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Mirrors > Home > ILE Home > Th. List > isummulc2 | GIF version |
Description: An infinite sum multiplied by a constant. (Contributed by NM, 12-Nov-2005.) (Revised by Mario Carneiro, 23-Apr-2014.) |
Ref | Expression |
---|---|
isumcl.1 | ⊢ 𝑍 = (ℤ≥‘𝑀) |
isumcl.2 | ⊢ (𝜑 → 𝑀 ∈ ℤ) |
isumcl.3 | ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) = 𝐴) |
isumcl.4 | ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝐴 ∈ ℂ) |
isumcl.5 | ⊢ (𝜑 → seq𝑀( + , 𝐹) ∈ dom ⇝ ) |
summulc.6 | ⊢ (𝜑 → 𝐵 ∈ ℂ) |
Ref | Expression |
---|---|
isummulc2 | ⊢ (𝜑 → (𝐵 · Σ𝑘 ∈ 𝑍 𝐴) = Σ𝑘 ∈ 𝑍 (𝐵 · 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | isumcl.1 | . . 3 ⊢ 𝑍 = (ℤ≥‘𝑀) | |
2 | isumcl.2 | . . 3 ⊢ (𝜑 → 𝑀 ∈ ℤ) | |
3 | eqidd 2141 | . . 3 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝑍) → ((𝑘 ∈ 𝑍 ↦ (𝐵 · 𝐴))‘𝑚) = ((𝑘 ∈ 𝑍 ↦ (𝐵 · 𝐴))‘𝑚)) | |
4 | summulc.6 | . . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ ℂ) | |
5 | 4 | adantr 274 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝐵 ∈ ℂ) |
6 | isumcl.4 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝐴 ∈ ℂ) | |
7 | 5, 6 | mulcld 7810 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐵 · 𝐴) ∈ ℂ) |
8 | 7 | fmpttd 5583 | . . . 4 ⊢ (𝜑 → (𝑘 ∈ 𝑍 ↦ (𝐵 · 𝐴)):𝑍⟶ℂ) |
9 | 8 | ffvelrnda 5563 | . . 3 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝑍) → ((𝑘 ∈ 𝑍 ↦ (𝐵 · 𝐴))‘𝑚) ∈ ℂ) |
10 | isumcl.3 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) = 𝐴) | |
11 | isumcl.5 | . . . . 5 ⊢ (𝜑 → seq𝑀( + , 𝐹) ∈ dom ⇝ ) | |
12 | 1, 2, 10, 6, 11 | isumclim2 11223 | . . . 4 ⊢ (𝜑 → seq𝑀( + , 𝐹) ⇝ Σ𝑘 ∈ 𝑍 𝐴) |
13 | 10, 6 | eqeltrd 2217 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℂ) |
14 | 13 | ralrimiva 2508 | . . . . 5 ⊢ (𝜑 → ∀𝑘 ∈ 𝑍 (𝐹‘𝑘) ∈ ℂ) |
15 | fveq2 5429 | . . . . . . 7 ⊢ (𝑘 = 𝑚 → (𝐹‘𝑘) = (𝐹‘𝑚)) | |
16 | 15 | eleq1d 2209 | . . . . . 6 ⊢ (𝑘 = 𝑚 → ((𝐹‘𝑘) ∈ ℂ ↔ (𝐹‘𝑚) ∈ ℂ)) |
17 | 16 | rspccva 2792 | . . . . 5 ⊢ ((∀𝑘 ∈ 𝑍 (𝐹‘𝑘) ∈ ℂ ∧ 𝑚 ∈ 𝑍) → (𝐹‘𝑚) ∈ ℂ) |
18 | 14, 17 | sylan 281 | . . . 4 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝑍) → (𝐹‘𝑚) ∈ ℂ) |
19 | simpr 109 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝑘 ∈ 𝑍) | |
