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Theorem ltexprlemlol 7664
Description: The lower cut of our constructed difference is lower. Lemma for ltexpri 7675. (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 7567 . . . . . . . . . . . 12 |- <P C_ ( P. X. P. )
76brel 4712 . . . . . . . . . . 11 |- ( A <P B -> ( A e. P. /\ B e. P. ) )
87simpld 112 . . . . . . . . . 10 |- ( A <P B -> A e. P. )
9 prop 7537 . . . . . . . . . . 11 |- ( A e. P. -> <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. )
10 elprnqu 7544 . . . . . . . . . . 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 7464 . . . . . . . 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 7537 . . . . . . . . 9 |- ( B e. P. -> <. ( 1st ` B ) , ( 2nd ` B ) >. e. P. )
20 prcdnql 7546 . . . . . . . . 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 1611 . . . 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 7660 . . . . . . . . 9 |- ( r e. ( 1st ` C ) <-> ( r e. Q. /\ E. y ( y e. ( 2nd ` A ) /\ ( y +Q r ) e. ( 1st ` B ) ) ) )
28 19.42v 1918 . . . . . . . . 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 1918 . . . . . . 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 1918 . . . . 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 7660 . . . . 5 |- ( q e. ( 1st ` C ) <-> ( q e. Q. /\ E. y ( y e. ( 2nd ` A ) /\ ( y +Q q ) e. ( 1st ` B ) ) ) )
37 19.42v 1918 . . . . 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 2615 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 1364   E.wex 1503   e. wcel 2164   E.wrex 2473   {crab 2476   <.cop 3622   class class class wbr 4030   ` cfv 5255  (class class class)co 5919   1stc1st 6193   2ndc2nd 6194   Q.cnq 7342   +Q cplq 7344   <Q cltq 7347   P.cnp 7353   <P cltp 7357
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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-coll 4145  ax-sep 4148  ax-nul 4156  ax-pow 4204  ax-pr 4239  ax-un 4465  ax-setind 4570  ax-iinf 4621
This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-ral 2477  df-rex 2478  df-reu 2479  df-rab 2481  df-v 2762  df-sbc 2987  df-csb 3082  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3448  df-pw 3604  df-sn 3625  df-pr 3626  df-op 3628  df-uni 3837  df-int 3872  df-iun 3915  df-br 4031  df-opab 4092  df-mpt 4093  df-tr 4129  df-eprel 4321  df-id 4325  df-iord 4398  df-on 4400  df-suc 4403  df-iom 4624  df-xp 4666  df-rel 4667  df-cnv 4668  df-co 4669  df-dm 4670  df-rn 4671  df-res 4672  df-ima 4673  df-iota 5216  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5922  df-oprab 5923  df-mpo 5924  df-1st 6195  df-2nd 6196  df-recs 6360  df-irdg 6425  df-oadd 6475  df-omul 6476  df-er 6589  df-ec 6591  df-qs 6595  df-ni 7366  df-pli 7367  df-mi 7368  df-lti 7369  df-plpq 7406  df-enq 7409  df-nqqs 7410  df-plqqs 7411  df-ltnqqs 7415  df-inp 7528  df-iltp 7532
This theorem is referenced by:  ltexprlemrnd  7667
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