geoserap - Intuitionistic Logic Explorer

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Theorem geoserap 11674

Description: The value of the finite geometric series 1 + 𝐴↑1 + 𝐴↑2 +... + 𝐴↑(𝑁 − 1). This is Metamath 100 proof #66. (Contributed by NM, 12-May-2006.) (Revised by Jim Kingdon, 24-Oct-2022.)

**Hypotheses**
Ref	Expression
geoser.1	⊢ (𝜑 → 𝐴 ∈ ℂ)
geoser.2	⊢ (𝜑 → 𝐴 # 1)
geoser.3	⊢ (𝜑 → 𝑁 ∈ ℕ₀)

**Assertion**
Ref	Expression
geoserap	⊢ (𝜑 → Σ𝑘 ∈ (0...(𝑁 − 1))(𝐴↑𝑘) = ((1 − (𝐴↑𝑁)) / (1 − 𝐴)))

Distinct variable groups: 𝐴,𝑘 𝑘,𝑁 𝜑,𝑘

**Proof of Theorem geoserap**
Step	Hyp	Ref	Expression
1		geoser.1	. . 3 ⊢ (𝜑 → 𝐴 ∈ ℂ)
2		geoser.2	. . 3 ⊢ (𝜑 → 𝐴 # 1)
3		0nn0 9266	. . . 4 ⊢ 0 ∈ ℕ₀
4	3	a1i 9	. . 3 ⊢ (𝜑 → 0 ∈ ℕ₀)
5		geoser.3	. . . 4 ⊢ (𝜑 → 𝑁 ∈ ℕ₀)
6		nn0uz 9638	. . . 4 ⊢ ℕ₀ = (ℤ_≥‘0)
7	5, 6	eleqtrdi 2289	. . 3 ⊢ (𝜑 → 𝑁 ∈ (ℤ_≥‘0))
8	1, 2, 4, 7	geosergap 11673	. 2 ⊢ (𝜑 → Σ𝑘 ∈ (0..^𝑁)(𝐴↑𝑘) = (((𝐴↑0) − (𝐴↑𝑁)) / (1 − 𝐴)))
9	5	nn0zd 9448	. . . 4 ⊢ (𝜑 → 𝑁 ∈ ℤ)
10		fzoval 10225	. . . 4 ⊢ (𝑁 ∈ ℤ → (0..^𝑁) = (0...(𝑁 − 1)))
11	9, 10	syl 14	. . 3 ⊢ (𝜑 → (0..^𝑁) = (0...(𝑁 − 1)))
12	11	sumeq1d 11533	. 2 ⊢ (𝜑 → Σ𝑘 ∈ (0..^𝑁)(𝐴↑𝑘) = Σ𝑘 ∈ (0...(𝑁 − 1))(𝐴↑𝑘))
13	1	exp0d 10761	. . . 4 ⊢ (𝜑 → (𝐴↑0) = 1)
14	13	oveq1d 5938	. . 3 ⊢ (𝜑 → ((𝐴↑0) − (𝐴↑𝑁)) = (1 − (𝐴↑𝑁)))
15	14	oveq1d 5938	. 2 ⊢ (𝜑 → (((𝐴↑0) − (𝐴↑𝑁)) / (1 − 𝐴)) = ((1 − (𝐴↑𝑁)) / (1 − 𝐴)))
16	8, 12, 15	3eqtr3d 2237	1 ⊢ (𝜑 → Σ𝑘 ∈ (0...(𝑁 − 1))(𝐴↑𝑘) = ((1 − (𝐴↑𝑁)) / (1 − 𝐴)))

Colors of variables: wff set class

Syntax hints: → wi 4 = wceq 1364 ∈ wcel 2167 class class class wbr 4034 ‘cfv 5259 (class class class)co 5923 ℂcc 7879 0cc0 7881 1c1 7882 − cmin 8199 # cap 8610 / cdiv 8701 ℕ₀cn0 9251 ℤcz 9328 ℤ_≥cuz 9603 ...cfz 10085 ..^cfzo 10219 ↑cexp 10632 Σcsu 11520

