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Theorem aaliou3lem3 26093

Description: Lemma for aaliou3 26100. (Contributed by Stefan O'Rear, 16-Nov-2014.)

**Hypotheses**
Ref	Expression
aaliou3lem.a	⊢ 𝐺 = (𝑐 ∈ (ℤ_≥‘𝐴) ↦ ((2↑-(!‘𝐴)) · ((1 / 2)↑(𝑐 − 𝐴))))
aaliou3lem.b	⊢ 𝐹 = (𝑎 ∈ ℕ ↦ (2↑-(!‘𝑎)))

**Assertion**
Ref	Expression
aaliou3lem3	⊢ (𝐴 ∈ ℕ → (seq𝐴( + , 𝐹) ∈ dom ⇝ ∧ Σ𝑏 ∈ (ℤ_≥‘𝐴)(𝐹‘𝑏) ∈ ℝ⁺ ∧ Σ𝑏 ∈ (ℤ_≥‘𝐴)(𝐹‘𝑏) ≤ (2 · (2↑-(!‘𝐴)))))

Distinct variable groups: 𝐹,𝑏,𝑐 𝐴,𝑎,𝑏,𝑐 𝐺,𝑎,𝑏 Allowed substitution hints: 𝐹(𝑎) 𝐺(𝑐)

