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 44029
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 2818 . . . . 5 (Base‘𝑀) = (Base‘𝑀)
2 eqid 2818 . . . . 5 (+g𝑀) = (+g𝑀)
31, 2mgmcl 17843 . . . 4 ((𝑀 ∈ Mgm ∧ 𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀)) → (𝑥(+g𝑀)𝑦) ∈ (Base‘𝑀))
433expb 1112 . . 3 ((𝑀 ∈ Mgm ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀))) → (𝑥(+g𝑀)𝑦) ∈ (Base‘𝑀))
54ralrimivva 3188 . 2 (𝑀 ∈ Mgm → ∀𝑥 ∈ (Base‘𝑀)∀𝑦 ∈ (Base‘𝑀)(𝑥(+g𝑀)𝑦) ∈ (Base‘𝑀))
6 fvex 6676 . . . 4 (+g𝑀) ∈ V
7 fvex 6676 . . . 4 (Base‘𝑀) ∈ V
86, 7pm3.2i 471 . . 3 ((+g𝑀) ∈ V ∧ (Base‘𝑀) ∈ V)
9 iscllaw 44024 . . 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 258 1 (𝑀 ∈ Mgm → (+g𝑀) clLaw (Base‘𝑀))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396  wcel 2105  wral 3135  Vcvv 3492   class class class wbr 5057  cfv 6348  (class class class)co 7145  Basecbs 16471  +gcplusg 16553  Mgmcmgm 17838   clLaw ccllaw 44018
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pr 5320
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-opab 5120  df-iota 6307  df-fv 6356  df-ov 7148  df-mgm 17840  df-cllaw 44021
This theorem is referenced by:  mgm2mgm  44062
  Copyright terms: Public domain W3C validator