Theorem cauappcvgprlemcl 7984

Description: Lemma for cauappcvgpr 7993. 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 7976	. . 3 |- ( ph -> ( E. s e. Q. s e. ( 1st ` L ) /\ E. r e. Q. r e. ( 2nd ` L ) ) )
6		ssrab2 3327	. . . . . 6 |- { l e. Q. | E. q e. Q. ( l +Q q ) <Q ( F ` q ) } C_ Q.
7		nqex 7694	. . . . . . 7 |- Q. e. _V
8	7	elpw2 4274	. . . . . 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 3327	. . . . . 6 |- { u e. Q. | E. q e. Q. ( ( F ` q ) +Q q ) <Q u } C_ Q.
11	7	elpw2 4274	. . . . . 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 4786	. . . . 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 2307	. . 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 7981	. . 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 7982	. . 3 |- ( ph -> A. s e. Q. -. ( s e. ( 1st ` L ) /\ s e. ( 2nd ` L ) ) )
19	1, 2, 3, 4	cauappcvgprlemloc 7983	. . 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 1204	. 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 7807	. 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 716   /\ w3a 1005   = wceq 1398   e. wcel 2205   A.wral 2522   E.wrex 2523   {crab 2526   C_ wss 3214   ~Pcpw 3674   <.cop 3697   class class class wbr 4114   X. cxp 4752   -->wf 5353   ` cfv 5357  (class class class)co 6058   1stc1st 6345   2ndc2nd 6346   Q.cnq 7611   +Q cplq 7613   <Q cltq 7616   P.cnp 7622

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 2207  ax-14 2208  ax-ext 2216  ax-coll 4230  ax-sep 4233  ax-nul 4241  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664  ax-iinf 4715

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 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-ral 2527  df-rex 2528  df-reu 2529  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-nul 3513  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-int 3955  df-iun 3998  df-br 4115  df-opab 4177  df-mpt 4178  df-tr 4214  df-eprel 4415  df-id 4419  df-po 4422  df-iso 4423  df-iord 4492  df-on 4494  df-suc 4497  df-iom 4718  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-f1 5362  df-fo 5363  df-f1o 5364  df-fv 5365  df-ov 6061  df-oprab 6062  df-mpo 6063  df-1st 6347  df-2nd 6348  df-recs 6549  df-irdg 6614  df-1o 6660  df-oadd 6664  df-omul 6665  df-er 6780  df-ec 6782  df-qs 6786  df-ni 7635  df-pli 7636  df-mi 7637  df-lti 7638  df-plpq 7675  df-mpq 7676  df-enq 7678  df-nqqs 7679  df-plqqs 7680  df-mqqs 7681  df-1nqqs 7682  df-rq 7683  df-ltnqqs 7684  df-inp 7797

This theorem is referenced by:  cauappcvgprlemladdfu  7985  cauappcvgprlemladdfl  7986  cauappcvgprlemladdru  7987  cauappcvgprlemladdrl  7988  cauappcvgprlemladd  7989  cauappcvgprlem1  7990  cauappcvgprlem2  7991  cauappcvgpr  7993