Theorem caucvgprlem2 7481

Description: Lemma for caucvgpr 7483. Part of showing the putative limit to be a limit. (Contributed by Jim Kingdon, 3-Oct-2020.)

Hypotheses

Ref	Expression
caucvgpr.f	|- ( ph -> F : N. --> Q. )
caucvgpr.cau	|- ( ph -> A. n e. N. A. k e. N. ( n <N k -> ( ( F ` n ) <Q ( ( F ` k ) +Q ( *Q ` [ <. n , 1o >. ] ~Q ) ) /\ ( F ` k ) <Q ( ( F ` n ) +Q ( *Q ` [ <. n , 1o >. ] ~Q ) ) ) ) )
caucvgpr.bnd	|- ( ph -> A. j e. N. A <Q ( F ` j ) )
caucvgpr.lim	|- L = <. { l e. Q. | E. j e. N. ( l +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( F ` j ) } , { u e. Q. | E. j e. N. ( ( F ` j ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q u } >.
caucvgprlemlim.q	|- ( ph -> Q e. Q. )
caucvgprlemlim.jk	|- ( ph -> J <N K )
caucvgprlemlim.jkq	|- ( ph -> ( *Q ` [ <. J , 1o >. ] ~Q ) <Q Q )

Assertion

Ref	Expression
caucvgprlem2	|- ( ph -> L <P <. { l | l <Q ( ( F ` K ) +Q Q ) } , { u | ( ( F ` K ) +Q Q ) <Q u } >. )

Distinct variable groups:   A, j   j, F, u, l   n, F, k   j, K, u, l   j, L, k   Q, l, u   j, l   j, k   k, n

Allowed substitution hints:   ph( u, j, k, n, l)   A( u, k, n, l)   Q( j, k, n)   J( u, j, k, n, l)   K( k, n)   L( u, n, l)

