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Mirrors > Home > MPE Home > Th. List > Mathboxes > climlec3 | Structured version Visualization version GIF version |
Description: Comparison of a constant to the limit of a sequence. (Contributed by Scott Fenton, 5-Jan-2018.) |
Ref | Expression |
---|---|
climlec3.1 | ⊢ 𝑍 = (ℤ≥‘𝑀) |
climlec3.2 | ⊢ (𝜑 → 𝑀 ∈ ℤ) |
climlec3.3 | ⊢ (𝜑 → 𝐵 ∈ ℝ) |
climlec3.4 | ⊢ (𝜑 → 𝐹 ⇝ 𝐴) |
climlec3.5 | ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℝ) |
climlec3.6 | ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ≤ 𝐵) |
Ref | Expression |
---|---|
climlec3 | ⊢ (𝜑 → 𝐴 ≤ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | climlec3.1 | . . 3 ⊢ 𝑍 = (ℤ≥‘𝑀) | |
2 | climlec3.2 | . . 3 ⊢ (𝜑 → 𝑀 ∈ ℤ) | |
3 | climlec3.3 | . . . 4 ⊢ (𝜑 → 𝐵 ∈ ℝ) | |
4 | 3 | renegcld 11066 | . . 3 ⊢ (𝜑 → -𝐵 ∈ ℝ) |
5 | climlec3.4 | . . . . 5 ⊢ (𝜑 → 𝐹 ⇝ 𝐴) | |
6 | 0cnd 10633 | . . . . 5 ⊢ (𝜑 → 0 ∈ ℂ) | |
7 | 1 | fvexi 6683 | . . . . . . 7 ⊢ 𝑍 ∈ V |
8 | 7 | mptex 6985 | . . . . . 6 ⊢ (𝑚 ∈ 𝑍 ↦ -(𝐹‘𝑚)) ∈ V |
9 | 8 | a1i 11 | . . . . 5 ⊢ (𝜑 → (𝑚 ∈ 𝑍 ↦ -(𝐹‘𝑚)) ∈ V) |
10 | climlec3.5 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℝ) | |
11 | 10 | recnd 10668 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℂ) |
12 | eqid 2821 | . . . . . . 7 ⊢ (𝑚 ∈ 𝑍 ↦ -(𝐹‘𝑚)) = (𝑚 ∈ 𝑍 ↦ -(𝐹‘𝑚)) | |
13 | fveq2 6669 | . . . . . . . 8 ⊢ (𝑚 = 𝑘 → (𝐹‘𝑚) = (𝐹‘𝑘)) | |
14 | 13 | negeqd 10879 | . . . . . . 7 ⊢ (𝑚 = 𝑘 → -(𝐹‘𝑚) = -(𝐹‘𝑘)) |
15 | simpr 487 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝑘 ∈ 𝑍) | |
16 | 10 | renegcld 11066 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → -(𝐹‘𝑘) ∈ ℝ) |
17 | 12, 14, 15, 16 | fvmptd3 6790 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → ((𝑚 ∈ 𝑍 ↦ -(𝐹‘𝑚))‘𝑘) = -(𝐹‘𝑘)) |
18 | df-neg 10872 | . . . . . 6 ⊢ -(𝐹‘𝑘) = (0 − (𝐹‘𝑘)) | |
19 | 17, 18 | syl6eq 2872 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → ((𝑚 ∈ 𝑍 ↦ -(𝐹‘𝑚))‘𝑘) = (0 − (𝐹‘𝑘))) |
20 | 1, 2, 5, 6, 9, 11, 19 | climsubc2 14994 | . . . 4 ⊢ (𝜑 → (𝑚 ∈ 𝑍 ↦ -(𝐹‘𝑚)) ⇝ (0 − 𝐴)) |
21 | df-neg 10872 | . . . 4 ⊢ -𝐴 = (0 − 𝐴) | |
22 | 20, 21 | breqtrrdi 5107 | . . 3 ⊢ (𝜑 → (𝑚 ∈ 𝑍 ↦ -(𝐹‘𝑚)) ⇝ -𝐴) |
23 | 17, 16 | eqeltrd 2913 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → ((𝑚 ∈ 𝑍 ↦ -(𝐹‘𝑚))‘𝑘) ∈ ℝ) |
24 | climlec3.6 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ≤ 𝐵) | |
25 | 3 | adantr 483 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝐵 ∈ ℝ) |
