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Mirrors > Home > MPE Home > Th. List > Mathboxes > sgrp2sgrp | Structured version Visualization version GIF version |
Description: Equivalence of the two definitions of a semigroup. (Contributed by AV, 16-Jan-2020.) |
Ref | Expression |
---|---|
sgrp2sgrp | ⊢ (𝑀 ∈ SGrpALT ↔ 𝑀 ∈ Smgrp) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mgm2mgm 44062 | . . . 4 ⊢ (𝑀 ∈ MgmALT ↔ 𝑀 ∈ Mgm) | |
2 | 1 | anbi1i 623 | . . 3 ⊢ ((𝑀 ∈ MgmALT ∧ (+g‘𝑀) assLaw (Base‘𝑀)) ↔ (𝑀 ∈ Mgm ∧ (+g‘𝑀) assLaw (Base‘𝑀))) |
3 | fvex 6676 | . . . . . 6 ⊢ (+g‘𝑀) ∈ V | |
4 | fvex 6676 | . . . . . 6 ⊢ (Base‘𝑀) ∈ V | |
5 | 3, 4 | pm3.2i 471 | . . . . 5 ⊢ ((+g‘𝑀) ∈ V ∧ (Base‘𝑀) ∈ V) |
6 | isasslaw 44027 | . . . . 5 ⊢ (((+g‘𝑀) ∈ V ∧ (Base‘𝑀) ∈ V) → ((+g‘𝑀) assLaw (Base‘𝑀) ↔ ∀𝑥 ∈ (Base‘𝑀)∀𝑦 ∈ (Base‘𝑀)∀𝑧 ∈ (Base‘𝑀)((𝑥(+g‘𝑀)𝑦)(+g‘𝑀)𝑧) = (𝑥(+g‘𝑀)(𝑦(+g‘𝑀)𝑧)))) | |
7 | 5, 6 | mp1i 13 | . . . 4 ⊢ (𝑀 ∈ Mgm → ((+g‘𝑀) assLaw (Base‘𝑀) ↔ ∀𝑥 ∈ (Base‘𝑀)∀𝑦 ∈ (Base‘𝑀)∀𝑧 ∈ (Base‘𝑀)((𝑥(+g‘𝑀)𝑦)(+g‘𝑀)𝑧) = (𝑥(+g‘𝑀)(𝑦(+g‘𝑀)𝑧)))) |
8 | 7 | pm5.32i 575 | . . 3 ⊢ ((𝑀 ∈ Mgm ∧ (+g‘𝑀) assLaw (Base‘𝑀)) ↔ (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)))) |
9 | 2, 8 | bitri 276 | . 2 ⊢ ((𝑀 ∈ MgmALT ∧ (+g‘𝑀) assLaw (Base‘𝑀)) ↔ (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)))) |
10 | eqid 2818 | . . 3 ⊢ (Base‘𝑀) = (Base‘𝑀) | |
11 | eqid 2818 | . . 3 ⊢ (+g‘𝑀) = (+g‘𝑀) | |
12 | 10, 11 | issgrpALT 44060 | . 2 ⊢ (𝑀 ∈ SGrpALT ↔ (𝑀 ∈ MgmALT ∧ (+g‘𝑀) assLaw (Base‘𝑀))) |
13 | 10, 11 | 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)))) |
14 | 9, 12, 13 | 3bitr4i 304 | 1 ⊢ (𝑀 ∈ SGrpALT ↔ 𝑀 ∈ Smgrp) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 207 ∧ wa 396 = wceq 1528 ∈ wcel 2105 ∀wral 3135 Vcvv 3492 class class class wbr 5057 ‘cfv 6348 (class class class)co 7145 Basecbs 16471 +gcplusg 16553 Mgmcmgm 17838 Smgrpcsgrp 17888 assLaw casslaw 44019 MgmALTcmgm2 44050 SGrpALTcsgrp2 44052 |
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-sep 5194 ax-nul 5201 ax-pr 5320 |
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-opab 5120 df-iota 6307 df-fv 6356 df-ov 7148 df-mgm 17840 df-sgrp 17889 df-cllaw 44021 df-asslaw 44023 df-mgm2 44054 df-sgrp2 44056 |
This theorem is referenced by: (None) |
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