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Theorem pj1eu 16198
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 16153 . . . 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 661 . . 3  |-  ( ph  ->  ( X  e.  ( T  .(+)  U )  <->  E. x  e.  T  E. y  e.  U  X  =  ( x  .+  y ) ) )
76biimpa 484 . 2  |-  ( (
ph  /\  X  e.  ( T  .(+)  U ) )  ->  E. x  e.  T  E. y  e.  U  X  =  ( x  .+  y ) )
8 reeanv 2893 . . . . 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 2461 . . . . . . 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 725 . . . . . . . . 9  |-  ( ( ( ph  /\  (
x  e.  T  /\  u  e.  T )
)  /\  ( y  e.  U  /\  v  e.  U ) )  ->  T  e.  (SubGrp `  G
) )
132ad2antrr 725 . . . . . . . . 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 725 . . . . . . . . 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 725 . . . . . . . . 9  |-  ( ( ( ph  /\  (
x  e.  T  /\  u  e.  T )
)  /\  ( y  e.  U  /\  v  e.  U ) )  ->  T  C_  ( Z `  U ) )
18 simplrl 759 . . . . . . . . 9  |-  ( ( ( ph  /\  (
x  e.  T  /\  u  e.  T )
)  /\  ( y  e.  U  /\  v  e.  U ) )  ->  x  e.  T )
19 simplrr 760 . . . . . . . . 9  |-  ( ( ( ph  /\  (
x  e.  T  /\  u  e.  T )
)  /\  ( y  e.  U  /\  v  e.  U ) )  ->  u  e.  T )
20 simprl 755 . . . . . . . . 9  |-  ( ( ( ph  /\  (
x  e.  T  /\  u  e.  T )
)  /\  ( y  e.  U  /\  v  e.  U ) )  -> 
y  e.  U )
21 simprr 756 . . . . . . . . 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 16195 . . . . . . . 8  |-  ( ( ( ph  /\  (
x  e.  T  /\  u  e.  T )
)  /\  ( y  e.  U  /\  v  e.  U ) )  -> 
( ( x  .+  y )  =  ( u  .+  v )  <-> 
( x  =  u  /\  y  =  v ) ) )
23 simpl 457 . . . . . . . 8  |-  ( ( x  =  u  /\  y  =  v )  ->  x  =  u )
2422, 23syl6bi 228 . . . . . . 7  |-  ( ( ( ph  /\  (
x  e.  T  /\  u  e.  T )
)  /\  ( y  e.  U  /\  v  e.  U ) )  -> 
( ( x  .+  y )  =  ( u  .+  v )  ->  x  =  u ) )
259, 24syl5 32 . . . . . 6  |-  ( ( ( ph  /\  (
x  e.  T  /\  u  e.  T )
)  /\  ( y  e.  U  /\  v  e.  U ) )  -> 
( ( X  =  ( x  .+  y
)  /\  X  =  ( u  .+  v ) )  ->  x  =  u ) )
2625rexlimdvva 2853 . . . . 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 218 . . . 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 2813 . . 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 465 . 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 6103 . . . . . 6  |-  ( x  =  u  ->  (
x  .+  y )  =  ( u  .+  y ) )
3130eqeq2d 2454 . . . . 5  |-  ( x  =  u  ->  ( X  =  ( x  .+  y )  <->  X  =  ( u  .+  y ) ) )
3231rexbidv 2741 . . . 4  |-  ( x  =  u  ->  ( E. y  e.  U  X  =  ( x  .+  y )  <->  E. y  e.  U  X  =  ( u  .+  y ) ) )
33 oveq2 6104 . . . . . 6  |-  ( y  =  v  ->  (
u  .+  y )  =  ( u  .+  v ) )
3433eqeq2d 2454 . . . . 5  |-  ( y  =  v  ->  ( X  =  ( u  .+  y )  <->  X  =  ( u  .+  v ) ) )
3534cbvrexv 2953 . . . 4  |-  ( E. y  e.  U  X  =  ( u  .+  y )  <->  E. v  e.  U  X  =  ( u  .+  v ) )
3632, 35syl6bb 261 . . 3  |-  ( x  =  u  ->  ( E. y  e.  U  X  =  ( x  .+  y )  <->  E. v  e.  U  X  =  ( u  .+  v ) ) )
3736reu4 3158 . 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 664 1  |-  ( (
ph  /\  X  e.  ( T  .(+)  U ) )  ->  E! x  e.  T  E. y  e.  U  X  =  ( x  .+  y ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1369    e. wcel 1756   A.wral 2720   E.wrex 2721   E!wreu 2722    i^i cin 3332    C_ wss 3333   {csn 3882   ` cfv 5423  (class class class)co 6096   +g cplusg 14243   0gc0g 14383  SubGrpcsubg 15680  Cntzccntz 15838   LSSumclsm 16138
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4408  ax-sep 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536  ax-un 6377  ax-cnex 9343  ax-resscn 9344  ax-1cn 9345  ax-icn 9346  ax-addcl 9347  ax-addrcl 9348  ax-mulcl 9349  ax-mulrcl 9350  ax-mulcom 9351  ax-addass 9352  ax-mulass 9353  ax-distr 9354  ax-i2m1 9355  ax-1ne0 9356  ax-1rid 9357  ax-rnegex 9358  ax-rrecex 9359  ax-cnre 9360  ax-pre-lttri 9361  ax-pre-lttrn 9362  ax-pre-ltadd 9363  ax-pre-mulgt0 9364
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-nel 2614  df-ral 2725  df-rex 2726  df-reu 2727  df-rmo 2728  df-rab 2729  df-v 2979  df-sbc 3192  df-csb 3294  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-pss 3349  df-nul 3643  df-if 3797  df-pw 3867  df-sn 3883  df-pr 3885  df-tp 3887  df-op 3889  df-uni 4097  df-iun 4178  df-br 4298  df-opab 4356  df-mpt 4357  df-tr 4391  df-eprel 4637  df-id 4641  df-po 4646  df-so 4647  df-fr 4684  df-we 4686  df-ord 4727  df-on 4728  df-lim 4729  df-suc 4730  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5386  df-fun 5425  df-fn 5426  df-f 5427  df-f1 5428  df-fo 5429  df-f1o 5430  df-fv 5431  df-riota 6057  df-ov 6099  df-oprab 6100  df-mpt2 6101  df-om 6482  df-1st 6582  df-2nd 6583  df-recs 6837  df-rdg 6871  df-er 7106  df-en 7316  df-dom 7317  df-sdom 7318  df-pnf 9425  df-mnf 9426  df-xr 9427  df-ltxr 9428  df-le 9429  df-sub 9602  df-neg 9603  df-nn 10328  df-2 10385  df-ndx 14182  df-slot 14183  df-base 14184  df-sets 14185  df-ress 14186  df-plusg 14256  df-0g 14385  df-mnd 15420  df-grp 15550  df-minusg 15551  df-sbg 15552  df-subg 15683  df-cntz 15840  df-lsm 16140
This theorem is referenced by:  pj1f  16199  pj1id  16201
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