Theorem binomlem6 8331

Description: Lemma for binomi 8332 (binomial theorem). Do the final induction.

Hypotheses

Ref	Expression
binomlem.1	|- A e. CC
binomlem.2	|- B e. CC

Assertion

Ref	Expression
binomlem6	|- (N e. NN -> ((A + B)^N) = sum_k e. (0...N)((N _C k) x. ((A^(N - k)) x. (B^k))))

Distinct variable groups:   A,k   B,k   k,N

Proof of Theorem binomlem6

Step	Hyp	Ref	Expression
1		opreq2 4890	. . 3 |- (x = 1 -> ((A + B)^x) = ((A + B)^1))
2		opreq2 4890	. . . 4 |- (x = 1 -> (0...x) = (0...1))
3		opreq1 4889	. . . . . 6 |- (x = 1 -> (x _C k) = (1 _C k))
4		opreq1 4889	. . . . . . . 8 |- (x = 1 -> (x - k) = (1 - k))
5	4	opreq2d 4898	. . . . . . 7 |- (x = 1 -> (A^(x - k)) = (A^(1 - k)))
6	5	opreq1d 4897	. . . . . 6 |- (x = 1 -> ((A^(x - k)) x. (B^k)) = ((A^(1 - k)) x. (B^k)))
7	3, 6	opreq12d 4900	. . . . 5 |- (x = 1 -> ((x _C k) x. ((A^(x - k)) x. (B^k))) = ((1 _C k) x. ((A^(1 - k)) x. (B^k))))
8	7	adantr 425	. . . 4 |- ((x = 1 /\ k e. (0...1)) -> ((x _C k) x. ((A^(x - k)) x. (B^k))) = ((1 _C k) x. ((A^(1 - k)) x. (B^k))))
9	2, 8	sumeq12rdv 8256	. . 3 |- (x = 1 -> sum_k e. (0...x)((x _C k) x. ((A^(x - k)) x. (B^k))) = sum_k e. (0...1)((1 _C k) x. ((A^(1 - k)) x. (B^k))))
10	1, 9	eqeq12d 1899	. 2 |- (x = 1 -> (((A + B)^x) = sum_k e. (0...x)((x _C k) x. ((A^(x - k)) x. (B^k))) <-> ((A + B)^1) = sum_k e. (0...1)((1 _C k) x. ((A^(1 - k)) x. (B^k)))))
11		opreq2 4890	. . 3 |- (x = j -> ((A + B)^x) = ((A + B)^j))
12		opreq2 4890	. . . 4 |- (x = j -> (0...x) = (0...j))
13		opreq1 4889	. . . . . 6 |- (x = j -> (x _C k) = (j _C k))
14		opreq1 4889	. . . . . . . 8 |- (x = j -> (x - k) = (j - k))
15	14	opreq2d 4898	. . . . . . 7 |- (x = j -> (A^(x - k)) = (A^(j - k)))
16	15	opreq1d 4897	. . . . . 6 |- (x = j -> ((A^(x - k)) x. (B^k)) = ((A^(j - k)) x. (B^k)))
17	13, 16	opreq12d 4900	. . . . 5 |- (x = j -> ((x _C k) x. ((A^(x - k)) x. (B^k))) = ((j _C k) x. ((A^(j - k)) x. (B^k))))
18	17	adantr 425	. . . 4 |- ((x = j /\ k e. (0...j)) -> ((x _C k) x. ((A^(x - k)) x. (B^k))) = ((j _C k) x. ((A^(j - k)) x. (B^k))))
19	12, 18	sumeq12rdv 8256	. . 3 |- (x = j -> sum_k e. (0...x)((x _C k) x. ((A^(x - k)) x. (B^k))) = sum_k e. (0...j)((j _C k) x. ((A^(j - k)) x. (B^k))))
20	11, 19	eqeq12d 1899	. 2 |- (x = j -> (((A + B)^x) = sum_k e. (0...x)((x _C k) x. ((A^(x - k)) x. (B^k))) <-> ((A + B)^j) = sum_k e. (0...j)((j _C k) x. ((A^(j - k)) x. (B^k)))))
