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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 48148 | . . . 4 ⊢ (𝑀 ∈ MgmALT ↔ 𝑀 ∈ Mgm) | |
| 2 | 1 | anbi1i 624 | . . 3 ⊢ ((𝑀 ∈ MgmALT ∧ (+g‘𝑀) assLaw (Base‘𝑀)) ↔ (𝑀 ∈ Mgm ∧ (+g‘𝑀) assLaw (Base‘𝑀))) | 
| 3 | fvex 6918 | . . . . . 6 ⊢ (+g‘𝑀) ∈ V | |
| 4 | fvex 6918 | . . . . . 6 ⊢ (Base‘𝑀) ∈ V | |
| 5 | 3, 4 | pm3.2i 470 | . . . . 5 ⊢ ((+g‘𝑀) ∈ V ∧ (Base‘𝑀) ∈ V) | 
| 6 | isasslaw 48113 | . . . . 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 574 | . . 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 2736 | . . 3 ⊢ (Base‘𝑀) = (Base‘𝑀) | |
| 11 | eqid 2736 | . . 3 ⊢ (+g‘𝑀) = (+g‘𝑀) | |
| 12 | 10, 11 | issgrpALT 48146 | . 2 ⊢ (𝑀 ∈ SGrpALT ↔ (𝑀 ∈ MgmALT ∧ (+g‘𝑀) assLaw (Base‘𝑀))) | 
| 13 | 10, 11 | issgrp 18734 | . 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 206 ∧ wa 395 = wceq 1539 ∈ wcel 2107 ∀wral 3060 Vcvv 3479 class class class wbr 5142 ‘cfv 6560 (class class class)co 7432 Basecbs 17248 +gcplusg 17298 Mgmcmgm 18652 Smgrpcsgrp 18732 assLaw casslaw 48105 MgmALTcmgm2 48136 SGrpALTcsgrp2 48138 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pr 5431 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-sb 2064 df-clab 2714 df-cleq 2728 df-clel 2815 df-ne 2940 df-ral 3061 df-rab 3436 df-v 3481 df-sbc 3788 df-dif 3953 df-un 3955 df-ss 3967 df-nul 4333 df-if 4525 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-br 5143 df-opab 5205 df-iota 6513 df-fv 6568 df-ov 7435 df-mgm 18654 df-sgrp 18733 df-cllaw 48107 df-asslaw 48109 df-mgm2 48140 df-sgrp2 48142 | 
| This theorem is referenced by: (None) | 
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