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Theorem pj1eu 15329
Description: Uniqueness of a left projection. (Contributed by Mario Carneiro, 15-Oct-2015.)
Hypotheses
Ref Expression
pj1eu.a  |-  .+  =  ( +g  `  G )
pj1eu.s  |-  .(+)  =  (
LSSum `  G )
pj1eu.o  |-  .0.  =  ( 0g `  G )
pj1eu.z  |-  Z  =  (Cntz `  G )
pj1eu.2  |-  ( ph  ->  T  e.  (SubGrp `  G ) )
pj1eu.3  |-  ( ph  ->  U  e.  (SubGrp `  G ) )
pj1eu.4  |-  ( ph  ->  ( T  i^i  U
)  =  {  .0.  } )
pj1eu.5  |-  ( ph  ->  T  C_  ( Z `  U ) )
Assertion
Ref Expression
pj1eu  |-  ( (
ph  /\  X  e.  ( T  .(+)  U ) )  ->  E! x  e.  T  E. y  e.  U  X  =  ( x  .+  y ) )
Distinct variable groups:    x, y,  .+    x,  .(+) , y    ph, x, y    x, G, y    x, T, y    x, U, y   
x, X, y
Allowed substitution hints:    .0. ( x, y)    Z( x, y)

Proof of Theorem pj1eu
Dummy variables  v  u are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 pj1eu.2 . . . 4  |-  ( ph  ->  T  e.  (SubGrp `  G ) )
2 pj1eu.3 . . . 4  |-  ( ph  ->  U  e.  (SubGrp `  G ) )
3 pj1eu.a . . . . 5  |-  .+  =  ( +g  `  G )
4 pj1eu.s . . . . 5  |-  .(+)  =  (
LSSum `  G )
53, 4lsmelval 15284 . . . 4  |-  ( ( T  e.  (SubGrp `  G )  /\  U  e.  (SubGrp `  G )
)  ->  ( X  e.  ( T  .(+)  U )  <->  E. x  e.  T  E. y  e.  U  X  =  ( x  .+  y ) ) )
61, 2, 5syl2anc 644 . . 3  |-  ( ph  ->  ( X  e.  ( T  .(+)  U )  <->  E. x  e.  T  E. y  e.  U  X  =  ( x  .+  y ) ) )
76biimpa 472 . 2  |-  ( (
ph  /\  X  e.  ( T  .(+)  U ) )  ->  E. x  e.  T  E. y  e.  U  X  =  ( x  .+  y ) )
8 reeanv 2876 . . . . 5  |-  ( E. y  e.  U  E. v  e.  U  ( X  =  ( x  .+  y )  /\  X  =  ( u  .+  v ) )  <->  ( E. y  e.  U  X  =  ( x  .+  y )  /\  E. v  e.  U  X  =  ( u  .+  v ) ) )
9 eqtr2 2455 . . . . . . 7  |-  ( ( X  =  ( x 
.+  y )  /\  X  =  ( u  .+  v ) )  -> 
( x  .+  y
)  =  ( u 
.+  v ) )
10 pj1eu.o . . . . . . . . 9  |-  .0.  =  ( 0g `  G )
11 pj1eu.z . . . . . . . . 9  |-  Z  =  (Cntz `  G )
121ad2antrr 708 . . . . . . . . 9  |-  ( ( ( ph  /\  (
x  e.  T  /\  u  e.  T )
)  /\  ( y  e.  U  /\  v  e.  U ) )  ->  T  e.  (SubGrp `  G
) )
132ad2antrr 708 . . . . . . . . 9  |-  ( ( ( ph  /\  (
x  e.  T  /\  u  e.  T )
)  /\  ( y  e.  U  /\  v  e.  U ) )  ->  U  e.  (SubGrp `  G
) )
14 pj1eu.4 . . . . . . . . . 10  |-  ( ph  ->  ( T  i^i  U
)  =  {  .0.  } )
1514ad2antrr 708 . . . . . . . . 9  |-  ( ( ( ph  /\  (
x  e.  T  /\  u  e.  T )
)  /\  ( y  e.  U  /\  v  e.  U ) )  -> 
( T  i^i  U
)  =  {  .0.  } )
16 pj1eu.5 . . . . . . . . . 10  |-  ( ph  ->  T  C_  ( Z `  U ) )
1716ad2antrr 708 . . . . . . . . 9  |-  ( ( ( ph  /\  (
x  e.  T  /\  u  e.  T )
