Theorem recexprlempr 6604

Description: B is a positive real. Lemma for recexpr 6610. (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 6596	. . 3 |- (A e. P. -> (E. q e. Q. q e. ( 1st ` B) /\ E. r e. Q. r e. ( 2nd ` B)))
3		ltrelnq 6349	. . . . . . . . . . 11 |- <Q C_ ( Q. X. Q.)
4	3	brel 4335	. . . . . . . . . 10 |- (x <Q y -> (x e. Q. /\ y e. Q.))
5	4	simpld 105	. . . . . . . . 9 |- (x <Q y -> x e. Q.)
6	5	adantr 261	. . . . . . . 8 |- ((x <Q y /\ ( *Q ` y) e. ( 2nd ` A)) -> x e. Q.)
7	6	exlimiv 1486	. . . . . . 7 |- (E.y(x <Q y /\ ( *Q ` y) e. ( 2nd ` A)) -> x e. Q.)
8	7	abssi 3009	. . . . . 6 |- {x | E.y(x <Q y /\ ( *Q ` y) e. ( 2nd ` A)) } C_ Q.
9		nqex 6347	. . . . . . 7 |- Q. e. _V
10	9	elpw2 3902	. . . . . 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 134	. . . . 5 |- {x | E.y(x <Q y /\ ( *Q ` y) e. ( 2nd ` A)) } e. ~P Q.
12	3	brel 4335	. . . . . . . . . 10 |- (y <Q x -> (y e. Q. /\ x e. Q.))
13	12	simprd 107	. . . . . . . . 9 |- (y <Q x -> x e. Q.)
14	13	adantr 261	. . . . . . . 8 |- ((y <Q x /\ ( *Q ` y) e. ( 1st ` A)) -> x e. Q.)
15	14	exlimiv 1486	. . . . . . 7 |- (E.y(y <Q x /\ ( *Q ` y) e. ( 1st ` A)) -> x e. Q.)
16	15	abssi 3009	. . . . . 6 |- {x | E.y(y <Q x /\ ( *Q ` y) e. ( 1st ` A)) } C_ Q.
17	9	elpw2 3902	. . . . . 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 134	. . . . 5 |- {x | E.y(y <Q x /\ ( *Q ` y) e. ( 1st ` A)) } e. ~P Q.
19		opelxpi 4319	. . . . 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 402	. . . 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 2107	. . 3 |- B e. ( ~P Q. X. ~P Q.)
22	2, 21	jctil 295	. 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 6601	. . 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 6602	. . 3 |- (A e. P. -> A. q e. Q. -. ( q e. ( 1st ` B) /\ q e. ( 2nd ` B)))
25	1	recexprlemloc 6603	. . 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 1083	. 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 6459	. 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 394	1 |- (A e. P. -> B e. P.)

Syntax hints:  -. wn 3   -> wi 4   /\ wa 97   <-> wb 98   \/ wo 628   /\ w3a 884   = wceq 1242  E.wex 1378   e. wcel 1390   {cab 2023  A.wral 2300  E.wrex 2301   C_ wss 2911   ~Pcpw 3351   <.cop 3370   class class class wbr 3755   X. cxp 4286   ` cfv 4845   1stc1st 5707   2ndc2nd 5708   Q.cnq 6264   *Qcrq 6268   <Q cltq 6269   P.cnp 6275

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-13 1401  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-coll 3863  ax-sep 3866  ax-nul 3874  ax-pow 3918  ax-pr 3935  ax-un 4136  ax-setind 4220  ax-iinf 4254

This theorem depends on definitions:  df-bi 110  df-dc 742  df-3or 885  df-3an 886  df-tru 1245  df-fal 1248  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ne 2203  df-ral 2305  df-rex 2306  df-reu 2307  df-rab 2309  df-v 2553  df-sbc 2759  df-csb 2847  df-dif 2914  df-un 2916  df-in 2918  df-ss 2925  df-nul 3219  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-int 3607  df-iun 3650  df-br 3756  df-opab 3810  df-mpt 3811  df-tr 3846  df-eprel 4017  df-id 4021  df-po 4024  df-iso 4025  df-iord 4069  df-on 4071  df-suc 4074  df-iom 4257  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-res 4300  df-ima 4301  df-iota 4810  df-fun 4847  df-fn 4848  df-f 4849  df-f1 4850  df-fo 4851  df-f1o 4852  df-fv 4853  df-ov 5458  df-oprab 5459  df-mpt2 5460  df-1st 5709  df-2nd 5710  df-recs 5861  df-irdg 5897  df-1o 5940  df-oadd 5944  df-omul 5945  df-er 6042  df-ec 6044  df-qs 6048  df-ni 6288  df-pli 6289  df-mi 6290  df-lti 6291  df-plpq 6328  df-mpq 6329  df-enq 6331  df-nqqs 6332  df-plqqs 6333  df-mqqs 6334  df-1nqqs 6335  df-rq 6336  df-ltnqqs 6337  df-inp 6449

This theorem is referenced by:  recexprlem1ssl  6605  recexprlem1ssu  6606  recexprlemss1l  6607  recexprlemss1u  6608  recexprlemex  6609  recexpr  6610