21		opreq2 4890	. . 3 |- (x = (j + 1) -> ((A + B)^x) = ((A + B)^(j + 1)))
22		opreq2 4890	. . . 4 |- (x = (j + 1) -> (0...x) = (0...(j + 1)))
23		opreq1 4889	. . . . . 6 |- (x = (j + 1) -> (x _C k) = ((j + 1) _C k))
24		opreq1 4889	. . . . . . . 8 |- (x = (j + 1) -> (x - k) = ((j + 1) - k))
25	24	opreq2d 4898	. . . . . . 7 |- (x = (j + 1) -> (A^(x - k)) = (A^((j + 1) - k)))
26	25	opreq1d 4897	. . . . . 6 |- (x = (j + 1) -> ((A^(x - k)) x. (B^k)) = ((A^((j + 1) - k)) x. (B^k)))
27	23, 26	opreq12d 4900	. . . . 5 |- (x = (j + 1) -> ((x _C k) x. ((A^(x - k)) x. (B^k))) = (((j + 1) _C k) x. ((A^((j + 1) - k)) x. (B^k))))
28	27	adantr 425	. . . 4 |- ((x = (j + 1) /\ k e. (0...(j + 1))) -> ((x _C k) x. ((A^(x - k)) x. (B^k))) = (((j + 1) _C k) x. ((A^((j + 1) - k)) x. (B^k))))
29	22, 28	sumeq12rdv 8256	. . 3 |- (x = (j + 1) -> sum_k e. (0...x)((x _C k) x. ((A^(x - k)) x. (B^k))) = sum_k e. (0...(j + 1))(((j + 1) _C k) x. ((A^((j + 1) - k)) x. (B^k))))
30	21, 29	eqeq12d 1899	. 2 |- (x = (j + 1) -> (((A + B)^x) = sum_k e. (0...x)((x _C k) x. ((A^(x - k)) x. (B^k))) <-> ((A + B)^(j + 1)) = sum_k e. (0...(j + 1))(((j + 1) _C k) x. ((A^((j + 1) - k)) x. (B^k)))))
31		opreq2 4890	. . 3 |- (x = N -> ((A + B)^x) = ((A + B)^N))
32		opreq2 4890	. . . 4 |- (x = N -> (0...x) = (0...N))
33		opreq1 4889	. . . . . 6 |- (x = N -> (x _C k) = (N _C k))
34		opreq1 4889	. . . . . . . 8 |- (x = N -> (x - k) = (N - k))
35	34	opreq2d 4898	. . . . . . 7 |- (x = N -> (A^(x - k)) = (A^(N - k)))
36	35	opreq1d 4897	. . . . . 6 |- (x = N -> ((A^(x - k)) x. (B^k)) = ((A^(N - k)) x. (B^k)))
37	33, 36	opreq12d 4900	. . . . 5 |- (x = N -> ((x _C k) x. ((A^(x - k)) x. (B^k))) = ((N _C k) x. ((A^(N - k)) x. (B^k))))
38	37	adantr 425	. . . 4 |- ((x = N /\ k e. (0...N)) -> ((x _C k) x. ((A^(x - k)) x. (B^k))) = ((N _C k) x. ((A^(N - k)) x. (B^k))))
39	32, 38	sumeq12rdv 8256	. . 3 |- (x = N -> sum_k e. (0...x)((x _C k) x. ((A^(x - k)) x. (B^k))) = sum_k e. (0...N)((N _C k) x. ((A^(N - k)) x. (B^k))))
40	31, 39	eqeq12d 1899	. 2 |- (x = N -> (((A + B)^x) = sum_k e. (0...x)((x _C k) x. ((A^(x - k)) x. (B^k))) <-> ((A + B)^N) = sum_k e. (0...N)((N _C k) x. ((A^(N - k)) x. (B^k)))))
41		binomlem.1	. . . . 5 |- A e. CC
42		binomlem.2	. . . . 5 |- B e. CC
43	41, 42	addcli 6473	. . . 4 |- (A + B) e. CC
44		exp1 7816	. . . 4 |- ((A + B) e. CC -> ((A + B)^1) = (A + B))
45	43, 44	ax-mp 7	. . 3 |- ((A + B)^1) = (A + B)
