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Theorem cmnpropd 15114
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 14414 . . 3  |-  ( ph  ->  ( K  e.  Mnd  <->  L  e.  Mnd ) )
53proplem 13608 . . . . . 6  |-  ( (
ph  /\  ( u  e.  B  /\  v  e.  B ) )  -> 
( u ( +g  `  K ) v )  =  ( u ( +g  `  L ) v ) )
63proplem 13608 . . . . . . 7  |-  ( (
ph  /\  ( v  e.  B  /\  u  e.  B ) )  -> 
( v ( +g  `  K ) u )  =  ( v ( +g  `  L ) u ) )
76ancom2s 777 . . . . . 6  |-  ( (
ph  /\  ( u  e.  B  /\  v  e.  B ) )  -> 
( v ( +g  `  K ) u )  =  ( v ( +g  `  L ) u ) )
85, 7eqeq12d 2310 . . . . 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 2596 . . . 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 2755 . . . . 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 2761 . . . 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 2755 . . . . 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 2761 . . . 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 274 . . 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 691 . 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 2296 . . 3  |-  ( Base `  K )  =  (
Base `  K )
17 eqid 2296 . . 3  |-  ( +g  `  K )  =  ( +g  `  K )
1816, 17iscmn 15112 . 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 2296 . . 3  |-  ( Base `  L )  =  (
Base `  L )
20 eqid 2296 . . 3  |-  ( +g  `  L )  =  ( +g  `  L )
2119, 20iscmn 15112 . 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 279 1  |-  ( ph  ->  ( K  e. CMnd  <->  L  e. CMnd ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1632    e. wcel 1696   A.wral 2556   ` cfv 5271  (class class class)co 5874   Basecbs 13164   +g cplusg 13224   Mndcmnd 14377  CMndccmn 15105
This theorem is referenced by:  ablpropd  15115  crngpropd  15389  prdscrngd  15412
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-nul 4165  ax-pow 4204
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-iota 5235  df-fv 5279  df-ov 5877  df-mnd 14383  df-cmn 15107
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