Theorem cauappcvgprlem2 7974

Description: Lemma for cauappcvgpr 7976. 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 7721	. . . . 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 5812	. . . 4 |- ( ph -> ( F ` Q ) e. Q. )
7		ltanqi 7716	. . . 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 7729	. . 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 531	. . . 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 541	. . . . . 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 5669	. . . . . . . . 9 |- ( q = Q -> ( F ` q ) = ( F ` Q ) )
15		id 19	. . . . . . . . 9 |- ( q = Q -> q = Q )
16	14, 15	oveq12d 6067	. . . . . . . 8 |- ( q = Q -> ( ( F ` q ) +Q q ) = ( ( F ` Q ) +Q Q ) )
17	16	breq1d 4118	. . . . . . 7 |- ( q = Q -> ( ( ( F ` q ) +Q q ) <Q x <-> ( ( F ` Q ) +Q Q ) <Q x ) )
18	17	rspcev 2920	. . . . . 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 4112	. . . . . . 7 |- ( u = x -> ( ( ( F ` q ) +Q q ) <Q u <-> ( ( F ` q ) +Q q ) <Q x ) )
21	20	rexbidv 2543	. . . . . 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 5672	. . . . . . 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 7677	. . . . . . . . 9 |- Q. e. _V
25	24	rabex 4255	. . . . . . . 8 |- { l e. Q. | E. q e. Q. ( l +Q q ) <Q ( F ` q ) } e. _V
26	24	rabex 4255	. . . . . . . 8 |- { u e. Q. | E. q e. Q. ( ( F ` q ) +Q q ) <Q u } e. _V
27	25, 26	op2nd 6340	. . . . . . 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 2253	. . . . . 6 |- ( 2nd ` L ) = { u e. Q. | E. q e. Q. ( ( F ` q ) +Q q ) <Q u }
29	21, 28	elrab2 2975	. . . . 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 542	. . . . . 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 2815	. . . . . . 7 |- x e. _V
33		breq1 4111	. . . . . . 7 |- ( l = x -> ( l <Q ( ( F ` Q ) +Q ( Q +Q R ) ) <-> x <Q ( ( F ` Q ) +Q ( Q +Q R ) ) ) )
34	32, 33	elab 2960	. . . . . 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 7863	. . . . . 6 |- { l | l <Q ( ( F ` Q ) +Q ( Q +Q R ) ) } e. _V
37		gtnqex 7864	. . . . . 6 |- { u | ( ( F ` Q ) +Q ( Q +Q R ) ) <Q u } e. _V
38	36, 37	op1st 6339	. . . . 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 2326	. . . 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 2591	. . . 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 1272	. . 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 7967	. . . . 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 7689	. . . . . . . 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 7689	. . . . . . 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 7861	. . . . . 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 7820	. . . 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 2665	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 1398   e. wcel 2203   {cab 2218   A.wral 2520   E.wrex 2521   {crab 2524   <.cop 3691   class class class wbr 4108   -->wf 5347   ` cfv 5351  (class class class)co 6049   1stc1st 6331   2ndc2nd 6332   Q.cnq 7594   +Q cplq 7596   <Q cltq 7599   P.cnp 7605   <P cltp 7609

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 2205  ax-14 2206  ax-ext 2214  ax-coll 4224  ax-sep 4227  ax-nul 4235  ax-pow 4286  ax-pr 4321  ax-un 4553  ax-setind 4658  ax-iinf 4709

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 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-ral 2525  df-rex 2526  df-reu 2527  df-rab 2529  df-v 2814  df-sbc 3042  df-csb 3138  df-dif 3212  df-un 3214  df-in 3216  df-ss 3223  df-nul 3508  df-pw 3670  df-sn 3694  df-pr 3695  df-op 3697  df-uni 3914  df-int 3949  df-iun 3992  df-br 4109  df-opab 4171  df-mpt 4172  df-tr 4208  df-eprel 4409  df-id 4413  df-po 4416  df-iso 4417  df-iord 4486  df-on 4488  df-suc 4491  df-iom 4712  df-xp 4754  df-rel 4755  df-cnv 4756  df-co 4757  df-dm 4758  df-rn 4759  df-res 4760  df-ima 4761  df-iota 5311  df-fun 5353  df-fn 5354  df-f 5355  df-f1 5356  df-fo 5357  df-f1o 5358  df-fv 5359  df-ov 6052  df-oprab 6053  df-mpo 6054  df-1st 6333  df-2nd 6334  df-recs 6535  df-irdg 6600  df-1o 6646  df-oadd 6650  df-omul 6651  df-er 6766  df-ec 6768  df-qs 6772  df-ni 7618  df-pli 7619  df-mi 7620  df-lti 7621  df-plpq 7658  df-mpq 7659  df-enq 7661  df-nqqs 7662  df-plqqs 7663  df-mqqs 7664  df-1nqqs 7665  df-rq 7666  df-ltnqqs 7667  df-inp 7780  df-iltp 7784

This theorem is referenced by:  cauappcvgprlemlim  7975