Theorem caucvgprprlemclphr 7853

Description: Lemma for caucvgprpr 7860. The putative limit is a positive real. Like caucvgprprlemcl 7852 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 3833	. . . . . . . . . . . . . 14 |- ( r = s -> <. r , 1o >. = <. s , 1o >. )
6	5	eceq1d 6679	. . . . . . . . . . . . 13 |- ( r = s -> [ <. r , 1o >. ] ~Q = [ <. s , 1o >. ] ~Q )
7	6	fveq2d 5603	. . . . . . . . . . . 12 |- ( r = s -> ( *Q ` [ <. r , 1o >. ] ~Q ) = ( *Q ` [ <. s , 1o >. ] ~Q ) )
8	7	oveq2d 5983	. . . . . . . . . . 11 |- ( r = s -> ( l +Q ( *Q ` [ <. r , 1o >. ] ~Q ) ) = ( l +Q ( *Q ` [ <. s , 1o >. ] ~Q ) ) )
9	8	breq2d 4071	. . . . . . . . . 10 |- ( r = s -> ( p <Q ( l +Q ( *Q ` [ <. r , 1o >. ] ~Q ) ) <-> p <Q ( l +Q ( *Q ` [ <. s , 1o >. ] ~Q ) ) ) )
10	9	abbidv 2325	. . . . . . . . 9 |- ( r = s -> { p | p <Q ( l +Q ( *Q ` [ <. r , 1o >. ] ~Q ) ) } = { p | p <Q ( l +Q ( *Q ` [ <. s , 1o >. ] ~Q ) ) } )
11	8	breq1d 4069	. . . . . . . . . 10 |- ( r = s -> ( ( l +Q ( *Q ` [ <. r , 1o >. ] ~Q ) ) <Q q <-> ( l +Q ( *Q ` [ <. s , 1o >. ] ~Q ) ) <Q q ) )
12	11	abbidv 2325	. . . . . . . . 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 3841	. . . . . . . 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 5599	. . . . . . . 8 |- ( r = s -> ( F ` r ) = ( F ` s ) )
15	13, 14	breq12d 4072	. . . . . . 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 2743	. . . . . 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 2761	. . . 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 4071	. . . . . . . . . . 11 |- ( r = s -> ( p <Q ( *Q ` [ <. r , 1o >. ] ~Q ) <-> p <Q ( *Q ` [ <. s , 1o >. ] ~Q ) ) )
20	19	abbidv 2325	. . . . . . . . . 10 |- ( r = s -> { p | p <Q ( *Q ` [ <. r , 1o >. ] ~Q ) } = { p | p <Q ( *Q ` [ <. s , 1o >. ] ~Q ) } )
21	7	breq1d 4069	. . . . . . . . . . 11 |- ( r = s -> ( ( *Q ` [ <. r , 1o >. ] ~Q ) <Q q <-> ( *Q ` [ <. s , 1o >. ] ~Q ) <Q q ) )
22	21	abbidv 2325	. . . . . . . . . 10 |- ( r = s -> { q | ( *Q ` [ <. r , 1o >. ] ~Q ) <Q q } = { q | ( *Q ` [ <. s , 1o >. ] ~Q ) <Q q } )
23	20, 22	opeq12d 3841	. . . . . . . . 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 5985	. . . . . . . 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 4069	. . . . . . 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 2743	. . . . . 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 2761	. . . 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 3838	. . 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 2228	. 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 7852	1 |- ( ph -> L e. P. )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1373   e. wcel 2178   {cab 2193   A.wral 2486   E.wrex 2487   {crab 2490   <.cop 3646   class class class wbr 4059   -->wf 5286   ` cfv 5290  (class class class)co 5967   1oc1o 6518   [cec 6641   N.cnpi 7420   <N clti 7423   ~Q ceq 7427   Q.cnq 7428   +Q cplq 7430   *Qcrq 7432   <Q cltq 7433   P.cnp 7439   +P. cpp 7441   <P cltp 7443

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 615  ax-in2 616  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2180  ax-14 2181  ax-ext 2189  ax-coll 4175  ax-sep 4178  ax-nul 4186  ax-pow 4234  ax-pr 4269  ax-un 4498  ax-setind 4603  ax-iinf 4654

This theorem depends on definitions:  df-bi 117  df-dc 837  df-3or 982  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ne 2379  df-ral 2491  df-rex 2492  df-reu 2493  df-rab 2495  df-v 2778  df-sbc 3006  df-csb 3102  df-dif 3176  df-un 3178  df-in 3180  df-ss 3187  df-nul 3469  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3865  df-int 3900  df-iun 3943  df-br 4060  df-opab 4122  df-mpt 4123  df-tr 4159  df-eprel 4354  df-id 4358  df-po 4361  df-iso 4362  df-iord 4431  df-on 4433  df-suc 4436  df-iom 4657  df-xp 4699  df-rel 4700  df-cnv 4701  df-co 4702  df-dm 4703  df-rn 4704  df-res 4705  df-ima 4706  df-iota 5251  df-fun 5292  df-fn 5293  df-f 5294  df-f1 5295  df-fo 5296  df-f1o 5297  df-fv 5298  df-ov 5970  df-oprab 5971  df-mpo 5972  df-1st 6249  df-2nd 6250  df-recs 6414  df-irdg 6479  df-1o 6525  df-2o 6526  df-oadd 6529  df-omul 6530  df-er 6643  df-ec 6645  df-qs 6649  df-ni 7452  df-pli 7453  df-mi 7454  df-lti 7455  df-plpq 7492  df-mpq 7493  df-enq 7495  df-nqqs 7496  df-plqqs 7497  df-mqqs 7498  df-1nqqs 7499  df-rq 7500  df-ltnqqs 7501  df-enq0 7572  df-nq0 7573  df-0nq0 7574  df-plq0 7575  df-mq0 7576  df-inp 7614  df-iplp 7616  df-iltp 7618

This theorem is referenced by:  caucvgprprlemexbt  7854  caucvgprprlemexb  7855