Theorem cauappcvgprlem2 7661

Description: Lemma for cauappcvgpr 7663. 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 7408	. . . . 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 5654	. . . 4 |- ( ph -> ( F ` Q ) e. Q. )
7		ltanqi 7403	. . . 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 7416	. . 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 5517	. . . . . . . . 9 |- ( q = Q -> ( F ` q ) = ( F ` Q ) )
15		id 19	. . . . . . . . 9 |- ( q = Q -> q = Q )
16	14, 15	oveq12d 5895	. . . . . . . 8 |- ( q = Q -> ( ( F ` q ) +Q q ) = ( ( F ` Q ) +Q Q ) )
17	16	breq1d 4015	. . . . . . 7 |- ( q = Q -> ( ( ( F ` q ) +Q q ) <Q x <-> ( ( F ` Q ) +Q Q ) <Q x ) )
18	17	rspcev 2843	. . . . . 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 4009	. . . . . . 7 |- ( u = x -> ( ( ( F ` q ) +Q q ) <Q u <-> ( ( F ` q ) +Q q ) <Q x ) )
21	20	rexbidv 2478	. . . . . 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 5520	. . . . . . 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 7364	. . . . . . . . 9 |- Q. e. _V
25	24	rabex 4149	. . . . . . . 8 |- { l e. Q. | E. q e. Q. ( l +Q q ) <Q ( F ` q ) } e. _V
26	24	rabex 4149	. . . . . . . 8 |- { u e. Q. | E. q e. Q. ( ( F ` q ) +Q q ) <Q u } e. _V
27	25, 26	op2nd 6150	. . . . . . 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 2198	. . . . . 6 |- ( 2nd ` L ) = { u e. Q. | E. q e. Q. ( ( F ` q ) +Q q ) <Q u }
29	21, 28	elrab2 2898	. . . . 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 2742	. . . . . . 7 |- x e. _V
33		breq1 4008	. . . . . . 7 |- ( l = x -> ( l <Q ( ( F ` Q ) +Q ( Q +Q R ) ) <-> x <Q ( ( F ` Q ) +Q ( Q +Q R ) ) ) )
34	32, 33	elab 2883	. . . . . 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 7550	. . . . . 6 |- { l | l <Q ( ( F ` Q ) +Q ( Q +Q R ) ) } e. _V
37		gtnqex 7551	. . . . . 6 |- { u | ( ( F ` Q ) +Q ( Q +Q R ) ) <Q u } e. _V
38	36, 37	op1st 6149	. . . . 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 2271	. . . 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 2526	. . . 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 1236	. . 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 7654	. . . . 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 7376	. . . . . . . 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 7376	. . . . . . 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 7548	. . . . . 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 7507	. . . 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 2599	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 1353   e. wcel 2148   {cab 2163   A.wral 2455   E.wrex 2456   {crab 2459   <.cop 3597   class class class wbr 4005   -->wf 5214   ` cfv 5218  (class class class)co 5877   1stc1st 6141   2ndc2nd 6142   Q.cnq 7281   +Q cplq 7283   <Q cltq 7286   P.cnp 7292   <P cltp 7296

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4120  ax-sep 4123  ax-nul 4131  ax-pow 4176  ax-pr 4211  ax-un 4435  ax-setind 4538  ax-iinf 4589

This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2741  df-sbc 2965  df-csb 3060  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-nul 3425  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-int 3847  df-iun 3890  df-br 4006  df-opab 4067  df-mpt 4068  df-tr 4104  df-eprel 4291  df-id 4295  df-po 4298  df-iso 4299  df-iord 4368  df-on 4370  df-suc 4373  df-iom 4592  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-ima 4641  df-iota 5180  df-fun 5220  df-fn 5221  df-f 5222  df-f1 5223  df-fo 5224  df-f1o 5225  df-fv 5226  df-ov 5880  df-oprab 5881  df-mpo 5882  df-1st 6143  df-2nd 6144  df-recs 6308  df-irdg 6373  df-1o 6419  df-oadd 6423  df-omul 6424  df-er 6537  df-ec 6539  df-qs 6543  df-ni 7305  df-pli 7306  df-mi 7307  df-lti 7308  df-plpq 7345  df-mpq 7346  df-enq 7348  df-nqqs 7349  df-plqqs 7350  df-mqqs 7351  df-1nqqs 7352  df-rq 7353  df-ltnqqs 7354  df-inp 7467  df-iltp 7471

This theorem is referenced by:  cauappcvgprlemlim  7662