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Theorem subgdisj1 16580
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 2467 . . . . . . 7  |-  ( -g `  G )  =  (
-g `  G )
54subgsubcl 16083 . . . . . 6  |-  ( ( T  e.  (SubGrp `  G )  /\  A  e.  T  /\  C  e.  T )  ->  ( A ( -g `  G
) C )  e.  T )
61, 2, 3, 5syl3anc 1228 . . . . 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 3510 . . . . . . . . . 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 16238 . . . . . . . . . 10  |-  ( ( C  e.  ( Z `
 U )  /\  B  e.  U )  ->  ( C  .+  B
)  =  ( B 
.+  C ) )
149, 10, 13syl2anc 661 . . . . . . . . 9  |-  ( ph  ->  ( C  .+  B
)  =  ( B 
.+  C ) )
157, 14oveq12d 6313 . . . . . . . 8  |-  ( ph  ->  ( ( A  .+  B ) ( -g `  G ) ( C 
.+  B ) )  =  ( ( C 
.+  D ) (
-g `  G )
( B  .+  C
) ) )
16 subgrcl 16077 . . . . . . . . . 10  |-  ( T  e.  (SubGrp `  G
)  ->  G  e.  Grp )
171, 16syl 16 . . . . . . . . 9  |-  ( ph  ->  G  e.  Grp )
18 eqid 2467 . . . . . . . . . . . . 13  |-  ( Base `  G )  =  (
Base `  G )
1918subgss 16073 . . . . . . . . . . . 12  |-  ( T  e.  (SubGrp `  G
)  ->  T  C_  ( Base `  G ) )
201, 19syl 16 . . . . . . . . . . 11  |-  ( ph  ->  T  C_  ( Base `  G ) )
2120, 2sseldd 3510 . . . . . . . . . 10  |-  ( ph  ->  A  e.  ( Base `  G ) )
22 subgdisj.u . . . . . . . . . . . 12  |-  ( ph  ->  U  e.  (SubGrp `  G ) )
2318subgss 16073 . . . . . . . . . . . 12  |-  ( U  e.  (SubGrp `  G
)  ->  U  C_  ( Base `  G ) )
2422, 23syl 16 . . . . . . . . . . 11  |-  ( ph  ->  U  C_  ( Base `  G ) )
2524, 10sseldd 3510 . . . . . . . . . 10  |-  ( ph  ->  B  e.  ( Base `  G ) )
2618, 11grpcl 15934 . . . . . . . . . 10  |-  ( ( G  e.  Grp  /\  A  e.  ( Base `  G )  /\  B  e.  ( Base `  G
) )  ->  ( A  .+  B )  e.  ( Base `  G
) )
2717, 21, 25, 26syl3anc 1228 . . . . . . . . 9  |-  ( ph  ->  ( A  .+  B
)  e.  ( Base `  G ) )
2820, 3sseldd 3510 . . . . . . . . 9  |-  ( ph  ->  C  e.  ( Base `  G ) )
2918, 11, 4grpsubsub4 16002 . . . . . . . . 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 1230 . . . . . . . 8  |-  ( ph  ->  ( ( ( A 
.+  B ) (
-g `  G ) B ) ( -g `  G ) C )  =  ( ( A 
.+  B ) (
-g `  G )
( C  .+  B
) ) )
317, 27eqeltrrd 2556 . . . . . . . . 9  |-  ( ph  ->  ( C  .+  D
)  e.  ( Base `  G ) )
3218, 11, 4grpsubsub4 16002 . . . . . . . . 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 1230 . . . . . . . 8  |-  ( ph  ->  ( ( ( C 
.+  D ) (
-g `  G ) C ) ( -g `  G ) B )  =  ( ( C 
.+  D ) (
-g `  G )
( B  .+  C
) ) )
3415, 30, 333eqtr4d 2518 . . . . . . 7  |-  ( ph  ->  ( ( ( A 
.+  B ) (
-g `  G ) B ) ( -g `  G ) C )  =  ( ( ( C  .+  D ) ( -g `  G
) C ) (
-g `  G ) B ) )
3518, 11, 4grppncan 16000 . . . . . . . . 9  |-  ( ( G  e.  Grp  /\  A  e.  ( Base `  G )  /\  B  e.  ( Base `  G
) )  ->  (
( A  .+  B
) ( -g `  G
) B )  =  A )
3617, 21, 25, 35syl3anc 1228 . . . . . . . 8  |-  ( ph  ->  ( ( A  .+  B ) ( -g `  G ) B )  =  A )
3736oveq1d 6310 . . . . . . 7  |-  ( ph  ->  ( ( ( A 
.+  B ) (
-g `  G ) B ) ( -g `  G ) C )  =  ( A (
-g `  G ) C ) )
38 subgdisj.d . . . . . . . . . . 11  |-  ( ph  ->  D  e.  U )
3911, 12cntzi 16238 . . . . . . . . . . 11  |-  ( ( C  e.  ( Z `
 U )  /\  D  e.  U )  ->  ( C  .+  D
)  =  ( D 
.+  C ) )
409, 38, 39syl2anc 661 . . . . . . . . . 10  |-  ( ph  ->  ( C  .+  D
)  =  ( D 
.+  C ) )
