Theorem fsump1s 6902

Description: The addition of the next term in a finite sum of A(k) is the previous term plus A(N + 1).

Assertion

Ref	Expression
fsump1s	⊢ ((N ∈ (ℤ≥ ‘M) ⋀ ∀k ∈ (M...(N + 1))A ∈ B) → Σk ∈ (M...(N + 1))A = (Σk ∈ (M...N)A + [(N + 1) / k]A))

Distinct variable groups:   k,M   k,N

Proof of Theorem fsump1s

Step	Hyp	Ref	Expression
1		class2set 2702	. . . . 5 ⊢ {x ∈ A∣A ∈ V} ∈ V
2	1	fsump1slem 6901	. . . 4 ⊢ (N ∈ (ℤ≥ ‘M) → Σk ∈ (M...(N + 1)){x ∈ A∣A ∈ V} = (Σk ∈ (M...N){x ∈ A∣A ∈ V} + [(N + 1) / k]{x ∈ A∣A ∈ V}))
3	2	adantr 389	. . 3 ⊢ ((N ∈ (ℤ≥ ‘M) ⋀ ∀k ∈ (M...(N + 1))A ∈ V) → Σk ∈ (M...(N + 1)){x ∈ A∣A ∈ V} = (Σk ∈ (M...N){x ∈ A∣A ∈ V} + [(N + 1) / k]{x ∈ A∣A ∈ V}))
4		class2seteq 2703	. . . . . 6 ⊢ (A ∈ V → {x ∈ A∣A ∈ V} = A)
5	4	r19.20si 1682	. . . . 5 ⊢ (∀k ∈ (M...(N + 1))A ∈ V → ∀k ∈ (M...(N + 1)){x ∈ A∣A ∈ V} = A)
6	5	sumeq2d 6880	. . . 4 ⊢ (∀k ∈ (M...(N + 1))A ∈ V → Σk ∈ (M...(N + 1)){x ∈ A∣A ∈ V} = Σk ∈ (M...(N + 1))A)
7	6	adantl 388	. . 3 ⊢ ((N ∈ (ℤ≥ ‘M) ⋀ ∀k ∈ (M...(N + 1))A ∈ V) → Σk ∈ (M...(N + 1)){x ∈ A∣A ∈ V} = Σk ∈ (M...(N + 1))A)
8		fzssp1t 6389	. . . . . . . . . 10 ⊢ ((M ∈ ℤ ⋀ N ∈ ℤ) → (M...N) ⊆ (M...(N + 1)))
9		eluzel2 6307	. . . . . . . . . 10 ⊢ (N ∈ (ℤ≥ ‘M) → M ∈ ℤ)
10		eluzelz 6306	. . . . . . . . . 10 ⊢ (N ∈ (ℤ≥ ‘M) → N ∈ ℤ)
11	8, 9, 10	sylanc 471	. . . . . . . . 9 ⊢ (N ∈ (ℤ≥ ‘M) → (M...N) ⊆ (M...(N + 1)))
12	11	sseld 2038	. . . . . . . 8 ⊢ (N ∈ (ℤ≥ ‘M) → (k ∈ (M...N) → k ∈ (M...(N + 1))))
13	4	a1i 8	. . . . . . . 8 ⊢ (N ∈ (ℤ≥ ‘M) → (A ∈ V → {x ∈ A∣A ∈ V} = A))
14	12, 13	imim12d 29	. . . . . . 7 ⊢ (N ∈ (ℤ≥ ‘M) → ((k ∈ (M...(N + 1)) → A ∈ V) → (k ∈ (M...N) → {x ∈ A∣A ∈ V} = A)))
15	14	r19.20dv2 1687	. . . . . 6 ⊢ (N ∈ (ℤ≥ ‘M) → (∀k ∈ (M...(N + 1))A ∈ V → ∀k ∈ (M...N){x ∈ A∣A ∈ V} = A))
16	15	imp 350	. . . . 5 ⊢ ((N ∈ (ℤ≥ ‘M) ⋀ ∀k ∈ (M...(N + 1))A ∈ V) → ∀k ∈ (M...N){x ∈ A∣A ∈ V} = A)
