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Theorem subgdisj2 16210
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 3564 . . 3  |-  ( T  i^i  U )  =  ( U  i^i  T
)
7 subgdisj.i . . 3  |-  ( ph  ->  ( T  i^i  U
)  =  {  .0.  } )
86, 7syl5eqr 2489 . 2  |-  ( ph  ->  ( U  i^i  T
)  =  {  .0.  } )
9 subgdisj.s . . 3  |-  ( ph  ->  T  C_  ( Z `  U ) )
103, 5, 4, 9cntzrecd 16196 . 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 3378 . . . 4  |-  ( ph  ->  A  e.  ( Z `
 U ) )
171, 3cntzi 15868 . . . 4  |-  ( ( A  e.  ( Z `
 U )  /\  B  e.  U )  ->  ( A  .+  B
)  =  ( B 
.+  A ) )
1816, 11, 17syl2anc 661 . . 3  |-  ( ph  ->  ( A  .+  B
)  =  ( B 
.+  A ) )
199, 14sseldd 3378 . . . 4  |-  ( ph  ->  C  e.  ( Z `
 U ) )
201, 3cntzi 15868 . . . 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 2483 . 2  |-  ( ph  ->  ( B  .+  A
)  =  ( D 
.+  C ) )
231, 2, 3, 4, 5, 8, 10, 11, 12, 13, 14, 22subgdisj1 16209 1  |-  ( ph  ->  B  =  D )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1369    e. wcel 1756    i^i cin 3348    C_ wss 3349   {csn 3898   ` cfv 5439  (class class class)co 6112   +g cplusg 14259   0gc0g 14399  SubGrpcsubg 15696  Cntzccntz 15854
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 4424  ax-sep 4434  ax-nul 4442  ax-pow 4491  ax-pr 4552  ax-un 6393  ax-cnex 9359  ax-resscn 9360  ax-1cn 9361  ax-icn 9362  ax-addcl 9363  ax-addrcl 9364  ax-mulcl 9365  ax-mulrcl 9366  ax-mulcom 9367  ax-addass 9368  ax-mulass 9369  ax-distr 9370  ax-i2m1 9371  ax-1ne0 9372  ax-1rid 9373  ax-rnegex 9374  ax-rrecex 9375  ax-cnre 9376  ax-pre-lttri 9377  ax-pre-lttrn 9378  ax-pre-ltadd 9379  ax-pre-mulgt0 9380
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 2622  df-nel 2623  df-ral 2741  df-rex 2742  df-reu 2743  df-rmo 2744  df-rab 2745  df-v 2995  df-sbc 3208  df-csb 3310  df-dif 3352  df-un 3354  df-in 3356  df-ss 3363  df-pss 3365  df-nul 3659  df-if 3813  df-pw 3883  df-sn 3899  df-pr 3901  df-tp 3903  df-op 3905  df-uni 4113  df-iun 4194  df-br 4314  df-opab 4372  df-mpt 4373  df-tr 4407  df-eprel 4653  df-id 4657  df-po 4662  df-so 4663  df-fr 4700  df-we 4702  df-ord 4743  df-on 4744  df-lim 4745  df-suc 4746  df-xp 4867  df-rel 4868  df-cnv 4869  df-co 4870  df-dm 4871  df-rn 4872  df-res 4873  df-ima 4874  df-iota 5402  df-fun 5441  df-fn 5442  df-f 5443  df-f1 5444  df-fo 5445  df-f1o 5446  df-fv 5447  df-riota 6073  df-ov 6115  df-oprab 6116  df-mpt2 6117  df-om 6498  df-1st 6598  df-2nd 6599  df-recs 6853  df-rdg 6887  df-er 7122  df-en 7332  df-dom 7333  df-sdom 7334  df-pnf 9441  df-mnf 9442  df-xr 9443  df-ltxr 9444  df-le 9445  df-sub 9618  df-neg 9619  df-nn 10344  df-2 10401  df-ndx 14198  df-slot 14199  df-base 14200  df-sets 14201  df-ress 14202  df-plusg 14272  df-0g 14401  df-mnd 15436  df-grp 15566  df-minusg 15567  df-sbg 15568  df-subg 15699  df-cntz 15856
This theorem is referenced by:  subgdisjb  16211  lvecindp  17241  lshpsmreu  32850  lshpkrlem5  32855
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