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Theorem pj1eu 15005
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 14960 . . . 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 2707 . . . . 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 2301 . . . . . . 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 15002 . . . . . . . 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 2674 . . . . 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 2635 . . 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 5865 . . . . . 6  |-  ( x  =  u  ->  (
x  .+  y )  =  ( u  .+  y ) )
3130eqeq2d 2294 . . . . 5  |-  ( x  =  u  ->  ( X  =  ( x  .+  y )  <->  X  =  ( u  .+  y ) ) )
3231rexbidv 2564 . . . 4  |-  ( x  =  u  ->  ( E. y  e.  U  X  =  ( x  .+  y )  <->  E. y  e.  U  X  =  ( u  .+  y ) ) )
33 oveq2 5866 . . . . . 6  |-  ( y  =  v  ->  (
u  .+  y )  =  ( u  .+  v ) )
3433eqeq2d 2294 . . . . 5  |-  ( y  =  v  ->  ( X  =  ( u  .+  y )  <->  X  =  ( u  .+  v ) ) )
3534cbvrexv 2765 . . . 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 2959 . 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 1623    e. wcel 1684   A.wral 2543   E.wrex 2544   E!wreu 2545    i^i cin 3151    C_ wss 3152   {csn 3640   ` cfv 5255  (class class class)co 5858   +g cplusg 13208   0gc0g 13400  SubGrpcsubg 14615  Cntzccntz 14791   LSSumclsm 14945
This theorem is referenced by:  pj1f  15006  pj1id  15008
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814
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 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-nn 9747  df-2 9804  df-ndx 13151  df-slot 13152  df-base 13153  df-sets 13154  df-ress 13155  df-plusg 13221  df-0g 13404  df-mnd 14367  df-grp 14489  df-minusg 14490  df-sbg 14491  df-subg 14618  df-cntz 14793  df-lsm 14947
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