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Mirrors > Home > MPE Home > Th. List > isermulc2 | Structured version Visualization version GIF version |
Description: Multiplication of an infinite series by a constant. (Contributed by Paul Chapman, 14-Nov-2007.) (Revised by Mario Carneiro, 1-Feb-2014.) |
Ref | Expression |
---|---|
clim2ser.1 | ⊢ 𝑍 = (ℤ≥‘𝑀) |
isermulc2.2 | ⊢ (𝜑 → 𝑀 ∈ ℤ) |
isermulc2.4 | ⊢ (𝜑 → 𝐶 ∈ ℂ) |
isermulc2.5 | ⊢ (𝜑 → seq𝑀( + , 𝐹) ⇝ 𝐴) |
isermulc2.6 | ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℂ) |
isermulc2.7 | ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐺‘𝑘) = (𝐶 · (𝐹‘𝑘))) |
Ref | Expression |
---|---|
isermulc2 | ⊢ (𝜑 → seq𝑀( + , 𝐺) ⇝ (𝐶 · 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | clim2ser.1 | . 2 ⊢ 𝑍 = (ℤ≥‘𝑀) | |
2 | isermulc2.2 | . 2 ⊢ (𝜑 → 𝑀 ∈ ℤ) | |
3 | isermulc2.5 | . 2 ⊢ (𝜑 → seq𝑀( + , 𝐹) ⇝ 𝐴) | |
4 | isermulc2.4 | . 2 ⊢ (𝜑 → 𝐶 ∈ ℂ) | |
5 | seqex 14023 | . . 3 ⊢ seq𝑀( + , 𝐺) ∈ V | |
6 | 5 | a1i 11 | . 2 ⊢ (𝜑 → seq𝑀( + , 𝐺) ∈ V) |
7 | isermulc2.6 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℂ) | |
8 | 1, 2, 7 | serf 14050 | . . 3 ⊢ (𝜑 → seq𝑀( + , 𝐹):𝑍⟶ℂ) |
9 | 8 | ffvelcdmda 7098 | . 2 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (seq𝑀( + , 𝐹)‘𝑗) ∈ ℂ) |
10 | addcl 11240 | . . . 4 ⊢ ((𝑘 ∈ ℂ ∧ 𝑥 ∈ ℂ) → (𝑘 + 𝑥) ∈ ℂ) | |
11 | 10 | adantl 480 | . . 3 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ (𝑘 ∈ ℂ ∧ 𝑥 ∈ ℂ)) → (𝑘 + 𝑥) ∈ ℂ) |
12 | 4 | adantr 479 | . . . 4 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → 𝐶 ∈ ℂ) |
13 | adddi 11247 | . . . . 5 ⊢ ((𝐶 ∈ ℂ ∧ 𝑘 ∈ ℂ ∧ 𝑥 ∈ ℂ) → (𝐶 · (𝑘 + 𝑥)) = ((𝐶 · 𝑘) + (𝐶 · 𝑥))) | |
14 | 13 | 3expb 1117 | . . . 4 ⊢ ((𝐶 ∈ ℂ ∧ (𝑘 ∈ ℂ ∧ 𝑥 ∈ ℂ)) → (𝐶 · (𝑘 + 𝑥)) = ((𝐶 · 𝑘) + (𝐶 · 𝑥))) |
15 | 12, 14 | sylan 578 | . . 3 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ (𝑘 ∈ ℂ ∧ 𝑥 ∈ ℂ)) → (𝐶 · (𝑘 + 𝑥)) = ((𝐶 · 𝑘) + (𝐶 · 𝑥))) |
16 | simpr 483 | . . . 4 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → 𝑗 ∈ 𝑍) | |
17 | 16, 1 | eleqtrdi 2836 | . . 3 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → 𝑗 ∈ (ℤ≥‘𝑀)) |
18 | elfzuz 13551 | . . . . . 6 ⊢ (𝑘 ∈ (𝑀...𝑗) → 𝑘 ∈ (ℤ≥‘𝑀)) | |
19 | 18, 1 | eleqtrrdi 2837 | . . . . 5 ⊢ (𝑘 ∈ (𝑀...𝑗) → 𝑘 ∈ 𝑍) |
20 | 19, 7 | sylan2 591 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑗)) → (𝐹‘𝑘) ∈ ℂ) |
21 | 20 | adantlr 713 | . . 3 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (𝑀...𝑗)) → (𝐹‘𝑘) ∈ ℂ) |
22 | isermulc2.7 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐺‘𝑘) = (𝐶 · (𝐹‘𝑘))) | |
