Users' Mathboxes Mathbox for Alexander van der Vekens < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  mgmplusgiopALT Structured version   Visualization version   GIF version

Theorem mgmplusgiopALT 42431
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 2765 . . . . 5 (Base‘𝑀) = (Base‘𝑀)
2 eqid 2765 . . . . 5 (+g𝑀) = (+g𝑀)
31, 2mgmcl 17513 . . . 4 ((𝑀 ∈ Mgm ∧ 𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀)) → (𝑥(+g𝑀)𝑦) ∈ (Base‘𝑀))
433expb 1149 . . 3 ((𝑀 ∈ Mgm ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀))) → (𝑥(+g𝑀)𝑦) ∈ (Base‘𝑀))
54ralrimivva 3118 . 2 (𝑀 ∈ Mgm → ∀𝑥 ∈ (Base‘𝑀)∀𝑦 ∈ (Base‘𝑀)(𝑥(+g𝑀)𝑦) ∈ (Base‘𝑀))
6 fvex 6388 . . . 4 (+g𝑀) ∈ V
7 fvex 6388 . . . 4 (Base‘𝑀) ∈ V
86, 7pm3.2i 462 . . 3 ((+g𝑀) ∈ V ∧ (Base‘𝑀) ∈ V)
9 iscllaw 42426 . . 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 248 1 (𝑀 ∈ Mgm → (+g𝑀) clLaw (Base‘𝑀))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wa 384  wcel 2155  wral 3055  Vcvv 3350   class class class wbr 4809  cfv 6068  (class class class)co 6842  Basecbs 16132  +gcplusg 16216  Mgmcmgm 17508   clLaw ccllaw 42420
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2069  ax-7 2105  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-sep 4941  ax-nul 4949  ax-pr 5062
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2062  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ral 3060  df-rex 3061  df-rab 3064  df-v 3352  df-sbc 3597  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-nul 4080  df-if 4244  df-sn 4335  df-pr 4337  df-op 4341  df-uni 4595  df-br 4810  df-opab 4872  df-iota 6031  df-fv 6076  df-ov 6845  df-mgm 17510  df-cllaw 42423
This theorem is referenced by:  mgm2mgm  42464
  Copyright terms: Public domain W3C validator