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Theorem subgdisjb 16291
Description: Vectors belonging to disjoint subgroups are uniquely determined by their sum. Analogous to opth 4661, this theorem shows a way of representing a pair of vectors. (Contributed by NM, 5-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 )
Assertion
Ref Expression
subgdisjb  |-  ( ph  ->  ( ( A  .+  B )  =  ( C  .+  D )  <-> 
( A  =  C  /\  B  =  D ) ) )

Proof of Theorem subgdisjb
StepHypRef Expression
1 subgdisj.p . . . . 5  |-  .+  =  ( +g  `  G )
2 subgdisj.o . . . . 5  |-  .0.  =  ( 0g `  G )
3 subgdisj.z . . . . 5  |-  Z  =  (Cntz `  G )
4 subgdisj.t . . . . . 6  |-  ( ph  ->  T  e.  (SubGrp `  G ) )
54adantr 465 . . . . 5  |-  ( (
ph  /\  ( A  .+  B )  =  ( C  .+  D ) )  ->  T  e.  (SubGrp `  G ) )
6 subgdisj.u . . . . . 6  |-  ( ph  ->  U  e.  (SubGrp `  G ) )
76adantr 465 . . . . 5  |-  ( (
ph  /\  ( A  .+  B )  =  ( C  .+  D ) )  ->  U  e.  (SubGrp `  G ) )
8 subgdisj.i . . . . . 6  |-  ( ph  ->  ( T  i^i  U
)  =  {  .0.  } )
98adantr 465 . . . . 5  |-  ( (
ph  /\  ( A  .+  B )  =  ( C  .+  D ) )  ->  ( T  i^i  U )  =  {  .0.  } )
10 subgdisj.s . . . . . 6  |-  ( ph  ->  T  C_  ( Z `  U ) )
1110adantr 465 . . . . 5  |-  ( (
ph  /\  ( A  .+  B )  =  ( C  .+  D ) )  ->  T  C_  ( Z `  U )
)
12 subgdisj.a . . . . . 6  |-  ( ph  ->  A  e.  T )
1312adantr 465 . . . . 5  |-  ( (
ph  /\  ( A  .+  B )  =  ( C  .+  D ) )  ->  A  e.  T )
14 subgdisj.c . . . . . 6  |-  ( ph  ->  C  e.  T )
1514adantr 465 . . . . 5  |-  ( (
ph  /\  ( A  .+  B )  =  ( C  .+  D ) )  ->  C  e.  T )
16 subgdisj.b . . . . . 6  |-  ( ph  ->  B  e.  U )
1716adantr 465 . . . . 5  |-  ( (
ph  /\  ( A  .+  B )  =  ( C  .+  D ) )  ->  B  e.  U )
18 subgdisj.d . . . . . 6  |-  ( ph  ->  D  e.  U )
1918adantr 465 . . . . 5  |-  ( (
ph  /\  ( A  .+  B )  =  ( C  .+  D ) )  ->  D  e.  U )
20 simpr 461 . . . . 5  |-  ( (
ph  /\  ( A  .+  B )  =  ( C  .+  D ) )  ->  ( A  .+  B )  =  ( C  .+  D ) )
211, 2, 3, 5, 7, 9, 11, 13, 15, 17, 19, 20subgdisj1 16289 . . . 4  |-  ( (
ph  /\  ( A  .+  B )  =  ( C  .+  D ) )  ->  A  =  C )
221, 2, 3, 5, 7, 9, 11, 13, 15, 17, 19, 20subgdisj2 16290 . . . 4  |-  ( (
ph  /\  ( A  .+  B )  =  ( C  .+  D ) )  ->  B  =  D )
2321, 22jca 532 . . 3  |-  ( (
ph  /\  ( A  .+  B )  =  ( C  .+  D ) )  ->  ( A  =  C  /\  B  =  D ) )
2423ex 434 . 2  |-  ( ph  ->  ( ( A  .+  B )  =  ( C  .+  D )  ->  ( A  =  C  /\  B  =  D ) ) )
25 oveq12 6196 . 2  |-  ( ( A  =  C  /\  B  =  D )  ->  ( A  .+  B
)  =  ( C 
.+  D ) )
2624, 25impbid1 203 1  |-  ( ph  ->  ( ( A  .+  B )  =  ( C  .+  D )  <-> 
( A  =  C  /\  B  =  D ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1370    e. wcel 1758    i^i cin 3422    C_ wss 3423   {csn 3972   ` cfv 5513  (class class class)co 6187   +g cplusg 14337   0gc0g 14477  SubGrpcsubg 15774  Cntzccntz 15932
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4498  ax-sep 4508  ax-nul 4516  ax-pow 4565  ax-pr 4626  ax-un 6469  ax-cnex 9436  ax-resscn 9437  ax-1cn 9438  ax-icn 9439  ax-addcl 9440  ax-addrcl 9441  ax-mulcl 9442  ax-mulrcl 9443  ax-mulcom 9444  ax-addass 9445  ax-mulass 9446  ax-distr 9447  ax-i2m1 9448  ax-1ne0 9449  ax-1rid 9450  ax-rnegex 9451  ax-rrecex 9452  ax-cnre 9453  ax-pre-lttri 9454  ax-pre-lttrn 9455  ax-pre-ltadd 9456  ax-pre-mulgt0 9457
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2599  df-ne 2644  df-nel 2645  df-ral 2798  df-rex 2799  df-reu 2800  df-rmo 2801  df-rab 2802  df-v 3067  df-sbc 3282  df-csb 3384  df-dif 3426  df-un 3428  df-in 3430  df-ss 3437  df-pss 3439  df-nul 3733  df-if 3887  df-pw 3957  df-sn 3973  df-pr 3975  df-tp 3977  df-op 3979  df-uni 4187  df-iun 4268  df-br 4388  df-opab 4446  df-mpt 4447  df-tr 4481  df-eprel 4727  df-id 4731  df-po 4736  df-so 4737  df-fr 4774  df-we 4776  df-ord 4817  df-on 4818  df-lim 4819  df-suc 4820  df-xp 4941  df-rel 4942  df-cnv 4943  df-co 4944  df-dm 4945  df-rn 4946  df-res 4947  df-ima 4948  df-iota 5476  df-fun 5515  df-fn 5516  df-f 5517  df-f1 5518  df-fo 5519  df-f1o 5520  df-fv 5521  df-riota 6148  df-ov 6190  df-oprab 6191  df-mpt2 6192  df-om 6574  df-1st 6674  df-2nd 6675  df-recs 6929  df-rdg 6963  df-er 7198  df-en 7408  df-dom 7409  df-sdom 7410  df-pnf 9518  df-mnf 9519  df-xr 9520  df-ltxr 9521  df-le 9522  df-sub 9695  df-neg 9696  df-nn 10421  df-2 10478  df-ndx 14276  df-slot 14277  df-base 14278  df-sets 14279  df-ress 14280  df-plusg 14350  df-0g 14479  df-mnd 15514  df-grp 15644  df-minusg 15645  df-sbg 15646  df-subg 15777  df-cntz 15934
This theorem is referenced by:  pj1eu  16294  pj1eq  16298  lvecindp2  17323
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