20 | eqid 2140 | . . . . . . . . 9 ⊢ (𝑘 ∈ 𝑍 ↦ (𝐵 · 𝐴)) = (𝑘 ∈ 𝑍 ↦ (𝐵 · 𝐴)) | |
21 | 20 | fvmpt2 5512 | . . . . . . . 8 ⊢ ((𝑘 ∈ 𝑍 ∧ (𝐵 · 𝐴) ∈ ℂ) → ((𝑘 ∈ 𝑍 ↦ (𝐵 · 𝐴))‘𝑘) = (𝐵 · 𝐴)) |
22 | 19, 7, 21 | syl2anc 409 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → ((𝑘 ∈ 𝑍 ↦ (𝐵 · 𝐴))‘𝑘) = (𝐵 · 𝐴)) |
23 | 10 | oveq2d 5798 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐵 · (𝐹‘𝑘)) = (𝐵 · 𝐴)) |
24 | 22, 23 | eqtr4d 2176 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → ((𝑘 ∈ 𝑍 ↦ (𝐵 · 𝐴))‘𝑘) = (𝐵 · (𝐹‘𝑘))) |
25 | 24 | ralrimiva 2508 | . . . . 5 ⊢ (𝜑 → ∀𝑘 ∈ 𝑍 ((𝑘 ∈ 𝑍 ↦ (𝐵 · 𝐴))‘𝑘) = (𝐵 · (𝐹‘𝑘))) |
26 | nffvmpt1 5440 | . . . . . . 7 ⊢ Ⅎ𝑘((𝑘 ∈ 𝑍 ↦ (𝐵 · 𝐴))‘𝑚) | |
27 | 26 | nfeq1 2292 | . . . . . 6 ⊢ Ⅎ𝑘((𝑘 ∈ 𝑍 ↦ (𝐵 · 𝐴))‘𝑚) = (𝐵 · (𝐹‘𝑚)) |
28 | fveq2 5429 | . . . . . . 7 ⊢ (𝑘 = 𝑚 → ((𝑘 ∈ 𝑍 ↦ (𝐵 · 𝐴))‘𝑘) = ((𝑘 ∈ 𝑍 ↦ (𝐵 · 𝐴))‘𝑚)) | |
29 | 15 | oveq2d 5798 | . . . . . . 7 ⊢ (𝑘 = 𝑚 → (𝐵 · (𝐹‘𝑘)) = (𝐵 · (𝐹‘𝑚))) |
30 | 28, 29 | eqeq12d 2155 | . . . . . 6 ⊢ (𝑘 = 𝑚 → (((𝑘 ∈ 𝑍 ↦ (𝐵 · 𝐴))‘𝑘) = (𝐵 · (𝐹‘𝑘)) ↔ ((𝑘 ∈ 𝑍 ↦ (𝐵 · 𝐴))‘𝑚) = (𝐵 · (𝐹‘𝑚)))) |
31 | 27, 30 | rspc 2787 | . . . . 5 ⊢ (𝑚 ∈ 𝑍 → (∀𝑘 ∈ 𝑍 ((𝑘 ∈ 𝑍 ↦ (𝐵 · 𝐴))‘𝑘) = (𝐵 · (𝐹‘𝑘)) → ((𝑘 ∈ 𝑍 ↦ (𝐵 · 𝐴))‘𝑚) = (𝐵 · (𝐹‘𝑚)))) |
32 | 25, 31 | mpan9 279 | . . . 4 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝑍) → ((𝑘 ∈ 𝑍 ↦ (𝐵 · 𝐴))‘𝑚) = (𝐵 · (𝐹‘𝑚))) |
33 | 1, 2, 4, 12, 18, 32 | isermulc2 11141 | . . 3 ⊢ (𝜑 → seq𝑀( + , (𝑘 ∈ 𝑍 ↦ (𝐵 · 𝐴))) ⇝ (𝐵 · Σ𝑘 ∈ 𝑍 𝐴)) |
34 | 1, 2, 3, 9, 33 | isumclim 11222 | . 2 ⊢ (𝜑 → Σ𝑚 ∈ 𝑍 ((𝑘 ∈ 𝑍 ↦ (𝐵 · 𝐴))‘𝑚) = (𝐵 · Σ𝑘 ∈ 𝑍 𝐴)) |
35 | 7 | ralrimiva 2508 | . . 3 ⊢ (𝜑 → ∀𝑘 ∈ 𝑍 (𝐵 · 𝐴) ∈ ℂ) |
36 | sumfct 11175 | . . 3 ⊢ (∀𝑘 ∈ 𝑍 (𝐵 · 𝐴) ∈ ℂ → Σ𝑚 ∈ 𝑍 ((𝑘 ∈ 𝑍 ↦ (𝐵 · 𝐴))‘𝑚) = Σ𝑘 ∈ 𝑍 (𝐵 · 𝐴)) | |
37 | 35, 36 | syl 14 | . 2 ⊢ (𝜑 → Σ𝑚 ∈ 𝑍 ((𝑘 ∈ 𝑍 ↦ (𝐵 · 𝐴))‘𝑚) = Σ𝑘 ∈ 𝑍 (𝐵 · 𝐴)) |
38 | 34, 37 | eqtr3d 2175 | 1 ⊢ (𝜑 → (𝐵 · Σ𝑘 ∈ 𝑍 𝐴) = Σ𝑘 ∈ 𝑍 (𝐵 · 𝐴)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 = wceq 1332 ∈ wcel 1481 ∀wral 2417 ↦ cmpt 3997 dom cdm 4547 ‘cfv 5131 (class class class)co 5782 ℂcc 7642 + caddc 7647 · cmul 7649 ℤcz 9078 ℤ≥cuz 9350 seqcseq 10249 ⇝ cli 11079 Σcsu 11154 