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Mirrors > Home > MPE Home > Th. List > climle | Structured version Visualization version GIF version |
Description: Comparison of the limits of two sequences. (Contributed by Paul Chapman, 10-Sep-2007.) (Revised by Mario Carneiro, 1-Feb-2014.) |
Ref | Expression |
---|---|
climadd.1 | ⊢ 𝑍 = (ℤ≥‘𝑀) |
climadd.2 | ⊢ (𝜑 → 𝑀 ∈ ℤ) |
climadd.4 | ⊢ (𝜑 → 𝐹 ⇝ 𝐴) |
climle.5 | ⊢ (𝜑 → 𝐺 ⇝ 𝐵) |
climle.6 | ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℝ) |
climle.7 | ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐺‘𝑘) ∈ ℝ) |
climle.8 | ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ≤ (𝐺‘𝑘)) |
Ref | Expression |
---|---|
climle | ⊢ (𝜑 → 𝐴 ≤ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | climadd.1 | . . 3 ⊢ 𝑍 = (ℤ≥‘𝑀) | |
2 | climadd.2 | . . 3 ⊢ (𝜑 → 𝑀 ∈ ℤ) | |
3 | climle.5 | . . . 4 ⊢ (𝜑 → 𝐺 ⇝ 𝐵) | |
4 | 1 | fvexi 6854 | . . . . . 6 ⊢ 𝑍 ∈ V |
5 | 4 | mptex 7170 | . . . . 5 ⊢ (𝑗 ∈ 𝑍 ↦ ((𝐺‘𝑗) − (𝐹‘𝑗))) ∈ V |
6 | 5 | a1i 11 | . . . 4 ⊢ (𝜑 → (𝑗 ∈ 𝑍 ↦ ((𝐺‘𝑗) − (𝐹‘𝑗))) ∈ V) |
7 | climadd.4 | . . . 4 ⊢ (𝜑 → 𝐹 ⇝ 𝐴) | |
8 | climle.7 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐺‘𝑘) ∈ ℝ) | |
9 | 8 | recnd 11142 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐺‘𝑘) ∈ ℂ) |
10 | climle.6 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℝ) | |
11 | 10 | recnd 11142 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℂ) |
12 | fveq2 6840 | . . . . . . 7 ⊢ (𝑗 = 𝑘 → (𝐺‘𝑗) = (𝐺‘𝑘)) | |
13 | fveq2 6840 | . . . . . . 7 ⊢ (𝑗 = 𝑘 → (𝐹‘𝑗) = (𝐹‘𝑘)) | |
14 | 12, 13 | oveq12d 7370 | . . . . . 6 ⊢ (𝑗 = 𝑘 → ((𝐺‘𝑗) − (𝐹‘𝑗)) = ((𝐺‘𝑘) − (𝐹‘𝑘))) |
15 | eqid 2738 | . . . . . 6 ⊢ (𝑗 ∈ 𝑍 ↦ ((𝐺‘𝑗) − (𝐹‘𝑗))) = (𝑗 ∈ 𝑍 ↦ ((𝐺‘𝑗) − (𝐹‘𝑗))) | |
16 | ovex 7385 | . . . . . 6 ⊢ ((𝐺‘𝑘) − (𝐹‘𝑘)) ∈ V | |
17 | 14, 15, 16 | fvmpt 6946 | . . . . 5 ⊢ (𝑘 ∈ 𝑍 → ((𝑗 ∈ 𝑍 ↦ ((𝐺‘𝑗) − (𝐹‘𝑗)))‘𝑘) = ((𝐺‘𝑘) − (𝐹‘𝑘))) |
18 | 17 | adantl 483 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → ((𝑗 ∈ 𝑍 ↦ ((𝐺‘𝑗) − (𝐹‘𝑗)))‘𝑘) = ((𝐺‘𝑘) − (𝐹‘𝑘))) |
19 | 1, 2, 3, 6, 7, 9, 11, 18 | climsub 15476 | . . 3 ⊢ (𝜑 → (𝑗 ∈ 𝑍 ↦ ((𝐺‘𝑗) − (𝐹‘𝑗))) ⇝ (𝐵 − 𝐴)) |
20 | 8, 10 | resubcld 11542 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → ((𝐺‘𝑘) − (𝐹‘𝑘)) ∈ ℝ) |
21 | 18, 20 | eqeltrd 2839 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → ((𝑗 ∈ 𝑍 ↦ ((𝐺‘𝑗) − (𝐹‘𝑗)))‘𝑘) ∈ ℝ) |
22 | climle.8 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ≤ (𝐺‘𝑘)) | |
23 | 8, 10 | subge0d 11704 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (0 ≤ ((𝐺‘𝑘) − (𝐹‘𝑘)) ↔ (𝐹‘𝑘) ≤ (𝐺‘𝑘))) |
24 | 22, 23 | mpbird 257 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 0 ≤ ((𝐺‘𝑘) − (𝐹‘𝑘))) |
