Theorem caucvgprlem1 7641

Description: Lemma for caucvgpr 7644. 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
caucvgprlem1	|- ( ph -> <. { l | l <Q ( F ` K ) } , { u | ( F ` K ) <Q u } >. <P ( L +P. <. { l | l <Q Q } , { u | Q <Q u } >. ) )

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

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

Proof of Theorem caucvgprlem1

Step	Hyp	Ref	Expression
1		caucvgprlemlim.jk	. . . . . 6 |- ( ph -> J <N K )
2		ltrelpi 7286	. . . . . . 7 |- <N C_ ( N. X. N. )
3	2	brel 4663	. . . . . 6 |- ( J <N K -> ( J e. N. /\ K e. N. ) )
4	1, 3	syl 14	. . . . 5 |- ( ph -> ( J e. N. /\ K e. N. ) )
5	4	simprd 113	. . . 4 |- ( ph -> K e. N. )
6		caucvgprlemlim.jkq	. . . . . 6 |- ( ph -> ( *Q ` [ <. J , 1o >. ] ~Q ) <Q Q )
7	1, 6	caucvgprlemk 7627	. . . . 5 |- ( ph -> ( *Q ` [ <. K , 1o >. ] ~Q ) <Q Q )
8		caucvgpr.f	. . . . . 6 |- ( ph -> F : N. --> Q. )
9	8, 5	ffvelrnd 5632	. . . . 5 |- ( ph -> ( F ` K ) e. Q. )
10		ltanqi 7364	. . . . 5 |- ( ( ( *Q ` [ <. K , 1o >. ] ~Q ) <Q Q /\ ( F ` K ) e. Q. ) -> ( ( F ` K ) +Q ( *Q ` [ <. K , 1o >. ] ~Q ) ) <Q ( ( F ` K ) +Q Q ) )
11	7, 9, 10	syl2anc 409	. . . 4 |- ( ph -> ( ( F ` K ) +Q ( *Q ` [ <. K , 1o >. ] ~Q ) ) <Q ( ( F ` K ) +Q Q ) )
12		opeq1 3765	. . . . . . . . 9 |- ( j = K -> <. j , 1o >. = <. K , 1o >. )
13	12	eceq1d 6549	. . . . . . . 8 |- ( j = K -> [ <. j , 1o >. ] ~Q = [ <. K , 1o >. ] ~Q )
14	13	fveq2d 5500	. . . . . . 7 |- ( j = K -> ( *Q ` [ <. j , 1o >. ] ~Q ) = ( *Q ` [ <. K , 1o >. ] ~Q ) )
15	14	oveq2d 5869	. . . . . 6 |- ( j = K -> ( ( F ` K ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) = ( ( F ` K ) +Q ( *Q ` [ <. K , 1o >. ] ~Q ) ) )
16		fveq2 5496	. . . . . . 7 |- ( j = K -> ( F ` j ) = ( F ` K ) )
17	16	oveq1d 5868	. . . . . 6 |- ( j = K -> ( ( F ` j ) +Q Q ) = ( ( F ` K ) +Q Q ) )
18	15, 17	breq12d 4002	. . . . 5 |- ( j = K -> ( ( ( F ` K ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( ( F ` j ) +Q Q ) <-> ( ( F ` K ) +Q ( *Q ` [ <. K , 1o >. ] ~Q ) ) <Q ( ( F ` K ) +Q Q ) ) )
19	18	rspcev 2834	. . . 4 |- ( ( K e. N. /\ ( ( F ` K ) +Q ( *Q ` [ <. K , 1o >. ] ~Q ) ) <Q ( ( F ` K ) +Q Q ) ) -> E. j e. N. ( ( F ` K ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( ( F ` j ) +Q Q ) )
20	5, 11, 19	syl2anc 409	. . 3 |- ( ph -> E. j e. N. ( ( F ` K ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( ( F ` j ) +Q Q ) )
21		oveq1 5860	. . . . . . . 8 |- ( l = ( F ` K ) -> ( l +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) = ( ( F ` K ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) )
22	21	breq1d 3999	. . . . . . 7 |- ( l = ( F ` K ) -> ( ( l +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( ( F ` j ) +Q Q ) <-> ( ( F ` K ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( ( F ` j ) +Q Q ) ) )
23	22	rexbidv 2471	. . . . . 6 |- ( l = ( F ` K ) -> ( E. j e. N. ( l +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( ( F ` j ) +Q Q ) <-> E. j e. N. ( ( F ` K ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( ( F ` j ) +Q Q ) ) )
24	23	elrab3 2887	. . . . 5 |- ( ( F ` K ) e. Q. -> ( ( F ` K ) e. { l e. Q. | E. j e. N. ( l +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( ( F ` j ) +Q Q ) } <-> E. j e. N. ( ( F ` K ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( ( F ` j ) +Q Q ) ) )
25	9, 24	syl 14	. . . 4 |- ( ph -> ( ( F ` K ) e. { l e. Q. | E. j e. N. ( l +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( ( F ` j ) +Q Q ) } <-> E. j e. N. ( ( F ` K ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( ( F ` j ) +Q Q ) ) )
26		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 ) ) ) ) )
27		caucvgpr.bnd	. . . . . 6 |- ( ph -> A. j e. N. A <Q ( F ` j ) )
28		caucvgpr.lim	. . . . . 6 |- 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 } >.
29		caucvgprlemlim.q	. . . . . 6 |- ( ph -> Q e. Q. )
30	8, 26, 27, 28, 29	caucvgprlemladdrl 7640	. . . . 5 |- ( ph -> { l e. Q. | E. j e. N. ( l +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( ( F ` j ) +Q Q ) } C_ ( 1st ` ( L +P. <. { l | l <Q Q } , { u | Q <Q u } >. ) ) )
31	30	sseld 3146	. . . 4 |- ( ph -> ( ( F ` K ) e. { l e. Q. | E. j e. N. ( l +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( ( F ` j ) +Q Q ) } -> ( F ` K ) e. ( 1st ` ( L +P. <. { l | l <Q Q } , { u | Q <Q u } >. ) ) ) )
32	25, 31	sylbird 169	. . 3 |- ( ph -> ( E. j e. N. ( ( F ` K ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( ( F ` j ) +Q Q ) -> ( F ` K ) e. ( 1st ` ( L +P. <. { l | l <Q Q } , { u | Q <Q u } >. ) ) ) )
33	20, 32	mpd 13	. 2 |- ( ph -> ( F ` K ) e. ( 1st ` ( L +P. <. { l | l <Q Q } , { u | Q <Q u } >. ) ) )
34	8, 26, 27, 28	caucvgprlemcl 7638	. . . 4 |- ( ph -> L e. P. )
35		nqprlu 7509	. . . . 5 |- ( Q e. Q. -> <. { l | l <Q Q } , { u | Q <Q u } >. e. P. )
36	29, 35	syl 14	. . . 4 |- ( ph -> <. { l | l <Q Q } , { u | Q <Q u } >. e. P. )
37		addclpr 7499	. . . 4 |- ( ( L e. P. /\ <. { l | l <Q Q } , { u | Q <Q u } >. e. P. ) -> ( L +P. <. { l | l <Q Q } , { u | Q <Q u } >. ) e. P. )
38	34, 36, 37	syl2anc 409	. . 3 |- ( ph -> ( L +P. <. { l | l <Q Q } , { u | Q <Q u } >. ) e. P. )
39		nqprl 7513	. . 3 |- ( ( ( F ` K ) e. Q. /\ ( L +P. <. { l | l <Q Q } , { u | Q <Q u } >. ) e. P. ) -> ( ( F ` K ) e. ( 1st ` ( L +P. <. { l | l <Q Q } , { u | Q <Q u } >. ) ) <-> <. { l | l <Q ( F ` K ) } , { u | ( F ` K ) <Q u } >. <P ( L +P. <. { l | l <Q Q } , { u | Q <Q u } >. ) ) )
40	9, 38, 39	syl2anc 409	. 2 |- ( ph -> ( ( F ` K ) e. ( 1st ` ( L +P. <. { l | l <Q Q } , { u | Q <Q u } >. ) ) <-> <. { l | l <Q ( F ` K ) } , { u | ( F ` K ) <Q u } >. <P ( L +P. <. { l | l <Q Q } , { u | Q <Q u } >. ) ) )
41	33, 40	mpbid 146	1 |- ( ph -> <. { l | l <Q ( F ` K ) } , { u | ( F ` K ) <Q u } >. <P ( L +P. <. { l | l <Q Q } , { u | Q <Q u } >. ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1348   e. wcel 2141   {cab 2156   A.wral 2448   E.wrex 2449   {crab 2452   <.cop 3586   class class class wbr 3989   -->wf 5194   ` cfv 5198  (class class class)co 5853   1stc1st 6117   1oc1o 6388   [cec 6511   N.cnpi 7234   <N clti 7237   ~Q ceq 7241   Q.cnq 7242   +Q cplq 7244   *Qcrq 7246   <Q cltq 7247   P.cnp 7253   +P. cpp 7255   <P cltp 7257

