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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 6902 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 6977). 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 6973). 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 6902 to get a limit (see caucvgsrlemgt1 6971). 4. Map the resulting limit from positive reals back to signed reals (see caucvgsrlemgt1 6971). 5. Offset that limit so that we get the limit of the original sequence rather than the limit of the offsetted sequence (see caucvgsrlemoffres 6976). (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 6505 | . . . . . . . . . . 11 | |
4 | breq1 3788 | . . . . . . . . . . . . . 14 | |
5 | fveq2 5198 | . . . . . . . . . . . . . . . 16 | |
6 | opeq1 3570 | . . . . . . . . . . . . . . . . . . . . . . . . 25 | |
7 | 6 | eceq1d 6165 | . . . . . . . . . . . . . . . . . . . . . . . 24 |
8 | 7 | fveq2d 5202 | . . . . . . . . . . . . . . . . . . . . . . 23 |
9 | 8 | breq2d 3797 | . . . . . . . . . . . . . . . . . . . . . 22 |
10 | 9 | abbidv 2196 | . . . . . . . . . . . . . . . . . . . . 21 |
11 | 8 | breq1d 3795 | . . . . . . . . . . . . . . . . . . . . . 22 |
12 | 11 | abbidv 2196 | . . . . . . . . . . . . . . . . . . . . 21 |
13 | 10, 12 | opeq12d 3578 | . . . . . . . . . . . . . . . . . . . 20 |
14 | 13 | oveq1d 5547 | . . . . . . . . . . . . . . . . . . 19 |
15 | 14 | opeq1d 3576 | . . . . . . . . . . . . . . . . . 18 |
16 | 15 | eceq1d 6165 | . . . . . . . . . . . . . . . . 17 |
17 | 16 | oveq2d 5548 | . . . . . . . . . . . . . . . 16 |
18 | 5, 17 | breq12d 3798 | . . . . . . . . . . . . . . 15 |
19 | 5, 16 | oveq12d 5550 | . . . . . . . . . . . . . . . 16 |
20 | 19 | breq2d 3797 | . . . . . . . . . . . . . . 15 |
21 | 18, 20 | anbi12d 456 | . . . . . . . . . . . . . 14 |
22 | 4, 21 | imbi12d 232 | . . . . . . . . . . . . 13 |
23 | 22 | ralbidv 2368 | . . . . . . . . . . . 12 |
24 | 23 | rspcv 2697 | . . . . . . . . . . 11 |
25 | 3, 2, 24 | mpsyl 64 | . . . . . . . . . 10 |
26 | simpl 107 | . . . . . . . . . . . 12 | |
27 | 26 | imim2i 12 | . . . . . . . . . . 11 |
28 | 27 | ralimi 2426 | . . . . . . . . . 10 |
29 | 25, 28 | syl 14 | . . . . . . . . 9 |
30 | breq2 3789 | . . . . . . . . . . 11 | |
31 | fveq2 5198 | . . . . . . . . . . . . 13 | |
32 | 31 | oveq1d 5547 | . . . . . . . . . . . 12 |
33 | 32 | breq2d 3797 | . . . . . . . . . . 11 |
34 | 30, 33 | imbi12d 232 | . . . . . . . . . 10 |
35 | 34 | rspcv 2697 | . . . . . . . . 9 |
36 | 29, 35 | mpan9 275 | . . . . . . . 8 |
37 | df-1nqqs 6541 | . . . . . . . . . . . . . . . . . . . 20 | |
38 | 37 | fveq2i 5201 | . . . . . . . . . . . . . . . . . . 19 |
39 | rec1nq 6585 | . . . . . . . . . . . . . . . . . . 19 | |
40 | 38, 39 | eqtr3i 2103 | . . . . . . . . . . . . . . . . . 18 |
41 | 40 | breq2i 3793 | . . . . . . . . . . . . . . . . 17 |
42 | 41 | abbii 2194 | . . . . . . . . . . . . . . . 16 |
43 | 40 | breq1i 3792 | . . . . . . . . . . . . . . . . 17 |
44 | 43 | abbii 2194 | . . . . . . . . . . . . . . . 16 |
45 | 42, 44 | opeq12i 3575 | . . . . . . . . . . . . . . 15 |
46 | df-i1p 6657 | . . . . . . . . . . . . . . 15 | |
47 | 45, 46 | eqtr4i 2104 | . . . . . . . . . . . . . 14 |
48 | 47 | oveq1i 5542 | . . . . . . . . . . . . 13 |
49 | 48 | opeq1i 3573 | . . . . . . . . . . . 12 |
50 | eceq1 6164 | . . . . . . . . . . . 12 | |
51 | 49, 50 | ax-mp 7 | . . . . . . . . . . 11 |
52 | df-1r 6909 | . . . . . . . . . . 11 | |
53 | 51, 52 | eqtr4i 2104 | . . . . . . . . . 10 |
54 | 53 | oveq2i 5543 | . . . . . . . . 9 |
55 | 54 | breq2i 3793 | . . . . . . . 8 |
56 | 36, 55 | syl6ib 159 | . . . . . . 7 |
57 | 56 | imp 122 | . . . . . 6 |
58 | 1 | adantr 270 | . . . . . . . . . 10 |
59 | 3 | a1i 9 | . . . . . . . . . 10 |
