Theorem cauappcvgprlem2 7601

Description: Lemma for cauappcvgpr 7603. 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 7348	. . . . 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 5621	. . . 4 |- ( ph -> ( F ` Q ) e. Q. )
7		ltanqi 7343	. . . 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 7356	. . 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 521	. . . 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 529	. . . . . 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 5486	. . . . . . . . 9 |- ( q = Q -> ( F ` q ) = ( F ` Q ) )
15		id 19	. . . . . . . . 9 |- ( q = Q -> q = Q )
16	14, 15	oveq12d 5860	. . . . . . . 8 |- ( q = Q -> ( ( F ` q ) +Q q ) = ( ( F ` Q ) +Q Q ) )
17	16	breq1d 3992	. . . . . . 7 |- ( q = Q -> ( ( ( F ` q ) +Q q ) <Q x <-> ( ( F ` Q ) +Q Q ) <Q x ) )
18	17	rspcev 2830	. . . . . 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 3986	. . . . . . 7 |- ( u = x -> ( ( ( F ` q ) +Q q ) <Q u <-> ( ( F ` q ) +Q q ) <Q x ) )
21	20	rexbidv 2467	. . . . . 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 5489	. . . . . . 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 7304	. . . . . . . . 9 |- Q. e. _V
25	24	rabex 4126	. . . . . . . 8 |- { l e. Q. | E. q e. Q. ( l +Q q ) <Q ( F ` q ) } e. _V
26	24	rabex 4126	. . . . . . . 8 |- { u e. Q. | E. q e. Q. ( ( F ` q ) +Q q ) <Q u } e. _V
27	25, 26	op2nd 6115	. . . . . . 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 2186	. . . . . 6 |- ( 2nd ` L ) = { u e. Q. | E. q e. Q. ( ( F ` q ) +Q q ) <Q u }
29	21, 28	elrab2 2885	. . . . 5 |- ( x e. ( 2nd ` L ) <-> ( x e. Q. /\ E. q e. Q. ( ( F ` q ) +Q q ) <Q x ) )
30	11, 19, 29	sylanbrc 414	. . . 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 530	. . . . . 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 2729	. . . . . . 7 |- x e. _V
33		breq1 3985	. . . . . . 7 |- ( l = x -> ( l <Q ( ( F ` Q ) +Q ( Q +Q R ) ) <-> x <Q ( ( F ` Q ) +Q ( Q +Q R ) ) ) )
34	32, 33	elab 2870	. . . . . 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 7490	. . . . . 6 |- { l | l <Q ( ( F ` Q ) +Q ( Q +Q R ) ) } e. _V
37		gtnqex 7491	. . . . . 6 |- { u | ( ( F ` Q ) +Q ( Q +Q R ) ) <Q u } e. _V
38	36, 37	op1st 6114	. . . . 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 2260	. . . 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 2515	. . . 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 1226	. . 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 7594	. . . . 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 7316	. . . . . . . 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 7316	. . . . . . 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 7488	. . . . . 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 7447	. . . 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 2588	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 1343   e. wcel 2136   {cab 2151   A.wral 2444   E.wrex 2445   {crab 2448   <.cop 3579   class class class wbr 3982   -->wf 5184   ` cfv 5188  (class class class)co 5842   1stc1st 6106   2ndc2nd 6107   Q.cnq 7221   +Q cplq 7223   <Q cltq 7226   P.cnp 7232   <P cltp 7236

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 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-coll 4097  ax-sep 4100  ax-nul 4108  ax-pow 4153  ax-pr 4187  ax-un 4411  ax-setind 4514  ax-iinf 4565

This theorem depends on definitions:  df-bi 116  df-dc 825  df-3or 969  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ne 2337  df-ral 2449  df-rex 2450  df-reu 2451  df-rab 2453  df-v 2728  df-sbc 2952  df-csb 3046  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-nul 3410  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-int 3825  df-iun 3868  df-br 3983  df-opab 4044  df-mpt 4045  df-tr 4081  df-eprel 4267  df-id 4271  df-po 4274  df-iso 4275  df-iord 4344  df-on 4346  df-suc 4349  df-iom 4568  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-res 4616  df-ima 4617  df-iota 5153  df-fun 5190  df-fn 5191  df-f 5192  df-f1 5193  df-fo 5194  df-f1o 5195  df-fv 5196  df-ov 5845  df-oprab 5846  df-mpo 5847  df-1st 6108  df-2nd 6109  df-recs 6273  df-irdg 6338  df-1o 6384  df-oadd 6388  df-omul 6389  df-er 6501  df-ec 6503  df-qs 6507  df-ni 7245  df-pli 7246  df-mi 7247  df-lti 7248  df-plpq 7285  df-mpq 7286  df-enq 7288  df-nqqs 7289  df-plqqs 7290  df-mqqs 7291  df-1nqqs 7292  df-rq 7293  df-ltnqqs 7294  df-inp 7407  df-iltp 7411

This theorem is referenced by:  cauappcvgprlemlim  7602