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Theorem mgmplusgiopALT 44248
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 2821 . . . . 5 (Base‘𝑀) = (Base‘𝑀)
2 eqid 2821 . . . . 5 (+g𝑀) = (+g𝑀)
31, 2mgmcl 17833 . . . 4 ((𝑀 ∈ Mgm ∧ 𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀)) → (𝑥(+g𝑀)𝑦) ∈ (Base‘𝑀))
433expb 1117 . . 3 ((𝑀 ∈ Mgm ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀))) → (𝑥(+g𝑀)𝑦) ∈ (Base‘𝑀))
54ralrimivva 3179 . 2 (𝑀 ∈ Mgm → ∀𝑥 ∈ (Base‘𝑀)∀𝑦 ∈ (Base‘𝑀)(𝑥(+g𝑀)𝑦) ∈ (Base‘𝑀))
6 fvex 6656 . . . 4 (+g𝑀) ∈ V
7 fvex 6656 . . . 4 (Base‘𝑀) ∈ V
86, 7pm3.2i 474 . . 3 ((+g𝑀) ∈ V ∧ (Base‘𝑀) ∈ V)
9 iscllaw 44243 . . 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 260 1 (𝑀 ∈ Mgm → (+g𝑀) clLaw (Base‘𝑀))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  wcel 2115  wral 3126  Vcvv 3471   class class class wbr 5039  cfv 6328  (class class class)co 7130  Basecbs 16461  +gcplusg 16543  Mgmcmgm 17828   clLaw ccllaw 44237
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 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2178  ax-ext 2793  ax-sep 5176  ax-nul 5183  ax-pr 5303
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2623  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2892  df-nfc 2960  df-ral 3131  df-rex 3132  df-rab 3135  df-v 3473  df-sbc 3750  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4267  df-if 4441  df-sn 4541  df-pr 4543  df-op 4547  df-uni 4812  df-br 5040  df-opab 5102  df-iota 6287  df-fv 6336  df-ov 7133  df-mgm 17830  df-cllaw 44240
This theorem is referenced by:  mgm2mgm  44281
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