Theorem mgmplusgiopALT 45276

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 2738	. . . . 5 ⊢ (Base‘𝑀) = (Base‘𝑀)
2		eqid 2738	. . . . 5 ⊢ (+g‘𝑀) = (+g‘𝑀)
3	1, 2	mgmcl 18244	. . . 4 ⊢ ((𝑀 ∈ Mgm ∧ 𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀)) → (𝑥(+g‘𝑀)𝑦) ∈ (Base‘𝑀))
4	3	3expb 1118	. . 3 ⊢ ((𝑀 ∈ Mgm ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀))) → (𝑥(+g‘𝑀)𝑦) ∈ (Base‘𝑀))
5	4	ralrimivva 3114	. 2 ⊢ (𝑀 ∈ Mgm → ∀𝑥 ∈ (Base‘𝑀)∀𝑦 ∈ (Base‘𝑀)(𝑥(+g‘𝑀)𝑦) ∈ (Base‘𝑀))
6		fvex 6769	. . . 4 ⊢ (+g‘𝑀) ∈ V
7		fvex 6769	. . . 4 ⊢ (Base‘𝑀) ∈ V
8	6, 7	pm3.2i 470	. . 3 ⊢ ((+g‘𝑀) ∈ V ∧ (Base‘𝑀) ∈ V)
9		iscllaw 45271	. . 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 256	1 ⊢ (𝑀 ∈ Mgm → (+g‘𝑀) clLaw (Base‘𝑀))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   ∈ wcel 2108  ∀wral 3063  Vcvv 3422   class class class wbr 5070  ‘cfv 6418  (class class class)co 7255  Basecbs 16840  +_gcplusg 16888  Mgmcmgm 18239   clLaw ccllaw 45265

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pr 5347

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  df-sbc 3712  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-br 5071  df-opab 5133  df-iota 6376  df-fv 6426  df-ov 7258  df-mgm 18241  df-cllaw 45268

This theorem is referenced by:  mgm2mgm  45309