Theorem caucvgpr 7962

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 7942 and caucvgprpr 7992. Reading cauappcvgpr 7942 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 3867	. . . . . . . . . . 11 |- ( z = j -> <. z , 1o >. = <. j , 1o >. )
5	4	eceq1d 6781	. . . . . . . . . 10 |- ( z = j -> [ <. z , 1o >. ] ~Q = [ <. j , 1o >. ] ~Q )
6	5	fveq2d 5652	. . . . . . . . 9 |- ( z = j -> ( *Q ` [ <. z , 1o >. ] ~Q ) = ( *Q ` [ <. j , 1o >. ] ~Q ) )
7	6	oveq2d 6044	. . . . . . . 8 |- ( z = j -> ( l +Q ( *Q ` [ <. z , 1o >. ] ~Q ) ) = ( l +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) )
8		fveq2 5648	. . . . . . . 8 |- ( z = j -> ( F ` z ) = ( F ` j ) )
9	7, 8	breq12d 4106	. . . . . . 7 |- ( z = j -> ( ( l +Q ( *Q ` [ <. z , 1o >. ] ~Q ) ) <Q ( F ` z ) <-> ( l +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( F ` j ) ) )
10	9	cbvrexv 2769	. . . . . 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 2789	. . . 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 6046	. . . . . . . 8 |- ( z = j -> ( ( F ` z ) +Q ( *Q ` [ <. z , 1o >. ] ~Q ) ) = ( ( F ` j ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) )
14	13	breq1d 4103	. . . . . . 7 |- ( z = j -> ( ( ( F ` z ) +Q ( *Q ` [ <. z , 1o >. ] ~Q ) ) <Q u <-> ( ( F ` j ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q u ) )
15	14	cbvrexv 2769	. . . . . 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 2789	. . . 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 3872	. . 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 7956	. 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 7961	. 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 6035	. . . . . . . 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 4105	. . . . . . 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 4096	. . . . . . 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 473	. . . . . 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 230	. . . . 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 2559	. . . 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 2533	. . 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 2911	. 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 411	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 104   <-> wb 105   = wceq 1398   e. wcel 2202   {cab 2217   A.wral 2511   E.wrex 2512   {crab 2515   <.cop 3676   class class class wbr 4093   -->wf 5329   ` cfv 5333  (class class class)co 6028   1oc1o 6618   [cec 6743   N.cnpi 7552   <N clti 7555   ~Q ceq 7559   Q.cnq 7560   +Q cplq 7562   *Qcrq 7564   <Q cltq 7565   P.cnp 7571   +P. cpp 7573   <P cltp 7575

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4209  ax-sep 4212  ax-nul 4220  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641  ax-iinf 4692

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-ral 2516  df-rex 2517  df-reu 2518  df-rab 2520  df-v 2805  df-sbc 3033  df-csb 3129  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-nul 3497  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-int 3934  df-iun 3977  df-br 4094  df-opab 4156  df-mpt 4157  df-tr 4193  df-eprel 4392  df-id 4396  df-po 4399  df-iso 4400  df-iord 4469  df-on 4471  df-suc 4474  df-iom 4695  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-f1 5338  df-fo 5339  df-f1o 5340  df-fv 5341  df-ov 6031  df-oprab 6032  df-mpo 6033  df-1st 6312  df-2nd 6313  df-recs 6514  df-irdg 6579  df-1o 6625  df-2o 6626  df-oadd 6629  df-omul 6630  df-er 6745  df-ec 6747  df-qs 6751  df-ni 7584  df-pli 7585  df-mi 7586  df-lti 7587  df-plpq 7624  df-mpq 7625  df-enq 7627  df-nqqs 7628  df-plqqs 7629  df-mqqs 7630  df-1nqqs 7631  df-rq 7632  df-ltnqqs 7633  df-enq0 7704  df-nq0 7705  df-0nq0 7706  df-plq0 7707  df-mq0 7708  df-inp 7746  df-iplp 7748  df-iltp 7750

This theorem is referenced by: (None)