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Theorem ltexpri 9327
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 9282 . . 3  |-  <P  C_  ( P.  X.  P. )
21brel 4998 . 2  |-  ( A 
<P  B  ->  ( A  e.  P.  /\  B  e.  P. ) )
3 ltprord 9314 . . 3  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( A  <P  B  <->  A  C.  B
) )
4 oveq2 6211 . . . . . . . . . . 11  |-  ( y  =  z  ->  (
w  +Q  y )  =  ( w  +Q  z ) )
54eleq1d 2523 . . . . . . . . . 10  |-  ( y  =  z  ->  (
( w  +Q  y
)  e.  B  <->  ( w  +Q  z )  e.  B
) )
65anbi2d 703 . . . . . . . . 9  |-  ( y  =  z  ->  (
( -.  w  e.  A  /\  ( w  +Q  y )  e.  B )  <->  ( -.  w  e.  A  /\  ( w  +Q  z
)  e.  B ) ) )
76exbidv 1681 . . . . . . . 8  |-  ( y  =  z  ->  ( E. w ( -.  w  e.  A  /\  (
w  +Q  y )  e.  B )  <->  E. w
( -.  w  e.  A  /\  ( w  +Q  z )  e.  B ) ) )
87cbvabv 2597 . . . . . . 7  |-  { y  |  E. w ( -.  w  e.  A  /\  ( w  +Q  y
)  e.  B ) }  =  { z  |  E. w ( -.  w  e.  A  /\  ( w  +Q  z
)  e.  B ) }
98ltexprlem5 9324 . . . . . 6  |-  ( ( B  e.  P.  /\  A  C.  B )  ->  { y  |  E. w ( -.  w  e.  A  /\  (
w  +Q  y )  e.  B ) }  e.  P. )
109adantll 713 . . . . 5  |-  ( ( ( A  e.  P.  /\  B  e.  P. )  /\  A  C.  B )  ->  { y  |  E. w ( -.  w  e.  A  /\  ( w  +Q  y
)  e.  B ) }  e.  P. )
118ltexprlem6 9325 . . . . . 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 9326 . . . . . 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 3484 . . . . 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 6211 . . . . . . 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 2456 . . . . . 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 3179 . . . . 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 661 . . . 4  |-  ( ( ( A  e.  P.  /\  B  e.  P. )  /\  A  C.  B )  ->  E. x  e.  P.  ( A  +P.  x )  =  B )
1817ex 434 . . 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 36 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 369    = wceq 1370   E.wex 1587    e. wcel 1758   {cab 2439   E.wrex 2800    C. wpss 3440   class class class wbr 4403  (class class class)co 6203    +Q cplq 9137   P.cnp 9141    +P. cpp 9143    <P cltp 9145
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485  ax-inf2 7962
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-reu 2806  df-rmo 2807  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-pss 3455  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-tp 3993  df-op 3995  df-uni 4203  df-int 4240  df-iun 4284  df-br 4404  df-opab 4462  df-mpt 4463  df-tr 4497  df-eprel 4743  df-id 4747  df-po 4752  df-so 4753  df-fr 4790  df-we 4792  df-ord 4833  df-on 4834  df-lim 4835  df-suc 4836  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-f1 5534  df-fo 5535  df-f1o 5536  df-fv 5537  df-ov 6206  df-oprab 6207  df-mpt2 6208  df-om 6590  df-1st 6690  df-2nd 6691  df-recs 6945  df-rdg 6979  df-1o 7033  df-oadd 7037  df-omul 7038  df-er 7214  df-ni 9156  df-pli 9157  df-mi 9158  df-lti 9159  df-plpq 9192  df-mpq 9193  df-ltpq 9194  df-enq 9195  df-nq 9196  df-erq 9197  df-plq 9198  df-mq 9199  df-1nq 9200  df-rq 9201  df-ltnq 9202  df-np 9265  df-plp 9267  df-ltp 9269
This theorem is referenced by:  ltaprlem  9328  recexsrlem  9385  mulgt0sr  9387  map2psrpr  9392
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