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-13 1492 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-coll 4051 ax-sep 4054 ax-nul 4062 ax-pow 4106 ax-pr 4139 ax-un 4363 ax-setind 4460 ax-iinf 4510 ax-cnex 7735 ax-resscn 7736 ax-1cn 7737 ax-1re 7738 ax-icn 7739 ax-addcl 7740 ax-addrcl 7741 ax-mulcl 7742 ax-mulrcl 7743 ax-addcom 7744 ax-mulcom 7745 ax-addass 7746 ax-mulass 7747 ax-distr 7748 ax-i2m1 7749 ax-0lt1 7750 ax-1rid 7751 ax-0id 7752 ax-rnegex 7753 ax-precex 7754 ax-cnre 7755 ax-pre-ltirr 7756 ax-pre-ltwlin 7757 ax-pre-lttrn 7758 ax-pre-apti 7759 ax-pre-ltadd 7760 ax-pre-mulgt0 7761 ax-pre-mulext 7762 ax-arch 7763 ax-caucvg 7764 |
This theorem depends on definitions: df-bi 116 df-dc 821 df-3or 964 df-3an 965 df-tru 1335 df-fal 1338 df-nf 1438 df-sb 1737 df-eu 2003 df-mo 2004 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ne 2310 df-nel 2405 df-ral 2422 df-rex 2423 df-reu 2424 df-rmo 2425 df-rab 2426 df-v 2691 df-sbc 2914 df-csb 3008 df-dif 3078 df-un 3080 df-in 3082 df-ss 3089 df-nul 3369 df-if 3480 df-pw 3517 df-sn 3538 df-pr 3539 df-op 3541 df-uni 3745 df-int 3780 df-iun 3823 df-br 3938 df-opab 3998 df-mpt 3999 df-tr 4035 df-id 4223 df-po 4226 df-iso 4227 df-iord 4296 df-on 4298 df-ilim 4299 df-suc 4301 df-iom 4513 df-xp 4553 df-rel 4554 df-cnv 4555 df-co 4556 df-dm 4557 df-rn 4558 df-res 4559 df-ima 4560 df-iota 5096 df-fun 5133 df-fn 5134 df-f 5135 df-f1 5136 df-fo 5137 df-f1o 5138 df-fv 5139 df-isom 5140 df-riota 5738 df-ov 5785 df-oprab 5786 df-mpo 5787 df-1st 6046 df-2nd 6047 df-recs 6210 df-irdg 6275 df-frec 6296 df-1o 6321 df-oadd 6325 df-er 6437 df-en 6643 df-dom 6644 df-fin 6645 df-pnf 7826 df-mnf 7827 df-xr 7828 df-ltxr 7829 df-le 7830 df-sub 7959 df-neg 7960 df-reap 8361 df-ap 8368 df-div 8457 df-inn 8745 df-2 8803 df-3 8804 df-4 8805 df-n0 9002 df-z 9079 df-uz 9351 df-q 9439 df-rp 9471 df-fz 9822 df-fzo 9951 df-seqfrec 10250 df-exp 10324 df-ihash 10554 df-cj 10646 df-re 10647 df-im 10648 df-rsqrt 10802 df-abs 10803 df-clim 11080 df-sumdc 11155 |
This theorem is referenced by: isummulc1 11228 trirecip 11302 geoisum1c 11321 |
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