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Mirrors > Home > MPE Home > Th. List > sgrpmgm | Structured version Visualization version GIF version |
Description: A semigroup is a magma. (Contributed by FL, 2-Nov-2009.) (Revised by AV, 6-Jan-2020.) |
Ref | Expression |
---|---|
sgrpmgm | ⊢ (𝑀 ∈ Smgrp → 𝑀 ∈ Mgm) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2818 | . . 3 ⊢ (Base‘𝑀) = (Base‘𝑀) | |
2 | eqid 2818 | . . 3 ⊢ (+g‘𝑀) = (+g‘𝑀) | |
3 | 1, 2 | issgrp 17890 | . 2 ⊢ (𝑀 ∈ Smgrp ↔ (𝑀 ∈ Mgm ∧ ∀𝑥 ∈ (Base‘𝑀)∀𝑦 ∈ (Base‘𝑀)∀𝑧 ∈ (Base‘𝑀)((𝑥(+g‘𝑀)𝑦)(+g‘𝑀)𝑧) = (𝑥(+g‘𝑀)(𝑦(+g‘𝑀)𝑧)))) |
4 | 3 | simplbi 498 | 1 ⊢ (𝑀 ∈ Smgrp → 𝑀 ∈ Mgm) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1528 ∈ wcel 2105 ∀wral 3135 ‘cfv 6348 (class class class)co 7145 Basecbs 16471 +gcplusg 16553 Mgmcmgm 17838 Smgrpcsgrp 17888 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-nul 5201 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-sbc 3770 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-br 5058 df-iota 6307 df-fv 6356 df-ov 7148 df-sgrp 17889 |
This theorem is referenced by: mndmgm 17906 gsumsgrpccat 17992 sgrpssmgm 18036 dfgrp2 18066 dfgrp3e 18137 mulgnndir 18194 mulgnnass 18200 rngcl 44082 isrnghmmul 44092 idrnghm 44107 c0rnghm 44112 |
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