Theorem cauappcvgprlem2 7855

Description: Lemma for cauappcvgpr 7857. Part of showing the putative limit to be a limit. (Contributed by Jim Kingdon, 23-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 } >.
cauappcvgprlem.q	|- ( ph -> Q e. Q. )
cauappcvgprlem.r	|- ( ph -> R e. Q. )

Assertion

Ref	Expression
cauappcvgprlem2	|- ( ph -> L <P <. { l | l <Q ( ( F ` Q ) +Q ( Q +Q R ) ) } , { u | ( ( F ` Q ) +Q ( Q +Q R ) ) <Q u } >. )

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

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

Proof of Theorem cauappcvgprlem2

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		cauappcvgprlem.q	. . . . 5 |- ( ph -> Q e. Q. )
2		cauappcvgprlem.r	. . . . 5 |- ( ph -> R e. Q. )
3		ltaddnq 7602	. . . . 5 |- ( ( Q e. Q. /\ R e. Q. ) -> Q <Q ( Q +Q R ) )
4	1, 2, 3	syl2anc 411	. . . 4 |- ( ph -> Q <Q ( Q +Q R ) )
5		cauappcvgpr.f	. . . . 5 |- ( ph -> F : Q. --> Q. )
6	5, 1	ffvelcdmd 5773	. . . 4 |- ( ph -> ( F ` Q ) e. Q. )
7		ltanqi 7597	. . . 4 |- ( ( Q <Q ( Q +Q R ) /\ ( F ` Q ) e. Q. ) -> ( ( F ` Q ) +Q Q ) <Q ( ( F ` Q ) +Q ( Q +Q R ) ) )
8	4, 6, 7	syl2anc 411	. . 3 |- ( ph -> ( ( F ` Q ) +Q Q ) <Q ( ( F ` Q ) +Q ( Q +Q R ) ) )
9		ltbtwnnqq 7610	. . 3 |- ( ( ( F ` Q ) +Q Q ) <Q ( ( F ` Q ) +Q ( Q +Q R ) ) <-> E. x e. Q. ( ( ( F ` Q ) +Q Q ) <Q x /\ x <Q ( ( F ` Q ) +Q ( Q +Q R ) ) ) )
10	8, 9	sylib 122	. 2 |- ( ph -> E. x e. Q. ( ( ( F ` Q ) +Q Q ) <Q x /\ x <Q ( ( F ` Q ) +Q ( Q +Q R ) ) ) )
11		simprl 529	. . . 4 |- ( ( ph /\ ( x e. Q. /\ ( ( ( F ` Q ) +Q Q ) <Q x /\ x <Q ( ( F ` Q ) +Q ( Q +Q R ) ) ) ) ) -> x e. Q. )
12	1	adantr 276	. . . . . 6 |- ( ( ph /\ ( x e. Q. /\ ( ( ( F ` Q ) +Q Q ) <Q x /\ x <Q ( ( F ` Q ) +Q ( Q +Q R ) ) ) ) ) -> Q e. Q. )
13		simprrl 539	. . . . . 6 |- ( ( ph /\ ( x e. Q. /\ ( ( ( F ` Q ) +Q Q ) <Q x /\ x <Q ( ( F ` Q ) +Q ( Q +Q R ) ) ) ) ) -> ( ( F ` Q ) +Q Q ) <Q x )
14		fveq2 5629	. . . . . . . . 9 |- ( q = Q -> ( F ` q ) = ( F ` Q ) )
15		id 19	. . . . . . . . 9 |- ( q = Q -> q = Q )
16	14, 15	oveq12d 6025	. . . . . . . 8 |- ( q = Q -> ( ( F ` q ) +Q q ) = ( ( F ` Q ) +Q Q ) )
17	16	breq1d 4093	. . . . . . 7 |- ( q = Q -> ( ( ( F ` q ) +Q q ) <Q x <-> ( ( F ` Q ) +Q Q ) <Q x ) )
18	17	rspcev 2907	. . . . . 6 |- ( ( Q e. Q. /\ ( ( F ` Q ) +Q Q ) <Q x ) -> E. q e. Q. ( ( F ` q ) +Q q ) <Q x )
