geoisum1c - Intuitionistic Logic Explorer

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Theorem geoisum1c 12303

Description: The infinite sum of 𝐴 · (𝑅↑1) + 𝐴 · (𝑅↑2)... is (𝐴 · 𝑅) / (1 − 𝑅). (Contributed by NM, 2-Nov-2007.) (Revised by Mario Carneiro, 26-Apr-2014.)

**Assertion**
Ref	Expression
geoisum1c	⊢ ((𝐴 ∈ ℂ ∧ 𝑅 ∈ ℂ ∧ (abs‘𝑅) < 1) → Σ𝑘 ∈ ℕ (𝐴 · (𝑅↑𝑘)) = ((𝐴 · 𝑅) / (1 − 𝑅)))

Distinct variable groups: 𝐴,𝑘 𝑅,𝑘

Proof of Theorem geoisum1c Dummy variable 𝑛 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		simp1 1028	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝑅 ∈ ℂ ∧ (abs‘𝑅) < 1) → 𝐴 ∈ ℂ)
2		simp2 1029	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝑅 ∈ ℂ ∧ (abs‘𝑅) < 1) → 𝑅 ∈ ℂ)
3		1cnd 8342	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝑅 ∈ ℂ ∧ (abs‘𝑅) < 1) → 1 ∈ ℂ)
4	3, 2	subcld 8638	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝑅 ∈ ℂ ∧ (abs‘𝑅) < 1) → (1 − 𝑅) ∈ ℂ)
5		1red 8341	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 𝑅 ∈ ℂ ∧ (abs‘𝑅) < 1) → 1 ∈ ℝ)
6		simp3 1030	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 𝑅 ∈ ℂ ∧ (abs‘𝑅) < 1) → (abs‘𝑅) < 1)
7	2, 5, 6	absltap 12292	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝑅 ∈ ℂ ∧ (abs‘𝑅) < 1) → 𝑅 # 1)
8		apsym 8936	. . . . . 6 ⊢ ((𝑅 ∈ ℂ ∧ 1 ∈ ℂ) → (𝑅 # 1 ↔ 1 # 𝑅))
9	2, 3, 8	syl2anc 415	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝑅 ∈ ℂ ∧ (abs‘𝑅) < 1) → (𝑅 # 1 ↔ 1 # 𝑅))
10	7, 9	mpbid 147	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝑅 ∈ ℂ ∧ (abs‘𝑅) < 1) → 1 # 𝑅)
11	3, 2, 10	subap0d 8974	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝑅 ∈ ℂ ∧ (abs‘𝑅) < 1) → (1 − 𝑅) # 0)
12	1, 2, 4, 11	divassapd 9158	. 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝑅 ∈ ℂ ∧ (abs‘𝑅) < 1) → ((𝐴 · 𝑅) / (1 − 𝑅)) = (𝐴 · (𝑅 / (1 − 𝑅))))
13		geoisum1 12302	. . . 4 ⊢ ((𝑅 ∈ ℂ ∧ (abs‘𝑅) < 1) → Σ𝑘 ∈ ℕ (𝑅↑𝑘) = (𝑅 / (1 − 𝑅)))
14	13	3adant1 1046	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝑅 ∈ ℂ ∧ (abs‘𝑅) < 1) → Σ𝑘 ∈ ℕ (𝑅↑𝑘) = (𝑅 / (1 − 𝑅)))
15	14	oveq2d 6101	. 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝑅 ∈ ℂ ∧ (abs‘𝑅) < 1) → (𝐴 · Σ𝑘 ∈ ℕ (𝑅↑𝑘)) = (𝐴 · (𝑅 / (1 − 𝑅))))
16		nnuz 9967	. . 3 ⊢ ℕ = (ℤ_≥‘1)
17		1zzd 9675	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝑅 ∈ ℂ ∧ (abs‘𝑅) < 1) → 1 ∈ ℤ)
