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Mathbox for Scott Fenton |
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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 11590 | . . 3 ⊢ (𝜑 → -𝐵 ∈ ℝ) |
5 | climlec3.4 | . . . . 5 ⊢ (𝜑 → 𝐹 ⇝ 𝐴) | |
6 | 0cnd 11156 | . . . . 5 ⊢ (𝜑 → 0 ∈ ℂ) | |
7 | 1 | fvexi 6860 | . . . . . . 7 ⊢ 𝑍 ∈ V |
8 | 7 | mptex 7177 | . . . . . 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 11191 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℂ) |
12 | eqid 2733 | . . . . . . 7 ⊢ (𝑚 ∈ 𝑍 ↦ -(𝐹‘𝑚)) = (𝑚 ∈ 𝑍 ↦ -(𝐹‘𝑚)) | |
13 | fveq2 6846 | . . . . . . . 8 ⊢ (𝑚 = 𝑘 → (𝐹‘𝑚) = (𝐹‘𝑘)) | |
14 | 13 | negeqd 11403 | . . . . . . 7 ⊢ (𝑚 = 𝑘 → -(𝐹‘𝑚) = -(𝐹‘𝑘)) |
15 | simpr 486 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝑘 ∈ 𝑍) | |
16 | 10 | renegcld 11590 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → -(𝐹‘𝑘) ∈ ℝ) |
17 | 12, 14, 15, 16 | fvmptd3 6975 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → ((𝑚 ∈ 𝑍 ↦ -(𝐹‘𝑚))‘𝑘) = -(𝐹‘𝑘)) |
18 | df-neg 11396 | . . . . . 6 ⊢ -(𝐹‘𝑘) = (0 − (𝐹‘𝑘)) | |
19 | 17, 18 | eqtrdi 2789 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → ((𝑚 ∈ 𝑍 ↦ -(𝐹‘𝑚))‘𝑘) = (0 − (𝐹‘𝑘))) |
20 | 1, 2, 5, 6, 9, 11, 19 | climsubc2 15530 | . . . 4 ⊢ (𝜑 → (𝑚 ∈ 𝑍 ↦ -(𝐹‘𝑚)) ⇝ (0 − 𝐴)) |
21 | df-neg 11396 | . . . 4 ⊢ -𝐴 = (0 − 𝐴) | |
22 | 20, 21 | breqtrrdi 5151 | . . 3 ⊢ (𝜑 → (𝑚 ∈ 𝑍 ↦ -(𝐹‘𝑚)) ⇝ -𝐴) |
23 | 17, 16 | eqeltrd 2834 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → ((𝑚 ∈ 𝑍 ↦ -(𝐹‘𝑚))‘𝑘) ∈ ℝ) |
24 | climlec3.6 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ≤ 𝐵) | |
25 | 3 | adantr 482 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝐵 ∈ ℝ) |
26 | 10, 25 | lenegd 11742 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → ((𝐹‘𝑘) ≤ 𝐵 ↔ -𝐵 ≤ -(𝐹‘𝑘))) |
27 | 24, 26 | mpbid 231 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → -𝐵 ≤ -(𝐹‘𝑘)) |
28 | 27, 17 | breqtrrd 5137 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → -𝐵 ≤ ((𝑚 ∈ 𝑍 ↦ -(𝐹‘𝑚))‘𝑘)) |
29 | 1, 2, 4, 22, 23, 28 | climlec2 15552 | . 2 ⊢ (𝜑 → -𝐵 ≤ -𝐴) |
30 | 1, 2, 5, 10 | climrecl 15474 | . . 3 ⊢ (𝜑 → 𝐴 ∈ ℝ) |
31 | 30, 3 | lenegd 11742 | . 2 ⊢ (𝜑 → (𝐴 ≤ 𝐵 ↔ -𝐵 ≤ -𝐴)) |
32 | 29, 31 | mpbird 257 | 1 ⊢ (𝜑 → 𝐴 ≤ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 = wceq 1542 ∈ wcel 2107 Vcvv 3447 class class class wbr 5109 ↦ cmpt 5192 ‘cfv 6500 (class class class)co 7361 ℝcr 11058 0cc0 11059 ≤ cle 11198 − cmin 11393 -cneg 11394 ℤcz 12507 ℤ≥cuz 12771 ⇝ cli 15375 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5246 ax-sep 5260 ax-nul 5267 ax-pow 5324 ax-pr 5388 ax-un 7676 ax-cnex 11115 ax-resscn 11116 ax-1cn 11117 ax-icn 11118 ax-addcl 11119 ax-addrcl 11120 ax-mulcl 11121 ax-mulrcl 11122 ax-mulcom 11123 ax-addass 11124 ax-mulass 11125 ax-distr 11126 ax-i2m1 11127 ax-1ne0 11128 ax-1rid 11129 ax-rnegex 11130 ax-rrecex 11131 ax-cnre 11132 ax-pre-lttri 11133 ax-pre-lttrn 11134 ax-pre-ltadd 11135 ax-pre-mulgt0 11136 ax-pre-sup 11137 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3352 df-reu 3353 df-rab 3407 df-v 3449 df-sbc 3744 df-csb 3860 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-pss 3933 df-nul 4287 df-if 4491 df-pw 4566 df-sn 4591 df-pr 4593 df-op 4597 df-uni 4870 df-iun 4960 df-br 5110 df-opab 5172 df-mpt 5193 df-tr 5227 df-id 5535 df-eprel 5541 df-po 5549 df-so 5550 df-fr 5592 df-we 5594 df-xp 5643 df-rel 5644 df-cnv 5645 df-co 5646 df-dm 5647 df-rn 5648 df-res 5649 df-ima 5650 df-pred 6257 df-ord 6324 df-on 6325 df-lim 6326 df-suc 6327 df-iota 6452 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-riota 7317 df-ov 7364 df-oprab 7365 df-mpo 7366 df-om 7807 df-2nd 7926 df-frecs 8216 df-wrecs 8247 df-recs 8321 df-rdg 8360 df-er 8654 df-pm 8774 df-en 8890 df-dom 8891 df-sdom 8892 df-sup 9386 df-inf 9387 df-pnf 11199 df-mnf 11200 df-xr 11201 df-ltxr 11202 df-le 11203 df-sub 11395 df-neg 11396 df-div 11821 df-nn 12162 df-2 12224 df-3 12225 df-n0 12422 df-z 12508 df-uz 12772 df-rp 12924 df-fl 13706 df-seq 13916 df-exp 13977 df-cj 14993 df-re 14994 df-im 14995 df-sqrt 15129 df-abs 15130 df-clim 15379 df-rlim 15380 |
This theorem is referenced by: (None) |
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