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Theorem subgdisj1 16200
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 2443 . . . . . . 7  |-  ( -g `  G )  =  (
-g `  G )
54subgsubcl 15704 . . . . . 6  |-  ( ( T  e.  (SubGrp `  G )  /\  A  e.  T  /\  C  e.  T )  ->  ( A ( -g `  G
) C )  e.  T )
61, 2, 3, 5syl3anc 1218 . . . . 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 3369 . . . . . . . . . 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 15859 . . . . . . . . . 10  |-  ( ( C  e.  ( Z `
 U )  /\  B  e.  U )  ->  ( C  .+  B
)  =  ( B 
.+  C ) )
149, 10, 13syl2anc 661 . . . . . . . . 9  |-  ( ph  ->  ( C  .+  B
)  =  ( B 
.+  C ) )
157, 14oveq12d 6121 . . . . . . . 8  |-  ( ph  ->  ( ( A  .+  B ) ( -g `  G ) ( C 
.+  B ) )  =  ( ( C 
.+  D ) (
-g `  G )
( B  .+  C
) ) )
16 subgrcl 15698 . . . . . . . . . 10  |-  ( T  e.  (SubGrp `  G
)  ->  G  e.  Grp )
171, 16syl 16 . . . . . . . . 9  |-  ( ph  ->  G  e.  Grp )
18 eqid 2443 . . . . . . . . . . . . 13  |-  ( Base `  G )  =  (
Base `  G )
1918subgss 15694 . . . . . . . . . . . 12  |-  ( T  e.  (SubGrp `  G
)  ->  T  C_  ( Base `  G ) )
201, 19syl 16 . . . . . . . . . . 11  |-  ( ph  ->  T  C_  ( Base `  G ) )
2120, 2sseldd 3369 . . . . . . . . . 10  |-  ( ph  ->  A  e.  ( Base `  G ) )
22 subgdisj.u . . . . . . . . . . . 12  |-  ( ph  ->  U  e.  (SubGrp `  G ) )
2318subgss 15694 . . . . . . . . . . . 12  |-  ( U  e.  (SubGrp `  G
)  ->  U  C_  ( Base `  G ) )
2422, 23syl 16 . . . . . . . . . . 11  |-  ( ph  ->  U  C_  ( Base `  G ) )
2524, 10sseldd 3369 . . . . . . . . . 10  |-  ( ph  ->  B  e.  ( Base `  G ) )
2618, 11grpcl 15563 . . . . . . . . . 10  |-  ( ( G  e.  Grp  /\  A  e.  ( Base `  G )  /\  B  e.  ( Base `  G
) )  ->  ( A  .+  B )  e.  ( Base `  G
) )
2717, 21, 25, 26syl3anc 1218 . . . . . . . . 9  |-  ( ph  ->  ( A  .+  B
)  e.  ( Base `  G ) )
2820, 3sseldd 3369 . . . . . . . . 9  |-  ( ph  ->  C  e.  ( Base `  G ) )
2918, 11, 4grpsubsub4 15630 . . . . . . . . 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 1220 . . . . . . . 8  |-  ( ph  ->  ( ( ( A 
.+  B ) (
-g `  G ) B ) ( -g `  G ) C )  =  ( ( A 
.+  B ) (
-g `  G )
( C  .+  B
) ) )
317, 27eqeltrrd 2518 . . . . . . . . 9  |-  ( ph  ->  ( C  .+  D
)  e.  ( Base `  G ) )
3218, 11, 4grpsubsub4 15630 . . . . . . . . 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 1220 . . . . . . . 8  |-  ( ph  ->  ( ( ( C 
.+  D ) (
-g `  G ) C ) ( -g `  G ) B )  =  ( ( C 
.+  D ) (
-g `  G )
( B  .+  C
) ) )
3415, 30, 333eqtr4d 2485 . . . . . . 7  |-  ( ph  ->  ( ( ( A 
.+  B ) (
-g `  G ) B ) ( -g `  G ) C )  =  ( ( ( C  .+  D ) ( -g `  G
) C ) (
-g `  G ) B ) )
3518, 11, 4grppncan 15628 . . . . . . . . 9  |-  ( ( G  e.  Grp  /\  A  e.  ( Base `  G )  /\  B  e.  ( Base `  G
) )  ->  (
( A  .+  B
) ( -g `  G
) B )  =  A )
3617, 21, 25, 35syl3anc 1218 . . . . . . . 8  |-  ( ph  ->  ( ( A  .+  B ) ( -g `  G ) B )  =  A )
3736oveq1d 6118 . . . . . . 7  |-  ( ph  ->  ( ( ( A 
.+  B ) (
-g `  G ) B ) ( -g `  G ) C )  =  ( A (
-g `  G ) C ) )
38 subgdisj.d . . . . . . . . . . 11  |-  ( ph  ->  D  e.  U )
3911, 12cntzi 15859 . . . . . . . . . . 11  |-  ( ( C  e.  ( Z `
 U )  /\  D  e.  U )  ->  ( C  .+  D
)  =  ( D 
.+  C ) )
409, 38, 39syl2anc 661 . . . . . . . . . 10  |-  ( ph  ->  ( C  .+  D
)  =  ( D 
.+  C ) )
4140oveq1d 6118 . . . . . . . . 9  |-  ( ph  ->  ( ( C  .+  D ) ( -g `  G ) C )  =  ( ( D 
.+  C ) (
-g `  G ) C ) )
