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Theorem subgdisj1 15016
Description: Vectors belonging to disjoint commuting subgroups are uniquely determined by their sum. (Contributed by NM, 2-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2016.)
Hypotheses
Ref Expression
subgdisj.p  |-  .+  =  ( +g  `  G )
subgdisj.o  |-  .0.  =  ( 0g `  G )
subgdisj.z  |-  Z  =  (Cntz `  G )
subgdisj.t  |-  ( ph  ->  T  e.  (SubGrp `  G ) )
subgdisj.u  |-  ( ph  ->  U  e.  (SubGrp `  G ) )
subgdisj.i  |-  ( ph  ->  ( T  i^i  U
)  =  {  .0.  } )
subgdisj.s  |-  ( ph  ->  T  C_  ( Z `  U ) )
subgdisj.a  |-  ( ph  ->  A  e.  T )
subgdisj.c  |-  ( ph  ->  C  e.  T )
subgdisj.b  |-  ( ph  ->  B  e.  U )
subgdisj.d  |-  ( ph  ->  D  e.  U )
subgdisj.j  |-  ( ph  ->  ( A  .+  B
)  =  ( C 
.+  D ) )
Assertion
Ref Expression
subgdisj1  |-  ( ph  ->  A  =  C )

Proof of Theorem subgdisj1
StepHypRef Expression
1 subgdisj.t . . . . . 6  |-  ( ph  ->  T  e.  (SubGrp `  G ) )
2 subgdisj.a . . . . . 6  |-  ( ph  ->  A  e.  T )
3 subgdisj.c . . . . . 6  |-  ( ph  ->  C  e.  T )
4 eqid 2296 . . . . . . 7  |-  ( -g `  G )  =  (
-g `  G )
54subgsubcl 14648 . . . . . 6  |-  ( ( T  e.  (SubGrp `  G )  /\  A  e.  T  /\  C  e.  T )  ->  ( A ( -g `  G
) C )  e.  T )
61, 2, 3, 5syl3anc 1182 . . . . 5  |-  ( ph  ->  ( A ( -g `  G ) C )  e.  T )
7 subgdisj.j . . . . . . . . 9  |-  ( ph  ->  ( A  .+  B
)  =  ( C 
.+  D ) )
8 subgdisj.s . . . . . . . . . . 11  |-  ( ph  ->  T  C_  ( Z `  U ) )
98, 3sseldd 3194 . . . . . . . . . 10  |-  ( ph  ->  C  e.  ( Z `
 U ) )
10 subgdisj.b . . . . . . . . . 10  |-  ( ph  ->  B  e.  U )
11 subgdisj.p . . . . . . . . . . 11  |-  .+  =  ( +g  `  G )
12 subgdisj.z . . . . . . . . . . 11  |-  Z  =  (Cntz `  G )
1311, 12cntzi 14821 . . . . . . . . . 10  |-  ( ( C  e.  ( Z `
 U )  /\  B  e.  U )  ->  ( C  .+  B
)  =  ( B 
.+  C ) )
149, 10, 13syl2anc 642 . . . . . . . . 9  |-  ( ph  ->  ( C  .+  B
)  =  ( B 
.+  C ) )
157, 14oveq12d 5892 . . . . . . . 8  |-  ( ph  ->  ( ( A  .+  B ) ( -g `  G ) ( C 
.+  B ) )  =  ( ( C 
.+  D ) (
-g `  G )
( B  .+  C
) ) )
16 subgrcl 14642 . . . . . . . . . 10  |-  ( T  e.  (SubGrp `  G
)  ->  G  e.  Grp )
171, 16syl 15 . . . . . . . . 9  |-  ( ph  ->  G  e.  Grp )
18 eqid 2296 . . . . . . . . . . . . 13  |-  ( Base `  G )  =  (
Base `  G )
1918subgss 14638 . . . . . . . . . . . 12  |-  ( T  e.  (SubGrp `  G
)  ->  T  C_  ( Base `  G ) )
201, 19syl 15 . . . . . . . . . . 11  |-  ( ph  ->  T  C_  ( Base `  G ) )
2120, 2sseldd 3194 . . . . . . . . . 10  |-  ( ph  ->  A  e.  ( Base `  G ) )