19	12, 13, 18	syl2anc 411	. . . . 5 |- ( ( ph /\ ( x e. Q. /\ ( ( ( F ` Q ) +Q Q ) <Q x /\ x <Q ( ( F ` Q ) +Q ( Q +Q R ) ) ) ) ) -> E. q e. Q. ( ( F ` q ) +Q q ) <Q x )
20		breq2 4087	. . . . . . 7 |- ( u = x -> ( ( ( F ` q ) +Q q ) <Q u <-> ( ( F ` q ) +Q q ) <Q x ) )
21	20	rexbidv 2531	. . . . . 6 |- ( u = x -> ( E. q e. Q. ( ( F ` q ) +Q q ) <Q u <-> E. q e. Q. ( ( F ` q ) +Q q ) <Q x ) )
22		cauappcvgpr.lim	. . . . . . . 8 |- 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 } >.
23	22	fveq2i 5632	. . . . . . 7 |- ( 2nd ` L ) = ( 2nd ` <. { 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 } >. )
24		nqex 7558	. . . . . . . . 9 |- Q. e. _V
25	24	rabex 4228	. . . . . . . 8 |- { l e. Q. | E. q e. Q. ( l +Q q ) <Q ( F ` q ) } e. _V
26	24	rabex 4228	. . . . . . . 8 |- { u e. Q. | E. q e. Q. ( ( F ` q ) +Q q ) <Q u } e. _V
27	25, 26	op2nd 6299	. . . . . . 7 |- ( 2nd ` <. { 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 } >. ) = { u e. Q. | E. q e. Q. ( ( F ` q ) +Q q ) <Q u }
28	23, 27	eqtri 2250	. . . . . 6 |- ( 2nd ` L ) = { u e. Q. | E. q e. Q. ( ( F ` q ) +Q q ) <Q u }
29	21, 28	elrab2 2962	. . . . 5 |- ( x e. ( 2nd ` L ) <-> ( x e. Q. /\ E. q e. Q. ( ( F ` q ) +Q q ) <Q x ) )
30	11, 19, 29	sylanbrc 417	. . . 4 |- ( ( ph /\ ( x e. Q. /\ ( ( ( F ` Q ) +Q Q ) <Q x /\ x <Q ( ( F ` Q ) +Q ( Q +Q R ) ) ) ) ) -> x e. ( 2nd ` L ) )
31		simprrr 540	. . . . . 6 |- ( ( ph /\ ( x e. Q. /\ ( ( ( F ` Q ) +Q Q ) <Q x /\ x <Q ( ( F ` Q ) +Q ( Q +Q R ) ) ) ) ) -> x <Q ( ( F ` Q ) +Q ( Q +Q R ) ) )
32		vex 2802	. . . . . . 7 |- x e. _V
33		breq1 4086	. . . . . . 7 |- ( l = x -> ( l <Q ( ( F ` Q ) +Q ( Q +Q R ) ) <-> x <Q ( ( F ` Q ) +Q ( Q +Q R ) ) ) )
34	32, 33	elab 2947	. . . . . 6 |- ( x e. { l | l <Q ( ( F ` Q ) +Q ( Q +Q R ) ) } <-> x <Q ( ( F ` Q ) +Q ( Q +Q R ) ) )
35	31, 34	sylibr 134	. . . . 5 |- ( ( ph /\ ( x e. Q. /\ ( ( ( F ` Q ) +Q Q ) <Q x /\ x <Q ( ( F ` Q ) +Q ( Q +Q R ) ) ) ) ) -> x e. { l | l <Q ( ( F ` Q ) +Q ( Q +Q R ) ) } )
36		ltnqex 7744	. . . . . 6 |- { l | l <Q ( ( F ` Q ) +Q ( Q +Q R ) ) } e. _V
37		gtnqex 7745	. . . . . 6 |- { u | ( ( F ` Q ) +Q ( Q +Q R ) ) <Q u } e. _V
38	36, 37	op1st 6298	. . . . 5 |- ( 1st ` <. { l | l <Q ( ( F ` Q ) +Q ( Q +Q R ) ) } , { u | ( ( F ` Q ) +Q ( Q +Q R ) ) <Q u } >. ) = { l | l <Q ( ( F ` Q ) +Q ( Q +Q R ) ) }
