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Mirrors > Home > ILE Home > Th. List > climlec2 | GIF version |
Description: Comparison of a constant to the limit of a sequence. (Contributed by NM, 28-Feb-2008.) (Revised by Mario Carneiro, 1-Feb-2014.) |
Ref | Expression |
---|---|
clim2iser.1 | ⊢ 𝑍 = (ℤ≥‘𝑀) |
climlec2.2 | ⊢ (𝜑 → 𝑀 ∈ ℤ) |
climlec2.3 | ⊢ (𝜑 → 𝐴 ∈ ℝ) |
climlec2.4 | ⊢ (𝜑 → 𝐹 ⇝ 𝐵) |
climlec2.5 | ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℝ) |
climlec2.6 | ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝐴 ≤ (𝐹‘𝑘)) |
Ref | Expression |
---|---|
climlec2 | ⊢ (𝜑 → 𝐴 ≤ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | clim2iser.1 | . 2 ⊢ 𝑍 = (ℤ≥‘𝑀) | |
2 | climlec2.2 | . 2 ⊢ (𝜑 → 𝑀 ∈ ℤ) | |
3 | climlec2.3 | . . . 4 ⊢ (𝜑 → 𝐴 ∈ ℝ) | |
4 | 3 | recnd 7787 | . . 3 ⊢ (𝜑 → 𝐴 ∈ ℂ) |
5 | 0z 9058 | . . 3 ⊢ 0 ∈ ℤ | |
6 | uzssz 9338 | . . . 4 ⊢ (ℤ≥‘0) ⊆ ℤ | |
7 | zex 9056 | . . . 4 ⊢ ℤ ∈ V | |
8 | 6, 7 | climconst2 11053 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ 0 ∈ ℤ) → (ℤ × {𝐴}) ⇝ 𝐴) |
9 | 4, 5, 8 | sylancl 409 | . 2 ⊢ (𝜑 → (ℤ × {𝐴}) ⇝ 𝐴) |
10 | climlec2.4 | . 2 ⊢ (𝜑 → 𝐹 ⇝ 𝐵) | |
11 | eluzelz 9328 | . . . . 5 ⊢ (𝑘 ∈ (ℤ≥‘𝑀) → 𝑘 ∈ ℤ) | |
12 | 11, 1 | eleq2s 2232 | . . . 4 ⊢ (𝑘 ∈ 𝑍 → 𝑘 ∈ ℤ) |
13 | fvconst2g 5627 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝑘 ∈ ℤ) → ((ℤ × {𝐴})‘𝑘) = 𝐴) | |
14 | 3, 12, 13 | syl2an 287 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → ((ℤ × {𝐴})‘𝑘) = 𝐴) |
15 | 3 | adantr 274 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝐴 ∈ ℝ) |
16 | 14, 15 | eqeltrd 2214 | . 2 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → ((ℤ × {𝐴})‘𝑘) ∈ ℝ) |
17 | climlec2.5 | . 2 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℝ) | |
18 | climlec2.6 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝐴 ≤ (𝐹‘𝑘)) | |
19 | 14, 18 | eqbrtrd 3945 | . 2 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → ((ℤ × {𝐴})‘𝑘) ≤ (𝐹‘𝑘)) |
20 | 1, 2, 9, 10, 16, 17, 19 | climle 11096 | 1 ⊢ (𝜑 → 𝐴 ≤ 𝐵) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 = wceq 1331 ∈ wcel 1480 {csn 3522 class class class wbr 3924 × cxp 4532 ‘cfv 5118 ℂcc 7611 ℝcr 7612 0cc0 7613 ≤ cle 7794 ℤcz 9047 ℤ≥cuz 9319 ⇝ cli 11040 |
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 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 ax-coll 4038 ax-sep 4041 ax-nul 4049 ax-pow 4093 ax-pr 4126 ax-un 4350 ax-setind 4447 ax-iinf 4497 ax-cnex 7704 ax-resscn 7705 ax-1cn 7706 ax-1re 7707 ax-icn 7708 ax-addcl 7709 ax-addrcl 7710 ax-mulcl 7711 ax-mulrcl 7712 ax-addcom 7713 ax-mulcom 7714 ax-addass 7715 ax-mulass 7716 ax-distr 7717 ax-i2m1 7718 ax-0lt1 7719 ax-1rid 7720 ax-0id 7721 ax-rnegex 7722 ax-precex 7723 ax-cnre 7724 ax-pre-ltirr 7725 ax-pre-ltwlin 7726 ax-pre-lttrn 7727 ax-pre-apti 7728 ax-pre-ltadd 7729 ax-pre-mulgt0 7730 ax-pre-mulext 7731 ax-arch 7732 ax-caucvg 7733 |
This theorem depends on definitions: df-bi 116 df-dc 820 df-3or 963 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2000 df-mo 2001 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ne 2307 df-nel 2402 df-ral 2419 df-rex 2420 df-reu 2421 df-rmo 2422 df-rab 2423 df-v 2683 df-sbc 2905 df-csb 2999 df-dif 3068 df-un 3070 df-in 3072 df-ss 3079 df-nul 3359 df-if 3470 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-uni 3732 df-int 3767 df-iun 3810 df-br 3925 df-opab 3985 df-mpt 3986 df-tr 4022 df-id 4210 df-po 4213 df-iso 4214 df-iord 4283 df-on 4285 df-ilim 4286 df-suc 4288 df-iom 4500 df-xp 4540 df-rel 4541 df-cnv 4542 df-co 4543 df-dm 4544 df-rn 4545 df-res 4546 df-ima 4547 df-iota 5083 df-fun 5120 df-fn 5121 df-f 5122 df-f1 5123 df-fo 5124 df-f1o 5125 df-fv 5126 df-riota 5723 df-ov 5770 df-oprab 5771 df-mpo 5772 df-1st 6031 df-2nd 6032 df-recs 6195 df-frec 6281 df-pnf 7795 df-mnf 7796 df-xr 7797 df-ltxr 7798 df-le 7799 df-sub 7928 df-neg 7929 df-reap 8330 df-ap 8337 df-div 8426 df-inn 8714 df-2 8772 df-3 8773 df-4 8774 df-n0 8971 df-z 9048 df-uz 9320 df-rp 9435 df-seqfrec 10212 df-exp 10286 df-cj 10607 df-re 10608 df-im 10609 df-rsqrt 10763 df-abs 10764 df-clim 11041 |
This theorem is referenced by: climub 11106 |
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