4224, 38sseldd 3369 . . . . . . . . . 10  |-  ( ph  ->  D  e.  ( Base `  G ) )
4318, 11, 4grppncan 15628 . . . . . . . . . 10  |-  ( ( G  e.  Grp  /\  D  e.  ( Base `  G )  /\  C  e.  ( Base `  G
) )  ->  (
( D  .+  C
) ( -g `  G
) C )  =  D )
4417, 42, 28, 43syl3anc 1218 . . . . . . . . 9  |-  ( ph  ->  ( ( D  .+  C ) ( -g `  G ) C )  =  D )
4541, 44eqtrd 2475 . . . . . . . 8  |-  ( ph  ->  ( ( C  .+  D ) ( -g `  G ) C )  =  D )
4645oveq1d 6118 . . . . . . 7  |-  ( ph  ->  ( ( ( C 
.+  D ) (
-g `  G ) C ) ( -g `  G ) B )  =  ( D (
-g `  G ) B ) )
4734, 37, 463eqtr3d 2483 . . . . . 6  |-  ( ph  ->  ( A ( -g `  G ) C )  =  ( D (
-g `  G ) B ) )
484subgsubcl 15704 . . . . . . 7  |-  ( ( U  e.  (SubGrp `  G )  /\  D  e.  U  /\  B  e.  U )  ->  ( D ( -g `  G
) B )  e.  U )
4922, 38, 10, 48syl3anc 1218 . . . . . 6  |-  ( ph  ->  ( D ( -g `  G ) B )  e.  U )
5047, 49eqeltrd 2517 . . . . 5  |-  ( ph  ->  ( A ( -g `  G ) C )  e.  U )
516, 50elind 3552 . . . 4  |-  ( ph  ->  ( A ( -g `  G ) C )  e.  ( T  i^i  U ) )
52 subgdisj.i . . . 4  |-  ( ph  ->  ( T  i^i  U
)  =  {  .0.  } )
5351, 52eleqtrd 2519 . . 3  |-  ( ph  ->  ( A ( -g `  G ) C )  e.  {  .0.  }
)
54 elsni 3914 . . 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 15624 . . 3  |-  ( ( G  e.  Grp  /\  A  e.  ( Base `  G )  /\  C  e.  ( Base `  G
) )  ->  (
( A ( -g `  G ) C )  =  .0.  <->  A  =  C ) )
5817, 21, 28, 57syl3anc 1218 . 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 1369    e. wcel 1756    i^i cin 3339    C_ wss 3340   {csn 3889   ` cfv 5430  (class class class)co 6103   Basecbs 14186   +g cplusg 14250   0gc0g 14390   Grpcgrp 15422   -gcsg 15425  SubGrpcsubg 15687  Cntzccntz 15845
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4415  ax-sep 4425  ax-nul 4433  ax-pow 4482  ax-pr 4543  ax-un 6384  ax-cnex 9350  ax-resscn 9351  ax-1cn 9352  ax-icn 9353  ax-addcl 9354  ax-addrcl 9355  ax-mulcl 9356  ax-mulrcl 9357  ax-mulcom 9358  ax-addass 9359  ax-mulass 9360  ax-distr 9361  ax-i2m1 9362  ax-1ne0 9363  ax-1rid 9364  ax-rnegex 9365  ax-rrecex 9366  ax-cnre 9367  ax-pre-lttri 9368  ax-pre-lttrn 9369  ax-pre-ltadd 9370  ax-pre-mulgt0 9371
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2620  df-nel 2621  df-ral 2732  df-rex 2733  df-reu 2734  df-rmo 2735  df-rab 2736  df-v 2986  df-sbc 3199  df-csb 3301  df-dif 3343  df-un 3345  df-in 3347  df-ss 3354  df-pss 3356  df-nul 3650  df-if 3804  df-pw 3874  df-sn 3890  df-pr 3892  df-tp 3894  df-op 3896  df-uni 4104  df-iun 4185  df-br 4305  df-opab 4363  df-mpt 4364  df-tr 4398  df-eprel 4644  df-id 4648  df-po 4653  df-so 4654  df-fr 4691  df-we 4693  df-ord 4734  df-on 4735  df-lim 4736  df-suc 4737  df-xp 4858  df-rel 4859  df-cnv 4860  df-co 4861  df-dm 4862  df-rn 4863  df-res 4864  df-ima 4865  df-iota 5393  df-fun 5432  df-fn 5433  df-f 5434  df-f1 5435  df-fo 5436  df-f1o 5437  df-fv 5438  df-riota 6064  df-ov 6106  df-oprab 6107  df-mpt2 6108  df-om 6489  df-1st 6589  df-2nd 6590  df-recs 6844  df-rdg 6878  df-er 7113  df-en 7323  df-dom 7324  df-sdom 7325  df-pnf 9432  df-mnf 9433  df-xr 9434  df-ltxr 9435  df-le 9436  df-sub 9609  df-neg 9610  df-nn 10335  df-2 10392  df-ndx 14189  df-slot 14190  df-base 14191  df-sets 14192  df-ress 14193  df-plusg 14263  df-0g 14392  df-mnd 15427  df-grp 15557  df-minusg 15558  df-sbg 15559  df-subg 15690  df-cntz 15847
This theorem is referenced by:  subgdisj2  16201  subgdisjb  16202  lvecindp  17231
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