Theorem caucvgprprlem1 7972

Description: Lemma for caucvgprpr 7975. Part of showing the putative limit to be a limit. (Contributed by Jim Kingdon, 25-Nov-2020.)

Hypotheses

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

Assertion

Ref	Expression
caucvgprprlem1	|- ( ph -> ( F ` K ) <P ( L +P. Q ) )

Distinct variable groups:   A, m   m, F   A, r   F, r, l, u, n, k   J, l, u   K, l, r, u   Q, r   k, L   ph, r   q, p, r, l, u   m, r   k, l, u, r, p, q   n, l, u, r

Allowed substitution hints:   ph( u, k, m, n, q, p, l)   A( u, k, n, q, p, l)   Q( u, k, m, n, q, p, l)   F( q, p)   J( k, m, n, r, q, p)   K( k, m, n, q, p)   L( u, m, n, r, q, p, l)

Proof of Theorem caucvgprprlem1

Step	Hyp	Ref	Expression
1		caucvgprpr.f	. 2 |- ( ph -> F : N. --> P. )
2		caucvgprpr.cau	. 2 |- ( ph -> A. n e. N. A. k e. N. ( n <N k -> ( ( F ` n ) <P ( ( F ` k ) +P. <. { l | l <Q ( *Q ` [ <. n , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. n , 1o >. ] ~Q ) <Q u } >. ) /\ ( F ` k ) <P ( ( F ` n ) +P. <. { l | l <Q ( *Q ` [ <. n , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. n , 1o >. ] ~Q ) <Q u } >. ) ) ) )
3		caucvgprpr.bnd	. 2 |- ( ph -> A. m e. N. A <P ( F ` m ) )
4		caucvgprpr.lim	. 2 |- L = <. { l e. Q. | E. r e. N. <. { p | p <Q ( l +Q ( *Q ` [ <. r , 1o >. ] ~Q ) ) } , { q | ( l +Q ( *Q ` [ <. r , 1o >. ] ~Q ) ) <Q q } >. <P ( F ` r ) } , { u e. Q. | E. r e. N. ( ( F ` r ) +P. <. { p | p <Q ( *Q ` [ <. r , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. r , 1o >. ] ~Q ) <Q q } >. ) <P <. { p | p <Q u } , { q | u <Q q } >. } >.
5		caucvgprprlemlim.jk	. . . . 5 |- ( ph -> J <N K )
6		ltrelpi 7587	. . . . . 6 |- <N C_ ( N. X. N. )
7	6	brel 4784	. . . . 5 |- ( J <N K -> ( J e. N. /\ K e. N. ) )
8	5, 7	syl 14	. . . 4 |- ( ph -> ( J e. N. /\ K e. N. ) )
9	8	simprd 114	. . 3 |- ( ph -> K e. N. )
10	1, 9	ffvelcdmd 5791	. 2 |- ( ph -> ( F ` K ) e. P. )
11		caucvgprprlemlim.q	. 2 |- ( ph -> Q e. P. )
12		caucvgprprlemlim.jkq	. . . . . 6 |- ( ph -> <. { l | l <Q ( *Q ` [ <. J , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. J , 1o >. ] ~Q ) <Q u } >. <P Q )
13	5, 12	caucvgprprlemk 7946	. . . . 5 |- ( ph -> <. { l | l <Q ( *Q ` [ <. K , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. K , 1o >. ] ~Q ) <Q u } >. <P Q )
14		nnnq 7685	. . . . . . . 8 |- ( K e. N. -> [ <. K , 1o >. ] ~Q e. Q. )
15	9, 14	syl 14	. . . . . . 7 |- ( ph -> [ <. K , 1o >. ] ~Q e. Q. )
16		recclnq 7655	. . . . . . 7 |- ( [ <. K , 1o >. ] ~Q e. Q. -> ( *Q ` [ <. K , 1o >. ] ~Q ) e. Q. )
17		nqprlu 7810	. . . . . . 7 |- ( ( *Q ` [ <. K , 1o >. ] ~Q ) e. Q. -> <. { l | l <Q ( *Q ` [ <. K , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. K , 1o >. ] ~Q ) <Q u } >. e. P. )
18	15, 16, 17	3syl 17	. . . . . 6 |- ( ph -> <. { l | l <Q ( *Q ` [ <. K , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. K , 1o >. ] ~Q ) <Q u } >. e. P. )
19		ltaprg 7882	. . . . . 6 |- ( ( <. { l | l <Q ( *Q ` [ <. K , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. K , 1o >. ] ~Q ) <Q u } >. e. P. /\ Q e. P. /\ ( F ` K ) e. P. ) -> ( <. { l | l <Q ( *Q ` [ <. K , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. K , 1o >. ] ~Q ) <Q u } >. <P Q <-> ( ( F ` K ) +P. <. { l | l <Q ( *Q ` [ <. K , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. K , 1o >. ] ~Q ) <Q u } >. ) <P ( ( F ` K ) +P. Q ) ) )
20	18, 11, 10, 19	syl3anc 1274	. . . . 5 |- ( ph -> ( <. { l | l <Q ( *Q ` [ <. K , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. K , 1o >. ] ~Q ) <Q u } >. <P Q <-> ( ( F ` K ) +P. <. { l | l <Q ( *Q ` [ <. K , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. K , 1o >. ] ~Q ) <Q u } >. ) <P ( ( F ` K ) +P. Q ) ) )
21	13, 20	mpbid 147	. . . 4 |- ( ph -> ( ( F ` K ) +P. <. { l | l <Q ( *Q ` [ <. K , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. K , 1o >. ] ~Q ) <Q u } >. ) <P ( ( F ` K ) +P. Q ) )
22		opeq1 3867	. . . . . . . . . . . 12 |- ( r = K -> <. r , 1o >. = <. K , 1o >. )
23	22	eceq1d 6781	. . . . . . . . . . 11 |- ( r = K -> [ <. r , 1o >. ] ~Q = [ <. K , 1o >. ] ~Q )
24	23	fveq2d 5652	. . . . . . . . . 10 |- ( r = K -> ( *Q ` [ <. r , 1o >. ] ~Q ) = ( *Q ` [ <. K , 1o >. ] ~Q ) )
25	24	breq2d 4105	. . . . . . . . 9 |- ( r = K -> ( l <Q ( *Q ` [ <. r , 1o >. ] ~Q ) <-> l <Q ( *Q ` [ <. K , 1o >. ] ~Q ) ) )
26	25	abbidv 2350	. . . . . . . 8 |- ( r = K -> { l | l <Q ( *Q ` [ <. r , 1o >. ] ~Q ) } = { l | l <Q ( *Q ` [ <. K , 1o >. ] ~Q ) } )
