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Theorem cmnpropd 12925
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 12733 . . 3  |-  ( ph  ->  ( K  e.  Mnd  <->  L  e.  Mnd ) )
53oveqrspc2v 5896 . . . . . 6  |-  ( (
ph  /\  ( u  e.  B  /\  v  e.  B ) )  -> 
( u ( +g  `  K ) v )  =  ( u ( +g  `  L ) v ) )
63oveqrspc2v 5896 . . . . . . 7  |-  ( (
ph  /\  ( v  e.  B  /\  u  e.  B ) )  -> 
( v ( +g  `  K ) u )  =  ( v ( +g  `  L ) u ) )
76ancom2s 566 . . . . . 6  |-  ( (
ph  /\  ( u  e.  B  /\  v  e.  B ) )  -> 
( v ( +g  `  K ) u )  =  ( v ( +g  `  L ) u ) )
85, 7eqeq12d 2192 . . . . 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 2499 . . . 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 2678 . . . . 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 2684 . . . 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 2678 . . . . 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 2684 . . . 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 218 . . 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 473 . 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 2177 . . 3  |-  ( Base `  K )  =  (
Base `  K )
17 eqid 2177 . . 3  |-  ( +g  `  K )  =  ( +g  `  K )
1816, 17iscmn 12923 . 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 2177 . . 3  |-  ( Base `  L )  =  (
Base `  L )
20 eqid 2177 . . 3  |-  ( +g  `  L )  =  ( +g  `  L )
2119, 20iscmn 12923 . 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 223 1  |-  ( ph  ->  ( K  e. CMnd  <->  L  e. CMnd ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    <-> wb 105    = wceq 1353    e. wcel 2148   A.wral 2455   ` cfv 5212  (class class class)co 5869   Basecbs 12445   +g cplusg 12518   Mndcmnd 12709  CMndccmn 12915
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4118  ax-pow 4171  ax-pr 4206  ax-un 4430  ax-cnex 7893  ax-resscn 7894  ax-1re 7896  ax-addrcl 7899
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2739  df-sbc 2963  df-un 3133  df-in 3135  df-ss 3142  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-int 3843  df-br 4001  df-opab 4062  df-mpt 4063  df-id 4290  df-xp 4629  df-rel 4630  df-cnv 4631  df-co 4632  df-dm 4633  df-rn 4634  df-res 4635  df-iota 5174  df-fun 5214  df-fn 5215  df-fv 5220  df-ov 5872  df-inn 8909  df-2 8967  df-ndx 12448  df-slot 12449  df-base 12451  df-plusg 12531  df-mgm 12667  df-sgrp 12700  df-mnd 12710  df-cmn 12917
This theorem is referenced by:  ablpropd  12926  crngpropd  13044
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