aaliou3lem6 - Metamath Proof Explorer

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Theorem aaliou3lem6 26408

Description: Lemma for aaliou3 26411. (Contributed by Stefan O'Rear, 16-Nov-2014.)

**Hypotheses**
Ref	Expression
aaliou3lem.c	⊢ 𝐹 = (𝑎 ∈ ℕ ↦ (2↑-(!‘𝑎)))
aaliou3lem.d	⊢ 𝐿 = Σ𝑏 ∈ ℕ (𝐹‘𝑏)
aaliou3lem.e	⊢ 𝐻 = (𝑐 ∈ ℕ ↦ Σ𝑏 ∈ (1...𝑐)(𝐹‘𝑏))

**Assertion**
Ref	Expression
aaliou3lem6	⊢ (𝐴 ∈ ℕ → ((𝐻‘𝐴) · (2↑(!‘𝐴))) ∈ ℤ)

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

**Proof of Theorem aaliou3lem6**
Step	Hyp	Ref	Expression
1		oveq2 7456	. . . . 5 ⊢ (𝑐 = 𝐴 → (1...𝑐) = (1...𝐴))
2	1	sumeq1d 15748	. . . 4 ⊢ (𝑐 = 𝐴 → Σ𝑏 ∈ (1...𝑐)(𝐹‘𝑏) = Σ𝑏 ∈ (1...𝐴)(𝐹‘𝑏))
3		aaliou3lem.e	. . . 4 ⊢ 𝐻 = (𝑐 ∈ ℕ ↦ Σ𝑏 ∈ (1...𝑐)(𝐹‘𝑏))
4		sumex 15736	. . . 4 ⊢ Σ𝑏 ∈ (1...𝐴)(𝐹‘𝑏) ∈ V
5	2, 3, 4	fvmpt 7029	. . 3 ⊢ (𝐴 ∈ ℕ → (𝐻‘𝐴) = Σ𝑏 ∈ (1...𝐴)(𝐹‘𝑏))
6	5	oveq1d 7463	. 2 ⊢ (𝐴 ∈ ℕ → ((𝐻‘𝐴) · (2↑(!‘𝐴))) = (Σ𝑏 ∈ (1...𝐴)(𝐹‘𝑏) · (2↑(!‘𝐴))))
7		fzfid 14024	. . . 4 ⊢ (𝐴 ∈ ℕ → (1...𝐴) ∈ Fin)
8		2rp 13062	. . . . . 6 ⊢ 2 ∈ ℝ⁺
9		nnnn0 12560	. . . . . . . 8 ⊢ (𝐴 ∈ ℕ → 𝐴 ∈ ℕ₀)
10	9	faccld 14333	. . . . . . 7 ⊢ (𝐴 ∈ ℕ → (!‘𝐴) ∈ ℕ)
11	10	nnzd 12666	. . . . . 6 ⊢ (𝐴 ∈ ℕ → (!‘𝐴) ∈ ℤ)
12		rpexpcl 14131	. . . . . 6 ⊢ ((2 ∈ ℝ⁺ ∧ (!‘𝐴) ∈ ℤ) → (2↑(!‘𝐴)) ∈ ℝ⁺)
13	8, 11, 12	sylancr 586	. . . . 5 ⊢ (𝐴 ∈ ℕ → (2↑(!‘𝐴)) ∈ ℝ⁺)
14	13	rpcnd 13101	. . . 4 ⊢ (𝐴 ∈ ℕ → (2↑(!‘𝐴)) ∈ ℂ)
15		elfznn 13613	. . . . . . 7 ⊢ (𝑏 ∈ (1...𝐴) → 𝑏 ∈ ℕ)
16		fveq2 6920	. . . . . . . . . 10 ⊢ (𝑎 = 𝑏 → (!‘𝑎) = (!‘𝑏))
17	16	negeqd 11530	. . . . . . . . 9 ⊢ (𝑎 = 𝑏 → -(!‘𝑎) = -(!‘𝑏))
18	17	oveq2d 7464	. . . . . . . 8 ⊢ (𝑎 = 𝑏 → (2↑-(!‘𝑎)) = (2↑-(!‘𝑏)))
19		aaliou3lem.c	. . . . . . . 8 ⊢ 𝐹 = (𝑎 ∈ ℕ ↦ (2↑-(!‘𝑎)))
20		ovex 7481	. . . . . . . 8 ⊢ (2↑-(!‘𝑏)) ∈ V
21	18, 19, 20	fvmpt 7029	. . . . . . 7 ⊢ (𝑏 ∈ ℕ → (𝐹‘𝑏) = (2↑-(!‘𝑏)))
22	15, 21	syl 17	. . . . . 6 ⊢ (𝑏 ∈ (1...𝐴) → (𝐹‘𝑏) = (2↑-(!‘𝑏)))
23	22	adantl 481	. . . . 5 ⊢ ((𝐴 ∈ ℕ ∧ 𝑏 ∈ (1...𝐴)) → (𝐹‘𝑏) = (2↑-(!‘𝑏)))
24	15	adantl 481	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℕ ∧ 𝑏 ∈ (1...𝐴)) → 𝑏 ∈ ℕ)