Proof of Theorem caucvgprlem2

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		caucvgprlemlim.jk	. . . . 5 |- ( ph -> J <N K )
2		caucvgprlemlim.jkq	. . . . 5 |- ( ph -> ( *Q ` [ <. J , 1o >. ] ~Q ) <Q Q )
3	1, 2	caucvgprlemk 7466	. . . 4 |- ( ph -> ( *Q ` [ <. K , 1o >. ] ~Q ) <Q Q )
4		caucvgpr.f	. . . . 5 |- ( ph -> F : N. --> Q. )
5		ltrelpi 7125	. . . . . . . 8 |- <N C_ ( N. X. N. )
6	5	brel 4586	. . . . . . 7 |- ( J <N K -> ( J e. N. /\ K e. N. ) )
7	1, 6	syl 14	. . . . . 6 |- ( ph -> ( J e. N. /\ K e. N. ) )
8	7	simprd 113	. . . . 5 |- ( ph -> K e. N. )
9	4, 8	ffvelrnd 5549	. . . 4 |- ( ph -> ( F ` K ) e. Q. )
10		ltanqi 7203	. . . 4 |- ( ( ( *Q ` [ <. K , 1o >. ] ~Q ) <Q Q /\ ( F ` K ) e. Q. ) -> ( ( F ` K ) +Q ( *Q ` [ <. K , 1o >. ] ~Q ) ) <Q ( ( F ` K ) +Q Q ) )
11	3, 9, 10	syl2anc 408	. . 3 |- ( ph -> ( ( F ` K ) +Q ( *Q ` [ <. K , 1o >. ] ~Q ) ) <Q ( ( F ` K ) +Q Q ) )
12		ltbtwnnqq 7216	. . 3 |- ( ( ( F ` K ) +Q ( *Q ` [ <. K , 1o >. ] ~Q ) ) <Q ( ( F ` K ) +Q Q ) <-> E. x e. Q. ( ( ( F ` K ) +Q ( *Q ` [ <. K , 1o >. ] ~Q ) ) <Q x /\ x <Q ( ( F ` K ) +Q Q ) ) )
13	11, 12	sylib 121	. 2 |- ( ph -> E. x e. Q. ( ( ( F ` K ) +Q ( *Q ` [ <. K , 1o >. ] ~Q ) ) <Q x /\ x <Q ( ( F ` K ) +Q Q ) ) )
14		simprl 520	. . . 4 |- ( ( ph /\ ( x e. Q. /\ ( ( ( F ` K ) +Q ( *Q ` [ <. K , 1o >. ] ~Q ) ) <Q x /\ x <Q ( ( F ` K ) +Q Q ) ) ) ) -> x e. Q. )
15	8	adantr 274	. . . . . 6 |- ( ( ph /\ ( x e. Q. /\ ( ( ( F ` K ) +Q ( *Q ` [ <. K , 1o >. ] ~Q ) ) <Q x /\ x <Q ( ( F ` K ) +Q Q ) ) ) ) -> K e. N. )
16		simprrl 528	. . . . . 6 |- ( ( ph /\ ( x e. Q. /\ ( ( ( F ` K ) +Q ( *Q ` [ <. K , 1o >. ] ~Q ) ) <Q x /\ x <Q ( ( F ` K ) +Q Q ) ) ) ) -> ( ( F ` K ) +Q ( *Q ` [ <. K , 1o >. ] ~Q ) ) <Q x )
17		fveq2 5414	. . . . . . . . 9 |- ( j = K -> ( F ` j ) = ( F ` K ) )
18		opeq1 3700	. . . . . . . . . . 11 |- ( j = K -> <. j , 1o >. = <. K , 1o >. )
19	18	eceq1d 6458	. . . . . . . . . 10 |- ( j = K -> [ <. j , 1o >. ] ~Q = [ <. K , 1o >. ] ~Q )
20	19	fveq2d 5418	. . . . . . . . 9 |- ( j = K -> ( *Q ` [ <. j , 1o >. ] ~Q ) = ( *Q ` [ <. K , 1o >. ] ~Q ) )
21	17, 20	oveq12d 5785	. . . . . . . 8 |- ( j = K -> ( ( F ` j ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) = ( ( F ` K ) +Q ( *Q ` [ <. K , 1o >. ] ~Q ) ) )
22	21	breq1d 3934	. . . . . . 7 |- ( j = K -> ( ( ( F ` j ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q x <-> ( ( F ` K ) +Q ( *Q ` [ <. K , 1o >. ] ~Q ) ) <Q x ) )
23	22	rspcev 2784	. . . . . 6 |- ( ( K e. N. /\ ( ( F ` K ) +Q ( *Q ` [ <. K , 1o >. ] ~Q ) ) <Q x ) -> E. j e. N. ( ( F ` j ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q x )
24	15, 16, 23	syl2anc 408	. . . . 5 |- ( ( ph /\ ( x e. Q. /\ ( ( ( F ` K ) +Q ( *Q ` [ <. K , 1o >. ] ~Q ) ) <Q x /\ x <Q ( ( F ` K ) +Q Q ) ) ) ) -> E. j e. N. ( ( F ` j ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q x )
25		breq2 3928	. . . . . . 7 |- ( u = x -> ( ( ( F ` j ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q u <-> ( ( F ` j ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q x ) )
26	25	rexbidv 2436	. . . . . 6 |- ( u = x -> ( E. j e. N. ( ( F ` j ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q u <-> E. j e. N. ( ( F ` j ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q x ) )
27		caucvgpr.lim	. . . . . . . 8 |- L = <. { l e. Q. | E. j e. N. ( l +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( F ` j ) } , { u e. Q. | E. j e. N. ( ( F ` j ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q u } >.
28	27	fveq2i 5417	. . . . . . 7 |- ( 2nd ` L ) = ( 2nd ` <. { l e. Q. | E. j e. N. ( l +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( F ` j ) } , { u e. Q. | E. j e. N. ( ( F ` j ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q u } >. )
29		nqex 7164	. . . . . . . . 9 |- Q. e. _V
30	29	rabex 4067	. . . . . . . 8 |- { l e. Q. | E. j e. N. ( l +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( F ` j ) } e. _V
31	29	rabex 4067	. . . . . . . 8 |- { u e. Q. | E. j e. N. ( ( F ` j ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q u } e. _V
32	30, 31	op2nd 6038	. . . . . . 7 |- ( 2nd ` <. { l e. Q. | E. j e. N. ( l +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( F ` j ) } , { u e. Q. | E. j e. N. ( ( F ` j ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q u } >. ) = { u e. Q. | E. j e. N. ( ( F ` j ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q u }
33	28, 32	eqtri 2158	. . . . . 6 |- ( 2nd ` L ) = { u e. Q. | E. j e. N. ( ( F ` j ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q u }
