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Mirrors > Home > ILE Home > Th. List > caucvgsr | Unicode version |
Description: A Cauchy sequence of
signed reals with a modulus of convergence
converges to a signed real. This is basically Corollary 11.2.13 of
[HoTT], p. (varies). 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).
This is similar to caucvgprpr 7634 but is for signed reals rather than positive reals. Here is an outline of how we prove it: 1. Choose a lower bound for the sequence (see caucvgsrlembnd 7723). 2. Offset each element of the sequence so that each element of the resulting sequence is greater than one (greater than zero would not suffice, because the limit as well as the elements of the sequence need to be positive) (see caucvgsrlemofff 7719). 3. Since a signed real (element of ) which is greater than zero can be mapped to a positive real (element of ), perform that mapping on each element of the sequence and invoke caucvgprpr 7634 to get a limit (see caucvgsrlemgt1 7717). 4. Map the resulting limit from positive reals back to signed reals (see caucvgsrlemgt1 7717). 5. Offset that limit so that we get the limit of the original sequence rather than the limit of the offsetted sequence (see caucvgsrlemoffres 7722). (Contributed by Jim Kingdon, 20-Jun-2021.) |
Ref | Expression |
---|---|
caucvgsr.f | |
caucvgsr.cau |
Ref | Expression |
---|---|
caucvgsr |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | caucvgsr.f | . 2 | |
2 | caucvgsr.cau | . 2 | |
3 | breq1 3970 | . . . . . . . . . . . . 13 | |
4 | fveq2 5470 | . . . . . . . . . . . . . . 15 | |
5 | opeq1 3743 | . . . . . . . . . . . . . . . . . . . . . . . 24 | |
6 | 5 | eceq1d 6518 | . . . . . . . . . . . . . . . . . . . . . . 23 |
7 | 6 | fveq2d 5474 | . . . . . . . . . . . . . . . . . . . . . 22 |
8 | 7 | breq2d 3979 | . . . . . . . . . . . . . . . . . . . . 21 |
9 | 8 | abbidv 2275 | . . . . . . . . . . . . . . . . . . . 20 |
10 | 7 | breq1d 3977 | . . . . . . . . . . . . . . . . . . . . 21 |
11 | 10 | abbidv 2275 | . . . . . . . . . . . . . . . . . . . 20 |
12 | 9, 11 | opeq12d 3751 | . . . . . . . . . . . . . . . . . . 19 |
13 | 12 | oveq1d 5841 | . . . . . . . . . . . . . . . . . 18 |
14 | 13 | opeq1d 3749 | . . . . . . . . . . . . . . . . 17 |
15 | 14 | eceq1d 6518 | . . . . . . . . . . . . . . . 16 |
16 | 15 | oveq2d 5842 | . . . . . . . . . . . . . . 15 |
17 | 4, 16 | breq12d 3980 | . . . . . . . . . . . . . 14 |
18 | 4, 15 | oveq12d 5844 | . . . . . . . . . . . . . . 15 |
19 | 18 | breq2d 3979 | . . . . . . . . . . . . . 14 |
20 | 17, 19 | anbi12d 465 | . . . . . . . . . . . . 13 |
21 | 3, 20 | imbi12d 233 | . . . . . . . . . . . 12 |
22 | 21 | ralbidv 2457 | . . . . . . . . . . 11 |
23 | 1pi 7237 | . . . . . . . . . . . 12 | |
