Theorem fsumcllem 6967

Description: - Lemma for finite sum closures. (The "-" before "Lemma" forces the math content to be displayed in the Statement List - NM 11-Feb-2008.)

Hypothesis

Ref	Expression
fsumcllem.1	⊢ ((x ∈ C ⋀ y ∈ C) → (x + y) ∈ C)

Assertion

Ref	Expression
fsumcllem	⊢ ((N ∈ (ℤ≥ ‘M) ⋀ ∀k ∈ (M...N)A ∈ C) → Σk ∈ (M...N)A ∈ C)

Distinct variable groups:   x,y,A   x,k,y,C   k,M,x,y   k,N

Proof of Theorem fsumcllem

Step	Hyp	Ref	Expression
1		opreq2 3964	. . . . 5 ⊢ (j = M → (M...j) = (M...M))
2	1	raleq1d 1787	. . . 4 ⊢ (j = M → (∀k ∈ (M...j)A ∈ C ↔ ∀k ∈ (M...M)A ∈ C))
3	1	sumeq1d 6943	. . . . 5 ⊢ (j = M → Σk ∈ (M...j)A = Σk ∈ (M...M)A)
4	3	eleq1d 1538	. . . 4 ⊢ (j = M → (Σk ∈ (M...j)A ∈ C ↔ Σk ∈ (M...M)A ∈ C))
5	2, 4	imbi12d 625	. . 3 ⊢ (j = M → ((∀k ∈ (M...j)A ∈ C → Σk ∈ (M...j)A ∈ C) ↔ (∀k ∈ (M...M)A ∈ C → Σk ∈ (M...M)A ∈ C)))
6		opreq2 3964	. . . . 5 ⊢ (j = m → (M...j) = (M...m))
7	6	raleq1d 1787	. . . 4 ⊢ (j = m → (∀k ∈ (M...j)A ∈ C ↔ ∀k ∈ (M...m)A ∈ C))
8	6	sumeq1d 6943	. . . . 5 ⊢ (j = m → Σk ∈ (M...j)A = Σk ∈ (M...m)A)
9	8	eleq1d 1538	. . . 4 ⊢ (j = m → (Σk ∈ (M...j)A ∈ C ↔ Σk ∈ (M...m)A ∈ C))
10	7, 9	imbi12d 625	. . 3 ⊢ (j = m → ((∀k ∈ (M...j)A ∈ C → Σk ∈ (M...j)A ∈ C) ↔ (∀k ∈ (M...m)A ∈ C → Σk ∈ (M...m)A ∈ C)))
11		opreq2 3964	. . . . 5 ⊢ (j = (m + 1) → (M...j) = (M...(m + 1)))
12	11	raleq1d 1787	. . . 4 ⊢ (j = (m + 1) → (∀k ∈ (M...j)A ∈ C ↔ ∀k ∈ (M...(m + 1))A ∈ C))
13	11	sumeq1d 6943	. . . . 5 ⊢ (j = (m + 1) → Σk ∈ (M...j)A = Σk ∈ (M...(m + 1))A)
14	13	eleq1d 1538	. . . 4 ⊢ (j = (m + 1) → (Σk ∈ (M...j)A ∈ C ↔ Σk ∈ (M...(m + 1))A ∈ C))
15	12, 14	imbi12d 625	. . 3 ⊢ (j = (m + 1) → ((∀k ∈ (M...j)A ∈ C → Σk ∈ (M...j)A ∈ C) ↔ (∀k ∈ (M...(m + 1))A ∈ C → Σk ∈ (M...(m + 1))A ∈ C)))
16		opreq2 3964	. . . . 5 ⊢ (j = N → (M...j) = (M...N))
17	16	raleq1d 1787	. . . 4 ⊢ (j = N → (∀k ∈ (M...j)A ∈ C ↔ ∀k ∈ (M...N)A ∈ C))
18	16	sumeq1d 6943	. . . . 5 ⊢ (j = N → Σk ∈ (M...j)A = Σk ∈ (M...N)A)
19	18	eleq1d 1538	. . . 4 ⊢ (j = N → (Σk ∈ (M...j)A ∈ C ↔ Σk ∈ (M...N)A ∈ C))
20	17, 19	imbi12d 625	. . 3 ⊢ (j = N → ((∀k ∈ (M...j)A ∈ C → Σk ∈ (M...j)A ∈ C) ↔ (∀k ∈ (M...N)A ∈ C → Σk ∈ (M...N)A ∈ C)))
21		fsum1s 6962	. . . . 5 ⊢ ((M ∈ ℤ ⋀ ∀k ∈ (M...M)A ∈ C) → Σk ∈ (M...M)A = [M / k]A)
22		ra4csbela 2039	. . . . . 6 ⊢ ((M ∈ (M...M) ⋀ ∀k ∈ (M...M)A ∈ C) → [M / k]A ∈ C)
23		elfz3t 6436	. . . . . 6 ⊢ (M ∈ ℤ → M ∈ (M...M))
24	22, 23	sylan 448	. . . . 5 ⊢ ((M ∈ ℤ ⋀ ∀k ∈ (M...M)A ∈ C) → [M / k]A ∈ C)
25	21, 24	eqeltrd 1546	. . . 4 ⊢ ((M ∈ ℤ ⋀ ∀k ∈ (M...M)A ∈ C) → Σk ∈ (M...M)A ∈ C)
26	25	ex 373	. . 3 ⊢ (M ∈ ℤ → (∀k ∈ (M...M)A ∈ C → Σk ∈ (M...M)A ∈ C))