34	26, 33	elrab2 2838	. . . . 5 |- ( x e. ( 2nd ` L ) <-> ( x e. Q. /\ E. j e. N. ( ( F ` j ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q x ) )
35	14, 24, 34	sylanbrc 413	. . . 4 |- ( ( ph /\ ( x e. Q. /\ ( ( ( F ` K ) +Q ( *Q ` [ <. K , 1o >. ] ~Q ) ) <Q x /\ x <Q ( ( F ` K ) +Q Q ) ) ) ) -> x e. ( 2nd ` L ) )
36		simprrr 529	. . . . . 6 |- ( ( ph /\ ( x e. Q. /\ ( ( ( F ` K ) +Q ( *Q ` [ <. K , 1o >. ] ~Q ) ) <Q x /\ x <Q ( ( F ` K ) +Q Q ) ) ) ) -> x <Q ( ( F ` K ) +Q Q ) )
37		vex 2684	. . . . . . 7 |- x e. _V
38		breq1 3927	. . . . . . 7 |- ( l = x -> ( l <Q ( ( F ` K ) +Q Q ) <-> x <Q ( ( F ` K ) +Q Q ) ) )
39	37, 38	elab 2823	. . . . . 6 |- ( x e. { l | l <Q ( ( F ` K ) +Q Q ) } <-> x <Q ( ( F ` K ) +Q Q ) )
40	36, 39	sylibr 133	. . . . 5 |- ( ( ph /\ ( x e. Q. /\ ( ( ( F ` K ) +Q ( *Q ` [ <. K , 1o >. ] ~Q ) ) <Q x /\ x <Q ( ( F ` K ) +Q Q ) ) ) ) -> x e. { l | l <Q ( ( F ` K ) +Q Q ) } )
41		ltnqex 7350	. . . . . 6 |- { l | l <Q ( ( F ` K ) +Q Q ) } e. _V
42		gtnqex 7351	. . . . . 6 |- { u | ( ( F ` K ) +Q Q ) <Q u } e. _V
43	41, 42	op1st 6037	. . . . 5 |- ( 1st ` <. { l | l <Q ( ( F ` K ) +Q Q ) } , { u | ( ( F ` K ) +Q Q ) <Q u } >. ) = { l | l <Q ( ( F ` K ) +Q Q ) }
44	40, 43	eleqtrrdi 2231	. . . 4 |- ( ( ph /\ ( x e. Q. /\ ( ( ( F ` K ) +Q ( *Q ` [ <. K , 1o >. ] ~Q ) ) <Q x /\ x <Q ( ( F ` K ) +Q Q ) ) ) ) -> x e. ( 1st ` <. { l | l <Q ( ( F ` K ) +Q Q ) } , { u | ( ( F ` K ) +Q Q ) <Q u } >. ) )
45		rspe 2479	. . . 4 |- ( ( x e. Q. /\ ( x e. ( 2nd ` L ) /\ x e. ( 1st ` <. { l | l <Q ( ( F ` K ) +Q Q ) } , { u | ( ( F ` K ) +Q Q ) <Q u } >. ) ) ) -> E. x e. Q. ( x e. ( 2nd ` L ) /\ x e. ( 1st ` <. { l | l <Q ( ( F ` K ) +Q Q ) } , { u | ( ( F ` K ) +Q Q ) <Q u } >. ) ) )
46	14, 35, 44, 45	syl12anc 1214	. . 3 |- ( ( ph /\ ( x e. Q. /\ ( ( ( F ` K ) +Q ( *Q ` [ <. K , 1o >. ] ~Q ) ) <Q x /\ x <Q ( ( F ` K ) +Q Q ) ) ) ) -> E. x e. Q. ( x e. ( 2nd ` L ) /\ x e. ( 1st ` <. { l | l <Q ( ( F ` K ) +Q Q ) } , { u | ( ( F ` K ) +Q Q ) <Q u } >. ) ) )
47		caucvgpr.cau	. . . . . 6 |- ( ph -> A. n e. N. A. k e. N. ( n <N k -> ( ( F ` n ) <Q ( ( F ` k ) +Q ( *Q ` [ <. n , 1o >. ] ~Q ) ) /\ ( F ` k ) <Q ( ( F ` n ) +Q ( *Q ` [ <. n , 1o >. ] ~Q ) ) ) ) )
48		caucvgpr.bnd	. . . . . 6 |- ( ph -> A. j e. N. A <Q ( F ` j ) )
49	4, 47, 48, 27	caucvgprlemcl 7477	. . . . 5 |- ( ph -> L e. P. )
50	49	adantr 274	. . . 4 |- ( ( ph /\ ( x e. Q. /\ ( ( ( F ` K ) +Q ( *Q ` [ <. K , 1o >. ] ~Q ) ) <Q x /\ x <Q ( ( F ` K ) +Q Q ) ) ) ) -> L e. P. )
51		caucvgprlemlim.q	. . . . . . 7 |- ( ph -> Q e. Q. )
52		addclnq 7176	. . . . . . 7 |- ( ( ( F ` K ) e. Q. /\ Q e. Q. ) -> ( ( F ` K ) +Q Q ) e. Q. )
53	9, 51, 52	syl2anc 408	. . . . . 6 |- ( ph -> ( ( F ` K ) +Q Q ) e. Q. )
54		nqprlu 7348	. . . . . 6 |- ( ( ( F ` K ) +Q Q ) e. Q. -> <. { l | l <Q ( ( F ` K ) +Q Q ) } , { u | ( ( F ` K ) +Q Q ) <Q u } >. e. P. )
55	53, 54	syl 14	. . . . 5 |- ( ph -> <. { l | l <Q ( ( F ` K ) +Q Q ) } , { u | ( ( F ` K ) +Q Q ) <Q u } >. e. P. )
56	55	adantr 274	. . . 4 |- ( ( ph /\ ( x e. Q. /\ ( ( ( F ` K ) +Q ( *Q ` [ <. K , 1o >. ] ~Q ) ) <Q x /\ x <Q ( ( F ` K ) +Q Q ) ) ) ) -> <. { l | l <Q ( ( F ` K ) +Q Q ) } , { u | ( ( F ` K ) +Q Q ) <Q u } >. e. P. )
57		ltdfpr 7307	. . . 4 |- ( ( L e. P. /\ <. { l | l <Q ( ( F ` K ) +Q Q ) } , { u | ( ( F ` K ) +Q Q ) <Q u } >. e. P. ) -> ( L <P <. { l | l <Q ( ( F ` K ) +Q Q ) } , { u | ( ( F ` K ) +Q Q ) <Q u } >. <-> E. x e. Q. ( x e. ( 2nd ` L ) /\ x e. ( 1st ` <. { l | l <Q ( ( F ` K ) +Q Q ) } , { u | ( ( F ` K ) +Q Q ) <Q u } >. ) ) ) )
58	50, 56, 57	syl2anc 408	. . 3 |- ( ( ph /\ ( x e. Q. /\ ( ( ( F ` K ) +Q ( *Q ` [ <. K , 1o >. ] ~Q ) ) <Q x /\ x <Q ( ( F ` K ) +Q Q ) ) ) ) -> ( L <P <. { l | l <Q ( ( F ` K ) +Q Q ) } , { u | ( ( F ` K ) +Q Q ) <Q u } >. <-> E. x e. Q. ( x e. ( 2nd ` L ) /\ x e. ( 1st ` <. { l | l <Q ( ( F ` K ) +Q Q ) } , { u | ( ( F ` K ) +Q Q ) <Q u } >. ) ) ) )
59	46, 58	mpbird 166	. 2 |- ( ( ph /\ ( x e. Q. /\ ( ( ( F ` K ) +Q ( *Q ` [ <. K , 1o >. ] ~Q ) ) <Q x /\ x <Q ( ( F ` K ) +Q Q ) ) ) ) -> L <P <. { l | l <Q ( ( F ` K ) +Q Q ) } , { u | ( ( F ` K ) +Q Q ) <Q u } >. )
60	13, 59	rexlimddv 2552	1 |- ( ph -> L <P <. { l | l <Q ( ( F ` K ) +Q Q ) } , { u | ( ( F ` K ) +Q Q ) <Q u } >. )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1331   e. wcel 1480   {cab 2123   A.wral 2414   E.wrex 2415   {crab 2418   <.cop 3525   class class class wbr 3924   -->wf 5114   ` cfv 5118  (class class class)co 5767   1stc1st 6029   2ndc2nd 6030   1oc1o 6299   [cec 6420   N.cnpi 7073   <N clti 7076   ~Q ceq 7080   Q.cnq 7081   +Q cplq 7083   *Qcrq 7085   <Q cltq 7086   P.cnp 7092   <P cltp 7096