**Proof of Theorem aaliou3lem3**
Step	Hyp	Ref	Expression
1		eqid 2730	. . 3 ⊢ (ℤ_≥‘𝐴) = (ℤ_≥‘𝐴)
2		nnz 12583	. . . 4 ⊢ (𝐴 ∈ ℕ → 𝐴 ∈ ℤ)
3		uzid 12841	. . . 4 ⊢ (𝐴 ∈ ℤ → 𝐴 ∈ (ℤ_≥‘𝐴))
4	2, 3	syl 17	. . 3 ⊢ (𝐴 ∈ ℕ → 𝐴 ∈ (ℤ_≥‘𝐴))
5		aaliou3lem.a	. . . 4 ⊢ 𝐺 = (𝑐 ∈ (ℤ_≥‘𝐴) ↦ ((2↑-(!‘𝐴)) · ((1 / 2)↑(𝑐 − 𝐴))))
6	5	aaliou3lem1 26091	. . 3 ⊢ ((𝐴 ∈ ℕ ∧ 𝑏 ∈ (ℤ_≥‘𝐴)) → (𝐺‘𝑏) ∈ ℝ)
7		aaliou3lem.b	. . . . . 6 ⊢ 𝐹 = (𝑎 ∈ ℕ ↦ (2↑-(!‘𝑎)))
8	5, 7	aaliou3lem2 26092	. . . . 5 ⊢ ((𝐴 ∈ ℕ ∧ 𝑏 ∈ (ℤ_≥‘𝐴)) → (𝐹‘𝑏) ∈ (0(,](𝐺‘𝑏)))
9		0xr 11265	. . . . . 6 ⊢ 0 ∈ ℝ^*
10		elioc2 13391	. . . . . 6 ⊢ ((0 ∈ ℝ^* ∧ (𝐺‘𝑏) ∈ ℝ) → ((𝐹‘𝑏) ∈ (0(,](𝐺‘𝑏)) ↔ ((𝐹‘𝑏) ∈ ℝ ∧ 0 < (𝐹‘𝑏) ∧ (𝐹‘𝑏) ≤ (𝐺‘𝑏))))
11	9, 6, 10	sylancr 585	. . . . 5 ⊢ ((𝐴 ∈ ℕ ∧ 𝑏 ∈ (ℤ_≥‘𝐴)) → ((𝐹‘𝑏) ∈ (0(,](𝐺‘𝑏)) ↔ ((𝐹‘𝑏) ∈ ℝ ∧ 0 < (𝐹‘𝑏) ∧ (𝐹‘𝑏) ≤ (𝐺‘𝑏))))
12	8, 11	mpbid 231	. . . 4 ⊢ ((𝐴 ∈ ℕ ∧ 𝑏 ∈ (ℤ_≥‘𝐴)) → ((𝐹‘𝑏) ∈ ℝ ∧ 0 < (𝐹‘𝑏) ∧ (𝐹‘𝑏) ≤ (𝐺‘𝑏)))
13	12	simp1d 1140	. . 3 ⊢ ((𝐴 ∈ ℕ ∧ 𝑏 ∈ (ℤ_≥‘𝐴)) → (𝐹‘𝑏) ∈ ℝ)
14		halfcn 12431	. . . . . 6 ⊢ (1 / 2) ∈ ℂ
15	14	a1i 11	. . . . 5 ⊢ (𝐴 ∈ ℕ → (1 / 2) ∈ ℂ)
16		halfre 12430	. . . . . . . . 9 ⊢ (1 / 2) ∈ ℝ
17		halfgt0 12432	. . . . . . . . 9 ⊢ 0 < (1 / 2)
18	16, 17	elrpii 12981	. . . . . . . 8 ⊢ (1 / 2) ∈ ℝ⁺
19		rprege0 12993	. . . . . . . 8 ⊢ ((1 / 2) ∈ ℝ⁺ → ((1 / 2) ∈ ℝ ∧ 0 ≤ (1 / 2)))
20		absid 15247	. . . . . . . 8 ⊢ (((1 / 2) ∈ ℝ ∧ 0 ≤ (1 / 2)) → (abs‘(1 / 2)) = (1 / 2))
21	18, 19, 20	mp2b 10	. . . . . . 7 ⊢ (abs‘(1 / 2)) = (1 / 2)
22		halflt1 12434	. . . . . . 7 ⊢ (1 / 2) < 1
23	21, 22	eqbrtri 5168	. . . . . 6 ⊢ (abs‘(1 / 2)) < 1
24	23	a1i 11	. . . . 5 ⊢ (𝐴 ∈ ℕ → (abs‘(1 / 2)) < 1)
25		2rp 12983	. . . . . . 7 ⊢ 2 ∈ ℝ⁺
26		nnnn0 12483	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℕ → 𝐴 ∈ ℕ₀)
27	26	faccld 14248	. . . . . . . . 9 ⊢ (𝐴 ∈ ℕ → (!‘𝐴) ∈ ℕ)
28	27	nnzd 12589	. . . . . . . 8 ⊢ (𝐴 ∈ ℕ → (!‘𝐴) ∈ ℤ)
29	28	znegcld 12672	. . . . . . 7 ⊢ (𝐴 ∈ ℕ → -(!‘𝐴) ∈ ℤ)
30		rpexpcl 14050	. . . . . . 7 ⊢ ((2 ∈ ℝ⁺ ∧ -(!‘𝐴) ∈ ℤ) → (2↑-(!‘𝐴)) ∈ ℝ⁺)
31	25, 29, 30	sylancr 585	. . . . . 6 ⊢ (𝐴 ∈ ℕ → (2↑-(!‘𝐴)) ∈ ℝ⁺)
32	31	rpcnd 13022	. . . . 5 ⊢ (𝐴 ∈ ℕ → (2↑-(!‘𝐴)) ∈ ℂ)
33	2, 15, 24, 32, 5	geolim3 26088	. . . 4 ⊢ (𝐴 ∈ ℕ → seq𝐴( + , 𝐺) ⇝ ((2↑-(!‘𝐴)) / (1 − (1 / 2))))
34		seqex 13972	. . . . 5 ⊢ seq𝐴( + , 𝐺) ∈ V
35		ovex 7444	. . . . 5 ⊢ ((2↑-(!‘𝐴)) / (1 − (1 / 2))) ∈ V
36	34, 35	breldm 5907	. . . 4 ⊢ (seq𝐴( + , 𝐺) ⇝ ((2↑-(!‘𝐴)) / (1 − (1 / 2))) → seq𝐴( + , 𝐺) ∈ dom ⇝ )
37	33, 36	syl 17	. . 3 ⊢ (𝐴 ∈ ℕ → seq𝐴( + , 𝐺) ∈ dom ⇝ )
38	12	simp2d 1141	. . . . 5 ⊢ ((𝐴 ∈ ℕ ∧ 𝑏 ∈ (ℤ_≥‘𝐴)) → 0 < (𝐹‘𝑏))
39	13, 38	elrpd 13017	. . . 4 ⊢ ((𝐴 ∈ ℕ ∧ 𝑏 ∈ (ℤ_≥‘𝐴)) → (𝐹‘𝑏) ∈ ℝ⁺)
40	39	rpge0d 13024	. . 3 ⊢ ((𝐴 ∈ ℕ ∧ 𝑏 ∈ (ℤ_≥‘𝐴)) → 0 ≤ (𝐹‘𝑏))
41	12	simp3d 1142	. . 3 ⊢ ((𝐴 ∈ ℕ ∧ 𝑏 ∈ (ℤ_≥‘𝐴)) → (𝐹‘𝑏) ≤ (𝐺‘𝑏))