)  /\  ( y  e.  U  /\  v  e.  U ) )  ->  T  C_  ( Z `  U ) )
18 simplrl 738 . . . . . . . . 9  |-  ( ( ( ph  /\  (
x  e.  T  /\  u  e.  T )
)  /\  ( y  e.  U  /\  v  e.  U ) )  ->  x  e.  T )
19 simplrr 739 . . . . . . . . 9  |-  ( ( ( ph  /\  (
x  e.  T  /\  u  e.  T )
)  /\  ( y  e.  U  /\  v  e.  U ) )  ->  u  e.  T )
20 simprl 734 . . . . . . . . 9  |-  ( ( ( ph  /\  (
x  e.  T  /\  u  e.  T )
)  /\  ( y  e.  U  /\  v  e.  U ) )  -> 
y  e.  U )
21 simprr 735 . . . . . . . . 9  |-  ( ( ( ph  /\  (
x  e.  T  /\  u  e.  T )
)  /\  ( y  e.  U  /\  v  e.  U ) )  -> 
v  e.  U )
223, 10, 11, 12, 13, 15, 17, 18, 19, 20, 21subgdisjb 15326 . . . . . . . 8  |-  ( ( ( ph  /\  (
x  e.  T  /\  u  e.  T )
)  /\  ( y  e.  U  /\  v  e.  U ) )  -> 
( ( x  .+  y )  =  ( u  .+  v )  <-> 
( x  =  u  /\  y  =  v ) ) )
23 simpl 445 . . . . . . . 8  |-  ( ( x  =  u  /\  y  =  v )  ->  x  =  u )
2422, 23syl6bi 221 . . . . . . 7  |-  ( ( ( ph  /\  (
x  e.  T  /\  u  e.  T )
)  /\  ( y  e.  U  /\  v  e.  U ) )  -> 
( ( x  .+  y )  =  ( u  .+  v )  ->  x  =  u ) )
259, 24syl5 31 . . . . . 6  |-  ( ( ( ph  /\  (
x  e.  T  /\  u  e.  T )
)  /\  ( y  e.  U  /\  v  e.  U ) )  -> 
( ( X  =  ( x  .+  y
)  /\  X  =  ( u  .+  v ) )  ->  x  =  u ) )
2625rexlimdvva 2838 . . . . 5  |-  ( (
ph  /\  ( x  e.  T  /\  u  e.  T ) )  -> 
( E. y  e.  U  E. v  e.  U  ( X  =  ( x  .+  y
)  /\  X  =  ( u  .+  v ) )  ->  x  =  u ) )
278, 26syl5bir 211 . . . 4  |-  ( (
ph  /\  ( x  e.  T  /\  u  e.  T ) )  -> 
( ( E. y  e.  U  X  =  ( x  .+  y )  /\  E. v  e.  U  X  =  ( u  .+  v ) )  ->  x  =  u ) )
2827ralrimivva 2799 . . 3  |-  ( ph  ->  A. x  e.  T  A. u  e.  T  ( ( E. y  e.  U  X  =  ( x  .+  y )  /\  E. v  e.  U  X  =  ( u  .+  v ) )  ->  x  =  u ) )
2928adantr 453 . 2  |-  ( (
ph  /\  X  e.  ( T  .(+)  U ) )  ->  A. x  e.  T  A. u  e.  T  ( ( E. y  e.  U  X  =  ( x  .+  y )  /\  E. v  e.  U  X  =  ( u  .+  v ) )  ->  x  =  u )
)
30 oveq1 6089 . . . . . 6  |-  ( x  =  u  ->  (
x  .+  y )  =  ( u  .+  y ) )
3130eqeq2d 2448 . . . . 5  |-  ( x  =  u  ->  ( X  =  ( x  .+  y )  <->  X  =  ( u  .+  y ) ) )
3231rexbidv 2727 . . . 4  |-  ( x  =  u  ->  ( E. y  e.  U  X  =  ( x  .+  y )  <->  E. y  e.  U  X  =  ( u  .+  y ) ) )
33 oveq2 6090 . . . . . 6  |-  ( y  =  v  ->  (
u  .+  y )  =  ( u  .+  v ) )
3433eqeq2d 2448 . . . . 5  |-  ( y  =  v  ->  ( X  =  ( u  .+  y )  <->  X  =  ( u  .+  v ) ) )
3534cbvrexv 2934 . . . 4  |-  ( E. y  e.  U  X  =  ( u  .+  y )  <->  E. v  e.  U  X  =  ( u  .+  v ) )
3632, 35syl6bb 254 . . 3  |-  ( x  =  u  ->  ( E. y  e.  U  X  =  ( x  .+  y )  <->  E. v  e.  U  X  =  ( u  .+  v ) ) )
