Theorem recexprlempr 7912

Description: B is a positive real. Lemma for recexpr 7918. (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 7904	. . 3 |- ( A e. P. -> ( E. q e. Q. q e. ( 1st ` B ) /\ E. r e. Q. r e. ( 2nd ` B ) ) )
3		ltrelnq 7645	. . . . . . . . . . 11 |- <Q C_ ( Q. X. Q. )
4	3	brel 4784	. . . . . . . . . 10 |- ( x <Q y -> ( x e. Q. /\ y e. Q. ) )
5	4	simpld 112	. . . . . . . . 9 |- ( x <Q y -> x e. Q. )
6	5	adantr 276	. . . . . . . 8 |- ( ( x <Q y /\ ( *Q ` y ) e. ( 2nd ` A ) ) -> x e. Q. )
7	6	exlimiv 1647	. . . . . . 7 |- ( E. y ( x <Q y /\ ( *Q ` y ) e. ( 2nd ` A ) ) -> x e. Q. )
8	7	abssi 3303	. . . . . 6 |- { x | E. y ( x <Q y /\ ( *Q ` y ) e. ( 2nd ` A ) ) } C_ Q.
9		nqex 7643	. . . . . . 7 |- Q. e. _V
10	9	elpw2 4252	. . . . . 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 146	. . . . 5 |- { x | E. y ( x <Q y /\ ( *Q ` y ) e. ( 2nd ` A ) ) } e. ~P Q.
12	3	brel 4784	. . . . . . . . . 10 |- ( y <Q x -> ( y e. Q. /\ x e. Q. ) )
13	12	simprd 114	. . . . . . . . 9 |- ( y <Q x -> x e. Q. )
14	13	adantr 276	. . . . . . . 8 |- ( ( y <Q x /\ ( *Q ` y ) e. ( 1st ` A ) ) -> x e. Q. )
15	14	exlimiv 1647	. . . . . . 7 |- ( E. y ( y <Q x /\ ( *Q ` y ) e. ( 1st ` A ) ) -> x e. Q. )
16	15	abssi 3303	. . . . . 6 |- { x | E. y ( y <Q x /\ ( *Q ` y ) e. ( 1st ` A ) ) } C_ Q.
17	9	elpw2 4252	. . . . . 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 146	. . . . 5 |- { x | E. y ( y <Q x /\ ( *Q ` y ) e. ( 1st ` A ) ) } e. ~P Q.
19		opelxpi 4763	. . . . 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 426	. . . 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 2304	. . 3 |- B e. ( ~P Q. X. ~P Q. )
22	2, 21	jctil 312	. 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 7909	. . 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 7910	. . 3 |- ( A e. P. -> A. q e. Q. -. ( q e. ( 1st ` B ) /\ q e. ( 2nd ` B ) ) )
25	1	recexprlemloc 7911	. . 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 1204	. 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 7756	. 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 417	1 |- ( A e. P. -> B e. P. )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 716   /\ w3a 1005   = wceq 1398   E.wex 1541   e. wcel 2202   {cab 2217   A.wral 2511   E.wrex 2512   C_ wss 3201   ~Pcpw 3656   <.cop 3676   class class class wbr 4093   X. cxp 4729   ` cfv 5333   1stc1st 6310   2ndc2nd 6311   Q.cnq 7560   *Qcrq 7564   <Q cltq 7565   P.cnp 7571

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4209  ax-sep 4212  ax-nul 4220  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641  ax-iinf 4692

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-ral 2516  df-rex 2517  df-reu 2518  df-rab 2520  df-v 2805  df-sbc 3033  df-csb 3129  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-nul 3497  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-int 3934  df-iun 3977  df-br 4094  df-opab 4156  df-mpt 4157  df-tr 4193  df-eprel 4392  df-id 4396  df-po 4399  df-iso 4400  df-iord 4469  df-on 4471  df-suc 4474  df-iom 4695  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-f1 5338  df-fo 5339  df-f1o 5340  df-fv 5341  df-ov 6031  df-oprab 6032  df-mpo 6033  df-1st 6312  df-2nd 6313  df-recs 6514  df-irdg 6579  df-1o 6625  df-oadd 6629  df-omul 6630  df-er 6745  df-ec 6747  df-qs 6751  df-ni 7584  df-pli 7585  df-mi 7586  df-lti 7587  df-plpq 7624  df-mpq 7625  df-enq 7627  df-nqqs 7628  df-plqqs 7629  df-mqqs 7630  df-1nqqs 7631  df-rq 7632  df-ltnqqs 7633  df-inp 7746

This theorem is referenced by:  recexprlem1ssl  7913  recexprlem1ssu  7914  recexprlemss1l  7915  recexprlemss1u  7916  recexprlemex  7917  recexpr  7918