17	16	sumeq2d 6880	. . . 4 ⊢ ((N ∈ (ℤ≥ ‘M) ⋀ ∀k ∈ (M...(N + 1))A ∈ V) → Σk ∈ (M...N){x ∈ A∣A ∈ V} = Σk ∈ (M...N)A)
18		ra4sbca 1969	. . . . . . 7 ⊢ (((N + 1) ∈ (M...(N + 1)) ⋀ ∀k ∈ (M...(N + 1))A ∈ V) → [(N + 1) / k]A ∈ V)
19		peano2uz 6330	. . . . . . . 8 ⊢ (N ∈ (ℤ≥ ‘M) → (N + 1) ∈ (ℤ≥ ‘M))
20		eluzfz2t 6372	. . . . . . . 8 ⊢ ((N + 1) ∈ (ℤ≥ ‘M) → (N + 1) ∈ (M...(N + 1)))
21	19, 20	syl 10	. . . . . . 7 ⊢ (N ∈ (ℤ≥ ‘M) → (N + 1) ∈ (M...(N + 1)))
22	18, 21	sylan 448	. . . . . 6 ⊢ ((N ∈ (ℤ≥ ‘M) ⋀ ∀k ∈ (M...(N + 1))A ∈ V) → [(N + 1) / k]A ∈ V)
23		equid 1113	. . . . . . 7 ⊢ x = x
24		oprex 3922	. . . . . . 7 ⊢ (N + 1) ∈ V
25	4	a1i 8	. . . . . . . 8 ⊢ (x = x → (A ∈ V → {x ∈ A∣A ∈ V} = A))
26	25	sbc19.20dv 1956	. . . . . . 7 ⊢ ((x = x ⋀ (N + 1) ∈ V) → ([(N + 1) / k]A ∈ V → [(N + 1) / k]{x ∈ A∣A ∈ V} = A))
27	23, 24, 26	mp2an 694	. . . . . 6 ⊢ ([(N + 1) / k]A ∈ V → [(N + 1) / k]{x ∈ A∣A ∈ V} = A)
28	22, 27	syl 10	. . . . 5 ⊢ ((N ∈ (ℤ≥ ‘M) ⋀ ∀k ∈ (M...(N + 1))A ∈ V) → [(N + 1) / k]{x ∈ A∣A ∈ V} = A)
29		sbceqdig 1983	. . . . . 6 ⊢ ((N + 1) ∈ V → ([(N + 1) / k]{x ∈ A∣A ∈ V} = A ↔ [(N + 1) / k]{x ∈ A∣A ∈ V} = [(N + 1) / k]A))
30	24, 29	ax-mp 7	. . . . 5 ⊢ ([(N + 1) / k]{x ∈ A∣A ∈ V} = A ↔ [(N + 1) / k]{x ∈ A∣A ∈ V} = [(N + 1) / k]A)
31	28, 30	sylib 198	. . . 4 ⊢ ((N ∈ (ℤ≥ ‘M) ⋀ ∀k ∈ (M...(N + 1))A ∈ V) → [(N + 1) / k]{x ∈ A∣A ∈ V} = [(N + 1) / k]A)
32	17, 31	opreq12d 3917	. . 3 ⊢ ((N ∈ (ℤ≥ ‘M) ⋀ ∀k ∈ (M...(N + 1))A ∈ V) → (Σk ∈ (M...N){x ∈ A∣A ∈ V} + [(N + 1) / k]{x ∈ A∣A ∈ V}) = (Σk ∈ (M...N)A + [(N + 1) / k]A))
33	3, 7, 32	3eqtr3d 1491	. 2 ⊢ ((N ∈ (ℤ≥ ‘M) ⋀ ∀k ∈ (M...(N + 1))A ∈ V) → Σk ∈ (M...(N + 1))A = (Σk ∈ (M...N)A + [(N + 1) / k]A))
34		elisset 1792	. . 3 ⊢ (A ∈ B → A ∈ V)
35	34	r19.20si 1682	. 2 ⊢ (∀k ∈ (M...(N + 1))A ∈ B → ∀k ∈ (M...(N + 1))A ∈ V)
36	33, 35	sylan2 451	1 ⊢ ((N ∈ (ℤ≥ ‘M) ⋀ ∀k ∈ (M...(N + 1))A ∈ B) → Σk ∈ (M...(N + 1))A = (Σk ∈ (M...N)A + [(N + 1) / k]A))

