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Theorem ltexprlemlol 7789
Description: The lower cut of our constructed difference is lower. Lemma for ltexpri 7800. (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
ltexprlemlol |- ( ( A <P B /\ q e. Q. ) -> ( E. r e. Q. ( q <Q r /\ r e. ( 1st ` C ) ) -> q e. ( 1st ` C ) ) )
Distinct variable groups:   x, y, q, r, A   x, B, y, q, r   x, C, y, q, r
Proof of Theorem ltexprlemlol
StepHypRef Expression
1 simplr 528 . . . . . 6 |- ( ( ( A <P B /\ q e. Q. ) /\ ( q <Q r /\ ( r e. Q. /\ ( y e. ( 2nd ` A ) /\ ( y +Q r ) e. ( 1st ` B ) ) ) ) ) -> q e. Q. )
2 simprrr 540 . . . . . . 7 |- ( ( ( A <P B /\ q e. Q. ) /\ ( q <Q r /\ ( r e. Q. /\ ( y e. ( 2nd ` A ) /\ ( y +Q r ) e. ( 1st ` B ) ) ) ) ) -> ( y e. ( 2nd ` A ) /\ ( y +Q r ) e. ( 1st ` B ) ) )
32simpld 112 . . . . . 6 |- ( ( ( A <P B /\ q e. Q. ) /\ ( q <Q r /\ ( r e. Q. /\ ( y e. ( 2nd ` A ) /\ ( y +Q r ) e. ( 1st ` B ) ) ) ) ) -> y e. ( 2nd ` A ) )
4 simprl 529 . . . . . . . 8 |- ( ( ( A <P B /\ q e. Q. ) /\ ( q <Q r /\ ( r e. Q. /\ ( y e. ( 2nd ` A ) /\ ( y +Q r ) e. ( 1st ` B ) ) ) ) ) -> q <Q r )
5 simpll 527 . . . . . . . . 9 |- ( ( ( A <P B /\ q e. Q. ) /\ ( q <Q r /\ ( r e. Q. /\ ( y e. ( 2nd ` A ) /\ ( y +Q r ) e. ( 1st ` B ) ) ) ) ) -> A <P B )
6 ltrelpr 7692 . . . . . . . . . . . 12 |- <P C_ ( P. X. P. )
76brel 4771 . . . . . . . . . . 11 |- ( A <P B -> ( A e. P. /\ B e. P. ) )
87simpld 112 . . . . . . . . . 10 |- ( A <P B -> A e. P. )
9 prop 7662 . . . . . . . . . . 11 |- ( A e. P. -> <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. )
10 elprnqu 7669 . . . . . . . . . . 11 |- ( ( <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. /\ y e. ( 2nd ` A ) ) -> y e. Q. )
119, 10sylan 283 . . . . . . . . . 10 |- ( ( A e. P. /\ y e. ( 2nd ` A ) ) -> y e. Q. )
128, 11sylan 283 . . . . . . . . 9 |- ( ( A <P B /\ y e. ( 2nd ` A ) ) -> y e. Q. )
135, 3, 12syl2anc 411 . . . . . . . 8 |- ( ( ( A <P B /\ q e. Q. ) /\ ( q <Q r /\ ( r e. Q. /\ ( y e. ( 2nd ` A ) /\ ( y +Q r ) e. ( 1st ` B ) ) ) ) ) -> y e. Q. )
14 ltanqi 7589 . . . . . . . 8 |- ( ( q <Q r /\ y e. Q. ) -> ( y +Q q ) <Q ( y +Q r ) )
154, 13, 14syl2anc 411 . . . . . . 7 |- ( ( ( A <P B /\ q e. Q. ) /\ ( q <Q r /\ ( r e. Q. /\ ( y e. ( 2nd ` A ) /\ ( y +Q r ) e. ( 1st ` B ) ) ) ) ) -> ( y +Q q ) <Q ( y +Q r ) )
167simprd 114 . . . . . . . . 9 |- ( A <P B -> B e. P. )
175, 16syl 14 . . . . . . . 8 |- ( ( ( A <P B /\ q e. Q. ) /\ ( q <Q r /\ ( r e. Q. /\ ( y e. ( 2nd ` A ) /\ ( y +Q r ) e. ( 1st ` B ) ) ) ) ) -> B e. P. )