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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-coll 4104  ax-sep 4107  ax-nul 4115  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-iinf 4572

This theorem depends on definitions:  df-bi 116  df-dc 830  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-ral 2453  df-rex 2454  df-reu 2455  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-int 3832  df-iun 3875  df-br 3990  df-opab 4051  df-mpt 4052  df-tr 4088  df-eprel 4274  df-id 4278  df-po 4281  df-iso 4282  df-iord 4351  df-on 4353  df-suc 4356  df-iom 4575  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-fo 5204  df-f1o 5205  df-fv 5206  df-ov 5856  df-oprab 5857  df-mpo 5858  df-1st 6119  df-2nd 6120  df-recs 6284  df-irdg 6349  df-1o 6395  df-2o 6396  df-oadd 6399  df-omul 6400  df-er 6513  df-ec 6515  df-qs 6519  df-ni 7266  df-pli 7267  df-mi 7268  df-lti 7269  df-plpq 7306  df-mpq 7307  df-enq 7309  df-nqqs 7310  df-plqqs 7311  df-mqqs 7312  df-1nqqs 7313  df-rq 7314  df-ltnqqs 7315  df-enq0 7386  df-nq0 7387  df-0nq0 7388  df-plq0 7389  df-mq0 7390  df-inp 7428  df-iplp 7430  df-iltp 7432

This theorem is referenced by:  caucvgprlemlim  7643