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Theorem subgdisj2 15252
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 3477 . . 3  |-  ( T  i^i  U )  =  ( U  i^i  T
)
7 subgdisj.i . . 3  |-  ( ph  ->  ( T  i^i  U
)  =  {  .0.  } )
86, 7syl5eqr 2434 . 2  |-  ( ph  ->  ( U  i^i  T
)  =  {  .0.  } )
9 subgdisj.s . . 3  |-  ( ph  ->  T  C_  ( Z `  U ) )
103, 5, 4, 9cntzrecd 15238 . 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 3293 . . . 4  |-  ( ph  ->  A  e.  ( Z `
 U ) )
171, 3cntzi 15056 . . . 4  |-  ( ( A  e.  ( Z `
 U )  /\  B  e.  U )  ->  ( A  .+  B
)  =  ( B 
.+  A ) )
1816, 11, 17syl2anc 643 . . 3  |-  ( ph  ->  ( A  .+  B
)  =  ( B 
.+  A ) )
199, 14sseldd 3293 . . . 4  |-  ( ph  ->  C  e.  ( Z `
 U ) )
201, 3cntzi 15056 . . . 4  |-  ( ( C  e.  ( Z `
 U )  /\  D  e.  U )  ->  ( C  .+  D
)  =  ( D 
.+  C ) )
2119, 12, 20syl2anc 643 . . 3  |-  ( ph  ->  ( C  .+  D
)  =  ( D 
.+  C ) )
2215, 18, 213eqtr3d 2428 . 2  |-  ( ph  ->  ( B  .+  A
)  =  ( D 
.+  C ) )
231, 2, 3, 4, 5, 8, 10, 11, 12, 13, 14, 22subgdisj1 15251 1  |-  ( ph  ->  B  =  D )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1649    e. wcel 1717    i^i cin 3263    C_ wss 3264   {csn 3758   ` cfv 5395  (class class class)co 6021   +g cplusg 13457   0gc0g 13651  SubGrpcsubg 14866  Cntzccntz 15042
This theorem is referenced by:  subgdisjb  15253  lvecindp  16138  lshpsmreu  29225  lshpkrlem5  29230
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369  ax-rep 4262  ax-sep 4272  ax-nul 4280  ax-pow 4319  ax-pr 4345  ax-un 4642  ax-cnex 8980  ax-resscn 8981  ax-1cn 8982  ax-icn 8983  ax-addcl 8984  ax-addrcl 8985  ax-mulcl 8986  ax-mulrcl 8987  ax-mulcom 8988  ax-addass 8989  ax-mulass 8990  ax-distr 8991  ax-i2m1 8992  ax-1ne0 8993  ax-1rid 8994  ax-rnegex 8995  ax-rrecex 8996  ax-cnre 8997  ax-pre-lttri 8998  ax-pre-lttrn 8999  ax-pre-ltadd 9000  ax-pre-mulgt0 9001
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-nel 2554  df-ral 2655  df-rex 2656  df-reu 2657  df-rmo 2658  df-rab 2659  df-v 2902  df-sbc 3106  df-csb 3196  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-pss 3280  df-nul 3573  df-if 3684  df-pw 3745  df-sn 3764  df-pr 3765  df-tp 3766  df-op 3767  df-uni 3959  df-iun 4038  df-br 4155  df-opab 4209  df-mpt 4210  df-tr 4245  df-eprel 4436  df-id 4440  df-po 4445  df-so 4446  df-fr 4483  df-we 4485  df-ord 4526  df-on 4527  df-lim 4528  df-suc 4529  df-om 4787  df-xp 4825  df-rel 4826  df-cnv 4827  df-co 4828  df-dm 4829  df-rn 4830  df-res 4831  df-ima 4832  df-iota 5359  df-fun 5397  df-fn 5398  df-f 5399  df-f1 5400  df-fo 5401  df-f1o 5402  df-fv 5403  df-ov 6024  df-oprab 6025  df-mpt2 6026  df-1st 6289  df-2nd 6290  df-riota 6486  df-recs 6570  df-rdg 6605  df-er 6842  df-en 7047  df-dom 7048  df-sdom 7049  df-pnf 9056  df-mnf 9057  df-xr 9058  df-ltxr 9059  df-le 9060  df-sub 9226  df-neg 9227  df-nn 9934  df-2 9991  df-ndx 13400  df-slot 13401  df-base 13402  df-sets 13403  df-ress 13404  df-plusg 13470  df-0g 13655  df-mnd 14618  df-grp 14740  df-minusg 14741  df-sbg 14742  df-subg 14869  df-cntz 15044
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