Theorem mgmplusgiopALT 48554

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.)

Assertion

Ref	Expression
mgmplusgiopALT	⊢ (𝑀 ∈ Mgm → (+g‘𝑀) clLaw (Base‘𝑀))

Proof of Theorem mgmplusgiopALT

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2737	. . . . 5 ⊢ (Base‘𝑀) = (Base‘𝑀)
2		eqid 2737	. . . . 5 ⊢ (+g‘𝑀) = (+g‘𝑀)
3	1, 2	mgmcl 18580	. . . 4 ⊢ ((𝑀 ∈ Mgm ∧ 𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀)) → (𝑥(+g‘𝑀)𝑦) ∈ (Base‘𝑀))
4	3	3expb 1121	. . 3 ⊢ ((𝑀 ∈ Mgm ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀))) → (𝑥(+g‘𝑀)𝑦) ∈ (Base‘𝑀))
5	4	ralrimivva 3181	. 2 ⊢ (𝑀 ∈ Mgm → ∀𝑥 ∈ (Base‘𝑀)∀𝑦 ∈ (Base‘𝑀)(𝑥(+g‘𝑀)𝑦) ∈ (Base‘𝑀))
6		fvex 6855	. . . 4 ⊢ (+g‘𝑀) ∈ V
7		fvex 6855	. . . 4 ⊢ (Base‘𝑀) ∈ V
8	6, 7	pm3.2i 470	. . 3 ⊢ ((+g‘𝑀) ∈ V ∧ (Base‘𝑀) ∈ V)
9		iscllaw 48549	. . 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 257	1 ⊢ (𝑀 ∈ Mgm → (+g‘𝑀) clLaw (Base‘𝑀))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∈ wcel 2114  ∀wral 3052  Vcvv 3442   class class class wbr 5100  ‘cfv 6500  (class class class)co 7368  Basecbs 17148  +_gcplusg 17189  Mgmcmgm 18575   clLaw ccllaw 48543

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709  ax-sep 5243  ax-nul 5253  ax-pr 5379

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-ne 2934  df-ral 3053  df-rab 3402  df-v 3444  df-sbc 3743  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-opab 5163  df-iota 6456  df-fv 6508  df-ov 7371  df-mgm 18577  df-cllaw 48546

This theorem is referenced by:  mgm2mgm  48587