Theorem caucvgprprlemclphr 7262

Description: Lemma for caucvgprpr 7269. The putative limit is a positive real. Like caucvgprprlemcl 7261 but without a distinct variable constraint 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 3622	. . . . . . . . . . . . . 14 |- ( r = s -> <. r , 1o >. = <. s , 1o >. )
6	5	eceq1d 6326	. . . . . . . . . . . . 13 |- ( r = s -> [ <. r , 1o >. ] ~Q = [ <. s , 1o >. ] ~Q )
7	6	fveq2d 5309	. . . . . . . . . . . 12 |- ( r = s -> ( *Q ` [ <. r , 1o >. ] ~Q ) = ( *Q ` [ <. s , 1o >. ] ~Q ) )
8	7	oveq2d 5668	. . . . . . . . . . 11 |- ( r = s -> ( l +Q ( *Q ` [ <. r , 1o >. ] ~Q ) ) = ( l +Q ( *Q ` [ <. s , 1o >. ] ~Q ) ) )
9	8	breq2d 3857	. . . . . . . . . 10 |- ( r = s -> ( p <Q ( l +Q ( *Q ` [ <. r , 1o >. ] ~Q ) ) <-> p <Q ( l +Q ( *Q ` [ <. s , 1o >. ] ~Q ) ) ) )
10	9	abbidv 2205	. . . . . . . . 9 |- ( r = s -> { p | p <Q ( l +Q ( *Q ` [ <. r , 1o >. ] ~Q ) ) } = { p | p <Q ( l +Q ( *Q ` [ <. s , 1o >. ] ~Q ) ) } )
11	8	breq1d 3855	. . . . . . . . . 10 |- ( r = s -> ( ( l +Q ( *Q ` [ <. r , 1o >. ] ~Q ) ) <Q q <-> ( l +Q ( *Q ` [ <. s , 1o >. ] ~Q ) ) <Q q ) )
12	11	abbidv 2205	. . . . . . . . 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 3630	. . . . . . . 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 5305	. . . . . . . 8 |- ( r = s -> ( F ` r ) = ( F ` s ) )
15	13, 14	breq12d 3858	. . . . . . 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 2591	. . . . . 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 2604	. . . 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 3857	. . . . . . . . . . 11 |- ( r = s -> ( p <Q ( *Q ` [ <. r , 1o >. ] ~Q ) <-> p <Q ( *Q ` [ <. s , 1o >. ] ~Q ) ) )
20	19	abbidv 2205	. . . . . . . . . 10 |- ( r = s -> { p | p <Q ( *Q ` [ <. r , 1o >. ] ~Q ) } = { p | p <Q ( *Q ` [ <. s , 1o >. ] ~Q ) } )
21	7	breq1d 3855	. . . . . . . . . . 11 |- ( r = s -> ( ( *Q ` [ <. r , 1o >. ] ~Q ) <Q q <-> ( *Q ` [ <. s , 1o >. ] ~Q ) <Q q ) )
22	21	abbidv 2205	. . . . . . . . . 10 |- ( r = s -> { q | ( *Q ` [ <. r , 1o >. ] ~Q ) <Q q } = { q | ( *Q ` [ <. s , 1o >. ] ~Q ) <Q q } )
23	20, 22	opeq12d 3630	. . . . . . . . 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 5670	. . . . . . . 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 3855	. . . . . . 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 2591	. . . . . 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 2604	. . . 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 3627	. . 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 2108	. 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 7261	1 |- ( ph -> L e. P. )

Syntax hints:   -> wi 4   /\ wa 102   <-> wb 103   = wceq 1289   e. wcel 1438   {cab 2074   A.wral 2359   E.wrex 2360   {crab 2363   <.cop 3449   class class class wbr 3845   -->wf 5011   ` cfv 5015  (class class class)co 5652   1oc1o 6174   [cec 6288   N.cnpi 6829   <N clti 6832   ~Q ceq 6836   Q.cnq 6837   +Q cplq 6839   *Qcrq 6841   <Q cltq 6842   P.cnp 6848   +P. cpp 6850   <P cltp 6852

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-coll 3954  ax-sep 3957  ax-nul 3965  ax-pow 4009  ax-pr 4036  ax-un 4260  ax-setind 4353  ax-iinf 4403

This theorem depends on definitions:  df-bi 115  df-dc 781  df-3or 925  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-ral 2364  df-rex 2365  df-reu 2366  df-rab 2368  df-v 2621  df-sbc 2841  df-csb 2934  df-dif 3001  df-un 3003  df-in 3005  df-ss 3012  df-nul 3287  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-uni 3654  df-int 3689  df-iun 3732  df-br 3846  df-opab 3900  df-mpt 3901  df-tr 3937  df-eprel 4116  df-id 4120  df-po 4123  df-iso 4124  df-iord 4193  df-on 4195  df-suc 4198  df-iom 4406  df-xp 4444  df-rel 4445  df-cnv 4446  df-co 4447  df-dm 4448  df-rn 4449  df-res 4450  df-ima 4451  df-iota 4980  df-fun 5017  df-fn 5018  df-f 5019  df-f1 5020  df-fo 5021  df-f1o 5022  df-fv 5023  df-ov 5655  df-oprab 5656  df-mpt2 5657  df-1st 5911  df-2nd 5912  df-recs 6070  df-irdg 6135  df-1o 6181  df-2o 6182  df-oadd 6185  df-omul 6186  df-er 6290  df-ec 6292  df-qs 6296  df-ni 6861  df-pli 6862  df-mi 6863  df-lti 6864  df-plpq 6901  df-mpq 6902  df-enq 6904  df-nqqs 6905  df-plqqs 6906  df-mqqs 6907  df-1nqqs 6908  df-rq 6909  df-ltnqqs 6910  df-enq0 6981  df-nq0 6982  df-0nq0 6983  df-plq0 6984  df-mq0 6985  df-inp 7023  df-iplp 7025  df-iltp 7027

This theorem is referenced by:  caucvgprprlemexbt  7263  caucvgprprlemexb  7264