reeftlclt - Metamath Proof Explorer

Metamath Proof Explorer

< Previous Next >
Related theorems
GIF version

Theorem reeftlclt 7380

Description: Closure of the sum of an infinite tail of the series defining the exponential function. (Contributed by Paul Chapman, 17-Jan-2008.)

**Hypothesis**
Ref	Expression
eftlext.1	⊢ F = {⟨j, y⟩∣(j ∈ ℕ₀ ⋀ y = ((A↑j) / (! ‘j)))}

**Assertion**
Ref	Expression
reeftlclt	⊢ ((A ∈ ℝ ⋀ M ∈ ℕ) → Σk ∈ (ℤ_≥ ‘M)(F ‘k) ∈ ℝ)

Distinct variable groups: A,j,k,y k,F k,M

**Proof of Theorem reeftlclt**
Step	Hyp	Ref	Expression
1		nn0ex 6107	. . . 4 ⊢ ℕ₀ ∈ V
2		eftlext.1	. . . 4 ⊢ F = {⟨j, y⟩∣(j ∈ ℕ₀ ⋀ y = ((A↑j) / (! ‘j)))}
3	1, 2	fopabex2 3618	. . 3 ⊢ F ∈ V
4	3	isumreclt 7210	. 2 ⊢ ((M ∈ ℤ ⋀ ∀k ∈ (ℤ_≥ ‘M)(F ‘k) ∈ ℝ ⋀ ∃x(⟨M, + ⟩seqF) ⇝ x) → Σk ∈ (ℤ_≥ ‘M)(F ‘k) ∈ ℝ)
5		nnzt 6155	. . 3 ⊢ (M ∈ ℕ → M ∈ ℤ)
6	5	adantl 390	. 2 ⊢ ((A ∈ ℝ ⋀ M ∈ ℕ) → M ∈ ℤ)
7	2	eftval 7316	. . . . . . . 8 ⊢ (k ∈ ℕ₀ → (F ‘k) = ((A↑k) / (! ‘k)))
8	7	adantl 390	. . . . . . 7 ⊢ ((A ∈ ℝ ⋀ k ∈ ℕ₀) → (F ‘k) = ((A↑k) / (! ‘k)))
9		reeftclt 7374	. . . . . . 7 ⊢ ((A ∈ ℝ ⋀ k ∈ ℕ₀) → ((A↑k) / (! ‘k)) ∈ ℝ)
10	8, 9	eqeltrd 1551	. . . . . 6 ⊢ ((A ∈ ℝ ⋀ k ∈ ℕ₀) → (F ‘k) ∈ ℝ)
11		uztrn 6429	. . . . . . . . 9 ⊢ ((k ∈ (ℤ_≥ ‘M) ⋀ M ∈ (ℤ_≥ ‘0)) → k ∈ (ℤ_≥ ‘0))
12		nnnn0t 6108	. . . . . . . . . 10 ⊢ (M ∈ ℕ → M ∈ ℕ₀)
13		elnn0uz 6442	. . . . . . . . . 10 ⊢ (M ∈ ℕ₀ ↔ M ∈ (ℤ_≥ ‘0))
14	12, 13	sylib 198	. . . . . . . . 9 ⊢ (M ∈ ℕ → M ∈ (ℤ_≥ ‘0))
15	11, 14	sylan2 453	. . . . . . . 8 ⊢ ((k ∈ (ℤ_≥ ‘M) ⋀ M ∈ ℕ) → k ∈ (ℤ_≥ ‘0))
16		elnn0uz 6442	. . . . . . . 8 ⊢ (k ∈ ℕ₀ ↔ k ∈ (ℤ_≥ ‘0))
17	15, 16	sylibr 200	. . . . . . 7 ⊢ ((k ∈ (ℤ_≥ ‘M) ⋀ M ∈ ℕ) → k ∈ ℕ₀)
18	17	ancoms 438	. . . . . 6 ⊢ ((M ∈ ℕ ⋀ k ∈ (ℤ_≥ ‘M)) → k ∈ ℕ₀)
19	10, 18	sylan2 453	. . . . 5 ⊢ ((A ∈ ℝ ⋀ (M ∈ ℕ ⋀ k ∈ (ℤ_≥ ‘M))) → (F ‘k) ∈ ℝ)
20	19	3impb 831	. . . 4 ⊢ ((A ∈ ℝ ⋀ M ∈ ℕ ⋀ k ∈ (ℤ_≥ ‘M)) → (F ‘k) ∈ ℝ)
21	20	3expia 837	. . 3 ⊢ ((A ∈ ℝ ⋀ M ∈ ℕ) → (k ∈ (ℤ_≥ ‘M) → (F ‘k) ∈ ℝ))
22	21	r19.21aiv 1716	. 2 ⊢ ((A ∈ ℝ ⋀ M ∈ ℕ) → ∀k ∈ (ℤ_≥ ‘M)(F ‘k) ∈ ℝ)
23	2	eftlext 7378	. . 3 ⊢ ((A ∈ ℂ ⋀ M ∈ ℕ) → ∃x(⟨M, + ⟩seqF) ⇝ x)
24		recnt 5325	. . 3 ⊢ (A ∈ ℝ → A ∈ ℂ)
25	23, 24	sylan 450	. 2 ⊢ ((A ∈ ℝ ⋀ M ∈ ℕ) → ∃x(⟨M, + ⟩seqF) ⇝ x)
26	4, 6, 22, 25	syl3anc 860	1 ⊢ ((A ∈ ℝ ⋀ M ∈ ℕ) → Σk ∈ (ℤ_≥ ‘M)(F ‘k) ∈ ℝ)

Colors of variables: wff set class

Syntax hints: → wi 3 ⋀ wa 223 = wceq 958 ∈ wcel 960 ∃wex 982 ∀wral 1648 ⟨cop 2415 class class class wbr 2624 {copab 2671 ‘cfv 3188 (class class class)co 3969 ℂcc 5244 ℝcr 5245 0cc0 5246 + caddc 5249 / cdiv 5306 ℕcn 5308 ℕ₀cn0 5309 ℤcz 5310 ℤ_≥cuz 6418 seqcseqz 6532 ↑cexp 6569 !cfa 6931 ⇝ cli 6974 Σcsu 6979

This theorem is referenced by: ef01tllem2 7384 ef01tllem2OLD 7385 absef01tllem 7387

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-fz 6469 df-seqz 6534 df-seq0 6535 df-exp 6570 df-sqr 6671 df-re 6752 df-im 6753 df-cj 6754 df-abs 6755 df-fac 6932 df-clim 6975 df-sum 6980 df-ef 7298