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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 ∈ SGrp → 𝑀 ∈ Mgm) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2777 | . . 3 ⊢ (Base‘𝑀) = (Base‘𝑀) | |
2 | eqid 2777 | . . 3 ⊢ (+g‘𝑀) = (+g‘𝑀) | |
3 | 1, 2 | issgrp 17671 | . 2 ⊢ (𝑀 ∈ SGrp ↔ (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 493 | 1 ⊢ (𝑀 ∈ SGrp → 𝑀 ∈ Mgm) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1601 ∈ wcel 2106 ∀wral 3089 ‘cfv 6135 (class class class)co 6922 Basecbs 16255 +gcplusg 16338 Mgmcmgm 17626 SGrpcsgrp 17669 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2054 ax-9 2115 ax-10 2134 ax-11 2149 ax-12 2162 ax-13 2333 ax-ext 2753 ax-nul 5025 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2550 df-eu 2586 df-clab 2763 df-cleq 2769 df-clel 2773 df-nfc 2920 df-ral 3094 df-rex 3095 df-rab 3098 df-v 3399 df-sbc 3652 df-dif 3794 df-un 3796 df-in 3798 df-ss 3805 df-nul 4141 df-if 4307 df-sn 4398 df-pr 4400 df-op 4404 df-uni 4672 df-br 4887 df-iota 6099 df-fv 6143 df-ov 6925 df-sgrp 17670 |
This theorem is referenced by: mndmgm 17686 sgrpssmgm 17807 dfgrp2 17834 dfgrp3e 17902 mulgnndir 17955 mulgnnass 17961 rngcl 42891 isrnghmmul 42901 idrnghm 42916 c0rnghm 42921 |
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