27		fsump1s 6966	. . . . . 6 ⊢ ((m ∈ (ℤ≥ ‘M) ⋀ ∀k ∈ (M...(m + 1))A ∈ C) → Σk ∈ (M...(m + 1))A = (Σk ∈ (M...m)A + [(m + 1) / k]A))
28	27	adantrl 394	. . . . 5 ⊢ ((m ∈ (ℤ≥ ‘M) ⋀ ((∀k ∈ (M...m)A ∈ C → Σk ∈ (M...m)A ∈ C) ⋀ ∀k ∈ (M...(m + 1))A ∈ C)) → Σk ∈ (M...(m + 1))A = (Σk ∈ (M...m)A + [(m + 1) / k]A))
29		fsumcllem.1	. . . . . . 7 ⊢ ((x ∈ C ⋀ y ∈ C) → (x + y) ∈ C)
30	29	caoprcl 4047	. . . . . 6 ⊢ ((Σk ∈ (M...m)A ∈ C ⋀ [(m + 1) / k]A ∈ C) → (Σk ∈ (M...m)A + [(m + 1) / k]A) ∈ C)
31		fzssp1t 6451	. . . . . . . . . . . 12 ⊢ ((M ∈ ℤ ⋀ m ∈ ℤ) → (M...m) ⊆ (M...(m + 1)))
32		eluzel2 6369	. . . . . . . . . . . 12 ⊢ (m ∈ (ℤ≥ ‘M) → M ∈ ℤ)
33		eluzelz 6368	. . . . . . . . . . . 12 ⊢ (m ∈ (ℤ≥ ‘M) → m ∈ ℤ)
34	31, 32, 33	sylanc 471	. . . . . . . . . . 11 ⊢ (m ∈ (ℤ≥ ‘M) → (M...m) ⊆ (M...(m + 1)))
35	34	sseld 2064	. . . . . . . . . 10 ⊢ (m ∈ (ℤ≥ ‘M) → (k ∈ (M...m) → k ∈ (M...(m + 1))))
36	35	imim1d 28	. . . . . . . . 9 ⊢ (m ∈ (ℤ≥ ‘M) → ((k ∈ (M...(m + 1)) → A ∈ C) → (k ∈ (M...m) → A ∈ C)))
37	36	r19.20dv2 1709	. . . . . . . 8 ⊢ (m ∈ (ℤ≥ ‘M) → (∀k ∈ (M...(m + 1))A ∈ C → ∀k ∈ (M...m)A ∈ C))
38	37	imim1d 28	. . . . . . 7 ⊢ (m ∈ (ℤ≥ ‘M) → ((∀k ∈ (M...m)A ∈ C → Σk ∈ (M...m)A ∈ C) → (∀k ∈ (M...(m + 1))A ∈ C → Σk ∈ (M...m)A ∈ C)))
39	38	imp32 363	. . . . . 6 ⊢ ((m ∈ (ℤ≥ ‘M) ⋀ ((∀k ∈ (M...m)A ∈ C → Σk ∈ (M...m)A ∈ C) ⋀ ∀k ∈ (M...(m + 1))A ∈ C)) → Σk ∈ (M...m)A ∈ C)
40		ra4csbela 2039	. . . . . . . 8 ⊢ (((m + 1) ∈ (M...(m + 1)) ⋀ ∀k ∈ (M...(m + 1))A ∈ C) → [(m + 1) / k]A ∈ C)
41		peano2uz 6392	. . . . . . . . 9 ⊢ (m ∈ (ℤ≥ ‘M) → (m + 1) ∈ (ℤ≥ ‘M))
42		eluzfz2t 6434	. . . . . . . . 9 ⊢ ((m + 1) ∈ (ℤ≥ ‘M) → (m + 1) ∈ (M...(m + 1)))
43	41, 42	syl 10	. . . . . . . 8 ⊢ (m ∈ (ℤ≥ ‘M) → (m + 1) ∈ (M...(m + 1)))
44	40, 43	sylan 448	. . . . . . 7 ⊢ ((m ∈ (ℤ≥ ‘M) ⋀ ∀k ∈ (M...(m + 1))A ∈ C) → [(m + 1) / k]A ∈ C)
45	44	adantrl 394	. . . . . 6 ⊢ ((m ∈ (ℤ≥ ‘M) ⋀ ((∀k ∈ (M...m)A ∈ C → Σk ∈ (M...m)A ∈ C) ⋀ ∀k ∈ (M...(m + 1))A ∈ C)) → [(m + 1) / k]A ∈ C)
46	30, 39, 45	sylanc 471	. . . . 5 ⊢ ((m ∈ (ℤ≥ ‘M) ⋀ ((∀k ∈ (M...m)A ∈ C → Σk ∈ (M...m)A ∈ C) ⋀ ∀k ∈ (M...(m + 1))A ∈ C)) → (Σk ∈ (M...m)A + [(m + 1) / k]A) ∈ C)
47	28, 46	eqeltrd 1546	. . . 4 ⊢ ((m ∈ (ℤ≥ ‘M) ⋀ ((∀k ∈ (M...m)A ∈ C → Σk ∈ (M...m)A ∈ C) ⋀ ∀k ∈ (M...(m + 1))A ∈ C)) → Σk ∈ (M...(m + 1))A ∈ C)
48	47	exp32 377	. . 3 ⊢ (m ∈ (ℤ≥ ‘M) → ((∀k ∈ (M...m)A ∈ C → Σk ∈ (M...m)A ∈ C) → (∀k ∈ (M...(m + 1))A ∈ C → Σk ∈ (M...(m + 1))A ∈ C)))
49	5, 10, 15, 20, 26, 48	uzind4 6395	. 2 ⊢ (N ∈ (ℤ≥ ‘M) → (∀k ∈ (M...N)A ∈ C → Σk ∈ (M...N)A ∈ C))
50	49	imp 350	1 ⊢ ((N ∈ (ℤ≥ ‘M) ⋀ ∀k ∈ (M...N)A ∈ C) → Σk ∈ (M...N)A ∈ C)

