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Mirrors > Home > MPE Home > Th. List > climlec2 | Structured version Visualization version 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 |
---|---|
clim2ser.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 | clim2ser.1 | . 2 ⊢ 𝑍 = (ℤ≥‘𝑀) | |
2 | climlec2.2 | . 2 ⊢ (𝜑 → 𝑀 ∈ ℤ) | |
3 | climlec2.3 | . . . 4 ⊢ (𝜑 → 𝐴 ∈ ℝ) | |
4 | 3 | recnd 11239 | . . 3 ⊢ (𝜑 → 𝐴 ∈ ℂ) |
5 | 0z 12566 | . . 3 ⊢ 0 ∈ ℤ | |
6 | uzssz 12840 | . . . 4 ⊢ (ℤ≥‘0) ⊆ ℤ | |
7 | zex 12564 | . . . 4 ⊢ ℤ ∈ V | |
8 | 6, 7 | climconst2 15489 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ 0 ∈ ℤ) → (ℤ × {𝐴}) ⇝ 𝐴) |
9 | 4, 5, 8 | sylancl 585 | . 2 ⊢ (𝜑 → (ℤ × {𝐴}) ⇝ 𝐴) |
10 | climlec2.4 | . 2 ⊢ (𝜑 → 𝐹 ⇝ 𝐵) | |
11 | eluzelz 12829 | . . . . 5 ⊢ (𝑘 ∈ (ℤ≥‘𝑀) → 𝑘 ∈ ℤ) | |
12 | 11, 1 | eleq2s 2843 | . . . 4 ⊢ (𝑘 ∈ 𝑍 → 𝑘 ∈ ℤ) |
13 | fvconst2g 7195 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝑘 ∈ ℤ) → ((ℤ × {𝐴})‘𝑘) = 𝐴) | |
14 | 3, 12, 13 | syl2an 595 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → ((ℤ × {𝐴})‘𝑘) = 𝐴) |
15 | 3 | adantr 480 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝐴 ∈ ℝ) |
16 | 14, 15 | eqeltrd 2825 | . 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 5160 | . 2 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → ((ℤ × {𝐴})‘𝑘) ≤ (𝐹‘𝑘)) |
20 | 1, 2, 9, 10, 16, 17, 19 | climle 15581 | 1 ⊢ (𝜑 → 𝐴 ≤ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1533 ∈ wcel 2098 {csn 4620 class class class wbr 5138 × cxp 5664 ‘cfv 6533 ℂcc 11104 ℝcr 11105 0cc0 11106 ≤ cle 11246 ℤcz 12555 ℤ≥cuz 12819 ⇝ cli 15425 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-rep 5275 ax-sep 5289 ax-nul 5296 ax-pow 5353 ax-pr 5417 ax-un 7718 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183 ax-pre-sup 11184 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-nel 3039 df-ral 3054 df-rex 3063 df-rmo 3368 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3770 df-csb 3886 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-pss 3959 df-nul 4315 df-if 4521 df-pw 4596 df-sn 4621 df-pr 4623 df-op 4627 df-uni 4900 df-iun 4989 df-br 5139 df-opab 5201 df-mpt 5222 df-tr 5256 df-id 5564 df-eprel 5570 df-po 5578 df-so 5579 df-fr 5621 df-we 5623 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-res 5678 df-ima 5679 df-pred 6290 df-ord 6357 df-on 6358 df-lim 6359 df-suc 6360 df-iota 6485 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-riota 7357 df-ov 7404 df-oprab 7405 df-mpo 7406 df-om 7849 df-2nd 7969 df-frecs 8261 df-wrecs 8292 df-recs 8366 df-rdg 8405 df-er 8699 df-pm 8819 df-en 8936 df-dom 8937 df-sdom 8938 df-sup 9433 df-inf 9434 df-pnf 11247 df-mnf 11248 df-xr 11249 df-ltxr 11250 df-le 11251 df-sub 11443 df-neg 11444 df-div 11869 df-nn 12210 df-2 12272 df-3 12273 df-n0 12470 df-z 12556 df-uz 12820 df-rp 12972 df-fl 13754 df-seq 13964 df-exp 14025 df-cj 15043 df-re 15044 df-im 15045 df-sqrt 15179 df-abs 15180 df-clim 15429 df-rlim 15430 |
This theorem is referenced by: climub 15605 climlec3 35198 dvgrat 43560 |
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