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Theorem ltexpri 8683
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 8638 . . 3  |-  <P  C_  ( P.  X.  P. )
21brel 4753 . 2  |-  ( A 
<P  B  ->  ( A  e.  P.  /\  B  e.  P. ) )
3 ltprord 8670 . . 3  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( A  <P  B  <->  A  C.  B ) )
4 oveq2 5882 . . . . . . . . . . 11  |-  ( y  =  z  ->  (
w  +Q  y )  =  ( w  +Q  z ) )
54eleq1d 2362 . . . . . . . . . 10  |-  ( y  =  z  ->  (
( w  +Q  y
)  e.  B  <->  ( w  +Q  z )  e.  B
) )
65anbi2d 684 . . . . . . . . 9  |-  ( y  =  z  ->  (
( -.  w  e.  A  /\  ( w  +Q  y )  e.  B )  <->  ( -.  w  e.  A  /\  ( w  +Q  z
)  e.  B ) ) )
76exbidv 1616 . . . . . . . 8  |-  ( y  =  z  ->  ( E. w ( -.  w  e.  A  /\  (
w  +Q  y )  e.  B )  <->  E. w
( -.  w  e.  A  /\  ( w  +Q  z )  e.  B ) ) )
87cbvabv 2415 . . . . . . 7  |-  { y  |  E. w ( -.  w  e.  A  /\  ( w  +Q  y
)  e.  B ) }  =  { z  |  E. w ( -.  w  e.  A  /\  ( w  +Q  z
)  e.  B ) }
98ltexprlem5 8680 . . . . . 6  |-  ( ( B  e.  P.  /\  A  C.  B )  ->  { y  |  E. w ( -.  w  e.  A  /\  (
w  +Q  y )  e.  B ) }  e.  P. )
109adantll 694 . . . . 5  |-  ( ( ( A  e.  P.  /\  B  e.  P. )  /\  A  C.  B )  ->  { y  |  E. w ( -.  w  e.  A  /\  ( w  +Q  y
)  e.  B ) }  e.  P. )
118ltexprlem6 8681 . . . . . 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 8682 . . . . . 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 3209 . . . . 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 5882 . . . . . . 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 2304 . . . . . 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 2897 . . . . 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 642 . . . 4  |-  ( ( ( A  e.  P.  /\  B  e.  P. )  /\  A  C.  B )  ->  E. x  e.  P.  ( A  +P.  x )  =  B )
1817ex 423 . . 3  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( A  C.  B  ->  E. x  e.  P.  ( A  +P.  x )  =  B ) )
193, 18sylbid 206 . 2  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( A  <P  B  ->  E. x  e.  P.  ( A  +P.  x )  =  B ) )
202, 19mpcom 32 1  |-  ( A 
<P  B  ->  E. x  e.  P.  ( A  +P.  x )  =  B )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 358   E.wex 1531    = wceq 1632    e. wcel 1696   {cab 2282   E.wrex 2557    C. wpss 3166   class class class wbr 4039  (class class class)co 5874    +Q cplq 8493   P.cnp 8497    +P. cpp 8499    <P cltp 8501
This theorem is referenced by:  ltaprlem  8684  recexsrlem  8741  mulgt0sr  8743  map2psrpr  8748
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-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-inf2 7358
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-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-int 3879  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-recs 6404  df-rdg 6439  df-1o 6495  df-oadd 6499  df-omul 6500  df-er 6676  df-ni 8512  df-pli 8513  df-mi 8514  df-lti 8515  df-plpq 8548  df-mpq 8549  df-ltpq 8550  df-enq 8551  df-nq 8552  df-erq 8553  df-plq 8554  df-mq 8555  df-1nq 8556  df-rq 8557  df-ltnq 8558  df-np 8621  df-plp 8623  df-ltp 8625
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