Theorem caucvgpr 6872

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 6852 and caucvgprpr 6902. Reading cauappcvgpr 6852 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 3570	. . . . . . . . . . 11 |- ( z = j -> <. z , 1o >. = <. j , 1o >. )
5	4	eceq1d 6165	. . . . . . . . . 10 |- ( z = j -> [ <. z , 1o >. ] ~Q = [ <. j , 1o >. ] ~Q )
6	5	fveq2d 5202	. . . . . . . . 9 |- ( z = j -> ( *Q ` [ <. z , 1o >. ] ~Q ) = ( *Q ` [ <. j , 1o >. ] ~Q ) )
7	6	oveq2d 5548	. . . . . . . 8 |- ( z = j -> ( l +Q ( *Q ` [ <. z , 1o >. ] ~Q ) ) = ( l +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) )
8		fveq2 5198	. . . . . . . 8 |- ( z = j -> ( F ` z ) = ( F ` j ) )
9	7, 8	breq12d 3798	. . . . . . 7 |- ( z = j -> ( ( l +Q ( *Q ` [ <. z , 1o >. ] ~Q ) ) <Q ( F ` z ) <-> ( l +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( F ` j ) ) )
10	9	cbvrexv 2578	. . . . . 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 2591	. . . 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 5550	. . . . . . . 8 |- ( z = j -> ( ( F ` z ) +Q ( *Q ` [ <. z , 1o >. ] ~Q ) ) = ( ( F ` j ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) )
14	13	breq1d 3795	. . . . . . 7 |- ( z = j -> ( ( ( F ` z ) +Q ( *Q ` [ <. z , 1o >. ] ~Q ) ) <Q u <-> ( ( F ` j ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q u ) )
15	14	cbvrexv 2578	. . . . . 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 2591	. . . 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 3575	. . 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 6866	. 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 6871	. 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 5539	. . . . . . . 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 3797	. . . . . . 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 3788	. . . . . . 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 456	. . . . . 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 228	. . . . 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 2392	. . . 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 2368	. . 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 2701	. 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 403	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 102   <-> wb 103   = wceq 1284   e. wcel 1433   {cab 2067   A.wral 2348   E.wrex 2349   {crab 2352   <.cop 3401   class class class wbr 3785   -->wf 4918   ` cfv 4922  (class class class)co 5532   1oc1o 6017   [cec 6127   N.cnpi 6462   <N clti 6465   ~Q ceq 6469   Q.cnq 6470   +Q cplq 6472   *Qcrq 6474   <Q cltq 6475   P.cnp 6481   +P. cpp 6483   <P cltp 6485

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 576  ax-in2 577  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-13 1444  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-coll 3893  ax-sep 3896  ax-nul 3904  ax-pow 3948  ax-pr 3964  ax-un 4188  ax-setind 4280  ax-iinf 4329

This theorem depends on definitions:  df-bi 115  df-dc 776  df-3or 920  df-3an 921  df-tru 1287  df-fal 1290  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ne 2246  df-ral 2353  df-rex 2354  df-reu 2355  df-rab 2357  df-v 2603  df-sbc 2816  df-csb 2909  df-dif 2975  df-un 2977  df-in 2979  df-ss 2986  df-nul 3252  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-uni 3602  df-int 3637  df-iun 3680  df-br 3786  df-opab 3840  df-mpt 3841  df-tr 3876  df-eprel 4044  df-id 4048  df-po 4051  df-iso 4052  df-iord 4121  df-on 4123  df-suc 4126  df-iom 4332  df-xp 4369  df-rel 4370  df-cnv 4371  df-co 4372  df-dm 4373  df-rn 4374  df-res 4375  df-ima 4376  df-iota 4887  df-fun 4924  df-fn 4925  df-f 4926  df-f1 4927  df-fo 4928  df-f1o 4929  df-fv 4930  df-ov 5535  df-oprab 5536  df-mpt2 5537  df-1st 5787  df-2nd 5788  df-recs 5943  df-irdg 5980  df-1o 6024  df-2o 6025  df-oadd 6028  df-omul 6029  df-er 6129  df-ec 6131  df-qs 6135  df-ni 6494  df-pli 6495  df-mi 6496  df-lti 6497  df-plpq 6534  df-mpq 6535  df-enq 6537  df-nqqs 6538  df-plqqs 6539  df-mqqs 6540  df-1nqqs 6541  df-rq 6542  df-ltnqqs 6543  df-enq0 6614  df-nq0 6615  df-0nq0 6616  df-plq0 6617  df-mq0 6618  df-inp 6656  df-iplp 6658  df-iltp 6660

This theorem is referenced by: (None)