Theorem geolim 7237

Description: The partial sums in the infinite series 1 + A↑1 + A↑2... converge to (1 / (1 − A)).

Hypothesis

Ref	Expression
geolim.1	⊢ F = {⟨k, y⟩∣(k ∈ ℕ0 ⋀ y = (A↑k))}

Assertion

Ref	Expression
geolim	⊢ ((A ∈ ℂ ⋀ (abs ‘A) < 1) → ( + seq0F) ⇝ (1 / (1 − A)))

Distinct variable group:   y,k,A

Proof of Theorem geolim

Step	Hyp	Ref	Expression
1		opreq1 3974	. . . . . . . 8 ⊢ (A = if((A ∈ ℂ ⋀ (abs ‘A) < 1), A, 0) → (A↑k) = ( if((A ∈ ℂ ⋀ (abs ‘A) < 1), A, 0)↑k))
2	1	eqeq2d 1489	. . . . . . 7 ⊢ (A = if((A ∈ ℂ ⋀ (abs ‘A) < 1), A, 0) → (y = (A↑k) ↔ y = ( if((A ∈ ℂ ⋀ (abs ‘A) < 1), A, 0)↑k)))
3	2	anbi2d 618	. . . . . 6 ⊢ (A = if((A ∈ ℂ ⋀ (abs ‘A) < 1), A, 0) → ((k ∈ ℕ0 ⋀ y = (A↑k)) ↔ (k ∈ ℕ0 ⋀ y = ( if((A ∈ ℂ ⋀ (abs ‘A) < 1), A, 0)↑k))))
4	3	opabbidv 2675	. . . . 5 ⊢ (A = if((A ∈ ℂ ⋀ (abs ‘A) < 1), A, 0) → {⟨k, y⟩∣(k ∈ ℕ0 ⋀ y = (A↑k))} = {⟨k, y⟩∣(k ∈ ℕ0 ⋀ y = ( if((A ∈ ℂ ⋀ (abs ‘A) < 1), A, 0)↑k))})
5	4	opreq2d 3982	. . . 4 ⊢ (A = if((A ∈ ℂ ⋀ (abs ‘A) < 1), A, 0) → ( + seq0{⟨k, y⟩∣(k ∈ ℕ0 ⋀ y = (A↑k))}) = ( + seq0{⟨k, y⟩∣(k ∈ ℕ0 ⋀ y = ( if((A ∈ ℂ ⋀ (abs ‘A) < 1), A, 0)↑k))}))
6		opreq2 3975	. . . . 5 ⊢ (A = if((A ∈ ℂ ⋀ (abs ‘A) < 1), A, 0) → (1 − A) = (1 − if((A ∈ ℂ ⋀ (abs ‘A) < 1), A, 0)))
7	6	opreq2d 3982	. . . 4 ⊢ (A = if((A ∈ ℂ ⋀ (abs ‘A) < 1), A, 0) → (1 / (1 − A)) = (1 / (1 − if((A ∈ ℂ ⋀ (abs ‘A) < 1), A, 0))))
8	5, 7	breq12d 2636	. . 3 ⊢ (A = if((A ∈ ℂ ⋀ (abs ‘A) < 1), A, 0) → (( + seq0{⟨k, y⟩∣(k ∈ ℕ0 ⋀ y = (A↑k))}) ⇝ (1 / (1 − A)) ↔ ( + seq0{⟨k, y⟩∣(k ∈ ℕ0 ⋀ y = ( if((A ∈ ℂ ⋀ (abs ‘A) < 1), A, 0)↑k))}) ⇝ (1 / (1 − if((A ∈ ℂ ⋀ (abs ‘A) < 1), A, 0)))))
9		eqid 1478	. . . 4 ⊢ {⟨k, y⟩∣(k ∈ ℕ0 ⋀ y = ( if((A ∈ ℂ ⋀ (abs ‘A) < 1), A, 0)↑k))} = {⟨k, y⟩∣(k ∈ ℕ0 ⋀ y = ( if((A ∈ ℂ ⋀ (abs ‘A) < 1), A, 0)↑k))}
10		eleq1 1537	. . . . . . 7 ⊢ (A = if((A ∈ ℂ ⋀ (abs ‘A) < 1), A, 0) → (A ∈ ℂ ↔ if((A ∈ ℂ ⋀ (abs ‘A) < 1), A, 0) ∈ ℂ))
