Theorem caucvgpr 7672

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 7652 and caucvgprpr 7702. Reading cauappcvgpr 7652 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 3776	. . . . . . . . . . 11 |- ( z = j -> <. z , 1o >. = <. j , 1o >. )
5	4	eceq1d 6565	. . . . . . . . . 10 |- ( z = j -> [ <. z , 1o >. ] ~Q = [ <. j , 1o >. ] ~Q )
6	5	fveq2d 5515	. . . . . . . . 9 |- ( z = j -> ( *Q ` [ <. z , 1o >. ] ~Q ) = ( *Q ` [ <. j , 1o >. ] ~Q ) )
7	6	oveq2d 5885	. . . . . . . 8 |- ( z = j -> ( l +Q ( *Q ` [ <. z , 1o >. ] ~Q ) ) = ( l +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) )
8		fveq2 5511	. . . . . . . 8 |- ( z = j -> ( F ` z ) = ( F ` j ) )
9	7, 8	breq12d 4013	. . . . . . 7 |- ( z = j -> ( ( l +Q ( *Q ` [ <. z , 1o >. ] ~Q ) ) <Q ( F ` z ) <-> ( l +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( F ` j ) ) )
10	9	cbvrexv 2704	. . . . . 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 2722	. . . 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 5887	. . . . . . . 8 |- ( z = j -> ( ( F ` z ) +Q ( *Q ` [ <. z , 1o >. ] ~Q ) ) = ( ( F ` j ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) )
14	13	breq1d 4010	. . . . . . 7 |- ( z = j -> ( ( ( F ` z ) +Q ( *Q ` [ <. z , 1o >. ] ~Q ) ) <Q u <-> ( ( F ` j ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q u ) )
15	14	cbvrexv 2704	. . . . . 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 2722	. . . 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 3781	. . 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 7666	. 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 7671	. 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 5876	. . . . . . . 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 4012	. . . . . . 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 4003	. . . . . . 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 2503	. . . 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 2477	. . 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 2841	. 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 1353   e. wcel 2148   {cab 2163   A.wral 2455   E.wrex 2456   {crab 2459   <.cop 3594   class class class wbr 4000   -->wf 5208   ` cfv 5212  (class class class)co 5869   1oc1o 6404   [cec 6527   N.cnpi 7262   <N clti 7265   ~Q ceq 7269   Q.cnq 7270   +Q cplq 7272   *Qcrq 7274   <Q cltq 7275   P.cnp 7281   +P. cpp 7283   <P cltp 7285

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4115  ax-sep 4118  ax-nul 4126  ax-pow 4171  ax-pr 4206  ax-un 4430  ax-setind 4533  ax-iinf 4584

This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-int 3843  df-iun 3886  df-br 4001  df-opab 4062  df-mpt 4063  df-tr 4099  df-eprel 4286  df-id 4290  df-po 4293  df-iso 4294  df-iord 4363  df-on 4365  df-suc 4368  df-iom 4587  df-xp 4629  df-rel 4630  df-cnv 4631  df-co 4632  df-dm 4633  df-rn 4634  df-res 4635  df-ima 4636  df-iota 5174  df-fun 5214  df-fn 5215  df-f 5216  df-f1 5217  df-fo 5218  df-f1o 5219  df-fv 5220  df-ov 5872  df-oprab 5873  df-mpo 5874  df-1st 6135  df-2nd 6136  df-recs 6300  df-irdg 6365  df-1o 6411  df-2o 6412  df-oadd 6415  df-omul 6416  df-er 6529  df-ec 6531  df-qs 6535  df-ni 7294  df-pli 7295  df-mi 7296  df-lti 7297  df-plpq 7334  df-mpq 7335  df-enq 7337  df-nqqs 7338  df-plqqs 7339  df-mqqs 7340  df-1nqqs 7341  df-rq 7342  df-ltnqqs 7343  df-enq0 7414  df-nq0 7415  df-0nq0 7416  df-plq0 7417  df-mq0 7418  df-inp 7456  df-iplp 7458  df-iltp 7460

This theorem is referenced by: (None)