Theorem caucvgprprlemclphr 7903

Description: Lemma for caucvgprpr 7910. The putative limit is a positive real. Like caucvgprprlemcl 7902 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 3857	. . . . . . . . . . . . . 14 |- ( r = s -> <. r , 1o >. = <. s , 1o >. )
6	5	eceq1d 6724	. . . . . . . . . . . . 13 |- ( r = s -> [ <. r , 1o >. ] ~Q = [ <. s , 1o >. ] ~Q )
7	6	fveq2d 5633	. . . . . . . . . . . 12 |- ( r = s -> ( *Q ` [ <. r , 1o >. ] ~Q ) = ( *Q ` [ <. s , 1o >. ] ~Q ) )
8	7	oveq2d 6023	. . . . . . . . . . 11 |- ( r = s -> ( l +Q ( *Q ` [ <. r , 1o >. ] ~Q ) ) = ( l +Q ( *Q ` [ <. s , 1o >. ] ~Q ) ) )
9	8	breq2d 4095	. . . . . . . . . 10 |- ( r = s -> ( p <Q ( l +Q ( *Q ` [ <. r , 1o >. ] ~Q ) ) <-> p <Q ( l +Q ( *Q ` [ <. s , 1o >. ] ~Q ) ) ) )
10	9	abbidv 2347	. . . . . . . . 9 |- ( r = s -> { p | p <Q ( l +Q ( *Q ` [ <. r , 1o >. ] ~Q ) ) } = { p | p <Q ( l +Q ( *Q ` [ <. s , 1o >. ] ~Q ) ) } )
11	8	breq1d 4093	. . . . . . . . . 10 |- ( r = s -> ( ( l +Q ( *Q ` [ <. r , 1o >. ] ~Q ) ) <Q q <-> ( l +Q ( *Q ` [ <. s , 1o >. ] ~Q ) ) <Q q ) )
12	11	abbidv 2347	. . . . . . . . 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 3865	. . . . . . . 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 5629	. . . . . . . 8 |- ( r = s -> ( F ` r ) = ( F ` s ) )
15	13, 14	breq12d 4096	. . . . . . 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 2766	. . . . . 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 2784	. . . 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 4095	. . . . . . . . . . 11 |- ( r = s -> ( p <Q ( *Q ` [ <. r , 1o >. ] ~Q ) <-> p <Q ( *Q ` [ <. s , 1o >. ] ~Q ) ) )
20	19	abbidv 2347	. . . . . . . . . 10 |- ( r = s -> { p | p <Q ( *Q ` [ <. r , 1o >. ] ~Q ) } = { p | p <Q ( *Q ` [ <. s , 1o >. ] ~Q ) } )
21	7	breq1d 4093	. . . . . . . . . . 11 |- ( r = s -> ( ( *Q ` [ <. r , 1o >. ] ~Q ) <Q q <-> ( *Q ` [ <. s , 1o >. ] ~Q ) <Q q ) )
22	21	abbidv 2347	. . . . . . . . . 10 |- ( r = s -> { q | ( *Q ` [ <. r , 1o >. ] ~Q ) <Q q } = { q | ( *Q ` [ <. s , 1o >. ] ~Q ) <Q q } )
23	20, 22	opeq12d 3865	. . . . . . . . 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 6025	. . . . . . . 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 4093	. . . . . . 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 2766	. . . . . 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 2784	. . . 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 3862	. . 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 2250	. 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 7902	1 |- ( ph -> L e. P. )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1395   e. wcel 2200   {cab 2215   A.wral 2508   E.wrex 2509   {crab 2512   <.cop 3669   class class class wbr 4083   -->wf 5314   ` cfv 5318  (class class class)co 6007   1oc1o 6561   [cec 6686   N.cnpi 7470   <N clti 7473   ~Q ceq 7477   Q.cnq 7478   +Q cplq 7480   *Qcrq 7482   <Q cltq 7483   P.cnp 7489   +P. cpp 7491   <P cltp 7493

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4199  ax-sep 4202  ax-nul 4210  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629  ax-iinf 4680

This theorem depends on definitions:  df-bi 117  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-int 3924  df-iun 3967  df-br 4084  df-opab 4146  df-mpt 4147  df-tr 4183  df-eprel 4380  df-id 4384  df-po 4387  df-iso 4388  df-iord 4457  df-on 4459  df-suc 4462  df-iom 4683  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-ima 4732  df-iota 5278  df-fun 5320  df-fn 5321  df-f 5322  df-f1 5323  df-fo 5324  df-f1o 5325  df-fv 5326  df-ov 6010  df-oprab 6011  df-mpo 6012  df-1st 6292  df-2nd 6293  df-recs 6457  df-irdg 6522  df-1o 6568  df-2o 6569  df-oadd 6572  df-omul 6573  df-er 6688  df-ec 6690  df-qs 6694  df-ni 7502  df-pli 7503  df-mi 7504  df-lti 7505  df-plpq 7542  df-mpq 7543  df-enq 7545  df-nqqs 7546  df-plqqs 7547  df-mqqs 7548  df-1nqqs 7549  df-rq 7550  df-ltnqqs 7551  df-enq0 7622  df-nq0 7623  df-0nq0 7624  df-plq0 7625  df-mq0 7626  df-inp 7664  df-iplp 7666  df-iltp 7668

This theorem is referenced by:  caucvgprprlemexbt  7904  caucvgprprlemexb  7905