26 | 10, 25 | lenegd 11218 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → ((𝐹‘𝑘) ≤ 𝐵 ↔ -𝐵 ≤ -(𝐹‘𝑘))) |
27 | 24, 26 | mpbid 234 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → -𝐵 ≤ -(𝐹‘𝑘)) |
28 | 27, 17 | breqtrrd 5093 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → -𝐵 ≤ ((𝑚 ∈ 𝑍 ↦ -(𝐹‘𝑚))‘𝑘)) |
29 | 1, 2, 4, 22, 23, 28 | climlec2 15014 | . 2 ⊢ (𝜑 → -𝐵 ≤ -𝐴) |
30 | 1, 2, 5, 10 | climrecl 14939 | . . 3 ⊢ (𝜑 → 𝐴 ∈ ℝ) |
31 | 30, 3 | lenegd 11218 | . 2 ⊢ (𝜑 → (𝐴 ≤ 𝐵 ↔ -𝐵 ≤ -𝐴)) |
32 | 29, 31 | mpbird 259 | 1 ⊢ (𝜑 → 𝐴 ≤ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1533 ∈ wcel 2110 Vcvv 3494 class class class wbr 5065 ↦ cmpt 5145 ‘cfv 6354 (class class class)co 7155 ℝcr 10535 0cc0 10536 ≤ cle 10675 − cmin 10869 -cneg 10870 ℤcz 11980 ℤ≥cuz 12242 ⇝ cli 14840 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5189 ax-sep 5202 ax-nul 5209 ax-pow 5265 ax-pr 5329 ax-un 7460 ax-cnex 10592 ax-resscn 10593 ax-1cn 10594 ax-icn 10595 ax-addcl 10596 ax-addrcl 10597 ax-mulcl 10598 ax-mulrcl 10599 ax-mulcom 10600 ax-addass 10601 ax-mulass 10602 ax-distr 10603 ax-i2m1 10604 ax-1ne0 10605 ax-1rid 10606 ax-rnegex 10607 ax-rrecex 10608 ax-cnre 10609 ax-pre-lttri 10610 ax-pre-lttrn 10611 ax-pre-ltadd 10612 ax-pre-mulgt0 10613 ax-pre-sup 10614 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4567 df-pr 4569 df-tp 4571 df-op 4573 df-uni 4838 df-iun 4920 df-br 5066 df-opab 5128 df-mpt 5146 df-tr 5172 df-id 5459 df-eprel 5464 df-po 5473 df-so 5474 df-fr 5513 df-we 5515 df-xp 5560 df-rel 5561 df-cnv 5562 df-co 5563 df-dm 5564 df-rn 5565 df-res 5566 df-ima 5567 df-pred 6147 df-ord 6193 df-on 6194 df-lim 6195 df-suc 6196 df-iota 6313 df-fun 6356 df-fn 6357 df-f 6358 df-f1 6359 df-fo 6360 df-f1o 6361 df-fv 6362 df-riota 7113 df-ov 7158 df-oprab 7159 df-mpo 7160 df-om 7580 df-2nd 7689 df-wrecs 7946 df-recs 8007 df-rdg 8045 df-er 8288 df-pm 8408 df-en 8509 df-dom 8510 df-sdom 8511 df-sup 8905 df-inf 8906 df-pnf 10676 df-mnf 10677 df-xr 10678 df-ltxr 10679 df-le 10680 df-sub 10871 df-neg 10872 df-div 11297 df-nn 11638 df-2 11699 df-3 11700 df-n0 11897 df-z 11981 df-uz 12243 df-rp 12389 df-fl 13161 df-seq 13369 df-exp 13429 df-cj 14457 df-re 14458 df-im 14459 df-sqrt 14593 df-abs 14594 df-clim 14844 df-rlim 14845 |
This theorem is referenced by: (None) |
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