25	24	nnnn0d 12613	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℕ ∧ 𝑏 ∈ (1...𝐴)) → 𝑏 ∈ ℕ₀)
26	25	faccld 14333	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℕ ∧ 𝑏 ∈ (1...𝐴)) → (!‘𝑏) ∈ ℕ)
27	26	nnzd 12666	. . . . . . . 8 ⊢ ((𝐴 ∈ ℕ ∧ 𝑏 ∈ (1...𝐴)) → (!‘𝑏) ∈ ℤ)
28	27	znegcld 12749	. . . . . . 7 ⊢ ((𝐴 ∈ ℕ ∧ 𝑏 ∈ (1...𝐴)) → -(!‘𝑏) ∈ ℤ)
29		rpexpcl 14131	. . . . . . 7 ⊢ ((2 ∈ ℝ⁺ ∧ -(!‘𝑏) ∈ ℤ) → (2↑-(!‘𝑏)) ∈ ℝ⁺)
30	8, 28, 29	sylancr 586	. . . . . 6 ⊢ ((𝐴 ∈ ℕ ∧ 𝑏 ∈ (1...𝐴)) → (2↑-(!‘𝑏)) ∈ ℝ⁺)
31	30	rpcnd 13101	. . . . 5 ⊢ ((𝐴 ∈ ℕ ∧ 𝑏 ∈ (1...𝐴)) → (2↑-(!‘𝑏)) ∈ ℂ)
32	23, 31	eqeltrd 2844	. . . 4 ⊢ ((𝐴 ∈ ℕ ∧ 𝑏 ∈ (1...𝐴)) → (𝐹‘𝑏) ∈ ℂ)
33	7, 14, 32	fsummulc1 15833	. . 3 ⊢ (𝐴 ∈ ℕ → (Σ𝑏 ∈ (1...𝐴)(𝐹‘𝑏) · (2↑(!‘𝐴))) = Σ𝑏 ∈ (1...𝐴)((𝐹‘𝑏) · (2↑(!‘𝐴))))
34	23	oveq1d 7463	. . . . 5 ⊢ ((𝐴 ∈ ℕ ∧ 𝑏 ∈ (1...𝐴)) → ((𝐹‘𝑏) · (2↑(!‘𝐴))) = ((2↑-(!‘𝑏)) · (2↑(!‘𝐴))))
35	11	adantr 480	. . . . . . 7 ⊢ ((𝐴 ∈ ℕ ∧ 𝑏 ∈ (1...𝐴)) → (!‘𝐴) ∈ ℤ)
36		2cnne0 12503	. . . . . . . 8 ⊢ (2 ∈ ℂ ∧ 2 ≠ 0)
37		expaddz 14157	. . . . . . . 8 ⊢ (((2 ∈ ℂ ∧ 2 ≠ 0) ∧ (-(!‘𝑏) ∈ ℤ ∧ (!‘𝐴) ∈ ℤ)) → (2↑(-(!‘𝑏) + (!‘𝐴))) = ((2↑-(!‘𝑏)) · (2↑(!‘𝐴))))
38	36, 37	mpan 689	. . . . . . 7 ⊢ ((-(!‘𝑏) ∈ ℤ ∧ (!‘𝐴) ∈ ℤ) → (2↑(-(!‘𝑏) + (!‘𝐴))) = ((2↑-(!‘𝑏)) · (2↑(!‘𝐴))))
39	28, 35, 38	syl2anc 583	. . . . . 6 ⊢ ((𝐴 ∈ ℕ ∧ 𝑏 ∈ (1...𝐴)) → (2↑(-(!‘𝑏) + (!‘𝐴))) = ((2↑-(!‘𝑏)) · (2↑(!‘𝐴))))
40		2z 12675	. . . . . . 7 ⊢ 2 ∈ ℤ
41	28	zcnd 12748	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℕ ∧ 𝑏 ∈ (1...𝐴)) → -(!‘𝑏) ∈ ℂ)
42	35	zcnd 12748	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℕ ∧ 𝑏 ∈ (1...𝐴)) → (!‘𝐴) ∈ ℂ)
43	41, 42	addcomd 11492	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℕ ∧ 𝑏 ∈ (1...𝐴)) → (-(!‘𝑏) + (!‘𝐴)) = ((!‘𝐴) + -(!‘𝑏)))
44	26	nncnd 12309	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℕ ∧ 𝑏 ∈ (1...𝐴)) → (!‘𝑏) ∈ ℂ)
45	42, 44	negsubd 11653	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℕ ∧ 𝑏 ∈ (1...𝐴)) → ((!‘𝐴) + -(!‘𝑏)) = ((!‘𝐴) − (!‘𝑏)))
46	43, 45	eqtrd 2780	. . . . . . . 8 ⊢ ((𝐴 ∈ ℕ ∧ 𝑏 ∈ (1...𝐴)) → (-(!‘𝑏) + (!‘𝐴)) = ((!‘𝐴) − (!‘𝑏)))
47	9	adantr 480	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℕ ∧ 𝑏 ∈ (1...𝐴)) → 𝐴 ∈ ℕ₀)
