Theorem caucvgpr 7623

Description: A Cauchy sequence of positive fractions with a modulus of convergence converges to a positive real. This is basically Corollary 11.2.13 of [HoTT], p. (varies) (one key difference being that this is for positive reals rather than signed reals). Also, the HoTT book theorem has a modulus of convergence (that is, a rate of convergence) specified by (11.2.9) in HoTT whereas this theorem fixes the rate of convergence to say that all terms after the nth term must be within 1 / n of the nth term (it should later be able to prove versions of this theorem with a different fixed rate or a modulus of convergence supplied as a hypothesis). We also specify that every term needs to be larger than a fraction A, to avoid the case where we have positive terms which "converge" to zero (which is not a positive real).

This proof (including its lemmas) is similar to the proofs of cauappcvgpr 7603 and caucvgprpr 7653. Reading cauappcvgpr 7603 first (the simplest of the three) might help understanding the other two.

(Contributed by Jim Kingdon, 18-Jun-2020.)

Hypotheses

Ref	Expression
caucvgpr.f	|- ( ph -> F : N. --> Q. )
caucvgpr.cau	|- ( ph -> A. n e. N. A. k e. N. ( n <N k -> ( ( F ` n ) <Q ( ( F ` k ) +Q ( *Q ` [ <. n , 1o >. ] ~Q ) ) /\ ( F ` k ) <Q ( ( F ` n ) +Q ( *Q ` [ <. n , 1o >. ] ~Q ) ) ) ) )
caucvgpr.bnd	|- ( ph -> A. j e. N. A <Q ( F ` j ) )

Assertion

Ref	Expression
caucvgpr	|- ( ph -> E. y e. P. A. x e. Q. E. j e. N. A. k e. N. ( j <N k -> ( <. { l | l <Q ( F ` k ) } , { u | ( F ` k ) <Q u } >. <P ( y +P. <. { l | l <Q x } , { u | x <Q u } >. ) /\ y <P <. { l | l <Q ( ( F ` k ) +Q x ) } , { u | ( ( F ` k ) +Q x ) <Q u } >. ) ) )

Distinct variable groups:   A, j   j, F, k, n, l, u, x, y   ph, j, k, x

Allowed substitution hints:   ph( y, u, n, l)   A( x, y, u, k, n, l)

