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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 12665 | . . 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 12691 | . . 3 ⊢ (𝜑 → seq𝑀( + , 𝐹):𝑍⟶ℂ) |
9 | 8 | ffvelrnda 6267 | . 2 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (seq𝑀( + , 𝐹)‘𝑗) ∈ ℂ) |
10 | addcl 9897 | . . . 4 ⊢ ((𝑘 ∈ ℂ ∧ 𝑥 ∈ ℂ) → (𝑘 + 𝑥) ∈ ℂ) | |
11 | 10 | adantl 481 | . . 3 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ (𝑘 ∈ ℂ ∧ 𝑥 ∈ ℂ)) → (𝑘 + 𝑥) ∈ ℂ) |
12 | 4 | adantr 480 | . . . 4 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → 𝐶 ∈ ℂ) |
13 | adddi 9904 | . . . . 5 ⊢ ((𝐶 ∈ ℂ ∧ 𝑘 ∈ ℂ ∧ 𝑥 ∈ ℂ) → (𝐶 · (𝑘 + 𝑥)) = ((𝐶 · 𝑘) + (𝐶 · 𝑥))) | |
14 | 13 | 3expb 1258 | . . . 4 ⊢ ((𝐶 ∈ ℂ ∧ (𝑘 ∈ ℂ ∧ 𝑥 ∈ ℂ)) → (𝐶 · (𝑘 + 𝑥)) = ((𝐶 · 𝑘) + (𝐶 · 𝑥))) |
15 | 12, 14 | sylan 487 | . . 3 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ (𝑘 ∈ ℂ ∧ 𝑥 ∈ ℂ)) → (𝐶 · (𝑘 + 𝑥)) = ((𝐶 · 𝑘) + (𝐶 · 𝑥))) |
16 | simpr 476 | . . . 4 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → 𝑗 ∈ 𝑍) | |
17 | 16, 1 | syl6eleq 2698 | . . 3 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → 𝑗 ∈ (ℤ≥‘𝑀)) |
18 | elfzuz 12209 | . . . . . 6 ⊢ (𝑘 ∈ (𝑀...𝑗) → 𝑘 ∈ (ℤ≥‘𝑀)) | |
19 | 18, 1 | syl6eleqr 2699 | . . . . 5 ⊢ (𝑘 ∈ (𝑀...𝑗) → 𝑘 ∈ 𝑍) |
20 | 19, 7 | sylan2 490 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑗)) → (𝐹‘𝑘) ∈ ℂ) |
21 | 20 | adantlr 747 | . . 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 490 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑗)) → (𝐺‘𝑘) = (𝐶 · (𝐹‘𝑘))) |
24 | 23 | adantlr 747 | . . 3 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (𝑀...𝑗)) → (𝐺‘𝑘) = (𝐶 · (𝐹‘𝑘))) |
25 | 11, 15, 17, 21, 24 | seqdistr 12714 | . 2 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (seq𝑀( + , 𝐺)‘𝑗) = (𝐶 · (seq𝑀( + , 𝐹)‘𝑗))) |
26 | 1, 2, 3, 4, 6, 9, 25 | climmulc2 14215 | 1 ⊢ (𝜑 → seq𝑀( + , 𝐺) ⇝ (𝐶 · 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1475 ∈ wcel 1977 Vcvv 3173 class class class wbr 4583 ‘cfv 5804 (class class class)co 6549 ℂcc 9813 + caddc 9818 · cmul 9820 ℤcz 11254 ℤ≥cuz 11563 ...cfz 12197 seqcseq 12663 ⇝ cli 14063 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-inf2 8421 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 ax-pre-sup 9893 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-sup 8231 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-div 10564 df-nn 10898 df-2 10956 df-3 10957 df-n0 11170 df-z 11255 df-uz 11564 df-rp 11709 df-fz 12198 df-seq 12664 df-exp 12723 df-cj 13687 df-re 13688 df-im 13689 df-sqrt 13823 df-abs 13824 df-clim 14067 |
This theorem is referenced by: isummulc2 14335 cvgcmpce 14391 mertens 14457 ege2le3 14659 eftlub 14678 geolim3 23898 abelthlem6 23994 abelthlem7 23996 logtayl2 24208 atantayl 24464 log2cnv 24471 log2tlbnd 24472 lgamgulmlem4 24558 geomcau 32725 binomcxplemnotnn0 37577 fouriersw 39124 |
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