Theorem caucvgprprlemclphr 8062

Description: Lemma for caucvgprpr 8069. The putative limit is a positive real. Like caucvgprprlemcl 8061 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 3899	. . . . . . . . . . . . . 14 |- ( r = s -> <. r , 1o >. = <. s , 1o >. )
6	5	eceq1d 6833	. . . . . . . . . . . . 13 |- ( r = s -> [ <. r , 1o >. ] ~Q = [ <. s , 1o >. ] ~Q )
7	6	fveq2d 5694	. . . . . . . . . . . 12 |- ( r = s -> ( *Q ` [ <. r , 1o >. ] ~Q ) = ( *Q ` [ <. s , 1o >. ] ~Q ) )
8	7	oveq2d 6091	. . . . . . . . . . 11 |- ( r = s -> ( l +Q ( *Q ` [ <. r , 1o >. ] ~Q ) ) = ( l +Q ( *Q ` [ <. s , 1o >. ] ~Q ) ) )
9	8	breq2d 4137	. . . . . . . . . 10 |- ( r = s -> ( p <Q ( l +Q ( *Q ` [ <. r , 1o >. ] ~Q ) ) <-> p <Q ( l +Q ( *Q ` [ <. s , 1o >. ] ~Q ) ) ) )
10	9	abbidv 2358	. . . . . . . . 9 |- ( r = s -> { p | p <Q ( l +Q ( *Q ` [ <. r , 1o >. ] ~Q ) ) } = { p | p <Q ( l +Q ( *Q ` [ <. s , 1o >. ] ~Q ) ) } )
11	8	breq1d 4135	. . . . . . . . . 10 |- ( r = s -> ( ( l +Q ( *Q ` [ <. r , 1o >. ] ~Q ) ) <Q q <-> ( l +Q ( *Q ` [ <. s , 1o >. ] ~Q ) ) <Q q ) )
12	11	abbidv 2358	. . . . . . . . 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 3907	. . . . . . . 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 5690	. . . . . . . 8 |- ( r = s -> ( F ` r ) = ( F ` s ) )
15	13, 14	breq12d 4138	. . . . . . 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 2787	. . . . . 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 2807	. . . 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 4137	. . . . . . . . . . 11 |- ( r = s -> ( p <Q ( *Q ` [ <. r , 1o >. ] ~Q ) <-> p <Q ( *Q ` [ <. s , 1o >. ] ~Q ) ) )
20	19	abbidv 2358	. . . . . . . . . 10 |- ( r = s -> { p | p <Q ( *Q ` [ <. r , 1o >. ] ~Q ) } = { p | p <Q ( *Q ` [ <. s , 1o >. ] ~Q ) } )
21	7	breq1d 4135	. . . . . . . . . . 11 |- ( r = s -> ( ( *Q ` [ <. r , 1o >. ] ~Q ) <Q q <-> ( *Q ` [ <. s , 1o >. ] ~Q ) <Q q ) )
22	21	abbidv 2358	. . . . . . . . . 10 |- ( r = s -> { q | ( *Q ` [ <. r , 1o >. ] ~Q ) <Q q } = { q | ( *Q ` [ <. s , 1o >. ] ~Q ) <Q q } )
23	20, 22	opeq12d 3907	. . . . . . . . 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 6093	. . . . . . . 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 4135	. . . . . . 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 2787	. . . . . 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 2807	. . . 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 3904	. . 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 2259	. 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 8061	1 |- ( ph -> L e. P. )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1402   e. wcel 2209   {cab 2224   A.wral 2528   E.wrex 2529   {crab 2532   <.cop 3708   class class class wbr 4125   -->wf 5368   ` cfv 5372  (class class class)co 6075   1oc1o 6670   [cec 6795   N.cnpi 7629   <N clti 7632   ~Q ceq 7636   Q.cnq 7637   +Q cplq 7639   *Qcrq 7641   <Q cltq 7642   P.cnp 7648   +P. cpp 7650   <P cltp 7652

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 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 4241  ax-sep 4244  ax-nul 4254  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-setind 4679  ax-iinf 4730

This theorem 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 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-int 3966  df-iun 4009  df-br 4126  df-opab 4188  df-mpt 4189  df-tr 4225  df-eprel 4429  df-id 4433  df-po 4436  df-iso 4437  df-iord 4506  df-on 4508  df-suc 4511  df-iom 4733  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782  df-iota 5332  df-fun 5374  df-fn 5375  df-f 5376  df-f1 5377  df-fo 5378  df-f1o 5379  df-fv 5380  df-ov 6078  df-oprab 6079  df-mpo 6080  df-1st 6364  df-2nd 6365  df-recs 6566  df-irdg 6631  df-1o 6677  df-2o 6678  df-oadd 6681  df-omul 6682  df-er 6797  df-ec 6799  df-qs 6803  df-ni 7661  df-pli 7662  df-mi 7663  df-lti 7664  df-plpq 7701  df-mpq 7702  df-enq 7704  df-nqqs 7705  df-plqqs 7706  df-mqqs 7707  df-1nqqs 7708  df-rq 7709  df-ltnqqs 7710  df-enq0 7781  df-nq0 7782  df-0nq0 7783  df-plq0 7784  df-mq0 7785  df-inp 7823  df-iplp 7825  df-iltp 7827

This theorem is referenced by:  caucvgprprlemexbt  8063  caucvgprprlemexb  8064