Proof of Theorem caucvgpr

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		caucvgpr.f	. . 3 |- ( ph -> F : N. --> Q. )
2		caucvgpr.cau	. . 3 |- ( ph -> A. n e. N. A. k e. N. ( n <N k -> ( ( F ` n ) <Q ( ( F ` k ) +Q ( *Q ` [ <. n , 1o >. ] ~Q ) ) /\ ( F ` k ) <Q ( ( F ` n ) +Q ( *Q ` [ <. n , 1o >. ] ~Q ) ) ) ) )
3		caucvgpr.bnd	. . 3 |- ( ph -> A. j e. N. A <Q ( F ` j ) )
4		opeq1 3758	. . . . . . . . . . 11 |- ( z = j -> <. z , 1o >. = <. j , 1o >. )
5	4	eceq1d 6537	. . . . . . . . . 10 |- ( z = j -> [ <. z , 1o >. ] ~Q = [ <. j , 1o >. ] ~Q )
6	5	fveq2d 5490	. . . . . . . . 9 |- ( z = j -> ( *Q ` [ <. z , 1o >. ] ~Q ) = ( *Q ` [ <. j , 1o >. ] ~Q ) )
7	6	oveq2d 5858	. . . . . . . 8 |- ( z = j -> ( l +Q ( *Q ` [ <. z , 1o >. ] ~Q ) ) = ( l +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) )
8		fveq2 5486	. . . . . . . 8 |- ( z = j -> ( F ` z ) = ( F ` j ) )
9	7, 8	breq12d 3995	. . . . . . 7 |- ( z = j -> ( ( l +Q ( *Q ` [ <. z , 1o >. ] ~Q ) ) <Q ( F ` z ) <-> ( l +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( F ` j ) ) )
10	9	cbvrexv 2693	. . . . . 6 |- ( E. z e. N. ( l +Q ( *Q ` [ <. z , 1o >. ] ~Q ) ) <Q ( F ` z ) <-> E. j e. N. ( l +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( F ` j ) )
11	10	a1i 9	. . . . 5 |- ( l e. Q. -> ( E. z e. N. ( l +Q ( *Q ` [ <. z , 1o >. ] ~Q ) ) <Q ( F ` z ) <-> E. j e. N. ( l +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( F ` j ) ) )
12	11	rabbiia 2711	. . . 4 |- { l e. Q. | E. z e. N. ( l +Q ( *Q ` [ <. z , 1o >. ] ~Q ) ) <Q ( F ` z ) } = { l e. Q. | E. j e. N. ( l +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( F ` j ) }
13	8, 6	oveq12d 5860	. . . . . . . 8 |- ( z = j -> ( ( F ` z ) +Q ( *Q ` [ <. z , 1o >. ] ~Q ) ) = ( ( F ` j ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) )
14	13	breq1d 3992	. . . . . . 7 |- ( z = j -> ( ( ( F ` z ) +Q ( *Q ` [ <. z , 1o >. ] ~Q ) ) <Q u <-> ( ( F ` j ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q u ) )
15	14	cbvrexv 2693	. . . . . 6 |- ( E. z e. N. ( ( F ` z ) +Q ( *Q ` [ <. z , 1o >. ] ~Q ) ) <Q u <-> E. j e. N. ( ( F ` j ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q u )
16	15	a1i 9	. . . . 5 |- ( u e. Q. -> ( E. z e. N. ( ( F ` z ) +Q ( *Q ` [ <. z , 1o >. ] ~Q ) ) <Q u <-> E. j e. N. ( ( F ` j ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q u ) )
17	16	rabbiia 2711	. . . 4 |- { u e. Q. | E. z e. N. ( ( F ` z ) +Q ( *Q ` [ <. z , 1o >. ] ~Q ) ) <Q u } = { u e. Q. | E. j e. N. ( ( F ` j ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q u }
18	12, 17	opeq12i 3763	. . 3 |- <. { l e. Q. | E. z e. N. ( l +Q ( *Q ` [ <. z , 1o >. ] ~Q ) ) <Q ( F ` z ) } , { u e. Q. | E. z e. N. ( ( F ` z ) +Q ( *Q ` [ <. z , 1o >. ] ~Q ) ) <Q u } >. = <. { l e. Q. | E. j e. N. ( l +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( F ` j ) } , { u e. Q. | E. j e. N. ( ( F ` j ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q u } >.
19	1, 2, 3, 18	caucvgprlemcl 7617	. 2 |- ( ph -> <. { l e. Q. | E. z e. N. ( l +Q ( *Q ` [ <. z , 1o >. ] ~Q ) ) <Q ( F ` z ) } , { u e. Q. | E. z e. N. ( ( F ` z ) +Q ( *Q ` [ <. z , 1o >. ] ~Q ) ) <Q u } >. e. P. )
20	1, 2, 3, 18	caucvgprlemlim 7622	. 2 |- ( ph -> A. x e. Q. E. j e. N. A. k e. N. ( j <N k -> ( <. { l | l <Q ( F ` k ) } , { u | ( F ` k ) <Q u } >. <P ( <. { l e. Q. | E. z e. N. ( l +Q ( *Q ` [ <. z , 1o >. ] ~Q ) ) <Q ( F ` z ) } , { u e. Q. | E. z e. N. ( ( F ` z ) +Q ( *Q ` [ <. z , 1o >. ] ~Q ) ) <Q u } >. +P. <. { l | l <Q x } , { u | x <Q u } >. ) /\ <. { l e. Q. | E. z e. N. ( l +Q ( *Q ` [ <. z , 1o >. ] ~Q ) ) <Q ( F ` z ) } , { u e. Q. | E. z e. N. ( ( F ` z ) +Q ( *Q ` [ <. z , 1o >. ] ~Q ) ) <Q u } >. <P <. { l | l <Q ( ( F ` k ) +Q x ) } , { u | ( ( F ` k ) +Q x ) <Q u } >. ) ) )
