Theorem caucvgprprlemclphr 8036

Description: Lemma for caucvgprpr 8043. The putative limit is a positive real. Like caucvgprprlemcl 8035 but without a disjoint variable condition between ph and r. (Contributed by Jim Kingdon, 19-Jun-2021.)

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 } >. } >.

Assertion

Ref	Expression
caucvgprprlemclphr	|- ( ph -> L e. P. )

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

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

Proof of Theorem caucvgprprlemclphr

Dummy variable s is distinct from all other variables.

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	. . 3 |- 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		opeq1 3888	. . . . . . . . . . . . . 14 |- ( r = s -> <. r , 1o >. = <. s , 1o >. )
6	5	eceq1d 6816	. . . . . . . . . . . . 13 |- ( r = s -> [ <. r , 1o >. ] ~Q = [ <. s , 1o >. ] ~Q )
7	6	fveq2d 5679	. . . . . . . . . . . 12 |- ( r = s -> ( *Q ` [ <. r , 1o >. ] ~Q ) = ( *Q ` [ <. s , 1o >. ] ~Q ) )
8	7	oveq2d 6074	. . . . . . . . . . 11 |- ( r = s -> ( l +Q ( *Q ` [ <. r , 1o >. ] ~Q ) ) = ( l +Q ( *Q ` [ <. s , 1o >. ] ~Q ) ) )
9	8	breq2d 4126	. . . . . . . . . 10 |- ( r = s -> ( p <Q ( l +Q ( *Q ` [ <. r , 1o >. ] ~Q ) ) <-> p <Q ( l +Q ( *Q ` [ <. s , 1o >. ] ~Q ) ) ) )
10	9	abbidv 2354	. . . . . . . . 9 |- ( r = s -> { p | p <Q ( l +Q ( *Q ` [ <. r , 1o >. ] ~Q ) ) } = { p | p <Q ( l +Q ( *Q ` [ <. s , 1o >. ] ~Q ) ) } )
11	8	breq1d 4124	. . . . . . . . . 10 |- ( r = s -> ( ( l +Q ( *Q ` [ <. r , 1o >. ] ~Q ) ) <Q q <-> ( l +Q ( *Q ` [ <. s , 1o >. ] ~Q ) ) <Q q ) )
12	11	abbidv 2354	. . . . . . . . 9 |- ( r = s -> { q | ( l +Q ( *Q ` [ <. r , 1o >. ] ~Q ) ) <Q q } = { q | ( l +Q ( *Q ` [ <. s , 1o >. ] ~Q ) ) <Q q } )
13	10, 12	opeq12d 3896	. . . . . . . 8 |- ( r = s -> <. { p | p <Q ( l +Q ( *Q ` [ <. r , 1o >. ] ~Q ) ) } , { q | ( l +Q ( *Q ` [ <. r , 1o >. ] ~Q ) ) <Q q } >. = <. { p | p <Q ( l +Q ( *Q ` [ <. s , 1o >. ] ~Q ) ) } , { q | ( l +Q ( *Q ` [ <. s , 1o >. ] ~Q ) ) <Q q } >. )
14		fveq2 5675	. . . . . . . 8 |- ( r = s -> ( F ` r ) = ( F ` s ) )
15	13, 14	breq12d 4127	. . . . . . 7 |- ( r = s -> ( <. { p | p <Q ( l +Q ( *Q ` [ <. r , 1o >. ] ~Q ) ) } , { q | ( l +Q ( *Q ` [ <. r , 1o >. ] ~Q ) ) <Q q } >. <P ( F ` r ) <-> <. { p | p <Q ( l +Q ( *Q ` [ <. s , 1o >. ] ~Q ) ) } , { q | ( l +Q ( *Q ` [ <. s , 1o >. ] ~Q ) ) <Q q } >. <P ( F ` s ) ) )
16	15	cbvrexv 2781	. . . . . 6 |- ( 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 ) <-> E. s e. N. <. { p | p <Q ( l +Q ( *Q ` [ <. s , 1o >. ] ~Q ) ) } , { q | ( l +Q ( *Q ` [ <. s , 1o >. ] ~Q ) ) <Q q } >. <P ( F ` s ) )
17	16	a1i 9	. . . . 5 |- ( 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 ) <-> E. s e. N. <. { p | p <Q ( l +Q ( *Q ` [ <. s , 1o >. ] ~Q ) ) } , { q | ( l +Q ( *Q ` [ <. s , 1o >. ] ~Q ) ) <Q q } >. <P ( F ` s ) ) )
18	17	rabbiia 2801	. . . 4 |- { 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 ) } = { l e. Q. | E. s e. N. <. { p | p <Q ( l +Q ( *Q ` [ <. s , 1o >. ] ~Q ) ) } , { q | ( l +Q ( *Q ` [ <. s , 1o >. ] ~Q ) ) <Q q } >. <P ( F ` s ) }
19	7	breq2d 4126	. . . . . . . . . . 11 |- ( r = s -> ( p <Q ( *Q ` [ <. r , 1o >. ] ~Q ) <-> p <Q ( *Q ` [ <. s , 1o >. ] ~Q ) ) )
20	19	abbidv 2354	. . . . . . . . . 10 |- ( r = s -> { p | p <Q ( *Q ` [ <. r , 1o >. ] ~Q ) } = { p | p <Q ( *Q ` [ <. s , 1o >. ] ~Q ) } )
21	7	breq1d 4124	. . . . . . . . . . 11 |- ( r = s -> ( ( *Q ` [ <. r , 1o >. ] ~Q ) <Q q <-> ( *Q ` [ <. s , 1o >. ] ~Q ) <Q q ) )
22	21	abbidv 2354	. . . . . . . . . 10 |- ( r = s -> { q | ( *Q ` [ <. r , 1o >. ] ~Q ) <Q q } = { q | ( *Q ` [ <. s , 1o >. ] ~Q ) <Q q } )
23	20, 22	opeq12d 3896	. . . . . . . . 9 |- ( r = s -> <. { p | p <Q ( *Q ` [ <. r , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. r , 1o >. ] ~Q ) <Q q } >. = <. { p | p <Q ( *Q ` [ <. s , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. s , 1o >. ] ~Q ) <Q q } >. )
24	14, 23	oveq12d 6076	. . . . . . . 8 |- ( r = s -> ( ( F ` r ) +P. <. { p | p <Q ( *Q ` [ <. r , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. r , 1o >. ] ~Q ) <Q q } >. ) = ( ( F ` s ) +P. <. { p | p <Q ( *Q ` [ <. s , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. s , 1o >. ] ~Q ) <Q q } >. ) )
25	24	breq1d 4124	. . . . . . 7 |- ( r = s -> ( ( ( 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 } >. <-> ( ( F ` s ) +P. <. { p | p <Q ( *Q ` [ <. s , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. s , 1o >. ] ~Q ) <Q q } >. ) <P <. { p | p <Q u } , { q | u <Q q } >. ) )
26	25	cbvrexv 2781	. . . . . 6 |- ( 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 } >. <-> E. s e. N. ( ( F ` s ) +P. <. { p | p <Q ( *Q ` [ <. s , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. s , 1o >. ] ~Q ) <Q q } >. ) <P <. { p | p <Q u } , { q | u <Q q } >. )
27	26	a1i 9	. . . . 5 |- ( 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 } >. <-> E. s e. N. ( ( F ` s ) +P. <. { p | p <Q ( *Q ` [ <. s , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. s , 1o >. ] ~Q ) <Q q } >. ) <P <. { p | p <Q u } , { q | u <Q q } >. ) )
28	27	rabbiia 2801	. . . 4 |- { 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 } >. } = { u e. Q. | E. s e. N. ( ( F ` s ) +P. <. { p | p <Q ( *Q ` [ <. s , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. s , 1o >. ] ~Q ) <Q q } >. ) <P <. { p | p <Q u } , { q | u <Q q } >. }
29	18, 28	opeq12i 3893	. . 3 |- <. { 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 } >. } >. = <. { l e. Q. | E. s e. N. <. { p | p <Q ( l +Q ( *Q ` [ <. s , 1o >. ] ~Q ) ) } , { q | ( l +Q ( *Q ` [ <. s , 1o >. ] ~Q ) ) <Q q } >. <P ( F ` s ) } , { u e. Q. | E. s e. N. ( ( F ` s ) +P. <. { p | p <Q ( *Q ` [ <. s , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. s , 1o >. ] ~Q ) <Q q } >. ) <P <. { p | p <Q u } , { q | u <Q q } >. } >.
30	4, 29	eqtri 2255	. 2 |- L = <. { l e. Q. | E. s e. N. <. { p | p <Q ( l +Q ( *Q ` [ <. s , 1o >. ] ~Q ) ) } , { q | ( l +Q ( *Q ` [ <. s , 1o >. ] ~Q ) ) <Q q } >. <P ( F ` s ) } , { u e. Q. | E. s e. N. ( ( F ` s ) +P. <. { p | p <Q ( *Q ` [ <. s , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. s , 1o >. ] ~Q ) <Q q } >. ) <P <. { p | p <Q u } , { q | u <Q q } >. } >.
31	1, 2, 3, 30	caucvgprprlemcl 8035	1 |- ( ph -> L e. P. )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1398   e. wcel 2205   {cab 2220   A.wral 2522   E.wrex 2523   {crab 2526   <.cop 3697   class class class wbr 4114   -->wf 5353   ` cfv 5357  (class class class)co 6058   1oc1o 6653   [cec 6778   N.cnpi 7603   <N clti 7606   ~Q ceq 7610   Q.cnq 7611   +Q cplq 7613   *Qcrq 7615   <Q cltq 7616   P.cnp 7622   +P. cpp 7624   <P cltp 7626