24 | 23 | a1i 9 | . . . . . . . . . . 11 |
25 | 22, 2, 24 | rspcdva 2821 | . . . . . . . . . 10 |
26 | simpl 108 | . . . . . . . . . . . 12 | |
27 | 26 | imim2i 12 | . . . . . . . . . . 11 |
28 | 27 | ralimi 2520 | . . . . . . . . . 10 |
29 | 25, 28 | syl 14 | . . . . . . . . 9 |
30 | breq2 3971 | . . . . . . . . . . 11 | |
31 | fveq2 5470 | . . . . . . . . . . . . 13 | |
32 | 31 | oveq1d 5841 | . . . . . . . . . . . 12 |
33 | 32 | breq2d 3979 | . . . . . . . . . . 11 |
34 | 30, 33 | imbi12d 233 | . . . . . . . . . 10 |
35 | 34 | rspcv 2812 | . . . . . . . . 9 |
36 | 29, 35 | mpan9 279 | . . . . . . . 8 |
37 | df-1nqqs 7273 | . . . . . . . . . . . . . . . . . . . 20 | |
38 | 37 | fveq2i 5473 | . . . . . . . . . . . . . . . . . . 19 |
39 | rec1nq 7317 | . . . . . . . . . . . . . . . . . . 19 | |
40 | 38, 39 | eqtr3i 2180 | . . . . . . . . . . . . . . . . . 18 |
41 | 40 | breq2i 3975 | . . . . . . . . . . . . . . . . 17 |
42 | 41 | abbii 2273 | . . . . . . . . . . . . . . . 16 |
43 | 40 | breq1i 3974 | . . . . . . . . . . . . . . . . 17 |
44 | 43 | abbii 2273 | . . . . . . . . . . . . . . . 16 |
45 | 42, 44 | opeq12i 3748 | . . . . . . . . . . . . . . 15 |
46 | df-i1p 7389 | . . . . . . . . . . . . . . 15 | |
47 | 45, 46 | eqtr4i 2181 | . . . . . . . . . . . . . 14 |
48 | 47 | oveq1i 5836 | . . . . . . . . . . . . 13 |
49 | 48 | opeq1i 3746 | . . . . . . . . . . . 12 |
50 | eceq1 6517 | . . . . . . . . . . . 12 | |
51 | 49, 50 | ax-mp 5 | . . . . . . . . . . 11 |
52 | df-1r 7654 | . . . . . . . . . . 11 | |
53 | 51, 52 | eqtr4i 2181 | . . . . . . . . . 10 |
54 | 53 | oveq2i 5837 | . . . . . . . . 9 |
55 | 54 | breq2i 3975 | . . . . . . . 8 |
56 | 36, 55 | syl6ib 160 | . . . . . . 7 |
57 | 56 | imp 123 | . . . . . 6 |
58 | 1 | adantr 274 | . . . . . . . . . 10 |
59 | 23 | a1i 9 | . . . . . . . . . 10 |
60 | 58, 59 | ffvelrnd 5605 | . . . . . . . . 9 |
61 | ltadd1sr 7698 | . . . . . . . . 9 | |
62 | 60, 61 | syl 14 | . . . . . . . 8 |
63 | 62 | adantr 274 | . . . . . . 7 |
64 | fveq2 5470 | . . . . . . . . 9 | |
65 | 64 | oveq1d 5841 | . . . . . . . 8 |
66 | 65 | adantl 275 | . . . . . . 7 |
67 | 63, 66 | breqtrd 3992 | . . . . . 6 |
68 | nlt1pig 7263 | . . . . . . . . 9 | |
69 | 68 | adantl 275 | . . . . . . . 8 |
70 | 69 | pm2.21d 609 | . . . . . . 7 |
71 | 70 | imp 123 | . . . . . 6 |
72 | pitri3or 7244 | . . . . . . . 8 | |
73 | 23, 72 | mpan 421 | . . . . . . 7 |
74 | 73 | adantl 275 | . . . . . 6 |
75 | 57, 67, 71, 74 | mpjao3dan 1289 | . . . . 5 |
76 | ltasrg 7692 | . . . . . . 7 | |
77 | 76 | adantl 275 | . . . . . 6 |
78 | 1 | ffvelrnda 5604 | . . . . . . 7 |
79 | 1sr 7673 | . . . . . . 7 | |
80 | addclsr 7675 | . . . . . . 7 | |
81 | 78, 79, 80 | sylancl 410 | . . . . . 6 |
82 | m1r 7674 | . . . . . . 7 | |
83 | 82 | a1i 9 | . . . . . 6 |
