geoisum1c - Intuitionistic Logic Explorer

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Theorem geoisum1c 11547

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 999	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝑅 ∈ ℂ ∧ (abs‘𝑅) < 1) → 𝐴 ∈ ℂ)
2		simp2 1000	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝑅 ∈ ℂ ∧ (abs‘𝑅) < 1) → 𝑅 ∈ ℂ)
3		1cnd 7992	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝑅 ∈ ℂ ∧ (abs‘𝑅) < 1) → 1 ∈ ℂ)
4	3, 2	subcld 8287	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝑅 ∈ ℂ ∧ (abs‘𝑅) < 1) → (1 − 𝑅) ∈ ℂ)
5		1red 7991	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 𝑅 ∈ ℂ ∧ (abs‘𝑅) < 1) → 1 ∈ ℝ)
6		simp3 1001	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 𝑅 ∈ ℂ ∧ (abs‘𝑅) < 1) → (abs‘𝑅) < 1)
7	2, 5, 6	absltap 11536	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝑅 ∈ ℂ ∧ (abs‘𝑅) < 1) → 𝑅 # 1)
8		apsym 8582	. . . . . 6 ⊢ ((𝑅 ∈ ℂ ∧ 1 ∈ ℂ) → (𝑅 # 1 ↔ 1 # 𝑅))
9	2, 3, 8	syl2anc 411	. . . . 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 8620	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝑅 ∈ ℂ ∧ (abs‘𝑅) < 1) → (1 − 𝑅) # 0)
12	1, 2, 4, 11	divassapd 8802	. 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝑅 ∈ ℂ ∧ (abs‘𝑅) < 1) → ((𝐴 · 𝑅) / (1 − 𝑅)) = (𝐴 · (𝑅 / (1 − 𝑅))))
13		geoisum1 11546	. . . 4 ⊢ ((𝑅 ∈ ℂ ∧ (abs‘𝑅) < 1) → Σ𝑘 ∈ ℕ (𝑅↑𝑘) = (𝑅 / (1 − 𝑅)))
14	13	3adant1 1017	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝑅 ∈ ℂ ∧ (abs‘𝑅) < 1) → Σ𝑘 ∈ ℕ (𝑅↑𝑘) = (𝑅 / (1 − 𝑅)))
15	14	oveq2d 5907	. 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝑅 ∈ ℂ ∧ (abs‘𝑅) < 1) → (𝐴 · Σ𝑘 ∈ ℕ (𝑅↑𝑘)) = (𝐴 · (𝑅 / (1 − 𝑅))))
16		nnuz 9582	. . 3 ⊢ ℕ = (ℤ_≥‘1)
17		1zzd 9299	. . 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 1003	. . . . 5 ⊢ (((𝐴 ∈ ℂ ∧ 𝑅 ∈ ℂ ∧ (abs‘𝑅) < 1) ∧ 𝑘 ∈ ℕ) → 𝑅 ∈ ℂ)
20	18	nnnn0d 9248	. . . . 5 ⊢ (((𝐴 ∈ ℂ ∧ 𝑅 ∈ ℂ ∧ (abs‘𝑅) < 1) ∧ 𝑘 ∈ ℕ) → 𝑘 ∈ ℕ₀)
21	19, 20	expcld 10673	. . . 4 ⊢ (((𝐴 ∈ ℂ ∧ 𝑅 ∈ ℂ ∧ (abs‘𝑅) < 1) ∧ 𝑘 ∈ ℕ) → (𝑅↑𝑘) ∈ ℂ)
22		oveq2 5899	. . . . 5 ⊢ (𝑛 = 𝑘 → (𝑅↑𝑛) = (𝑅↑𝑘))
23		eqid 2189	. . . . 5 ⊢ (𝑛 ∈ ℕ ↦ (𝑅↑𝑛)) = (𝑛 ∈ ℕ ↦ (𝑅↑𝑛))
24	22, 23	fvmptg 5608	. . . 4 ⊢ ((𝑘 ∈ ℕ ∧ (𝑅↑𝑘) ∈ ℂ) → ((𝑛 ∈ ℕ ↦ (𝑅↑𝑛))‘𝑘) = (𝑅↑𝑘))
25	18, 21, 24	syl2anc 411	. . 3 ⊢ (((𝐴 ∈ ℂ ∧ 𝑅 ∈ ℂ ∧ (abs‘𝑅) < 1) ∧ 𝑘 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ (𝑅↑𝑛))‘𝑘) = (𝑅↑𝑘))
26		nnnn0 9202	. . . . 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 10673	. . 3 ⊢ (((𝐴 ∈ ℂ ∧ 𝑅 ∈ ℂ ∧ (abs‘𝑅) < 1) ∧ 𝑘 ∈ ℕ) → (𝑅↑𝑘) ∈ ℂ)
29		seqex 10466	. . . 4 ⊢ seq1( + , (𝑛 ∈ ℕ ↦ (𝑅↑𝑛))) ∈ V
30		1nn0 9211	. . . . . . 7 ⊢ 1 ∈ ℕ₀
31	30	a1i 9	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 𝑅 ∈ ℂ ∧ (abs‘𝑅) < 1) → 1 ∈ ℕ₀)
32		elnnuz 9583	. . . . . . 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 11539	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝑅 ∈ ℂ ∧ (abs‘𝑅) < 1) → seq1( + , (𝑛 ∈ ℕ ↦ (𝑅↑𝑛))) ⇝ ((𝑅↑1) / (1 − 𝑅)))
35		climcl 11309	. . . . 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 4848	. . . 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 1353	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝑅 ∈ ℂ ∧ (abs‘𝑅) < 1) → seq1( + , (𝑛 ∈ ℕ ↦ (𝑅↑𝑛))) ∈ dom ⇝ )
39	16, 17, 25, 28, 38, 1	isummulc2 11453	. 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝑅 ∈ ℂ ∧ (abs‘𝑅) < 1) → (𝐴 · Σ𝑘 ∈ ℕ (𝑅↑𝑘)) = Σ𝑘 ∈ ℕ (𝐴 · (𝑅↑𝑘)))
40	12, 15, 39	3eqtr2rd 2229	1 ⊢ ((𝐴 ∈ ℂ ∧ 𝑅 ∈ ℂ ∧ (abs‘𝑅) < 1) → Σ𝑘 ∈ ℕ (𝐴 · (𝑅↑𝑘)) = ((𝐴 · 𝑅) / (1 − 𝑅)))

