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Theorem subgdisj1 15325
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 2438 . . . . . . 7  |-  ( -g `  G )  =  (
-g `  G )
54subgsubcl 14957 . . . . . 6  |-  ( ( T  e.  (SubGrp `  G )  /\  A  e.  T  /\  C  e.  T )  ->  ( A ( -g `  G
) C )  e.  T )
61, 2, 3, 5syl3anc 1185 . . . . 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 3351 . . . . . . . . . 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 15130 . . . . . . . . . 10  |-  ( ( C  e.  ( Z `
 U )  /\  B  e.  U )  ->  ( C  .+  B
)  =  ( B 
.+  C ) )
149, 10, 13syl2anc 644 . . . . . . . . 9  |-  ( ph  ->  ( C  .+  B
)  =  ( B 
.+  C ) )
157, 14oveq12d 6101 . . . . . . . 8  |-  ( ph  ->  ( ( A  .+  B ) ( -g `  G ) ( C 
.+  B ) )  =  ( ( C 
.+  D ) (
-g `  G )
( B  .+  C
) ) )
16 subgrcl 14951 . . . . . . . . . 10  |-  ( T  e.  (SubGrp `  G
)  ->  G  e.  Grp )
171, 16syl 16 . . . . . . . . 9  |-  ( ph  ->  G  e.  Grp )
18 eqid 2438 . . . . . . . . . . . . 13  |-  ( Base `  G )  =  (
Base `  G )
1918subgss 14947 . . . . . . . . . . . 12  |-  ( T  e.  (SubGrp `  G
)  ->  T  C_  ( Base `  G ) )
201, 19syl 16 . . . . . . . . . . 11  |-  ( ph  ->  T  C_  ( Base `  G ) )
2120, 2sseldd 3351 . . . . . . . . . 10  |-  ( ph  ->  A  e.  ( Base `  G ) )
22 subgdisj.u . . . . . . . . . . . 12  |-  ( ph  ->  U  e.  (SubGrp `  G ) )
2318subgss 14947 . . . . . . . . . . . 12  |-  ( U  e.  (SubGrp `  G
)  ->  U  C_  ( Base `  G ) )
2422, 23syl 16 . . . . . . . . . . 11  |-  ( ph  ->  U  C_  ( Base `  G ) )
2524, 10sseldd 3351 . . . . . . . . . 10  |-  ( ph  ->  B  e.  ( Base `  G ) )
2618, 11grpcl 14820 . . . . . . . . . 10  |-  ( ( G  e.  Grp  /\  A  e.  ( Base `  G )  /\  B  e.  ( Base `  G
) )  ->  ( A  .+  B )  e.  ( Base `  G
) )
2717, 21, 25, 26syl3anc 1185 . . . . . . . . 9  |-  ( ph  ->  ( A  .+  B
)  e.  ( Base `  G ) )
2820, 3sseldd 3351 . . . . . . . . 9  |-  ( ph  ->  C  e.  ( Base `  G ) )
2918, 11, 4grpsubsub4 14883 . . . . . . . . 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 1187 . . . . . . . 8  |-  ( ph  ->  ( ( ( A 
.+  B ) (
-g `  G ) B ) ( -g `  G ) C )  =  ( ( A 
.+  B ) (
-g `  G )
( C  .+  B
) ) )
317, 27eqeltrrd 2513 . . . . . . . . 9  |-  ( ph  ->  ( C  .+  D
)  e.  ( Base `  G ) )
3218, 11, 4grpsubsub4 14883 . . . . . . . . 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 1187 . . . . . . . 8  |-  ( ph  ->  ( ( ( C 
.+  D ) (
-g `  G ) C ) ( -g `  G ) B )  =  ( ( C 
.+  D ) (
-g `  G )
( B  .+  C
) ) )
3415, 30, 333eqtr4d 2480 . . . . . . 7  |-  ( ph  ->  ( ( ( A 
.+  B ) (
-g `  G ) B ) ( -g `  G ) C )  =  ( ( ( C  .+  D ) ( -g `  G
) C ) (
-g `  G ) B ) )
3518, 11, 4grppncan 14881 . . . . . . . . 9  |-  ( ( G  e.  Grp  /\  A  e.  ( Base `  G )  /\  B  e.  ( Base `  G
) )  ->  (
( A  .+  B
) ( -g `  G
) B )  =  A )
3617, 21, 25, 35syl3anc 1185 . . . . . . . 8  |-  ( ph  ->  ( ( A  .+  B ) ( -g `  G ) B )  =  A )
3736oveq1d 6098 . . . . . . 7  |-  ( ph  ->  ( ( ( A 
.+  B ) (
-g `  G ) B ) ( -g `  G ) C )  =  ( A (
-g `  G ) C ) )
38 subgdisj.d . . . . . . . . . . 11  |-  ( ph  ->  D  e.  U )
3911, 12cntzi 15130 . . . . . . . . . . 11  |-  ( ( C  e.  ( Z `
 U )  /\  D  e.  U )  ->  ( C  .+  D
)  =  ( D 
.+  C ) )
409, 38, 39syl2anc 644 . . . . . . . . . 10  |-  ( ph  ->  ( C  .+  D
)  =  ( D 
.+  C ) )
4140oveq1d 6098 . . . . . . . . 9  |-  ( ph  ->  ( ( C  .+  D ) ( -g `  G ) C )  =  ( ( D 
