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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 2622 | . . . . 5 ⊢ (Base‘𝑀) = (Base‘𝑀) | |
2 | eqid 2622 | . . . . 5 ⊢ (+g‘𝑀) = (+g‘𝑀) | |
3 | 1, 2 | ismgmALT 41859 | . . . 4 ⊢ (𝑀 ∈ MgmALT → (𝑀 ∈ MgmALT ↔ (+g‘𝑀) clLaw (Base‘𝑀))) |
4 | fvex 6201 | . . . . . 6 ⊢ (+g‘𝑀) ∈ V | |
5 | fvex 6201 | . . . . . 6 ⊢ (Base‘𝑀) ∈ V | |
6 | iscllaw 41825 | . . . . . 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 708 | . . . . 5 ⊢ ((+g‘𝑀) clLaw (Base‘𝑀) ↔ ∀𝑥 ∈ (Base‘𝑀)∀𝑦 ∈ (Base‘𝑀)(𝑥(+g‘𝑀)𝑦) ∈ (Base‘𝑀)) |
8 | 1, 2 | ismgm 17243 | . . . . . 6 ⊢ (𝑀 ∈ MgmALT → (𝑀 ∈ Mgm ↔ ∀𝑥 ∈ (Base‘𝑀)∀𝑦 ∈ (Base‘𝑀)(𝑥(+g‘𝑀)𝑦) ∈ (Base‘𝑀))) |
9 | 8 | biimprd 238 | . . . . 5 ⊢ (𝑀 ∈ MgmALT → (∀𝑥 ∈ (Base‘𝑀)∀𝑦 ∈ (Base‘𝑀)(𝑥(+g‘𝑀)𝑦) ∈ (Base‘𝑀) → 𝑀 ∈ Mgm)) |
10 | 7, 9 | syl5bi 232 | . . . 4 ⊢ (𝑀 ∈ MgmALT → ((+g‘𝑀) clLaw (Base‘𝑀) → 𝑀 ∈ Mgm)) |
11 | 3, 10 | sylbid 230 | . . 3 ⊢ (𝑀 ∈ MgmALT → (𝑀 ∈ MgmALT → 𝑀 ∈ Mgm)) |
12 | 11 | pm2.43i 52 | . 2 ⊢ (𝑀 ∈ MgmALT → 𝑀 ∈ Mgm) |
13 | mgmplusgiopALT 41830 | . . 3 ⊢ (𝑀 ∈ Mgm → (+g‘𝑀) clLaw (Base‘𝑀)) | |
14 | 1, 2 | ismgmALT 41859 | . . 3 ⊢ (𝑀 ∈ Mgm → (𝑀 ∈ MgmALT ↔ (+g‘𝑀) clLaw (Base‘𝑀))) |
15 | 13, 14 | mpbird 247 | . 2 ⊢ (𝑀 ∈ Mgm → 𝑀 ∈ MgmALT) |
16 | 12, 15 | impbii 199 | 1 ⊢ (𝑀 ∈ MgmALT ↔ 𝑀 ∈ Mgm) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 196 ∈ wcel 1990 ∀wral 2912 Vcvv 3200 class class class wbr 4653 ‘cfv 5888 (class class class)co 6650 Basecbs 15857 +gcplusg 15941 Mgmcmgm 17240 clLaw ccllaw 41819 MgmALTcmgm2 41851 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pr 4906 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-iota 5851 df-fv 5896 df-ov 6653 df-mgm 17242 df-cllaw 41822 df-mgm2 41855 |
This theorem is referenced by: sgrp2sgrp 41864 |
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