46		0z 7355	. . . . . 6 |- 0 e. ZZ
47		uzid 7596	. . . . . 6 |- (0 e. ZZ -> 0 e. (ZZ>=` 0))
48	46, 47	ax-mp 7	. . . . 5 |- 0 e. (ZZ>=` 0)
49		oprex 4907	. . . . . 6 |- ((1 _C k) x. ((A^(1 - k)) x. (B^k))) e. _V
50	42	elisseti 2301	. . . . . 6 |- B e. _V
51		ax1cn 6422	. . . . . . . . 9 |- 1 e. CC
52	51	addid2i 6484	. . . . . . . 8 |- (0 + 1) = 1
53	52	eqeq2i 1894	. . . . . . 7 |- (k = (0 + 1) <-> k = 1)
54		opreq2 4890	. . . . . . . . . 10 |- (k = 1 -> (1 _C k) = (1 _C 1))
55		1nn0 7323	. . . . . . . . . . 11 |- 1 e. NN0
56		bcnn 8216	. . . . . . . . . . 11 |- (1 e. NN0 -> (1 _C 1) = 1)
57	55, 56	ax-mp 7	. . . . . . . . . 10 |- (1 _C 1) = 1
58	54, 57	syl6eq 1944	. . . . . . . . 9 |- (k = 1 -> (1 _C k) = 1)
59		opreq2 4890	. . . . . . . . . . . . . 14 |- (k = 1 -> (1 - k) = (1 - 1))
60	51	subidi 6551	. . . . . . . . . . . . . 14 |- (1 - 1) = 0
61	59, 60	syl6eq 1944	. . . . . . . . . . . . 13 |- (k = 1 -> (1 - k) = 0)
62	61	opreq2d 4898	. . . . . . . . . . . 12 |- (k = 1 -> (A^(1 - k)) = (A^0))
63		exp0 7814	. . . . . . . . . . . . 13 |- (A e. CC -> (A^0) = 1)
64	41, 63	ax-mp 7	. . . . . . . . . . . 12 |- (A^0) = 1
65	62, 64	syl6eq 1944	. . . . . . . . . . 11 |- (k = 1 -> (A^(1 - k)) = 1)
66		opreq2 4890	. . . . . . . . . . . 12 |- (k = 1 -> (B^k) = (B^1))
67		exp1 7816	. . . . . . . . . . . . 13 |- (B e. CC -> (B^1) = B)
68	42, 67	ax-mp 7	. . . . . . . . . . . 12 |- (B^1) = B
69	66, 68	syl6eq 1944	. . . . . . . . . . 11 |- (k = 1 -> (B^k) = B)
70	65, 69	opreq12d 4900	. . . . . . . . . 10 |- (k = 1 -> ((A^(1 - k)) x. (B^k)) = (1 x. B))
71	42	mulid2i 6486	. . . . . . . . . 10 |- (1 x. B) = B
72	70, 71	syl6eq 1944	. . . . . . . . 9 |- (k = 1 -> ((A^(1 - k)) x. (B^k)) = B)
73	58, 72	opreq12d 4900	. . . . . . . 8 |- (k = 1 -> ((1 _C k) x. ((A^(1 - k)) x. (B^k))) = (1 x. B))
74	73, 71	syl6eq 1944	. . . . . . 7 |- (k = 1 -> ((1 _C k) x. ((A^(1 - k)) x. (B^k))) = B)
75	53, 74	sylbi 216	. . . . . 6 |- (k = (0 + 1) -> ((1 _C k) x. ((A^(1 - k)) x. (B^k))) = B)
76	49, 50, 75	fsump1i 8266	. . . . 5 |- (0 e. (ZZ>=` 0) -> sum_k e. (0...(0 + 1))((1 _C k) x. ((A^(1 - k)) x. (B^k))) = (sum_k e. (0...0)((1 _C k) x. ((A^(1 - k)) x. (B^k))) + B))
77	48, 76	ax-mp 7	. . . 4 |- sum_k e. (0...(0 + 1))((1 _C k) x. ((A^(1 - k)) x. (B^k))) = (sum_k e. (0...0)((1 _C k) x. ((A^(1 - k)) x. (B^k))) + B)
78		opreq2 4890	. . . . . . . . . 10 |- (k = 0 -> (1 _C k) = (1 _C 0))
