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Theorem ltexprlemupu 7594
Description: The upper cut of our constructed difference is upper. Lemma for ltexpri 7603. (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 528 . . . . . 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 540 . . . . . . 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 529 . . . . . . . 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 539 . . . . . . . . . 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 7495 . . . . . . . . . . . . 13 |- <P C_ ( P. X. P. )
98brel 4675 . . . . . . . . . . . 12 |- ( A <P B -> ( A e. P. /\ B e. P. ) )
109simpld 112 . . . . . . . . . . 11 |- ( A <P B -> A e. P. )
11 prop 7465 . . . . . . . . . . 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 7471 . . . . . . . . . 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 7392 . . . . . . . 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 7465 . . . . . . . . 9 |- ( B e. P. -> <. ( 1st ` B ) , ( 2nd ` B ) >. e. P. )
22 prcunqu 7475 . . . . . . . . 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 1600 . . . 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 7589 . . . . . . . . 9 |- ( q e. ( 2nd ` C ) <-> ( q e. Q. /\ E. y ( y e. ( 1st ` A ) /\ ( y +Q q ) e. ( 2nd ` B ) ) ) )
30 19.42v 1906 . . . . . . . . 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 1906 . . . . . . 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 1906 . . . . 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 7589 . . . . 5 |- ( r e. ( 2nd ` C ) <-> ( r e. Q. /\ E. y ( y e. ( 1st ` A ) /\ ( y +Q r ) e. ( 2nd ` B ) ) ) )
39 19.42v 1906 . . . . 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 2598 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 1353   E.wex 1492   e. wcel 2148   E.wrex 2456   {crab 2459   <.cop 3594   class class class wbr 4000   ` cfv 5212  (class class class)co 5869   1stc1st 6133   2ndc2nd 6134   Q.cnq 7270   +Q cplq 7272   <Q cltq 7275   P.cnp 7281   <P cltp 7285
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 4115  ax-sep 4118  ax-nul 4126  ax-pow 4171  ax-pr 4206  ax-un 4430  ax-setind 4533  ax-iinf 4584
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 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-int 3843  df-iun 3886  df-br 4001  df-opab 4062  df-mpt 4063  df-tr 4099  df-eprel 4286  df-id 4290  df-iord 4363  df-on 4365  df-suc 4368  df-iom 4587  df-xp 4629  df-rel 4630  df-cnv 4631  df-co 4632  df-dm 4633  df-rn 4634  df-res 4635  df-ima 4636  df-iota 5174  df-fun 5214  df-fn 5215  df-f 5216  df-f1 5217  df-fo 5218  df-f1o 5219  df-fv 5220  df-ov 5872  df-oprab 5873  df-mpo 5874  df-1st 6135  df-2nd 6136  df-recs 6300  df-irdg 6365  df-oadd 6415  df-omul 6416  df-er 6529  df-ec 6531  df-qs 6535  df-ni 7294  df-pli 7295  df-mi 7296  df-lti 7297  df-plpq 7334  df-enq 7337  df-nqqs 7338  df-plqqs 7339  df-ltnqqs 7343  df-inp 7456  df-iltp 7460
This theorem is referenced by:  ltexprlemrnd  7595
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