21		oveq1 5849	. . . . . . . 8 |- ( y = <. { l e. Q. | E. z e. N. ( l +Q ( *Q ` [ <. z , 1o >. ] ~Q ) ) <Q ( F ` z ) } , { u e. Q. | E. z e. N. ( ( F ` z ) +Q ( *Q ` [ <. z , 1o >. ] ~Q ) ) <Q u } >. -> ( y +P. <. { l | l <Q x } , { u | x <Q u } >. ) = ( <. { l e. Q. | E. z e. N. ( l +Q ( *Q ` [ <. z , 1o >. ] ~Q ) ) <Q ( F ` z ) } , { u e. Q. | E. z e. N. ( ( F ` z ) +Q ( *Q ` [ <. z , 1o >. ] ~Q ) ) <Q u } >. +P. <. { l | l <Q x } , { u | x <Q u } >. ) )
22	21	breq2d 3994	. . . . . . 7 |- ( y = <. { l e. Q. | E. z e. N. ( l +Q ( *Q ` [ <. z , 1o >. ] ~Q ) ) <Q ( F ` z ) } , { u e. Q. | E. z e. N. ( ( F ` z ) +Q ( *Q ` [ <. z , 1o >. ] ~Q ) ) <Q u } >. -> ( <. { l | l <Q ( F ` k ) } , { u | ( F ` k ) <Q u } >. <P ( y +P. <. { l | l <Q x } , { u | x <Q u } >. ) <-> <. { l | l <Q ( F ` k ) } , { u | ( F ` k ) <Q u } >. <P ( <. { l e. Q. | E. z e. N. ( l +Q ( *Q ` [ <. z , 1o >. ] ~Q ) ) <Q ( F ` z ) } , { u e. Q. | E. z e. N. ( ( F ` z ) +Q ( *Q ` [ <. z , 1o >. ] ~Q ) ) <Q u } >. +P. <. { l | l <Q x } , { u | x <Q u } >. ) ) )
23		breq1 3985	. . . . . . 7 |- ( y = <. { l e. Q. | E. z e. N. ( l +Q ( *Q ` [ <. z , 1o >. ] ~Q ) ) <Q ( F ` z ) } , { u e. Q. | E. z e. N. ( ( F ` z ) +Q ( *Q ` [ <. z , 1o >. ] ~Q ) ) <Q u } >. -> ( y <P <. { l | l <Q ( ( F ` k ) +Q x ) } , { u | ( ( F ` k ) +Q x ) <Q u } >. <-> <. { l e. Q. | E. z e. N. ( l +Q ( *Q ` [ <. z , 1o >. ] ~Q ) ) <Q ( F ` z ) } , { u e. Q. | E. z e. N. ( ( F ` z ) +Q ( *Q ` [ <. z , 1o >. ] ~Q ) ) <Q u } >. <P <. { l | l <Q ( ( F ` k ) +Q x ) } , { u | ( ( F ` k ) +Q x ) <Q u } >. ) )
24	22, 23	anbi12d 465	. . . . . 6 |- ( y = <. { l e. Q. | E. z e. N. ( l +Q ( *Q ` [ <. z , 1o >. ] ~Q ) ) <Q ( F ` z ) } , { u e. Q. | E. z e. N. ( ( F ` z ) +Q ( *Q ` [ <. z , 1o >. ] ~Q ) ) <Q u } >. -> ( ( <. { l | l <Q ( F ` k ) } , { u | ( F ` k ) <Q u } >. <P ( y +P. <. { l | l <Q x } , { u | x <Q u } >. ) /\ y <P <. { l | l <Q ( ( F ` k ) +Q x ) } , { u | ( ( F ` k ) +Q x ) <Q u } >. ) <-> ( <. { l | l <Q ( F ` k ) } , { u | ( F ` k ) <Q u } >. <P ( <. { l e. Q. | E. z e. N. ( l +Q ( *Q ` [ <. z , 1o >. ] ~Q ) ) <Q ( F ` z ) } , { u e. Q. | E. z e. N. ( ( F ` z ) +Q ( *Q ` [ <. z , 1o >. ] ~Q ) ) <Q u } >. +P. <. { l | l <Q x } , { u | x <Q u } >. ) /\ <. { l e. Q. | E. z e. N. ( l +Q ( *Q ` [ <. z , 1o >. ] ~Q ) ) <Q ( F ` z ) } , { u e. Q. | E. z e. N. ( ( F ` z ) +Q ( *Q ` [ <. z , 1o >. ] ~Q ) ) <Q u } >. <P <. { l | l <Q ( ( F ` k ) +Q x ) } , { u | ( ( F ` k ) +Q x ) <Q u } >. ) ) )
25	24	imbi2d 229	. . . . 5 |- ( y = <. { l e. Q. | E. z e. N. ( l +Q ( *Q ` [ <. z , 1o >. ] ~Q ) ) <Q ( F ` z ) } , { u e. Q. | E. z e. N. ( ( F ` z ) +Q ( *Q ` [ <. z , 1o >. ] ~Q ) ) <Q u } >. -> ( ( j <N k -> ( <. { l | l <Q ( F ` k ) } , { u | ( F ` k ) <Q u } >. <P ( y +P. <. { l | l <Q x } , { u | x <Q u } >. ) /\ y <P <. { l | l <Q ( ( F ` k ) +Q x ) } , { u | ( ( F ` k ) +Q x ) <Q u } >. ) ) <-> ( j <N k -> ( <. { l | l <Q ( F ` k ) } , { u | ( F ` k ) <Q u } >. <P ( <. { l e. Q. | E. z e. N. ( l +Q ( *Q ` [ <. z , 1o >. ] ~Q ) ) <Q ( F ` z ) } , { u e. Q. | E. z e. N. ( ( F ` z ) +Q ( *Q ` [ <. z , 1o >. ] ~Q ) ) <Q u } >. +P. <. { l | l <Q x } , { u | x <Q u } >. ) /\ <. { l e. Q. | E. z e. N. ( l +Q ( *Q ` [ <. z , 1o >. ] ~Q ) ) <Q ( F ` z ) } , { u e. Q. | E. z e. N. ( ( F ` z ) +Q ( *Q ` [ <. z , 1o >. ] ~Q ) ) <Q u } >. <P <. { l | l <Q ( ( F ` k ) +Q x ) } , { u | ( ( F ` k ) +Q x ) <Q u } >. ) ) ) )
