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Theorem subgdisj2 16583
Description: Vectors belonging to disjoint subgroups are uniquely determined by their sum. (Contributed by NM, 12-Jul-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
subgdisj2  |-  ( ph  ->  B  =  D )

Proof of Theorem subgdisj2
StepHypRef Expression
1 subgdisj.p . 2  |-  .+  =  ( +g  `  G )
2 subgdisj.o . 2  |-  .0.  =  ( 0g `  G )
3 subgdisj.z . 2  |-  Z  =  (Cntz `  G )
4 subgdisj.u . 2  |-  ( ph  ->  U  e.  (SubGrp `  G ) )
5 subgdisj.t . 2  |-  ( ph  ->  T  e.  (SubGrp `  G ) )
6 incom 3696 . . 3  |-  ( T  i^i  U )  =  ( U  i^i  T
)
7 subgdisj.i . . 3  |-  ( ph  ->  ( T  i^i  U
)  =  {  .0.  } )
86, 7syl5eqr 2522 . 2  |-  ( ph  ->  ( U  i^i  T
)  =  {  .0.  } )
9 subgdisj.s . . 3  |-  ( ph  ->  T  C_  ( Z `  U ) )
103, 5, 4, 9cntzrecd 16569 . 2  |-  ( ph  ->  U  C_  ( Z `  T ) )
11 subgdisj.b . 2  |-  ( ph  ->  B  e.  U )
12 subgdisj.d . 2  |-  ( ph  ->  D  e.  U )
13 subgdisj.a . 2  |-  ( ph  ->  A  e.  T )
14 subgdisj.c . 2  |-  ( ph  ->  C  e.  T )
15 subgdisj.j . . 3  |-  ( ph  ->  ( A  .+  B
)  =  ( C 
.+  D ) )
169, 13sseldd 3510 . . . 4  |-  ( ph  ->  A  e.  ( Z `
 U ) )
171, 3cntzi 16239 . . . 4  |-  ( ( A  e.  ( Z `
 U )  /\  B  e.  U )  ->  ( A  .+  B
)  =  ( B 
.+  A ) )
1816, 11, 17syl2anc 661 . . 3  |-  ( ph  ->  ( A  .+  B
)  =  ( B 
.+  A ) )
199, 14sseldd 3510 . . . 4  |-  ( ph  ->  C  e.  ( Z `
 U ) )
201, 3cntzi 16239 . . . 4  |-  ( ( C  e.  ( Z `
 U )  /\  D  e.  U )  ->  ( C  .+  D
)  =  ( D 
.+  C ) )
2119, 12, 20syl2anc 661 . . 3  |-  ( ph  ->  ( C  .+  D
)  =  ( D 
.+  C ) )
2215, 18, 213eqtr3d 2516 . 2  |-  ( ph  ->  ( B  .+  A
)  =  ( D 
.+  C ) )
231, 2, 3, 4, 5, 8, 10, 11, 12, 13, 14, 22subgdisj1 16582 1  |-  ( ph  ->  B  =  D )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1379    e. wcel 1767    i^i cin 3480    C_ wss 3481   {csn 4033   ` cfv 5594  (class class class)co 6295   +g cplusg 14572   0gc0g 14712  SubGrpcsubg 16067  Cntzccntz 16225
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 14510  df-slot 14511  df-base 14512  df-sets 14513  df-ress 14514  df-plusg 14585  df-0g 14714  df-mgm 15746  df-sgrp 15785  df-mnd 15795  df-grp 15929  df-minusg 15930  df-sbg 15931  df-subg 16070  df-cntz 16227
This theorem is referenced by:  subgdisjb  16584  lvecindp  17655  lshpsmreu  34307  lshpkrlem5  34312
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