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Mathbox for Alexander van der Vekens |
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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 46247 | . . . 4 ⊢ (𝑀 ∈ MgmALT ↔ 𝑀 ∈ Mgm) | |
2 | 1 | anbi1i 625 | . . 3 ⊢ ((𝑀 ∈ MgmALT ∧ (+g‘𝑀) assLaw (Base‘𝑀)) ↔ (𝑀 ∈ Mgm ∧ (+g‘𝑀) assLaw (Base‘𝑀))) |
3 | fvex 6856 | . . . . . 6 ⊢ (+g‘𝑀) ∈ V | |
4 | fvex 6856 | . . . . . 6 ⊢ (Base‘𝑀) ∈ V | |
5 | 3, 4 | pm3.2i 472 | . . . . 5 ⊢ ((+g‘𝑀) ∈ V ∧ (Base‘𝑀) ∈ V) |
6 | isasslaw 46212 | . . . . 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 576 | . . 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 275 | . 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 2733 | . . 3 ⊢ (Base‘𝑀) = (Base‘𝑀) | |
11 | eqid 2733 | . . 3 ⊢ (+g‘𝑀) = (+g‘𝑀) | |
12 | 10, 11 | issgrpALT 46245 | . 2 ⊢ (𝑀 ∈ SGrpALT ↔ (𝑀 ∈ MgmALT ∧ (+g‘𝑀) assLaw (Base‘𝑀))) |
13 | 10, 11 | issgrp 18552 | . 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 303 | 1 ⊢ (𝑀 ∈ SGrpALT ↔ 𝑀 ∈ Smgrp) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∧ wa 397 = wceq 1542 ∈ wcel 2107 ∀wral 3061 Vcvv 3444 class class class wbr 5106 ‘cfv 6497 (class class class)co 7358 Basecbs 17088 +gcplusg 17138 Mgmcmgm 18500 Smgrpcsgrp 18550 assLaw casslaw 46204 MgmALTcmgm2 46235 SGrpALTcsgrp2 46237 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2704 ax-sep 5257 ax-nul 5264 ax-pr 5385 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-ne 2941 df-ral 3062 df-rab 3407 df-v 3446 df-sbc 3741 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-br 5107 df-opab 5169 df-iota 6449 df-fv 6505 df-ov 7361 df-mgm 18502 df-sgrp 18551 df-cllaw 46206 df-asslaw 46208 df-mgm2 46239 df-sgrp2 46241 |
This theorem is referenced by: (None) |
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