![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > isumle | Structured version Visualization version GIF version |
Description: Comparison of two infinite sums. (Contributed by Paul Chapman, 13-Nov-2007.) (Revised by Mario Carneiro, 24-Apr-2014.) |
Ref | Expression |
---|---|
isumle.1 | ⊢ 𝑍 = (ℤ≥‘𝑀) |
isumle.2 | ⊢ (𝜑 → 𝑀 ∈ ℤ) |
isumle.3 | ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) = 𝐴) |
isumle.4 | ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝐴 ∈ ℝ) |
isumle.5 | ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐺‘𝑘) = 𝐵) |
isumle.6 | ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝐵 ∈ ℝ) |
isumle.7 | ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝐴 ≤ 𝐵) |
isumle.8 | ⊢ (𝜑 → seq𝑀( + , 𝐹) ∈ dom ⇝ ) |
isumle.9 | ⊢ (𝜑 → seq𝑀( + , 𝐺) ∈ dom ⇝ ) |
Ref | Expression |
---|---|
isumle | ⊢ (𝜑 → Σ𝑘 ∈ 𝑍 𝐴 ≤ Σ𝑘 ∈ 𝑍 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | isumle.1 | . . 3 ⊢ 𝑍 = (ℤ≥‘𝑀) | |
2 | isumle.2 | . . 3 ⊢ (𝜑 → 𝑀 ∈ ℤ) | |
3 | isumle.8 | . . . 4 ⊢ (𝜑 → seq𝑀( + , 𝐹) ∈ dom ⇝ ) | |
4 | climdm 14669 | . . . 4 ⊢ (seq𝑀( + , 𝐹) ∈ dom ⇝ ↔ seq𝑀( + , 𝐹) ⇝ ( ⇝ ‘seq𝑀( + , 𝐹))) | |
5 | 3, 4 | sylib 210 | . . 3 ⊢ (𝜑 → seq𝑀( + , 𝐹) ⇝ ( ⇝ ‘seq𝑀( + , 𝐹))) |
6 | isumle.9 | . . . 4 ⊢ (𝜑 → seq𝑀( + , 𝐺) ∈ dom ⇝ ) | |
7 | climdm 14669 | . . . 4 ⊢ (seq𝑀( + , 𝐺) ∈ dom ⇝ ↔ seq𝑀( + , 𝐺) ⇝ ( ⇝ ‘seq𝑀( + , 𝐺))) | |
8 | 6, 7 | sylib 210 | . . 3 ⊢ (𝜑 → seq𝑀( + , 𝐺) ⇝ ( ⇝ ‘seq𝑀( + , 𝐺))) |
9 | isumle.3 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) = 𝐴) | |
10 | isumle.4 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝐴 ∈ ℝ) | |
11 | 9, 10 | eqeltrd 2906 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℝ) |
12 | isumle.5 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐺‘𝑘) = 𝐵) | |
13 | isumle.6 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝐵 ∈ ℝ) | |
14 | 12, 13 | eqeltrd 2906 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐺‘𝑘) ∈ ℝ) |
15 | isumle.7 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝐴 ≤ 𝐵) | |
16 | 15, 9, 12 | 3brtr4d 4907 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ≤ (𝐺‘𝑘)) |
17 | 1, 2, 5, 8, 11, 14, 16 | iserle 14774 | . 2 ⊢ (𝜑 → ( ⇝ ‘seq𝑀( + , 𝐹)) ≤ ( ⇝ ‘seq𝑀( + , 𝐺))) |
18 | 10 | recnd 10392 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝐴 ∈ ℂ) |
19 | 1, 2, 9, 18 | isum 14834 | . 2 ⊢ (𝜑 → Σ𝑘 ∈ 𝑍 𝐴 = ( ⇝ ‘seq𝑀( + , 𝐹))) |
20 | 13 | recnd 10392 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝐵 ∈ ℂ) |
21 | 1, 2, 12, 20 | isum 14834 | . 2 ⊢ (𝜑 → Σ𝑘 ∈ 𝑍 𝐵 = ( ⇝ ‘seq𝑀( + , 𝐺))) |
