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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 6810 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 6885). 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 6881). 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 6810 to get a limit (see caucvgsrlemgt1 6879). 4. Map the resulting limit from positive reals back to signed reals (see caucvgsrlemgt1 6879). 5. Offset that limit so that we get the limit of the original sequence rather than the limit of the offsetted sequence (see caucvgsrlemoffres 6884). (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 | 1pi 6413 | . . . . . . . . . . 11 | |
4 | breq1 3767 | . . . . . . . . . . . . . 14 | |
5 | fveq2 5178 | . . . . . . . . . . . . . . . 16 | |
6 | opeq1 3549 | . . . . . . . . . . . . . . . . . . . . . . . . 25 | |
7 | 6 | eceq1d 6142 | . . . . . . . . . . . . . . . . . . . . . . . 24 |
8 | 7 | fveq2d 5182 | . . . . . . . . . . . . . . . . . . . . . . 23 |
9 | 8 | breq2d 3776 | . . . . . . . . . . . . . . . . . . . . . 22 |
10 | 9 | abbidv 2155 | . . . . . . . . . . . . . . . . . . . . 21 |
11 | 8 | breq1d 3774 | . . . . . . . . . . . . . . . . . . . . . 22 |
12 | 11 | abbidv 2155 | . . . . . . . . . . . . . . . . . . . . 21 |
13 | 10, 12 | opeq12d 3557 | . . . . . . . . . . . . . . . . . . . 20 |
14 | 13 | oveq1d 5527 | . . . . . . . . . . . . . . . . . . 19 |
15 | 14 | opeq1d 3555 | . . . . . . . . . . . . . . . . . 18 |
16 | 15 | eceq1d 6142 | . . . . . . . . . . . . . . . . 17 |
17 | 16 | oveq2d 5528 | . . . . . . . . . . . . . . . 16 |
18 | 5, 17 | breq12d 3777 | . . . . . . . . . . . . . . 15 |
19 | 5, 16 | oveq12d 5530 | . . . . . . . . . . . . . . . 16 |
20 | 19 | breq2d 3776 | . . . . . . . . . . . . . . 15 |
21 | 18, 20 | anbi12d 442 | . . . . . . . . . . . . . 14 |
22 | 4, 21 | imbi12d 223 | . . . . . . . . . . . . 13 |
23 | 22 | ralbidv 2326 | . . . . . . . . . . . 12 |
24 | 23 | rspcv 2652 | . . . . . . . . . . 11 |
25 | 3, 2, 24 | mpsyl 59 | . . . . . . . . . 10 |
26 | simpl 102 | . . . . . . . . . . . 12 | |
27 | 26 | imim2i 12 | . . . . . . . . . . 11 |
28 | 27 | ralimi 2384 | . . . . . . . . . 10 |
29 | 25, 28 | syl 14 | . . . . . . . . 9 |
30 | breq2 3768 | . . . . . . . . . . 11 | |
31 | fveq2 5178 | . . . . . . . . . . . . 13 | |
32 | 31 | oveq1d 5527 | . . . . . . . . . . . 12 |
33 | 32 | breq2d 3776 | . . . . . . . . . . 11 |
34 | 30, 33 | imbi12d 223 | . . . . . . . . . 10 |
35 | 34 | rspcv 2652 | . . . . . . . . 9 |
36 | 29, 35 | mpan9 265 | . . . . . . . 8 |
37 | df-1nqqs 6449 | . . . . . . . . . . . . . . . . . . . 20 | |
38 | 37 | fveq2i 5181 | . . . . . . . . . . . . . . . . . . 19 |
39 | rec1nq 6493 | . . . . . . . . . . . . . . . . . . 19 | |
40 | 38, 39 | eqtr3i 2062 | . . . . . . . . . . . . . . . . . 18 |
41 | 40 | breq2i 3772 | . . . . . . . . . . . . . . . . 17 |
42 | 41 | abbii 2153 | . . . . . . . . . . . . . . . 16 |
43 | 40 | breq1i 3771 | . . . . . . . . . . . . . . . . 17 |
44 | 43 | abbii 2153 | . . . . . . . . . . . . . . . 16 |
45 | 42, 44 | opeq12i 3554 | . . . . . . . . . . . . . . 15 |
46 | df-i1p 6565 | . . . . . . . . . . . . . . 15 | |
47 | 45, 46 | eqtr4i 2063 | . . . . . . . . . . . . . 14 |
48 | 47 | oveq1i 5522 | . . . . . . . . . . . . 13 |
49 | 48 | opeq1i 3552 | . . . . . . . . . . . 12 |
50 | eceq1 6141 | . . . . . . . . . . . 12 | |
51 | 49, 50 | ax-mp 7 | . . . . . . . . . . 11 |
52 | df-1r 6817 | . . . . . . . . . . 11 | |
53 | 51, 52 | eqtr4i 2063 | . . . . . . . . . 10 |
54 | 53 | oveq2i 5523 | . . . . . . . . 9 |
55 | 54 | breq2i 3772 | . . . . . . . 8 |
56 | 36, 55 | syl6ib 150 | . . . . . . 7 |
57 | 56 | imp 115 | . . . . . 6 |
58 | 1 | adantr 261 | . . . . . . . . . 10 |
59 | 3 | a1i 9 | . . . . . . . . . 10 |
