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Theorem ltexprlemupu 7977
Description: The upper cut of our constructed difference is upper. Lemma for ltexpri 7986. (Contributed by Jim Kingdon, 21-Dec-2019.)
Hypothesis
Ref Expression
ltexprlem.1 |- C = <. { x e. Q. | E. y ( y e. ( 2nd ` A ) /\ ( y +Q x ) e. ( 1st ` B ) ) } , { x e. Q. | E. y ( y e. ( 1st ` A ) /\ ( y +Q x ) e. ( 2nd ` B ) ) } >.
Assertion
Ref Expression
ltexprlemupu |- ( ( A <P B /\ r e. Q. ) -> ( E. q e. Q. ( q <Q r /\ q e. ( 2nd ` C ) ) -> r e. ( 2nd ` C ) ) )
Distinct variable groups:   x, y, q, r, A   x, B, y, q, r   x, C, y, q, r
Proof of Theorem ltexprlemupu
StepHypRef Expression
1 simplr 533 . . . . . 6 |- ( ( ( A <P B /\ r e. Q. ) /\ ( q <Q r /\ ( q e. Q. /\ ( y e. ( 1st ` A ) /\ ( y +Q q ) e. ( 2nd ` B ) ) ) ) ) -> r e. Q. )
2 simprrr 546 . . . . . . 7 |- ( ( ( A <P B /\ r e. Q. ) /\ ( q <Q r /\ ( q e. Q. /\ ( y e. ( 1st ` A ) /\ ( y +Q q ) e. ( 2nd ` B ) ) ) ) ) -> ( y e. ( 1st ` A ) /\ ( y +Q q ) e. ( 2nd ` B ) ) )
32simpld 112 . . . . . 6 |- ( ( ( A <P B /\ r e. Q. ) /\ ( q <Q r /\ ( q e. Q. /\ ( y e. ( 1st ` A ) /\ ( y +Q q ) e. ( 2nd ` B ) ) ) ) ) -> y e. ( 1st ` A ) )
4 simprl 535 . . . . . . . 8 |- ( ( ( A <P B /\ r e. Q. ) /\ ( q <Q r /\ ( q e. Q. /\ ( y e. ( 1st ` A ) /\ ( y +Q q ) e. ( 2nd ` B ) ) ) ) ) -> q <Q r )
5 simpll 531 . . . . . . . . 9 |- ( ( ( A <P B /\ r e. Q. ) /\ ( q <Q r /\ ( q e. Q. /\ ( y e. ( 1st ` A ) /\ ( y +Q q ) e. ( 2nd ` B ) ) ) ) ) -> A <P B )
6 simprrl 545 . . . . . . . . . 10 |- ( ( q <Q r /\ ( q e. Q. /\ ( y e. ( 1st ` A ) /\ ( y +Q q ) e. ( 2nd ` B ) ) ) ) -> y e. ( 1st ` A ) )
76adantl 277 . . . . . . . . 9 |- ( ( ( A <P B /\ r e. Q. ) /\ ( q <Q r /\ ( q e. Q. /\ ( y e. ( 1st ` A ) /\ ( y +Q q ) e. ( 2nd ` B ) ) ) ) ) -> y e. ( 1st ` A ) )
8 ltrelpr 7878 . . . . . . . . . . . . 13 |- <P C_ ( P. X. P. )
98brel 4829 . . . . . . . . . . . 12 |- ( A <P B -> ( A e. P. /\ B e. P. ) )
109simpld 112 . . . . . . . . . . 11 |- ( A <P B -> A e. P. )
11 prop 7848 . . . . . . . . . . 11 |- ( A e. P. -> <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. )
1210, 11syl 14 . . . . . . . . . 10 |- ( A <P B -> <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. )
13 elprnql 7854 . . . . . . . . . 10 |- ( ( <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. /\ y e. ( 1st ` A ) ) -> y e. Q. )
1412, 13sylan 283 . . . . . . . . 9 |- ( ( A <P B /\ y e. ( 1st ` A ) ) -> y e. Q. )
155, 7, 14syl2anc 415 . . . . . . . 8 |- ( ( ( A <P B /\ r e. Q. ) /\ ( q <Q r /\ ( q e. Q. /\ ( y e. ( 1st ` A ) /\ ( y +Q q ) e. ( 2nd ` B ) ) ) ) ) -> y e. Q. )
16 ltanqi 7775 . . . . . . . 8 |- ( ( q <Q r /\ y e. Q. ) -> ( y +Q q ) <Q ( y +Q r ) )
174, 15, 16syl2anc 415 . . . . . . 7 |- ( ( ( A <P B /\ r e. Q. ) /\ ( q <Q r /\ ( q e. Q. /\ ( y e. ( 1st ` A ) /\ ( y +Q q ) e. ( 2nd ` B ) ) ) ) ) -> ( y +Q q ) <Q ( y +Q r ) )
