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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 7494 and caucvgprpr 7544. Reading cauappcvgpr 7494 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 3713 | . . . . . . . . . . 11 | |
5 | 4 | eceq1d 6473 | . . . . . . . . . 10 |
6 | 5 | fveq2d 5433 | . . . . . . . . 9 |
7 | 6 | oveq2d 5798 | . . . . . . . 8 |
8 | fveq2 5429 | . . . . . . . 8 | |
9 | 7, 8 | breq12d 3950 | . . . . . . 7 |
10 | 9 | cbvrexv 2658 | . . . . . 6 |
11 | 10 | a1i 9 | . . . . 5 |
12 | 11 | rabbiia 2674 | . . . 4 |
13 | 8, 6 | oveq12d 5800 | . . . . . . . 8 |
14 | 13 | breq1d 3947 | . . . . . . 7 |
15 | 14 | cbvrexv 2658 | . . . . . 6 |
16 | 15 | a1i 9 | . . . . 5 |
17 | 16 | rabbiia 2674 | . . . 4 |
18 | 12, 17 | opeq12i 3718 | . . 3 |
19 | 1, 2, 3, 18 | caucvgprlemcl 7508 | . 2 |
20 | 1, 2, 3, 18 | caucvgprlemlim 7513 | . 2 |
21 | oveq1 5789 | . . . . . . . 8 | |
22 | 21 | breq2d 3949 | . . . . . . 7 |
23 | breq1 3940 | . . . . . . 7 | |
24 | 22, 23 | anbi12d 465 | . . . . . 6 |
25 | 24 | imbi2d 229 | . . . . 5 |
26 | 25 | rexralbidv 2464 | . . . 4 |
27 | 26 | ralbidv 2438 | . . 3 |
28 | 27 | rspcev 2793 | . 2 |
29 | 19, 20, 28 | syl2anc 409 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wceq 1332 wcel 1481 cab 2126 wral 2417 wrex 2418 crab 2421 cop 3535 class class class wbr 3937 wf 5127 cfv 5131 (class class class)co 5782 c1o 6314 cec 6435 cnpi 7104 clti 7107 ceq 7111 cnq 7112 cplq 7114 crq 7116 cltq 7117 cnp 7123 cpp 7125 cltp 7127 |
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 604 ax-in2 605 ax-io 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-13 1492 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-coll 4051 ax-sep 4054 ax-nul 4062 ax-pow 4106 ax-pr 4139 ax-un 4363 ax-setind 4460 ax-iinf 4510 |
This theorem depends on definitions: df-bi 116 df-dc 821 df-3or 964 df-3an 965 df-tru 1335 df-fal 1338 df-nf 1438 df-sb 1737 df-eu 2003 df-mo 2004 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ne 2310 df-ral 2422 df-rex 2423 df-reu 2424 df-rab 2426 df-v 2691 df-sbc 2914 df-csb 3008 df-dif 3078 df-un 3080 df-in 3082 df-ss 3089 df-nul 3369 df-pw 3517 df-sn 3538 df-pr 3539 df-op 3541 df-uni 3745 df-int 3780 df-iun 3823 df-br 3938 df-opab 3998 df-mpt 3999 df-tr 4035 df-eprel 4219 df-id 4223 df-po 4226 df-iso 4227 df-iord 4296 df-on 4298 df-suc 4301 df-iom 4513 df-xp 4553 df-rel 4554 df-cnv 4555 df-co 4556 df-dm 4557 df-rn 4558 df-res 4559 df-ima 4560 df-iota 5096 df-fun 5133 df-fn 5134 df-f 5135 df-f1 5136 df-fo 5137 df-f1o 5138 df-fv 5139 df-ov 5785 df-oprab 5786 df-mpo 5787 df-1st 6046 df-2nd 6047 df-recs 6210 df-irdg 6275 df-1o 6321 df-2o 6322 df-oadd 6325 df-omul 6326 df-er 6437 df-ec 6439 df-qs 6443 df-ni 7136 df-pli 7137 df-mi 7138 df-lti 7139 df-plpq 7176 df-mpq 7177 df-enq 7179 df-nqqs 7180 df-plqqs 7181 df-mqqs 7182 df-1nqqs 7183 df-rq 7184 df-ltnqqs 7185 df-enq0 7256 df-nq0 7257 df-0nq0 7258 df-plq0 7259 df-mq0 7260 df-inp 7298 df-iplp 7300 df-iltp 7302 |
This theorem is referenced by: (None) |
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