39	35, 38	eleqtrrdi 2323	. . . 4 |- ( ( ph /\ ( x e. Q. /\ ( ( ( F ` Q ) +Q Q ) <Q x /\ x <Q ( ( F ` Q ) +Q ( Q +Q R ) ) ) ) ) -> x e. ( 1st ` <. { l | l <Q ( ( F ` Q ) +Q ( Q +Q R ) ) } , { u | ( ( F ` Q ) +Q ( Q +Q R ) ) <Q u } >. ) )
40		rspe 2579	. . . 4 |- ( ( x e. Q. /\ ( x e. ( 2nd ` L ) /\ x e. ( 1st ` <. { l | l <Q ( ( F ` Q ) +Q ( Q +Q R ) ) } , { u | ( ( F ` Q ) +Q ( Q +Q R ) ) <Q u } >. ) ) ) -> E. x e. Q. ( x e. ( 2nd ` L ) /\ x e. ( 1st ` <. { l | l <Q ( ( F ` Q ) +Q ( Q +Q R ) ) } , { u | ( ( F ` Q ) +Q ( Q +Q R ) ) <Q u } >. ) ) )
41	11, 30, 39, 40	syl12anc 1269	. . 3 |- ( ( ph /\ ( x e. Q. /\ ( ( ( F ` Q ) +Q Q ) <Q x /\ x <Q ( ( F ` Q ) +Q ( Q +Q R ) ) ) ) ) -> E. x e. Q. ( x e. ( 2nd ` L ) /\ x e. ( 1st ` <. { l | l <Q ( ( F ` Q ) +Q ( Q +Q R ) ) } , { u | ( ( F ` Q ) +Q ( Q +Q R ) ) <Q u } >. ) ) )
42		cauappcvgpr.app	. . . . . 6 |- ( 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 ) ) ) )
43		cauappcvgpr.bnd	. . . . . 6 |- ( ph -> A. p e. Q. A <Q ( F ` p ) )
44	5, 42, 43, 22	cauappcvgprlemcl 7848	. . . . 5 |- ( ph -> L e. P. )
45	44	adantr 276	. . . 4 |- ( ( ph /\ ( x e. Q. /\ ( ( ( F ` Q ) +Q Q ) <Q x /\ x <Q ( ( F ` Q ) +Q ( Q +Q R ) ) ) ) ) -> L e. P. )
46		addclnq 7570	. . . . . . . 8 |- ( ( Q e. Q. /\ R e. Q. ) -> ( Q +Q R ) e. Q. )
47	1, 2, 46	syl2anc 411	. . . . . . 7 |- ( ph -> ( Q +Q R ) e. Q. )
48		addclnq 7570	. . . . . . 7 |- ( ( ( F ` Q ) e. Q. /\ ( Q +Q R ) e. Q. ) -> ( ( F ` Q ) +Q ( Q +Q R ) ) e. Q. )
49	6, 47, 48	syl2anc 411	. . . . . 6 |- ( ph -> ( ( F ` Q ) +Q ( Q +Q R ) ) e. Q. )
50		nqprlu 7742	. . . . . 6 |- ( ( ( F ` Q ) +Q ( Q +Q R ) ) e. Q. -> <. { l | l <Q ( ( F ` Q ) +Q ( Q +Q R ) ) } , { u | ( ( F ` Q ) +Q ( Q +Q R ) ) <Q u } >. e. P. )
51	49, 50	syl 14	. . . . 5 |- ( ph -> <. { l | l <Q ( ( F ` Q ) +Q ( Q +Q R ) ) } , { u | ( ( F ` Q ) +Q ( Q +Q R ) ) <Q u } >. e. P. )
52	51	adantr 276	. . . 4 |- ( ( ph /\ ( x e. Q. /\ ( ( ( F ` Q ) +Q Q ) <Q x /\ x <Q ( ( F ` Q ) +Q ( Q +Q R ) ) ) ) ) -> <. { l | l <Q ( ( F ` Q ) +Q ( Q +Q R ) ) } , { u | ( ( F ` Q ) +Q ( Q +Q R ) ) <Q u } >. e. P. )