.+  C ) (
-g `  G ) C ) )
4224, 38sseldd 3351 . . . . . . . . . 10  |-  ( ph  ->  D  e.  ( Base `  G ) )
4318, 11, 4grppncan 14881 . . . . . . . . . 10  |-  ( ( G  e.  Grp  /\  D  e.  ( Base `  G )  /\  C  e.  ( Base `  G
) )  ->  (
( D  .+  C
) ( -g `  G
) C )  =  D )
4417, 42, 28, 43syl3anc 1185 . . . . . . . . 9  |-  ( ph  ->  ( ( D  .+  C ) ( -g `  G ) C )  =  D )
4541, 44eqtrd 2470 . . . . . . . 8  |-  ( ph  ->  ( ( C  .+  D ) ( -g `  G ) C )  =  D )
4645oveq1d 6098 . . . . . . 7  |-  ( ph  ->  ( ( ( C 
.+  D ) (
-g `  G ) C ) ( -g `  G ) B )  =  ( D (
-g `  G ) B ) )
4734, 37, 463eqtr3d 2478 . . . . . 6  |-  ( ph  ->  ( A ( -g `  G ) C )  =  ( D (
-g `  G ) B ) )
484subgsubcl 14957 . . . . . . 7  |-  ( ( U  e.  (SubGrp `  G )  /\  D  e.  U  /\  B  e.  U )  ->  ( D ( -g `  G
) B )  e.  U )
4922, 38, 10, 48syl3anc 1185 . . . . . 6  |-  ( ph  ->  ( D ( -g `  G ) B )  e.  U )
5047, 49eqeltrd 2512 . . . . 5  |-  ( ph  ->  ( A ( -g `  G ) C )  e.  U )
51 elin 3532 . . . . 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 647 . . . 4  |-  ( ph  ->  ( A ( -g `  G ) C )  e.  ( T  i^i  U ) )
53 subgdisj.i . . . 4  |-  ( ph  ->  ( T  i^i  U
)  =  {  .0.  } )
5452, 53eleqtrd 2514 . . 3  |-  ( ph  ->  ( A ( -g `  G ) C )  e.  {  .0.  }
)
55 elsni 3840 . . 3  |-  ( ( A ( -g `  G
) C )  e. 
{  .0.  }  ->  ( A ( -g `  G
) C )  =  .0.  )
5654, 55syl 16 . 2  |-  ( ph  ->  ( A ( -g `  G ) C )  =  .0.  )
57 subgdisj.o . . . 4  |-  .0.  =  ( 0g `  G )
5818, 57, 4grpsubeq0 14877 . . 3  |-  ( ( G  e.  Grp  /\  A  e.  ( Base `  G )  /\  C  e.  ( Base `  G
) )  ->  (
( A ( -g `  G ) C )  =  .0.  <->  A  =  C ) )
5917, 21, 28, 58syl3anc 1185 . 2  |-  ( ph  ->  ( ( A (
-g `  G ) C )  =  .0.  <->  A  =  C ) )
6056, 59mpbid 203 1  |-  ( ph  ->  A  =  C )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    = wceq 1653    e. wcel 1726    i^i cin 3321    C_ wss 3322   {csn 3816   ` cfv 5456  (class class class)co 6083   Basecbs 13471   +g cplusg 13531   0gc0g 13725   Grpcgrp 14687   -gcsg 14690  SubGrpcsubg 14940  Cntzccntz 15116
This theorem is referenced by:  subgdisj2  15326  subgdisjb  15327  lvecindp  16212
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4322  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703  ax-cnex 9048  ax-resscn 9049  ax-1cn 9050  ax-icn 9051  ax-addcl 9052  ax-addrcl 9053  ax-mulcl 9054  ax-mulrcl 9055  ax-mulcom 9056  ax-addass 9057  ax-mulass 9058  ax-distr 9059  ax-i2m1 9060  ax-1ne0 9061  ax-1rid 9062  ax-rnegex 9063  ax-rrecex 9064  ax-cnre 9065  ax-pre-lttri 9066  ax-pre-lttrn 9067  ax-pre-ltadd 9068  ax-pre-mulgt0 9069
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-iun 4097  df-br 4215  df-opab 4269  df-mpt 4270  df-tr 4305  df-eprel 4496  df-id 4500  df-po 4505  df-so 4506  df-fr 4543  df-we 4545  df-ord 4586  df-on 4587  df-lim 4588  df-suc 4589  df-om 4848  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-1st 6351  df-2nd 6352  df-riota 6551  df-recs 6635  df-rdg 6670  df-er 6907  df-en 7112  df-dom 7113  df-sdom 7114  df-pnf 9124  df-mnf 9125  df-xr 9126  df-ltxr 9127  df-le 9128  df-sub 9295  df-neg 9296  df-nn 10003  df-2 10060  df-ndx 13474  df-slot 13475  df-base 13476  df-sets 13477  df-ress 13478  df-plusg 13544  df-0g 13729  df-mnd 14692  df-grp 14814  df-minusg 14815  df-sbg 14816  df-subg 14943  df-cntz 15118
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