aaliou3lem3 - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > aaliou3lem3		Structured version Visualization version GIF version

Theorem aaliou3lem3 25857

Description: Lemma for aaliou3 25864. (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 2733	. . 3 ⊢ (ℤ_≥‘𝐴) = (ℤ_≥‘𝐴)
2		nnz 12579	. . . 4 ⊢ (𝐴 ∈ ℕ → 𝐴 ∈ ℤ)
3		uzid 12837	. . . 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 25855	. . 3 ⊢ ((𝐴 ∈ ℕ ∧ 𝑏 ∈ (ℤ_≥‘𝐴)) → (𝐺‘𝑏) ∈ ℝ)
7		aaliou3lem.b	. . . . . 6 ⊢ 𝐹 = (𝑎 ∈ ℕ ↦ (2↑-(!‘𝑎)))
8	5, 7	aaliou3lem2 25856	. . . . 5 ⊢ ((𝐴 ∈ ℕ ∧ 𝑏 ∈ (ℤ_≥‘𝐴)) → (𝐹‘𝑏) ∈ (0(,](𝐺‘𝑏)))
9		0xr 11261	. . . . . 6 ⊢ 0 ∈ ℝ^*
10		elioc2 13387	. . . . . 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 588	. . . . 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 1143	. . 3 ⊢ ((𝐴 ∈ ℕ ∧ 𝑏 ∈ (ℤ_≥‘𝐴)) → (𝐹‘𝑏) ∈ ℝ)
14		halfcn 12427	. . . . . 6 ⊢ (1 / 2) ∈ ℂ
15	14	a1i 11	. . . . 5 ⊢ (𝐴 ∈ ℕ → (1 / 2) ∈ ℂ)
16		halfre 12426	. . . . . . . . 9 ⊢ (1 / 2) ∈ ℝ
17		halfgt0 12428	. . . . . . . . 9 ⊢ 0 < (1 / 2)
18	16, 17	elrpii 12977	. . . . . . . 8 ⊢ (1 / 2) ∈ ℝ⁺
19		rprege0 12989	. . . . . . . 8 ⊢ ((1 / 2) ∈ ℝ⁺ → ((1 / 2) ∈ ℝ ∧ 0 ≤ (1 / 2)))
20		absid 15243	. . . . . . . 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 12430	. . . . . . 7 ⊢ (1 / 2) < 1
23	21, 22	eqbrtri 5170	. . . . . 6 ⊢ (abs‘(1 / 2)) < 1
24	23	a1i 11	. . . . 5 ⊢ (𝐴 ∈ ℕ → (abs‘(1 / 2)) < 1)
25		2rp 12979	. . . . . . 7 ⊢ 2 ∈ ℝ⁺
26		nnnn0 12479	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℕ → 𝐴 ∈ ℕ₀)
27	26	faccld 14244	. . . . . . . . 9 ⊢ (𝐴 ∈ ℕ → (!‘𝐴) ∈ ℕ)
28	27	nnzd 12585	. . . . . . . 8 ⊢ (𝐴 ∈ ℕ → (!‘𝐴) ∈ ℤ)
29	28	znegcld 12668	. . . . . . 7 ⊢ (𝐴 ∈ ℕ → -(!‘𝐴) ∈ ℤ)
30		rpexpcl 14046	. . . . . . 7 ⊢ ((2 ∈ ℝ⁺ ∧ -(!‘𝐴) ∈ ℤ) → (2↑-(!‘𝐴)) ∈ ℝ⁺)
31	25, 29, 30	sylancr 588	. . . . . 6 ⊢ (𝐴 ∈ ℕ → (2↑-(!‘𝐴)) ∈ ℝ⁺)
32	31	rpcnd 13018	. . . . 5 ⊢ (𝐴 ∈ ℕ → (2↑-(!‘𝐴)) ∈ ℂ)
33	2, 15, 24, 32, 5	geolim3 25852	. . . 4 ⊢ (𝐴 ∈ ℕ → seq𝐴( + , 𝐺) ⇝ ((2↑-(!‘𝐴)) / (1 − (1 / 2))))
34		seqex 13968	. . . . 5 ⊢ seq𝐴( + , 𝐺) ∈ V
35		ovex 7442	. . . . 5 ⊢ ((2↑-(!‘𝐴)) / (1 − (1 / 2))) ∈ V
36	34, 35	breldm 5909	. . . 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 1144	. . . . 5 ⊢ ((𝐴 ∈ ℕ ∧ 𝑏 ∈ (ℤ_≥‘𝐴)) → 0 < (𝐹‘𝑏))
39	13, 38	elrpd 13013	. . . 4 ⊢ ((𝐴 ∈ ℕ ∧ 𝑏 ∈ (ℤ_≥‘𝐴)) → (𝐹‘𝑏) ∈ ℝ⁺)
40	39	rpge0d 13020	. . 3 ⊢ ((𝐴 ∈ ℕ ∧ 𝑏 ∈ (ℤ_≥‘𝐴)) → 0 ≤ (𝐹‘𝑏))
41	12	simp3d 1145	. . 3 ⊢ ((𝐴 ∈ ℕ ∧ 𝑏 ∈ (ℤ_≥‘𝐴)) → (𝐹‘𝑏) ≤ (𝐺‘𝑏))
