Theorem caucvgprlem2 7679

Description: Lemma for caucvgpr 7681. 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 7664	. . . 4 |- ( ph -> ( *Q ` [ <. K , 1o >. ] ~Q ) <Q Q )
4		caucvgpr.f	. . . . 5 |- ( ph -> F : N. --> Q. )
5		ltrelpi 7323	. . . . . . . 8 |- <N C_ ( N. X. N. )
6	5	brel 4679	. . . . . . 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 114	. . . . 5 |- ( ph -> K e. N. )
9	4, 8	ffvelcdmd 5653	. . . 4 |- ( ph -> ( F ` K ) e. Q. )
10		ltanqi 7401	. . . 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 411	. . 3 |- ( ph -> ( ( F ` K ) +Q ( *Q ` [ <. K , 1o >. ] ~Q ) ) <Q ( ( F ` K ) +Q Q ) )
12		ltbtwnnqq 7414	. . 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 122	. 2 |- ( ph -> E. x e. Q. ( ( ( F ` K ) +Q ( *Q ` [ <. K , 1o >. ] ~Q ) ) <Q x /\ x <Q ( ( F ` K ) +Q Q ) ) )
14		simprl 529	. . . 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 276	. . . . . 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 539	. . . . . 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 5516	. . . . . . . . 9 |- ( j = K -> ( F ` j ) = ( F ` K ) )
18		opeq1 3779	. . . . . . . . . . 11 |- ( j = K -> <. j , 1o >. = <. K , 1o >. )
19	18	eceq1d 6571	. . . . . . . . . 10 |- ( j = K -> [ <. j , 1o >. ] ~Q = [ <. K , 1o >. ] ~Q )
20	19	fveq2d 5520	. . . . . . . . 9 |- ( j = K -> ( *Q ` [ <. j , 1o >. ] ~Q ) = ( *Q ` [ <. K , 1o >. ] ~Q ) )
21	17, 20	oveq12d 5893	. . . . . . . 8 |- ( j = K -> ( ( F ` j ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) = ( ( F ` K ) +Q ( *Q ` [ <. K , 1o >. ] ~Q ) ) )
22	21	breq1d 4014	. . . . . . 7 |- ( j = K -> ( ( ( F ` j ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q x <-> ( ( F ` K ) +Q ( *Q ` [ <. K , 1o >. ] ~Q ) ) <Q x ) )
23	22	rspcev 2842	. . . . . 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 411	. . . . 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 4008	. . . . . . 7 |- ( u = x -> ( ( ( F ` j ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q u <-> ( ( F ` j ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q x ) )
26	25	rexbidv 2478	. . . . . 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 5519	. . . . . . 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 7362	. . . . . . . . 9 |- Q. e. _V
30	29	rabex 4148	. . . . . . . 8 |- { l e. Q. | E. j e. N. ( l +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( F ` j ) } e. _V
31	29	rabex 4148	. . . . . . . 8 |- { u e. Q. | E. j e. N. ( ( F ` j ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q u } e. _V
32	30, 31	op2nd 6148	. . . . . . 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 2198	. . . . . 6 |- ( 2nd ` L ) = { u e. Q. | E. j e. N. ( ( F ` j ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q u }
34	26, 33	elrab2 2897	. . . . 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 417	. . . 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 540	. . . . . 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 2741	. . . . . . 7 |- x e. _V
38		breq1 4007	. . . . . . 7 |- ( l = x -> ( l <Q ( ( F ` K ) +Q Q ) <-> x <Q ( ( F ` K ) +Q Q ) ) )
39	37, 38	elab 2882	. . . . . 6 |- ( x e. { l | l <Q ( ( F ` K ) +Q Q ) } <-> x <Q ( ( F ` K ) +Q Q ) )
40	36, 39	sylibr 134	. . . . 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 7548	. . . . . 6 |- { l | l <Q ( ( F ` K ) +Q Q ) } e. _V
42		gtnqex 7549	. . . . . 6 |- { u | ( ( F ` K ) +Q Q ) <Q u } e. _V
43	41, 42	op1st 6147	. . . . 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 2271	. . . 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 2526	. . . 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 1236	. . 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 7675	. . . . 5 |- ( ph -> L e. P. )
50	49	adantr 276	. . . 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 7374	. . . . . . 7 |- ( ( ( F ` K ) e. Q. /\ Q e. Q. ) -> ( ( F ` K ) +Q Q ) e. Q. )
53	9, 51, 52	syl2anc 411	. . . . . 6 |- ( ph -> ( ( F ` K ) +Q Q ) e. Q. )
54		nqprlu 7546	. . . . . 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 276	. . . 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 7505	. . . 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 411	. . 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 167	. 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 2599	1 |- ( ph -> L <P <. { l | l <Q ( ( F ` K ) +Q Q ) } , { u | ( ( F ` K ) +Q Q ) <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 3596   class class class wbr 4004   -->wf 5213   ` cfv 5217  (class class class)co 5875   1stc1st 6139   2ndc2nd 6140   1oc1o 6410   [cec 6533   N.cnpi 7271   <N clti 7274   ~Q ceq 7278   Q.cnq 7279   +Q cplq 7281   *Qcrq 7283   <Q cltq 7284   P.cnp 7290   <P cltp 7294

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 4119  ax-sep 4122  ax-nul 4130  ax-pow 4175  ax-pr 4210  ax-un 4434  ax-setind 4537  ax-iinf 4588

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 2740  df-sbc 2964  df-csb 3059  df-dif 3132  df-un 3134  df-in 3136  df-ss 3143  df-nul 3424  df-pw 3578  df-sn 3599  df-pr 3600  df-op 3602  df-uni 3811  df-int 3846  df-iun 3889  df-br 4005  df-opab 4066  df-mpt 4067  df-tr 4103  df-eprel 4290  df-id 4294  df-po 4297  df-iso 4298  df-iord 4367  df-on 4369  df-suc 4372  df-iom 4591  df-xp 4633  df-rel 4634  df-cnv 4635  df-co 4636  df-dm 4637  df-rn 4638  df-res 4639  df-ima 4640  df-iota 5179  df-fun 5219  df-fn 5220  df-f 5221  df-f1 5222  df-fo 5223  df-f1o 5224  df-fv 5225  df-ov 5878  df-oprab 5879  df-mpo 5880  df-1st 6141  df-2nd 6142  df-recs 6306  df-irdg 6371  df-1o 6417  df-oadd 6421  df-omul 6422  df-er 6535  df-ec 6537  df-qs 6541  df-ni 7303  df-pli 7304  df-mi 7305  df-lti 7306  df-plpq 7343  df-mpq 7344  df-enq 7346  df-nqqs 7347  df-plqqs 7348  df-mqqs 7349  df-1nqqs 7350  df-rq 7351  df-ltnqqs 7352  df-inp 7465  df-iltp 7469

This theorem is referenced by:  caucvgprlemlim  7680