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Theorem cmnpropd 17129
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 16268 . . 3  |-  ( ph  ->  ( K  e.  Mnd  <->  L  e.  Mnd ) )
53oveqrspc2v 6300 . . . . . 6  |-  ( (
ph  /\  ( u  e.  B  /\  v  e.  B ) )  -> 
( u ( +g  `  K ) v )  =  ( u ( +g  `  L ) v ) )
63oveqrspc2v 6300 . . . . . . 7  |-  ( (
ph  /\  ( v  e.  B  /\  u  e.  B ) )  -> 
( v ( +g  `  K ) u )  =  ( v ( +g  `  L ) u ) )
76ancom2s 803 . . . . . 6  |-  ( (
ph  /\  ( u  e.  B  /\  v  e.  B ) )  -> 
( v ( +g  `  K ) u )  =  ( v ( +g  `  L ) u ) )
85, 7eqeq12d 2424 . . . . 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 2845 . . . 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 3009 . . . . 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 3017 . . . 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 3009 . . . . 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 3017 . . . 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 709 . 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 2402 . . 3  |-  ( Base `  K )  =  (
Base `  K )
17 eqid 2402 . . 3  |-  ( +g  `  K )  =  ( +g  `  K )
1816, 17iscmn 17127 . 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 2402 . . 3  |-  ( Base `  L )  =  (
Base `  L )
20 eqid 2402 . . 3  |-  ( +g  `  L )  =  ( +g  `  L )
2119, 20iscmn 17127 . 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 367    = wceq 1405    e. wcel 1842   A.wral 2753   ` cfv 5568  (class class class)co 6277   Basecbs 14839   +g cplusg 14907   Mndcmnd 16241  CMndccmn 17120
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-nul 4524  ax-pow 4571
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2758  df-rex 2759  df-rab 2762  df-v 3060  df-sbc 3277  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-nul 3738  df-if 3885  df-sn 3972  df-pr 3974  df-op 3978  df-uni 4191  df-br 4395  df-iota 5532  df-fv 5576  df-ov 6280  df-mgm 16194  df-sgrp 16233  df-mnd 16243  df-cmn 17122
This theorem is referenced by:  ablpropd  17130  crngpropd  17549  prdscrngd  17580  resvcmn  28267
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