Theorem cauappcvgprlemcl 7828

Description: Lemma for cauappcvgpr 7837. The putative limit is a positive real. (Contributed by Jim Kingdon, 20-Jun-2020.)

Hypotheses

Ref	Expression
cauappcvgpr.f	|- ( ph -> F : Q. --> Q. )
cauappcvgpr.app	|- ( ph -> A. p e. Q. A. q e. Q. ( ( F ` p ) <Q ( ( F ` q ) +Q ( p +Q q ) ) /\ ( F ` q ) <Q ( ( F ` p ) +Q ( p +Q q ) ) ) )
cauappcvgpr.bnd	|- ( ph -> A. p e. Q. A <Q ( F ` p ) )
cauappcvgpr.lim	|- L = <. { l e. Q. | E. q e. Q. ( l +Q q ) <Q ( F ` q ) } , { u e. Q. | E. q e. Q. ( ( F ` q ) +Q q ) <Q u } >.

Assertion

Ref	Expression
cauappcvgprlemcl	|- ( ph -> L e. P. )

Distinct variable groups:   A, p   L, p, q   ph, p, q   F, l, u, p, q

Allowed substitution hints:   ph( u, l)   A( u, q, l)   L( u, l)

Proof of Theorem cauappcvgprlemcl

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

Step	Hyp	Ref	Expression
1		cauappcvgpr.f	. . . 4 |- ( ph -> F : Q. --> Q. )
2		cauappcvgpr.app	. . . 4 |- ( ph -> A. p e. Q. A. q e. Q. ( ( F ` p ) <Q ( ( F ` q ) +Q ( p +Q q ) ) /\ ( F ` q ) <Q ( ( F ` p ) +Q ( p +Q q ) ) ) )
3		cauappcvgpr.bnd	. . . 4 |- ( ph -> A. p e. Q. A <Q ( F ` p ) )
4		cauappcvgpr.lim	. . . 4 |- L = <. { l e. Q. | E. q e. Q. ( l +Q q ) <Q ( F ` q ) } , { u e. Q. | E. q e. Q. ( ( F ` q ) +Q q ) <Q u } >.
5	1, 2, 3, 4	cauappcvgprlemm 7820	. . 3 |- ( ph -> ( E. s e. Q. s e. ( 1st ` L ) /\ E. r e. Q. r e. ( 2nd ` L ) ) )
6		ssrab2 3309	. . . . . 6 |- { l e. Q. | E. q e. Q. ( l +Q q ) <Q ( F ` q ) } C_ Q.
7		nqex 7538	. . . . . . 7 |- Q. e. _V
8	7	elpw2 4240	. . . . . 6 |- ( { l e. Q. | E. q e. Q. ( l +Q q ) <Q ( F ` q ) } e. ~P Q. <-> { l e. Q. | E. q e. Q. ( l +Q q ) <Q ( F ` q ) } C_ Q. )
9	6, 8	mpbir 146	. . . . 5 |- { l e. Q. | E. q e. Q. ( l +Q q ) <Q ( F ` q ) } e. ~P Q.
10		ssrab2 3309	. . . . . 6 |- { u e. Q. | E. q e. Q. ( ( F ` q ) +Q q ) <Q u } C_ Q.
11	7	elpw2 4240	. . . . . 6 |- ( { u e. Q. | E. q e. Q. ( ( F ` q ) +Q q ) <Q u } e. ~P Q. <-> { u e. Q. | E. q e. Q. ( ( F ` q ) +Q q ) <Q u } C_ Q. )
12	10, 11	mpbir 146	. . . . 5 |- { u e. Q. | E. q e. Q. ( ( F ` q ) +Q q ) <Q u } e. ~P Q.
13		opelxpi 4748	. . . . 5 |- ( ( { l e. Q. | E. q e. Q. ( l +Q q ) <Q ( F ` q ) } e. ~P Q. /\ { u e. Q. | E. q e. Q. ( ( F ` q ) +Q q ) <Q u } e. ~P Q. ) -> <. { l e. Q. | E. q e. Q. ( l +Q q ) <Q ( F ` q ) } , { u e. Q. | E. q e. Q. ( ( F ` q ) +Q q ) <Q u } >. e. ( ~P Q. X. ~P Q. ) )
14	9, 12, 13	mp2an 426	. . . 4 |- <. { l e. Q. | E. q e. Q. ( l +Q q ) <Q ( F ` q ) } , { u e. Q. | E. q e. Q. ( ( F ` q ) +Q q ) <Q u } >. e. ( ~P Q. X. ~P Q. )
15	4, 14	eqeltri 2302	. . 3 |- L e. ( ~P Q. X. ~P Q. )
16	5, 15	jctil 312	. 2 |- ( ph -> ( L e. ( ~P Q. X. ~P Q. ) /\ ( E. s e. Q. s e. ( 1st ` L ) /\ E. r e. Q. r e. ( 2nd ` L ) ) ) )
17	1, 2, 3, 4	cauappcvgprlemrnd 7825	. . 3 |- ( ph -> ( A. s e. Q. ( s e. ( 1st ` L ) <-> E. r e. Q. ( s <Q r /\ r e. ( 1st ` L ) ) ) /\ A. r e. Q. ( r e. ( 2nd ` L ) <-> E. s e. Q. ( s <Q r /\ s e. ( 2nd ` L ) ) ) ) )
18	1, 2, 3, 4	cauappcvgprlemdisj 7826	. . 3 |- ( ph -> A. s e. Q. -. ( s e. ( 1st ` L ) /\ s e. ( 2nd ` L ) ) )
19	1, 2, 3, 4	cauappcvgprlemloc 7827	. . 3 |- ( ph -> A. s e. Q. A. r e. Q. ( s <Q r -> ( s e. ( 1st ` L ) \/ r e. ( 2nd ` L ) ) ) )
20	17, 18, 19	3jca 1201	. 2 |- ( ph -> ( ( A. s e. Q. ( s e. ( 1st ` L ) <-> E. r e. Q. ( s <Q r /\ r e. ( 1st ` L ) ) ) /\ A. r e. Q. ( r e. ( 2nd ` L ) <-> E. s e. Q. ( s <Q r /\ s e. ( 2nd ` L ) ) ) ) /\ A. s e. Q. -. ( s e. ( 1st ` L ) /\ s e. ( 2nd ` L ) ) /\ A. s e. Q. A. r e. Q. ( s <Q r -> ( s e. ( 1st ` L ) \/ r e. ( 2nd ` L ) ) ) ) )
21		elnp1st2nd 7651	. 2 |- ( L e. P. <-> ( ( L e. ( ~P Q. X. ~P Q. ) /\ ( E. s e. Q. s e. ( 1st ` L ) /\ E. r e. Q. r e. ( 2nd ` L ) ) ) /\ ( ( A. s e. Q. ( s e. ( 1st ` L ) <-> E. r e. Q. ( s <Q r /\ r e. ( 1st ` L ) ) ) /\ A. r e. Q. ( r e. ( 2nd ` L ) <-> E. s e. Q. ( s <Q r /\ s e. ( 2nd ` L ) ) ) ) /\ A. s e. Q. -. ( s e. ( 1st ` L ) /\ s e. ( 2nd ` L ) ) /\ A. s e. Q. A. r e. Q. ( s <Q r -> ( s e. ( 1st ` L ) \/ r e. ( 2nd ` L ) ) ) ) ) )
22	16, 20, 21	sylanbrc 417	1 |- ( ph -> L e. P. )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 713   /\ w3a 1002   = wceq 1395   e. wcel 2200   A.wral 2508   E.wrex 2509   {crab 2512   C_ wss 3197   ~Pcpw 3649   <.cop 3669   class class class wbr 4082   X. cxp 4714   -->wf 5310   ` cfv 5314  (class class class)co 5994   1stc1st 6274   2ndc2nd 6275   Q.cnq 7455   +Q cplq 7457   <Q cltq 7460   P.cnp 7466