189simprd 114 . . . . . . . . 9 |- ( A <P B -> B e. P. )
195, 18syl 14 . . . . . . . 8 |- ( ( ( A <P B /\ r e. Q. ) /\ ( q <Q r /\ ( q e. Q. /\ ( y e. ( 1st ` A ) /\ ( y +Q q ) e. ( 2nd ` B ) ) ) ) ) -> B e. P. )
202simprd 114 . . . . . . . 8 |- ( ( ( A <P B /\ r e. Q. ) /\ ( q <Q r /\ ( q e. Q. /\ ( y e. ( 1st ` A ) /\ ( y +Q q ) e. ( 2nd ` B ) ) ) ) ) -> ( y +Q q ) e. ( 2nd ` B ) )
21 prop 7848 . . . . . . . . 9 |- ( B e. P. -> <. ( 1st ` B ) , ( 2nd ` B ) >. e. P. )
22 prcunqu 7858 . . . . . . . . 9 |- ( ( <. ( 1st ` B ) , ( 2nd ` B ) >. e. P. /\ ( y +Q q ) e. ( 2nd ` B ) ) -> ( ( y +Q q ) <Q ( y +Q r ) -> ( y +Q r ) e. ( 2nd ` B ) ) )
2321, 22sylan 283 . . . . . . . 8 |- ( ( B e. P. /\ ( y +Q q ) e. ( 2nd ` B ) ) -> ( ( y +Q q ) <Q ( y +Q r ) -> ( y +Q r ) e. ( 2nd ` B ) ) )
2419, 20, 23syl2anc 415 . . . . . . 7 |- ( ( ( A <P B /\ r e. Q. ) /\ ( q <Q r /\ ( q e. Q. /\ ( y e. ( 1st ` A ) /\ ( y +Q q ) e. ( 2nd ` B ) ) ) ) ) -> ( ( y +Q q ) <Q ( y +Q r ) -> ( y +Q r ) e. ( 2nd ` B ) ) )
2517, 24mpd 13 . . . . . 6 |- ( ( ( A <P B /\ r e. Q. ) /\ ( q <Q r /\ ( q e. Q. /\ ( y e. ( 1st ` A ) /\ ( y +Q q ) e. ( 2nd ` B ) ) ) ) ) -> ( y +Q r ) e. ( 2nd ` B ) )
261, 3, 25jca32 310 . . . . 5 |- ( ( ( A <P B /\ r e. Q. ) /\ ( q <Q r /\ ( q e. Q. /\ ( y e. ( 1st ` A ) /\ ( y +Q q ) e. ( 2nd ` B ) ) ) ) ) -> ( r e. Q. /\ ( y e. ( 1st ` A ) /\ ( y +Q r ) e. ( 2nd ` B ) ) ) )
2726eximi 1653 . . . 4 |- ( E. y ( ( A <P B /\ r e. Q. ) /\ ( q <Q r /\ ( q e. Q. /\ ( y e. ( 1st ` A ) /\ ( y +Q q ) e. ( 2nd ` B ) ) ) ) ) -> E. y ( r e. Q. /\ ( y e. ( 1st ` A ) /\ ( y +Q r ) e. ( 2nd ` B ) ) ) )
28 ltexprlem.1 . . . . . . . . . 10 |- C = <. { x e. Q. | E. y ( y e. ( 2nd ` A ) /\ ( y +Q x ) e. ( 1st ` B ) ) } , { x e. Q. | E. y ( y e. ( 1st ` A ) /\ ( y +Q x ) e. ( 2nd ` B ) ) } >.
2928ltexprlemelu 7972 . . . . . . . . 9 |- ( q e. ( 2nd ` C ) <-> ( q e. Q. /\ E. y ( y e. ( 1st ` A ) /\ ( y +Q q ) e. ( 2nd ` B ) ) ) )
30 19.42v 1962 . . . . . . . . 9 |- ( E. y ( q e. Q. /\ ( y e. ( 1st ` A ) /\ ( y +Q q ) e. ( 2nd ` B ) ) ) <-> ( q e. Q. /\ E. y ( y e. ( 1st ` A ) /\ ( y +Q q ) e. ( 2nd ` B ) ) ) )
3129, 30bitr4i 187 . . . . . . . 8 |- ( q e. ( 2nd ` C ) <-> E. y ( q e. Q. /\ ( y e. ( 1st ` A ) /\ ( y +Q q ) e. ( 2nd ` B ) ) ) )
3231anbi2i 461 . . . . . . 7 |- ( ( q <Q r /\ q e. ( 2nd ` C ) ) <-> ( q <Q r /\ E. y ( q e. Q. /\ ( y e. ( 1st ` A ) /\ ( y +Q q ) e. ( 2nd ` B ) ) ) ) )
33 19.42v 1962 . . . . . . 7 |- ( E. y ( q <Q r /\ ( q e. Q. /\ ( y e. ( 1st ` A ) /\ ( y +Q q ) e. ( 2nd ` B ) ) ) ) <-> ( q <Q r /\ E. y ( q e. Q. /\ ( y e. ( 1st ` A ) /\ ( y +Q q ) e. ( 2nd ` B ) ) ) ) )
3432, 33bitr4i 187 . . . . . 6 |- ( ( q <Q r /\ q e. ( 2nd ` C ) ) <-> E. y ( q <Q r /\ ( q e. Q. /\ ( y e. ( 1st ` A ) /\ ( y +Q q ) e. ( 2nd ` B ) ) ) ) )