79		bcn0 8215	. . . . . . . . . . 11 |- (1 e. NN0 -> (1 _C 0) = 1)
80	55, 79	ax-mp 7	. . . . . . . . . 10 |- (1 _C 0) = 1
81	78, 80	syl6eq 1944	. . . . . . . . 9 |- (k = 0 -> (1 _C k) = 1)
82		opreq2 4890	. . . . . . . . . . . . . 14 |- (k = 0 -> (1 - k) = (1 - 0))
83	51	subid1i 6552	. . . . . . . . . . . . . 14 |- (1 - 0) = 1
84	82, 83	syl6eq 1944	. . . . . . . . . . . . 13 |- (k = 0 -> (1 - k) = 1)
85	84	opreq2d 4898	. . . . . . . . . . . 12 |- (k = 0 -> (A^(1 - k)) = (A^1))
86		exp1 7816	. . . . . . . . . . . . 13 |- (A e. CC -> (A^1) = A)
87	41, 86	ax-mp 7	. . . . . . . . . . . 12 |- (A^1) = A
88	85, 87	syl6eq 1944	. . . . . . . . . . 11 |- (k = 0 -> (A^(1 - k)) = A)
89		opreq2 4890	. . . . . . . . . . . 12 |- (k = 0 -> (B^k) = (B^0))
90		exp0 7814	. . . . . . . . . . . . 13 |- (B e. CC -> (B^0) = 1)
91	42, 90	ax-mp 7	. . . . . . . . . . . 12 |- (B^0) = 1
92	89, 91	syl6eq 1944	. . . . . . . . . . 11 |- (k = 0 -> (B^k) = 1)
93	88, 92	opreq12d 4900	. . . . . . . . . 10 |- (k = 0 -> ((A^(1 - k)) x. (B^k)) = (A x. 1))
94	41	mulid1i 6485	. . . . . . . . . 10 |- (A x. 1) = A
95	93, 94	syl6eq 1944	. . . . . . . . 9 |- (k = 0 -> ((A^(1 - k)) x. (B^k)) = A)
96	81, 95	opreq12d 4900	. . . . . . . 8 |- (k = 0 -> ((1 _C k) x. ((A^(1 - k)) x. (B^k))) = (1 x. A))
97	41	mulid2i 6486	. . . . . . . 8 |- (1 x. A) = A
98	96, 97	syl6eq 1944	. . . . . . 7 |- (k = 0 -> ((1 _C k) x. ((A^(1 - k)) x. (B^k))) = A)
99	98	fsum1i 8265	. . . . . 6 |- ((A e. CC /\ 0 e. ZZ) -> sum_k e. (0...0)((1 _C k) x. ((A^(1 - k)) x. (B^k))) = A)
100	41, 46, 99	mp2an 761	. . . . 5 |- sum_k e. (0...0)((1 _C k) x. ((A^(1 - k)) x. (B^k))) = A
101	100	opreq1i 4892	. . . 4 |- (sum_k e. (0...0)((1 _C k) x. ((A^(1 - k)) x. (B^k))) + B) = (A + B)
102	77, 101	eqtr2i 1909	. . 3 |- (A + B) = sum_k e. (0...(0 + 1))((1 _C k) x. ((A^(1 - k)) x. (B^k)))
103	52	opreq2i 4893	. . . 4 |- (0...(0 + 1)) = (0...1)
104	103	sumeq1i 8247	. . 3 |- sum_k e. (0...(0 + 1))((1 _C k) x. ((A^(1 - k)) x. (B^k))) = sum_k e. (0...1)((1 _C k) x. ((A^(1 - k)) x. (B^k)))
105	45, 102, 104	3eqtri 1912	. 2 |- ((A + B)^1) = sum_k e. (0...1)((1 _C k) x. ((A^(1 - k)) x. (B^k)))
106		expp1 7817	. . . . . 6 |- (((A + B) e. CC /\ j e. NN0) -> ((A + B)^(j + 1)) = (((A + B)^j) x. (A + B)))
107		nnnn0 7315	. . . . . 6 |- (j e. NN -> j e. NN0)
108	106, 43, 107	sylancr 526	. . . . 5 |- (j e. NN -> ((A + B)^(j + 1)) = (((A + B)^j) x. (A + B)))
