Theorem ef1tlub 7382

Description: An upper bound on the infinite tail of the series expansion of the exponential function at 1. (Contributed by Paul Chapman, 19-Jan-2008.)

Hypothesis

Ref	Expression
ef1tllem.1	⊢ F = {⟨j, y⟩∣(j ∈ ℕ0 ⋀ y = ((A↑j) / (! ‘j)))}

Assertion

Ref	Expression
ef1tlub	⊢ ((M ∈ ℕ ⋀ A = 1) → Σk ∈ (ℤ≥ ‘M)(F ‘k) ≤ ((M + 1) / ((! ‘M) · M)))

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

Proof of Theorem ef1tlub

Step	Hyp	Ref	Expression
1		fveq2 3730	. . . 4 ⊢ (M = if(M ∈ ℕ, M, 1) → (ℤ≥ ‘M) = (ℤ≥ ‘ if(M ∈ ℕ, M, 1)))
2	1	sumeq1d 6990	. . 3 ⊢ (M = if(M ∈ ℕ, M, 1) → Σk ∈ (ℤ≥ ‘M)(F ‘k) = Σk ∈ (ℤ≥ ‘ if(M ∈ ℕ, M, 1))(F ‘k))
3		opreq1 3974	. . . 4 ⊢ (M = if(M ∈ ℕ, M, 1) → (M + 1) = ( if(M ∈ ℕ, M, 1) + 1))
4		fveq2 3730	. . . . 5 ⊢ (M = if(M ∈ ℕ, M, 1) → (! ‘M) = (! ‘ if(M ∈ ℕ, M, 1)))
5		id 59	. . . . 5 ⊢ (M = if(M ∈ ℕ, M, 1) → M = if(M ∈ ℕ, M, 1))
6	4, 5	opreq12d 3984	. . . 4 ⊢ (M = if(M ∈ ℕ, M, 1) → ((! ‘M) · M) = ((! ‘ if(M ∈ ℕ, M, 1)) · if(M ∈ ℕ, M, 1)))
7	3, 6	opreq12d 3984	. . 3 ⊢ (M = if(M ∈ ℕ, M, 1) → ((M + 1) / ((! ‘M) · M)) = (( if(M ∈ ℕ, M, 1) + 1) / ((! ‘ if(M ∈ ℕ, M, 1)) · if(M ∈ ℕ, M, 1))))
8	2, 7	breq12d 2636	. 2 ⊢ (M = if(M ∈ ℕ, M, 1) → (Σk ∈ (ℤ≥ ‘M)(F ‘k) ≤ ((M + 1) / ((! ‘M) · M)) ↔ Σk ∈ (ℤ≥ ‘ if(M ∈ ℕ, M, 1))(F ‘k) ≤ (( if(M ∈ ℕ, M, 1) + 1) / ((! ‘ if(M ∈ ℕ, M, 1)) · if(M ∈ ℕ, M, 1)))))
9		opreq1 3974	. . . . . . . . . 10 ⊢ (A = if(A = 1, A, 1) → (A↑j) = ( if(A = 1, A, 1)↑j))
10	9	opreq1d 3981	. . . . . . . . 9 ⊢ (A = if(A = 1, A, 1) → ((A↑j) / (! ‘j)) = (( if(A = 1, A, 1)↑j) / (! ‘j)))
11	10	eqeq2d 1489	. . . . . . . 8 ⊢ (A = if(A = 1, A, 1) → (y = ((A↑j) / (! ‘j)) ↔ y = (( if(A = 1, A, 1)↑j) / (! ‘j))))
12	11	anbi2d 618	. . . . . . 7 ⊢ (A = if(A = 1, A, 1) → ((j ∈ ℕ0 ⋀ y = ((A↑j) / (! ‘j))) ↔ (j ∈ ℕ0 ⋀ y = (( if(A = 1, A, 1)↑j) / (! ‘j)))))
13	12	opabbidv 2675	. . . . . 6 ⊢ (A = if(A = 1, A, 1) → {⟨j, y⟩∣(j ∈ ℕ0 ⋀ y = ((A↑j) / (! ‘j)))} = {⟨j, y⟩∣(j ∈ ℕ0 ⋀ y = (( if(A = 1, A, 1)↑j) / (! ‘j)))})
14		ef1tllem.1	. . . . . 6 ⊢ F = {⟨j, y⟩∣(j ∈ ℕ0 ⋀ y = ((A↑j) / (! ‘j)))}
15	13, 14	syl5eq 1522	. . . . 5 ⊢ (A = if(A = 1, A, 1) → F = {⟨j, y⟩∣(j ∈ ℕ0 ⋀ y = (( if(A = 1, A, 1)↑j) / (! ‘j)))})