182simprd 114 . . . . . . . 8 |- ( ( ( A <P B /\ q e. Q. ) /\ ( q <Q r /\ ( r e. Q. /\ ( y e. ( 2nd ` A ) /\ ( y +Q r ) e. ( 1st ` B ) ) ) ) ) -> ( y +Q r ) e. ( 1st ` B ) )
19 prop 7662 . . . . . . . . 9 |- ( B e. P. -> <. ( 1st ` B ) , ( 2nd ` B ) >. e. P. )
20 prcdnql 7671 . . . . . . . . 9 |- ( ( <. ( 1st ` B ) , ( 2nd ` B ) >. e. P. /\ ( y +Q r ) e. ( 1st ` B ) ) -> ( ( y +Q q ) <Q ( y +Q r ) -> ( y +Q q ) e. ( 1st ` B ) ) )
2119, 20sylan 283 . . . . . . . 8 |- ( ( B e. P. /\ ( y +Q r ) e. ( 1st ` B ) ) -> ( ( y +Q q ) <Q ( y +Q r ) -> ( y +Q q ) e. ( 1st ` B ) ) )
2217, 18, 21syl2anc 411 . . . . . . 7 |- ( ( ( A <P B /\ q e. Q. ) /\ ( q <Q r /\ ( r e. Q. /\ ( y e. ( 2nd ` A ) /\ ( y +Q r ) e. ( 1st ` B ) ) ) ) ) -> ( ( y +Q q ) <Q ( y +Q r ) -> ( y +Q q ) e. ( 1st ` B ) ) )
2315, 22mpd 13 . . . . . 6 |- ( ( ( A <P B /\ q e. Q. ) /\ ( q <Q r /\ ( r e. Q. /\ ( y e. ( 2nd ` A ) /\ ( y +Q r ) e. ( 1st ` B ) ) ) ) ) -> ( y +Q q ) e. ( 1st ` B ) )
241, 3, 23jca32 310 . . . . 5 |- ( ( ( A <P B /\ q e. Q. ) /\ ( q <Q r /\ ( r e. Q. /\ ( y e. ( 2nd ` A ) /\ ( y +Q r ) e. ( 1st ` B ) ) ) ) ) -> ( q e. Q. /\ ( y e. ( 2nd ` A ) /\ ( y +Q q ) e. ( 1st ` B ) ) ) )
2524eximi 1646 . . . 4 |- ( E. y ( ( A <P B /\ q e. Q. ) /\ ( q <Q r /\ ( r e. Q. /\ ( y e. ( 2nd ` A ) /\ ( y +Q r ) e. ( 1st ` B ) ) ) ) ) -> E. y ( q e. Q. /\ ( y e. ( 2nd ` A ) /\ ( y +Q q ) e. ( 1st ` B ) ) ) )
26 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 ) ) } >.
2726ltexprlemell 7785 . . . . . . . . 9 |- ( r e. ( 1st ` C ) <-> ( r e. Q. /\ E. y ( y e. ( 2nd ` A ) /\ ( y +Q r ) e. ( 1st ` B ) ) ) )
28 19.42v 1953 . . . . . . . . 9 |- ( E. y ( r e. Q. /\ ( y e. ( 2nd ` A ) /\ ( y +Q r ) e. ( 1st ` B ) ) ) <-> ( r e. Q. /\ E. y ( y e. ( 2nd ` A ) /\ ( y +Q r ) e. ( 1st ` B ) ) ) )
2927, 28bitr4i 187 . . . . . . . 8 |- ( r e. ( 1st ` C ) <-> E. y ( r e. Q. /\ ( y e. ( 2nd ` A ) /\ ( y +Q r ) e. ( 1st ` B ) ) ) )
3029anbi2i 457 . . . . . . 7 |- ( ( q <Q r /\ r e. ( 1st ` C ) ) <-> ( q <Q r /\ E. y ( r e. Q. /\ ( y e. ( 2nd ` A ) /\ ( y +Q r ) e. ( 1st ` B ) ) ) ) )
31 19.42v 1953 . . . . . . 7 |- ( E. y ( q <Q r /\ ( r e. Q. /\ ( y e. ( 2nd ` A ) /\ ( y +Q r ) e. ( 1st ` B ) ) ) ) <-> ( q <Q r /\ E. y ( r e. Q. /\ ( y e. ( 2nd ` A ) /\ ( y +Q r ) e. ( 1st ` B ) ) ) ) )
3230, 31bitr4i 187 . . . . . 6 |- ( ( q <Q r /\ r e. ( 1st ` C ) ) <-> E. y ( q <Q r /\ ( r e. Q. /\ ( y e. ( 2nd ` A ) /\ ( y +Q r ) e. ( 1st ` B ) ) ) ) )
3332anbi2i 457 . . . . 5 |- ( ( ( A <P B /\ q e. Q. ) /\ ( q <Q r /\ r e. ( 1st ` C ) ) ) <-> ( ( A <P B /\ q e. Q. ) /\ E. y ( q <Q r /\ ( r e. Q. /\ ( y e. ( 2nd ` A ) /\ ( y +Q r ) e. ( 1st ` B ) ) ) ) ) )
