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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 7624 and caucvgprpr 7674. Reading cauappcvgpr 7624 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 3765 | . . . . . . . . . . 11 | |
5 | 4 | eceq1d 6549 | . . . . . . . . . 10 |
6 | 5 | fveq2d 5500 | . . . . . . . . 9 |
7 | 6 | oveq2d 5869 | . . . . . . . 8 |
8 | fveq2 5496 | . . . . . . . 8 | |
9 | 7, 8 | breq12d 4002 | . . . . . . 7 |
10 | 9 | cbvrexv 2697 | . . . . . 6 |
11 | 10 | a1i 9 | . . . . 5 |
12 | 11 | rabbiia 2715 | . . . 4 |
13 | 8, 6 | oveq12d 5871 | . . . . . . . 8 |
14 | 13 | breq1d 3999 | . . . . . . 7 |
15 | 14 | cbvrexv 2697 | . . . . . 6 |
16 | 15 | a1i 9 | . . . . 5 |
17 | 16 | rabbiia 2715 | . . . 4 |
18 | 12, 17 | opeq12i 3770 | . . 3 |
19 | 1, 2, 3, 18 | caucvgprlemcl 7638 | . 2 |
20 | 1, 2, 3, 18 | caucvgprlemlim 7643 | . 2 |
21 | oveq1 5860 | . . . . . . . 8 | |
22 | 21 | breq2d 4001 | . . . . . . 7 |
23 | breq1 3992 | . . . . . . 7 | |
24 | 22, 23 | anbi12d 470 | . . . . . 6 |
25 | 24 | imbi2d 229 | . . . . 5 |
26 | 25 | rexralbidv 2496 | . . . 4 |
27 | 26 | ralbidv 2470 | . . 3 |
28 | 27 | rspcev 2834 | . 2 |
29 | 19, 20, 28 | syl2anc 409 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wceq 1348 wcel 2141 cab 2156 wral 2448 wrex 2449 crab 2452 cop 3586 class class class wbr 3989 wf 5194 cfv 5198 (class class class)co 5853 c1o 6388 cec 6511 cnpi 7234 clti 7237 ceq 7241 cnq 7242 cplq 7244 crq 7246 cltq 7247 cnp 7253 cpp 7255 cltp 7257 |
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 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-13 2143 ax-14 2144 ax-ext 2152 ax-coll 4104 ax-sep 4107 ax-nul 4115 ax-pow 4160 ax-pr 4194 ax-un 4418 ax-setind 4521 ax-iinf 4572 |
This theorem depends on definitions: df-bi 116 df-dc 830 df-3or 974 df-3an 975 df-tru 1351 df-fal 1354 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ne 2341 df-ral 2453 df-rex 2454 df-reu 2455 df-rab 2457 df-v 2732 df-sbc 2956 df-csb 3050 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-nul 3415 df-pw 3568 df-sn 3589 df-pr 3590 df-op 3592 df-uni 3797 df-int 3832 df-iun 3875 df-br 3990 df-opab 4051 df-mpt 4052 df-tr 4088 df-eprel 4274 df-id 4278 df-po 4281 df-iso 4282 df-iord 4351 df-on 4353 df-suc 4356 df-iom 4575 df-xp 4617 df-rel 4618 df-cnv 4619 df-co 4620 df-dm 4621 df-rn 4622 df-res 4623 df-ima 4624 df-iota 5160 df-fun 5200 df-fn 5201 df-f 5202 df-f1 5203 df-fo 5204 df-f1o 5205 df-fv 5206 df-ov 5856 df-oprab 5857 df-mpo 5858 df-1st 6119 df-2nd 6120 df-recs 6284 df-irdg 6349 df-1o 6395 df-2o 6396 df-oadd 6399 df-omul 6400 df-er 6513 df-ec 6515 df-qs 6519 df-ni 7266 df-pli 7267 df-mi 7268 df-lti 7269 df-plpq 7306 df-mpq 7307 df-enq 7309 df-nqqs 7310 df-plqqs 7311 df-mqqs 7312 df-1nqqs 7313 df-rq 7314 df-ltnqqs 7315 df-enq0 7386 df-nq0 7387 df-0nq0 7388 df-plq0 7389 df-mq0 7390 df-inp 7428 df-iplp 7430 df-iltp 7432 |
This theorem is referenced by: (None) |
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