Theorem recexprlempr 7433

Description: B is a positive real. Lemma for recexpr 7439. (Contributed by Jim Kingdon, 27-Dec-2019.)

Hypothesis

Ref	Expression
recexpr.1	|- B = <. { x | E. y ( x <Q y /\ ( *Q ` y ) e. ( 2nd ` A ) ) } , { x | E. y ( y <Q x /\ ( *Q ` y ) e. ( 1st ` A ) ) } >.

Assertion

Ref	Expression
recexprlempr	|- ( A e. P. -> B e. P. )

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

Proof of Theorem recexprlempr

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

Step	Hyp	Ref	Expression
1		recexpr.1	. . . 4 |- B = <. { x | E. y ( x <Q y /\ ( *Q ` y ) e. ( 2nd ` A ) ) } , { x | E. y ( y <Q x /\ ( *Q ` y ) e. ( 1st ` A ) ) } >.
2	1	recexprlemm 7425	. . 3 |- ( A e. P. -> ( E. q e. Q. q e. ( 1st ` B ) /\ E. r e. Q. r e. ( 2nd ` B ) ) )
3		ltrelnq 7166	. . . . . . . . . . 11 |- <Q C_ ( Q. X. Q. )
4	3	brel 4586	. . . . . . . . . 10 |- ( x <Q y -> ( x e. Q. /\ y e. Q. ) )
5	4	simpld 111	. . . . . . . . 9 |- ( x <Q y -> x e. Q. )
6	5	adantr 274	. . . . . . . 8 |- ( ( x <Q y /\ ( *Q ` y ) e. ( 2nd ` A ) ) -> x e. Q. )
7	6	exlimiv 1577	. . . . . . 7 |- ( E. y ( x <Q y /\ ( *Q ` y ) e. ( 2nd ` A ) ) -> x e. Q. )
8	7	abssi 3167	. . . . . 6 |- { x | E. y ( x <Q y /\ ( *Q ` y ) e. ( 2nd ` A ) ) } C_ Q.
9		nqex 7164	. . . . . . 7 |- Q. e. _V
10	9	elpw2 4077	. . . . . 6 |- ( { x | E. y ( x <Q y /\ ( *Q ` y ) e. ( 2nd ` A ) ) } e. ~P Q. <-> { x | E. y ( x <Q y /\ ( *Q ` y ) e. ( 2nd ` A ) ) } C_ Q. )
11	8, 10	mpbir 145	. . . . 5 |- { x | E. y ( x <Q y /\ ( *Q ` y ) e. ( 2nd ` A ) ) } e. ~P Q.
12	3	brel 4586	. . . . . . . . . 10 |- ( y <Q x -> ( y e. Q. /\ x e. Q. ) )
13	12	simprd 113	. . . . . . . . 9 |- ( y <Q x -> x e. Q. )
14	13	adantr 274	. . . . . . . 8 |- ( ( y <Q x /\ ( *Q ` y ) e. ( 1st ` A ) ) -> x e. Q. )
15	14	exlimiv 1577	. . . . . . 7 |- ( E. y ( y <Q x /\ ( *Q ` y ) e. ( 1st ` A ) ) -> x e. Q. )
16	15	abssi 3167	. . . . . 6 |- { x | E. y ( y <Q x /\ ( *Q ` y ) e. ( 1st ` A ) ) } C_ Q.
17	9	elpw2 4077	. . . . . 6 |- ( { x | E. y ( y <Q x /\ ( *Q ` y ) e. ( 1st ` A ) ) } e. ~P Q. <-> { x | E. y ( y <Q x /\ ( *Q ` y ) e. ( 1st ` A ) ) } C_ Q. )
18	16, 17	mpbir 145	. . . . 5 |- { x | E. y ( y <Q x /\ ( *Q ` y ) e. ( 1st ` A ) ) } e. ~P Q.
19		opelxpi 4566	. . . . 5 |- ( ( { x | E. y ( x <Q y /\ ( *Q ` y ) e. ( 2nd ` A ) ) } e. ~P Q. /\ { x | E. y ( y <Q x /\ ( *Q ` y ) e. ( 1st ` A ) ) } e. ~P Q. ) -> <. { x | E. y ( x <Q y /\ ( *Q ` y ) e. ( 2nd ` A ) ) } , { x | E. y ( y <Q x /\ ( *Q ` y ) e. ( 1st ` A ) ) } >. e. ( ~P Q. X. ~P Q. ) )
20	11, 18, 19	mp2an 422	. . . 4 |- <. { x | E. y ( x <Q y /\ ( *Q ` y ) e. ( 2nd ` A ) ) } , { x | E. y ( y <Q x /\ ( *Q ` y ) e. ( 1st ` A ) ) } >. e. ( ~P Q. X. ~P Q. )
21	1, 20	eqeltri 2210	. . 3 |- B e. ( ~P Q. X. ~P Q. )
22	2, 21	jctil 310	. 2 |- ( A e. P. -> ( B e. ( ~P Q. X. ~P Q. ) /\ ( E. q e. Q. q e. ( 1st ` B ) /\ E. r e. Q. r e. ( 2nd ` B ) ) ) )
23	1	recexprlemrnd 7430	. . 3 |- ( A e. P. -> ( A. q e. Q. ( q e. ( 1st ` B ) <-> E. r e. Q. ( q <Q r /\ r e. ( 1st ` B ) ) ) /\ A. r e. Q. ( r e. ( 2nd ` B ) <-> E. q e. Q. ( q <Q r /\ q e. ( 2nd ` B ) ) ) ) )
24	1	recexprlemdisj 7431	. . 3 |- ( A e. P. -> A. q e. Q. -. ( q e. ( 1st ` B ) /\ q e. ( 2nd ` B ) ) )
25	1	recexprlemloc 7432	. . 3 |- ( A e. P. -> A. q e. Q. A. r e. Q. ( q <Q r -> ( q e. ( 1st ` B ) \/ r e. ( 2nd ` B ) ) ) )
26	23, 24, 25	3jca 1161	. 2 |- ( A e. P. -> ( ( A. q e. Q. ( q e. ( 1st ` B ) <-> E. r e. Q. ( q <Q r /\ r e. ( 1st ` B ) ) ) /\ A. r e. Q. ( r e. ( 2nd ` B ) <-> E. q e. Q. ( q <Q r /\ q e. ( 2nd ` B ) ) ) ) /\ A. q e. Q. -. ( q e. ( 1st ` B ) /\ q e. ( 2nd ` B ) ) /\ A. q e. Q. A. r e. Q. ( q <Q r -> ( q e. ( 1st ` B ) \/ r e. ( 2nd ` B ) ) ) ) )
27		elnp1st2nd 7277	. 2 |- ( B e. P. <-> ( ( B e. ( ~P Q. X. ~P Q. ) /\ ( E. q e. Q. q e. ( 1st ` B ) /\ E. r e. Q. r e. ( 2nd ` B ) ) ) /\ ( ( A. q e. Q. ( q e. ( 1st ` B ) <-> E. r e. Q. ( q <Q r /\ r e. ( 1st ` B ) ) ) /\ A. r e. Q. ( r e. ( 2nd ` B ) <-> E. q e. Q. ( q <Q r /\ q e. ( 2nd ` B ) ) ) ) /\ A. q e. Q. -. ( q e. ( 1st ` B ) /\ q e. ( 2nd ` B ) ) /\ A. q e. Q. A. r e. Q. ( q <Q r -> ( q e. ( 1st ` B ) \/ r e. ( 2nd ` B ) ) ) ) ) )
28	22, 26, 27	sylanbrc 413	1 |- ( A e. P. -> B e. P. )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 103   <-> wb 104   \/ wo 697   /\ w3a 962   = wceq 1331   E.wex 1468   e. wcel 1480   {cab 2123   A.wral 2414   E.wrex 2415   C_ wss 3066   ~Pcpw 3505   <.cop 3525   class class class wbr 3924   X. cxp 4532   ` cfv 5118   1stc1st 6029   2ndc2nd 6030   Q.cnq 7081   *Qcrq 7085   <Q cltq 7086   P.cnp 7092

