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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 6659 | . . . . . 6 ⊢ 𝑍 ∈ V |
5 | 4 | mptex 6963 | . . . . 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 10658 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐺‘𝑘) ∈ ℂ) |
10 | climle.6 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℝ) | |
11 | 10 | recnd 10658 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℂ) |
12 | fveq2 6645 | . . . . . . 7 ⊢ (𝑗 = 𝑘 → (𝐺‘𝑗) = (𝐺‘𝑘)) | |
13 | fveq2 6645 | . . . . . . 7 ⊢ (𝑗 = 𝑘 → (𝐹‘𝑗) = (𝐹‘𝑘)) | |
14 | 12, 13 | oveq12d 7153 | . . . . . 6 ⊢ (𝑗 = 𝑘 → ((𝐺‘𝑗) − (𝐹‘𝑗)) = ((𝐺‘𝑘) − (𝐹‘𝑘))) |
15 | eqid 2798 | . . . . . 6 ⊢ (𝑗 ∈ 𝑍 ↦ ((𝐺‘𝑗) − (𝐹‘𝑗))) = (𝑗 ∈ 𝑍 ↦ ((𝐺‘𝑗) − (𝐹‘𝑗))) | |
16 | ovex 7168 | . . . . . 6 ⊢ ((𝐺‘𝑘) − (𝐹‘𝑘)) ∈ V | |
17 | 14, 15, 16 | fvmpt 6745 | . . . . 5 ⊢ (𝑘 ∈ 𝑍 → ((𝑗 ∈ 𝑍 ↦ ((𝐺‘𝑗) − (𝐹‘𝑗)))‘𝑘) = ((𝐺‘𝑘) − (𝐹‘𝑘))) |
18 | 17 | adantl 485 | . . . 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 14982 | . . 3 ⊢ (𝜑 → (𝑗 ∈ 𝑍 ↦ ((𝐺‘𝑗) − (𝐹‘𝑗))) ⇝ (𝐵 − 𝐴)) |
20 | 8, 10 | resubcld 11057 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → ((𝐺‘𝑘) − (𝐹‘𝑘)) ∈ ℝ) |
21 | 18, 20 | eqeltrd 2890 | . . 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 11219 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (0 ≤ ((𝐺‘𝑘) − (𝐹‘𝑘)) ↔ (𝐹‘𝑘) ≤ (𝐺‘𝑘))) |
24 | 22, 23 | mpbird 260 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 0 ≤ ((𝐺‘𝑘) − (𝐹‘𝑘))) |
25 | 24, 18 | breqtrrd 5058 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 0 ≤ ((𝑗 ∈ 𝑍 ↦ ((𝐺‘𝑗) − (𝐹‘𝑗)))‘𝑘)) |
26 | 1, 2, 19, 21, 25 | climge0 14933 | . 2 ⊢ (𝜑 → 0 ≤ (𝐵 − 𝐴)) |
27 | 1, 2, 3, 8 | climrecl 14932 | . . 3 ⊢ (𝜑 → 𝐵 ∈ ℝ) |
28 | 1, 2, 7, 10 | climrecl 14932 | . . 3 ⊢ (𝜑 → 𝐴 ∈ ℝ) |
29 | 27, 28 | subge0d 11219 | . 2 ⊢ (𝜑 → (0 ≤ (𝐵 − 𝐴) ↔ 𝐴 ≤ 𝐵)) |
30 | 26, 29 | mpbid 235 | 1 ⊢ (𝜑 → 𝐴 ≤ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1538 ∈ wcel 2111 Vcvv 3441 class class class wbr 5030 ↦ cmpt 5110 ‘cfv 6324 (class class class)co 7135 ℝcr 10525 0cc0 10526 ≤ cle 10665 − cmin 10859 ℤcz 11969 ℤ≥cuz 12231 ⇝ cli 14833 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 ax-pre-sup 10604 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-2nd 7672 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-er 8272 df-pm 8392 df-en 8493 df-dom 8494 df-sdom 8495 df-sup 8890 df-inf 8891 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-div 11287 df-nn 11626 df-2 11688 df-3 11689 df-n0 11886 df-z 11970 df-uz 12232 df-rp 12378 df-fl 13157 df-seq 13365 df-exp 13426 df-cj 14450 df-re 14451 df-im 14452 df-sqrt 14586 df-abs 14587 df-clim 14837 df-rlim 14838 |
This theorem is referenced by: climlec2 15007 iserle 15008 iseraltlem1 15030 iserabs 15162 cvgcmpub 15164 itg2monolem1 24354 ulmdvlem1 24995 dchrisumlema 26072 dchrisumlem3 26075 stirlinglem10 42725 |
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