Theorem fsumcnlem 9267

Description: Lemma for fsumcn 9268. Warning: The HTML proof page is 0.4MB in size.

Hypotheses

Ref	Expression
fsumcn.1	|- C e. Met
fsumcn.2	|- X = dom dom C
fsumcn.7	|- D = (abs o. - )
fsumcn.j	|- J = (Open` C)
fsumcn.k	|- K = (Open` D)
fsumcn.11	|- (k e. NN -> F e. (J Cn K))
fsumcn.13	|- G = {<.w, v>. | (w e. X /\ v = sum_k e. (1...N)(F` w))}
fsumcnlem.14	|- B = {<.<.u, t>., s>. | ((u e. (CC X. CC) /\ t e. (CC X. CC)) /\ s = sup({((1st` u)D(1st` t)), ((2nd` u)D(2nd` t))}, RR, < ))}

Assertion

Ref	Expression
fsumcnlem	|- (N e. NN -> G e. (J Cn K))

Distinct variable groups:   w,C   t,s,u,w,D   v,s,F,t,u,w   v,N,w   w,k,J   k,K,w   k,s,t,u,v,X,w

Proof of Theorem fsumcnlem

Step	Hyp	Ref	Expression
1		elnnuz 7609	. . . . . . . . . . . 12 |- (m e. NN <-> m e. (ZZ>=` 1))
2		fvex 4689	. . . . . . . . . . . . 13 |- (F` w) e. _V
3	2	fsump1slem 8272	. . . . . . . . . . . 12 |- (m e. (ZZ>=` 1) -> sum_k e. (1...(m + 1))(F` w) = (sum_k e. (1...m)(F` w) + [_(m + 1) / k]_(F` w)))
4	1, 3	sylbi 216	. . . . . . . . . . 11 |- (m e. NN -> sum_k e. (1...(m + 1))(F` w) = (sum_k e. (1...m)(F` w) + [_(m + 1) / k]_(F` w)))
5	4	adantr 425	. . . . . . . . . 10 |- ((m e. NN /\ w e. X) -> sum_k e. (1...(m + 1))(F` w) = (sum_k e. (1...m)(F` w) + [_(m + 1) / k]_(F` w)))
6		sumex 8241	. . . . . . . . . . . . 13 |- sum_k e. (1...m)(F` w) e. _V
7		fvopab2 4754	. . . . . . . . . . . . 13 |- ((w e. X /\ sum_k e. (1...m)(F` w) e. _V) -> ({<.w, v>. | (w e. X /\ v = sum_k e. (1...m)(F` w))}` w) = sum_k e. (1...m)(F` w))
8	6, 7	mpan2 760	. . . . . . . . . . . 12 |- (w e. X -> ({<.w, v>. | (w e. X /\ v = sum_k e. (1...m)(F` w))}` w) = sum_k e. (1...m)(F` w))
9		oprex 4907	. . . . . . . . . . . . . . 15 |- (m + 1) e. _V
10		csbfv12g 4699	. . . . . . . . . . . . . . 15 |- ((m + 1) e. _V -> [_(m + 1) / k]_(F` w) = ([_(m + 1) / k]_F` [_(m + 1) / k]_w))
11	9, 10	ax-mp 7	. . . . . . . . . . . . . 14 |- [_(m + 1) / k]_(F` w) = ([_(m + 1) / k]_F` [_(m + 1) / k]_w)
12		ax-17 1317	. . . . . . . . . . . . . . . . 17 |- (v e. w -> A.k v e. w)
13	12	csbconstgf 2551	. . . . . . . . . . . . . . . 16 |- ((m + 1) e. _V -> [_(m + 1) / k]_w = w)
14	9, 13	ax-mp 7	. . . . . . . . . . . . . . 15 |- [_(m + 1) / k]_w = w
15	14	fveq2i 4684	. . . . . . . . . . . . . 14 |- ([_(m + 1) / k]_F` [_(m + 1) / k]_w) = ([_(m + 1) / k]_F` w)
16	11, 15	eqtr2i 1909	. . . . . . . . . . . . 13 |- ([_(m + 1) / k]_F` w) = [_(m + 1) / k]_(F` w)
17	16	a1i 8	. . . . . . . . . . . 12 |- (w e. X -> ([_(m + 1) / k]_F` w) = [_(m + 1) / k]_(F` w))