34 19.42v 1953 . . . . 5 |- ( E. y ( ( A <P B /\ q e. Q. ) /\ ( q <Q r /\ ( r e. Q. /\ ( y e. ( 2nd ` A ) /\ ( y +Q r ) e. ( 1st ` B ) ) ) ) ) <-> ( ( A <P B /\ q e. Q. ) /\ E. y ( q <Q r /\ ( r e. Q. /\ ( y e. ( 2nd ` A ) /\ ( y +Q r ) e. ( 1st ` B ) ) ) ) ) )
3533, 34bitr4i 187 . . . 4 |- ( ( ( A <P B /\ q e. Q. ) /\ ( q <Q r /\ r e. ( 1st ` C ) ) ) <-> E. y ( ( A <P B /\ q e. Q. ) /\ ( q <Q r /\ ( r e. Q. /\ ( y e. ( 2nd ` A ) /\ ( y +Q r ) e. ( 1st ` B ) ) ) ) ) )
3626ltexprlemell 7785 . . . . 5 |- ( q e. ( 1st ` C ) <-> ( q e. Q. /\ E. y ( y e. ( 2nd ` A ) /\ ( y +Q q ) e. ( 1st ` B ) ) ) )
37 19.42v 1953 . . . . 5 |- ( E. y ( q e. Q. /\ ( y e. ( 2nd ` A ) /\ ( y +Q q ) e. ( 1st ` B ) ) ) <-> ( q e. Q. /\ E. y ( y e. ( 2nd ` A ) /\ ( y +Q q ) e. ( 1st ` B ) ) ) )
3836, 37bitr4i 187 . . . 4 |- ( q e. ( 1st ` C ) <-> E. y ( q e. Q. /\ ( y e. ( 2nd ` A ) /\ ( y +Q q ) e. ( 1st ` B ) ) ) )
3925, 35, 383imtr4i 201 . . 3 |- ( ( ( A <P B /\ q e. Q. ) /\ ( q <Q r /\ r e. ( 1st ` C ) ) ) -> q e. ( 1st ` C ) )
4039ex 115 . 2 |- ( ( A <P B /\ q e. Q. ) -> ( ( q <Q r /\ r e. ( 1st ` C ) ) -> q e. ( 1st ` C ) ) )
4140rexlimdvw 2652 1 |- ( ( A <P B /\ q e. Q. ) -> ( E. r e. Q. ( q <Q r /\ r e. ( 1st ` C ) ) -> q e. ( 1st ` C ) ) )
Colors of variables: wff set class
Syntax hints:   -> wi 4   /\ wa 104   = wceq 1395   E.wex 1538   e. wcel 2200   E.wrex 2509   {crab 2512   <.cop 3669   class class class wbr 4083   ` cfv 5318  (class class class)co 6001   1stc1st 6284   2ndc2nd 6285   Q.cnq 7467   +Q cplq 7469   <Q cltq 7472   P.cnp 7478   <P cltp 7482
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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4199  ax-sep 4202  ax-nul 4210  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629  ax-iinf 4680
This theorem depends on definitions:  df-bi 117  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-int 3924  df-iun 3967  df-br 4084  df-opab 4146  df-mpt 4147  df-tr 4183  df-eprel 4380  df-id 4384  df-iord 4457  df-on 4459  df-suc 4462  df-iom 4683  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-ima 4732  df-iota 5278  df-fun 5320  df-fn 5321  df-f 5322  df-f1 5323  df-fo 5324  df-f1o 5325  df-fv 5326  df-ov 6004  df-oprab 6005  df-mpo 6006  df-1st 6286  df-2nd 6287  df-recs 6451  df-irdg 6516  df-oadd 6566  df-omul 6567  df-er 6680  df-ec 6682  df-qs 6686  df-ni 7491  df-pli 7492  df-mi 7493  df-lti 7494  df-plpq 7531  df-enq 7534  df-nqqs 7535  df-plqqs 7536  df-ltnqqs 7540  df-inp 7653  df-iltp 7657
This theorem is referenced by:  ltexprlemrnd  7792
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