Theorem cauappcvgprlem2 7622

Description: Lemma for cauappcvgpr 7624. 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 7369	. . . . 5 |- ( ( Q e. Q. /\ R e. Q. ) -> Q <Q ( Q +Q R ) )
4	1, 2, 3	syl2anc 409	. . . 4 |- ( ph -> Q <Q ( Q +Q R ) )
5		cauappcvgpr.f	. . . . 5 |- ( ph -> F : Q. --> Q. )
6	5, 1	ffvelrnd 5632	. . . 4 |- ( ph -> ( F ` Q ) e. Q. )
7		ltanqi 7364	. . . 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 409	. . 3 |- ( ph -> ( ( F ` Q ) +Q Q ) <Q ( ( F ` Q ) +Q ( Q +Q R ) ) )
9		ltbtwnnqq 7377	. . 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 121	. 2 |- ( ph -> E. x e. Q. ( ( ( F ` Q ) +Q Q ) <Q x /\ x <Q ( ( F ` Q ) +Q ( Q +Q R ) ) ) )
11		simprl 526	. . . 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 274	. . . . . 6 |- ( ( ph /\ ( x e. Q. /\ ( ( ( F ` Q ) +Q Q ) <Q x /\ x <Q ( ( F ` Q ) +Q ( Q +Q R ) ) ) ) ) -> Q e. Q. )
13		simprrl 534	. . . . . 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 5496	. . . . . . . . 9 |- ( q = Q -> ( F ` q ) = ( F ` Q ) )
15		id 19	. . . . . . . . 9 |- ( q = Q -> q = Q )
16	14, 15	oveq12d 5871	. . . . . . . 8 |- ( q = Q -> ( ( F ` q ) +Q q ) = ( ( F ` Q ) +Q Q ) )
17	16	breq1d 3999	. . . . . . 7 |- ( q = Q -> ( ( ( F ` q ) +Q q ) <Q x <-> ( ( F ` Q ) +Q Q ) <Q x ) )
18	17	rspcev 2834	. . . . . 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 409	. . . . 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 3993	. . . . . . 7 |- ( u = x -> ( ( ( F ` q ) +Q q ) <Q u <-> ( ( F ` q ) +Q q ) <Q x ) )
21	20	rexbidv 2471	. . . . . 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 5499	. . . . . . 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 7325	. . . . . . . . 9 |- Q. e. _V
25	24	rabex 4133	. . . . . . . 8 |- { l e. Q. | E. q e. Q. ( l +Q q ) <Q ( F ` q ) } e. _V
26	24	rabex 4133	. . . . . . . 8 |- { u e. Q. | E. q e. Q. ( ( F ` q ) +Q q ) <Q u } e. _V
27	25, 26	op2nd 6126	. . . . . . 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 2191	. . . . . 6 |- ( 2nd ` L ) = { u e. Q. | E. q e. Q. ( ( F ` q ) +Q q ) <Q u }
29	21, 28	elrab2 2889	. . . . 5 |- ( x e. ( 2nd ` L ) <-> ( x e. Q. /\ E. q e. Q. ( ( F ` q ) +Q q ) <Q x ) )
30	11, 19, 29	sylanbrc 415	. . . 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 535	. . . . . 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 2733	. . . . . . 7 |- x e. _V
33		breq1 3992	. . . . . . 7 |- ( l = x -> ( l <Q ( ( F ` Q ) +Q ( Q +Q R ) ) <-> x <Q ( ( F ` Q ) +Q ( Q +Q R ) ) ) )
34	32, 33	elab 2874	. . . . . 6 |- ( x e. { l | l <Q ( ( F ` Q ) +Q ( Q +Q R ) ) } <-> x <Q ( ( F ` Q ) +Q ( Q +Q R ) ) )
35	31, 34	sylibr 133	. . . . 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 7511	. . . . . 6 |- { l | l <Q ( ( F ` Q ) +Q ( Q +Q R ) ) } e. _V
37		gtnqex 7512	. . . . . 6 |- { u | ( ( F ` Q ) +Q ( Q +Q R ) ) <Q u } e. _V
38	36, 37	op1st 6125	. . . . 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 2264	. . . 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 2519	. . . 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 1231	. . 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 7615	. . . . 5 |- ( ph -> L e. P. )
45	44	adantr 274	. . . 4 |- ( ( ph /\ ( x e. Q. /\ ( ( ( F ` Q ) +Q Q ) <Q x /\ x <Q ( ( F ` Q ) +Q ( Q +Q R ) ) ) ) ) -> L e. P. )
46		addclnq 7337	. . . . . . . 8 |- ( ( Q e. Q. /\ R e. Q. ) -> ( Q +Q R ) e. Q. )
47	1, 2, 46	syl2anc 409	. . . . . . 7 |- ( ph -> ( Q +Q R ) e. Q. )
48		addclnq 7337	. . . . . . 7 |- ( ( ( F ` Q ) e. Q. /\ ( Q +Q R ) e. Q. ) -> ( ( F ` Q ) +Q ( Q +Q R ) ) e. Q. )
49	6, 47, 48	syl2anc 409	. . . . . 6 |- ( ph -> ( ( F ` Q ) +Q ( Q +Q R ) ) e. Q. )
50		nqprlu 7509	. . . . . 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 274	. . . 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 7468	. . . 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 409	. . 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 166	. 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 2592	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 103   <-> wb 104   = wceq 1348   e. wcel 2141   {cab 2156   A.wral 2448   E.wrex 2449   {crab 2452   <.cop 3586   class class class wbr 3989   -->wf 5194   ` cfv 5198  (class class class)co 5853   1stc1st 6117   2ndc2nd 6118   Q.cnq 7242   +Q cplq 7244   <Q cltq 7247   P.cnp 7253   <P cltp 7257

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-coll 4104  ax-sep 4107  ax-nul 4115  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-iinf 4572

This theorem depends on definitions:  df-bi 116  df-dc 830  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-ral 2453  df-rex 2454  df-reu 2455  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-int 3832  df-iun 3875  df-br 3990  df-opab 4051  df-mpt 4052  df-tr 4088  df-eprel 4274  df-id 4278  df-po 4281  df-iso 4282  df-iord 4351  df-on 4353  df-suc 4356  df-iom 4575  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-fo 5204  df-f1o 5205  df-fv 5206  df-ov 5856  df-oprab 5857  df-mpo 5858  df-1st 6119  df-2nd 6120  df-recs 6284  df-irdg 6349  df-1o 6395  df-oadd 6399  df-omul 6400  df-er 6513  df-ec 6515  df-qs 6519  df-ni 7266  df-pli 7267  df-mi 7268  df-lti 7269  df-plpq 7306  df-mpq 7307  df-enq 7309  df-nqqs 7310  df-plqqs 7311  df-mqqs 7312  df-1nqqs 7313  df-rq 7314  df-ltnqqs 7315  df-inp 7428  df-iltp 7432

This theorem is referenced by:  cauappcvgprlemlim  7623