Theorem recexprlempr 7573

Description: B is a positive real. Lemma for recexpr 7579. (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 7565	. . 3 |- ( A e. P. -> ( E. q e. Q. q e. ( 1st ` B ) /\ E. r e. Q. r e. ( 2nd ` B ) ) )
3		ltrelnq 7306	. . . . . . . . . . 11 |- <Q C_ ( Q. X. Q. )
4	3	brel 4656	. . . . . . . . . 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 1586	. . . . . . 7 |- ( E. y ( x <Q y /\ ( *Q ` y ) e. ( 2nd ` A ) ) -> x e. Q. )
8	7	abssi 3217	. . . . . 6 |- { x | E. y ( x <Q y /\ ( *Q ` y ) e. ( 2nd ` A ) ) } C_ Q.
9		nqex 7304	. . . . . . 7 |- Q. e. _V
10	9	elpw2 4136	. . . . . 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 4656	. . . . . . . . . 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 1586	. . . . . . 7 |- ( E. y ( y <Q x /\ ( *Q ` y ) e. ( 1st ` A ) ) -> x e. Q. )
16	15	abssi 3217	. . . . . 6 |- { x | E. y ( y <Q x /\ ( *Q ` y ) e. ( 1st ` A ) ) } C_ Q.
17	9	elpw2 4136	. . . . . 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 4636	. . . . 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 423	. . . 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 2239	. . 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 7570	. . 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 7571	. . 3 |- ( A e. P. -> A. q e. Q. -. ( q e. ( 1st ` B ) /\ q e. ( 2nd ` B ) ) )
25	1	recexprlemloc 7572	. . 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 1167	. 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 7417	. 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 414	1 |- ( A e. P. -> B e. P. )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 103   <-> wb 104   \/ wo 698   /\ w3a 968   = wceq 1343   E.wex 1480   e. wcel 2136   {cab 2151   A.wral 2444   E.wrex 2445   C_ wss 3116   ~Pcpw 3559   <.cop 3579   class class class wbr 3982   X. cxp 4602   ` cfv 5188   1stc1st 6106   2ndc2nd 6107   Q.cnq 7221   *Qcrq 7225   <Q cltq 7226   P.cnp 7232

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 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-coll 4097  ax-sep 4100  ax-nul 4108  ax-pow 4153  ax-pr 4187  ax-un 4411  ax-setind 4514  ax-iinf 4565

This theorem depends on definitions:  df-bi 116  df-dc 825  df-3or 969  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ne 2337  df-ral 2449  df-rex 2450  df-reu 2451  df-rab 2453  df-v 2728  df-sbc 2952  df-csb 3046  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-nul 3410  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-int 3825  df-iun 3868  df-br 3983  df-opab 4044  df-mpt 4045  df-tr 4081  df-eprel 4267  df-id 4271  df-po 4274  df-iso 4275  df-iord 4344  df-on 4346  df-suc 4349  df-iom 4568  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-res 4616  df-ima 4617  df-iota 5153  df-fun 5190  df-fn 5191  df-f 5192  df-f1 5193  df-fo 5194  df-f1o 5195  df-fv 5196  df-ov 5845  df-oprab 5846  df-mpo 5847  df-1st 6108  df-2nd 6109  df-recs 6273  df-irdg 6338  df-1o 6384  df-oadd 6388  df-omul 6389  df-er 6501  df-ec 6503  df-qs 6507  df-ni 7245  df-pli 7246  df-mi 7247  df-lti 7248  df-plpq 7285  df-mpq 7286  df-enq 7288  df-nqqs 7289  df-plqqs 7290  df-mqqs 7291  df-1nqqs 7292  df-rq 7293  df-ltnqqs 7294  df-inp 7407

This theorem is referenced by:  recexprlem1ssl  7574  recexprlem1ssu  7575  recexprlemss1l  7576  recexprlemss1u  7577  recexprlemex  7578  recexpr  7579