42	1, 4, 6, 13, 37, 40, 41	cvgcmp 15762	. 2 ⊢ (𝐴 ∈ ℕ → seq𝐴( + , 𝐹) ∈ dom ⇝ )
43		eqidd 2734	. . 3 ⊢ ((𝐴 ∈ ℕ ∧ 𝑏 ∈ (ℤ_≥‘𝐴)) → (𝐹‘𝑏) = (𝐹‘𝑏))
44	1, 1, 4, 43, 39, 42	isumrpcl 15789	. 2 ⊢ (𝐴 ∈ ℕ → Σ𝑏 ∈ (ℤ_≥‘𝐴)(𝐹‘𝑏) ∈ ℝ⁺)
45		eqidd 2734	. . . 4 ⊢ ((𝐴 ∈ ℕ ∧ 𝑏 ∈ (ℤ_≥‘𝐴)) → (𝐺‘𝑏) = (𝐺‘𝑏))
46	1, 2, 43, 13, 45, 6, 41, 42, 37	isumle 15790	. . 3 ⊢ (𝐴 ∈ ℕ → Σ𝑏 ∈ (ℤ_≥‘𝐴)(𝐹‘𝑏) ≤ Σ𝑏 ∈ (ℤ_≥‘𝐴)(𝐺‘𝑏))
47	6	recnd 11242	. . . . 5 ⊢ ((𝐴 ∈ ℕ ∧ 𝑏 ∈ (ℤ_≥‘𝐴)) → (𝐺‘𝑏) ∈ ℂ)
48	1, 2, 45, 47, 33	isumclim 15703	. . . 4 ⊢ (𝐴 ∈ ℕ → Σ𝑏 ∈ (ℤ_≥‘𝐴)(𝐺‘𝑏) = ((2↑-(!‘𝐴)) / (1 − (1 / 2))))
49		1mhlfehlf 12431	. . . . . 6 ⊢ (1 − (1 / 2)) = (1 / 2)
50	49	oveq2i 7420	. . . . 5 ⊢ ((2↑-(!‘𝐴)) / (1 − (1 / 2))) = ((2↑-(!‘𝐴)) / (1 / 2))
51		2cn 12287	. . . . . . . 8 ⊢ 2 ∈ ℂ
52		mulcl 11194	. . . . . . . 8 ⊢ (((2↑-(!‘𝐴)) ∈ ℂ ∧ 2 ∈ ℂ) → ((2↑-(!‘𝐴)) · 2) ∈ ℂ)
53	32, 51, 52	sylancl 587	. . . . . . 7 ⊢ (𝐴 ∈ ℕ → ((2↑-(!‘𝐴)) · 2) ∈ ℂ)
54	53	div1d 11982	. . . . . 6 ⊢ (𝐴 ∈ ℕ → (((2↑-(!‘𝐴)) · 2) / 1) = ((2↑-(!‘𝐴)) · 2))
55		1rp 12978	. . . . . . . . 9 ⊢ 1 ∈ ℝ⁺
56		rpcnne0 12992	. . . . . . . . 9 ⊢ (1 ∈ ℝ⁺ → (1 ∈ ℂ ∧ 1 ≠ 0))
57	55, 56	ax-mp 5	. . . . . . . 8 ⊢ (1 ∈ ℂ ∧ 1 ≠ 0)
58		2cnne0 12422	. . . . . . . 8 ⊢ (2 ∈ ℂ ∧ 2 ≠ 0)
59		divdiv2 11926	. . . . . . . 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 1454	. . . . . . 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 11196	. . . . . . 7 ⊢ ((2 ∈ ℂ ∧ (2↑-(!‘𝐴)) ∈ ℂ) → (2 · (2↑-(!‘𝐴))) = ((2↑-(!‘𝐴)) · 2))
63	51, 32, 62	sylancr 588	. . . . . 6 ⊢ (𝐴 ∈ ℕ → (2 · (2↑-(!‘𝐴))) = ((2↑-(!‘𝐴)) · 2))
64	54, 61, 63	3eqtr4d 2783	. . . . 5 ⊢ (𝐴 ∈ ℕ → ((2↑-(!‘𝐴)) / (1 / 2)) = (2 · (2↑-(!‘𝐴))))
65	50, 64	eqtrid 2785	. . . 4 ⊢ (𝐴 ∈ ℕ → ((2↑-(!‘𝐴)) / (1 − (1 / 2))) = (2 · (2↑-(!‘𝐴))))
66	48, 65	eqtrd 2773	. . 3 ⊢ (𝐴 ∈ ℕ → Σ𝑏 ∈ (ℤ_≥‘𝐴)(𝐺‘𝑏) = (2 · (2↑-(!‘𝐴))))
67	46, 66	breqtrd 5175	. 2 ⊢ (𝐴 ∈ ℕ → Σ𝑏 ∈ (ℤ_≥‘𝐴)(𝐹‘𝑏) ≤ (2 · (2↑-(!‘𝐴))))
68	42, 44, 67	3jca 1129	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 397 ∧ w3a 1088 = wceq 1542 ∈ wcel 2107 ≠ wne 2941 class class class wbr 5149 ↦ cmpt 5232 dom cdm 5677 ‘cfv 6544 (class class class)co 7409 ℂcc 11108 ℝcr 11109 0cc0 11110 1c1 11111 + caddc 11113 · cmul 11115 ℝ^*cxr 11247 < clt 11248 ≤ cle 11249 − cmin 11444 -cneg 11445 / cdiv 11871 ℕcn 12212 2c2 12267 ℤcz 12558 ℤ_≥cuz 12822 ℝ⁺crp 12974 (,]cioc 13325 seqcseq 13966 ↑cexp 14027 !cfa 14233 abscabs 15181 ⇝ cli 15428 Σcsu 15632