48		elfzle2 13588	. . . . . . . . . . 11 ⊢ (𝑏 ∈ (1...𝐴) → 𝑏 ≤ 𝐴)
49	48	adantl 481	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℕ ∧ 𝑏 ∈ (1...𝐴)) → 𝑏 ≤ 𝐴)
50		facwordi 14338	. . . . . . . . . 10 ⊢ ((𝑏 ∈ ℕ₀ ∧ 𝐴 ∈ ℕ₀ ∧ 𝑏 ≤ 𝐴) → (!‘𝑏) ≤ (!‘𝐴))
51	25, 47, 49, 50	syl3anc 1371	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℕ ∧ 𝑏 ∈ (1...𝐴)) → (!‘𝑏) ≤ (!‘𝐴))
52	26	nnnn0d 12613	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℕ ∧ 𝑏 ∈ (1...𝐴)) → (!‘𝑏) ∈ ℕ₀)
53	47	faccld 14333	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℕ ∧ 𝑏 ∈ (1...𝐴)) → (!‘𝐴) ∈ ℕ)
54	53	nnnn0d 12613	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℕ ∧ 𝑏 ∈ (1...𝐴)) → (!‘𝐴) ∈ ℕ₀)
55		nn0sub 12603	. . . . . . . . . 10 ⊢ (((!‘𝑏) ∈ ℕ₀ ∧ (!‘𝐴) ∈ ℕ₀) → ((!‘𝑏) ≤ (!‘𝐴) ↔ ((!‘𝐴) − (!‘𝑏)) ∈ ℕ₀))
56	52, 54, 55	syl2anc 583	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℕ ∧ 𝑏 ∈ (1...𝐴)) → ((!‘𝑏) ≤ (!‘𝐴) ↔ ((!‘𝐴) − (!‘𝑏)) ∈ ℕ₀))
57	51, 56	mpbid 232	. . . . . . . 8 ⊢ ((𝐴 ∈ ℕ ∧ 𝑏 ∈ (1...𝐴)) → ((!‘𝐴) − (!‘𝑏)) ∈ ℕ₀)
58	46, 57	eqeltrd 2844	. . . . . . 7 ⊢ ((𝐴 ∈ ℕ ∧ 𝑏 ∈ (1...𝐴)) → (-(!‘𝑏) + (!‘𝐴)) ∈ ℕ₀)
59		zexpcl 14127	. . . . . . 7 ⊢ ((2 ∈ ℤ ∧ (-(!‘𝑏) + (!‘𝐴)) ∈ ℕ₀) → (2↑(-(!‘𝑏) + (!‘𝐴))) ∈ ℤ)
60	40, 58, 59	sylancr 586	. . . . . 6 ⊢ ((𝐴 ∈ ℕ ∧ 𝑏 ∈ (1...𝐴)) → (2↑(-(!‘𝑏) + (!‘𝐴))) ∈ ℤ)
61	39, 60	eqeltrrd 2845	. . . . 5 ⊢ ((𝐴 ∈ ℕ ∧ 𝑏 ∈ (1...𝐴)) → ((2↑-(!‘𝑏)) · (2↑(!‘𝐴))) ∈ ℤ)
62	34, 61	eqeltrd 2844	. . . 4 ⊢ ((𝐴 ∈ ℕ ∧ 𝑏 ∈ (1...𝐴)) → ((𝐹‘𝑏) · (2↑(!‘𝐴))) ∈ ℤ)
63	7, 62	fsumzcl 15783	. . 3 ⊢ (𝐴 ∈ ℕ → Σ𝑏 ∈ (1...𝐴)((𝐹‘𝑏) · (2↑(!‘𝐴))) ∈ ℤ)
64	33, 63	eqeltrd 2844	. 2 ⊢ (𝐴 ∈ ℕ → (Σ𝑏 ∈ (1...𝐴)(𝐹‘𝑏) · (2↑(!‘𝐴))) ∈ ℤ)
65	6, 64	eqeltrd 2844	1 ⊢ (𝐴 ∈ ℕ → ((𝐻‘𝐴) · (2↑(!‘𝐴))) ∈ ℤ)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1537 ∈ wcel 2108 ≠ wne 2946 class class class wbr 5166 ↦ cmpt 5249 ‘cfv 6573 (class class class)co 7448 ℂcc 11182 0cc0 11184 1c1 11185 + caddc 11187 · cmul 11189 ≤ cle 11325 − cmin 11520 -cneg 11521 ℕcn 12293 2c2 12348 ℕ₀cn0 12553 ℤcz 12639 ℝ⁺crp 13057 ...cfz 13567 ↑cexp 14112 !cfa 14322 Σcsu 15734