109	108	adantr 425	. . . 4 |- ((j e. NN /\ ((A + B)^j) = sum_k e. (0...j)((j _C k) x. ((A^(j - k)) x. (B^k)))) -> ((A + B)^(j + 1)) = (((A + B)^j) x. (A + B)))
110		simpr 350	. . . . 5 |- ((j e. NN /\ ((A + B)^j) = sum_k e. (0...j)((j _C k) x. ((A^(j - k)) x. (B^k)))) -> ((A + B)^j) = sum_k e. (0...j)((j _C k) x. ((A^(j - k)) x. (B^k))))
111	110	opreq1d 4897	. . . 4 |- ((j e. NN /\ ((A + B)^j) = sum_k e. (0...j)((j _C k) x. ((A^(j - k)) x. (B^k)))) -> (((A + B)^j) x. (A + B)) = (sum_k e. (0...j)((j _C k) x. ((A^(j - k)) x. (B^k))) x. (A + B)))
112	41, 42	binomlem5 8330	. . . . 5 |- (j e. NN -> (sum_k e. (0...j)((j _C k) x. ((A^(j - k)) x. (B^k))) x. (A + B)) = sum_k e. (0...(j + 1))(((j + 1) _C k) x. ((A^((j + 1) - k)) x. (B^k))))
113	112	adantr 425	. . . 4 |- ((j e. NN /\ ((A + B)^j) = sum_k e. (0...j)((j _C k) x. ((A^(j - k)) x. (B^k)))) -> (sum_k e. (0...j)((j _C k) x. ((A^(j - k)) x. (B^k))) x. (A + B)) = sum_k e. (0...(j + 1))(((j + 1) _C k) x. ((A^((j + 1) - k)) x. (B^k))))
114	109, 111, 113	3eqtrd 1929	. . 3 |- ((j e. NN /\ ((A + B)^j) = sum_k e. (0...j)((j _C k) x. ((A^(j - k)) x. (B^k)))) -> ((A + B)^(j + 1)) = sum_k e. (0...(j + 1))(((j + 1) _C k) x. ((A^((j + 1) - k)) x. (B^k))))
115	114	ex 402	. 2 |- (j e. NN -> (((A + B)^j) = sum_k e. (0...j)((j _C k) x. ((A^(j - k)) x. (B^k))) -> ((A + B)^(j + 1)) = sum_k e. (0...(j + 1))(((j + 1) _C k) x. ((A^((j + 1) - k)) x. (B^k)))))
116	10, 20, 30, 40, 105, 115	nnind 7120	1 |- (N e. NN -> ((A + B)^N) = sum_k e. (0...N)((N _C k) x. ((A^(N - k)) x. (B^k))))

Syntax hints:   -> wi 3   /\ wa 240   = wceq 1298   e. wcel 1300  ` cfv 3998  (class class class)co 4884  CCcc 6384  0cc0 6386  1c1 6387   + caddc 6389   x. cmul 6391   - cmin 6445  NNcn 6449  NN0cn0 6450  ZZcz 6451  ZZ>=cuz 7586  ...cfz 7637  ^cexp 7811   _C cbc 8208  sum_csu 8239

This theorem is referenced by:  binomi 8332

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-inf2 5731

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-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-en 5427  df-dom 5428  df-sdom 5429  df-undef 5556  df-riota 5560  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-n0 7309  df-z 7345  df-uz 7587  df-fz 7638  df-seq1 7721  df-shft 7754  df-seqz 7776  df-exp 7812  df-fac 8184  df-bc 8209  df-sum 8240

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