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Theorem ltexprlemlol 7686
Description: The lower cut of our constructed difference is lower. Lemma for ltexpri 7697. (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 7589 . . . . . . . . . . . 12 |- <P C_ ( P. X. P. )
76brel 4716 . . . . . . . . . . 11 |- ( A <P B -> ( A e. P. /\ B e. P. ) )
87simpld 112 . . . . . . . . . 10 |- ( A <P B -> A e. P. )
9 prop 7559 . . . . . . . . . . 11 |- ( A e. P. -> <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. )
10 elprnqu 7566 . . . . . . . . . . 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 7486 . . . . . . . 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 7559 . . . . . . . . 9 |- ( B e. P. -> <. ( 1st ` B ) , ( 2nd ` B ) >. e. P. )
20 prcdnql 7568 . . . . . . . . 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 1614 . . . 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 7682 . . . . . . . . 9 |- ( r e. ( 1st ` C ) <-> ( r e. Q. /\ E. y ( y e. ( 2nd ` A ) /\ ( y +Q r ) e. ( 1st ` B ) ) ) )
28 19.42v 1921 . . . . . . . . 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 1921 . . . . . . 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 1921 . . . . 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 7682 . . . . 5 |- ( q e. ( 1st ` C ) <-> ( q e. Q. /\ E. y ( y e. ( 2nd ` A ) /\ ( y +Q q ) e. ( 1st ` B ) ) ) )
37 19.42v 1921 . . . . 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 2618 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 1506   e. wcel 2167   E.wrex 2476   {crab 2479   <.cop 3626   class class class wbr 4034   ` cfv 5259  (class class class)co 5925   1stc1st 6205   2ndc2nd 6206   Q.cnq 7364   +Q cplq 7366   <Q cltq 7369   P.cnp 7375   <P cltp 7379
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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-coll 4149  ax-sep 4152  ax-nul 4160  ax-pow 4208  ax-pr 4243  ax-un 4469  ax-setind 4574  ax-iinf 4625
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 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-ral 2480  df-rex 2481  df-reu 2482  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3452  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-int 3876  df-iun 3919  df-br 4035  df-opab 4096  df-mpt 4097  df-tr 4133  df-eprel 4325  df-id 4329  df-iord 4402  df-on 4404  df-suc 4407  df-iom 4628  df-xp 4670  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-rn 4675  df-res 4676  df-ima 4677  df-iota 5220  df-fun 5261  df-fn 5262  df-f 5263  df-f1 5264  df-fo 5265  df-f1o 5266  df-fv 5267  df-ov 5928  df-oprab 5929  df-mpo 5930  df-1st 6207  df-2nd 6208  df-recs 6372  df-irdg 6437  df-oadd 6487  df-omul 6488  df-er 6601  df-ec 6603  df-qs 6607  df-ni 7388  df-pli 7389  df-mi 7390  df-lti 7391  df-plpq 7428  df-enq 7431  df-nqqs 7432  df-plqqs 7433  df-ltnqqs 7437  df-inp 7550  df-iltp 7554
This theorem is referenced by:  ltexprlemrnd  7689
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