18		simpr 110	. . . 4 ⊢ (((𝐴 ∈ ℂ ∧ 𝑅 ∈ ℂ ∧ (abs‘𝑅) < 1) ∧ 𝑘 ∈ ℕ) → 𝑘 ∈ ℕ)
19		simpl2 1032	. . . . 5 ⊢ (((𝐴 ∈ ℂ ∧ 𝑅 ∈ ℂ ∧ (abs‘𝑅) < 1) ∧ 𝑘 ∈ ℕ) → 𝑅 ∈ ℂ)
20	18	nnnn0d 9624	. . . . 5 ⊢ (((𝐴 ∈ ℂ ∧ 𝑅 ∈ ℂ ∧ (abs‘𝑅) < 1) ∧ 𝑘 ∈ ℕ) → 𝑘 ∈ ℕ₀)
21	19, 20	expcld 11124	. . . 4 ⊢ (((𝐴 ∈ ℂ ∧ 𝑅 ∈ ℂ ∧ (abs‘𝑅) < 1) ∧ 𝑘 ∈ ℕ) → (𝑅↑𝑘) ∈ ℂ)
22		oveq2 6093	. . . . 5 ⊢ (𝑛 = 𝑘 → (𝑅↑𝑛) = (𝑅↑𝑘))
23		eqid 2238	. . . . 5 ⊢ (𝑛 ∈ ℕ ↦ (𝑅↑𝑛)) = (𝑛 ∈ ℕ ↦ (𝑅↑𝑛))
24	22, 23	fvmptg 5781	. . . 4 ⊢ ((𝑘 ∈ ℕ ∧ (𝑅↑𝑘) ∈ ℂ) → ((𝑛 ∈ ℕ ↦ (𝑅↑𝑛))‘𝑘) = (𝑅↑𝑘))
25	18, 21, 24	syl2anc 415	. . 3 ⊢ (((𝐴 ∈ ℂ ∧ 𝑅 ∈ ℂ ∧ (abs‘𝑅) < 1) ∧ 𝑘 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ (𝑅↑𝑛))‘𝑘) = (𝑅↑𝑘))
26		nnnn0 9574	. . . . 5 ⊢ (𝑘 ∈ ℕ → 𝑘 ∈ ℕ₀)
27	26	adantl 277	. . . 4 ⊢ (((𝐴 ∈ ℂ ∧ 𝑅 ∈ ℂ ∧ (abs‘𝑅) < 1) ∧ 𝑘 ∈ ℕ) → 𝑘 ∈ ℕ₀)
28	19, 27	expcld 11124	. . 3 ⊢ (((𝐴 ∈ ℂ ∧ 𝑅 ∈ ℂ ∧ (abs‘𝑅) < 1) ∧ 𝑘 ∈ ℕ) → (𝑅↑𝑘) ∈ ℂ)
29		seqex 10899	. . . 4 ⊢ seq1( + , (𝑛 ∈ ℕ ↦ (𝑅↑𝑛))) ∈ V
30		1nn0 9583	. . . . . . 7 ⊢ 1 ∈ ℕ₀
31	30	a1i 9	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 𝑅 ∈ ℂ ∧ (abs‘𝑅) < 1) → 1 ∈ ℕ₀)
32		elnnuz 9968	. . . . . . 7 ⊢ (𝑘 ∈ ℕ ↔ 𝑘 ∈ (ℤ_≥‘1))
33	32, 25	sylan2br 288	. . . . . 6 ⊢ (((𝐴 ∈ ℂ ∧ 𝑅 ∈ ℂ ∧ (abs‘𝑅) < 1) ∧ 𝑘 ∈ (ℤ_≥‘1)) → ((𝑛 ∈ ℕ ↦ (𝑅↑𝑛))‘𝑘) = (𝑅↑𝑘))
34	2, 6, 31, 33	geolim2 12295	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝑅 ∈ ℂ ∧ (abs‘𝑅) < 1) → seq1( + , (𝑛 ∈ ℕ ↦ (𝑅↑𝑛))) ⇝ ((𝑅↑1) / (1 − 𝑅)))
35		climcl 12064	. . . . 5 ⊢ (seq1( + , (𝑛 ∈ ℕ ↦ (𝑅↑𝑛))) ⇝ ((𝑅↑1) / (1 − 𝑅)) → ((𝑅↑1) / (1 − 𝑅)) ∈ ℂ)
36	34, 35	syl 14	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝑅 ∈ ℂ ∧ (abs‘𝑅) < 1) → ((𝑅↑1) / (1 − 𝑅)) ∈ ℂ)
37		breldmg 4987	. . . 4 ⊢ ((seq1( + , (𝑛 ∈ ℕ ↦ (𝑅↑𝑛))) ∈ V ∧ ((𝑅↑1) / (1 − 𝑅)) ∈ ℂ ∧ seq1( + , (𝑛 ∈ ℕ ↦ (𝑅↑𝑛))) ⇝ ((𝑅↑1) / (1 − 𝑅))) → seq1( + , (𝑛 ∈ ℕ ↦ (𝑅↑𝑛))) ∈ dom ⇝ )
38	29, 36, 34, 37	mp3an2i 1383	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝑅 ∈ ℂ ∧ (abs‘𝑅) < 1) → seq1( + , (𝑛 ∈ ℕ ↦ (𝑅↑𝑛))) ∈ dom ⇝ )
39	16, 17, 25, 28, 38, 1	isummulc2 12209	. 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝑅 ∈ ℂ ∧ (abs‘𝑅) < 1) → (𝐴 · Σ𝑘 ∈ ℕ (𝑅↑𝑘)) = Σ𝑘 ∈ ℕ (𝐴 · (𝑅↑𝑘)))
40	12, 15, 39	3eqtr2rd 2278	1 ⊢ ((𝐴 ∈ ℂ ∧ 𝑅 ∈ ℂ ∧ (abs‘𝑅) < 1) → Σ𝑘 ∈ ℕ (𝐴 · (𝑅↑𝑘)) = ((𝐴 · 𝑅) / (1 − 𝑅)))