16	15	fveq1d 3732	. . . 4 ⊢ (A = if(A = 1, A, 1) → (F ‘k) = ({⟨j, y⟩∣(j ∈ ℕ0 ⋀ y = (( if(A = 1, A, 1)↑j) / (! ‘j)))} ‘k))
17	16	sumeq2sdv 6993	. . 3 ⊢ (A = if(A = 1, A, 1) → Σk ∈ (ℤ≥ ‘ if(M ∈ ℕ, M, 1))(F ‘k) = Σk ∈ (ℤ≥ ‘ if(M ∈ ℕ, M, 1))({⟨j, y⟩∣(j ∈ ℕ0 ⋀ y = (( if(A = 1, A, 1)↑j) / (! ‘j)))} ‘k))
18	17	breq1d 2634	. 2 ⊢ (A = if(A = 1, A, 1) → (Σk ∈ (ℤ≥ ‘ if(M ∈ ℕ, M, 1))(F ‘k) ≤ (( if(M ∈ ℕ, M, 1) + 1) / ((! ‘ if(M ∈ ℕ, M, 1)) · if(M ∈ ℕ, M, 1))) ↔ Σk ∈ (ℤ≥ ‘ if(M ∈ ℕ, M, 1))({⟨j, y⟩∣(j ∈ ℕ0 ⋀ y = (( if(A = 1, A, 1)↑j) / (! ‘j)))} ‘k) ≤ (( if(M ∈ ℕ, M, 1) + 1) / ((! ‘ if(M ∈ ℕ, M, 1)) · if(M ∈ ℕ, M, 1)))))
19		eqid 1478	. . 3 ⊢ {⟨j, y⟩∣(j ∈ ℕ0 ⋀ y = (( if(A = 1, A, 1)↑j) / (! ‘j)))} = {⟨j, y⟩∣(j ∈ ℕ0 ⋀ y = (( if(A = 1, A, 1)↑j) / (! ‘j)))}
20		eqid 1478	. . 3 ⊢ {⟨j, y⟩∣(j ∈ ℕ0 ⋀ y = ((1 / ( if(M ∈ ℕ, M, 1) + 1))↑j))} = {⟨j, y⟩∣(j ∈ ℕ0 ⋀ y = ((1 / ( if(M ∈ ℕ, M, 1) + 1))↑j))}
21		eqid 1478	. . 3 ⊢ {⟨j, y⟩∣(j ∈ ℕ0 ⋀ y = ((1 / (! ‘ if(M ∈ ℕ, M, 1))) · ((1 / ( if(M ∈ ℕ, M, 1) + 1))↑j)))} = {⟨j, y⟩∣(j ∈ ℕ0 ⋀ y = ((1 / (! ‘ if(M ∈ ℕ, M, 1))) · ((1 / ( if(M ∈ ℕ, M, 1) + 1))↑j)))}
22		eqid 1478	. . 3 ⊢ {⟨j, y⟩∣(j ∈ ℕ0 ⋀ y = (1 / (! ‘( if(M ∈ ℕ, M, 1) + j))))} = {⟨j, y⟩∣(j ∈ ℕ0 ⋀ y = (1 / (! ‘( if(M ∈ ℕ, M, 1) + j))))}
23		1nn 5936	. . . 4 ⊢ 1 ∈ ℕ
24	23	elimel 2398	. . 3 ⊢ if(M ∈ ℕ, M, 1) ∈ ℕ
25		eqeq1 1484	. . . 4 ⊢ (A = if(A = 1, A, 1) → (A = 1 ↔ if(A = 1, A, 1) = 1))
26		eqeq1 1484	. . . 4 ⊢ (1 = if(A = 1, A, 1) → (1 = 1 ↔ if(A = 1, A, 1) = 1))
27		eqid 1478	. . . 4 ⊢ 1 = 1
28	25, 26, 27	elimhyp 2394	. . 3 ⊢ if(A = 1, A, 1) = 1
29	19, 20, 21, 22, 24, 28	ef1tllem 7381	. 2 ⊢ Σk ∈ (ℤ≥ ‘ if(M ∈ ℕ, M, 1))({⟨j, y⟩∣(j ∈ ℕ0 ⋀ y = (( if(A = 1, A, 1)↑j) / (! ‘j)))} ‘k) ≤ (( if(M ∈ ℕ, M, 1) + 1) / ((! ‘ if(M ∈ ℕ, M, 1)) · if(M ∈ ℕ, M, 1)))
30	8, 18, 29	dedth2h 2391	1 ⊢ ((M ∈ ℕ ⋀ A = 1) → Σk ∈ (ℤ≥ ‘M)(F ‘k) ≤ ((M + 1) / ((! ‘M) · M)))

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  1c1 5247   + caddc 5249   · cmul 5251   / cdiv 5306   ≤ cle 5307  ℕcn 5308  ℕ₀cn0 5309  ℤ_≥cuz 6418  ↑cexp 6569  !cfa 6931  Σcsu 6979

This theorem is referenced by:  ef01tllem2 7384  ef01tllem2OLD 7385  eirrlem3 7391

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

Copyright terms: Public domain