11		fveq2 3730	. . . . . . . 8 ⊢ (A = if((A ∈ ℂ ⋀ (abs ‘A) < 1), A, 0) → (abs ‘A) = (abs ‘ if((A ∈ ℂ ⋀ (abs ‘A) < 1), A, 0)))
12	11	breq1d 2634	. . . . . . 7 ⊢ (A = if((A ∈ ℂ ⋀ (abs ‘A) < 1), A, 0) → ((abs ‘A) < 1 ↔ (abs ‘ if((A ∈ ℂ ⋀ (abs ‘A) < 1), A, 0)) < 1))
13	10, 12	anbi12d 630	. . . . . 6 ⊢ (A = if((A ∈ ℂ ⋀ (abs ‘A) < 1), A, 0) → ((A ∈ ℂ ⋀ (abs ‘A) < 1) ↔ ( if((A ∈ ℂ ⋀ (abs ‘A) < 1), A, 0) ∈ ℂ ⋀ (abs ‘ if((A ∈ ℂ ⋀ (abs ‘A) < 1), A, 0)) < 1)))
14		eleq1 1537	. . . . . . 7 ⊢ (0 = if((A ∈ ℂ ⋀ (abs ‘A) < 1), A, 0) → (0 ∈ ℂ ↔ if((A ∈ ℂ ⋀ (abs ‘A) < 1), A, 0) ∈ ℂ))
15		fveq2 3730	. . . . . . . 8 ⊢ (0 = if((A ∈ ℂ ⋀ (abs ‘A) < 1), A, 0) → (abs ‘0) = (abs ‘ if((A ∈ ℂ ⋀ (abs ‘A) < 1), A, 0)))
16	15	breq1d 2634	. . . . . . 7 ⊢ (0 = if((A ∈ ℂ ⋀ (abs ‘A) < 1), A, 0) → ((abs ‘0) < 1 ↔ (abs ‘ if((A ∈ ℂ ⋀ (abs ‘A) < 1), A, 0)) < 1))
17	14, 16	anbi12d 630	. . . . . 6 ⊢ (0 = if((A ∈ ℂ ⋀ (abs ‘A) < 1), A, 0) → ((0 ∈ ℂ ⋀ (abs ‘0) < 1) ↔ ( if((A ∈ ℂ ⋀ (abs ‘A) < 1), A, 0) ∈ ℂ ⋀ (abs ‘ if((A ∈ ℂ ⋀ (abs ‘A) < 1), A, 0)) < 1)))
18		0cn 5340	. . . . . . 7 ⊢ 0 ∈ ℂ
19		abs0 6877	. . . . . . . 8 ⊢ (abs ‘0) = 0
20		lt01 5692	. . . . . . . 8 ⊢ 0 < 1
21	19, 20	eqbrtr 2639	. . . . . . 7 ⊢ (abs ‘0) < 1
22	18, 21	pm3.2i 285	. . . . . 6 ⊢ (0 ∈ ℂ ⋀ (abs ‘0) < 1)
23	13, 17, 22	elimhyp 2394	. . . . 5 ⊢ ( if((A ∈ ℂ ⋀ (abs ‘A) < 1), A, 0) ∈ ℂ ⋀ (abs ‘ if((A ∈ ℂ ⋀ (abs ‘A) < 1), A, 0)) < 1)
24	23	pm3.26i 320	. . . 4 ⊢ if((A ∈ ℂ ⋀ (abs ‘A) < 1), A, 0) ∈ ℂ
25	23	pm3.27i 324	. . . 4 ⊢ (abs ‘ if((A ∈ ℂ ⋀ (abs ‘A) < 1), A, 0)) < 1
26	9, 24, 25	geolimi 7236	. . 3 ⊢ ( + seq0{⟨k, y⟩∣(k ∈ ℕ0 ⋀ y = ( if((A ∈ ℂ ⋀ (abs ‘A) < 1), A, 0)↑k))}) ⇝ (1 / (1 − if((A ∈ ℂ ⋀ (abs ‘A) < 1), A, 0)))
27	8, 26	dedth 2387	. 2 ⊢ ((A ∈ ℂ ⋀ (abs ‘A) < 1) → ( + seq0{⟨k, y⟩∣(k ∈ ℕ0 ⋀ y = (A↑k))}) ⇝ (1 / (1 − A)))
28		geolim.1	. . 3 ⊢ F = {⟨k, y⟩∣(k ∈ ℕ0 ⋀ y = (A↑k))}
29	28	opreq2i 3978	. 2 ⊢ ( + seq0F) = ( + seq0{⟨k, y⟩∣(k ∈ ℕ0 ⋀ y = (A↑k))})
30	27, 29	syl5eqbr 2653	1 ⊢ ((A ∈ ℂ ⋀ (abs ‘A) < 1) → ( + seq0F) ⇝ (1 / (1 − A)))