53		ltdfpr 7701	. . . 4 |- ( ( L e. P. /\ <. { l | l <Q ( ( F ` Q ) +Q ( Q +Q R ) ) } , { u | ( ( F ` Q ) +Q ( Q +Q R ) ) <Q u } >. e. P. ) -> ( L <P <. { l | l <Q ( ( F ` Q ) +Q ( Q +Q R ) ) } , { u | ( ( F ` Q ) +Q ( Q +Q R ) ) <Q u } >. <-> E. x e. Q. ( x e. ( 2nd ` L ) /\ x e. ( 1st ` <. { l | l <Q ( ( F ` Q ) +Q ( Q +Q R ) ) } , { u | ( ( F ` Q ) +Q ( Q +Q R ) ) <Q u } >. ) ) ) )
54	45, 52, 53	syl2anc 411	. . 3 |- ( ( ph /\ ( x e. Q. /\ ( ( ( F ` Q ) +Q Q ) <Q x /\ x <Q ( ( F ` Q ) +Q ( Q +Q R ) ) ) ) ) -> ( L <P <. { l | l <Q ( ( F ` Q ) +Q ( Q +Q R ) ) } , { u | ( ( F ` Q ) +Q ( Q +Q R ) ) <Q u } >. <-> E. x e. Q. ( x e. ( 2nd ` L ) /\ x e. ( 1st ` <. { l | l <Q ( ( F ` Q ) +Q ( Q +Q R ) ) } , { u | ( ( F ` Q ) +Q ( Q +Q R ) ) <Q u } >. ) ) ) )
55	41, 54	mpbird 167	. 2 |- ( ( ph /\ ( x e. Q. /\ ( ( ( F ` Q ) +Q Q ) <Q x /\ x <Q ( ( F ` Q ) +Q ( Q +Q R ) ) ) ) ) -> L <P <. { l | l <Q ( ( F ` Q ) +Q ( Q +Q R ) ) } , { u | ( ( F ` Q ) +Q ( Q +Q R ) ) <Q u } >. )
56	10, 55	rexlimddv 2653	1 |- ( ph -> L <P <. { l | l <Q ( ( F ` Q ) +Q ( Q +Q R ) ) } , { u | ( ( F ` Q ) +Q ( Q +Q R ) ) <Q u } >. )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1395   e. wcel 2200   {cab 2215   A.wral 2508   E.wrex 2509   {crab 2512   <.cop 3669   class class class wbr 4083   -->wf 5314   ` cfv 5318  (class class class)co 6007   1stc1st 6290   2ndc2nd 6291   Q.cnq 7475   +Q cplq 7477   <Q cltq 7480   P.cnp 7486   <P cltp 7490

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 4199  ax-sep 4202  ax-nul 4210  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629  ax-iinf 4680

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 3889  df-int 3924  df-iun 3967  df-br 4084  df-opab 4146  df-mpt 4147  df-tr 4183  df-eprel 4380  df-id 4384  df-po 4387  df-iso 4388  df-iord 4457  df-on 4459  df-suc 4462  df-iom 4683  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-ima 4732  df-iota 5278  df-fun 5320  df-fn 5321  df-f 5322  df-f1 5323  df-fo 5324  df-f1o 5325  df-fv 5326  df-ov 6010  df-oprab 6011  df-mpo 6012  df-1st 6292  df-2nd 6293  df-recs 6457  df-irdg 6522  df-1o 6568  df-oadd 6572  df-omul 6573  df-er 6688  df-ec 6690  df-qs 6694  df-ni 7499  df-pli 7500  df-mi 7501  df-lti 7502  df-plpq 7539  df-mpq 7540  df-enq 7542  df-nqqs 7543  df-plqqs 7544  df-mqqs 7545  df-1nqqs 7546  df-rq 7547  df-ltnqqs 7548  df-inp 7661  df-iltp 7665

This theorem is referenced by:  cauappcvgprlemlim  7856