25 | 24, 18 | breqtrrd 5132 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 0 ≤ ((𝑗 ∈ 𝑍 ↦ ((𝐺‘𝑗) − (𝐹‘𝑗)))‘𝑘)) |
26 | 1, 2, 19, 21, 25 | climge0 15426 | . 2 ⊢ (𝜑 → 0 ≤ (𝐵 − 𝐴)) |
27 | 1, 2, 3, 8 | climrecl 15425 | . . 3 ⊢ (𝜑 → 𝐵 ∈ ℝ) |
28 | 1, 2, 7, 10 | climrecl 15425 | . . 3 ⊢ (𝜑 → 𝐴 ∈ ℝ) |
29 | 27, 28 | subge0d 11704 | . 2 ⊢ (𝜑 → (0 ≤ (𝐵 − 𝐴) ↔ 𝐴 ≤ 𝐵)) |
30 | 26, 29 | mpbid 231 | 1 ⊢ (𝜑 → 𝐴 ≤ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 = wceq 1542 ∈ wcel 2107 Vcvv 3444 class class class wbr 5104 ↦ cmpt 5187 ‘cfv 6494 (class class class)co 7352 ℝcr 11009 0cc0 11010 ≤ cle 11149 − cmin 11344 ℤcz 12458 ℤ≥cuz 12722 ⇝ cli 15326 |
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 2709 ax-rep 5241 ax-sep 5255 ax-nul 5262 ax-pow 5319 ax-pr 5383 ax-un 7665 ax-cnex 11066 ax-resscn 11067 ax-1cn 11068 ax-icn 11069 ax-addcl 11070 ax-addrcl 11071 ax-mulcl 11072 ax-mulrcl 11073 ax-mulcom 11074 ax-addass 11075 ax-mulass 11076 ax-distr 11077 ax-i2m1 11078 ax-1ne0 11079 ax-1rid 11080 ax-rnegex 11081 ax-rrecex 11082 ax-cnre 11083 ax-pre-lttri 11084 ax-pre-lttrn 11085 ax-pre-ltadd 11086 ax-pre-mulgt0 11087 ax-pre-sup 11088 |
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 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2888 df-ne 2943 df-nel 3049 df-ral 3064 df-rex 3073 df-rmo 3352 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3739 df-csb 3855 df-dif 3912 df-un 3914 df-in 3916 df-ss 3926 df-pss 3928 df-nul 4282 df-if 4486 df-pw 4561 df-sn 4586 df-pr 4588 df-op 4592 df-uni 4865 df-iun 4955 df-br 5105 df-opab 5167 df-mpt 5188 df-tr 5222 df-id 5530 df-eprel 5536 df-po 5544 df-so 5545 df-fr 5587 df-we 5589 df-xp 5638 df-rel 5639 df-cnv 5640 df-co 5641 df-dm 5642 df-rn 5643 df-res 5644 df-ima 5645 df-pred 6252 df-ord 6319 df-on 6320 df-lim 6321 df-suc 6322 df-iota 6446 df-fun 6496 df-fn 6497 df-f 6498 df-f1 6499 df-fo 6500 df-f1o 6501 df-fv 6502 df-riota 7308 df-ov 7355 df-oprab 7356 df-mpo 7357 df-om 7796 df-2nd 7915 df-frecs 8205 df-wrecs 8236 df-recs 8310 df-rdg 8349 df-er 8607 df-pm 8727 df-en 8843 df-dom 8844 df-sdom 8845 df-sup 9337 df-inf 9338 df-pnf 11150 df-mnf 11151 df-xr 11152 df-ltxr 11153 df-le 11154 df-sub 11346 df-neg 11347 df-div 11772 df-nn 12113 df-2 12175 df-3 12176 df-n0 12373 df-z 12459 df-uz 12723 df-rp 12871 df-fl 13652 df-seq 13862 df-exp 13923 df-cj 14944 df-re 14945 df-im 14946 df-sqrt 15080 df-abs 15081 df-clim 15330 df-rlim 15331 |
This theorem is referenced by: climlec2 15503 iserle 15504 iseraltlem1 15526 iserabs 15660 cvgcmpub 15662 itg2monolem1 25067 ulmdvlem1 25711 dchrisumlema 26788 dchrisumlem3 26791 stirlinglem10 44219 |
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