Syntax hints:   → wi 3   ⋀ wa 223   = wceq 958   ∈ wcel 960   ifcif 2365   class class class wbr 2624  {copab 2671   ‘cfv 3188  (class class class)co 3969  ℂcc 5244  0cc0 5246  1c1 5247   + caddc 5249   − cmin 5304   / cdiv 5306  ℕ₀cn0 5309   < clt 5498  seq₀cseq0 6533  ↑cexp 6569  abscabs 6751   ⇝ cli 6974

This theorem is referenced by:  georeclim 7240  geoisum 7242

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-9 967  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-rep 2698  ax-sep 2708  ax-nul 2715  ax-pow 2748  ax-pr 2785  ax-un 2872  ax-inf2 4634

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 778  df-3an 779  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-nel 1591  df-ral 1652  df-rex 1653  df-reu 1654  df-rab 1655  df-v 1815  df-sbc 1945  df-csb 2005  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-pss 2058  df-nul 2284  df-if 2366  df-pw 2406  df-sn 2416  df-pr 2417  df-tp 2419  df-op 2420  df-uni 2508  df-int 2538  df-iun 2572  df-br 2625  df-opab 2672  df-tr 2686  df-eprel 2838  df-id 2841  df-po 2846  df-so 2856  df-fr 2923  df-we 2940  df-ord 2957  df-on 2958  df-lim 2959  df-suc 2960  df-om 3138  df-xp 3190  df-rel 3191  df-cnv 3192  df-co 3193  df-dm 3194  df-rn 3195  df-res 3196  df-ima 3197  df-fun 3198  df-fn 3199  df-f 3200  df-f1 3201  df-fo 3202  df-f1o 3203  df-fv 3204  df-rdg 3938  df-opr 3971  df-oprab 3972  df-1st 4085  df-2nd 4086  df-1o 4139  df-oadd 4141  df-omul 4142  df-er 4267  df-ec 4269  df-qs 4272  df-en 4374  df-dom 4375  df-sdom 4376  df-sup 4583  df-ni 5012  df-pli 5013  df-mi 5014  df-lti 5015  df-plpq 5047  df-mpq 5048  df-enq 5049  df-nq 5050  df-plq 5051  df-mq 5052  df-rq 5053  df-ltq 5054  df-1q 5055  df-np 5098  df-1p 5099  df-plp 5100  df-mp 5101  df-ltp 5102  df-plpr 5176  df-mpr 5177  df-enr 5178  df-nr 5179  df-plr 5180  df-mr 5181  df-ltr 5182  df-0r 5183  df-1r 5184  df-m1r 5185  df-c 5252  df-0 5253  df-1 5254  df-i 5255  df-r 5256  df-plus 5257  df-mul 5258  df-lt 5259  df-sub 5368  df-neg 5370  df-pnf 5499  df-mnf 5500  df-xr 5501  df-ltxr 5502  df-le 5503  df-div 5715  df-n 5927  df-2 5972  df-n0 6102  df-z 6138  df-fl 6226  df-seq1 6309  df-shft 6342  df-uz 6419  df-seq0 6535  df-exp 6570  df-sqr 6671  df-re 6752  df-im 6753  df-cj 6754  df-abs 6755  df-clim 6975

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