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Mirrors > Home > ILE Home > Th. List > caucvgpr | Unicode version |
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
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 , 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 7463 and caucvgprpr 7513. Reading cauappcvgpr 7463 first (the simplest of the three) might help understanding the other two. (Contributed by Jim Kingdon, 18-Jun-2020.) |
Ref | Expression |
---|---|
caucvgpr.f | |
caucvgpr.cau | |
caucvgpr.bnd |
Ref | Expression |
---|---|
caucvgpr |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | caucvgpr.f | . . 3 | |
2 | caucvgpr.cau | . . 3 | |
3 | caucvgpr.bnd | . . 3 | |
4 | opeq1 3700 | . . . . . . . . . . 11 | |
5 | 4 | eceq1d 6458 | . . . . . . . . . 10 |
6 | 5 | fveq2d 5418 | . . . . . . . . 9 |
7 | 6 | oveq2d 5783 | . . . . . . . 8 |
8 | fveq2 5414 | . . . . . . . 8 | |
9 | 7, 8 | breq12d 3937 | . . . . . . 7 |
10 | 9 | cbvrexv 2653 | . . . . . 6 |
11 | 10 | a1i 9 | . . . . 5 |
12 | 11 | rabbiia 2666 | . . . 4 |
13 | 8, 6 | oveq12d 5785 | . . . . . . . 8 |
14 | 13 | breq1d 3934 | . . . . . . 7 |
15 | 14 | cbvrexv 2653 | . . . . . 6 |
16 | 15 | a1i 9 | . . . . 5 |
17 | 16 | rabbiia 2666 | . . . 4 |
18 | 12, 17 | opeq12i 3705 | . . 3 |
19 | 1, 2, 3, 18 | caucvgprlemcl 7477 | . 2 |
20 | 1, 2, 3, 18 | caucvgprlemlim 7482 | . 2 |
21 | oveq1 5774 | . . . . . . . 8 | |
22 | 21 | breq2d 3936 | . . . . . . 7 |
23 | breq1 3927 | . . . . . . 7 | |
24 | 22, 23 | anbi12d 464 | . . . . . 6 |
25 | 24 | imbi2d 229 | . . . . 5 |
26 | 25 | rexralbidv 2459 | . . . 4 |
27 | 26 | ralbidv 2435 | . . 3 |
28 | 27 | rspcev 2784 | . 2 |
29 | 19, 20, 28 | syl2anc 408 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wceq 1331 wcel 1480 cab 2123 wral 2414 wrex 2415 crab 2418 cop 3525 class class class wbr 3924 wf 5114 cfv 5118 (class class class)co 5767 c1o 6299 cec 6420 cnpi 7073 clti 7076 ceq 7080 cnq 7081 cplq 7083 crq 7085 cltq 7086 cnp 7092 cpp 7094 cltp 7096 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 ax-coll 4038 ax-sep 4041 ax-nul 4049 ax-pow 4093 ax-pr 4126 ax-un 4350 ax-setind 4447 ax-iinf 4497 |
This theorem depends on definitions: df-bi 116 df-dc 820 df-3or 963 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2000 df-mo 2001 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ne 2307 df-ral 2419 df-rex 2420 df-reu 2421 df-rab 2423 df-v 2683 df-sbc 2905 df-csb 2999 df-dif 3068 df-un 3070 df-in 3072 df-ss 3079 df-nul 3359 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-uni 3732 df-int 3767 df-iun 3810 df-br 3925 df-opab 3985 df-mpt 3986 df-tr 4022 df-eprel 4206 df-id 4210 df-po 4213 df-iso 4214 df-iord 4283 df-on 4285 df-suc 4288 df-iom 4500 df-xp 4540 df-rel 4541 df-cnv 4542 df-co 4543 df-dm 4544 df-rn 4545 df-res 4546 df-ima 4547 df-iota 5083 df-fun 5120 df-fn 5121 df-f 5122 df-f1 5123 df-fo 5124 df-f1o 5125 df-fv 5126 df-ov 5770 df-oprab 5771 df-mpo 5772 df-1st 6031 df-2nd 6032 df-recs 6195 df-irdg 6260 df-1o 6306 df-2o 6307 df-oadd 6310 df-omul 6311 df-er 6422 df-ec 6424 df-qs 6428 df-ni 7105 df-pli 7106 df-mi 7107 df-lti 7108 df-plpq 7145 df-mpq 7146 df-enq 7148 df-nqqs 7149 df-plqqs 7150 df-mqqs 7151 df-1nqqs 7152 df-rq 7153 df-ltnqqs 7154 df-enq0 7225 df-nq0 7226 df-0nq0 7227 df-plq0 7228 df-mq0 7229 df-inp 7267 df-iplp 7269 df-iltp 7271 |
This theorem is referenced by: (None) |
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