3534anbi2i 461 . . . . 5 |- ( ( ( A <P B /\ r e. Q. ) /\ ( q <Q r /\ q e. ( 2nd ` C ) ) ) <-> ( ( A <P B /\ r e. Q. ) /\ E. y ( q <Q r /\ ( q e. Q. /\ ( y e. ( 1st ` A ) /\ ( y +Q q ) e. ( 2nd ` B ) ) ) ) ) )
36 19.42v 1962 . . . . 5 |- ( E. y ( ( A <P B /\ r e. Q. ) /\ ( q <Q r /\ ( q e. Q. /\ ( y e. ( 1st ` A ) /\ ( y +Q q ) e. ( 2nd ` B ) ) ) ) ) <-> ( ( A <P B /\ r e. Q. ) /\ E. y ( q <Q r /\ ( q e. Q. /\ ( y e. ( 1st ` A ) /\ ( y +Q q ) e. ( 2nd ` B ) ) ) ) ) )
3735, 36bitr4i 187 . . . 4 |- ( ( ( A <P B /\ r e. Q. ) /\ ( q <Q r /\ q e. ( 2nd ` C ) ) ) <-> E. y ( ( A <P B /\ r e. Q. ) /\ ( q <Q r /\ ( q e. Q. /\ ( y e. ( 1st ` A ) /\ ( y +Q q ) e. ( 2nd ` B ) ) ) ) ) )
3828ltexprlemelu 7972 . . . . 5 |- ( r e. ( 2nd ` C ) <-> ( r e. Q. /\ E. y ( y e. ( 1st ` A ) /\ ( y +Q r ) e. ( 2nd ` B ) ) ) )
39 19.42v 1962 . . . . 5 |- ( E. y ( r e. Q. /\ ( y e. ( 1st ` A ) /\ ( y +Q r ) e. ( 2nd ` B ) ) ) <-> ( r e. Q. /\ E. y ( y e. ( 1st ` A ) /\ ( y +Q r ) e. ( 2nd ` B ) ) ) )
4038, 39bitr4i 187 . . . 4 |- ( r e. ( 2nd ` C ) <-> E. y ( r e. Q. /\ ( y e. ( 1st ` A ) /\ ( y +Q r ) e. ( 2nd ` B ) ) ) )
4127, 37, 403imtr4i 201 . . 3 |- ( ( ( A <P B /\ r e. Q. ) /\ ( q <Q r /\ q e. ( 2nd ` C ) ) ) -> r e. ( 2nd ` C ) )
4241ex 115 . 2 |- ( ( A <P B /\ r e. Q. ) -> ( ( q <Q r /\ q e. ( 2nd ` C ) ) -> r e. ( 2nd ` C ) ) )
4342rexlimdvw 2672 1 |- ( ( A <P B /\ r e. Q. ) -> ( E. q e. Q. ( q <Q r /\ q e. ( 2nd ` C ) ) -> r e. ( 2nd ` C ) ) )
Colors of variables:    wff set class
This proof depends on syntax axioms:   -> wi 4   /\ wa 104   = wceq 1402   E.wex 1545   e. wcel 2209   E.wrex 2529   {crab 2532   <.cop 3712   class class class wbr 4131   ` cfv 5380  (class class class)co 6088   1stc1st 6376   2ndc2nd 6377   Q.cnq 7653   +Q cplq 7655   <Q cltq 7658   P.cnp 7664   <P cltp 7668
This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4248  ax-sep 4251  ax-nul 4261  ax-pow 4313  ax-pr 4348  ax-un 4580  ax-setind 4686  ax-iinf 4737
This proof depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4132  df-opab 4194  df-mpt 4195  df-tr 4232  df-eprel 4436  df-id 4440  df-iord 4513  df-on 4515  df-suc 4518  df-iom 4740  df-xp 4782  df-rel 4783  df-cnv 4784  df-co 4785  df-dm 4786  df-rn 4787  df-res 4788  df-ima 4789  df-iota 5340  df-fun 5382  df-fn 5383  df-f 5384  df-f1 5385  df-fo 5386  df-f1o 5387  df-fv 5388  df-ov 6091  df-oprab 6092  df-mpo 6093  df-1st 6378  df-2nd 6379  df-recs 6580  df-irdg 6645  df-oadd 6695  df-omul 6696  df-er 6811  df-ec 6813  df-qs 6817  df-ni 7677  df-pli 7678  df-mi 7679  df-lti 7680  df-plpq 7717  df-enq 7720  df-nqqs 7721  df-plqqs 7722  df-ltnqqs 7726  df-inp 7839  df-iltp 7843
This theorem is used by:  ltexprlemrnd  7978
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