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Theorem pj1eu 15021
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 14976 . . . 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 642 . . 3  |-  ( ph  ->  ( X  e.  ( T  .(+)  U )  <->  E. x  e.  T  E. y  e.  U  X  =  ( x  .+  y ) ) )
76biimpa 470 . 2  |-  ( (
ph  /\  X  e.  ( T  .(+)  U ) )  ->  E. x  e.  T  E. y  e.  U  X  =  ( x  .+  y ) )
8 reeanv 2720 . . . . 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 2314 . . . . . . 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 706 . . . . . . . . 9  |-  ( ( ( ph  /\  (
x  e.  T  /\  u  e.  T )
)  /\  ( y  e.  U  /\  v  e.  U ) )  ->  T  e.  (SubGrp `  G
) )
132ad2antrr 706 . . . . . . . . 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 706 . . . . . . . . 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 706 . . . . . . . . 9  |-  ( ( ( ph  /\  (
x  e.  T  /\  u  e.  T )
)  /\  ( y  e.  U  /\  v  e.  U ) )  ->  T  C_  ( Z `  U ) )
18 simplrl 736 . . . . . . . . 9  |-  ( ( ( ph  /\  (
x  e.  T  /\  u  e.  T )
)  /\  ( y  e.  U  /\  v  e.  U ) )  ->  x  e.  T )
19 simplrr 737 . . . . . . . . 9  |-  ( ( ( ph  /\  (
x  e.  T  /\  u  e.  T )
)  /\  ( y  e.  U  /\  v  e.  U ) )  ->  u  e.  T )
20 simprl 732 . . . . . . . . 9  |-  ( ( ( ph  /\  (
x  e.  T  /\  u  e.  T )
)  /\  ( y  e.  U  /\  v  e.  U ) )  -> 
y  e.  U )
21 simprr 733 . . . . . . . . 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 15018 . . . . . . . 8  |-  ( ( ( ph  /\  (
x  e.  T  /\  u  e.  T )
)  /\  ( y  e.  U  /\  v  e.  U ) )  -> 
( ( x  .+  y )  =  ( u  .+  v )  <-> 
( x  =  u  /\  y  =  v ) ) )
23 simpl 443 . . . . . . . 8  |-  ( ( x  =  u  /\  y  =  v )  ->  x  =  u )
2422, 23syl6bi 219 . . . . . . 7  |-  ( ( ( ph  /\  (
x  e.  T  /\  u  e.  T )
)  /\  ( y  e.  U  /\  v  e.  U ) )  -> 
( ( x  .+  y )  =  ( u  .+  v )  ->  x  =  u ) )
259, 24syl5 28 . . . . . 6  |-  ( ( ( ph  /\  (
x  e.  T  /\  u  e.  T )
)  /\  ( y  e.  U  /\  v  e.  U ) )  -> 
( ( X  =  ( x  .+  y
)  /\  X  =  ( u  .+  v ) )  ->  x  =  u ) )
2625rexlimdvva 2687 . . . . 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 209 . . . 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 2648 . . 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 451 . 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 5881 . . . . . 6  |-  ( x  =  u  ->  (
x  .+  y )  =  ( u  .+  y ) )
3130eqeq2d 2307 . . . . 5  |-  ( x  =  u  ->  ( X  =  ( x  .+  y )  <->  X  =  ( u  .+  y ) ) )
3231rexbidv 2577 . . . 4  |-  ( x  =  u  ->  ( E. y  e.  U  X  =  ( x  .+  y )  <->  E. y  e.  U  X  =  ( u  .+  y ) ) )
33 oveq2 5882 . . . . . 6  |-  ( y  =  v  ->  (
u  .+  y )  =  ( u  .+  v ) )
3433eqeq2d 2307 . . . . 5  |-  ( y  =  v  ->  ( X  =  ( u  .+  y )  <->  X  =  ( u  .+  v ) ) )
3534cbvrexv 2778 . . . 4  |-  ( E. y  e.  U  X  =  ( u  .+  y )  <->  E. v  e.  U  X  =  ( u  .+  v ) )
3632, 35syl6bb 252 . . 3  |-  ( x  =  u  ->  ( E. y  e.  U  X  =  ( x  .+  y )  <->  E. v  e.  U  X  =  ( u  .+  v ) ) )
3736reu4 2972 . 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 645 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 176    /\ wa 358    = wceq 1632    e. wcel 1696   A.wral 2556   E.wrex 2557   E!wreu 2558    i^i cin 3164    C_ wss 3165   {csn 3653   ` cfv 5271  (class class class)co 5874   +g cplusg 13224   0gc0g 13416  SubGrpcsubg 14631  Cntzccntz 14807   LSSumclsm 14961
This theorem is referenced by:  pj1f  15022  pj1id  15024
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-nn 9763  df-2 9820  df-ndx 13167  df-slot 13168  df-base 13169  df-sets 13170  df-ress 13171  df-plusg 13237  df-0g 13420  df-mnd 14383  df-grp 14505  df-minusg 14506  df-sbg 14507  df-subg 14634  df-cntz 14809  df-lsm 14963
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