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Theorem pj1eu 16840
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 16795 . . . 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 3025 . . . . 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 2484 . . . . . . 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 761 . . . . . . . . 9  |-  ( ( ( ph  /\  (
x  e.  T  /\  u  e.  T )
)  /\  ( y  e.  U  /\  v  e.  U ) )  ->  x  e.  T )
19 simplrr 762 . . . . . . . . 9  |-  ( ( ( ph  /\  (
x  e.  T  /\  u  e.  T )
)  /\  ( y  e.  U  /\  v  e.  U ) )  ->  u  e.  T )
20 simprl 756 . . . . . . . . 9  |-  ( ( ( ph  /\  (
x  e.  T  /\  u  e.  T )
)  /\  ( y  e.  U  /\  v  e.  U ) )  -> 
y  e.  U )
21 simprr 757 . . . . . . . . 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 16837 . . . . . . . 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 2956 . . . . 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 2878 . . 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 6303 . . . . . 6  |-  ( x  =  u  ->  (
x  .+  y )  =  ( u  .+  y ) )
3130eqeq2d 2471 . . . . 5  |-  ( x  =  u  ->  ( X  =  ( x  .+  y )  <->  X  =  ( u  .+  y ) ) )
3231rexbidv 2968 . . . 4  |-  ( x  =  u  ->  ( E. y  e.  U  X  =  ( x  .+  y )  <->  E. y  e.  U  X  =  ( u  .+  y ) ) )
33 oveq2 6304 . . . . . 6  |-  ( y  =  v  ->  (
u  .+  y )  =  ( u  .+  v ) )
3433eqeq2d 2471 . . . . 5  |-  ( y  =  v  ->  ( X  =  ( u  .+  y )  <->  X  =  ( u  .+  v ) ) )
3534cbvrexv 3085 . . . 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 3293 . 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 1395    e. wcel 1819   A.wral 2807   E.wrex 2808   E!wreu 2809    i^i cin 3470    C_ wss 3471   {csn 4032   ` cfv 5594  (class class class)co 6296   +g cplusg 14711   0gc0g 14856  SubGrpcsubg 16321  Cntzccntz 16479   LSSumclsm 16780
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6700  df-1st 6799  df-2nd 6800  df-recs 7060  df-rdg 7094  df-er 7329  df-en 7536  df-dom 7537  df-sdom 7538  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-nn 10557  df-2 10615  df-ndx 14646  df-slot 14647  df-base 14648  df-sets 14649  df-ress 14650  df-plusg 14724  df-0g 14858  df-mgm 15998  df-sgrp 16037  df-mnd 16047  df-grp 16183  df-minusg 16184  df-sbg 16185  df-subg 16324  df-cntz 16481  df-lsm 16782
This theorem is referenced by:  pj1f  16841  pj1id  16843
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