Theorem caucvgprprlem1 7541

Description: Lemma for caucvgprpr 7544. 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 7156	. . . . . 6 |- <N C_ ( N. X. N. )
7	6	brel 4599	. . . . 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 113	. . 3 |- ( ph -> K e. N. )
10	1, 9	ffvelrnd 5564	. 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 7515	. . . . 5 |- ( ph -> <. { l | l <Q ( *Q ` [ <. K , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. K , 1o >. ] ~Q ) <Q u } >. <P Q )
14		nnnq 7254	. . . . . . . 8 |- ( K e. N. -> [ <. K , 1o >. ] ~Q e. Q. )
15	9, 14	syl 14	. . . . . . 7 |- ( ph -> [ <. K , 1o >. ] ~Q e. Q. )
16		recclnq 7224	. . . . . . 7 |- ( [ <. K , 1o >. ] ~Q e. Q. -> ( *Q ` [ <. K , 1o >. ] ~Q ) e. Q. )
17		nqprlu 7379	. . . . . . 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 7451	. . . . . 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 1217	. . . . 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 146	. . . 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 3713	. . . . . . . . . . . 12 |- ( r = K -> <. r , 1o >. = <. K , 1o >. )
23	22	eceq1d 6473	. . . . . . . . . . 11 |- ( r = K -> [ <. r , 1o >. ] ~Q = [ <. K , 1o >. ] ~Q )
24	23	fveq2d 5433	. . . . . . . . . 10 |- ( r = K -> ( *Q ` [ <. r , 1o >. ] ~Q ) = ( *Q ` [ <. K , 1o >. ] ~Q ) )
25	24	breq2d 3949	. . . . . . . . 9 |- ( r = K -> ( l <Q ( *Q ` [ <. r , 1o >. ] ~Q ) <-> l <Q ( *Q ` [ <. K , 1o >. ] ~Q ) ) )
26	25	abbidv 2258	. . . . . . . 8 |- ( r = K -> { l | l <Q ( *Q ` [ <. r , 1o >. ] ~Q ) } = { l | l <Q ( *Q ` [ <. K , 1o >. ] ~Q ) } )
27	24	breq1d 3947	. . . . . . . . 9 |- ( r = K -> ( ( *Q ` [ <. r , 1o >. ] ~Q ) <Q u <-> ( *Q ` [ <. K , 1o >. ] ~Q ) <Q u ) )
28	27	abbidv 2258	. . . . . . . 8 |- ( r = K -> { u | ( *Q ` [ <. r , 1o >. ] ~Q ) <Q u } = { u | ( *Q ` [ <. K , 1o >. ] ~Q ) <Q u } )
29	26, 28	opeq12d 3721	. . . . . . 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 5798	. . . . . 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 5429	. . . . . . 7 |- ( r = K -> ( F ` r ) = ( F ` K ) )
32	31	oveq1d 5797	. . . . . 6 |- ( r = K -> ( ( F ` r ) +P. Q ) = ( ( F ` K ) +P. Q ) )
33	30, 32	breq12d 3950	. . . . 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 2793	. . . 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 409	. . 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 3940	. . . . . . . 8 |- ( l = p -> ( l <Q ( *Q ` [ <. r , 1o >. ] ~Q ) <-> p <Q ( *Q ` [ <. r , 1o >. ] ~Q ) ) )
37	36	cbvabv 2265	. . . . . . 7 |- { l | l <Q ( *Q ` [ <. r , 1o >. ] ~Q ) } = { p | p <Q ( *Q ` [ <. r , 1o >. ] ~Q ) }
38		breq2 3941	. . . . . . . 8 |- ( u = q -> ( ( *Q ` [ <. r , 1o >. ] ~Q ) <Q u <-> ( *Q ` [ <. r , 1o >. ] ~Q ) <Q q ) )
39	38	cbvabv 2265	. . . . . . 7 |- { u | ( *Q ` [ <. r , 1o >. ] ~Q ) <Q u } = { q | ( *Q ` [ <. r , 1o >. ] ~Q ) <Q q }
40	37, 39	opeq12i 3718	. . . . . 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 5793	. . . . 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 3944	. . . 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 2445	. . 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 121	. 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 7540	1 |- ( ph -> ( F ` K ) <P ( L +P. Q ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1332   e. wcel 1481   {cab 2126   A.wral 2417   E.wrex 2418   {crab 2421   <.cop 3535   class class class wbr 3937   -->wf 5127   ` cfv 5131  (class class class)co 5782   1oc1o 6314   [cec 6435   N.cnpi 7104   <N clti 7107   ~Q ceq 7111   Q.cnq 7112   +Q cplq 7114   *Qcrq 7116   <Q cltq 7117   P.cnp 7123   +P. cpp 7125   <P cltp 7127

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 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-coll 4051  ax-sep 4054  ax-nul 4062  ax-pow 4106  ax-pr 4139  ax-un 4363  ax-setind 4460  ax-iinf 4510

This theorem depends on definitions:  df-bi 116  df-dc 821  df-3or 964  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-ral 2422  df-rex 2423  df-reu 2424  df-rab 2426  df-v 2691  df-sbc 2914  df-csb 3008  df-dif 3078  df-un 3080  df-in 3082  df-ss 3089  df-nul 3369  df-pw 3517  df-sn 3538  df-pr 3539  df-op 3541  df-uni 3745  df-int 3780  df-iun 3823  df-br 3938  df-opab 3998  df-mpt 3999  df-tr 4035  df-eprel 4219  df-id 4223  df-po 4226  df-iso 4227  df-iord 4296  df-on 4298  df-suc 4301  df-iom 4513  df-xp 4553  df-rel 4554  df-cnv 4555  df-co 4556  df-dm 4557  df-rn 4558  df-res 4559  df-ima 4560  df-iota 5096  df-fun 5133  df-fn 5134  df-f 5135  df-f1 5136  df-fo 5137  df-f1o 5138  df-fv 5139  df-ov 5785  df-oprab 5786  df-mpo 5787  df-1st 6046  df-2nd 6047  df-recs 6210  df-irdg 6275  df-1o 6321  df-2o 6322  df-oadd 6325  df-omul 6326  df-er 6437  df-ec 6439  df-qs 6443  df-ni 7136  df-pli 7137  df-mi 7138  df-lti 7139  df-plpq 7176  df-mpq 7177  df-enq 7179  df-nqqs 7180  df-plqqs 7181  df-mqqs 7182  df-1nqqs 7183  df-rq 7184  df-ltnqqs 7185  df-enq0 7256  df-nq0 7257  df-0nq0 7258  df-plq0 7259  df-mq0 7260  df-inp 7298  df-iplp 7300  df-iltp 7302

This theorem is referenced by:  caucvgprprlemlim  7543