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Mirrors > Home > ILE Home > Th. List > cau3 | GIF version |
Description: Convert between three-quantifier and four-quantifier versions of the Cauchy criterion. (In particular, the four-quantifier version has no occurrence of 𝑗 in the assertion, so it can be used with rexanuz 10552 and friends.) (Contributed by Mario Carneiro, 15-Feb-2014.) |
Ref | Expression |
---|---|
cau3.1 | ⊢ 𝑍 = (ℤ≥‘𝑀) |
Ref | Expression |
---|---|
cau3 | ⊢ (∀𝑥 ∈ ℝ+ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑥) ↔ ∀𝑥 ∈ ℝ+ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)((𝐹‘𝑘) ∈ ℂ ∧ ∀𝑚 ∈ (ℤ≥‘𝑘)(abs‘((𝐹‘𝑘) − (𝐹‘𝑚))) < 𝑥)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cau3.1 | . . . 4 ⊢ 𝑍 = (ℤ≥‘𝑀) | |
2 | uzssz 9137 | . . . 4 ⊢ (ℤ≥‘𝑀) ⊆ ℤ | |
3 | 1, 2 | eqsstri 3071 | . . 3 ⊢ 𝑍 ⊆ ℤ |
4 | id 19 | . . 3 ⊢ ((𝐹‘𝑘) ∈ ℂ → (𝐹‘𝑘) ∈ ℂ) | |
5 | eleq1 2157 | . . 3 ⊢ ((𝐹‘𝑘) = (𝐹‘𝑗) → ((𝐹‘𝑘) ∈ ℂ ↔ (𝐹‘𝑗) ∈ ℂ)) | |
6 | eleq1 2157 | . . 3 ⊢ ((𝐹‘𝑘) = (𝐹‘𝑚) → ((𝐹‘𝑘) ∈ ℂ ↔ (𝐹‘𝑚) ∈ ℂ)) | |
7 | abssub 10665 | . . . 4 ⊢ (((𝐹‘𝑗) ∈ ℂ ∧ (𝐹‘𝑘) ∈ ℂ) → (abs‘((𝐹‘𝑗) − (𝐹‘𝑘))) = (abs‘((𝐹‘𝑘) − (𝐹‘𝑗)))) | |
8 | 7 | 3adant1 964 | . . 3 ⊢ ((⊤ ∧ (𝐹‘𝑗) ∈ ℂ ∧ (𝐹‘𝑘) ∈ ℂ) → (abs‘((𝐹‘𝑗) − (𝐹‘𝑘))) = (abs‘((𝐹‘𝑘) − (𝐹‘𝑗)))) |
9 | abssub 10665 | . . . 4 ⊢ (((𝐹‘𝑚) ∈ ℂ ∧ (𝐹‘𝑗) ∈ ℂ) → (abs‘((𝐹‘𝑚) − (𝐹‘𝑗))) = (abs‘((𝐹‘𝑗) − (𝐹‘𝑚)))) | |
10 | 9 | 3adant1 964 | . . 3 ⊢ ((⊤ ∧ (𝐹‘𝑚) ∈ ℂ ∧ (𝐹‘𝑗) ∈ ℂ) → (abs‘((𝐹‘𝑚) − (𝐹‘𝑗))) = (abs‘((𝐹‘𝑗) − (𝐹‘𝑚)))) |
11 | abs3lem 10675 | . . . 4 ⊢ ((((𝐹‘𝑘) ∈ ℂ ∧ (𝐹‘𝑚) ∈ ℂ) ∧ ((𝐹‘𝑗) ∈ ℂ ∧ 𝑥 ∈ ℝ)) → (((abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < (𝑥 / 2) ∧ (abs‘((𝐹‘𝑗) − (𝐹‘𝑚))) < (𝑥 / 2)) → (abs‘((𝐹‘𝑘) − (𝐹‘𝑚))) < 𝑥)) | |
12 | 11 | 3adant1 964 | . . 3 ⊢ ((⊤ ∧ ((𝐹‘𝑘) ∈ ℂ ∧ (𝐹‘𝑚) ∈ ℂ) ∧ ((𝐹‘𝑗) ∈ ℂ ∧ 𝑥 ∈ ℝ)) → (((abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < (𝑥 / 2) ∧ (abs‘((𝐹‘𝑗) − (𝐹‘𝑚))) < (𝑥 / 2)) → (abs‘((𝐹‘𝑘) − (𝐹‘𝑚))) < 𝑥)) |
13 | 3, 4, 5, 6, 8, 10, 12 | cau3lem 10678 | . 2 ⊢ (⊤ → (∀𝑥 ∈ ℝ+ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑥) ↔ ∀𝑥 ∈ ℝ+ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)((𝐹‘𝑘) ∈ ℂ ∧ ∀𝑚 ∈ (ℤ≥‘𝑘)(abs‘((𝐹‘𝑘) − (𝐹‘𝑚))) < 𝑥))) |