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4198  ax-sep 4201  ax-nul 4209  ax-pow 4257  ax-pr 4292  ax-un 4521  ax-setind 4626  ax-iinf 4677

This theorem depends on definitions:  df-bi 117  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3888  df-int 3923  df-iun 3966  df-br 4083  df-opab 4145  df-mpt 4146  df-tr 4182  df-eprel 4377  df-id 4381  df-po 4384  df-iso 4385  df-iord 4454  df-on 4456  df-suc 4459  df-iom 4680  df-xp 4722  df-rel 4723  df-cnv 4724  df-co 4725  df-dm 4726  df-rn 4727  df-res 4728  df-ima 4729  df-iota 5274  df-fun 5316  df-fn 5317  df-f 5318  df-f1 5319  df-fo 5320  df-f1o 5321  df-fv 5322  df-ov 5997  df-oprab 5998  df-mpo 5999  df-1st 6276  df-2nd 6277  df-recs 6441  df-irdg 6506  df-1o 6552  df-oadd 6556  df-omul 6557  df-er 6670  df-ec 6672  df-qs 6676  df-ni 7479  df-pli 7480  df-mi 7481  df-lti 7482  df-plpq 7519  df-mpq 7520  df-enq 7522  df-nqqs 7523  df-plqqs 7524  df-mqqs 7525  df-1nqqs 7526  df-rq 7527  df-ltnqqs 7528  df-inp 7641

This theorem is referenced by:  cauappcvgprlemladdfu  7829  cauappcvgprlemladdfl  7830  cauappcvgprlemladdru  7831  cauappcvgprlemladdrl  7832  cauappcvgprlemladd  7833  cauappcvgprlem1  7834  cauappcvgprlem2  7835  cauappcvgpr  7837