26	25	rexralbidv 2492	. . . 4 |- ( y = <. { l e. Q. | E. z e. N. ( l +Q ( *Q ` [ <. z , 1o >. ] ~Q ) ) <Q ( F ` z ) } , { u e. Q. | E. z e. N. ( ( F ` z ) +Q ( *Q ` [ <. z , 1o >. ] ~Q ) ) <Q u } >. -> ( E. j e. N. A. k e. N. ( j <N k -> ( <. { l | l <Q ( F ` k ) } , { u | ( F ` k ) <Q u } >. <P ( y +P. <. { l | l <Q x } , { u | x <Q u } >. ) /\ y <P <. { l | l <Q ( ( F ` k ) +Q x ) } , { u | ( ( F ` k ) +Q x ) <Q u } >. ) ) <-> E. j e. N. A. k e. N. ( j <N k -> ( <. { l | l <Q ( F ` k ) } , { u | ( F ` k ) <Q u } >. <P ( <. { l e. Q. | E. z e. N. ( l +Q ( *Q ` [ <. z , 1o >. ] ~Q ) ) <Q ( F ` z ) } , { u e. Q. | E. z e. N. ( ( F ` z ) +Q ( *Q ` [ <. z , 1o >. ] ~Q ) ) <Q u } >. +P. <. { l | l <Q x } , { u | x <Q u } >. ) /\ <. { l e. Q. | E. z e. N. ( l +Q ( *Q ` [ <. z , 1o >. ] ~Q ) ) <Q ( F ` z ) } , { u e. Q. | E. z e. N. ( ( F ` z ) +Q ( *Q ` [ <. z , 1o >. ] ~Q ) ) <Q u } >. <P <. { l | l <Q ( ( F ` k ) +Q x ) } , { u | ( ( F ` k ) +Q x ) <Q u } >. ) ) ) )
27	26	ralbidv 2466	. . 3 |- ( y = <. { l e. Q. | E. z e. N. ( l +Q ( *Q ` [ <. z , 1o >. ] ~Q ) ) <Q ( F ` z ) } , { u e. Q. | E. z e. N. ( ( F ` z ) +Q ( *Q ` [ <. z , 1o >. ] ~Q ) ) <Q u } >. -> ( A. x e. Q. E. j e. N. A. k e. N. ( j <N k -> ( <. { l | l <Q ( F ` k ) } , { u | ( F ` k ) <Q u } >. <P ( y +P. <. { l | l <Q x } , { u | x <Q u } >. ) /\ y <P <. { l | l <Q ( ( F ` k ) +Q x ) } , { u | ( ( F ` k ) +Q x ) <Q u } >. ) ) <-> A. x e. Q. E. j e. N. A. k e. N. ( j <N k -> ( <. { l | l <Q ( F ` k ) } , { u | ( F ` k ) <Q u } >. <P ( <. { l e. Q. | E. z e. N. ( l +Q ( *Q ` [ <. z , 1o >. ] ~Q ) ) <Q ( F ` z ) } , { u e. Q. | E. z e. N. ( ( F ` z ) +Q ( *Q ` [ <. z , 1o >. ] ~Q ) ) <Q u } >. +P. <. { l | l <Q x } , { u | x <Q u } >. ) /\ <. { l e. Q. | E. z e. N. ( l +Q ( *Q ` [ <. z , 1o >. ] ~Q ) ) <Q ( F ` z ) } , { u e. Q. | E. z e. N. ( ( F ` z ) +Q ( *Q ` [ <. z , 1o >. ] ~Q ) ) <Q u } >. <P <. { l | l <Q ( ( F ` k ) +Q x ) } , { u | ( ( F ` k ) +Q x ) <Q u } >. ) ) ) )
28	27	rspcev 2830	. 2 |- ( ( <. { l e. Q. | E. z e. N. ( l +Q ( *Q ` [ <. z , 1o >. ] ~Q ) ) <Q ( F ` z ) } , { u e. Q. | E. z e. N. ( ( F ` z ) +Q ( *Q ` [ <. z , 1o >. ] ~Q ) ) <Q u } >. e. P. /\ A. x e. Q. E. j e. N. A. k e. N. ( j <N k -> ( <. { l | l <Q ( F ` k ) } , { u | ( F ` k ) <Q u } >. <P ( <. { l e. Q. | E. z e. N. ( l +Q ( *Q ` [ <. z , 1o >. ] ~Q ) ) <Q ( F ` z ) } , { u e. Q. | E. z e. N. ( ( F ` z ) +Q ( *Q ` [ <. z , 1o >. ] ~Q ) ) <Q u } >. +P. <. { l | l <Q x } , { u | x <Q u } >. ) /\ <. { l e. Q. | E. z e. N. ( l +Q ( *Q ` [ <. z , 1o >. ] ~Q ) ) <Q ( F ` z ) } , { u e. Q. | E. z e. N. ( ( F ` z ) +Q ( *Q ` [ <. z , 1o >. ] ~Q ) ) <Q u } >. <P <. { l | l <Q ( ( F ` k ) +Q x ) } , { u | ( ( F ` k ) +Q x ) <Q u } >. ) ) ) -> E. y e. P. A. x e. Q. E. j e. N. A. k e. N. ( j <N k -> ( <. { l | l <Q ( F ` k ) } , { u | ( F ` k ) <Q u } >. <P ( y +P. <. { l | l <Q x } , { u | x <Q u } >. ) /\ y <P <. { l | l <Q ( ( F ` k ) +Q x ) } , { u | ( ( F ` k ) +Q x ) <Q u } >. ) ) )
29	19, 20, 28	syl2anc 409	1 |- ( ph -> E. y e. P. A. x e. Q. E. j e. N. A. k e. N. ( j <N k -> ( <. { l | l <Q ( F ` k ) } , { u | ( F ` k ) <Q u } >. <P ( y +P. <. { l | l <Q x } , { u | x <Q u } >. ) /\ y <P <. { l | l <Q ( ( F ` k ) +Q x ) } , { u | ( ( F ` k ) +Q x ) <Q u } >. ) ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1343   e. wcel 2136   {cab 2151   A.wral 2444   E.wrex 2445   {crab 2448   <.cop 3579   class class class wbr 3982   -->wf 5184   ` cfv 5188  (class class class)co 5842   1oc1o 6377   [cec 6499   N.cnpi 7213   <N clti 7216   ~Q ceq 7220   Q.cnq 7221   +Q cplq 7223   *Qcrq 7225   <Q cltq 7226   P.cnp 7232   +P. cpp 7234   <P cltp 7236

