Theorem cauappcvgpr 7722

Description: A Cauchy approximation has a limit. A Cauchy approximation, here F, is similar to a Cauchy sequence but is indexed by the desired tolerance (that is, how close together terms needs to be) rather than by natural numbers. This is basically Theorem 11.2.12 of [HoTT], p. (varies) with a few differences such as that we are proving the existence of a limit without anything about how fast it converges (that is, mere existence instead of existence, in HoTT terms), and that the codomain of F is Q. rather than P.. 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 caucvgpr 7742 and caucvgprpr 7772 but is somewhat simpler, so reading this one first may help understanding the other two.

(Contributed by Jim Kingdon, 19-Jun-2020.)

Hypotheses

Ref	Expression
cauappcvgpr.f	|- ( ph -> F : Q. --> Q. )
cauappcvgpr.app	|- ( ph -> A. p e. Q. A. q e. Q. ( ( F ` p ) <Q ( ( F ` q ) +Q ( p +Q q ) ) /\ ( F ` q ) <Q ( ( F ` p ) +Q ( p +Q q ) ) ) )
cauappcvgpr.bnd	|- ( ph -> A. p e. Q. A <Q ( F ` p ) )

Assertion

Ref	Expression
cauappcvgpr	|- ( ph -> E. y e. P. A. q e. Q. A. r e. Q. ( <. { l | l <Q ( F ` q ) } , { u | ( F ` q ) <Q u } >. <P ( y +P. <. { l | l <Q ( q +Q r ) } , { u | ( q +Q r ) <Q u } >. ) /\ y <P <. { l | l <Q ( ( F ` q ) +Q ( q +Q r ) ) } , { u | ( ( F ` q ) +Q ( q +Q r ) ) <Q u } >. ) )

Distinct variable groups:   A, p   F, q, y, r, u   F, p, l, q   y, l, r   u, q, y, r   u, p, r, q, l   ph, q, p

Allowed substitution hints:   ph( y, u, r, l)   A( y, u, r, q, l)