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-coll 4038  ax-sep 4041  ax-nul 4049  ax-pow 4093  ax-pr 4126  ax-un 4350  ax-setind 4447  ax-iinf 4497

This theorem depends on definitions:  df-bi 116  df-dc 820  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ne 2307  df-ral 2419  df-rex 2420  df-reu 2421  df-rab 2423  df-v 2683  df-sbc 2905  df-csb 2999  df-dif 3068  df-un 3070  df-in 3072  df-ss 3079  df-nul 3359  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-uni 3732  df-int 3767  df-iun 3810  df-br 3925  df-opab 3985  df-mpt 3986  df-tr 4022  df-eprel 4206  df-id 4210  df-po 4213  df-iso 4214  df-iord 4283  df-on 4285  df-suc 4288  df-iom 4500  df-xp 4540  df-rel 4541  df-cnv 4542  df-co 4543  df-dm 4544  df-rn 4545  df-res 4546  df-ima 4547  df-iota 5083  df-fun 5120  df-fn 5121  df-f 5122  df-f1 5123  df-fo 5124  df-f1o 5125  df-fv 5126  df-ov 5770  df-oprab 5771  df-mpo 5772  df-1st 6031  df-2nd 6032  df-recs 6195  df-irdg 6260  df-1o 6306  df-oadd 6310  df-omul 6311  df-er 6422  df-ec 6424  df-qs 6428  df-ni 7105  df-pli 7106  df-mi 7107  df-lti 7108  df-plpq 7145  df-mpq 7146  df-enq 7148  df-nqqs 7149  df-plqqs 7150  df-mqqs 7151  df-1nqqs 7152  df-rq 7153  df-ltnqqs 7154  df-inp 7267

This theorem is referenced by:  recexprlem1ssl  7434  recexprlem1ssu  7435  recexprlemss1l  7436  recexprlemss1u  7437  recexprlemex  7438  recexpr  7439