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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-coll 4038  ax-sep 4041  ax-nul 4049  ax-pow 4093  ax-pr 4126  ax-un 4350  ax-setind 4447  ax-iinf 4497

This theorem depends on definitions:  df-bi 116  df-dc 820  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ne 2307  df-ral 2419  df-rex 2420  df-reu 2421  df-rab 2423  df-v 2683  df-sbc 2905  df-csb 2999  df-dif 3068  df-un 3070  df-in 3072  df-ss 3079  df-nul 3359  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-uni 3732  df-int 3767  df-iun 3810  df-br 3925  df-opab 3985  df-mpt 3986  df-tr 4022  df-eprel 4206  df-id 4210  df-po 4213  df-iso 4214  df-iord 4283  df-on 4285  df-suc 4288  df-iom 4500  df-xp 4540  df-rel 4541  df-cnv 4542  df-co 4543  df-dm 4544  df-rn 4545  df-res 4546  df-ima 4547  df-iota 5083  df-fun 5120  df-fn 5121  df-f 5122  df-f1 5123  df-fo 5124  df-f1o 5125  df-fv 5126  df-ov 5770  df-oprab 5771  df-mpo 5772  df-1st 6031  df-2nd 6032  df-recs 6195  df-irdg 6260  df-1o 6306  df-oadd 6310  df-omul 6311  df-er 6422  df-ec 6424  df-qs 6428  df-ni 7105  df-pli 7106  df-mi 7107  df-lti 7108  df-plpq 7145  df-mpq 7146  df-enq 7148  df-nqqs 7149  df-plqqs 7150  df-mqqs 7151  df-1nqqs 7152  df-rq 7153  df-ltnqqs 7154  df-inp 7267  df-iltp 7271

This theorem is referenced by:  caucvgprlemlim  7482