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Theorem ltexpri 9371
Description: Proposition 9-3.5(iv) of [Gleason] p. 123. (Contributed by NM, 13-May-1996.) (Revised by Mario Carneiro, 14-Jun-2013.) (New usage is discouraged.)
Assertion
Ref Expression
ltexpri  |-  ( A 
<P  B  ->  E. x  e.  P.  ( A  +P.  x )  =  B )
Distinct variable groups:    x, A    x, B

Proof of Theorem ltexpri
Dummy variables  y 
z  w are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ltrelpr 9326 . . 3  |-  <P  C_  ( P.  X.  P. )
21brel 4991 . 2  |-  ( A 
<P  B  ->  ( A  e.  P.  /\  B  e.  P. ) )
3 ltprord 9358 . . 3  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( A  <P  B  <->  A  C.  B
) )
4 oveq2 6242 . . . . . . . . . . 11  |-  ( y  =  z  ->  (
w  +Q  y )  =  ( w  +Q  z ) )
54eleq1d 2471 . . . . . . . . . 10  |-  ( y  =  z  ->  (
( w  +Q  y
)  e.  B  <->  ( w  +Q  z )  e.  B
) )
65anbi2d 702 . . . . . . . . 9  |-  ( y  =  z  ->  (
( -.  w  e.  A  /\  ( w  +Q  y )  e.  B )  <->  ( -.  w  e.  A  /\  ( w  +Q  z
)  e.  B ) ) )
76exbidv 1735 . . . . . . . 8  |-  ( y  =  z  ->  ( E. w ( -.  w  e.  A  /\  (
w  +Q  y )  e.  B )  <->  E. w
( -.  w  e.  A  /\  ( w  +Q  z )  e.  B ) ) )
87cbvabv 2545 . . . . . . 7  |-  { y  |  E. w ( -.  w  e.  A  /\  ( w  +Q  y
)  e.  B ) }  =  { z  |  E. w ( -.  w  e.  A  /\  ( w  +Q  z
)  e.  B ) }
98ltexprlem5 9368 . . . . . 6  |-  ( ( B  e.  P.  /\  A  C.  B )  ->  { y  |  E. w ( -.  w  e.  A  /\  (
w  +Q  y )  e.  B ) }  e.  P. )
109adantll 712 . . . . 5  |-  ( ( ( A  e.  P.  /\  B  e.  P. )  /\  A  C.  B )  ->  { y  |  E. w ( -.  w  e.  A  /\  ( w  +Q  y
)  e.  B ) }  e.  P. )
118ltexprlem6 9369 . . . . . 6  |-  ( ( ( A  e.  P.  /\  B  e.  P. )  /\  A  C.  B )  ->  ( A  +P.  { y  |  E. w
( -.  w  e.  A  /\  ( w  +Q  y )  e.  B ) } ) 
C_  B )
128ltexprlem7 9370 . . . . . 6  |-  ( ( ( A  e.  P.  /\  B  e.  P. )  /\  A  C.  B )  ->  B  C_  ( A  +P.  { y  |  E. w ( -.  w  e.  A  /\  ( w  +Q  y
)  e.  B ) } ) )
1311, 12eqssd 3458 . . . . 5  |-  ( ( ( A  e.  P.  /\  B  e.  P. )  /\  A  C.  B )  ->  ( A  +P.  { y  |  E. w
( -.  w  e.  A  /\  ( w  +Q  y )  e.  B ) } )  =  B )
14 oveq2 6242 . . . . . . 7  |-  ( x  =  { y  |  E. w ( -.  w  e.  A  /\  ( w  +Q  y
)  e.  B ) }  ->  ( A  +P.  x )  =  ( A  +P.  { y  |  E. w ( -.  w  e.  A  /\  ( w  +Q  y
)  e.  B ) } ) )
1514eqeq1d 2404 . . . . . 6  |-  ( x  =  { y  |  E. w ( -.  w  e.  A  /\  ( w  +Q  y
)  e.  B ) }  ->  ( ( A  +P.  x )  =  B  <->  ( A  +P.  { y  |  E. w
( -.  w  e.  A  /\  ( w  +Q  y )  e.  B ) } )  =  B ) )
1615rspcev 3159 . . . . 5  |-  ( ( { y  |  E. w ( -.  w  e.  A  /\  (
w  +Q  y )  e.  B ) }  e.  P.  /\  ( A  +P.  { y  |  E. w ( -.  w  e.  A  /\  ( w  +Q  y
)  e.  B ) } )  =  B )  ->  E. x  e.  P.  ( A  +P.  x )  =  B )
1710, 13, 16syl2anc 659 . . . 4  |-  ( ( ( A  e.  P.  /\  B  e.  P. )  /\  A  C.  B )  ->  E. x  e.  P.  ( A  +P.  x )  =  B )
1817ex 432 . . 3  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( A  C.  B  ->  E. x  e.  P.  ( A  +P.  x )  =  B ) )
193, 18sylbid 215 . 2  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( A  <P  B  ->  E. x  e.  P.  ( A  +P.  x )  =  B ) )
202, 19mpcom 34 1  |-  ( A 
<P  B  ->  E. x  e.  P.  ( A  +P.  x )  =  B )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 367    = wceq 1405   E.wex 1633    e. wcel 1842   {cab 2387   E.wrex 2754    C. wpss 3414   class class class wbr 4394  (class class class)co 6234    +Q cplq 9183   P.cnp 9187    +P. cpp 9189    <P cltp 9191
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4516  ax-nul 4524  ax-pow 4571  ax-pr 4629  ax-un 6530  ax-inf2 8011
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2758  df-rex 2759  df-reu 2760  df-rmo 2761  df-rab 2762  df-v 3060  df-sbc 3277  df-csb 3373  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-pss 3429  df-nul 3738  df-if 3885  df-pw 3956  df-sn 3972  df-pr 3974  df-tp 3976  df-op 3978  df-uni 4191  df-int 4227  df-iun 4272  df-br 4395  df-opab 4453  df-mpt 4454  df-tr 4489  df-eprel 4733  df-id 4737  df-po 4743  df-so 4744  df-fr 4781  df-we 4783  df-ord 4824  df-on 4825  df-lim 4826  df-suc 4827  df-xp 4948  df-rel 4949  df-cnv 4950  df-co 4951  df-dm 4952  df-rn 4953  df-res 4954  df-ima 4955  df-iota 5489  df-fun 5527  df-fn 5528  df-f 5529  df-f1 5530  df-fo 5531  df-f1o 5532  df-fv 5533  df-ov 6237  df-oprab 6238  df-mpt2 6239  df-om 6639  df-1st 6738  df-2nd 6739  df-recs 6999  df-rdg 7033  df-1o 7087  df-oadd 7091  df-omul 7092  df-er 7268  df-ni 9200  df-pli 9201  df-mi 9202  df-lti 9203  df-plpq 9236  df-mpq 9237  df-ltpq 9238  df-enq 9239  df-nq 9240  df-erq 9241  df-plq 9242  df-mq 9243  df-1nq 9244  df-rq 9245  df-ltnq 9246  df-np 9309  df-plp 9311  df-ltp 9313
This theorem is referenced by:  ltaprlem  9372  recexsrlem  9430  mulgt0sr  9432  map2psrpr  9437
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