42	1, 4, 6, 13, 37, 40, 41	cvgcmp 15766	. 2 ⊢ (𝐴 ∈ ℕ → seq𝐴( + , 𝐹) ∈ dom ⇝ )
43		eqidd 2731	. . 3 ⊢ ((𝐴 ∈ ℕ ∧ 𝑏 ∈ (ℤ_≥‘𝐴)) → (𝐹‘𝑏) = (𝐹‘𝑏))
44	1, 1, 4, 43, 39, 42	isumrpcl 15793	. 2 ⊢ (𝐴 ∈ ℕ → Σ𝑏 ∈ (ℤ_≥‘𝐴)(𝐹‘𝑏) ∈ ℝ⁺)
45		eqidd 2731	. . . 4 ⊢ ((𝐴 ∈ ℕ ∧ 𝑏 ∈ (ℤ_≥‘𝐴)) → (𝐺‘𝑏) = (𝐺‘𝑏))
46	1, 2, 43, 13, 45, 6, 41, 42, 37	isumle 15794	. . 3 ⊢ (𝐴 ∈ ℕ → Σ𝑏 ∈ (ℤ_≥‘𝐴)(𝐹‘𝑏) ≤ Σ𝑏 ∈ (ℤ_≥‘𝐴)(𝐺‘𝑏))
47	6	recnd 11246	. . . . 5 ⊢ ((𝐴 ∈ ℕ ∧ 𝑏 ∈ (ℤ_≥‘𝐴)) → (𝐺‘𝑏) ∈ ℂ)
48	1, 2, 45, 47, 33	isumclim 15707	. . . 4 ⊢ (𝐴 ∈ ℕ → Σ𝑏 ∈ (ℤ_≥‘𝐴)(𝐺‘𝑏) = ((2↑-(!‘𝐴)) / (1 − (1 / 2))))
49		1mhlfehlf 12435	. . . . . 6 ⊢ (1 − (1 / 2)) = (1 / 2)
50	49	oveq2i 7422	. . . . 5 ⊢ ((2↑-(!‘𝐴)) / (1 − (1 / 2))) = ((2↑-(!‘𝐴)) / (1 / 2))
51		2cn 12291	. . . . . . . 8 ⊢ 2 ∈ ℂ
52		mulcl 11196	. . . . . . . 8 ⊢ (((2↑-(!‘𝐴)) ∈ ℂ ∧ 2 ∈ ℂ) → ((2↑-(!‘𝐴)) · 2) ∈ ℂ)
53	32, 51, 52	sylancl 584	. . . . . . 7 ⊢ (𝐴 ∈ ℕ → ((2↑-(!‘𝐴)) · 2) ∈ ℂ)
54	53	div1d 11986	. . . . . 6 ⊢ (𝐴 ∈ ℕ → (((2↑-(!‘𝐴)) · 2) / 1) = ((2↑-(!‘𝐴)) · 2))
55		1rp 12982	. . . . . . . . 9 ⊢ 1 ∈ ℝ⁺
56		rpcnne0 12996	. . . . . . . . 9 ⊢ (1 ∈ ℝ⁺ → (1 ∈ ℂ ∧ 1 ≠ 0))
57	55, 56	ax-mp 5	. . . . . . . 8 ⊢ (1 ∈ ℂ ∧ 1 ≠ 0)
58		2cnne0 12426	. . . . . . . 8 ⊢ (2 ∈ ℂ ∧ 2 ≠ 0)
59		divdiv2 11930	. . . . . . . 8 ⊢ (((2↑-(!‘𝐴)) ∈ ℂ ∧ (1 ∈ ℂ ∧ 1 ≠ 0) ∧ (2 ∈ ℂ ∧ 2 ≠ 0)) → ((2↑-(!‘𝐴)) / (1 / 2)) = (((2↑-(!‘𝐴)) · 2) / 1))
60	57, 58, 59	mp3an23 1451	. . . . . . 7 ⊢ ((2↑-(!‘𝐴)) ∈ ℂ → ((2↑-(!‘𝐴)) / (1 / 2)) = (((2↑-(!‘𝐴)) · 2) / 1))
61	32, 60	syl 17	. . . . . 6 ⊢ (𝐴 ∈ ℕ → ((2↑-(!‘𝐴)) / (1 / 2)) = (((2↑-(!‘𝐴)) · 2) / 1))
62		mulcom 11198	. . . . . . 7 ⊢ ((2 ∈ ℂ ∧ (2↑-(!‘𝐴)) ∈ ℂ) → (2 · (2↑-(!‘𝐴))) = ((2↑-(!‘𝐴)) · 2))
63	51, 32, 62	sylancr 585	. . . . . 6 ⊢ (𝐴 ∈ ℕ → (2 · (2↑-(!‘𝐴))) = ((2↑-(!‘𝐴)) · 2))
64	54, 61, 63	3eqtr4d 2780	. . . . 5 ⊢ (𝐴 ∈ ℕ → ((2↑-(!‘𝐴)) / (1 / 2)) = (2 · (2↑-(!‘𝐴))))
65	50, 64	eqtrid 2782	. . . 4 ⊢ (𝐴 ∈ ℕ → ((2↑-(!‘𝐴)) / (1 − (1 / 2))) = (2 · (2↑-(!‘𝐴))))
66	48, 65	eqtrd 2770	. . 3 ⊢ (𝐴 ∈ ℕ → Σ𝑏 ∈ (ℤ_≥‘𝐴)(𝐺‘𝑏) = (2 · (2↑-(!‘𝐴))))
67	46, 66	breqtrd 5173	. 2 ⊢ (𝐴 ∈ ℕ → Σ𝑏 ∈ (ℤ_≥‘𝐴)(𝐹‘𝑏) ≤ (2 · (2↑-(!‘𝐴))))
68	42, 44, 67	3jca 1126	1 ⊢ (𝐴 ∈ ℕ → (seq𝐴( + , 𝐹) ∈ dom ⇝ ∧ Σ𝑏 ∈ (ℤ_≥‘𝐴)(𝐹‘𝑏) ∈ ℝ⁺ ∧ Σ𝑏 ∈ (ℤ_≥‘𝐴)(𝐹‘𝑏) ≤ (2 · (2↑-(!‘𝐴)))))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 394 ∧ w3a 1085 = wceq 1539 ∈ wcel 2104 ≠ wne 2938 class class class wbr 5147 ↦ cmpt 5230 dom cdm 5675 ‘cfv 6542 (class class class)co 7411 ℂcc 11110 ℝcr 11111 0cc0 11112 1c1 11113 + caddc 11115 · cmul 11117 ℝ^*cxr 11251 < clt 11252 ≤ cle 11253 − cmin 11448 -cneg 11449 / cdiv 11875 ℕcn 12216 2c2 12271 ℤcz 12562 ℤ_≥cuz 12826 ℝ⁺crp 12978 (,]cioc 13329 seqcseq 13970 ↑cexp 14031 !cfa 14237 abscabs 15185 ⇝ cli 15432 Σcsu 15636

