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Theorem mgmplusgiopALT 42358
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 2771 . . . . 5 (Base‘𝑀) = (Base‘𝑀)
2 eqid 2771 . . . . 5 (+g𝑀) = (+g𝑀)
31, 2mgmcl 17453 . . . 4 ((𝑀 ∈ Mgm ∧ 𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀)) → (𝑥(+g𝑀)𝑦) ∈ (Base‘𝑀))
433expb 1113 . . 3 ((𝑀 ∈ Mgm ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀))) → (𝑥(+g𝑀)𝑦) ∈ (Base‘𝑀))
54ralrimivva 3120 . 2 (𝑀 ∈ Mgm → ∀𝑥 ∈ (Base‘𝑀)∀𝑦 ∈ (Base‘𝑀)(𝑥(+g𝑀)𝑦) ∈ (Base‘𝑀))
6 fvex 6342 . . . 4 (+g𝑀) ∈ V
7 fvex 6342 . . . 4 (Base‘𝑀) ∈ V
86, 7pm3.2i 447 . . 3 ((+g𝑀) ∈ V ∧ (Base‘𝑀) ∈ V)
9 iscllaw 42353 . . 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 247 1 (𝑀 ∈ Mgm → (+g𝑀) clLaw (Base‘𝑀))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 382  wcel 2145  wral 3061  Vcvv 3351   class class class wbr 4786  cfv 6031  (class class class)co 6793  Basecbs 16064  +gcplusg 16149  Mgmcmgm 17448   clLaw ccllaw 42347
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pr 5034
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 835  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-br 4787  df-opab 4847  df-iota 5994  df-fv 6039  df-ov 6796  df-mgm 17450  df-cllaw 42350
This theorem is referenced by:  mgm2mgm  42391
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