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-coll 4097  ax-sep 4100  ax-nul 4108  ax-pow 4153  ax-pr 4187  ax-un 4411  ax-setind 4514  ax-iinf 4565

This theorem depends on definitions:  df-bi 116  df-dc 825  df-3or 969  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ne 2337  df-ral 2449  df-rex 2450  df-reu 2451  df-rab 2453  df-v 2728  df-sbc 2952  df-csb 3046  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-nul 3410  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-int 3825  df-iun 3868  df-br 3983  df-opab 4044  df-mpt 4045  df-tr 4081  df-eprel 4267  df-id 4271  df-po 4274  df-iso 4275  df-iord 4344  df-on 4346  df-suc 4349  df-iom 4568  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-res 4616  df-ima 4617  df-iota 5153  df-fun 5190  df-fn 5191  df-f 5192  df-f1 5193  df-fo 5194  df-f1o 5195  df-fv 5196  df-ov 5845  df-oprab 5846  df-mpo 5847  df-1st 6108  df-2nd 6109  df-recs 6273  df-irdg 6338  df-1o 6384  df-2o 6385  df-oadd 6388  df-omul 6389  df-er 6501  df-ec 6503  df-qs 6507  df-ni 7245  df-pli 7246  df-mi 7247  df-lti 7248  df-plpq 7285  df-mpq 7286  df-enq 7288  df-nqqs 7289  df-plqqs 7290  df-mqqs 7291  df-1nqqs 7292  df-rq 7293  df-ltnqqs 7294  df-enq0 7365  df-nq0 7366  df-0nq0 7367  df-plq0 7368  df-mq0 7369  df-inp 7407  df-iplp 7409  df-iltp 7411

This theorem is referenced by: (None)