4140oveq1d 6310 . . . . . . . . 9  |-  ( ph  ->  ( ( C  .+  D ) ( -g `  G ) C )  =  ( ( D 
.+  C ) (
-g `  G ) C ) )
4224, 38sseldd 3510 . . . . . . . . . 10  |-  ( ph  ->  D  e.  ( Base `  G ) )
4318, 11, 4grppncan 16000 . . . . . . . . . 10  |-  ( ( G  e.  Grp  /\  D  e.  ( Base `  G )  /\  C  e.  ( Base `  G
) )  ->  (
( D  .+  C
) ( -g `  G
) C )  =  D )
4417, 42, 28, 43syl3anc 1228 . . . . . . . . 9  |-  ( ph  ->  ( ( D  .+  C ) ( -g `  G ) C )  =  D )
4541, 44eqtrd 2508 . . . . . . . 8  |-  ( ph  ->  ( ( C  .+  D ) ( -g `  G ) C )  =  D )
4645oveq1d 6310 . . . . . . 7  |-  ( ph  ->  ( ( ( C 
.+  D ) (
-g `  G ) C ) ( -g `  G ) B )  =  ( D (
-g `  G ) B ) )
4734, 37, 463eqtr3d 2516 . . . . . 6  |-  ( ph  ->  ( A ( -g `  G ) C )  =  ( D (
-g `  G ) B ) )
484subgsubcl 16083 . . . . . . 7  |-  ( ( U  e.  (SubGrp `  G )  /\  D  e.  U  /\  B  e.  U )  ->  ( D ( -g `  G
) B )  e.  U )
4922, 38, 10, 48syl3anc 1228 . . . . . 6  |-  ( ph  ->  ( D ( -g `  G ) B )  e.  U )
5047, 49eqeltrd 2555 . . . . 5  |-  ( ph  ->  ( A ( -g `  G ) C )  e.  U )
516, 50elind 3693 . . . 4  |-  ( ph  ->  ( A ( -g `  G ) C )  e.  ( T  i^i  U ) )
52 subgdisj.i . . . 4  |-  ( ph  ->  ( T  i^i  U
)  =  {  .0.  } )
5351, 52eleqtrd 2557 . . 3  |-  ( ph  ->  ( A ( -g `  G ) C )  e.  {  .0.  }
)
54 elsni 4058 . . 3  |-  ( ( A ( -g `  G
) C )  e. 
{  .0.  }  ->  ( A ( -g `  G
) C )  =  .0.  )
5553, 54syl 16 . 2  |-  ( ph  ->  ( A ( -g `  G ) C )  =  .0.  )
56 subgdisj.o . . . 4  |-  .0.  =  ( 0g `  G )
5718, 56, 4grpsubeq0 15995 . . 3  |-  ( ( G  e.  Grp  /\  A  e.  ( Base `  G )  /\  C  e.  ( Base `  G
) )  ->  (
( A ( -g `  G ) C )  =  .0.  <->  A  =  C ) )
5817, 21, 28, 57syl3anc 1228 . 2  |-  ( ph  ->  ( ( A (
-g `  G ) C )  =  .0.  <->  A  =  C ) )
5955, 58mpbid 210 1  |-  ( ph  ->  A  =  C )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    = wceq 1379    e. wcel 1767    i^i cin 3480    C_ wss 3481   {csn 4033   ` cfv 5594  (class class class)co 6295   Basecbs 14506   +g cplusg 14571   0gc0g 14711   Grpcgrp 15924   -gcsg 15926  SubGrpcsubg 16066  Cntzccntz 16224
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587  ax-cnex 9560  ax-resscn 9561  ax-1cn 9562  ax-icn 9563  ax-addcl 9564  ax-addrcl 9565  ax-mulcl 9566  ax-mulrcl 9567  ax-mulcom 9568  ax-addass 9569  ax-mulass 9570  ax-distr 9571  ax-i2m1 9572  ax-1ne0 9573  ax-1rid 9574  ax-rnegex 9575  ax-rrecex 9576  ax-cnre 9577  ax-pre-lttri 9578  ax-pre-lttrn 9579  ax-pre-ltadd 9580  ax-pre-mulgt0 9581
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2822  df-rex 2823  df-reu 2824  df-rmo 2825  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-we 4846  df-ord 4887  df-on 4888  df-lim 4889  df-suc 4890  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6256  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-om 6696  df-1st 6795  df-2nd 6796  df-recs 7054  df-rdg 7088  df-er 7323  df-en 7529  df-dom 7530  df-sdom 7531  df-pnf 9642  df-mnf 9643  df-xr 9644  df-ltxr 9645  df-le 9646  df-sub 9819  df-neg 9820  df-nn 10549  df-2 10606  df-ndx 14509  df-slot 14510  df-base 14511  df-sets 14512  df-ress 14513  df-plusg 14584  df-0g 14713  df-mgm 15745  df-sgrp 15784  df-mnd 15794  df-grp 15928  df-minusg 15929  df-sbg 15930  df-subg 16069  df-cntz 16226
This theorem is referenced by:  subgdisj2  16581  subgdisjb  16582  lvecindp  17653
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