Theorem caucvgprlem1 7620

Description: Lemma for caucvgpr 7623. 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 7265	. . . . . . 7 |- <N C_ ( N. X. N. )
3	2	brel 4656	. . . . . 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 7606	. . . . 5 |- ( ph -> ( *Q ` [ <. K , 1o >. ] ~Q ) <Q Q )
8		caucvgpr.f	. . . . . 6 |- ( ph -> F : N. --> Q. )
9	8, 5	ffvelrnd 5621	. . . . 5 |- ( ph -> ( F ` K ) e. Q. )
10		ltanqi 7343	. . . . 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 3758	. . . . . . . . 9 |- ( j = K -> <. j , 1o >. = <. K , 1o >. )
13	12	eceq1d 6537	. . . . . . . 8 |- ( j = K -> [ <. j , 1o >. ] ~Q = [ <. K , 1o >. ] ~Q )
14	13	fveq2d 5490	. . . . . . 7 |- ( j = K -> ( *Q ` [ <. j , 1o >. ] ~Q ) = ( *Q ` [ <. K , 1o >. ] ~Q ) )
15	14	oveq2d 5858	. . . . . 6 |- ( j = K -> ( ( F ` K ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) = ( ( F ` K ) +Q ( *Q ` [ <. K , 1o >. ] ~Q ) ) )
16		fveq2 5486	. . . . . . 7 |- ( j = K -> ( F ` j ) = ( F ` K ) )
17	16	oveq1d 5857	. . . . . 6 |- ( j = K -> ( ( F ` j ) +Q Q ) = ( ( F ` K ) +Q Q ) )
18	15, 17	breq12d 3995	. . . . 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 2830	. . . 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 5849	. . . . . . . 8 |- ( l = ( F ` K ) -> ( l +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) = ( ( F ` K ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) )
22	21	breq1d 3992	. . . . . . 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 2467	. . . . . 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 2883	. . . . 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 7619	. . . . 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 3141	. . . 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 7617	. . . 4 |- ( ph -> L e. P. )
35		nqprlu 7488	. . . . 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 7478	. . . 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 7492	. . 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 1343   e. wcel 2136   {cab 2151   A.wral 2444   E.wrex 2445   {crab 2448   <.cop 3579   class class class wbr 3982   -->wf 5184   ` cfv 5188  (class class class)co 5842   1stc1st 6106   1oc1o 6377   [cec 6499   N.cnpi 7213   <N clti 7216   ~Q ceq 7220   Q.cnq 7221   +Q cplq 7223   *Qcrq 7225   <Q cltq 7226   P.cnp 7232   +P. cpp 7234   <P cltp 7236

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 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-coll 4097  ax-sep 4100  ax-nul 4108  ax-pow 4153  ax-pr 4187  ax-un 4411  ax-setind 4514  ax-iinf 4565

This theorem depends on definitions:  df-bi 116  df-dc 825  df-3or 969  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ne 2337  df-ral 2449  df-rex 2450  df-reu 2451  df-rab 2453  df-v 2728  df-sbc 2952  df-csb 3046  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-nul 3410  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-int 3825  df-iun 3868  df-br 3983  df-opab 4044  df-mpt 4045  df-tr 4081  df-eprel 4267  df-id 4271  df-po 4274  df-iso 4275  df-iord 4344  df-on 4346  df-suc 4349  df-iom 4568  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-res 4616  df-ima 4617  df-iota 5153  df-fun 5190  df-fn 5191  df-f 5192  df-f1 5193  df-fo 5194  df-f1o 5195  df-fv 5196  df-ov 5845  df-oprab 5846  df-mpo 5847  df-1st 6108  df-2nd 6109  df-recs 6273  df-irdg 6338  df-1o 6384  df-2o 6385  df-oadd 6388  df-omul 6389  df-er 6501  df-ec 6503  df-qs 6507  df-ni 7245  df-pli 7246  df-mi 7247  df-lti 7248  df-plpq 7285  df-mpq 7286  df-enq 7288  df-nqqs 7289  df-plqqs 7290  df-mqqs 7291  df-1nqqs 7292  df-rq 7293  df-ltnqqs 7294  df-enq0 7365  df-nq0 7366  df-0nq0 7367  df-plq0 7368  df-mq0 7369  df-inp 7407  df-iplp 7409  df-iltp 7411

This theorem is referenced by:  caucvgprlemlim  7622