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 2207  ax-14 2208  ax-ext 2216  ax-coll 4230  ax-sep 4233  ax-nul 4241  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664  ax-iinf 4715

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 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-ral 2527  df-rex 2528  df-reu 2529  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-nul 3513  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-int 3955  df-iun 3998  df-br 4115  df-opab 4177  df-mpt 4178  df-tr 4214  df-eprel 4415  df-id 4419  df-po 4422  df-iso 4423  df-iord 4492  df-on 4494  df-suc 4497  df-iom 4718  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-f1 5362  df-fo 5363  df-f1o 5364  df-fv 5365  df-ov 6061  df-oprab 6062  df-mpo 6063  df-1st 6347  df-2nd 6348  df-recs 6549  df-irdg 6614  df-1o 6660  df-2o 6661  df-oadd 6664  df-omul 6665  df-er 6780  df-ec 6782  df-qs 6786  df-ni 7635  df-pli 7636  df-mi 7637  df-lti 7638  df-plpq 7675  df-mpq 7676  df-enq 7678  df-nqqs 7679  df-plqqs 7680  df-mqqs 7681  df-1nqqs 7682  df-rq 7683  df-ltnqqs 7684  df-enq0 7755  df-nq0 7756  df-0nq0 7757  df-plq0 7758  df-mq0 7759  df-inp 7797  df-iplp 7799  df-iltp 7801

This theorem is referenced by:  caucvgprprlemexbt  8037  caucvgprprlemexb  8038