Theorem cauappcvgpr 7661

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 7681 and caucvgprpr 7711 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 5883	. . . . . . . 8 |- ( z = q -> ( l +Q z ) = ( l +Q q ) )
5		fveq2 5516	. . . . . . . 8 |- ( z = q -> ( F ` z ) = ( F ` q ) )
6	4, 5	breq12d 4017	. . . . . . 7 |- ( z = q -> ( ( l +Q z ) <Q ( F ` z ) <-> ( l +Q q ) <Q ( F ` q ) ) )
7	6	cbvrexv 2705	. . . . . 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 2723	. . . 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 5893	. . . . . . . 8 |- ( z = q -> ( ( F ` z ) +Q z ) = ( ( F ` q ) +Q q ) )
12	11	breq1d 4014	. . . . . . 7 |- ( z = q -> ( ( ( F ` z ) +Q z ) <Q u <-> ( ( F ` q ) +Q q ) <Q u ) )
13	12	cbvrexv 2705	. . . . . 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 2723	. . . 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 3784	. . 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 7652	. 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 7660	. 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 5882	. . . . . 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 4016	. . . . 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 4007	. . . . 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 2501	. . 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 2842	. 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 1353   e. wcel 2148   {cab 2163   A.wral 2455   E.wrex 2456   {crab 2459   <.cop 3596   class class class wbr 4004   -->wf 5213   ` cfv 5217  (class class class)co 5875   Q.cnq 7279   +Q cplq 7281   <Q cltq 7284   P.cnp 7290   +P. cpp 7292   <P cltp 7294

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 4119  ax-sep 4122  ax-nul 4130  ax-pow 4175  ax-pr 4210  ax-un 4434  ax-setind 4537  ax-iinf 4588

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 2740  df-sbc 2964  df-csb 3059  df-dif 3132  df-un 3134  df-in 3136  df-ss 3143  df-nul 3424  df-pw 3578  df-sn 3599  df-pr 3600  df-op 3602  df-uni 3811  df-int 3846  df-iun 3889  df-br 4005  df-opab 4066  df-mpt 4067  df-tr 4103  df-eprel 4290  df-id 4294  df-po 4297  df-iso 4298  df-iord 4367  df-on 4369  df-suc 4372  df-iom 4591  df-xp 4633  df-rel 4634  df-cnv 4635  df-co 4636  df-dm 4637  df-rn 4638  df-res 4639  df-ima 4640  df-iota 5179  df-fun 5219  df-fn 5220  df-f 5221  df-f1 5222  df-fo 5223  df-f1o 5224  df-fv 5225  df-ov 5878  df-oprab 5879  df-mpo 5880  df-1st 6141  df-2nd 6142  df-recs 6306  df-irdg 6371  df-1o 6417  df-2o 6418  df-oadd 6421  df-omul 6422  df-er 6535  df-ec 6537  df-qs 6541  df-ni 7303  df-pli 7304  df-mi 7305  df-lti 7306  df-plpq 7343  df-mpq 7344  df-enq 7346  df-nqqs 7347  df-plqqs 7348  df-mqqs 7349  df-1nqqs 7350  df-rq 7351  df-ltnqqs 7352  df-enq0 7423  df-nq0 7424  df-0nq0 7425  df-plq0 7426  df-mq0 7427  df-inp 7465  df-iplp 7467  df-iltp 7469

This theorem is referenced by: (None)