27	24	breq1d 4103	. . . . . . . . 9 |- ( r = K -> ( ( *Q ` [ <. r , 1o >. ] ~Q ) <Q u <-> ( *Q ` [ <. K , 1o >. ] ~Q ) <Q u ) )
28	27	abbidv 2350	. . . . . . . 8 |- ( r = K -> { u | ( *Q ` [ <. r , 1o >. ] ~Q ) <Q u } = { u | ( *Q ` [ <. K , 1o >. ] ~Q ) <Q u } )
29	26, 28	opeq12d 3875	. . . . . . 7 |- ( r = K -> <. { l | l <Q ( *Q ` [ <. r , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. r , 1o >. ] ~Q ) <Q u } >. = <. { l | l <Q ( *Q ` [ <. K , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. K , 1o >. ] ~Q ) <Q u } >. )
30	29	oveq2d 6044	. . . . . 6 |- ( r = K -> ( ( F ` K ) +P. <. { l | l <Q ( *Q ` [ <. r , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. r , 1o >. ] ~Q ) <Q u } >. ) = ( ( F ` K ) +P. <. { l | l <Q ( *Q ` [ <. K , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. K , 1o >. ] ~Q ) <Q u } >. ) )
31		fveq2 5648	. . . . . . 7 |- ( r = K -> ( F ` r ) = ( F ` K ) )
32	31	oveq1d 6043	. . . . . 6 |- ( r = K -> ( ( F ` r ) +P. Q ) = ( ( F ` K ) +P. Q ) )
33	30, 32	breq12d 4106	. . . . 5 |- ( r = K -> ( ( ( F ` K ) +P. <. { l | l <Q ( *Q ` [ <. r , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. r , 1o >. ] ~Q ) <Q u } >. ) <P ( ( F ` r ) +P. Q ) <-> ( ( F ` K ) +P. <. { l | l <Q ( *Q ` [ <. K , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. K , 1o >. ] ~Q ) <Q u } >. ) <P ( ( F ` K ) +P. Q ) ) )
34	33	rspcev 2911	. . . 4 |- ( ( K e. N. /\ ( ( F ` K ) +P. <. { l | l <Q ( *Q ` [ <. K , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. K , 1o >. ] ~Q ) <Q u } >. ) <P ( ( F ` K ) +P. Q ) ) -> E. r e. N. ( ( F ` K ) +P. <. { l | l <Q ( *Q ` [ <. r , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. r , 1o >. ] ~Q ) <Q u } >. ) <P ( ( F ` r ) +P. Q ) )
35	9, 21, 34	syl2anc 411	. . 3 |- ( ph -> E. r e. N. ( ( F ` K ) +P. <. { l | l <Q ( *Q ` [ <. r , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. r , 1o >. ] ~Q ) <Q u } >. ) <P ( ( F ` r ) +P. Q ) )
36		breq1 4096	. . . . . . . 8 |- ( l = p -> ( l <Q ( *Q ` [ <. r , 1o >. ] ~Q ) <-> p <Q ( *Q ` [ <. r , 1o >. ] ~Q ) ) )
37	36	cbvabv 2357	. . . . . . 7 |- { l | l <Q ( *Q ` [ <. r , 1o >. ] ~Q ) } = { p | p <Q ( *Q ` [ <. r , 1o >. ] ~Q ) }
38		breq2 4097	. . . . . . . 8 |- ( u = q -> ( ( *Q ` [ <. r , 1o >. ] ~Q ) <Q u <-> ( *Q ` [ <. r , 1o >. ] ~Q ) <Q q ) )
39	38	cbvabv 2357	. . . . . . 7 |- { u | ( *Q ` [ <. r , 1o >. ] ~Q ) <Q u } = { q | ( *Q ` [ <. r , 1o >. ] ~Q ) <Q q }
40	37, 39	opeq12i 3872	. . . . . 6 |- <. { l | l <Q ( *Q ` [ <. r , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. r , 1o >. ] ~Q ) <Q u } >. = <. { p | p <Q ( *Q ` [ <. r , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. r , 1o >. ] ~Q ) <Q q } >.
41	40	oveq2i 6039	. . . . 5 |- ( ( F ` K ) +P. <. { l | l <Q ( *Q ` [ <. r , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. r , 1o >. ] ~Q ) <Q u } >. ) = ( ( F ` K ) +P. <. { p | p <Q ( *Q ` [ <. r , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. r , 1o >. ] ~Q ) <Q q } >. )
42	41	breq1i 4100	. . . 4 |- ( ( ( F ` K ) +P. <. { l | l <Q ( *Q ` [ <. r , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. r , 1o >. ] ~Q ) <Q u } >. ) <P ( ( F ` r ) +P. Q ) <-> ( ( F ` K ) +P. <. { p | p <Q ( *Q ` [ <. r , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. r , 1o >. ] ~Q ) <Q q } >. ) <P ( ( F ` r ) +P. Q ) )
43	42	rexbii 2540	. . 3 |- ( E. r e. N. ( ( F ` K ) +P. <. { l | l <Q ( *Q ` [ <. r , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. r , 1o >. ] ~Q ) <Q u } >. ) <P ( ( F ` r ) +P. Q ) <-> E. r e. N. ( ( F ` K ) +P. <. { p | p <Q ( *Q ` [ <. r , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. r , 1o >. ] ~Q ) <Q q } >. ) <P ( ( F ` r ) +P. Q ) )
44	35, 43	sylib 122	. 2 |- ( ph -> E. r e. N. ( ( F ` K ) +P. <. { p | p <Q ( *Q ` [ <. r , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. r , 1o >. ] ~Q ) <Q q } >. ) <P ( ( F ` r ) +P. Q ) )
45	1, 2, 3, 4, 10, 11, 44	caucvgprprlemaddq 7971	1 |- ( ph -> ( F ` K ) <P ( L +P. Q ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1398   e. wcel 2202   {cab 2217   A.wral 2511   E.wrex 2512   {crab 2515   <.cop 3676   class class class wbr 4093   -->wf 5329   ` cfv 5333  (class class class)co 6028   1oc1o 6618   [cec 6743   N.cnpi 7535   <N clti 7538   ~Q ceq 7542   Q.cnq 7543   +Q cplq 7545   *Qcrq 7547   <Q cltq 7548   P.cnp 7554   +P. cpp 7556   <P cltp 7558

