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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 2736 | . . . . 5 ⊢ (Base‘𝑀) = (Base‘𝑀) | |
| 2 | eqid 2736 | . . . . 5 ⊢ (+g‘𝑀) = (+g‘𝑀) | |
| 3 | 1, 2 | ismgmALT 48144 | . . . 4 ⊢ (𝑀 ∈ MgmALT → (𝑀 ∈ MgmALT ↔ (+g‘𝑀) clLaw (Base‘𝑀))) | 
| 4 | fvex 6918 | . . . . . 6 ⊢ (+g‘𝑀) ∈ V | |
| 5 | fvex 6918 | . . . . . 6 ⊢ (Base‘𝑀) ∈ V | |
| 6 | iscllaw 48110 | . . . . . 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 18655 | . . . . . 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 48115 | . . 3 ⊢ (𝑀 ∈ Mgm → (+g‘𝑀) clLaw (Base‘𝑀)) | |
| 14 | 1, 2 | ismgmALT 48144 | . . 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 2107 ∀wral 3060 Vcvv 3479 class class class wbr 5142 ‘cfv 6560 (class class class)co 7432 Basecbs 17248 +gcplusg 17298 Mgmcmgm 18652 clLaw ccllaw 48104 MgmALTcmgm2 48136 | 
| 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-cllaw 48107 df-mgm2 48140 | 
| This theorem is referenced by: sgrp2sgrp 48149 | 
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