Theorem recexprlempr 7699

Description: B is a positive real. Lemma for recexpr 7705. (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 7691	. . 3 |- ( A e. P. -> ( E. q e. Q. q e. ( 1st ` B ) /\ E. r e. Q. r e. ( 2nd ` B ) ) )
3		ltrelnq 7432	. . . . . . . . . . 11 |- <Q C_ ( Q. X. Q. )
4	3	brel 4715	. . . . . . . . . 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 1612	. . . . . . 7 |- ( E. y ( x <Q y /\ ( *Q ` y ) e. ( 2nd ` A ) ) -> x e. Q. )
8	7	abssi 3258	. . . . . 6 |- { x | E. y ( x <Q y /\ ( *Q ` y ) e. ( 2nd ` A ) ) } C_ Q.
9		nqex 7430	. . . . . . 7 |- Q. e. _V
10	9	elpw2 4190	. . . . . 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 4715	. . . . . . . . . 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 1612	. . . . . . 7 |- ( E. y ( y <Q x /\ ( *Q ` y ) e. ( 1st ` A ) ) -> x e. Q. )
16	15	abssi 3258	. . . . . 6 |- { x | E. y ( y <Q x /\ ( *Q ` y ) e. ( 1st ` A ) ) } C_ Q.
17	9	elpw2 4190	. . . . . 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 4695	. . . . 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 2269	. . 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 7696	. . 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 7697	. . 3 |- ( A e. P. -> A. q e. Q. -. ( q e. ( 1st ` B ) /\ q e. ( 2nd ` B ) ) )
25	1	recexprlemloc 7698	. . 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 1179	. 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 7543	. 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 709   /\ w3a 980   = wceq 1364   E.wex 1506   e. wcel 2167   {cab 2182   A.wral 2475   E.wrex 2476   C_ wss 3157   ~Pcpw 3605   <.cop 3625   class class class wbr 4033   X. cxp 4661   ` cfv 5258   1stc1st 6196   2ndc2nd 6197   Q.cnq 7347   *Qcrq 7351   <Q cltq 7352   P.cnp 7358

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 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-coll 4148  ax-sep 4151  ax-nul 4159  ax-pow 4207  ax-pr 4242  ax-un 4468  ax-setind 4573  ax-iinf 4624

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-ral 2480  df-rex 2481  df-reu 2482  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3451  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-int 3875  df-iun 3918  df-br 4034  df-opab 4095  df-mpt 4096  df-tr 4132  df-eprel 4324  df-id 4328  df-po 4331  df-iso 4332  df-iord 4401  df-on 4403  df-suc 4406  df-iom 4627  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-res 4675  df-ima 4676  df-iota 5219  df-fun 5260  df-fn 5261  df-f 5262  df-f1 5263  df-fo 5264  df-f1o 5265  df-fv 5266  df-ov 5925  df-oprab 5926  df-mpo 5927  df-1st 6198  df-2nd 6199  df-recs 6363  df-irdg 6428  df-1o 6474  df-oadd 6478  df-omul 6479  df-er 6592  df-ec 6594  df-qs 6598  df-ni 7371  df-pli 7372  df-mi 7373  df-lti 7374  df-plpq 7411  df-mpq 7412  df-enq 7414  df-nqqs 7415  df-plqqs 7416  df-mqqs 7417  df-1nqqs 7418  df-rq 7419  df-ltnqqs 7420  df-inp 7533

This theorem is referenced by:  recexprlem1ssl  7700  recexprlem1ssu  7701  recexprlemss1l  7702  recexprlemss1u  7703  recexprlemex  7704  recexpr  7705