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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 10336 | . . 3 ⊢ (𝜑 → -𝐵 ∈ ℝ) |
5 | climlec3.4 | . . . . 5 ⊢ (𝜑 → 𝐹 ⇝ 𝐴) | |
6 | 0cnd 9912 | . . . . 5 ⊢ (𝜑 → 0 ∈ ℂ) | |
7 | fvex 6113 | . . . . . . . 8 ⊢ (ℤ≥‘𝑀) ∈ V | |
8 | 1, 7 | eqeltri 2684 | . . . . . . 7 ⊢ 𝑍 ∈ V |
9 | 8 | mptex 6390 | . . . . . 6 ⊢ (𝑚 ∈ 𝑍 ↦ -(𝐹‘𝑚)) ∈ V |
10 | 9 | a1i 11 | . . . . 5 ⊢ (𝜑 → (𝑚 ∈ 𝑍 ↦ -(𝐹‘𝑚)) ∈ V) |
11 | climlec3.5 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℝ) | |
12 | 11 | recnd 9947 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℂ) |
13 | simpr 476 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝑘 ∈ 𝑍) | |
14 | 11 | renegcld 10336 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → -(𝐹‘𝑘) ∈ ℝ) |
15 | fveq2 6103 | . . . . . . . . 9 ⊢ (𝑚 = 𝑘 → (𝐹‘𝑚) = (𝐹‘𝑘)) | |
16 | 15 | negeqd 10154 | . . . . . . . 8 ⊢ (𝑚 = 𝑘 → -(𝐹‘𝑚) = -(𝐹‘𝑘)) |
17 | eqid 2610 | . . . . . . . 8 ⊢ (𝑚 ∈ 𝑍 ↦ -(𝐹‘𝑚)) = (𝑚 ∈ 𝑍 ↦ -(𝐹‘𝑚)) | |
18 | 16, 17 | fvmptg 6189 | . . . . . . 7 ⊢ ((𝑘 ∈ 𝑍 ∧ -(𝐹‘𝑘) ∈ ℝ) → ((𝑚 ∈ 𝑍 ↦ -(𝐹‘𝑚))‘𝑘) = -(𝐹‘𝑘)) |
19 | 13, 14, 18 | syl2anc 691 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → ((𝑚 ∈ 𝑍 ↦ -(𝐹‘𝑚))‘𝑘) = -(𝐹‘𝑘)) |
20 | df-neg 10148 | . . . . . 6 ⊢ -(𝐹‘𝑘) = (0 − (𝐹‘𝑘)) | |
21 | 19, 20 | syl6eq 2660 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → ((𝑚 ∈ 𝑍 ↦ -(𝐹‘𝑚))‘𝑘) = (0 − (𝐹‘𝑘))) |
22 | 1, 2, 5, 6, 10, 12, 21 | climsubc2 14217 | . . . 4 ⊢ (𝜑 → (𝑚 ∈ 𝑍 ↦ -(𝐹‘𝑚)) ⇝ (0 − 𝐴)) |
23 | df-neg 10148 | . . . 4 ⊢ -𝐴 = (0 − 𝐴) | |
24 | 22, 23 | syl6breqr 4625 | . . 3 ⊢ (𝜑 → (𝑚 ∈ 𝑍 ↦ -(𝐹‘𝑚)) ⇝ -𝐴) |
25 | 19, 14 | eqeltrd 2688 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → ((𝑚 ∈ 𝑍 ↦ -(𝐹‘𝑚))‘𝑘) ∈ ℝ) |
26 | climlec3.6 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ≤ 𝐵) | |
27 | 3 | adantr 480 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝐵 ∈ ℝ) |
28 | 11, 27 | lenegd 10485 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → ((𝐹‘𝑘) ≤ 𝐵 ↔ -𝐵 ≤ -(𝐹‘𝑘))) |
29 | 26, 28 | mpbid 221 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → -𝐵 ≤ -(𝐹‘𝑘)) |
30 | 29, 19 | breqtrrd 4611 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → -𝐵 ≤ ((𝑚 ∈ 𝑍 ↦ -(𝐹‘𝑚))‘𝑘)) |
31 | 1, 2, 4, 24, 25, 30 | climlec2 14237 | . 2 ⊢ (𝜑 → -𝐵 ≤ -𝐴) |
32 | 1, 2, 5, 11 | climrecl 14162 | . . 3 ⊢ (𝜑 → 𝐴 ∈ ℝ) |
33 | 32, 3 | lenegd 10485 | . 2 ⊢ (𝜑 → (𝐴 ≤ 𝐵 ↔ -𝐵 ≤ -𝐴)) |
34 | 31, 33 | mpbird 246 | 1 ⊢ (𝜑 → 𝐴 ≤ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1475 ∈ wcel 1977 Vcvv 3173 class class class wbr 4583 ↦ cmpt 4643 ‘cfv 5804 (class class class)co 6549 ℝcr 9814 0cc0 9815 ≤ cle 9954 − cmin 10145 -cneg 10146 ℤcz 11254 ℤ≥cuz 11563 ⇝ cli 14063 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 ax-pre-sup 9893 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-er 7629 df-pm 7747 df-en 7842 df-dom 7843 df-sdom 7844 df-sup 8231 df-inf 8232 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-div 10564 df-nn 10898 df-2 10956 df-3 10957 df-n0 11170 df-z 11255 df-uz 11564 df-rp 11709 df-fl 12455 df-seq 12664 df-exp 12723 df-cj 13687 df-re 13688 df-im 13689 df-sqrt 13823 df-abs 13824 df-clim 14067 df-rlim 14068 |
This theorem is referenced by: (None) |
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