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Theorem ltexprlemupu 7935
Description: The upper cut of our constructed difference is upper. Lemma for ltexpri 7944. (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 529 . . . . . 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 542 . . . . . . 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 531 . . . . . . . 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 527 . . . . . . . . 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 541 . . . . . . . . . 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 7836 . . . . . . . . . . . . 13 |- <P C_ ( P. X. P. )
98brel 4807 . . . . . . . . . . . 12 |- ( A <P B -> ( A e. P. /\ B e. P. ) )
109simpld 112 . . . . . . . . . . 11 |- ( A <P B -> A e. P. )
11 prop 7806 . . . . . . . . . . 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 7812 . . . . . . . . . 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 411 . . . . . . . 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 7733 . . . . . . . 8 |- ( ( q <Q r /\ y e. Q. ) -> ( y +Q q ) <Q ( y +Q r ) )
174, 15, 16syl2anc 411 . . . . . . 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 7806 . . . . . . . . 9 |- ( B e. P. -> <. ( 1st ` B ) , ( 2nd ` B ) >. e. P. )
22 prcunqu 7816 . . . . . . . . 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 411 . . . . . . 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 1649 . . . 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 7930 . . . . . . . . 9 |- ( q e. ( 2nd ` C ) <-> ( q e. Q. /\ E. y ( y e. ( 1st ` A ) /\ ( y +Q q ) e. ( 2nd ` B ) ) ) )
30 19.42v 1958 . . . . . . . . 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 457 . . . . . . 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 1958 . . . . . . 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 457 . . . . 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 1958 . . . . 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 7930 . . . . 5 |- ( r e. ( 2nd ` C ) <-> ( r e. Q. /\ E. y ( y e. ( 1st ` A ) /\ ( y +Q r ) e. ( 2nd ` B ) ) ) )
39 19.42v 1958 . . . . 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 2666 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
Syntax hints:   -> wi 4   /\ wa 104   = wceq 1398   E.wex 1541   e. wcel 2205   E.wrex 2523   {crab 2526   <.cop 3697   class class class wbr 4114   ` cfv 5357  (class class class)co 6058   1stc1st 6345   2ndc2nd 6346   Q.cnq 7611   +Q cplq 7613   <Q cltq 7616   P.cnp 7622   <P cltp 7626
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-coll 4230  ax-sep 4233  ax-nul 4241  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664  ax-iinf 4715
This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-ral 2527  df-rex 2528  df-reu 2529  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-nul 3513  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-int 3955  df-iun 3998  df-br 4115  df-opab 4177  df-mpt 4178  df-tr 4214  df-eprel 4415  df-id 4419  df-iord 4492  df-on 4494  df-suc 4497  df-iom 4718  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-f1 5362  df-fo 5363  df-f1o 5364  df-fv 5365  df-ov 6061  df-oprab 6062  df-mpo 6063  df-1st 6347  df-2nd 6348  df-recs 6549  df-irdg 6614  df-oadd 6664  df-omul 6665  df-er 6780  df-ec 6782  df-qs 6786  df-ni 7635  df-pli 7636  df-mi 7637  df-lti 7638  df-plpq 7675  df-enq 7678  df-nqqs 7679  df-plqqs 7680  df-ltnqqs 7684  df-inp 7797  df-iltp 7801
This theorem is referenced by:  ltexprlemrnd  7936
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