60 | 58, 59 | ffvelrnd 5303 | . . . . . . . . 9 |
61 | ltadd1sr 6861 | . . . . . . . . 9 | |
62 | 60, 61 | syl 14 | . . . . . . . 8 |
63 | 62 | adantr 261 | . . . . . . 7 |
64 | fveq2 5178 | . . . . . . . . 9 | |
65 | 64 | oveq1d 5527 | . . . . . . . 8 |
66 | 65 | adantl 262 | . . . . . . 7 |
67 | 63, 66 | breqtrd 3788 | . . . . . 6 |
68 | nlt1pig 6439 | . . . . . . . . 9 | |
69 | 68 | adantl 262 | . . . . . . . 8 |
70 | 69 | pm2.21d 549 | . . . . . . 7 |
71 | 70 | imp 115 | . . . . . 6 |
72 | pitri3or 6420 | . . . . . . . 8 | |
73 | 3, 72 | mpan 400 | . . . . . . 7 |
74 | 73 | adantl 262 | . . . . . 6 |
75 | 57, 67, 71, 74 | mpjao3dan 1202 | . . . . 5 |
76 | ltasrg 6855 | . . . . . . 7 | |
77 | 76 | adantl 262 | . . . . . 6 |
78 | 1 | ffvelrnda 5302 | . . . . . . 7 |
79 | 1sr 6836 | . . . . . . 7 | |
80 | addclsr 6838 | . . . . . . 7 | |
81 | 78, 79, 80 | sylancl 392 | . . . . . 6 |
82 | m1r 6837 | . . . . . . 7 | |
83 | 82 | a1i 9 | . . . . . 6 |
84 | addcomsrg 6840 | . . . . . . 7 | |
85 | 84 | adantl 262 | . . . . . 6 |
86 | 77, 60, 81, 83, 85 | caovord2d 5670 | . . . . 5 |
87 | 75, 86 | mpbid 135 | . . . 4 |
88 | 79 | a1i 9 | . . . . . 6 |
89 | addasssrg 6841 | . . . . . 6 | |
90 | 78, 88, 83, 89 | syl3anc 1135 | . . . . 5 |
91 | addcomsrg 6840 | . . . . . . . . 9 | |
92 | 79, 82, 91 | mp2an 402 | . . . . . . . 8 |
93 | m1p1sr 6845 | . . . . . . . 8 | |
94 | 92, 93 | eqtri 2060 | . . . . . . 7 |
95 | 94 | oveq2i 5523 | . . . . . 6 |
96 | 0idsr 6852 | . . . . . . 7 | |
97 | 78, 96 | syl 14 | . . . . . 6 |
98 | 95, 97 | syl5eq 2084 | . . . . 5 |
99 | 90, 98 | eqtrd 2072 | . . . 4 |
100 | 87, 99 | breqtrd 3788 | . . 3 |
101 | 100 | ralrimiva 2392 | . 2 |
102 | 1, 2, 101 | caucvgsrlembnd 6885 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 97 wb 98 w3o 884 w3a 885 wceq 1243 wcel 1393 cab 2026 wral 2306 wrex 2307 cop 3378 class class class wbr 3764 wf 4898 cfv 4902 (class class class)co 5512 c1o 5994 cec 6104 cnpi 6370 clti 6373 ceq 6377 c1q 6379 crq 6382 cltq 6383 c1p 6390 cpp 6391 cer 6394 cnr 6395 c0r 6396 c1r 6397 cm1r 6398 cplr 6399 cltr 6401 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-in1 544 ax-in2 545 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-13 1404 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-coll 3872 ax-sep 3875 ax-nul 3883 ax-pow 3927 ax-pr 3944 ax-un 4170 ax-setind 4262 ax-iinf 4311 |
This theorem depends on definitions: df-bi 110 df-dc 743 df-3or 886 df-3an 887 df-tru 1246 df-fal 1249 df-nf 1350 df-sb 1646 df-eu 1903 df-mo 1904 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ne 2206 df-ral 2311 df-rex 2312 df-reu 2313 df-rmo 2314 df-rab 2315 df-v 2559 df-sbc 2765 df-csb 2853 df-dif 2920 df-un 2922 df-in 2924 df-ss 2931 df-nul 3225 df-pw 3361 df-sn 3381 df-pr 3382 df-op 3384 df-uni 3581 df-int 3616 df-iun 3659 df-br 3765 df-opab 3819 df-mpt 3820 df-tr 3855 df-eprel 4026 df-id 4030 df-po 4033 df-iso 4034 df-iord 4103 df-on 4105 df-suc 4108 df-iom 4314 df-xp 4351 df-rel 4352 df-cnv 4353 df-co 4354 df-dm 4355 df-rn 4356 df-res 4357 df-ima 4358 df-iota 4867 df-fun 4904 df-fn 4905 df-f 4906 df-f1 4907 df-fo 4908 df-f1o 4909 df-fv 4910 df-riota 5468 df-ov 5515 df-oprab 5516 df-mpt2 5517 df-1st 5767 df-2nd 5768 df-recs 5920 df-irdg 5957 df-1o 6001 df-2o 6002 df-oadd 6005 df-omul 6006 df-er 6106 df-ec 6108 df-qs 6112 df-ni 6402 df-pli 6403 df-mi 6404 df-lti 6405 df-plpq 6442 df-mpq 6443 df-enq 6445 df-nqqs 6446 df-plqqs 6447 df-mqqs 6448 df-1nqqs 6449 df-rq 6450 df-ltnqqs 6451 df-enq0 6522 df-nq0 6523 df-0nq0 6524 df-plq0 6525 df-mq0 6526 df-inp 6564 df-i1p 6565 df-iplp 6566 df-imp 6567 df-iltp 6568 df-enr 6811 df-nr 6812 df-plr 6813 df-mr 6814 df-ltr 6815 df-0r 6816 df-1r 6817 df-m1r 6818 |
This theorem is referenced by: axcaucvglemres 6973 |
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