Proof of Theorem cauappcvgpr

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		cauappcvgpr.f	. . 3 |- ( ph -> F : Q. --> Q. )
2		cauappcvgpr.app	. . 3 |- ( ph -> A. p e. Q. A. q e. Q. ( ( F ` p ) <Q ( ( F ` q ) +Q ( p +Q q ) ) /\ ( F ` q ) <Q ( ( F ` p ) +Q ( p +Q q ) ) ) )
3		cauappcvgpr.bnd	. . 3 |- ( ph -> A. p e. Q. A <Q ( F ` p ) )
4		oveq2 5926	. . . . . . . 8 |- ( z = q -> ( l +Q z ) = ( l +Q q ) )
5		fveq2 5554	. . . . . . . 8 |- ( z = q -> ( F ` z ) = ( F ` q ) )
6	4, 5	breq12d 4042	. . . . . . 7 |- ( z = q -> ( ( l +Q z ) <Q ( F ` z ) <-> ( l +Q q ) <Q ( F ` q ) ) )
7	6	cbvrexv 2727	. . . . . 6 |- ( E. z e. Q. ( l +Q z ) <Q ( F ` z ) <-> E. q e. Q. ( l +Q q ) <Q ( F ` q ) )
8	7	a1i 9	. . . . 5 |- ( l e. Q. -> ( E. z e. Q. ( l +Q z ) <Q ( F ` z ) <-> E. q e. Q. ( l +Q q ) <Q ( F ` q ) ) )
9	8	rabbiia 2745	. . . 4 |- { l e. Q. | E. z e. Q. ( l +Q z ) <Q ( F ` z ) } = { l e. Q. | E. q e. Q. ( l +Q q ) <Q ( F ` q ) }
10		id 19	. . . . . . . . 9 |- ( z = q -> z = q )
11	5, 10	oveq12d 5936	. . . . . . . 8 |- ( z = q -> ( ( F ` z ) +Q z ) = ( ( F ` q ) +Q q ) )
12	11	breq1d 4039	. . . . . . 7 |- ( z = q -> ( ( ( F ` z ) +Q z ) <Q u <-> ( ( F ` q ) +Q q ) <Q u ) )
13	12	cbvrexv 2727	. . . . . 6 |- ( E. z e. Q. ( ( F ` z ) +Q z ) <Q u <-> E. q e. Q. ( ( F ` q ) +Q q ) <Q u )
14	13	a1i 9	. . . . 5 |- ( u e. Q. -> ( E. z e. Q. ( ( F ` z ) +Q z ) <Q u <-> E. q e. Q. ( ( F ` q ) +Q q ) <Q u ) )
15	14	rabbiia 2745	. . . 4 |- { u e. Q. | E. z e. Q. ( ( F ` z ) +Q z ) <Q u } = { u e. Q. | E. q e. Q. ( ( F ` q ) +Q q ) <Q u }
16	9, 15	opeq12i 3809	. . 3 |- <. { l e. Q. | E. z e. Q. ( l +Q z ) <Q ( F ` z ) } , { u e. Q. | E. z e. Q. ( ( F ` z ) +Q z ) <Q u } >. = <. { l e. Q. | E. q e. Q. ( l +Q q ) <Q ( F ` q ) } , { u e. Q. | E. q e. Q. ( ( F ` q ) +Q q ) <Q u } >.
17	1, 2, 3, 16	cauappcvgprlemcl 7713	. 2 |- ( ph -> <. { l e. Q. | E. z e. Q. ( l +Q z ) <Q ( F ` z ) } , { u e. Q. | E. z e. Q. ( ( F ` z ) +Q z ) <Q u } >. e. P. )
18	1, 2, 3, 16	cauappcvgprlemlim 7721	. 2 |- ( ph -> A. q e. Q. A. r e. Q. ( <. { l | l <Q ( F ` q ) } , { u | ( F ` q ) <Q u } >. <P ( <. { l e. Q. | E. z e. Q. ( l +Q z ) <Q ( F ` z ) } , { u e. Q. | E. z e. Q. ( ( F ` z ) +Q z ) <Q u } >. +P. <. { l | l <Q ( q +Q r ) } , { u | ( q +Q r ) <Q u } >. ) /\ <. { l e. Q. | E. z e. Q. ( l +Q z ) <Q ( F ` z ) } , { u e. Q. | E. z e. Q. ( ( F ` z ) +Q z ) <Q u } >. <P <. { l | l <Q ( ( F ` q ) +Q ( q +Q r ) ) } , { u | ( ( F ` q ) +Q ( q +Q r ) ) <Q u } >. ) )
19		oveq1 5925	. . . . . 6 |- ( y = <. { l e. Q. | E. z e. Q. ( l +Q z ) <Q ( F ` z ) } , { u e. Q. | E. z e. Q. ( ( F ` z ) +Q z ) <Q u } >. -> ( y +P. <. { l | l <Q ( q +Q r ) } , { u | ( q +Q r ) <Q u } >. ) = ( <. { l e. Q. | E. z e. Q. ( l +Q z ) <Q ( F ` z ) } , { u e. Q. | E. z e. Q. ( ( F ` z ) +Q z ) <Q u } >. +P. <. { l | l <Q ( q +Q r ) } , { u | ( q +Q r ) <Q u } >. ) )
20	19	breq2d 4041	. . . . 5 |- ( y = <. { l e. Q. | E. z e. Q. ( l +Q z ) <Q ( F ` z ) } , { u e. Q. | E. z e. Q. ( ( F ` z ) +Q z ) <Q u } >. -> ( <. { l | l <Q ( F ` q ) } , { u | ( F ` q ) <Q u } >. <P ( y +P. <. { l | l <Q ( q +Q r ) } , { u | ( q +Q r ) <Q u } >. ) <-> <. { l | l <Q ( F ` q ) } , { u | ( F ` q ) <Q u } >. <P ( <. { l e. Q. | E. z e. Q. ( l +Q z ) <Q ( F ` z ) } , { u e. Q. | E. z e. Q. ( ( F ` z ) +Q z ) <Q u } >. +P. <. { l | l <Q ( q +Q r ) } , { u | ( q +Q r ) <Q u } >. ) ) )
