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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 2821 | . . . . 5 ⊢ (Base‘𝑀) = (Base‘𝑀) | |
2 | eqid 2821 | . . . . 5 ⊢ (+g‘𝑀) = (+g‘𝑀) | |
3 | 1, 2 | ismgmALT 44150 | . . . 4 ⊢ (𝑀 ∈ MgmALT → (𝑀 ∈ MgmALT ↔ (+g‘𝑀) clLaw (Base‘𝑀))) |
4 | fvex 6683 | . . . . . 6 ⊢ (+g‘𝑀) ∈ V | |
5 | fvex 6683 | . . . . . 6 ⊢ (Base‘𝑀) ∈ V | |
6 | iscllaw 44116 | . . . . . 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 690 | . . . . 5 ⊢ ((+g‘𝑀) clLaw (Base‘𝑀) ↔ ∀𝑥 ∈ (Base‘𝑀)∀𝑦 ∈ (Base‘𝑀)(𝑥(+g‘𝑀)𝑦) ∈ (Base‘𝑀)) |
8 | 1, 2 | ismgm 17853 | . . . . . 6 ⊢ (𝑀 ∈ MgmALT → (𝑀 ∈ Mgm ↔ ∀𝑥 ∈ (Base‘𝑀)∀𝑦 ∈ (Base‘𝑀)(𝑥(+g‘𝑀)𝑦) ∈ (Base‘𝑀))) |
9 | 8 | biimprd 250 | . . . . 5 ⊢ (𝑀 ∈ MgmALT → (∀𝑥 ∈ (Base‘𝑀)∀𝑦 ∈ (Base‘𝑀)(𝑥(+g‘𝑀)𝑦) ∈ (Base‘𝑀) → 𝑀 ∈ Mgm)) |
10 | 7, 9 | syl5bi 244 | . . . 4 ⊢ (𝑀 ∈ MgmALT → ((+g‘𝑀) clLaw (Base‘𝑀) → 𝑀 ∈ Mgm)) |
11 | 3, 10 | sylbid 242 | . . 3 ⊢ (𝑀 ∈ MgmALT → (𝑀 ∈ MgmALT → 𝑀 ∈ Mgm)) |
12 | 11 | pm2.43i 52 | . 2 ⊢ (𝑀 ∈ MgmALT → 𝑀 ∈ Mgm) |
13 | mgmplusgiopALT 44121 | . . 3 ⊢ (𝑀 ∈ Mgm → (+g‘𝑀) clLaw (Base‘𝑀)) | |
14 | 1, 2 | ismgmALT 44150 | . . 3 ⊢ (𝑀 ∈ Mgm → (𝑀 ∈ MgmALT ↔ (+g‘𝑀) clLaw (Base‘𝑀))) |
15 | 13, 14 | mpbird 259 | . 2 ⊢ (𝑀 ∈ Mgm → 𝑀 ∈ MgmALT) |
16 | 12, 15 | impbii 211 | 1 ⊢ (𝑀 ∈ MgmALT ↔ 𝑀 ∈ Mgm) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 ∈ wcel 2114 ∀wral 3138 Vcvv 3494 class class class wbr 5066 ‘cfv 6355 (class class class)co 7156 Basecbs 16483 +gcplusg 16565 Mgmcmgm 17850 clLaw ccllaw 44110 MgmALTcmgm2 44142 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pr 5330 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3773 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-br 5067 df-opab 5129 df-iota 6314 df-fv 6363 df-ov 7159 df-mgm 17852 df-cllaw 44113 df-mgm2 44146 |
This theorem is referenced by: sgrp2sgrp 44155 |
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