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Mirrors > Home > ILE Home > Th. List > axcaucvg | Unicode version |
Description: Real number completeness
axiom. A Cauchy sequence with a modulus of
convergence converges. 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).
Because we are stating this axiom before we have introduced notations for or division, we use for the natural numbers and express a reciprocal in terms of . This construction-dependent theorem should not be referenced directly; instead, use ax-caucvg 7096. (Contributed by Jim Kingdon, 8-Jul-2021.) (New usage is discouraged.) |
Ref | Expression |
---|---|
axcaucvg.n | |
axcaucvg.f | |
axcaucvg.cau |
Ref | Expression |
---|---|
axcaucvg |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | axcaucvg.n | . 2 | |
2 | axcaucvg.f | . 2 | |
3 | axcaucvg.cau | . 2 | |
4 | breq1 3788 | . . . . . . . . . . . . 13 | |
5 | 4 | cbvabv 2202 | . . . . . . . . . . . 12 |
6 | breq2 3789 | . . . . . . . . . . . . 13 | |
7 | 6 | cbvabv 2202 | . . . . . . . . . . . 12 |
8 | 5, 7 | opeq12i 3575 | . . . . . . . . . . 11 |
9 | 8 | oveq1i 5542 | . . . . . . . . . 10 |
10 | 9 | opeq1i 3573 | . . . . . . . . 9 |
11 | eceq1 6164 | . . . . . . . . 9 | |
12 | 10, 11 | ax-mp 7 | . . . . . . . 8 |
13 | 12 | opeq1i 3573 | . . . . . . 7 |
14 | 13 | fveq2i 5201 | . . . . . 6 |
15 | 14 | a1i 9 | . . . . 5 |
16 | opeq1 3570 | . . . . 5 | |
17 | 15, 16 | eqeq12d 2095 | . . . 4 |
18 | 17 | cbvriotav 5499 | . . 3 |
19 | 18 | mpteq2i 3865 | . 2 |
20 | 1, 2, 3, 19 | axcaucvglemres 7065 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 102 wceq 1284 wcel 1433 cab 2067 wral 2348 wrex 2349 cop 3401 cint 3636 class class class wbr 3785 cmpt 3839 wf 4918 cfv 4922 crio 5487 (class class class)co 5532 c1o 6017 cec 6127 cnpi 6462 ceq 6469 cltq 6475 c1p 6482 cpp 6483 cer 6486 cnr 6487 c0r 6488 cr 6980 cc0 6981 c1 6982 caddc 6984 cltrr 6985 cmul 6986 |
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 df-c 6987 df-0 6988 df-1 6989 df-r 6991 df-add 6992 df-mul 6993 df-lt 6994 |
This theorem is referenced by: (None) |
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