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2701 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7727 ax-inf2 9638 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189 ax-pre-sup 11190

This theorem depends on definitions: df-bi 206 df-an 395 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2532 df-eu 2561 df-clab 2708 df-cleq 2722 df-clel 2808 df-nfc 2883 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3474 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-int 4950 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-se 5631 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-isom 6551 df-riota 7367 df-ov 7414 df-oprab 7415 df-mpo 7416 df-om 7858 df-1st 7977 df-2nd 7978 df-frecs 8268 df-wrecs 8299 df-recs 8373 df-rdg 8412 df-1o 8468 df-er 8705 df-pm 8825 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-sup 9439 df-inf 9440 df-oi 9507 df-card 9936 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-div 11876 df-nn 12217 df-2 12279 df-3 12280 df-n0 12477 df-z 12563 df-uz 12827 df-rp 12979 df-ioc 13333 df-ico 13334 df-fz 13489 df-fzo 13632 df-fl 13761 df-seq 13971 df-exp 14032 df-fac 14238 df-hash 14295 df-shft 15018 df-cj 15050 df-re 15051 df-im 15052 df-sqrt 15186 df-abs 15187 df-limsup 15419 df-clim 15436 df-rlim 15437 df-sum 15637

This theorem is referenced by: aaliou3lem4 26095 aaliou3lem7 26098

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