21		breq1 4032	. . . . 5 |- ( y = <. { l e. Q. | E. z e. Q. ( l +Q z ) <Q ( F ` z ) } , { u e. Q. | E. z e. Q. ( ( F ` z ) +Q z ) <Q u } >. -> ( y <P <. { l | l <Q ( ( F ` q ) +Q ( q +Q r ) ) } , { u | ( ( F ` q ) +Q ( q +Q r ) ) <Q u } >. <-> <. { l e. Q. | E. z e. Q. ( l +Q z ) <Q ( F ` z ) } , { u e. Q. | E. z e. Q. ( ( F ` z ) +Q z ) <Q u } >. <P <. { l | l <Q ( ( F ` q ) +Q ( q +Q r ) ) } , { u | ( ( F ` q ) +Q ( q +Q r ) ) <Q u } >. ) )
22	20, 21	anbi12d 473	. . . 4 |- ( y = <. { l e. Q. | E. z e. Q. ( l +Q z ) <Q ( F ` z ) } , { u e. Q. | E. z e. Q. ( ( F ` z ) +Q z ) <Q u } >. -> ( ( <. { l | l <Q ( F ` q ) } , { u | ( F ` q ) <Q u } >. <P ( y +P. <. { l | l <Q ( q +Q r ) } , { u | ( q +Q r ) <Q u } >. ) /\ y <P <. { l | l <Q ( ( F ` q ) +Q ( q +Q r ) ) } , { u | ( ( F ` q ) +Q ( q +Q r ) ) <Q u } >. ) <-> ( <. { l | l <Q ( F ` q ) } , { u | ( F ` q ) <Q u } >. <P ( <. { l e. Q. | E. z e. Q. ( l +Q z ) <Q ( F ` z ) } , { u e. Q. | E. z e. Q. ( ( F ` z ) +Q z ) <Q u } >. +P. <. { l | l <Q ( q +Q r ) } , { u | ( q +Q r ) <Q u } >. ) /\ <. { l e. Q. | E. z e. Q. ( l +Q z ) <Q ( F ` z ) } , { u e. Q. | E. z e. Q. ( ( F ` z ) +Q z ) <Q u } >. <P <. { l | l <Q ( ( F ` q ) +Q ( q +Q r ) ) } , { u | ( ( F ` q ) +Q ( q +Q r ) ) <Q u } >. ) ) )
23	22	2ralbidv 2518	. . 3 |- ( y = <. { l e. Q. | E. z e. Q. ( l +Q z ) <Q ( F ` z ) } , { u e. Q. | E. z e. Q. ( ( F ` z ) +Q z ) <Q u } >. -> ( A. q e. Q. A. r e. Q. ( <. { l | l <Q ( F ` q ) } , { u | ( F ` q ) <Q u } >. <P ( y +P. <. { l | l <Q ( q +Q r ) } , { u | ( q +Q r ) <Q u } >. ) /\ y <P <. { l | l <Q ( ( F ` q ) +Q ( q +Q r ) ) } , { u | ( ( F ` q ) +Q ( q +Q r ) ) <Q u } >. ) <-> A. q e. Q. A. r e. Q. ( <. { l | l <Q ( F ` q ) } , { u | ( F ` q ) <Q u } >. <P ( <. { l e. Q. | E. z e. Q. ( l +Q z ) <Q ( F ` z ) } , { u e. Q. | E. z e. Q. ( ( F ` z ) +Q z ) <Q u } >. +P. <. { l | l <Q ( q +Q r ) } , { u | ( q +Q r ) <Q u } >. ) /\ <. { l e. Q. | E. z e. Q. ( l +Q z ) <Q ( F ` z ) } , { u e. Q. | E. z e. Q. ( ( F ` z ) +Q z ) <Q u } >. <P <. { l | l <Q ( ( F ` q ) +Q ( q +Q r ) ) } , { u | ( ( F ` q ) +Q ( q +Q r ) ) <Q u } >. ) ) )
24	23	rspcev 2864	. 2 |- ( ( <. { l e. Q. | E. z e. Q. ( l +Q z ) <Q ( F ` z ) } , { u e. Q. | E. z e. Q. ( ( F ` z ) +Q z ) <Q u } >. e. P. /\ A. q e. Q. A. r e. Q. ( <. { l | l <Q ( F ` q ) } , { u | ( F ` q ) <Q u } >. <P ( <. { l e. Q. | E. z e. Q. ( l +Q z ) <Q ( F ` z ) } , { u e. Q. | E. z e. Q. ( ( F ` z ) +Q z ) <Q u } >. +P. <. { l | l <Q ( q +Q r ) } , { u | ( q +Q r ) <Q u } >. ) /\ <. { l e. Q. | E. z e. Q. ( l +Q z ) <Q ( F ` z ) } , { u e. Q. | E. z e. Q. ( ( F ` z ) +Q z ) <Q u } >. <P <. { l | l <Q ( ( F ` q ) +Q ( q +Q r ) ) } , { u | ( ( F ` q ) +Q ( q +Q r ) ) <Q u } >. ) ) -> E. y e. P. A. q e. Q. A. r e. Q. ( <. { l | l <Q ( F ` q ) } , { u | ( F ` q ) <Q u } >. <P ( y +P. <. { l | l <Q ( q +Q r ) } , { u | ( q +Q r ) <Q u } >. ) /\ y <P <. { l | l <Q ( ( F ` q ) +Q ( q +Q r ) ) } , { u | ( ( F ` q ) +Q ( q +Q r ) ) <Q u } >. ) )
25	17, 18, 24	syl2anc 411	1 |- ( ph -> E. y e. P. A. q e. Q. A. r e. Q. ( <. { l | l <Q ( F ` q ) } , { u | ( F ` q ) <Q u } >. <P ( y +P. <. { l | l <Q ( q +Q r ) } , { u | ( q +Q r ) <Q u } >. ) /\ y <P <. { l | l <Q ( ( F ` q ) +Q ( q +Q r ) ) } , { u | ( ( F ` q ) +Q ( q +Q r ) ) <Q u } >. ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1364   e. wcel 2164   {cab 2179   A.wral 2472   E.wrex 2473   {crab 2476   <.cop 3621   class class class wbr 4029   -->wf 5250   ` cfv 5254  (class class class)co 5918   Q.cnq 7340   +Q cplq 7342   <Q cltq 7345   P.cnp 7351   +P. cpp 7353   <P cltp 7355