84 | addcomsrg 7677 | . . . . . . 7 | |
85 | 84 | adantl 275 | . . . . . 6 |
86 | 77, 60, 81, 83, 85 | caovord2d 5992 | . . . . 5 |
87 | 75, 86 | mpbid 146 | . . . 4 |
88 | 79 | a1i 9 | . . . . . 6 |
89 | addasssrg 7678 | . . . . . 6 | |
90 | 78, 88, 83, 89 | syl3anc 1220 | . . . . 5 |
91 | addcomsrg 7677 | . . . . . . . . 9 | |
92 | 79, 82, 91 | mp2an 423 | . . . . . . . 8 |
93 | m1p1sr 7682 | . . . . . . . 8 | |
94 | 92, 93 | eqtri 2178 | . . . . . . 7 |
95 | 94 | oveq2i 5837 | . . . . . 6 |
96 | 0idsr 7689 | . . . . . . 7 | |
97 | 78, 96 | syl 14 | . . . . . 6 |
98 | 95, 97 | syl5eq 2202 | . . . . 5 |
99 | 90, 98 | eqtrd 2190 | . . . 4 |
100 | 87, 99 | breqtrd 3992 | . . 3 |
101 | 100 | ralrimiva 2530 | . 2 |
102 | 1, 2, 101 | caucvgsrlembnd 7723 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 103 wb 104 w3o 962 w3a 963 wceq 1335 wcel 2128 cab 2143 wral 2435 wrex 2436 cop 3564 class class class wbr 3967 wf 5168 cfv 5172 (class class class)co 5826 c1o 6358 cec 6480 cnpi 7194 clti 7197 ceq 7201 c1q 7203 crq 7206 cltq 7207 c1p 7214 cpp 7215 cer 7218 cnr 7219 c0r 7220 c1r 7221 cm1r 7222 cplr 7223 cltr 7225 |
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 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-13 2130 ax-14 2131 ax-ext 2139 ax-coll 4081 ax-sep 4084 ax-nul 4092 ax-pow 4137 ax-pr 4171 ax-un 4395 ax-setind 4498 ax-iinf 4549 |
This theorem depends on definitions: df-bi 116 df-dc 821 df-3or 964 df-3an 965 df-tru 1338 df-fal 1341 df-nf 1441 df-sb 1743 df-eu 2009 df-mo 2010 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ne 2328 df-ral 2440 df-rex 2441 df-reu 2442 df-rmo 2443 df-rab 2444 df-v 2714 df-sbc 2938 df-csb 3032 df-dif 3104 df-un 3106 df-in 3108 df-ss 3115 df-nul 3396 df-pw 3546 df-sn 3567 df-pr 3568 df-op 3570 df-uni 3775 df-int 3810 df-iun 3853 df-br 3968 df-opab 4028 df-mpt 4029 df-tr 4065 df-eprel 4251 df-id 4255 df-po 4258 df-iso 4259 df-iord 4328 df-on 4330 df-suc 4333 df-iom 4552 df-xp 4594 df-rel 4595 df-cnv 4596 df-co 4597 df-dm 4598 df-rn 4599 df-res 4600 df-ima 4601 df-iota 5137 df-fun 5174 df-fn 5175 df-f 5176 df-f1 5177 df-fo 5178 df-f1o 5179 df-fv 5180 df-riota 5782 df-ov 5829 df-oprab 5830 df-mpo 5831 df-1st 6090 df-2nd 6091 df-recs 6254 df-irdg 6319 df-1o 6365 df-2o 6366 df-oadd 6369 df-omul 6370 df-er 6482 df-ec 6484 df-qs 6488 df-ni 7226 df-pli 7227 df-mi 7228 df-lti 7229 df-plpq 7266 df-mpq 7267 df-enq 7269 df-nqqs 7270 df-plqqs 7271 df-mqqs 7272 df-1nqqs 7273 df-rq 7274 df-ltnqqs 7275 df-enq0 7346 df-nq0 7347 df-0nq0 7348 df-plq0 7349 df-mq0 7350 df-inp 7388 df-i1p 7389 df-iplp 7390 df-imp 7391 df-iltp 7392 df-enr 7648 df-nr 7649 df-plr 7650 df-mr 7651 df-ltr 7652 df-0r 7653 df-1r 7654 df-m1r 7655 |
This theorem is referenced by: axcaucvglemres 7821 |
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