18	8, 17	opreq12d 4900	. . . . . . . . . . 11 |- (w e. X -> (({<.w, v>. | (w e. X /\ v = sum_k e. (1...m)(F` w))}` w) + ([_(m + 1) / k]_F` w)) = (sum_k e. (1...m)(F` w) + [_(m + 1) / k]_(F` w)))
19	18	adantl 424	. . . . . . . . . 10 |- ((m e. NN /\ w e. X) -> (({<.w, v>. | (w e. X /\ v = sum_k e. (1...m)(F` w))}` w) + ([_(m + 1) / k]_F` w)) = (sum_k e. (1...m)(F` w) + [_(m + 1) / k]_(F` w)))
20	5, 19	eqtr4d 1928	. . . . . . . . 9 |- ((m e. NN /\ w e. X) -> sum_k e. (1...(m + 1))(F` w) = (({<.w, v>. | (w e. X /\ v = sum_k e. (1...m)(F` w))}` w) + ([_(m + 1) / k]_F` w)))
21	20	eqeq2d 1895	. . . . . . . 8 |- ((m e. NN /\ w e. X) -> (v = sum_k e. (1...(m + 1))(F` w) <-> v = (({<.w, v>. | (w e. X /\ v = sum_k e. (1...m)(F` w))}` w) + ([_(m + 1) / k]_F` w))))
22	21	pm5.32da 711	. . . . . . 7 |- (m e. NN -> ((w e. X /\ v = sum_k e. (1...(m + 1))(F` w)) <-> (w e. X /\ v = (({<.w, v>. | (w e. X /\ v = sum_k e. (1...m)(F` w))}` w) + ([_(m + 1) / k]_F` w)))))
23	22	opabbidv 3401	. . . . . 6 |- (m e. NN -> {<.w, v>. | (w e. X /\ v = sum_k e. (1...(m + 1))(F` w))} = {<.w, v>. | (w e. X /\ v = (({<.w, v>. | (w e. X /\ v = sum_k e. (1...m)(F` w))}` w) + ([_(m + 1) / k]_F` w)))})
24	23	adantr 425	. . . . 5 |- ((m e. NN /\ {<.w, v>. | (w e. X /\ v = sum_k e. (1...m)(F` w))} e. (J Cn K)) -> {<.w, v>. | (w e. X /\ v = sum_k e. (1...(m + 1))(F` w))} = {<.w, v>. | (w e. X /\ v = (({<.w, v>. | (w e. X /\ v = sum_k e. (1...m)(F` w))}` w) + ([_(m + 1) / k]_F` w)))})
25		simpr 350	. . . . . 6 |- ((m e. NN /\ {<.w, v>. | (w e. X /\ v = sum_k e. (1...m)(F` w))} e. (J Cn K)) -> {<.w, v>. | (w e. X /\ v = sum_k e. (1...m)(F` w))} e. (J Cn K))
26		ra4csbela 2587	. . . . . . . 8 |- (((m + 1) e. NN /\ A.k e. NN F e. (J Cn K)) -> [_(m + 1) / k]_F e. (J Cn K))
27		peano2nn 7118	. . . . . . . 8 |- (m e. NN -> (m + 1) e. NN)
28		fsumcn.11	. . . . . . . . 9 |- (k e. NN -> F e. (J Cn K))
29	28	rgen 2159	. . . . . . . 8 |- A.k e. NN F e. (J Cn K)
30	26, 27, 29	sylancl 525	. . . . . . 7 |- (m e. NN -> [_(m + 1) / k]_F e. (J Cn K))
31	30	adantr 425	. . . . . 6 |- ((m e. NN /\ {<.w, v>. | (w e. X /\ v = sum_k e. (1...m)(F` w))} e. (J Cn K)) -> [_(m + 1) / k]_F e. (J Cn K))
32		fsumcn.2	. . . . . . . 8 |- X = dom dom C
33		fsumcn.7	. . . . . . . . 9 |- D = (abs o. - )
34	33	cnmetba 9181	. . . . . . . 8 |- CC = dom dom D
35		fsumcn.1	. . . . . . . 8 |- C e. Met
36	33	cnmet 9182	. . . . . . . 8 |- D e. Met
37		fsumcn.j	. . . . . . . 8 |- J = (Open` C)