3736reu4 3129 . 2  |-  ( E! x  e.  T  E. y  e.  U  X  =  ( x  .+  y )  <->  ( E. x  e.  T  E. y  e.  U  X  =  ( x  .+  y )  /\  A. x  e.  T  A. u  e.  T  (
( E. y  e.  U  X  =  ( x  .+  y )  /\  E. v  e.  U  X  =  ( u  .+  v ) )  ->  x  =  u ) ) )
387, 29, 37sylanbrc 647 1  |-  ( (
ph  /\  X  e.  ( T  .(+)  U ) )  ->  E! x  e.  T  E. y  e.  U  X  =  ( x  .+  y ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    /\ wa 360    = wceq 1653    e. wcel 1726   A.wral 2706   E.wrex 2707   E!wreu 2708    i^i cin 3320    C_ wss 3321   {csn 3815   ` cfv 5455  (class class class)co 6082   +g cplusg 13530   0gc0g 13724  SubGrpcsubg 14939  Cntzccntz 15115   LSSumclsm 15269
This theorem is referenced by:  pj1f  15330  pj1id  15332
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2418  ax-rep 4321  ax-sep 4331  ax-nul 4339  ax-pow 4378  ax-pr 4404  ax-un 4702  ax-cnex 9047  ax-resscn 9048  ax-1cn 9049  ax-icn 9050  ax-addcl 9051  ax-addrcl 9052  ax-mulcl 9053  ax-mulrcl 9054  ax-mulcom 9055  ax-addass 9056  ax-mulass 9057  ax-distr 9058  ax-i2m1 9059  ax-1ne0 9060  ax-1rid 9061  ax-rnegex 9062  ax-rrecex 9063  ax-cnre 9064  ax-pre-lttri 9065  ax-pre-lttrn 9066  ax-pre-ltadd 9067  ax-pre-mulgt0 9068
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2286  df-mo 2287  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-nel 2603  df-ral 2711  df-rex 2712  df-reu 2713  df-rmo 2714  df-rab 2715  df-v 2959  df-sbc 3163  df-csb 3253  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-pss 3337  df-nul 3630  df-if 3741  df-pw 3802  df-sn 3821  df-pr 3822  df-tp 3823  df-op 3824  df-uni 4017  df-iun 4096  df-br 4214  df-opab 4268  df-mpt 4269  df-tr 4304  df-eprel 4495  df-id 4499  df-po 4504  df-so 4505  df-fr 4542  df-we 4544  df-ord 4585  df-on 4586  df-lim 4587  df-suc 4588  df-om 4847  df-xp 4885  df-rel 4886  df-cnv 4887  df-co 4888  df-dm 4889  df-rn 4890  df-res 4891  df-ima 4892  df-iota 5419  df-fun 5457  df-fn 5458  df-f 5459  df-f1 5460  df-fo 5461  df-f1o 5462  df-fv 5463  df-ov 6085  df-oprab 6086  df-mpt2 6087  df-1st 6350  df-2nd 6351  df-riota 6550  df-recs 6634  df-rdg 6669  df-er 6906  df-en 7111  df-dom 7112  df-sdom 7113  df-pnf 9123  df-mnf 9124  df-xr 9125  df-ltxr 9126  df-le 9127  df-sub 9294  df-neg 9295  df-nn 10002  df-2 10059  df-ndx 13473  df-slot 13474  df-base 13475  df-sets 13476  df-ress 13477  df-plusg 13543  df-0g 13728  df-mnd 14691  df-grp 14813  df-minusg 14814  df-sbg 14815  df-subg 14942  df-cntz 15117  df-lsm 15271
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