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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 2771 | . . . . 5 ⊢ (Base‘𝑀) = (Base‘𝑀) | |
2 | eqid 2771 | . . . . 5 ⊢ (+g‘𝑀) = (+g‘𝑀) | |
3 | 1, 2 | mgmcl 17453 | . . . 4 ⊢ ((𝑀 ∈ Mgm ∧ 𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀)) → (𝑥(+g‘𝑀)𝑦) ∈ (Base‘𝑀)) |
4 | 3 | 3expb 1113 | . . 3 ⊢ ((𝑀 ∈ Mgm ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀))) → (𝑥(+g‘𝑀)𝑦) ∈ (Base‘𝑀)) |
5 | 4 | ralrimivva 3120 | . 2 ⊢ (𝑀 ∈ Mgm → ∀𝑥 ∈ (Base‘𝑀)∀𝑦 ∈ (Base‘𝑀)(𝑥(+g‘𝑀)𝑦) ∈ (Base‘𝑀)) |
6 | fvex 6342 | . . . 4 ⊢ (+g‘𝑀) ∈ V | |
7 | fvex 6342 | . . . 4 ⊢ (Base‘𝑀) ∈ V | |
8 | 6, 7 | pm3.2i 447 | . . 3 ⊢ ((+g‘𝑀) ∈ V ∧ (Base‘𝑀) ∈ V) |
9 | iscllaw 42353 | . . 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 247 | 1 ⊢ (𝑀 ∈ Mgm → (+g‘𝑀) clLaw (Base‘𝑀)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∧ wa 382 ∈ wcel 2145 ∀wral 3061 Vcvv 3351 class class class wbr 4786 ‘cfv 6031 (class class class)co 6793 Basecbs 16064 +gcplusg 16149 Mgmcmgm 17448 clLaw ccllaw 42347 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-9 2154 ax-10 2174 ax-11 2190 ax-12 2203 ax-13 2408 ax-ext 2751 ax-sep 4915 ax-nul 4923 ax-pr 5034 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 835 df-3an 1073 df-tru 1634 df-ex 1853 df-nf 1858 df-sb 2050 df-eu 2622 df-mo 2623 df-clab 2758 df-cleq 2764 df-clel 2767 df-nfc 2902 df-ral 3066 df-rex 3067 df-rab 3070 df-v 3353 df-sbc 3588 df-dif 3726 df-un 3728 df-in 3730 df-ss 3737 df-nul 4064 df-if 4226 df-sn 4317 df-pr 4319 df-op 4323 df-uni 4575 df-br 4787 df-opab 4847 df-iota 5994 df-fv 6039 df-ov 6796 df-mgm 17450 df-cllaw 42350 |
This theorem is referenced by: mgm2mgm 42391 |
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