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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 47385 | . . . 4 ⊢ (𝑀 ∈ MgmALT ↔ 𝑀 ∈ Mgm) | |
2 | 1 | anbi1i 622 | . . 3 ⊢ ((𝑀 ∈ MgmALT ∧ (+g‘𝑀) assLaw (Base‘𝑀)) ↔ (𝑀 ∈ Mgm ∧ (+g‘𝑀) assLaw (Base‘𝑀))) |
3 | fvex 6915 | . . . . . 6 ⊢ (+g‘𝑀) ∈ V | |
4 | fvex 6915 | . . . . . 6 ⊢ (Base‘𝑀) ∈ V | |
5 | 3, 4 | pm3.2i 469 | . . . . 5 ⊢ ((+g‘𝑀) ∈ V ∧ (Base‘𝑀) ∈ V) |
6 | isasslaw 47350 | . . . . 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 573 | . . 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 274 | . 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 2728 | . . 3 ⊢ (Base‘𝑀) = (Base‘𝑀) | |
11 | eqid 2728 | . . 3 ⊢ (+g‘𝑀) = (+g‘𝑀) | |
12 | 10, 11 | issgrpALT 47383 | . 2 ⊢ (𝑀 ∈ SGrpALT ↔ (𝑀 ∈ MgmALT ∧ (+g‘𝑀) assLaw (Base‘𝑀))) |
13 | 10, 11 | issgrp 18689 | . 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 302 | 1 ⊢ (𝑀 ∈ SGrpALT ↔ 𝑀 ∈ Smgrp) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∧ wa 394 = wceq 1533 ∈ wcel 2098 ∀wral 3058 Vcvv 3473 class class class wbr 5152 ‘cfv 6553 (class class class)co 7426 Basecbs 17189 +gcplusg 17242 Mgmcmgm 18607 Smgrpcsgrp 18687 assLaw casslaw 47342 MgmALTcmgm2 47373 SGrpALTcsgrp2 47375 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-ext 2699 ax-sep 5303 ax-nul 5310 ax-pr 5433 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-sb 2060 df-clab 2706 df-cleq 2720 df-clel 2806 df-ne 2938 df-ral 3059 df-rab 3431 df-v 3475 df-sbc 3779 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4327 df-if 4533 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-br 5153 df-opab 5215 df-iota 6505 df-fv 6561 df-ov 7429 df-mgm 18609 df-sgrp 18688 df-cllaw 47344 df-asslaw 47346 df-mgm2 47377 df-sgrp2 47379 |
This theorem is referenced by: (None) |
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