38		fsumcn.k	. . . . . . . 8 |- K = (Open` D)
39		eqid 1884	. . . . . . . 8 |- (Open` B) = (Open` B)
40		fsumcnlem.14	. . . . . . . 8 |- B = {<.<.u, t>., s>. | ((u e. (CC X. CC) /\ t e. (CC X. CC)) /\ s = sup({((1st` u)D(1st` t)), ((2nd` u)D(2nd` t))}, RR, < ))}
41	33, 40, 38, 39	addcn 9264	. . . . . . . 8 |- + e. ((Open` B) Cn K)
42		eleq1 1957	. . . . . . . . . . . . . . . 16 |- (w = r -> (w e. X <-> r e. X))
43	42	adantr 425	. . . . . . . . . . . . . . 15 |- ((w = r /\ v = q) -> (w e. X <-> r e. X))
44		id 73	. . . . . . . . . . . . . . . 16 |- (v = q -> v = q)
45		fveq2 4681	. . . . . . . . . . . . . . . . 17 |- (w = r -> (F` w) = (F` r))
46	45	sumeq2sdv 8253	. . . . . . . . . . . . . . . 16 |- (w = r -> sum_k e. (1...m)(F` w) = sum_k e. (1...m)(F` r))
47	44, 46	eqeqan12rd 1903	. . . . . . . . . . . . . . 15 |- ((w = r /\ v = q) -> (v = sum_k e. (1...m)(F` w) <-> q = sum_k e. (1...m)(F` r)))
48	43, 47	anbi12d 690	. . . . . . . . . . . . . 14 |- ((w = r /\ v = q) -> ((w e. X /\ v = sum_k e. (1...m)(F` w)) <-> (r e. X /\ q = sum_k e. (1...m)(F` r))))
49	48	cbvopabv 3404	. . . . . . . . . . . . 13 |- {<.w, v>. | (w e. X /\ v = sum_k e. (1...m)(F` w))} = {<.r, q>. | (r e. X /\ q = sum_k e. (1...m)(F` r))}
50	49	fveq1i 4682	. . . . . . . . . . . 12 |- ({<.w, v>. | (w e. X /\ v = sum_k e. (1...m)(F` w))}` w) = ({<.r, q>. | (r e. X /\ q = sum_k e. (1...m)(F` r))}` w)
51	50	opreq1i 4892	. . . . . . . . . . 11 |- (({<.w, v>. | (w e. X /\ v = sum_k e. (1...m)(F` w))}` w) + ([_(m + 1) / k]_F` w)) = (({<.r, q>. | (r e. X /\ q = sum_k e. (1...m)(F` r))}` w) + ([_(m + 1) / k]_F` w))
52	51	eqeq2i 1894	. . . . . . . . . 10 |- (v = (({<.w, v>. | (w e. X /\ v = sum_k e. (1...m)(F` w))}` w) + ([_(m + 1) / k]_F` w)) <-> v = (({<.r, q>. | (r e. X /\ q = sum_k e. (1...m)(F` r))}` w) + ([_(m + 1) / k]_F` w)))
53	52	anbi2i 538	. . . . . . . . 9 |- ((w e. X /\ v = (({<.w, v>. | (w e. X /\ v = sum_k e. (1...m)(F` w))}` w) + ([_(m + 1) / k]_F` w))) <-> (w e. X /\ v = (({<.r, q>. | (r e. X /\ q = sum_k e. (1...m)(F` r))}` w) + ([_(m + 1) / k]_F` w))))
54	53	opabbii 3402	. . . . . . . 8 |- {<.w, v>. | (w e. X /\ v = (({<.w, v>. | (w e. X /\ v = sum_k e. (1...m)(F` w))}` w) + ([_(m + 1) / k]_F` w)))} = {<.w, v>. | (w e. X /\ v = (({<.r, q>. | (r e. X /\ q = sum_k e. (1...m)(F` r))}` w) + ([_(m + 1) / k]_F` w)))}
55	32, 34, 34, 35, 36, 36, 36, 37, 38, 38, 39, 38, 40, 41, 54	oprcn 9255	. . . . . . 7 |- (({<.r, q>. | (r e. X /\ q = sum_k e. (1...m)(F` r))} e. (J Cn K) /\ [_(m + 1) / k]_F e. (J Cn K)) -> {<.w, v>. | (w e. X /\ v = (({<.w, v>. | (w e. X /\ v = sum_k e. (1...m)(F` w))}` w) + ([_(m + 1) / k]_F` w)))} e. (J Cn K))
