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Theorem subgdisj2 15001
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 3361 . . 3  |-  ( T  i^i  U )  =  ( U  i^i  T
)
7 subgdisj.i . . 3  |-  ( ph  ->  ( T  i^i  U
)  =  {  .0.  } )
86, 7syl5eqr 2329 . 2  |-  ( ph  ->  ( U  i^i  T
)  =  {  .0.  } )
9 subgdisj.s . . 3  |-  ( ph  ->  T  C_  ( Z `  U ) )
103, 5, 4, 9cntzrecd 14987 . 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 3181 . . . 4  |-  ( ph  ->  A  e.  ( Z `
 U ) )
171, 3cntzi 14805 . . . 4  |-  ( ( A  e.  ( Z `
 U )  /\  B  e.  U )  ->  ( A  .+  B
)  =  ( B 
.+  A ) )
1816, 11, 17syl2anc 642 . . 3  |-  ( ph  ->  ( A  .+  B
)  =  ( B 
.+  A ) )
199, 14sseldd 3181 . . . 4  |-  ( ph  ->  C  e.  ( Z `
 U ) )
201, 3cntzi 14805 . . . 4  |-  ( ( C  e.  ( Z `
 U )  /\  D  e.  U )  ->  ( C  .+  D
)  =  ( D 
.+  C ) )
2119, 12, 20syl2anc 642 . . 3  |-  ( ph  ->  ( C  .+  D
)  =  ( D 
.+  C ) )
2215, 18, 213eqtr3d 2323 . 2  |-  ( ph  ->  ( B  .+  A
)  =  ( D 
.+  C ) )
231, 2, 3, 4, 5, 8, 10, 11, 12, 13, 14, 22subgdisj1 15000 1  |-  ( ph  ->  B  =  D )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1623    e. wcel 1684    i^i cin 3151    C_ wss 3152   {csn 3640   ` cfv 5255  (class class class)co 5858   +g cplusg 13208   0gc0g 13400  SubGrpcsubg 14615  Cntzccntz 14791
This theorem is referenced by:  subgdisjb  15002  lvecindp  15891  lshpsmreu  29299  lshpkrlem5  29304
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-nn 9747  df-2 9804  df-ndx 13151  df-slot 13152  df-base 13153  df-sets 13154  df-ress 13155  df-plusg 13221  df-0g 13404  df-mnd 14367  df-grp 14489  df-minusg 14490  df-sbg 14491  df-subg 14618  df-cntz 14793
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