Syntax hints:   → wi 3   ↔ wb 146   ⋀ wa 223   = wceq 1099   ∈ wcel 1105  [wsbc 1153  ∀wral 1621  {crab 1624  Vcvv 1786  [csb 1972   ⊆ wss 2018   ‘cfv 3145  (class class class)co 3902  1c1 5158   + caddc 5160  ℤcz 5221  ℤ_≥cuz 6300  ...cfz 6350  Σcsu 6868

This theorem is referenced by:  fsumcllem 6903  fsum1ps 6907  fsumsplit 6909  fsumadd 6911  fsumcom 6917  fsumrev 6918  fsummulc1 6922  fsumconst 6927  fsumcmp 6929  fsumabs 6932

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-4 951  ax-5 952  ax-6 953  ax-7 954  ax-gen 955  ax-8 1101  ax-9 1102  ax-10 1103  ax-12 1104  ax-13 1107  ax-14 1108  ax-11 1180  ax-17 1190  ax-16 1194  ax-11o 1202  ax-ext 1436  ax-rep 2661  ax-sep 2671  ax-nul 2678  ax-pow 2710  ax-pr 2747  ax-un 2830  ax-inf2 4549

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 773  df-3an 774  df-ex 957  df-sb 1155  df-eu 1359  df-mo 1360  df-clab 1441  df-cleq 1446  df-clel 1449  df-ne 1563  df-nel 1564  df-ral 1625  df-rex 1626  df-reu 1627  df-rab 1628  df-v 1787  df-sbc 1913  df-csb 1973  df-dif 2020  df-un 2021  df-in 2022  df-ss 2024  df-pss 2026  df-nul 2252  df-if 2333  df-pw 2373  df-sn 2383  df-pr 2384  df-tp 2386  df-op 2387  df-uni 2472  df-int 2502  df-iun 2536  df-br 2588  df-opab 2635  df-tr 2649  df-eprel 2794  df-id 2797  df-po 2804  df-so 2814  df-fr 2880  df-we 2897  df-ord 2914  df-on 2915  df-lim 2916  df-suc 2917  df-om 3095  df-xp 3147  df-rel 3148  df-cnv 3149  df-co 3150  df-dm 3151  df-rn 3152  df-res 3153  df-ima 3154  df-fun 3155  df-fn 3156  df-f 3157  df-f1 3158  df-fo 3159  df-f1o 3160  df-fv 3161  df-rdg 3871  df-opr 3904  df-oprab 3905  df-1st 4017  df-2nd 4018  df-1o 4071  df-oadd 4073  df-omul 4074  df-er 4199  df-ec 4201  df-qs 4204  df-en 4305  df-dom 4306  df-sdom 4307  df-ni 4923  df-pli 4924  df-mi 4925  df-lti 4926  df-plpq 4958  df-mpq 4959  df-enq 4960  df-nq 4961  df-plq 4962  df-mq 4963  df-rq 4964  df-ltq 4965  df-1q 4966  df-np 5009  df-1p 5010  df-plp 5011  df-mp 5012  df-ltp 5013  df-plpr 5087  df-mpr 5088  df-enr 5089  df-nr 5090  df-plr 5091  df-mr 5092  df-ltr 5093  df-0r 5094  df-1r 5095  df-m1r 5096  df-c 5163  df-0 5164  df-1 5165  df-i 5166  df-r 5167  df-plus 5168  df-mul 5169  df-lt 5170  df-sub 5279  df-neg 5281  df-pnf 5410  df-mnf 5411  df-xr 5412  df-ltxr 5413  df-le 5414  df-n 5824  df-n0 5998  df-z 6034  df-seq1 6196  df-shft 6229  df-uz 6301  df-fz 6351  df-seqz 6416  df-sum 6869

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