56	49	eleq1i 1960	. . . . . . 7 |- ({<.w, v>. | (w e. X /\ v = sum_k e. (1...m)(F` w))} e. (J Cn K) <-> {<.r, q>. | (r e. X /\ q = sum_k e. (1...m)(F` r))} e. (J Cn K))
57	55, 56	sylanb 498	. . . . . 6 |- (({<.w, v>. | (w e. X /\ v = sum_k e. (1...m)(F` w))} e. (J Cn K) /\ [_(m + 1) / k]_F e. (J Cn K)) -> {<.w, v>. | (w e. X /\ v = (({<.w, v>. | (w e. X /\ v = sum_k e. (1...m)(F` w))}` w) + ([_(m + 1) / k]_F` w)))} e. (J Cn K))
58	25, 31, 57	syl11anc 524	. . . . 5 |- ((m e. NN /\ {<.w, v>. | (w e. X /\ v = sum_k e. (1...m)(F` w))} e. (J Cn K)) -> {<.w, v>. | (w e. X /\ v = (({<.w, v>. | (w e. X /\ v = sum_k e. (1...m)(F` w))}` w) + ([_(m + 1) / k]_F` w)))} e. (J Cn K))
59	24, 58	eqeltrd 1971	. . . 4 |- ((m e. NN /\ {<.w, v>. | (w e. X /\ v = sum_k e. (1...m)(F` w))} e. (J Cn K)) -> {<.w, v>. | (w e. X /\ v = sum_k e. (1...(m + 1))(F` w))} e. (J Cn K))
60	59	ex 402	. . 3 |- (m e. NN -> ({<.w, v>. | (w e. X /\ v = sum_k e. (1...m)(F` w))} e. (J Cn K) -> {<.w, v>. | (w e. X /\ v = sum_k e. (1...(m + 1))(F` w))} e. (J Cn K)))
61		1nn 7117	. . . . . 6 |- 1 e. NN
62		csbopabg 3409	. . . . . 6 |- (1 e. NN -> [_1 / k]_{<.w, v>. | (w e. X /\ v = (F` w))} = {<.w, v>. | [1 / k](w e. X /\ v = (F` w))})
63	61, 62	ax-mp 7	. . . . 5 |- [_1 / k]_{<.w, v>. | (w e. X /\ v = (F` w))} = {<.w, v>. | [1 / k](w e. X /\ v = (F` w))}
64		sbcang 2497	. . . . . . . 8 |- (1 e. NN -> ([1 / k](w e. X /\ v = (F` w)) <-> ([1 / k]w e. X /\ [1 / k]v = (F` w))))
65	61, 64	ax-mp 7	. . . . . . 7 |- ([1 / k](w e. X /\ v = (F` w)) <-> ([1 / k]w e. X /\ [1 / k]v = (F` w)))
66	61	elisseti 2301	. . . . . . . . 9 |- 1 e. _V
67		ax-17 1317	. . . . . . . . . 10 |- (w e. X -> A.k w e. X)
68	67	sbcgf 2520	. . . . . . . . 9 |- (1 e. _V -> ([1 / k]w e. X <-> w e. X))
69	66, 68	ax-mp 7	. . . . . . . 8 |- ([1 / k]w e. X <-> w e. X)
70		sbceq2dig 2559	. . . . . . . . . 10 |- (1 e. _V -> ([1 / k]v = (F` w) <-> v = [_1 / k]_(F` w)))
71	66, 70	ax-mp 7	. . . . . . . . 9 |- ([1 / k]v = (F` w) <-> v = [_1 / k]_(F` w))
72		1z 7368	. . . . . . . . . . 11 |- 1 e. ZZ
73	2	a1i 8	. . . . . . . . . . . 12 |- (k e. (1...1) -> (F` w) e. _V)
74	73	rgen 2159	. . . . . . . . . . 11 |- A.k e. (1...1)(F` w) e. _V
75		fsum1s 8269	. . . . . . . . . . 11 |- ((1 e. ZZ /\ A.k e. (1...1)(F` w) e. _V) -> sum_k e. (1...1)(F` w) = [_1 / k]_(F` w))
76	72, 74, 75	mp2an 761	. . . . . . . . . 10 |- sum_k e. (1...1)(F` w) = [_1 / k]_(F` w)
