Theorem mgmplusgiopALT 48882

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 2769	. . . . 5 ⊢ (Base‘𝑀) = (Base‘𝑀)
2		eqid 2769	. . . . 5 ⊢ (+g‘𝑀) = (+g‘𝑀)
3	1, 2	mgmcl 18701	. . . 4 ⊢ ((𝑀 ∈ Mgm ∧ 𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀)) → (𝑥(+g‘𝑀)𝑦) ∈ (Base‘𝑀))
4	3	3expb 1136	. . 3 ⊢ ((𝑀 ∈ Mgm ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀))) → (𝑥(+g‘𝑀)𝑦) ∈ (Base‘𝑀))
5	4	ralrimivva 3214	. 2 ⊢ (𝑀 ∈ Mgm → ∀𝑥 ∈ (Base‘𝑀)∀𝑦 ∈ (Base‘𝑀)(𝑥(+g‘𝑀)𝑦) ∈ (Base‘𝑀))
6		fvex 6895	. . . 4 ⊢ (+g‘𝑀) ∈ V
7		fvex 6895	. . . 4 ⊢ (Base‘𝑀) ∈ V
8	6, 7	pm3.2i 475	. . 3 ⊢ ((+g‘𝑀) ∈ V ∧ (Base‘𝑀) ∈ V)
9		iscllaw 48877	. . 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 14	. 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))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   ∈ wcel 2149  ∀wral 3085  Vcvv 3463   class class class wbr 5113  ‘cfv 6537  (class class class)co 7411  Basecbs 17269  +_gcplusg 17310  Mgmcmgm 18696   clLaw ccllaw 48871

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-ext 2741  ax-sep 5261  ax-nul 5271  ax-pr 5405

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-ne 2965  df-ral 3086  df-rab 3424  df-v 3465  df-sbc 3754  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-br 5114  df-opab 5178  df-iota 6493  df-fv 6545  df-ov 7414  df-mgm 18698  df-cllaw 48874

This theorem is referenced by:  mgm2mgm  48915