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 2204  ax-14 2205  ax-ext 2213  ax-coll 4209  ax-sep 4212  ax-nul 4220  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641  ax-iinf 4692

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 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-ral 2516  df-rex 2517  df-reu 2518  df-rab 2520  df-v 2805  df-sbc 3033  df-csb 3129  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-nul 3497  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-int 3934  df-iun 3977  df-br 4094  df-opab 4156  df-mpt 4157  df-tr 4193  df-eprel 4392  df-id 4396  df-po 4399  df-iso 4400  df-iord 4469  df-on 4471  df-suc 4474  df-iom 4695  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-f1 5338  df-fo 5339  df-f1o 5340  df-fv 5341  df-ov 6031  df-oprab 6032  df-mpo 6033  df-1st 6312  df-2nd 6313  df-recs 6514  df-irdg 6579  df-1o 6625  df-2o 6626  df-oadd 6629  df-omul 6630  df-er 6745  df-ec 6747  df-qs 6751  df-ni 7567  df-pli 7568  df-mi 7569  df-lti 7570  df-plpq 7607  df-mpq 7608  df-enq 7610  df-nqqs 7611  df-plqqs 7612  df-mqqs 7613  df-1nqqs 7614  df-rq 7615  df-ltnqqs 7616  df-enq0 7687  df-nq0 7688  df-0nq0 7689  df-plq0 7690  df-mq0 7691  df-inp 7729  df-iplp 7731  df-iltp 7733

This theorem is referenced by:  caucvgprprlemlim  7974