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Mirrors > Home > ILE Home > Th. List > isumle | 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 11345 | . . . 4 ⊢ (seq𝑀( + , 𝐹) ∈ dom ⇝ ↔ seq𝑀( + , 𝐹) ⇝ ( ⇝ ‘seq𝑀( + , 𝐹))) | |
5 | 3, 4 | sylib 122 | . . 3 ⊢ (𝜑 → seq𝑀( + , 𝐹) ⇝ ( ⇝ ‘seq𝑀( + , 𝐹))) |
6 | isumle.9 | . . . 4 ⊢ (𝜑 → seq𝑀( + , 𝐺) ∈ dom ⇝ ) | |
7 | climdm 11345 | . . . 4 ⊢ (seq𝑀( + , 𝐺) ∈ dom ⇝ ↔ seq𝑀( + , 𝐺) ⇝ ( ⇝ ‘seq𝑀( + , 𝐺))) | |
8 | 6, 7 | sylib 122 | . . 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 2266 | . . 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 2266 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐺‘𝑘) ∈ ℝ) |
15 | isumle.7 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝐴 ≤ 𝐵) | |
16 | 15, 9, 12 | 3brtr4d 4053 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ≤ (𝐺‘𝑘)) |
17 | 1, 2, 5, 8, 11, 14, 16 | iserle 11392 | . 2 ⊢ (𝜑 → ( ⇝ ‘seq𝑀( + , 𝐹)) ≤ ( ⇝ ‘seq𝑀( + , 𝐺))) |
18 | 10 | recnd 8022 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝐴 ∈ ℂ) |
19 | 1, 2, 9, 18 | isum 11435 | . 2 ⊢ (𝜑 → Σ𝑘 ∈ 𝑍 𝐴 = ( ⇝ ‘seq𝑀( + , 𝐹))) |
20 | 13 | recnd 8022 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝐵 ∈ ℂ) |
21 | 1, 2, 12, 20 | isum 11435 | . 2 ⊢ (𝜑 → Σ𝑘 ∈ 𝑍 𝐵 = ( ⇝ ‘seq𝑀( + , 𝐺))) |
22 | 17, 19, 21 | 3brtr4d 4053 | 1 ⊢ (𝜑 → Σ𝑘 ∈ 𝑍 𝐴 ≤ Σ𝑘 ∈ 𝑍 𝐵) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 = wceq 1364 ∈ wcel 2160 class class class wbr 4021 dom cdm 4647 ‘cfv 5238 ℝcr 7845 + caddc 7849 ≤ cle 8029 ℤcz 9289 ℤ≥cuz 9564 seqcseq 10485 ⇝ cli 11328 Σcsu 11403 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2162 ax-14 2163 ax-ext 2171 ax-coll 4136 ax-sep 4139 ax-nul 4147 ax-pow 4195 ax-pr 4230 ax-un 4454 ax-setind 4557 ax-iinf 4608 ax-cnex 7937 ax-resscn 7938 ax-1cn 7939 ax-1re 7940 ax-icn 7941 ax-addcl 7942 ax-addrcl 7943 ax-mulcl 7944 ax-mulrcl 7945 ax-addcom 7946 ax-mulcom 7947 ax-addass 7948 ax-mulass 7949 ax-distr 7950 ax-i2m1 7951 ax-0lt1 7952 ax-1rid 7953 ax-0id 7954 ax-rnegex 7955 ax-precex 7956 ax-cnre 7957 ax-pre-ltirr 7958 ax-pre-ltwlin 7959 ax-pre-lttrn 7960 ax-pre-apti 7961 ax-pre-ltadd 7962 ax-pre-mulgt0 7963 ax-pre-mulext 7964 ax-arch 7965 ax-caucvg 7966 |
This theorem depends on definitions: df-bi 117 df-dc 836 df-3or 981 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ne 2361 df-nel 2456 df-ral 2473 df-rex 2474 df-reu 2475 df-rmo 2476 df-rab 2477 df-v 2754 df-sbc 2978 df-csb 3073 df-dif 3146 df-un 3148 df-in 3150 df-ss 3157 df-nul 3438 df-if 3550 df-pw 3595 df-sn 3616 df-pr 3617 df-op 3619 df-uni 3828 df-int 3863 df-iun 3906 df-br 4022 df-opab 4083 df-mpt 4084 df-tr 4120 df-id 4314 df-po 4317 df-iso 4318 df-iord 4387 df-on 4389 df-ilim 4390 df-suc 4392 df-iom 4611 df-xp 4653 df-rel 4654 df-cnv 4655 df-co 4656 df-dm 4657 df-rn 4658 df-res 4659 df-ima 4660 df-iota 5199 df-fun 5240 df-fn 5241 df-f 5242 df-f1 5243 df-fo 5244 df-f1o 5245 df-fv 5246 df-isom 5247 df-riota 5855 df-ov 5903 df-oprab 5904 df-mpo 5905 df-1st 6169 df-2nd 6170 df-recs 6334 df-irdg 6399 df-frec 6420 df-1o 6445 df-oadd 6449 df-er 6563 df-en 6771 df-dom 6772 df-fin 6773 df-pnf 8030 df-mnf 8031 df-xr 8032 df-ltxr 8033 df-le 8034 df-sub 8166 df-neg 8167 df-reap 8568 df-ap 8575 df-div 8666 df-inn 8956 df-2 9014 df-3 9015 df-4 9016 df-n0 9213 df-z 9290 df-uz 9565 df-q 9657 df-rp 9691 df-fz 10046 df-fzo 10180 df-seqfrec 10486 df-exp 10561 df-ihash 10798 df-cj 10893 df-re 10894 df-im 10895 df-rsqrt 11049 df-abs 11050 df-clim 11329 df-sumdc 11404 |
This theorem is referenced by: isumlessdc 11546 eftlub 11740 eflegeo 11751 trilpolemisumle 15292 |
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