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Theorem mgmplusgiopALT 41622
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 → (+g𝑀) clLaw (Base‘𝑀))

Proof of Theorem mgmplusgiopALT
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2609 . . . . 5 (Base‘𝑀) = (Base‘𝑀)
2 eqid 2609 . . . . 5 (+g𝑀) = (+g𝑀)
31, 2mgmcl 17014 . . . 4 ((𝑀 ∈ Mgm ∧ 𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀)) → (𝑥(+g𝑀)𝑦) ∈ (Base‘𝑀))
433expb 1257 . . 3 ((𝑀 ∈ Mgm ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀))) → (𝑥(+g𝑀)𝑦) ∈ (Base‘𝑀))
54ralrimivva 2953 . 2 (𝑀 ∈ Mgm → ∀𝑥 ∈ (Base‘𝑀)∀𝑦 ∈ (Base‘𝑀)(𝑥(+g𝑀)𝑦) ∈ (Base‘𝑀))
6 fvex 6098 . . . 4 (+g𝑀) ∈ V
7 fvex 6098 . . . 4 (Base‘𝑀) ∈ V
86, 7pm3.2i 469 . . 3 ((+g𝑀) ∈ V ∧ (Base‘𝑀) ∈ V)
9 iscllaw 41617 . . 3 (((+g𝑀) ∈ V ∧ (Base‘𝑀) ∈ V) → ((+g𝑀) clLaw (Base‘𝑀) ↔ ∀𝑥 ∈ (Base‘𝑀)∀𝑦 ∈ (Base‘𝑀)(𝑥(+g𝑀)𝑦) ∈ (Base‘𝑀)))
108, 9mp1i 13 . 2 (𝑀 ∈ Mgm → ((+g𝑀) clLaw (Base‘𝑀) ↔ ∀𝑥 ∈ (Base‘𝑀)∀𝑦 ∈ (Base‘𝑀)(𝑥(+g𝑀)𝑦) ∈ (Base‘𝑀)))
115, 10mpbird 245 1 (𝑀 ∈ Mgm → (+g𝑀) clLaw (Base‘𝑀))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 194  wa 382  wcel 1976  wral 2895  Vcvv 3172   class class class wbr 4577  cfv 5790  (class class class)co 6527  Basecbs 15641  +gcplusg 15714  Mgmcmgm 17009   clLaw ccllaw 41611
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2032  ax-13 2232  ax-ext 2589  ax-sep 4703  ax-nul 4712  ax-pr 4828
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ral 2900  df-rex 2901  df-rab 2904  df-v 3174  df-sbc 3402  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-nul 3874  df-if 4036  df-sn 4125  df-pr 4127  df-op 4131  df-uni 4367  df-br 4578  df-opab 4638  df-iota 5754  df-fv 5798  df-ov 6530  df-mgm 17011  df-cllaw 41614
This theorem is referenced by:  mgm2mgm  41655
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