22 subgdisj.u . . . . . . . . . . . 12  |-  ( ph  ->  U  e.  (SubGrp `  G ) )
2318subgss 14638 . . . . . . . . . . . 12  |-  ( U  e.  (SubGrp `  G
)  ->  U  C_  ( Base `  G ) )
2422, 23syl 15 . . . . . . . . . . 11  |-  ( ph  ->  U  C_  ( Base `  G ) )
2524, 10sseldd 3194 . . . . . . . . . 10  |-  ( ph  ->  B  e.  ( Base `  G ) )
2618, 11grpcl 14511 . . . . . . . . . 10  |-  ( ( G  e.  Grp  /\  A  e.  ( Base `  G )  /\  B  e.  ( Base `  G
) )  ->  ( A  .+  B )  e.  ( Base `  G
) )
2717, 21, 25, 26syl3anc 1182 . . . . . . . . 9  |-  ( ph  ->  ( A  .+  B
)  e.  ( Base `  G ) )
2820, 3sseldd 3194 . . . . . . . . 9  |-  ( ph  ->  C  e.  ( Base `  G ) )
2918, 11, 4grpsubsub4 14574 . . . . . . . . 9  |-  ( ( G  e.  Grp  /\  ( ( A  .+  B )  e.  (
Base `  G )  /\  B  e.  ( Base `  G )  /\  C  e.  ( Base `  G ) ) )  ->  ( ( ( A  .+  B ) ( -g `  G
) B ) (
-g `  G ) C )  =  ( ( A  .+  B
) ( -g `  G
) ( C  .+  B ) ) )
3017, 27, 25, 28, 29syl13anc 1184 . . . . . . . 8  |-  ( ph  ->  ( ( ( A 
.+  B ) (
-g `  G ) B ) ( -g `  G ) C )  =  ( ( A 
.+  B ) (
-g `  G )
( C  .+  B
) ) )
317, 27eqeltrrd 2371 . . . . . . . . 9  |-  ( ph  ->  ( C  .+  D
)  e.  ( Base `  G ) )
3218, 11, 4grpsubsub4 14574 . . . . . . . . 9  |-  ( ( G  e.  Grp  /\  ( ( C  .+  D )  e.  (
Base `  G )  /\  C  e.  ( Base `  G )  /\  B  e.  ( Base `  G ) ) )  ->  ( ( ( C  .+  D ) ( -g `  G
) C ) (
-g `  G ) B )  =  ( ( C  .+  D
) ( -g `  G
) ( B  .+  C ) ) )
3317, 31, 28, 25, 32syl13anc 1184 . . . . . . . 8  |-  ( ph  ->  ( ( ( C 
.+  D ) (
-g `  G ) C ) ( -g `  G ) B )  =  ( ( C 
.+  D ) (
-g `  G )
( B  .+  C
) ) )
3415, 30, 333eqtr4d 2338 . . . . . . 7  |-  ( ph  ->  ( ( ( A 
.+  B ) (
-g `  G ) B ) ( -g `  G ) C )  =  ( ( ( C  .+  D ) ( -g `  G
) C ) (
-g `  G ) B ) )
3518, 11, 4grppncan 14572 . . . . . . . . 9  |-  ( ( G  e.  Grp  /\  A  e.  ( Base `  G )  /\  B  e.  ( Base `  G
) )  ->  (
( A  .+  B
) ( -g `  G
) B )  =  A )
3617, 21, 25, 35syl3anc 1182 . . . . . . . 8  |-  ( ph  ->  ( ( A  .+  B ) ( -g `  G ) B )  =  A )
3736oveq1d 5889 . . . . . . 7  |-  ( ph  ->  ( ( ( A 
.+  B ) (
-g `  G ) B ) ( -g `  G ) C )  =  ( A (
-g `  G ) C ) )
38 subgdisj.d . . . . . . . . . . 11  |-  ( ph  ->  D  e.  U )
3911, 12cntzi 14821 . . . . . . . . . . 11  |-  ( ( C  e.  ( Z `
 U )  /\  D  e.  U )  ->  ( C  .+  D
)  =  ( D 
.+  C ) )
409, 38, 39syl2anc 642 . . . . . . . . . 10  |-  ( ph  ->  ( C  .+  D
)  =  ( D 
.+  C ) )
4140oveq1d 5889 . . . . . . . . 9  |-  ( ph  ->  ( ( C  .+  D ) ( -g `  G ) C )  =  ( ( D 
.+  C ) (
-g `  G ) C ) )
4224, 38sseldd 3194 . . . . . . . . . 10  |-  ( ph  ->  D  e.  ( Base `  G ) )
