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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 2798 | . . . . 5 ⊢ (Base‘𝑀) = (Base‘𝑀) | |
2 | eqid 2798 | . . . . 5 ⊢ (+g‘𝑀) = (+g‘𝑀) | |
3 | 1, 2 | ismgmALT 44483 | . . . 4 ⊢ (𝑀 ∈ MgmALT → (𝑀 ∈ MgmALT ↔ (+g‘𝑀) clLaw (Base‘𝑀))) |
4 | fvex 6658 | . . . . . 6 ⊢ (+g‘𝑀) ∈ V | |
5 | fvex 6658 | . . . . . 6 ⊢ (Base‘𝑀) ∈ V | |
6 | iscllaw 44449 | . . . . . 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 691 | . . . . 5 ⊢ ((+g‘𝑀) clLaw (Base‘𝑀) ↔ ∀𝑥 ∈ (Base‘𝑀)∀𝑦 ∈ (Base‘𝑀)(𝑥(+g‘𝑀)𝑦) ∈ (Base‘𝑀)) |
8 | 1, 2 | ismgm 17845 | . . . . . 6 ⊢ (𝑀 ∈ MgmALT → (𝑀 ∈ Mgm ↔ ∀𝑥 ∈ (Base‘𝑀)∀𝑦 ∈ (Base‘𝑀)(𝑥(+g‘𝑀)𝑦) ∈ (Base‘𝑀))) |
9 | 8 | biimprd 251 | . . . . 5 ⊢ (𝑀 ∈ MgmALT → (∀𝑥 ∈ (Base‘𝑀)∀𝑦 ∈ (Base‘𝑀)(𝑥(+g‘𝑀)𝑦) ∈ (Base‘𝑀) → 𝑀 ∈ Mgm)) |
10 | 7, 9 | syl5bi 245 | . . . 4 ⊢ (𝑀 ∈ MgmALT → ((+g‘𝑀) clLaw (Base‘𝑀) → 𝑀 ∈ Mgm)) |
11 | 3, 10 | sylbid 243 | . . 3 ⊢ (𝑀 ∈ MgmALT → (𝑀 ∈ MgmALT → 𝑀 ∈ Mgm)) |
12 | 11 | pm2.43i 52 | . 2 ⊢ (𝑀 ∈ MgmALT → 𝑀 ∈ Mgm) |
13 | mgmplusgiopALT 44454 | . . 3 ⊢ (𝑀 ∈ Mgm → (+g‘𝑀) clLaw (Base‘𝑀)) | |
14 | 1, 2 | ismgmALT 44483 | . . 3 ⊢ (𝑀 ∈ Mgm → (𝑀 ∈ MgmALT ↔ (+g‘𝑀) clLaw (Base‘𝑀))) |
15 | 13, 14 | mpbird 260 | . 2 ⊢ (𝑀 ∈ Mgm → 𝑀 ∈ MgmALT) |
16 | 12, 15 | impbii 212 | 1 ⊢ (𝑀 ∈ MgmALT ↔ 𝑀 ∈ Mgm) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 209 ∈ wcel 2111 ∀wral 3106 Vcvv 3441 class class class wbr 5030 ‘cfv 6324 (class class class)co 7135 Basecbs 16475 +gcplusg 16557 Mgmcmgm 17842 clLaw ccllaw 44443 MgmALTcmgm2 44475 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pr 5295 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ral 3111 df-rex 3112 df-v 3443 df-sbc 3721 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-iota 6283 df-fv 6332 df-ov 7138 df-mgm 17844 df-cllaw 44446 df-mgm2 44479 |
This theorem is referenced by: sgrp2sgrp 44488 |
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