77	76	eqeq2i 1894	. . . . . . . . 9 |- (v = sum_k e. (1...1)(F` w) <-> v = [_1 / k]_(F` w))
78	71, 77	bitr4i 193	. . . . . . . 8 |- ([1 / k]v = (F` w) <-> v = sum_k e. (1...1)(F` w))
79	69, 78	anbi12i 540	. . . . . . 7 |- (([1 / k]w e. X /\ [1 / k]v = (F` w)) <-> (w e. X /\ v = sum_k e. (1...1)(F` w)))
80	65, 79	bitri 190	. . . . . 6 |- ([1 / k](w e. X /\ v = (F` w)) <-> (w e. X /\ v = sum_k e. (1...1)(F` w)))
81	80	opabbii 3402	. . . . 5 |- {<.w, v>. | [1 / k](w e. X /\ v = (F` w))} = {<.w, v>. | (w e. X /\ v = sum_k e. (1...1)(F` w))}
82	63, 81	eqtr2i 1909	. . . 4 |- {<.w, v>. | (w e. X /\ v = sum_k e. (1...1)(F` w))} = [_1 / k]_{<.w, v>. | (w e. X /\ v = (F` w))}
83	32, 34, 37, 38	metcnf 9162	. . . . . . . . . 10 |- ((C e. Met /\ D e. Met /\ F e. (J Cn K)) -> F:X-->CC)
84	35, 36, 83	mp3an12 1181	. . . . . . . . 9 |- (F e. (J Cn K) -> F:X-->CC)
85	28, 84	syl 12	. . . . . . . 8 |- (k e. NN -> F:X-->CC)
86		fopabfv 4804	. . . . . . . . 9 |- (F:X-->CC <-> (F = {<.w, v>. | (w e. X /\ v = (F` w))} /\ A.w e. X (F` w) e. CC))
87	86	simplbi 349	. . . . . . . 8 |- (F:X-->CC -> F = {<.w, v>. | (w e. X /\ v = (F` w))})
88	85, 87	syl 12	. . . . . . 7 |- (k e. NN -> F = {<.w, v>. | (w e. X /\ v = (F` w))})
89	88, 28	eqeltrrd 1972	. . . . . 6 |- (k e. NN -> {<.w, v>. | (w e. X /\ v = (F` w))} e. (J Cn K))
90	89	rgen 2159	. . . . 5 |- A.k e. NN {<.w, v>. | (w e. X /\ v = (F` w))} e. (J Cn K)
91		ra4csbela 2587	. . . . 5 |- ((1 e. NN /\ A.k e. NN {<.w, v>. | (w e. X /\ v = (F` w))} e. (J Cn K)) -> [_1 / k]_{<.w, v>. | (w e. X /\ v = (F` w))} e. (J Cn K))
92	61, 90, 91	mp2an 761	. . . 4 |- [_1 / k]_{<.w, v>. | (w e. X /\ v = (F` w))} e. (J Cn K)
93	82, 92	eqeltri 1967	. . 3 |- {<.w, v>. | (w e. X /\ v = sum_k e. (1...1)(F` w))} e. (J Cn K)
94		opreq2 4890	. . . . . . . 8 |- (j = 1 -> (1...j) = (1...1))
95	94	sumeq1d 8250	. . . . . . 7 |- (j = 1 -> sum_k e. (1...j)(F` w) = sum_k e. (1...1)(F` w))
96	95	eqeq2d 1895	. . . . . 6 |- (j = 1 -> (v = sum_k e. (1...j)(F` w) <-> v = sum_k e. (1...1)(F` w)))
97	96	anbi2d 678	. . . . 5 |- (j = 1 -> ((w e. X /\ v = sum_k e. (1...j)(F` w)) <-> (w e. X /\ v = sum_k e. (1...1)(F` w))))
98	97	opabbidv 3401	. . . 4 |- (j = 1 -> {<.w, v>. | (w e. X /\ v = sum_k e. (1...j)(F` w))} = {<.w, v>. | (w e. X /\ v = sum_k e. (1...1)(F` w))})
99	98	eleq1d 1963	. . 3 |- (j = 1 -> ({<.w, v>. | (w e. X /\ v = sum_k e. (1...j)(F` w))} e. (J Cn K) <-> {<.w, v>. | (w e. X /\ v = sum_k e. (1...1)(F` w))} e. (J Cn K)))
