Theorem ltexprlemrnd 7700

Description: Our constructed difference is rounded. Lemma for ltexpri 7708. (Contributed by Jim Kingdon, 17-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
ltexprlemrnd	|- ( A <P B -> ( A. q e. Q. ( q e. ( 1st ` C ) <-> E. r e. Q. ( q <Q r /\ r e. ( 1st ` C ) ) ) /\ A. r e. Q. ( r e. ( 2nd ` C ) <-> E. q e. Q. ( q <Q r /\ q e. ( 2nd ` C ) ) ) ) )

Distinct variable groups:   x, y, q, r, A   x, B, y, q, r   x, C, y, q, r

Proof of Theorem ltexprlemrnd

Step	Hyp	Ref	Expression
1		ltexprlem.1	. . . . . 6 |- 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 ) ) } >.
2	1	ltexprlemopl 7696	. . . . 5 |- ( ( A <P B /\ q e. Q. /\ q e. ( 1st ` C ) ) -> E. r e. Q. ( q <Q r /\ r e. ( 1st ` C ) ) )
3	2	3expia 1207	. . . 4 |- ( ( A <P B /\ q e. Q. ) -> ( q e. ( 1st ` C ) -> E. r e. Q. ( q <Q r /\ r e. ( 1st ` C ) ) ) )
4	1	ltexprlemlol 7697	. . . 4 |- ( ( A <P B /\ q e. Q. ) -> ( E. r e. Q. ( q <Q r /\ r e. ( 1st ` C ) ) -> q e. ( 1st ` C ) ) )
5	3, 4	impbid 129	. . 3 |- ( ( A <P B /\ q e. Q. ) -> ( q e. ( 1st ` C ) <-> E. r e. Q. ( q <Q r /\ r e. ( 1st ` C ) ) ) )
6	5	ralrimiva 2578	. 2 |- ( A <P B -> A. q e. Q. ( q e. ( 1st ` C ) <-> E. r e. Q. ( q <Q r /\ r e. ( 1st ` C ) ) ) )
7	1	ltexprlemopu 7698	. . . . 5 |- ( ( A <P B /\ r e. Q. /\ r e. ( 2nd ` C ) ) -> E. q e. Q. ( q <Q r /\ q e. ( 2nd ` C ) ) )
8	7	3expia 1207	. . . 4 |- ( ( A <P B /\ r e. Q. ) -> ( r e. ( 2nd ` C ) -> E. q e. Q. ( q <Q r /\ q e. ( 2nd ` C ) ) ) )
9	1	ltexprlemupu 7699	. . . 4 |- ( ( A <P B /\ r e. Q. ) -> ( E. q e. Q. ( q <Q r /\ q e. ( 2nd ` C ) ) -> r e. ( 2nd ` C ) ) )
10	8, 9	impbid 129	. . 3 |- ( ( A <P B /\ r e. Q. ) -> ( r e. ( 2nd ` C ) <-> E. q e. Q. ( q <Q r /\ q e. ( 2nd ` C ) ) ) )
11	10	ralrimiva 2578	. 2 |- ( A <P B -> A. r e. Q. ( r e. ( 2nd ` C ) <-> E. q e. Q. ( q <Q r /\ q e. ( 2nd ` C ) ) ) )
12	6, 11	jca 306	1 |- ( A <P B -> ( A. q e. Q. ( q e. ( 1st ` C ) <-> E. r e. Q. ( q <Q r /\ r e. ( 1st ` C ) ) ) /\ A. r e. Q. ( r e. ( 2nd ` C ) <-> E. q e. Q. ( q <Q r /\ q e. ( 2nd ` C ) ) ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1372   E.wex 1514   e. wcel 2175   A.wral 2483   E.wrex 2484   {crab 2487   <.cop 3635   class class class wbr 4043   ` cfv 5268  (class class class)co 5934   1stc1st 6214   2ndc2nd 6215   Q.cnq 7375   +Q cplq 7377   <Q cltq 7380   <P cltp 7390

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 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-13 2177  ax-14 2178  ax-ext 2186  ax-coll 4158  ax-sep 4161  ax-nul 4169  ax-pow 4217  ax-pr 4252  ax-un 4478  ax-setind 4583  ax-iinf 4634

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1375  df-fal 1378  df-nf 1483  df-sb 1785  df-eu 2056  df-mo 2057  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-ne 2376  df-ral 2488  df-rex 2489  df-reu 2490  df-rab 2492  df-v 2773  df-sbc 2998  df-csb 3093  df-dif 3167  df-un 3169  df-in 3171  df-ss 3178  df-nul 3460  df-pw 3617  df-sn 3638  df-pr 3639  df-op 3641  df-uni 3850  df-int 3885  df-iun 3928  df-br 4044  df-opab 4105  df-mpt 4106  df-tr 4142  df-eprel 4334  df-id 4338  df-po 4341  df-iso 4342  df-iord 4411  df-on 4413  df-suc 4416  df-iom 4637  df-xp 4679  df-rel 4680  df-cnv 4681  df-co 4682  df-dm 4683  df-rn 4684  df-res 4685  df-ima 4686  df-iota 5229  df-fun 5270  df-fn 5271  df-f 5272  df-f1 5273  df-fo 5274  df-f1o 5275  df-fv 5276  df-ov 5937  df-oprab 5938  df-mpo 5939  df-1st 6216  df-2nd 6217  df-recs 6381  df-irdg 6446  df-1o 6492  df-oadd 6496  df-omul 6497  df-er 6610  df-ec 6612  df-qs 6616  df-ni 7399  df-pli 7400  df-mi 7401  df-lti 7402  df-plpq 7439  df-mpq 7440  df-enq 7442  df-nqqs 7443  df-plqqs 7444  df-mqqs 7445  df-1nqqs 7446  df-ltnqqs 7448  df-inp 7561  df-iltp 7565

This theorem is referenced by:  ltexprlempr  7703