Colors of variables: wff set class

Syntax hints: → wi 4 ∧ wa 104 ↔ wb 105 ∧ w3a 980 = wceq 1364 ∈ wcel 2160 Vcvv 2752 class class class wbr 4018 ↦ cmpt 4079 dom cdm 4641 ‘cfv 5231 (class class class)co 5891 ℂcc 7828 1c1 7831 + caddc 7833 · cmul 7835 < clt 8011 − cmin 8147 # cap 8557 / cdiv 8648 ℕcn 8938 ℕ₀cn0 9195 ℤ_≥cuz 9547 seqcseq 10464 ↑cexp 10538 abscabs 11025 ⇝ cli 11305 Σcsu 11380

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 4133 ax-sep 4136 ax-nul 4144 ax-pow 4189 ax-pr 4224 ax-un 4448 ax-setind 4551 ax-iinf 4602 ax-cnex 7921 ax-resscn 7922 ax-1cn 7923 ax-1re 7924 ax-icn 7925 ax-addcl 7926 ax-addrcl 7927 ax-mulcl 7928 ax-mulrcl 7929 ax-addcom 7930 ax-mulcom 7931 ax-addass 7932 ax-mulass 7933 ax-distr 7934 ax-i2m1 7935 ax-0lt1 7936 ax-1rid 7937 ax-0id 7938 ax-rnegex 7939 ax-precex 7940 ax-cnre 7941 ax-pre-ltirr 7942 ax-pre-ltwlin 7943 ax-pre-lttrn 7944 ax-pre-apti 7945 ax-pre-ltadd 7946 ax-pre-mulgt0 7947 ax-pre-mulext 7948 ax-arch 7949 ax-caucvg 7950

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 3592 df-sn 3613 df-pr 3614 df-op 3616 df-uni 3825 df-int 3860 df-iun 3903 df-br 4019 df-opab 4080 df-mpt 4081 df-tr 4117 df-id 4308 df-po 4311 df-iso 4312 df-iord 4381 df-on 4383 df-ilim 4384 df-suc 4386 df-iom 4605 df-xp 4647 df-rel 4648 df-cnv 4649 df-co 4650 df-dm 4651 df-rn 4652 df-res 4653 df-ima 4654 df-iota 5193 df-fun 5233 df-fn 5234 df-f 5235 df-f1 5236 df-fo 5237 df-f1o 5238 df-fv 5239 df-isom 5240 df-riota 5847 df-ov 5894 df-oprab 5895 df-mpo 5896 df-1st 6159 df-2nd 6160 df-recs 6324 df-irdg 6389 df-frec 6410 df-1o 6435 df-oadd 6439 df-er 6553 df-en 6759 df-dom 6760 df-fin 6761 df-pnf 8013 df-mnf 8014 df-xr 8015 df-ltxr 8016 df-le 8017 df-sub 8149 df-neg 8150 df-reap 8551 df-ap 8558 df-div 8649 df-inn 8939 df-2 8997 df-3 8998 df-4 8999 df-n0 9196 df-z 9273 df-uz 9548 df-q 9639 df-rp 9673 df-fz 10028 df-fzo 10162 df-seqfrec 10465 df-exp 10539 df-ihash 10775 df-cj 10870 df-re 10871 df-im 10872 df-rsqrt 11026 df-abs 11027 df-clim 11306 df-sumdc 11381

This theorem is referenced by: 0.999... 11548

W3C validator