23 | 19, 22 | sylan2 591 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑗)) → (𝐺‘𝑘) = (𝐶 · (𝐹‘𝑘))) |
24 | 23 | adantlr 713 | . . 3 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (𝑀...𝑗)) → (𝐺‘𝑘) = (𝐶 · (𝐹‘𝑘))) |
25 | 11, 15, 17, 21, 24 | seqdistr 14073 | . 2 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (seq𝑀( + , 𝐺)‘𝑗) = (𝐶 · (seq𝑀( + , 𝐹)‘𝑗))) |
26 | 1, 2, 3, 4, 6, 9, 25 | climmulc2 15639 | 1 ⊢ (𝜑 → seq𝑀( + , 𝐺) ⇝ (𝐶 · 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 = wceq 1534 ∈ wcel 2099 Vcvv 3462 class class class wbr 5153 ‘cfv 6554 (class class class)co 7424 ℂcc 11156 + caddc 11161 · cmul 11163 ℤcz 12610 ℤ≥cuz 12874 ...cfz 13538 seqcseq 14021 ⇝ cli 15486 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-rep 5290 ax-sep 5304 ax-nul 5311 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-inf2 9684 ax-cnex 11214 ax-resscn 11215 ax-1cn 11216 ax-icn 11217 ax-addcl 11218 ax-addrcl 11219 ax-mulcl 11220 ax-mulrcl 11221 ax-mulcom 11222 ax-addass 11223 ax-mulass 11224 ax-distr 11225 ax-i2m1 11226 ax-1ne0 11227 ax-1rid 11228 ax-rnegex 11229 ax-rrecex 11230 ax-cnre 11231 ax-pre-lttri 11232 ax-pre-lttrn 11233 ax-pre-ltadd 11234 ax-pre-mulgt0 11235 ax-pre-sup 11236 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3967 df-nul 4326 df-if 4534 df-pw 4609 df-sn 4634 df-pr 4636 df-op 4640 df-uni 4914 df-iun 5003 df-br 5154 df-opab 5216 df-mpt 5237 df-tr 5271 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6312 df-ord 6379 df-on 6380 df-lim 6381 df-suc 6382 df-iota 6506 df-fun 6556 df-fn 6557 df-f 6558 df-f1 6559 df-fo 6560 df-f1o 6561 df-fv 6562 df-riota 7380 df-ov 7427 df-oprab 7428 df-mpo 7429 df-om 7877 df-1st 8003 df-2nd 8004 df-frecs 8296 df-wrecs 8327 df-recs 8401 df-rdg 8440 df-er 8734 df-en 8975 df-dom 8976 df-sdom 8977 df-sup 9485 df-pnf 11300 df-mnf 11301 df-xr 11302 df-ltxr 11303 df-le 11304 df-sub 11496 df-neg 11497 df-div 11922 df-nn 12265 df-2 12327 df-3 12328 df-n0 12525 df-z 12611 df-uz 12875 df-rp 13029 df-fz 13539 df-seq 14022 df-exp 14082 df-cj 15104 df-re 15105 df-im 15106 df-sqrt 15240 df-abs 15241 df-clim 15490 |
This theorem is referenced by: isummulc2 15766 cvgcmpce 15822 mertens 15890 ege2le3 16092 eftlub 16111 geolim3 26367 abelthlem6 26466 abelthlem7 26468 logtayl2 26689 atantayl 26965 log2cnv 26972 log2tlbnd 26973 lgamgulmlem4 27060 geomcau 37460 binomcxplemnotnn0 44030 fouriersw 45852 |
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