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725 ax-inf2 9636 ax-cnex 11166 ax-resscn 11167 ax-1cn 11168 ax-icn 11169 ax-addcl 11170 ax-addrcl 11171 ax-mulcl 11172 ax-mulrcl 11173 ax-mulcom 11174 ax-addass 11175 ax-mulass 11176 ax-distr 11177 ax-i2m1 11178 ax-1ne0 11179 ax-1rid 11180 ax-rnegex 11181 ax-rrecex 11182 ax-cnre 11183 ax-pre-lttri 11184 ax-pre-lttrn 11185 ax-pre-ltadd 11186 ax-pre-mulgt0 11187 ax-pre-sup 11188

This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-int 4952 df-iun 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5575 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-se 5633 df-we 5634 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-isom 6553 df-riota 7365 df-ov 7412 df-oprab 7413 df-mpo 7414 df-om 7856 df-1st 7975 df-2nd 7976 df-frecs 8266 df-wrecs 8297 df-recs 8371 df-rdg 8410 df-1o 8466 df-er 8703 df-pm 8823 df-en 8940 df-dom 8941 df-sdom 8942 df-fin 8943 df-sup 9437 df-inf 9438 df-oi 9505 df-card 9934 df-pnf 11250 df-mnf 11251 df-xr 11252 df-ltxr 11253 df-le 11254 df-sub 11446 df-neg 11447 df-div 11872 df-nn 12213 df-2 12275 df-3 12276 df-n0 12473 df-z 12559 df-uz 12823 df-rp 12975 df-ioc 13329 df-ico 13330 df-fz 13485 df-fzo 13628 df-fl 13757 df-seq 13967 df-exp 14028 df-fac 14234 df-hash 14291 df-shft 15014 df-cj 15046 df-re 15047 df-im 15048 df-sqrt 15182 df-abs 15183 df-limsup 15415 df-clim 15432 df-rlim 15433 df-sum 15633

This theorem is referenced by: aaliou3lem4 25859 aaliou3lem7 25862

W3C validator