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Mathbox for Alexander van der Vekens |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > mgm2mgm | Structured version Visualization version GIF version |
Description: Equivalence of the two definitions of a magma. (Contributed by AV, 16-Jan-2020.) |
Ref | Expression |
---|---|
mgm2mgm | ⊢ (𝑀 ∈ MgmALT ↔ 𝑀 ∈ Mgm) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2735 | . . . . 5 ⊢ (Base‘𝑀) = (Base‘𝑀) | |
2 | eqid 2735 | . . . . 5 ⊢ (+g‘𝑀) = (+g‘𝑀) | |
3 | 1, 2 | ismgmALT 48067 | . . . 4 ⊢ (𝑀 ∈ MgmALT → (𝑀 ∈ MgmALT ↔ (+g‘𝑀) clLaw (Base‘𝑀))) |
4 | fvex 6920 | . . . . . 6 ⊢ (+g‘𝑀) ∈ V | |
5 | fvex 6920 | . . . . . 6 ⊢ (Base‘𝑀) ∈ V | |
6 | iscllaw 48033 | . . . . . 6 ⊢ (((+g‘𝑀) ∈ V ∧ (Base‘𝑀) ∈ V) → ((+g‘𝑀) clLaw (Base‘𝑀) ↔ ∀𝑥 ∈ (Base‘𝑀)∀𝑦 ∈ (Base‘𝑀)(𝑥(+g‘𝑀)𝑦) ∈ (Base‘𝑀))) | |
7 | 4, 5, 6 | mp2an 692 | . . . . 5 ⊢ ((+g‘𝑀) clLaw (Base‘𝑀) ↔ ∀𝑥 ∈ (Base‘𝑀)∀𝑦 ∈ (Base‘𝑀)(𝑥(+g‘𝑀)𝑦) ∈ (Base‘𝑀)) |
8 | 1, 2 | ismgm 18667 | . . . . . 6 ⊢ (𝑀 ∈ MgmALT → (𝑀 ∈ Mgm ↔ ∀𝑥 ∈ (Base‘𝑀)∀𝑦 ∈ (Base‘𝑀)(𝑥(+g‘𝑀)𝑦) ∈ (Base‘𝑀))) |
9 | 8 | biimprd 248 | . . . . 5 ⊢ (𝑀 ∈ MgmALT → (∀𝑥 ∈ (Base‘𝑀)∀𝑦 ∈ (Base‘𝑀)(𝑥(+g‘𝑀)𝑦) ∈ (Base‘𝑀) → 𝑀 ∈ Mgm)) |
10 | 7, 9 | biimtrid 242 | . . . 4 ⊢ (𝑀 ∈ MgmALT → ((+g‘𝑀) clLaw (Base‘𝑀) → 𝑀 ∈ Mgm)) |
11 | 3, 10 | sylbid 240 | . . 3 ⊢ (𝑀 ∈ MgmALT → (𝑀 ∈ MgmALT → 𝑀 ∈ Mgm)) |
12 | 11 | pm2.43i 52 | . 2 ⊢ (𝑀 ∈ MgmALT → 𝑀 ∈ Mgm) |
13 | mgmplusgiopALT 48038 | . . 3 ⊢ (𝑀 ∈ Mgm → (+g‘𝑀) clLaw (Base‘𝑀)) | |
14 | 1, 2 | ismgmALT 48067 | . . 3 ⊢ (𝑀 ∈ Mgm → (𝑀 ∈ MgmALT ↔ (+g‘𝑀) clLaw (Base‘𝑀))) |
15 | 13, 14 | mpbird 257 | . 2 ⊢ (𝑀 ∈ Mgm → 𝑀 ∈ MgmALT) |
16 | 12, 15 | impbii 209 | 1 ⊢ (𝑀 ∈ MgmALT ↔ 𝑀 ∈ Mgm) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 206 ∈ wcel 2106 ∀wral 3059 Vcvv 3478 class class class wbr 5148 ‘cfv 6563 (class class class)co 7431 Basecbs 17245 +gcplusg 17298 Mgmcmgm 18664 clLaw ccllaw 48027 MgmALTcmgm2 48059 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-ne 2939 df-ral 3060 df-rab 3434 df-v 3480 df-sbc 3792 df-dif 3966 df-un 3968 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-iota 6516 df-fv 6571 df-ov 7434 df-mgm 18666 df-cllaw 48030 df-mgm2 48063 |
This theorem is referenced by: sgrp2sgrp 48072 |
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