4318, 11, 4grppncan 14572 . . . . . . . . . 10  |-  ( ( G  e.  Grp  /\  D  e.  ( Base `  G )  /\  C  e.  ( Base `  G
) )  ->  (
( D  .+  C
) ( -g `  G
) C )  =  D )
4417, 42, 28, 43syl3anc 1182 . . . . . . . . 9  |-  ( ph  ->  ( ( D  .+  C ) ( -g `  G ) C )  =  D )
4541, 44eqtrd 2328 . . . . . . . 8  |-  ( ph  ->  ( ( C  .+  D ) ( -g `  G ) C )  =  D )
4645oveq1d 5889 . . . . . . 7  |-  ( ph  ->  ( ( ( C 
.+  D ) (
-g `  G ) C ) ( -g `  G ) B )  =  ( D (
-g `  G ) B ) )
4734, 37, 463eqtr3d 2336 . . . . . 6  |-  ( ph  ->  ( A ( -g `  G ) C )  =  ( D (
-g `  G ) B ) )
484subgsubcl 14648 . . . . . . 7  |-  ( ( U  e.  (SubGrp `  G )  /\  D  e.  U  /\  B  e.  U )  ->  ( D ( -g `  G
) B )  e.  U )
4922, 38, 10, 48syl3anc 1182 . . . . . 6  |-  ( ph  ->  ( D ( -g `  G ) B )  e.  U )
5047, 49eqeltrd 2370 . . . . 5  |-  ( ph  ->  ( A ( -g `  G ) C )  e.  U )
51 elin 3371 . . . . 5  |-  ( ( A ( -g `  G
) C )  e.  ( T  i^i  U
)  <->  ( ( A ( -g `  G
) C )  e.  T  /\  ( A ( -g `  G
) C )  e.  U ) )
526, 50, 51sylanbrc 645 . . . 4  |-  ( ph  ->  ( A ( -g `  G ) C )  e.  ( T  i^i  U ) )
53 subgdisj.i . . . 4  |-  ( ph  ->  ( T  i^i  U
)  =  {  .0.  } )
5452, 53eleqtrd 2372 . . 3  |-  ( ph  ->  ( A ( -g `  G ) C )  e.  {  .0.  }
)
55 elsni 3677 . . 3  |-  ( ( A ( -g `  G
) C )  e. 
{  .0.  }  ->  ( A ( -g `  G
) C )  =  .0.  )
5654, 55syl 15 . 2  |-  ( ph  ->  ( A ( -g `  G ) C )  =  .0.  )
57 subgdisj.o . . . 4  |-  .0.  =  ( 0g `  G )
5818, 57, 4grpsubeq0 14568 . . 3  |-  ( ( G  e.  Grp  /\  A  e.  ( Base `  G )  /\  C  e.  ( Base `  G
) )  ->  (
( A ( -g `  G ) C )  =  .0.  <->  A  =  C ) )
5917, 21, 28, 58syl3anc 1182 . 2  |-  ( ph  ->  ( ( A (
-g `  G ) C )  =  .0.  <->  A  =  C ) )
6056, 59mpbid 201 1  |-  ( ph  ->  A  =  C )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    = wceq 1632    e. wcel 1696    i^i cin 3164    C_ wss 3165   {csn 3653   ` cfv 5271  (class class class)co 5874   Basecbs 13164   +g cplusg 13224   0gc0g 13416   Grpcgrp 14378   -gcsg 14381  SubGrpcsubg 14631  Cntzccntz 14807
This theorem is referenced by:  subgdisj2  15017  subgdisjb  15018  lvecindp  15907
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-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  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-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-nn 9763  df-2 9820  df-ndx 13167  df-slot 13168  df-base 13169  df-sets 13170  df-ress 13171  df-plusg 13237  df-0g 13420  df-mnd 14383  df-grp 14505  df-minusg 14506  df-sbg 14507  df-subg 14634  df-cntz 14809
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