Theorem caucvgpr 7644

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 7624 and caucvgprpr 7674. Reading cauappcvgpr 7624 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 3765	. . . . . . . . . . 11 |- ( z = j -> <. z , 1o >. = <. j , 1o >. )
5	4	eceq1d 6549	. . . . . . . . . 10 |- ( z = j -> [ <. z , 1o >. ] ~Q = [ <. j , 1o >. ] ~Q )
6	5	fveq2d 5500	. . . . . . . . 9 |- ( z = j -> ( *Q ` [ <. z , 1o >. ] ~Q ) = ( *Q ` [ <. j , 1o >. ] ~Q ) )
7	6	oveq2d 5869	. . . . . . . 8 |- ( z = j -> ( l +Q ( *Q ` [ <. z , 1o >. ] ~Q ) ) = ( l +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) )
8		fveq2 5496	. . . . . . . 8 |- ( z = j -> ( F ` z ) = ( F ` j ) )
9	7, 8	breq12d 4002	. . . . . . 7 |- ( z = j -> ( ( l +Q ( *Q ` [ <. z , 1o >. ] ~Q ) ) <Q ( F ` z ) <-> ( l +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( F ` j ) ) )
10	9	cbvrexv 2697	. . . . . 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 2715	. . . 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 5871	. . . . . . . 8 |- ( z = j -> ( ( F ` z ) +Q ( *Q ` [ <. z , 1o >. ] ~Q ) ) = ( ( F ` j ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) )
14	13	breq1d 3999	. . . . . . 7 |- ( z = j -> ( ( ( F ` z ) +Q ( *Q ` [ <. z , 1o >. ] ~Q ) ) <Q u <-> ( ( F ` j ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q u ) )
15	14	cbvrexv 2697	. . . . . 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 2715	. . . 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 3770	. . 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 7638	. 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 7643	. 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 5860	. . . . . . . 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 4001	. . . . . . 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 3992	. . . . . . 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 470	. . . . . 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 2496	. . . 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 2470	. . 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 2834	. 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 1348   e. wcel 2141   {cab 2156   A.wral 2448   E.wrex 2449   {crab 2452   <.cop 3586   class class class wbr 3989   -->wf 5194   ` cfv 5198  (class class class)co 5853   1oc1o 6388   [cec 6511   N.cnpi 7234   <N clti 7237   ~Q ceq 7241   Q.cnq 7242   +Q cplq 7244   *Qcrq 7246   <Q cltq 7247   P.cnp 7253   +P. cpp 7255   <P cltp 7257

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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-coll 4104  ax-sep 4107  ax-nul 4115  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-iinf 4572

This theorem depends on definitions:  df-bi 116  df-dc 830  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-ral 2453  df-rex 2454  df-reu 2455  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-int 3832  df-iun 3875  df-br 3990  df-opab 4051  df-mpt 4052  df-tr 4088  df-eprel 4274  df-id 4278  df-po 4281  df-iso 4282  df-iord 4351  df-on 4353  df-suc 4356  df-iom 4575  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-fo 5204  df-f1o 5205  df-fv 5206  df-ov 5856  df-oprab 5857  df-mpo 5858  df-1st 6119  df-2nd 6120  df-recs 6284  df-irdg 6349  df-1o 6395  df-2o 6396  df-oadd 6399  df-omul 6400  df-er 6513  df-ec 6515  df-qs 6519  df-ni 7266  df-pli 7267  df-mi 7268  df-lti 7269  df-plpq 7306  df-mpq 7307  df-enq 7309  df-nqqs 7310  df-plqqs 7311  df-mqqs 7312  df-1nqqs 7313  df-rq 7314  df-ltnqqs 7315  df-enq0 7386  df-nq0 7387  df-0nq0 7388  df-plq0 7389  df-mq0 7390  df-inp 7428  df-iplp 7430  df-iltp 7432

This theorem is referenced by: (None)