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Theorem subgdisjb 16910
Description: Vectors belonging to disjoint subgroups are uniquely determined by their sum. Analogous to opth 4711, 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 463 . . . . 5  |-  ( (
ph  /\  ( A  .+  B )  =  ( C  .+  D ) )  ->  T  e.  (SubGrp `  G ) )
6 subgdisj.u . . . . . 6  |-  ( ph  ->  U  e.  (SubGrp `  G ) )
76adantr 463 . . . . 5  |-  ( (
ph  /\  ( A  .+  B )  =  ( C  .+  D ) )  ->  U  e.  (SubGrp `  G ) )
8 subgdisj.i . . . . . 6  |-  ( ph  ->  ( T  i^i  U
)  =  {  .0.  } )
98adantr 463 . . . . 5  |-  ( (
ph  /\  ( A  .+  B )  =  ( C  .+  D ) )  ->  ( T  i^i  U )  =  {  .0.  } )
10 subgdisj.s . . . . . 6  |-  ( ph  ->  T  C_  ( Z `  U ) )
1110adantr 463 . . . . 5  |-  ( (
ph  /\  ( A  .+  B )  =  ( C  .+  D ) )  ->  T  C_  ( Z `  U )
)
12 subgdisj.a . . . . . 6  |-  ( ph  ->  A  e.  T )
1312adantr 463 . . . . 5  |-  ( (
ph  /\  ( A  .+  B )  =  ( C  .+  D ) )  ->  A  e.  T )
14 subgdisj.c . . . . . 6  |-  ( ph  ->  C  e.  T )
1514adantr 463 . . . . 5  |-  ( (
ph  /\  ( A  .+  B )  =  ( C  .+  D ) )  ->  C  e.  T )
16 subgdisj.b . . . . . 6  |-  ( ph  ->  B  e.  U )
1716adantr 463 . . . . 5  |-  ( (
ph  /\  ( A  .+  B )  =  ( C  .+  D ) )  ->  B  e.  U )
18 subgdisj.d . . . . . 6  |-  ( ph  ->  D  e.  U )
1918adantr 463 . . . . 5  |-  ( (
ph  /\  ( A  .+  B )  =  ( C  .+  D ) )  ->  D  e.  U )
20 simpr 459 . . . . 5  |-  ( (
ph  /\  ( A  .+  B )  =  ( C  .+  D ) )  ->  ( A  .+  B )  =  ( C  .+  D ) )
211, 2, 3, 5, 7, 9, 11, 13, 15, 17, 19, 20subgdisj1 16908 . . . 4  |-  ( (
ph  /\  ( A  .+  B )  =  ( C  .+  D ) )  ->  A  =  C )
221, 2, 3, 5, 7, 9, 11, 13, 15, 17, 19, 20subgdisj2 16909 . . . 4  |-  ( (
ph  /\  ( A  .+  B )  =  ( C  .+  D ) )  ->  B  =  D )
2321, 22jca 530 . . 3  |-  ( (
ph  /\  ( A  .+  B )  =  ( C  .+  D ) )  ->  ( A  =  C  /\  B  =  D ) )
2423ex 432 . 2  |-  ( ph  ->  ( ( A  .+  B )  =  ( C  .+  D )  ->  ( A  =  C  /\  B  =  D ) ) )
25 oveq12 6279 . 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 367    = wceq 1398    e. wcel 1823    i^i cin 3460    C_ wss 3461   {csn 4016   ` cfv 5570  (class class class)co 6270   +g cplusg 14784   0gc0g 14929  SubGrpcsubg 16394  Cntzccntz 16552
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-om 6674  df-1st 6773  df-2nd 6774  df-recs 7034  df-rdg 7068  df-er 7303  df-en 7510  df-dom 7511  df-sdom 7512  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9798  df-neg 9799  df-nn 10532  df-2 10590  df-ndx 14719  df-slot 14720  df-base 14721  df-sets 14722  df-ress 14723  df-plusg 14797  df-0g 14931  df-mgm 16071  df-sgrp 16110  df-mnd 16120  df-grp 16256  df-minusg 16257  df-sbg 16258  df-subg 16397  df-cntz 16554
This theorem is referenced by:  pj1eu  16913  pj1eq  16917  lvecindp2  17980
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