Colors of variables: wff set class

This proof depends on syntax axioms: → wi 4 ∧ wa 104 ↔ wb 105 ∧ w3a 1009 = wceq 1402 ∈ wcel 2209 Vcvv 2821 class class class wbr 4130 ↦ cmpt 4192 dom cdm 4774 ‘cfv 5377 (class class class)co 6085 ℂcc 8177 1c1 8180 + caddc 8182 · cmul 8184 < clt 8360 − cmin 8498 # cap 8911 / cdiv 9004 ℕcn 9306 ℕ₀cn0 9567 ℤ_≥cuz 9930 seqcseq 10897 ↑cexp 10988 abscabs 11777 ⇝ cli 12060 Σcsu 12135

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 623 ax-in2 624 ax-io 721 ax-5 1500 ax-7 1501 ax-gen 1502 ax-ie1 1546 ax-ie2 1547 ax-8 1557 ax-10 1558 ax-11 1559 ax-i12 1560 ax-bndl 1562 ax-4 1563 ax-17 1579 ax-i9 1583 ax-ial 1587 ax-i5r 1588 ax-14 2212 ax-ext 2220 ax-coll 4246 ax-sep 4249 ax-nul 4259 ax-pow 4311 ax-pr 4346 ax-un 4578 ax-setind 4684 ax-iinf 4735 ax-cnex 8270 ax-resscn 8271 ax-1cn 8272 ax-1re 8273 ax-icn 8274 ax-addcl 8275 ax-addrcl 8276 ax-mulcl 8277 ax-mulrcl 8278 ax-addcom 8279 ax-mulcom 8280 ax-addass 8281 ax-mulass 8282 ax-distr 8283 ax-i2m1 8284 ax-0lt1 8285 ax-1rid 8286 ax-0id 8287 ax-rnegex 8288 ax-precex 8289 ax-cnre 8290 ax-pre-ltirr 8291 ax-pre-ltwlin 8292 ax-pre-lttrn 8293 ax-pre-apti 8294 ax-pre-ltadd 8295 ax-pre-mulgt0 8296 ax-pre-mulext 8297 ax-arch 8298 ax-caucvg 8299

This proof depends on definitions: df-bi 117 df-dc 847 df-3or 1010 df-3an 1011 df-tru 1405 df-fal 1408 df-nf 1514 df-sb 1816 df-eu 2089 df-mo 2090 df-clab 2225 df-cleq 2231 df-clel 2234 df-nfc 2381 df-ne 2421 df-nel 2516 df-ral 2533 df-rex 2534 df-reu 2535 df-rmo 2536 df-rab 2537 df-v 2823 df-sbc 3052 df-csb 3148 df-dif 3222 df-un 3224 df-in 3226 df-ss 3233 df-nul 3521 df-if 3639 df-pw 3690 df-sn 3715 df-pr 3716 df-op 3718 df-uni 3936 df-int 3971 df-iun 4014 df-br 4131 df-opab 4193 df-mpt 4194 df-tr 4230 df-id 4438 df-po 4441 df-iso 4442 df-iord 4511 df-on 4513 df-ilim 4514 df-suc 4516 df-iom 4738 df-xp 4780 df-rel 4781 df-cnv 4782 df-co 4783 df-dm 4784 df-rn 4785 df-res 4786 df-ima 4787 df-iota 5337 df-fun 5379 df-fn 5380 df-f 5381 df-f1 5382 df-fo 5383 df-f1o 5384 df-fv 5385 df-isom 5386 df-riota 6038 df-ov 6088 df-oprab 6089 df-mpo 6090 df-1st 6374 df-2nd 6375 df-recs 6576 df-irdg 6641 df-frec 6662 df-1o 6687 df-oadd 6691 df-er 6807 df-en 7023 df-dom 7024 df-fin 7025 df-pnf 8362 df-mnf 8363 df-xr 8364 df-ltxr 8365 df-le 8366 df-sub 8500 df-neg 8501 df-reap 8905 df-ap 8912 df-div 9005 df-inn 9307 df-2 9365 df-3 9366 df-4 9367 df-n0 9568 df-z 9649 df-uz 9931 df-q 10029 df-rp 10065 df-fz 10422 df-fzo 10560 df-seqfrec 10898 df-exp 10989 df-ihash 11229 df-cj 11621 df-re 11622 df-im 11623 df-rsqrt 11778 df-abs 11779 df-clim 12061 df-sumdc 12136

This theorem is used by: 0.999... 12304

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