Syntax hints:   → wi 3   ⋀ wa 223   = wceq 955   ∈ wcel 957  ∀wral 1643  [csb 1998   ⊆ wss 2044   ‘cfv 3178  (class class class)co 3958  1c1 5218   + caddc 5220  ℤcz 5281  ℤ_≥cuz 6362  ...cfz 6412  Σcsu 6932

This theorem is referenced by:  fsumclt 6968  fsumreclt 6970  eirrlem2 7348

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 961  ax-gen 962  ax-8 963  ax-9 964  ax-10 965  ax-11 966  ax-12 967  ax-13 968  ax-14 969  ax-17 970  ax-4 972  ax-5o 974  ax-6o 977  ax-9o 1122  ax-10o 1139  ax-16 1209  ax-11o 1217  ax-ext 1458  ax-rep 2689  ax-sep 2699  ax-nul 2706  ax-pow 2738  ax-pr 2775  ax-un 2862  ax-inf2 4608

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 775  df-3an 776  df-ex 980  df-sb 1171  df-eu 1381  df-mo 1382  df-clab 1463  df-cleq 1468  df-clel 1471  df-ne 1585  df-nel 1586  df-ral 1647  df-rex 1648  df-reu 1649  df-rab 1650  df-v 1809  df-sbc 1939  df-csb 1999  df-dif 2046  df-un 2047  df-in 2048  df-ss 2050  df-pss 2052  df-nul 2278  df-if 2359  df-pw 2399  df-sn 2409  df-pr 2410  df-tp 2412  df-op 2413  df-uni 2500  df-int 2530  df-iun 2564  df-br 2616  df-opab 2663  df-tr 2677  df-eprel 2828  df-id 2831  df-po 2836  df-so 2846  df-fr 2913  df-we 2930  df-ord 2947  df-on 2948  df-lim 2949  df-suc 2950  df-om 3128  df-xp 3180  df-rel 3181  df-cnv 3182  df-co 3183  df-dm 3184  df-rn 3185  df-res 3186  df-ima 3187  df-fun 3188  df-fn 3189  df-f 3190  df-f1 3191  df-fo 3192  df-f1o 3193  df-fv 3194  df-rdg 3927  df-opr 3960  df-oprab 3961  df-1st 4072  df-2nd 4073  df-1o 4126  df-oadd 4128  df-omul 4129  df-er 4254  df-ec 4256  df-qs 4259  df-en 4360  df-dom 4361  df-sdom 4362  df-ni 4983  df-pli 4984  df-mi 4985  df-lti 4986  df-plpq 5018  df-mpq 5019  df-enq 5020  df-nq 5021  df-plq 5022  df-mq 5023  df-rq 5024  df-ltq 5025  df-1q 5026  df-np 5069  df-1p 5070  df-plp 5071  df-mp 5072  df-ltp 5073  df-plpr 5147  df-mpr 5148  df-enr 5149  df-nr 5150  df-plr 5151  df-mr 5152  df-ltr 5153  df-0r 5154  df-1r 5155  df-m1r 5156  df-c 5223  df-0 5224  df-1 5225  df-i 5226  df-r 5227  df-plus 5228  df-mul 5229  df-lt 5230  df-sub 5339  df-neg 5341  df-pnf 5470  df-mnf 5471  df-xr 5472  df-ltxr 5473  df-le 5474  df-n 5883  df-n0 6057  df-z 6093  df-seq1 6258  df-shft 6291  df-uz 6363  df-fz 6413  df-seqz 6478  df-sum 6933

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