![]() |
Mathbox for Alexander van der Vekens |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > mgmplusgiopALT | Structured version Visualization version GIF version |
Description: Slot 2 (group operation) of a magma as extensible structure is a closed operation on the base set. (Contributed by AV, 13-Jan-2020.) (New usage is discouraged.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
mgmplusgiopALT | ⊢ (𝑀 ∈ Mgm → (+g‘𝑀) clLaw (Base‘𝑀)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2798 | . . . . 5 ⊢ (Base‘𝑀) = (Base‘𝑀) | |
2 | eqid 2798 | . . . . 5 ⊢ (+g‘𝑀) = (+g‘𝑀) | |
3 | 1, 2 | mgmcl 17847 | . . . 4 ⊢ ((𝑀 ∈ Mgm ∧ 𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀)) → (𝑥(+g‘𝑀)𝑦) ∈ (Base‘𝑀)) |
4 | 3 | 3expb 1117 | . . 3 ⊢ ((𝑀 ∈ Mgm ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀))) → (𝑥(+g‘𝑀)𝑦) ∈ (Base‘𝑀)) |
5 | 4 | ralrimivva 3156 | . 2 ⊢ (𝑀 ∈ Mgm → ∀𝑥 ∈ (Base‘𝑀)∀𝑦 ∈ (Base‘𝑀)(𝑥(+g‘𝑀)𝑦) ∈ (Base‘𝑀)) |
6 | fvex 6658 | . . . 4 ⊢ (+g‘𝑀) ∈ V | |
7 | fvex 6658 | . . . 4 ⊢ (Base‘𝑀) ∈ V | |
8 | 6, 7 | pm3.2i 474 | . . 3 ⊢ ((+g‘𝑀) ∈ V ∧ (Base‘𝑀) ∈ V) |
9 | iscllaw 44449 | . . 3 ⊢ (((+g‘𝑀) ∈ V ∧ (Base‘𝑀) ∈ V) → ((+g‘𝑀) clLaw (Base‘𝑀) ↔ ∀𝑥 ∈ (Base‘𝑀)∀𝑦 ∈ (Base‘𝑀)(𝑥(+g‘𝑀)𝑦) ∈ (Base‘𝑀))) | |
10 | 8, 9 | mp1i 13 | . 2 ⊢ (𝑀 ∈ Mgm → ((+g‘𝑀) clLaw (Base‘𝑀) ↔ ∀𝑥 ∈ (Base‘𝑀)∀𝑦 ∈ (Base‘𝑀)(𝑥(+g‘𝑀)𝑦) ∈ (Base‘𝑀))) |
11 | 5, 10 | mpbird 260 | 1 ⊢ (𝑀 ∈ Mgm → (+g‘𝑀) clLaw (Base‘𝑀)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 ∈ 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 |
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 |
This theorem is referenced by: mgm2mgm 44487 |
Copyright terms: Public domain | W3C validator |