100		opreq2 4890	. . . . . . . 8 |- (j = m -> (1...j) = (1...m))
101	100	sumeq1d 8250	. . . . . . 7 |- (j = m -> sum_k e. (1...j)(F` w) = sum_k e. (1...m)(F` w))
102	101	eqeq2d 1895	. . . . . 6 |- (j = m -> (v = sum_k e. (1...j)(F` w) <-> v = sum_k e. (1...m)(F` w)))
103	102	anbi2d 678	. . . . 5 |- (j = m -> ((w e. X /\ v = sum_k e. (1...j)(F` w)) <-> (w e. X /\ v = sum_k e. (1...m)(F` w))))
104	103	opabbidv 3401	. . . 4 |- (j = m -> {<.w, v>. | (w e. X /\ v = sum_k e. (1...j)(F` w))} = {<.w, v>. | (w e. X /\ v = sum_k e. (1...m)(F` w))})
105	104	eleq1d 1963	. . 3 |- (j = m -> ({<.w, v>. | (w e. X /\ v = sum_k e. (1...j)(F` w))} e. (J Cn K) <-> {<.w, v>. | (w e. X /\ v = sum_k e. (1...m)(F` w))} e. (J Cn K)))
106		opreq2 4890	. . . . . . . 8 |- (j = (m + 1) -> (1...j) = (1...(m + 1)))
107	106	sumeq1d 8250	. . . . . . 7 |- (j = (m + 1) -> sum_k e. (1...j)(F` w) = sum_k e. (1...(m + 1))(F` w))
108	107	eqeq2d 1895	. . . . . 6 |- (j = (m + 1) -> (v = sum_k e. (1...j)(F` w) <-> v = sum_k e. (1...(m + 1))(F` w)))
109	108	anbi2d 678	. . . . 5 |- (j = (m + 1) -> ((w e. X /\ v = sum_k e. (1...j)(F` w)) <-> (w e. X /\ v = sum_k e. (1...(m + 1))(F` w))))
110	109	opabbidv 3401	. . . 4 |- (j = (m + 1) -> {<.w, v>. | (w e. X /\ v = sum_k e. (1...j)(F` w))} = {<.w, v>. | (w e. X /\ v = sum_k e. (1...(m + 1))(F` w))})
111	110	eleq1d 1963	. . 3 |- (j = (m + 1) -> ({<.w, v>. | (w e. X /\ v = sum_k e. (1...j)(F` w))} e. (J Cn K) <-> {<.w, v>. | (w e. X /\ v = sum_k e. (1...(m + 1))(F` w))} e. (J Cn K)))
112		opreq2 4890	. . . . . . . 8 |- (j = N -> (1...j) = (1...N))
113	112	sumeq1d 8250	. . . . . . 7 |- (j = N -> sum_k e. (1...j)(F` w) = sum_k e. (1...N)(F` w))
114	113	eqeq2d 1895	. . . . . 6 |- (j = N -> (v = sum_k e. (1...j)(F` w) <-> v = sum_k e. (1...N)(F` w)))
115	114	anbi2d 678	. . . . 5 |- (j = N -> ((w e. X /\ v = sum_k e. (1...j)(F` w)) <-> (w e. X /\ v = sum_k e. (1...N)(F` w))))
116	115	opabbidv 3401	. . . 4 |- (j = N -> {<.w, v>. | (w e. X /\ v = sum_k e. (1...j)(F` w))} = {<.w, v>. | (w e. X /\ v = sum_k e. (1...N)(F` w))})
117	116	eleq1d 1963	. . 3 |- (j = N -> ({<.w, v>. | (w e. X /\ v = sum_k e. (1...j)(F` w))} e. (J Cn K) <-> {<.w, v>. | (w e. X /\ v = sum_k e. (1...N)(F` w))} e. (J Cn K)))
118	60, 93, 99, 105, 111, 117	nnindALT 7121	. 2 |- (N e. NN -> {<.w, v>. | (w e. X /\ v = sum_k e. (1...N)(F` w))} e. (J Cn K))
119		fsumcn.13	. 2 |- G = {<.w, v>. | (w e. X /\ v = sum_k e. (1...N)(F` w))}
120	118, 119	syl5eqel 1975	1 |- (N e. NN -> G e. (J Cn K))