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-rep 5303 ax-sep 5317 ax-nul 5324 ax-pow 5383 ax-pr 5447 ax-un 7770 ax-inf2 9710 ax-cnex 11240 ax-resscn 11241 ax-1cn 11242 ax-icn 11243 ax-addcl 11244 ax-addrcl 11245 ax-mulcl 11246 ax-mulrcl 11247 ax-mulcom 11248 ax-addass 11249 ax-mulass 11250 ax-distr 11251 ax-i2m1 11252 ax-1ne0 11253 ax-1rid 11254 ax-rnegex 11255 ax-rrecex 11256 ax-cnre 11257 ax-pre-lttri 11258 ax-pre-lttrn 11259 ax-pre-ltadd 11260 ax-pre-mulgt0 11261 ax-pre-sup 11262

This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1088 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ne 2947 df-nel 3053 df-ral 3068 df-rex 3077 df-rmo 3388 df-reu 3389 df-rab 3444 df-v 3490 df-sbc 3805 df-csb 3922 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-pss 3996 df-nul 4353 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-op 4655 df-uni 4932 df-int 4971 df-iun 5017 df-br 5167 df-opab 5229 df-mpt 5250 df-tr 5284 df-id 5593 df-eprel 5599 df-po 5607 df-so 5608 df-fr 5652 df-se 5653 df-we 5654 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-rn 5711 df-res 5712 df-ima 5713 df-pred 6332 df-ord 6398 df-on 6399 df-lim 6400 df-suc 6401 df-iota 6525 df-fun 6575 df-fn 6576 df-f 6577 df-f1 6578 df-fo 6579 df-f1o 6580 df-fv 6581 df-isom 6582 df-riota 7404 df-ov 7451 df-oprab 7452 df-mpo 7453 df-om 7904 df-1st 8030 df-2nd 8031 df-frecs 8322 df-wrecs 8353 df-recs 8427 df-rdg 8466 df-1o 8522 df-er 8763 df-en 9004 df-dom 9005 df-sdom 9006 df-fin 9007 df-sup 9511 df-oi 9579 df-card 10008 df-pnf 11326 df-mnf 11327 df-xr 11328 df-ltxr 11329 df-le 11330 df-sub 11522 df-neg 11523 df-div 11948 df-nn 12294 df-2 12356 df-3 12357 df-n0 12554 df-z 12640 df-uz 12904 df-rp 13058 df-fz 13568 df-fzo 13712 df-seq 14053 df-exp 14113 df-fac 14323 df-hash 14380 df-cj 15148 df-re 15149 df-im 15150 df-sqrt 15284 df-abs 15285 df-clim 15534 df-sum 15735

This theorem is referenced by: aaliou3lem9 26410

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