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Theorem ltexprlemlol 7601
Description: The lower cut of our constructed difference is lower. Lemma for ltexpri 7612. (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 7504 . . . . . . . . . . . 12 |- <P C_ ( P. X. P. )
76brel 4679 . . . . . . . . . . 11 |- ( A <P B -> ( A e. P. /\ B e. P. ) )
87simpld 112 . . . . . . . . . 10 |- ( A <P B -> A e. P. )
9 prop 7474 . . . . . . . . . . 11 |- ( A e. P. -> <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. )
10 elprnqu 7481 . . . . . . . . . . 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 7401 . . . . . . . 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 7474 . . . . . . . . 9 |- ( B e. P. -> <. ( 1st ` B ) , ( 2nd ` B ) >. e. P. )
20 prcdnql 7483 . . . . . . . . 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 1600 . . . 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 7597 . . . . . . . . 9 |- ( r e. ( 1st ` C ) <-> ( r e. Q. /\ E. y ( y e. ( 2nd ` A ) /\ ( y +Q r ) e. ( 1st ` B ) ) ) )
28 19.42v 1906 . . . . . . . . 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 1906 . . . . . . 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 1906 . . . . 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 7597 . . . . 5 |- ( q e. ( 1st ` C ) <-> ( q e. Q. /\ E. y ( y e. ( 2nd ` A ) /\ ( y +Q q ) e. ( 1st ` B ) ) ) )
37 19.42v 1906 . . . . 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 2598 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 1353   E.wex 1492   e. wcel 2148   E.wrex 2456   {crab 2459   <.cop 3596   class class class wbr 4004   ` cfv 5217  (class class class)co 5875   1stc1st 6139   2ndc2nd 6140   Q.cnq 7279   +Q cplq 7281   <Q cltq 7284   P.cnp 7290   <P cltp 7294
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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4119  ax-sep 4122  ax-nul 4130  ax-pow 4175  ax-pr 4210  ax-un 4434  ax-setind 4537  ax-iinf 4588
This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2740  df-sbc 2964  df-csb 3059  df-dif 3132  df-un 3134  df-in 3136  df-ss 3143  df-nul 3424  df-pw 3578  df-sn 3599  df-pr 3600  df-op 3602  df-uni 3811  df-int 3846  df-iun 3889  df-br 4005  df-opab 4066  df-mpt 4067  df-tr 4103  df-eprel 4290  df-id 4294  df-iord 4367  df-on 4369  df-suc 4372  df-iom 4591  df-xp 4633  df-rel 4634  df-cnv 4635  df-co 4636  df-dm 4637  df-rn 4638  df-res 4639  df-ima 4640  df-iota 5179  df-fun 5219  df-fn 5220  df-f 5221  df-f1 5222  df-fo 5223  df-f1o 5224  df-fv 5225  df-ov 5878  df-oprab 5879  df-mpo 5880  df-1st 6141  df-2nd 6142  df-recs 6306  df-irdg 6371  df-oadd 6421  df-omul 6422  df-er 6535  df-ec 6537  df-qs 6541  df-ni 7303  df-pli 7304  df-mi 7305  df-lti 7306  df-plpq 7343  df-enq 7346  df-nqqs 7347  df-plqqs 7348  df-ltnqqs 7352  df-inp 7465  df-iltp 7469
This theorem is referenced by:  ltexprlemrnd  7604
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