MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  cmnpropd Structured version   Unicode version

Theorem cmnpropd 16598
Description: If two structures have the same group components (properties), one is a commutative monoid iff the other one is. (Contributed by Mario Carneiro, 6-Jan-2015.)
Hypotheses
Ref Expression
ablpropd.1  |-  ( ph  ->  B  =  ( Base `  K ) )
ablpropd.2  |-  ( ph  ->  B  =  ( Base `  L ) )
ablpropd.3  |-  ( (
ph  /\  ( x  e.  B  /\  y  e.  B ) )  -> 
( x ( +g  `  K ) y )  =  ( x ( +g  `  L ) y ) )
Assertion
Ref Expression
cmnpropd  |-  ( ph  ->  ( K  e. CMnd  <->  L  e. CMnd ) )
Distinct variable groups:    x, y, B    x, K, y    x, L, y    ph, x, y

Proof of Theorem cmnpropd
Dummy variables  v  u are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ablpropd.1 . . . 4  |-  ( ph  ->  B  =  ( Base `  K ) )
2 ablpropd.2 . . . 4  |-  ( ph  ->  B  =  ( Base `  L ) )
3 ablpropd.3 . . . 4  |-  ( (
ph  /\  ( x  e.  B  /\  y  e.  B ) )  -> 
( x ( +g  `  K ) y )  =  ( x ( +g  `  L ) y ) )
41, 2, 3mndpropd 15754 . . 3  |-  ( ph  ->  ( K  e.  Mnd  <->  L  e.  Mnd ) )
53proplem 14936 . . . . . 6  |-  ( (
ph  /\  ( u  e.  B  /\  v  e.  B ) )  -> 
( u ( +g  `  K ) v )  =  ( u ( +g  `  L ) v ) )
63proplem 14936 . . . . . . 7  |-  ( (
ph  /\  ( v  e.  B  /\  u  e.  B ) )  -> 
( v ( +g  `  K ) u )  =  ( v ( +g  `  L ) u ) )
76ancom2s 800 . . . . . 6  |-  ( (
ph  /\  ( u  e.  B  /\  v  e.  B ) )  -> 
( v ( +g  `  K ) u )  =  ( v ( +g  `  L ) u ) )
85, 7eqeq12d 2484 . . . . 5  |-  ( (
ph  /\  ( u  e.  B  /\  v  e.  B ) )  -> 
( ( u ( +g  `  K ) v )  =  ( v ( +g  `  K
) u )  <->  ( u
( +g  `  L ) v )  =  ( v ( +g  `  L
) u ) ) )
982ralbidva 2901 . . . 4  |-  ( ph  ->  ( A. u  e.  B  A. v  e.  B  ( u ( +g  `  K ) v )  =  ( v ( +g  `  K
) u )  <->  A. u  e.  B  A. v  e.  B  ( u
( +g  `  L ) v )  =  ( v ( +g  `  L
) u ) ) )
101raleqdv 3059 . . . . 5  |-  ( ph  ->  ( A. v  e.  B  ( u ( +g  `  K ) v )  =  ( v ( +g  `  K
) u )  <->  A. v  e.  ( Base `  K
) ( u ( +g  `  K ) v )  =  ( v ( +g  `  K
) u ) ) )
111, 10raleqbidv 3067 . . . 4  |-  ( ph  ->  ( A. u  e.  B  A. v  e.  B  ( u ( +g  `  K ) v )  =  ( v ( +g  `  K
) u )  <->  A. u  e.  ( Base `  K
) A. v  e.  ( Base `  K
) ( u ( +g  `  K ) v )  =  ( v ( +g  `  K
) u ) ) )
122raleqdv 3059 . . . . 5  |-  ( ph  ->  ( A. v  e.  B  ( u ( +g  `  L ) v )  =  ( v ( +g  `  L
) u )  <->  A. v  e.  ( Base `  L
) ( u ( +g  `  L ) v )  =  ( v ( +g  `  L
) u ) ) )
132, 12raleqbidv 3067 . . . 4  |-  ( ph  ->  ( A. u  e.  B  A. v  e.  B  ( u ( +g  `  L ) v )  =  ( v ( +g  `  L
) u )  <->  A. u  e.  ( Base `  L
) A. v  e.  ( Base `  L
) ( u ( +g  `  L ) v )  =  ( v ( +g  `  L
) u ) ) )
149, 11, 133bitr3d 283 . . 3  |-  ( ph  ->  ( A. u  e.  ( Base `  K
) A. v  e.  ( Base `  K
) ( u ( +g  `  K ) v )  =  ( v ( +g  `  K
) u )  <->  A. u  e.  ( Base `  L
) A. v  e.  ( Base `  L
) ( u ( +g  `  L ) v )  =  ( v ( +g  `  L
) u ) ) )
154, 14anbi12d 710 . 2  |-  ( ph  ->  ( ( K  e. 
Mnd  /\  A. u  e.  ( Base `  K
) A. v  e.  ( Base `  K
) ( u ( +g  `  K ) v )  =  ( v ( +g  `  K
) u ) )  <-> 
( L  e.  Mnd  /\ 
A. u  e.  (
Base `  L ) A. v  e.  ( Base `  L ) ( u ( +g  `  L
) v )  =  ( v ( +g  `  L ) u ) ) ) )
16 eqid 2462 . . 3  |-  ( Base `  K )  =  (
Base `  K )
17 eqid 2462 . . 3  |-  ( +g  `  K )  =  ( +g  `  K )
1816, 17iscmn 16596 . 2  |-  ( K  e. CMnd 
<->  ( K  e.  Mnd  /\ 
A. u  e.  (
Base `  K ) A. v  e.  ( Base `  K ) ( u ( +g  `  K
) v )  =  ( v ( +g  `  K ) u ) ) )
19 eqid 2462 . . 3  |-  ( Base `  L )  =  (
Base `  L )
20 eqid 2462 . . 3  |-  ( +g  `  L )  =  ( +g  `  L )
2119, 20iscmn 16596 . 2  |-  ( L  e. CMnd 
<->  ( L  e.  Mnd  /\ 
A. u  e.  (
Base `  L ) A. v  e.  ( Base `  L ) ( u ( +g  `  L
) v )  =  ( v ( +g  `  L ) u ) ) )
2215, 18, 213bitr4g 288 1  |-  ( ph  ->  ( K  e. CMnd  <->  L  e. CMnd ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1374    e. wcel 1762   A.wral 2809   ` cfv 5581  (class class class)co 6277   Basecbs 14481   +g cplusg 14546   Mndcmnd 15717  CMndccmn 16589
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1963  ax-ext 2440  ax-nul 4571  ax-pow 4620
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2274  df-mo 2275  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-ne 2659  df-ral 2814  df-rex 2815  df-rab 2818  df-v 3110  df-sbc 3327  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3781  df-if 3935  df-sn 4023  df-pr 4025  df-op 4029  df-uni 4241  df-br 4443  df-iota 5544  df-fv 5589  df-ov 6280  df-mnd 15723  df-cmn 16591
This theorem is referenced by:  ablpropd  16599  crngpropd  17013  prdscrngd  17041  resvcmn  27479
  Copyright terms: Public domain W3C validator