Theorem caucvgprprlem1 8082

Description: Lemma for caucvgprpr 8085. 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 7697	. . . . . 6 |- <N C_ ( N. X. N. )
7	6	brel 4829	. . . . 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 5847	. 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 8056	. . . . 5 |- ( ph -> <. { l | l <Q ( *Q ` [ <. K , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. K , 1o >. ] ~Q ) <Q u } >. <P Q )
14		nnnq 7795	. . . . . . . 8 |- ( K e. N. -> [ <. K , 1o >. ] ~Q e. Q. )
15	9, 14	syl 14	. . . . . . 7 |- ( ph -> [ <. K , 1o >. ] ~Q e. Q. )
16		recclnq 7765	. . . . . . 7 |- ( [ <. K , 1o >. ] ~Q e. Q. -> ( *Q ` [ <. K , 1o >. ] ~Q ) e. Q. )
17		nqprlu 7920	. . . . . . 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 7992	. . . . . 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 1278	. . . . 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 3904	. . . . . . . . . . . 12 |- ( r = K -> <. r , 1o >. = <. K , 1o >. )
23	22	eceq1d 6847	. . . . . . . . . . 11 |- ( r = K -> [ <. r , 1o >. ] ~Q = [ <. K , 1o >. ] ~Q )
24	23	fveq2d 5702	. . . . . . . . . 10 |- ( r = K -> ( *Q ` [ <. r , 1o >. ] ~Q ) = ( *Q ` [ <. K , 1o >. ] ~Q ) )
25	24	breq2d 4143	. . . . . . . . 9 |- ( r = K -> ( l <Q ( *Q ` [ <. r , 1o >. ] ~Q ) <-> l <Q ( *Q ` [ <. K , 1o >. ] ~Q ) ) )
26	25	abbidv 2358	. . . . . . . 8 |- ( r = K -> { l | l <Q ( *Q ` [ <. r , 1o >. ] ~Q ) } = { l | l <Q ( *Q ` [ <. K , 1o >. ] ~Q ) } )
27	24	breq1d 4141	. . . . . . . . 9 |- ( r = K -> ( ( *Q ` [ <. r , 1o >. ] ~Q ) <Q u <-> ( *Q ` [ <. K , 1o >. ] ~Q ) <Q u ) )
28	27	abbidv 2358	. . . . . . . 8 |- ( r = K -> { u | ( *Q ` [ <. r , 1o >. ] ~Q ) <Q u } = { u | ( *Q ` [ <. K , 1o >. ] ~Q ) <Q u } )
29	26, 28	opeq12d 3912	. . . . . . 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 6104	. . . . . 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 5698	. . . . . . 7 |- ( r = K -> ( F ` r ) = ( F ` K ) )
32	31	oveq1d 6103	. . . . . 6 |- ( r = K -> ( ( F ` r ) +P. Q ) = ( ( F ` K ) +P. Q ) )
33	30, 32	breq12d 4144	. . . . 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 2929	. . . 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 415	. . 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 4134	. . . . . . . 8 |- ( l = p -> ( l <Q ( *Q ` [ <. r , 1o >. ] ~Q ) <-> p <Q ( *Q ` [ <. r , 1o >. ] ~Q ) ) )
37	36	cbvabv 2365	. . . . . . 7 |- { l | l <Q ( *Q ` [ <. r , 1o >. ] ~Q ) } = { p | p <Q ( *Q ` [ <. r , 1o >. ] ~Q ) }
38		breq2 4135	. . . . . . . 8 |- ( u = q -> ( ( *Q ` [ <. r , 1o >. ] ~Q ) <Q u <-> ( *Q ` [ <. r , 1o >. ] ~Q ) <Q q ) )
39	38	cbvabv 2365	. . . . . . 7 |- { u | ( *Q ` [ <. r , 1o >. ] ~Q ) <Q u } = { q | ( *Q ` [ <. r , 1o >. ] ~Q ) <Q q }
40	37, 39	opeq12i 3909	. . . . . 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 6099	. . . . 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 4138	. . . 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 2557	. . 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 8081	1 |- ( ph -> ( F ` K ) <P ( L +P. Q ) )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1402   e. wcel 2209   {cab 2224   A.wral 2528   E.wrex 2529   {crab 2532   <.cop 3712   class class class wbr 4131   -->wf 5376   ` cfv 5380  (class class class)co 6088   1oc1o 6684   [cec 6809   N.cnpi 7645   <N clti 7648   ~Q ceq 7652   Q.cnq 7653   +Q cplq 7655   *Qcrq 7657   <Q cltq 7658   P.cnp 7664   +P. cpp 7666   <P cltp 7668

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4248  ax-sep 4251  ax-nul 4261  ax-pow 4313  ax-pr 4348  ax-un 4580  ax-setind 4686  ax-iinf 4737

This proof depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4132  df-opab 4194  df-mpt 4195  df-tr 4232  df-eprel 4436  df-id 4440  df-po 4443  df-iso 4444  df-iord 4513  df-on 4515  df-suc 4518  df-iom 4740  df-xp 4782  df-rel 4783  df-cnv 4784  df-co 4785  df-dm 4786  df-rn 4787  df-res 4788  df-ima 4789  df-iota 5340  df-fun 5382  df-fn 5383  df-f 5384  df-f1 5385  df-fo 5386  df-f1o 5387  df-fv 5388  df-ov 6091  df-oprab 6092  df-mpo 6093  df-1st 6378  df-2nd 6379  df-recs 6580  df-irdg 6645  df-1o 6691  df-2o 6692  df-oadd 6695  df-omul 6696  df-er 6811  df-ec 6813  df-qs 6817  df-ni 7677  df-pli 7678  df-mi 7679  df-lti 7680  df-plpq 7717  df-mpq 7718  df-enq 7720  df-nqqs 7721  df-plqqs 7722  df-mqqs 7723  df-1nqqs 7724  df-rq 7725  df-ltnqqs 7726  df-enq0 7797  df-nq0 7798  df-0nq0 7799  df-plq0 7800  df-mq0 7801  df-inp 7839  df-iplp 7841  df-iltp 7843

This theorem is used by:  caucvgprprlemlim  8084