Theorem cauappcvgpr 8029

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 8049 and caucvgprpr 8079 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 6093	. . . . . . . 8 |- ( z = q -> ( l +Q z ) = ( l +Q q ) )
5		fveq2 5695	. . . . . . . 8 |- ( z = q -> ( F ` z ) = ( F ` q ) )
6	4, 5	breq12d 4143	. . . . . . 7 |- ( z = q -> ( ( l +Q z ) <Q ( F ` z ) <-> ( l +Q q ) <Q ( F ` q ) ) )
7	6	cbvrexv 2787	. . . . . 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 2807	. . . 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 6103	. . . . . . . 8 |- ( z = q -> ( ( F ` z ) +Q z ) = ( ( F ` q ) +Q q ) )
12	11	breq1d 4140	. . . . . . 7 |- ( z = q -> ( ( ( F ` z ) +Q z ) <Q u <-> ( ( F ` q ) +Q q ) <Q u ) )
13	12	cbvrexv 2787	. . . . . 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 2807	. . . 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 3909	. . 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 8020	. 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 8028	. 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 6092	. . . . . 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 4142	. . . . 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 4133	. . . . 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 477	. . . 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 2574	. . 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 2929	. 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 415	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 } >. ) )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1402   e. wcel 2209   {cab 2224   A.wral 2528   E.wrex 2529   {crab 2532   <.cop 3712   class class class wbr 4130   -->wf 5373   ` cfv 5377  (class class class)co 6085   Q.cnq 7647   +Q cplq 7649   <Q cltq 7652   P.cnp 7658   +P. cpp 7660   <P cltp 7662

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4246  ax-sep 4249  ax-nul 4259  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-iinf 4735

This proof depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4131  df-opab 4193  df-mpt 4194  df-tr 4230  df-eprel 4434  df-id 4438  df-po 4441  df-iso 4442  df-iord 4511  df-on 4513  df-suc 4516  df-iom 4738  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-fo 5383  df-f1o 5384  df-fv 5385  df-ov 6088  df-oprab 6089  df-mpo 6090  df-1st 6374  df-2nd 6375  df-recs 6576  df-irdg 6641  df-1o 6687  df-2o 6688  df-oadd 6691  df-omul 6692  df-er 6807  df-ec 6809  df-qs 6813  df-ni 7671  df-pli 7672  df-mi 7673  df-lti 7674  df-plpq 7711  df-mpq 7712  df-enq 7714  df-nqqs 7715  df-plqqs 7716  df-mqqs 7717  df-1nqqs 7718  df-rq 7719  df-ltnqqs 7720  df-enq0 7791  df-nq0 7792  df-0nq0 7793  df-plq0 7794  df-mq0 7795  df-inp 7833  df-iplp 7835  df-iltp 7837

This theorem is used by: (None)