mndsgrp | GIF version |
Description: A monoid is a semigroup. (Contributed by FL, 2-Nov-2009.) (Revised by AV, 6-Jan-2020.) (Proof shortened by AV, 6-Feb-2020.) |
Ref | Expression |
---|---|
mndsgrp | ⊢ (𝐺 ∈ Mnd → 𝐺 ∈ Smgrp) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2177 | . . 3 ⊢ (Base‘𝐺) = (Base‘𝐺) | |
2 | eqid 2177 | . . 3 ⊢ (+g‘𝐺) = (+g‘𝐺) | |
3 | 1, 2 | ismnddef 12711 | . 2 ⊢ (𝐺 ∈ Mnd ↔ (𝐺 ∈ Smgrp ∧ ∃𝑒 ∈ (Base‘𝐺)∀𝑥 ∈ (Base‘𝐺)((𝑒(+g‘𝐺)𝑥) = 𝑥 ∧ (𝑥(+g‘𝐺)𝑒) = 𝑥))) |
4 | 3 | simplbi 274 | 1 ⊢ (𝐺 ∈ Mnd → 𝐺 ∈ Smgrp) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 = wceq 1353 ∈ wcel 2148 ∀wral 2455 ∃wrex 2456 ‘cfv 5212 (class class class)co 5869 Basecbs 12445 +gcplusg 12518 Smgrpcsgrp 12699 Mndcmnd 12709 |
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 7893 ax-resscn 7894 ax-1re 7896 ax-addrcl 7899 |
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 8909 df-2 8967 df-ndx 12448 df-slot 12449 df-base 12451 df-plusg 12531 df-mnd 12710 |
This theorem is referenced by: mndmgm 12715 mndass 12717 grpsgrp 12791 mulgnn0dir 12901 mulgnn0ass 12907 |
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