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 1461 ax-7 1462 ax-gen 1463 ax-ie1 1507 ax-ie2 1508 ax-8 1518 ax-10 1519 ax-11 1520 ax-i12 1521 ax-bndl 1523 ax-4 1524 ax-17 1540 ax-i9 1544 ax-ial 1548 ax-i5r 1549 ax-13 2169 ax-14 2170 ax-ext 2178 ax-coll 4149 ax-sep 4152 ax-nul 4160 ax-pow 4208 ax-pr 4243 ax-un 4469 ax-setind 4574 ax-iinf 4625 ax-cnex 7972 ax-resscn 7973 ax-1cn 7974 ax-1re 7975 ax-icn 7976 ax-addcl 7977 ax-addrcl 7978 ax-mulcl 7979 ax-mulrcl 7980 ax-addcom 7981 ax-mulcom 7982 ax-addass 7983 ax-mulass 7984 ax-distr 7985 ax-i2m1 7986 ax-0lt1 7987 ax-1rid 7988 ax-0id 7989 ax-rnegex 7990 ax-precex 7991 ax-cnre 7992 ax-pre-ltirr 7993 ax-pre-ltwlin 7994 ax-pre-lttrn 7995 ax-pre-apti 7996 ax-pre-ltadd 7997 ax-pre-mulgt0 7998 ax-pre-mulext 7999 ax-arch 8000 ax-caucvg 8001

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 1475 df-sb 1777 df-eu 2048 df-mo 2049 df-clab 2183 df-cleq 2189 df-clel 2192 df-nfc 2328 df-ne 2368 df-nel 2463 df-ral 2480 df-rex 2481 df-reu 2482 df-rmo 2483 df-rab 2484 df-v 2765 df-sbc 2990 df-csb 3085 df-dif 3159 df-un 3161 df-in 3163 df-ss 3170 df-nul 3452 df-if 3563 df-pw 3608 df-sn 3629 df-pr 3630 df-op 3632 df-uni 3841 df-int 3876 df-iun 3919 df-br 4035 df-opab 4096 df-mpt 4097 df-tr 4133 df-id 4329 df-po 4332 df-iso 4333 df-iord 4402 df-on 4404 df-ilim 4405 df-suc 4407 df-iom 4628 df-xp 4670 df-rel 4671 df-cnv 4672 df-co 4673 df-dm 4674 df-rn 4675 df-res 4676 df-ima 4677 df-iota 5220 df-fun 5261 df-fn 5262 df-f 5263 df-f1 5264 df-fo 5265 df-f1o 5266 df-fv 5267 df-isom 5268 df-riota 5878 df-ov 5926 df-oprab 5927 df-mpo 5928 df-1st 6199 df-2nd 6200 df-recs 6364 df-irdg 6429 df-frec 6450 df-1o 6475 df-oadd 6479 df-er 6593 df-en 6801 df-dom 6802 df-fin 6803 df-pnf 8065 df-mnf 8066 df-xr 8067 df-ltxr 8068 df-le 8069 df-sub 8201 df-neg 8202 df-reap 8604 df-ap 8611 df-div 8702 df-inn 8993 df-2 9051 df-3 9052 df-4 9053 df-n0 9252 df-z 9329 df-uz 9604 df-q 9696 df-rp 9731 df-fz 10086 df-fzo 10220 df-seqfrec 10542 df-exp 10633 df-ihash 10870 df-cj 11009 df-re 11010 df-im 11011 df-rsqrt 11165 df-abs 11166 df-clim 11446 df-sumdc 11521

This theorem is referenced by: pwm1geoserap1 11675 geolim 11678 geolim2 11679 geo2sum 11681 geo2sum2 11682 3dvds 12031 1sgm2ppw 15241 mersenne 15243

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