Theorem caucvgprlem1 7993

Description: Lemma for caucvgpr 7996. 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 7638	. . . . . . 7 |- <N C_ ( N. X. N. )
3	2	brel 4801	. . . . . 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 114	. . . 4 |- ( ph -> K e. N. )
6		caucvgprlemlim.jkq	. . . . . 6 |- ( ph -> ( *Q ` [ <. J , 1o >. ] ~Q ) <Q Q )
7	1, 6	caucvgprlemk 7979	. . . . 5 |- ( ph -> ( *Q ` [ <. K , 1o >. ] ~Q ) <Q Q )
8		caucvgpr.f	. . . . . 6 |- ( ph -> F : N. --> Q. )
9	8, 5	ffvelcdmd 5812	. . . . 5 |- ( ph -> ( F ` K ) e. Q. )
10		ltanqi 7716	. . . . 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 411	. . . 4 |- ( ph -> ( ( F ` K ) +Q ( *Q ` [ <. K , 1o >. ] ~Q ) ) <Q ( ( F ` K ) +Q Q ) )
12		opeq1 3882	. . . . . . . . 9 |- ( j = K -> <. j , 1o >. = <. K , 1o >. )
13	12	eceq1d 6802	. . . . . . . 8 |- ( j = K -> [ <. j , 1o >. ] ~Q = [ <. K , 1o >. ] ~Q )
14	13	fveq2d 5673	. . . . . . 7 |- ( j = K -> ( *Q ` [ <. j , 1o >. ] ~Q ) = ( *Q ` [ <. K , 1o >. ] ~Q ) )
15	14	oveq2d 6065	. . . . . 6 |- ( j = K -> ( ( F ` K ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) = ( ( F ` K ) +Q ( *Q ` [ <. K , 1o >. ] ~Q ) ) )
16		fveq2 5669	. . . . . . 7 |- ( j = K -> ( F ` j ) = ( F ` K ) )
17	16	oveq1d 6064	. . . . . 6 |- ( j = K -> ( ( F ` j ) +Q Q ) = ( ( F ` K ) +Q Q ) )
18	15, 17	breq12d 4121	. . . . 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 2920	. . . 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 411	. . 3 |- ( ph -> E. j e. N. ( ( F ` K ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( ( F ` j ) +Q Q ) )
21		oveq1 6056	. . . . . . . 8 |- ( l = ( F ` K ) -> ( l +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) = ( ( F ` K ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) )
22	21	breq1d 4118	. . . . . . 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 2543	. . . . . 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 2973	. . . . 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 7992	. . . . 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 3236	. . . 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 170	. . 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 7990	. . . 4 |- ( ph -> L e. P. )
35		nqprlu 7861	. . . . 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 7851	. . . 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 411	. . 3 |- ( ph -> ( L +P. <. { l | l <Q Q } , { u | Q <Q u } >. ) e. P. )
39		nqprl 7865	. . 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 411	. 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 147	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 104   <-> wb 105   = wceq 1398   e. wcel 2203   {cab 2218   A.wral 2520   E.wrex 2521   {crab 2524   <.cop 3691   class class class wbr 4108   -->wf 5347   ` cfv 5351  (class class class)co 6049   1stc1st 6331   1oc1o 6639   [cec 6764   N.cnpi 7586   <N clti 7589   ~Q ceq 7593   Q.cnq 7594   +Q cplq 7596   *Qcrq 7598   <Q cltq 7599   P.cnp 7605   +P. cpp 7607   <P cltp 7609

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-coll 4224  ax-sep 4227  ax-nul 4235  ax-pow 4286  ax-pr 4321  ax-un 4553  ax-setind 4658  ax-iinf 4709

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-ral 2525  df-rex 2526  df-reu 2527  df-rab 2529  df-v 2814  df-sbc 3042  df-csb 3138  df-dif 3212  df-un 3214  df-in 3216  df-ss 3223  df-nul 3508  df-pw 3670  df-sn 3694  df-pr 3695  df-op 3697  df-uni 3914  df-int 3949  df-iun 3992  df-br 4109  df-opab 4171  df-mpt 4172  df-tr 4208  df-eprel 4409  df-id 4413  df-po 4416  df-iso 4417  df-iord 4486  df-on 4488  df-suc 4491  df-iom 4712  df-xp 4754  df-rel 4755  df-cnv 4756  df-co 4757  df-dm 4758  df-rn 4759  df-res 4760  df-ima 4761  df-iota 5311  df-fun 5353  df-fn 5354  df-f 5355  df-f1 5356  df-fo 5357  df-f1o 5358  df-fv 5359  df-ov 6052  df-oprab 6053  df-mpo 6054  df-1st 6333  df-2nd 6334  df-recs 6535  df-irdg 6600  df-1o 6646  df-2o 6647  df-oadd 6650  df-omul 6651  df-er 6766  df-ec 6768  df-qs 6772  df-ni 7618  df-pli 7619  df-mi 7620  df-lti 7621  df-plpq 7658  df-mpq 7659  df-enq 7661  df-nqqs 7662  df-plqqs 7663  df-mqqs 7664  df-1nqqs 7665  df-rq 7666  df-ltnqqs 7667  df-enq0 7738  df-nq0 7739  df-0nq0 7740  df-plq0 7741  df-mq0 7742  df-inp 7780  df-iplp 7782  df-iltp 7784

This theorem is referenced by:  caucvgprlemlim  7995