MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  subgdisj2 Structured version   Unicode version

Theorem subgdisj2 15316
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 3525 . . 3  |-  ( T  i^i  U )  =  ( U  i^i  T
)
7 subgdisj.i . . 3  |-  ( ph  ->  ( T  i^i  U
)  =  {  .0.  } )
86, 7syl5eqr 2481 . 2  |-  ( ph  ->  ( U  i^i  T
)  =  {  .0.  } )
9 subgdisj.s . . 3  |-  ( ph  ->  T  C_  ( Z `  U ) )
103, 5, 4, 9cntzrecd 15302 . 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 3341 . . . 4  |-  ( ph  ->  A  e.  ( Z `
 U ) )
171, 3cntzi 15120 . . . 4  |-  ( ( A  e.  ( Z `
 U )  /\  B  e.  U )  ->  ( A  .+  B
)  =  ( B 
.+  A ) )
1816, 11, 17syl2anc 643 . . 3  |-  ( ph  ->  ( A  .+  B
)  =  ( B 
.+  A ) )
199, 14sseldd 3341 . . . 4  |-  ( ph  ->  C  e.  ( Z `
 U ) )
201, 3cntzi 15120 . . . 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 2475 . 2  |-  ( ph  ->  ( B  .+  A
)  =  ( D 
.+  C ) )
231, 2, 3, 4, 5, 8, 10, 11, 12, 13, 14, 22subgdisj1 15315 1  |-  ( ph  ->  B  =  D )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1652    e. wcel 1725    i^i cin 3311    C_ wss 3312   {csn 3806   ` cfv 5446  (class class class)co 6073   +g cplusg 13521   0gc0g 13715  SubGrpcsubg 14930  Cntzccntz 15106
This theorem is referenced by:  subgdisjb  15317  lvecindp  16202  lshpsmreu  29844  lshpkrlem5  29849
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-cnex 9038  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-mulcom 9046  ax-addass 9047  ax-mulass 9048  ax-distr 9049  ax-i2m1 9050  ax-1ne0 9051  ax-1rid 9052  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057  ax-pre-ltadd 9058  ax-pre-mulgt0 9059
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-riota 6541  df-recs 6625  df-rdg 6660  df-er 6897  df-en 7102  df-dom 7103  df-sdom 7104  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118  df-sub 9285  df-neg 9286  df-nn 9993  df-2 10050  df-ndx 13464  df-slot 13465  df-base 13466  df-sets 13467  df-ress 13468  df-plusg 13534  df-0g 13719  df-mnd 14682  df-grp 14804  df-minusg 14805  df-sbg 14806  df-subg 14933  df-cntz 15108
  Copyright terms: Public domain W3C validator