22 | 17, 19, 21 | 3brtr4d 4907 | 1 ⊢ (𝜑 → Σ𝑘 ∈ 𝑍 𝐴 ≤ Σ𝑘 ∈ 𝑍 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 386 = wceq 1656 ∈ wcel 2164 class class class wbr 4875 dom cdm 5346 ‘cfv 6127 ℝcr 10258 + caddc 10262 ≤ cle 10399 ℤcz 11711 ℤ≥cuz 11975 seqcseq 13102 ⇝ cli 14599 Σcsu 14800 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-8 2166 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-13 2389 ax-ext 2803 ax-rep 4996 ax-sep 5007 ax-nul 5015 ax-pow 5067 ax-pr 5129 ax-un 7214 ax-inf2 8822 ax-cnex 10315 ax-resscn 10316 ax-1cn 10317 ax-icn 10318 ax-addcl 10319 ax-addrcl 10320 ax-mulcl 10321 ax-mulrcl 10322 ax-mulcom 10323 ax-addass 10324 ax-mulass 10325 ax-distr 10326 ax-i2m1 10327 ax-1ne0 10328 ax-1rid 10329 ax-rnegex 10330 ax-rrecex 10331 ax-cnre 10332 ax-pre-lttri 10333 ax-pre-lttrn 10334 ax-pre-ltadd 10335 ax-pre-mulgt0 10336 ax-pre-sup 10337 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-3or 1112 df-3an 1113 df-tru 1660 df-ex 1879 df-nf 1883 df-sb 2068 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-nel 3103 df-ral 3122 df-rex 3123 df-reu 3124 df-rmo 3125 df-rab 3126 df-v 3416 df-sbc 3663 df-csb 3758 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-pss 3814 df-nul 4147 df-if 4309 df-pw 4382 df-sn 4400 df-pr 4402 df-tp 4404 df-op 4406 df-uni 4661 df-int 4700 df-iun 4744 df-br 4876 df-opab 4938 df-mpt 4955 df-tr 4978 df-id 5252 df-eprel 5257 df-po 5265 df-so 5266 df-fr 5305 df-se 5306 df-we 5307 df-xp 5352 df-rel 5353 df-cnv 5354 df-co 5355 df-dm 5356 df-rn 5357 df-res 5358 df-ima 5359 df-pred 5924 df-ord 5970 df-on 5971 df-lim 5972 df-suc 5973 df-iota 6090 df-fun 6129 df-fn 6130 df-f 6131 df-f1 6132 df-fo 6133 df-f1o 6134 df-fv 6135 df-isom 6136 df-riota 6871 df-ov 6913 df-oprab 6914 df-mpt2 6915 df-om 7332 df-1st 7433 df-2nd 7434 df-wrecs 7677 df-recs 7739 df-rdg 7777 df-1o 7831 df-oadd 7835 df-er 8014 df-pm 8130 df-en 8229 df-dom 8230 df-sdom 8231 df-fin 8232 df-sup 8623 df-inf 8624 df-oi 8691 df-card 9085 df-pnf 10400 df-mnf 10401 df-xr 10402 df-ltxr 10403 df-le 10404 df-sub 10594 df-neg 10595 df-div 11017 df-nn 11358 df-2 11421 df-3 11422 df-n0 11626 df-z 11712 df-uz 11976 df-rp 12120 df-fz 12627 df-fzo 12768 df-fl 12895 df-seq 13103 df-exp 13162 df-hash 13418 df-cj 14223 df-re 14224 df-im 14225 df-sqrt 14359 df-abs 14360 df-clim 14603 df-rlim 14604 df-sum 14801 |
This theorem is referenced by: isumless 14958 eftlub 15218 eflegeo 15230 rpnnen2lem7 15330 aaliou3lem3 24505 abelthlem7 24598 log2tlbnd 25092 |
Copyright terms: Public domain | W3C validator |