60 | 58, 59 | ffvelrnd 5324 | . . . . . . . . 9 |
61 | ltadd1sr 6953 | . . . . . . . . 9 | |
62 | 60, 61 | syl 14 | . . . . . . . 8 |
63 | 62 | adantr 270 | . . . . . . 7 |
64 | fveq2 5198 | . . . . . . . . 9 | |
65 | 64 | oveq1d 5547 | . . . . . . . 8 |
66 | 65 | adantl 271 | . . . . . . 7 |
67 | 63, 66 | breqtrd 3809 | . . . . . 6 |
68 | nlt1pig 6531 | . . . . . . . . 9 | |
69 | 68 | adantl 271 | . . . . . . . 8 |
70 | 69 | pm2.21d 581 | . . . . . . 7 |
71 | 70 | imp 122 | . . . . . 6 |
72 | pitri3or 6512 | . . . . . . . 8 | |
73 | 3, 72 | mpan 414 | . . . . . . 7 |
74 | 73 | adantl 271 | . . . . . 6 |
75 | 57, 67, 71, 74 | mpjao3dan 1238 | . . . . 5 |
76 | ltasrg 6947 | . . . . . . 7 | |
77 | 76 | adantl 271 | . . . . . 6 |
78 | 1 | ffvelrnda 5323 | . . . . . . 7 |
79 | 1sr 6928 | . . . . . . 7 | |
80 | addclsr 6930 | . . . . . . 7 | |
81 | 78, 79, 80 | sylancl 404 | . . . . . 6 |
82 | m1r 6929 | . . . . . . 7 | |
83 | 82 | a1i 9 | . . . . . 6 |
84 | addcomsrg 6932 | . . . . . . 7 | |
85 | 84 | adantl 271 | . . . . . 6 |
86 | 77, 60, 81, 83, 85 | caovord2d 5690 | . . . . 5 |
87 | 75, 86 | mpbid 145 | . . . 4 |
88 | 79 | a1i 9 | . . . . . 6 |
89 | addasssrg 6933 | . . . . . 6 | |
90 | 78, 88, 83, 89 | syl3anc 1169 | . . . . 5 |
91 | addcomsrg 6932 | . . . . . . . . 9 | |
92 | 79, 82, 91 | mp2an 416 | . . . . . . . 8 |
93 | m1p1sr 6937 | . . . . . . . 8 | |
94 | 92, 93 | eqtri 2101 | . . . . . . 7 |
95 | 94 | oveq2i 5543 | . . . . . 6 |
96 | 0idsr 6944 | . . . . . . 7 | |
97 | 78, 96 | syl 14 | . . . . . 6 |
98 | 95, 97 | syl5eq 2125 | . . . . 5 |
99 | 90, 98 | eqtrd 2113 | . . . 4 |
100 | 87, 99 | breqtrd 3809 | . . 3 |
101 | 100 | ralrimiva 2434 | . 2 |
102 | 1, 2, 101 | caucvgsrlembnd 6977 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 102 wb 103 w3o 918 w3a 919 wceq 1284 wcel 1433 cab 2067 wral 2348 wrex 2349 cop 3401 class class class wbr 3785 wf 4918 cfv 4922 (class class class)co 5532 c1o 6017 cec 6127 cnpi 6462 clti 6465 ceq 6469 c1q 6471 crq 6474 cltq 6475 c1p 6482 cpp 6483 cer 6486 cnr 6487 c0r 6488 c1r 6489 cm1r 6490 cplr 6491 cltr 6493 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 576 ax-in2 577 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-13 1444 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-coll 3893 ax-sep 3896 ax-nul 3904 ax-pow 3948 ax-pr 3964 ax-un 4188 ax-setind 4280 ax-iinf 4329 |
This theorem depends on definitions: df-bi 115 df-dc 776 df-3or 920 df-3an 921 df-tru 1287 df-fal 1290 df-nf 1390 df-sb 1686 df-eu 1944 df-mo 1945 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ne 2246 df-ral 2353 df-rex 2354 df-reu 2355 df-rmo 2356 df-rab 2357 df-v 2603 df-sbc 2816 df-csb 2909 df-dif 2975 df-un 2977 df-in 2979 df-ss 2986 df-nul 3252 df-pw 3384 df-sn 3404 df-pr 3405 df-op 3407 df-uni 3602 df-int 3637 df-iun 3680 df-br 3786 df-opab 3840 df-mpt 3841 df-tr 3876 df-eprel 4044 df-id 4048 df-po 4051 df-iso 4052 df-iord 4121 df-on 4123 df-suc 4126 df-iom 4332 df-xp 4369 df-rel 4370 df-cnv 4371 df-co 4372 df-dm 4373 df-rn 4374 df-res 4375 df-ima 4376 df-iota 4887 df-fun 4924 df-fn 4925 df-f 4926 df-f1 4927 df-fo 4928 df-f1o 4929 df-fv 4930 df-riota 5488 df-ov 5535 df-oprab 5536 df-mpt2 5537 df-1st 5787 df-2nd 5788 df-recs 5943 df-irdg 5980 df-1o 6024 df-2o 6025 df-oadd 6028 df-omul 6029 df-er 6129 df-ec 6131 df-qs 6135 df-ni 6494 df-pli 6495 df-mi 6496 df-lti 6497 df-plpq 6534 df-mpq 6535 df-enq 6537 df-nqqs 6538 df-plqqs 6539 df-mqqs 6540 df-1nqqs 6541 df-rq 6542 df-ltnqqs 6543 df-enq0 6614 df-nq0 6615 df-0nq0 6616 df-plq0 6617 df-mq0 6618 df-inp 6656 df-i1p 6657 df-iplp 6658 df-imp 6659 df-iltp 6660 df-enr 6903 df-nr 6904 df-plr 6905 df-mr 6906 df-ltr 6907 df-0r 6908 df-1r 6909 df-m1r 6910 |
This theorem is referenced by: axcaucvglemres 7065 |
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