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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 ∈ SGrp) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mgm2mgm 43504 | . . . 4 ⊢ (𝑀 ∈ MgmALT ↔ 𝑀 ∈ Mgm) | |
2 | 1 | anbi1i 614 | . . 3 ⊢ ((𝑀 ∈ MgmALT ∧ (+g‘𝑀) assLaw (Base‘𝑀)) ↔ (𝑀 ∈ Mgm ∧ (+g‘𝑀) assLaw (Base‘𝑀))) |
3 | fvex 6512 | . . . . . 6 ⊢ (+g‘𝑀) ∈ V | |
4 | fvex 6512 | . . . . . 6 ⊢ (Base‘𝑀) ∈ V | |
5 | 3, 4 | pm3.2i 463 | . . . . 5 ⊢ ((+g‘𝑀) ∈ V ∧ (Base‘𝑀) ∈ V) |
6 | isasslaw 43469 | . . . . 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 567 | . . 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 267 | . 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 2778 | . . 3 ⊢ (Base‘𝑀) = (Base‘𝑀) | |
11 | eqid 2778 | . . 3 ⊢ (+g‘𝑀) = (+g‘𝑀) | |
12 | 10, 11 | issgrpALT 43502 | . 2 ⊢ (𝑀 ∈ SGrpALT ↔ (𝑀 ∈ MgmALT ∧ (+g‘𝑀) assLaw (Base‘𝑀))) |
13 | 10, 11 | issgrp 17753 | . 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)))) |
14 | 9, 12, 13 | 3bitr4i 295 | 1 ⊢ (𝑀 ∈ SGrpALT ↔ 𝑀 ∈ SGrp) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 198 ∧ wa 387 = wceq 1507 ∈ wcel 2050 ∀wral 3088 Vcvv 3415 class class class wbr 4929 ‘cfv 6188 (class class class)co 6976 Basecbs 16339 +gcplusg 16421 Mgmcmgm 17708 SGrpcsgrp 17751 assLaw casslaw 43461 MgmALTcmgm2 43492 SGrpALTcsgrp2 43494 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1965 ax-8 2052 ax-9 2059 ax-10 2079 ax-11 2093 ax-12 2106 ax-13 2301 ax-ext 2750 ax-sep 5060 ax-nul 5067 ax-pr 5186 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3an 1070 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2016 df-mo 2547 df-eu 2584 df-clab 2759 df-cleq 2771 df-clel 2846 df-nfc 2918 df-ral 3093 df-rex 3094 df-rab 3097 df-v 3417 df-sbc 3682 df-dif 3832 df-un 3834 df-in 3836 df-ss 3843 df-nul 4179 df-if 4351 df-sn 4442 df-pr 4444 df-op 4448 df-uni 4713 df-br 4930 df-opab 4992 df-iota 6152 df-fv 6196 df-ov 6979 df-mgm 17710 df-sgrp 17752 df-cllaw 43463 df-asslaw 43465 df-mgm2 43496 df-sgrp2 43498 |
This theorem is referenced by: (None) |
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