14 | 13 | mptru 1305 | 1 ⊢ (∀𝑥 ∈ ℝ+ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑥) ↔ ∀𝑥 ∈ ℝ+ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)((𝐹‘𝑘) ∈ ℂ ∧ ∀𝑚 ∈ (ℤ≥‘𝑘)(abs‘((𝐹‘𝑘) − (𝐹‘𝑚))) < 𝑥)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ↔ wb 104 = wceq 1296 ⊤wtru 1297 ∈ wcel 1445 ∀wral 2370 ∃wrex 2371 class class class wbr 3867 ‘cfv 5049 (class class class)co 5690 ℂcc 7445 ℝcr 7446 < clt 7619 − cmin 7750 / cdiv 8236 2c2 8571 ℤcz 8848 ℤ≥cuz 9118 ℝ+crp 9233 abscabs 10561 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 582 ax-in2 583 ax-io 668 ax-5 1388 ax-7 1389 ax-gen 1390 ax-ie1 1434 ax-ie2 1435 ax-8 1447 ax-10 1448 ax-11 1449 ax-i12 1450 ax-bndl 1451 ax-4 1452 ax-13 1456 ax-14 1457 ax-17 1471 ax-i9 1475 ax-ial 1479 ax-i5r 1480 ax-ext 2077 ax-coll 3975 ax-sep 3978 ax-nul 3986 ax-pow 4030 ax-pr 4060 ax-un 4284 ax-setind 4381 ax-iinf 4431 ax-cnex 7533 ax-resscn 7534 ax-1cn 7535 ax-1re 7536 ax-icn 7537 ax-addcl 7538 ax-addrcl 7539 ax-mulcl 7540 ax-mulrcl 7541 ax-addcom 7542 ax-mulcom 7543 ax-addass 7544 ax-mulass 7545 ax-distr 7546 ax-i2m1 7547 ax-0lt1 7548 ax-1rid 7549 ax-0id 7550 ax-rnegex 7551 ax-precex 7552 ax-cnre 7553 ax-pre-ltirr 7554 ax-pre-ltwlin 7555 ax-pre-lttrn 7556 ax-pre-apti 7557 ax-pre-ltadd 7558 ax-pre-mulgt0 7559 ax-pre-mulext 7560 ax-arch 7561 ax-caucvg 7562 |
This theorem depends on definitions: df-bi 116 df-dc 784 df-3or 928 df-3an 929 df-tru 1299 df-fal 1302 df-nf 1402 df-sb 1700 df-eu 1958 df-mo 1959 df-clab 2082 df-cleq 2088 df-clel 2091 df-nfc 2224 df-ne 2263 df-nel 2358 df-ral 2375 df-rex 2376 df-reu 2377 df-rmo 2378 df-rab 2379 df-v 2635 df-sbc 2855 df-csb 2948 df-dif 3015 df-un 3017 df-in 3019 df-ss 3026 df-nul 3303 df-if 3414 df-pw 3451 df-sn 3472 df-pr 3473 df-op 3475 df-uni 3676 df-int 3711 df-iun 3754 df-br 3868 df-opab 3922 df-mpt 3923 df-tr 3959 df-id 4144 df-po 4147 df-iso 4148 df-iord 4217 df-on 4219 df-ilim 4220 df-suc 4222 df-iom 4434 df-xp 4473 df-rel 4474 df-cnv 4475 df-co 4476 df-dm 4477 df-rn 4478 df-res 4479 df-ima 4480 df-iota 5014 df-fun 5051 df-fn 5052 df-f 5053 df-f1 5054 df-fo 5055 df-f1o 5056 df-fv 5057 df-riota 5646 df-ov 5693 df-oprab 5694 df-mpt2 5695 df-1st 5949 df-2nd 5950 df-recs 6108 df-frec 6194 df-pnf 7621 df-mnf 7622 df-xr 7623 df-ltxr 7624 df-le 7625 df-sub 7752 df-neg 7753 df-reap 8149 df-ap 8156 df-div 8237 df-inn 8521 df-2 8579 df-3 8580 df-4 8581 df-n0 8772 df-z 8849 df-uz 9119 df-rp 9234 df-iseq 10002 df-seq3 10003 df-exp 10086 df-cj 10407 df-re 10408 df-im 10409 df-rsqrt 10562 df-abs 10563 |
This theorem is referenced by: cau4 10680 serf0 10910 |
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