Theorem cauappcvgpr 7211

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 7231 and caucvgprpr 7261 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 5652	. . . . . . . 8 |- ( z = q -> ( l +Q z ) = ( l +Q q ) )
5		fveq2 5299	. . . . . . . 8 |- ( z = q -> ( F ` z ) = ( F ` q ) )
6	4, 5	breq12d 3856	. . . . . . 7 |- ( z = q -> ( ( l +Q z ) <Q ( F ` z ) <-> ( l +Q q ) <Q ( F ` q ) ) )
7	6	cbvrexv 2591	. . . . . 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 2604	. . . 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 5662	. . . . . . . 8 |- ( z = q -> ( ( F ` z ) +Q z ) = ( ( F ` q ) +Q q ) )
12	11	breq1d 3853	. . . . . . 7 |- ( z = q -> ( ( ( F ` z ) +Q z ) <Q u <-> ( ( F ` q ) +Q q ) <Q u ) )
13	12	cbvrexv 2591	. . . . . 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 2604	. . . 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 3625	. . 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 7202	. 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 7210	. 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 5651	. . . . . 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 3855	. . . . 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 3846	. . . . 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 457	. . . 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 2402	. . 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 2722	. 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 403	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 102   <-> wb 103   = wceq 1289   e. wcel 1438   {cab 2074   A.wral 2359   E.wrex 2360   {crab 2363   <.cop 3447   class class class wbr 3843   -->wf 5006   ` cfv 5010  (class class class)co 5644   Q.cnq 6829   +Q cplq 6831   <Q cltq 6834   P.cnp 6840   +P. cpp 6842   <P cltp 6844

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 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-coll 3952  ax-sep 3955  ax-nul 3963  ax-pow 4007  ax-pr 4034  ax-un 4258  ax-setind 4351  ax-iinf 4401

This theorem depends on definitions:  df-bi 115  df-dc 781  df-3or 925  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-ral 2364  df-rex 2365  df-reu 2366  df-rab 2368  df-v 2621  df-sbc 2841  df-csb 2934  df-dif 3001  df-un 3003  df-in 3005  df-ss 3012  df-nul 3287  df-pw 3429  df-sn 3450  df-pr 3451  df-op 3453  df-uni 3652  df-int 3687  df-iun 3730  df-br 3844  df-opab 3898  df-mpt 3899  df-tr 3935  df-eprel 4114  df-id 4118  df-po 4121  df-iso 4122  df-iord 4191  df-on 4193  df-suc 4196  df-iom 4404  df-xp 4442  df-rel 4443  df-cnv 4444  df-co 4445  df-dm 4446  df-rn 4447  df-res 4448  df-ima 4449  df-iota 4975  df-fun 5012  df-fn 5013  df-f 5014  df-f1 5015  df-fo 5016  df-f1o 5017  df-fv 5018  df-ov 5647  df-oprab 5648  df-mpt2 5649  df-1st 5903  df-2nd 5904  df-recs 6062  df-irdg 6127  df-1o 6173  df-2o 6174  df-oadd 6177  df-omul 6178  df-er 6282  df-ec 6284  df-qs 6288  df-ni 6853  df-pli 6854  df-mi 6855  df-lti 6856  df-plpq 6893  df-mpq 6894  df-enq 6896  df-nqqs 6897  df-plqqs 6898  df-mqqs 6899  df-1nqqs 6900  df-rq 6901  df-ltnqqs 6902  df-enq0 6973  df-nq0 6974  df-0nq0 6975  df-plq0 6976  df-mq0 6977  df-inp 7015  df-iplp 7017  df-iltp 7019

This theorem is referenced by: (None)