Theorem ltexprlempr 7795

Description: Our constructed difference is a positive real. Lemma for ltexpri 7800. (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
ltexprlempr	|- ( A <P B -> C e. P. )

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

Proof of Theorem ltexprlempr

Dummy variables q r are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ltexprlem.1	. . . 4 |- 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	ltexprlemm 7787	. . 3 |- ( A <P B -> ( E. q e. Q. q e. ( 1st ` C ) /\ E. r e. Q. r e. ( 2nd ` C ) ) )
3		ssrab2 3309	. . . . . 6 |- { x e. Q. | E. y ( y e. ( 2nd ` A ) /\ ( y +Q x ) e. ( 1st ` B ) ) } C_ Q.
4		nqex 7550	. . . . . . 7 |- Q. e. _V
5	4	elpw2 4241	. . . . . 6 |- ( { x e. Q. | E. y ( y e. ( 2nd ` A ) /\ ( y +Q x ) e. ( 1st ` B ) ) } e. ~P Q. <-> { x e. Q. | E. y ( y e. ( 2nd ` A ) /\ ( y +Q x ) e. ( 1st ` B ) ) } C_ Q. )
6	3, 5	mpbir 146	. . . . 5 |- { x e. Q. | E. y ( y e. ( 2nd ` A ) /\ ( y +Q x ) e. ( 1st ` B ) ) } e. ~P Q.
7		ssrab2 3309	. . . . . 6 |- { x e. Q. | E. y ( y e. ( 1st ` A ) /\ ( y +Q x ) e. ( 2nd ` B ) ) } C_ Q.
8	4	elpw2 4241	. . . . . 6 |- ( { x e. Q. | E. y ( y e. ( 1st ` A ) /\ ( y +Q x ) e. ( 2nd ` B ) ) } e. ~P Q. <-> { x e. Q. | E. y ( y e. ( 1st ` A ) /\ ( y +Q x ) e. ( 2nd ` B ) ) } C_ Q. )
9	7, 8	mpbir 146	. . . . 5 |- { x e. Q. | E. y ( y e. ( 1st ` A ) /\ ( y +Q x ) e. ( 2nd ` B ) ) } e. ~P Q.
10		opelxpi 4751	. . . . 5 |- ( ( { x e. Q. | E. y ( y e. ( 2nd ` A ) /\ ( y +Q x ) e. ( 1st ` B ) ) } e. ~P Q. /\ { x e. Q. | E. y ( y e. ( 1st ` A ) /\ ( y +Q x ) e. ( 2nd ` B ) ) } e. ~P Q. ) -> <. { 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 ) ) } >. e. ( ~P Q. X. ~P Q. ) )
11	6, 9, 10	mp2an 426	. . . 4 |- <. { 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 ) ) } >. e. ( ~P Q. X. ~P Q. )
12	1, 11	eqeltri 2302	. . 3 |- C e. ( ~P Q. X. ~P Q. )
13	2, 12	jctil 312	. 2 |- ( A <P B -> ( C e. ( ~P Q. X. ~P Q. ) /\ ( E. q e. Q. q e. ( 1st ` C ) /\ E. r e. Q. r e. ( 2nd ` C ) ) ) )
14	1	ltexprlemrnd 7792	. . 3 |- ( 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 ) ) ) ) )
15	1	ltexprlemdisj 7793	. . 3 |- ( A <P B -> A. q e. Q. -. ( q e. ( 1st ` C ) /\ q e. ( 2nd ` C ) ) )
16	1	ltexprlemloc 7794	. . 3 |- ( A <P B -> A. q e. Q. A. r e. Q. ( q <Q r -> ( q e. ( 1st ` C ) \/ r e. ( 2nd ` C ) ) ) )
17	14, 15, 16	3jca 1201	. 2 |- ( 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 ) ) ) ) /\ A. q e. Q. -. ( q e. ( 1st ` C ) /\ q e. ( 2nd ` C ) ) /\ A. q e. Q. A. r e. Q. ( q <Q r -> ( q e. ( 1st ` C ) \/ r e. ( 2nd ` C ) ) ) ) )
18		elnp1st2nd 7663	. 2 |- ( C e. P. <-> ( ( C e. ( ~P Q. X. ~P Q. ) /\ ( E. q e. Q. q e. ( 1st ` C ) /\ E. r e. Q. r e. ( 2nd ` C ) ) ) /\ ( ( 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 ) ) ) ) /\ A. q e. Q. -. ( q e. ( 1st ` C ) /\ q e. ( 2nd ` C ) ) /\ A. q e. Q. A. r e. Q. ( q <Q r -> ( q e. ( 1st ` C ) \/ r e. ( 2nd ` C ) ) ) ) ) )
19	13, 17, 18	sylanbrc 417	1 |- ( A <P B -> C e. P. )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 713   /\ w3a 1002   = wceq 1395   E.wex 1538   e. wcel 2200   A.wral 2508   E.wrex 2509   {crab 2512   C_ wss 3197   ~Pcpw 3649   <.cop 3669   class class class wbr 4083   X. cxp 4717   ` cfv 5318  (class class class)co 6001   1stc1st 6284   2ndc2nd 6285   Q.cnq 7467   +Q cplq 7469   <Q cltq 7472   P.cnp 7478   <P cltp 7482

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4199  ax-sep 4202  ax-nul 4210  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629  ax-iinf 4680

This theorem depends on definitions:  df-bi 117  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-int 3924  df-iun 3967  df-br 4084  df-opab 4146  df-mpt 4147  df-tr 4183  df-eprel 4380  df-id 4384  df-po 4387  df-iso 4388  df-iord 4457  df-on 4459  df-suc 4462  df-iom 4683  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-ima 4732  df-iota 5278  df-fun 5320  df-fn 5321  df-f 5322  df-f1 5323  df-fo 5324  df-f1o 5325  df-fv 5326  df-ov 6004  df-oprab 6005  df-mpo 6006  df-1st 6286  df-2nd 6287  df-recs 6451  df-irdg 6516  df-1o 6562  df-2o 6563  df-oadd 6566  df-omul 6567  df-er 6680  df-ec 6682  df-qs 6686  df-ni 7491  df-pli 7492  df-mi 7493  df-lti 7494  df-plpq 7531  df-mpq 7532  df-enq 7534  df-nqqs 7535  df-plqqs 7536  df-mqqs 7537  df-1nqqs 7538  df-rq 7539  df-ltnqqs 7540  df-enq0 7611  df-nq0 7612  df-0nq0 7613  df-plq0 7614  df-mq0 7615  df-inp 7653  df-iltp 7657

This theorem is referenced by:  ltexprlemfl  7796  ltexprlemrl  7797  ltexprlemfu  7798  ltexprlemru  7799  ltexpri  7800