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   = wceq 1298   e. wcel 1300  [wsbc 1534  A.wral 2105  _Vcvv 2292  [_csb 2540  {cpr 3045  {copab 3395   X. cxp 3984  dom cdm 3986   o. ccom 3990  -->wf 3994  ` cfv 3998  (class class class)co 4884  {copab2 4885  1stc1st 5018  2ndc2nd 5019  supcsup 5663  CCcc 6384  RRcr 6385  1c1 6387   + caddc 6389   - cmin 6445  NNcn 6449  ZZcz 6451   < clt 6653  ZZ>=cuz 7586  ...cfz 7637  abscabs 8000  sum_csu 8239   Cn ccn 9028  Metcme 9066  Opencopn 9069

This theorem is referenced by:  fsumcn 9268

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-5 1302  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-rep 3428  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790  ax-reg 5695  ax-inf2 5731  ax-ac 5906

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3or 859  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-nel 2020  df-ral 2109  df-rex 2110  df-reu 2111  df-rab 2112  df-v 2294  df-sbc 2454  df-csb 2541  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-pss 2607  df-nul 2876  df-if 2983  df-pw 3035  df-sn 3049  df-pr 3050  df-tp 3052  df-op 3053  df-uni 3178  df-int 3215  df-iun 3257  df-iin 3258  df-br 3339  df-opab 3396  df-tr 3412  df-eprel 3583  df-id 3586  df-po 3591  df-so 3604  df-fr 3625  df-we 3644  df-ord 3660  df-on 3661  df-lim 3662  df-suc 3663  df-om 3950  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009  df-f 4010  df-f1 4011  df-fo 4012  df-f1o 4013  df-fv 4014  df-opr 4886  df-oprab 4887  df-mpt 5006  df-1st 5020  df-2nd 5021  df-iota 5089  df-rdg 5140  df-1o 5177  df-oadd 5179  df-omul 5180  df-er 5318  df-ec 5320  df-qs 5323  df-map 5383  df-en 5427  df-dom 5428  df-sdom 5429  df-undef 5556  df-riota 5560  df-sup 5664  df-r1 5750  df-rank 5751  df-ni 6152  df-pli 6153  df-mi 6154  df-lti 6155  df-plpq 6187  df-mpq 6188  df-enq 6189  df-nq 6190  df-plq 6191  df-mq 6192  df-rq 6193  df-ltq 6194  df-1q 6195  df-np 6238  df-1p 6239  df-plp 6240  df-mp 6241  df-ltp 6242  df-plpr 6316  df-mpr 6317  df-enr 6318  df-nr 6319  df-plr 6320  df-mr 6321  df-ltr 6322  df-0r 6323  df-1r 6324  df-m1r 6325  df-c 6392  df-0 6393  df-1 6394  df-i 6395  df-r 6396  df-plus 6397  df-mul 6398  df-lt 6399  df-sub 6511  df-neg 6513  df-pnf 6654  df-mnf 6655  df-xr 6656  df-ltxr 6657  df-le 6658  df-div 6892  df-n 7108  df-2 7154  df-n0 7309  df-z 7345  df-fl 7463  df-uz 7587  df-fz 7638  df-seq1 7721  df-shft 7754  df-seqz 7776  df-exp 7812  df-sqr 7920  df-re 8001  df-im 8002  df-cj 8003  df-abs 8004  df-clim 8235  df-sum 8240  df-top 8861  df-cn 9030  df-cnp 9031  df-met 9070  df-bl 9072  df-opn 9073  df-lm 9200

Copyright terms: Public domain