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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-coll 4144  ax-sep 4147  ax-nul 4155  ax-pow 4203  ax-pr 4238  ax-un 4464  ax-setind 4569  ax-iinf 4620

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-ral 2477  df-rex 2478  df-reu 2479  df-rab 2481  df-v 2762  df-sbc 2986  df-csb 3081  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3447  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-int 3871  df-iun 3914  df-br 4030  df-opab 4091  df-mpt 4092  df-tr 4128  df-eprel 4320  df-id 4324  df-po 4327  df-iso 4328  df-iord 4397  df-on 4399  df-suc 4402  df-iom 4623  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-res 4671  df-ima 4672  df-iota 5215  df-fun 5256  df-fn 5257  df-f 5258  df-f1 5259  df-fo 5260  df-f1o 5261  df-fv 5262  df-ov 5921  df-oprab 5922  df-mpo 5923  df-1st 6193  df-2nd 6194  df-recs 6358  df-irdg 6423  df-1o 6469  df-2o 6470  df-oadd 6473  df-omul 6474  df-er 6587  df-ec 6589  df-qs 6593  df-ni 7364  df-pli 7365  df-mi 7366  df-lti 7367  df-plpq 7404  df-mpq 7405  df-enq 7407  df-nqqs 7408  df-plqqs 7409  df-mqqs 7410  df-1nqqs 7411  df-rq 7412  df-ltnqqs 7413  df-enq0 7484  df-nq0 7485  df-0nq0 7486  df-plq0 7487  df-mq0 7488  df-inp 7526  df-iplp 7528  df-iltp 7530

This theorem is referenced by: (None)