Theorem binomlem6 7274

Description: Lemma for binomi 7275 (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 4027	. . 3 |- (x = 1 -> ((A + B)^x) = ((A + B)^1))
2		opreq2 4027	. . . 4 |- (x = 1 -> (0...x) = (0...1))
3		opreq1 4026	. . . . . 6 |- (x = 1 -> (x C. k) = (1 C. k))
4		opreq1 4026	. . . . . . . 8 |- (x = 1 -> (x - k) = (1 - k))
5	4	opreq2d 4034	. . . . . . 7 |- (x = 1 -> (A^(x - k)) = (A^(1 - k)))
6	5	opreq1d 4033	. . . . . 6 |- (x = 1 -> ((A^(x - k)) x. (B^k)) = ((A^(1 - k)) x. (B^k)))
7	3, 6	opreq12d 4036	. . . . 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 389	. . . 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 7199	. . 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 1532	. 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 4027	. . 3 |- (x = j -> ((A + B)^x) = ((A + B)^j))
12		opreq2 4027	. . . 4 |- (x = j -> (0...x) = (0...j))
13		opreq1 4026	. . . . . 6 |- (x = j -> (x C. k) = (j C. k))
14		opreq1 4026	. . . . . . . 8 |- (x = j -> (x - k) = (j - k))
15	14	opreq2d 4034	. . . . . . 7 |- (x = j -> (A^(x - k)) = (A^(j - k)))
16	15	opreq1d 4033	. . . . . 6 |- (x = j -> ((A^(x - k)) x. (B^k)) = ((A^(j - k)) x. (B^k)))
17	13, 16	opreq12d 4036	. . . . 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 389	. . . 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 7199	. . 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 1532	. 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 4027	. . 3 |- (x = (j + 1) -> ((A + B)^x) = ((A + B)^(j + 1)))
22		opreq2 4027	. . . 4 |- (x = (j + 1) -> (0...x) = (0...(j + 1)))
23		opreq1 4026	. . . . . 6 |- (x = (j + 1) -> (x C. k) = ((j + 1) C. k))
24		opreq1 4026	. . . . . . . 8 |- (x = (j + 1) -> (x - k) = ((j + 1) - k))
25	24	opreq2d 4034	. . . . . . 7 |- (x = (j + 1) -> (A^(x - k)) = (A^((j + 1) - k)))
26	25	opreq1d 4033	. . . . . 6 |- (x = (j + 1) -> ((A^(x - k)) x. (B^k)) = ((A^((j + 1) - k)) x. (B^k)))
27	23, 26	opreq12d 4036	. . . . 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 389	. . . 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 7199	. . 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 1532	. 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 4027	. . 3 |- (x = N -> ((A + B)^x) = ((A + B)^N))
32		opreq2 4027	. . . 4 |- (x = N -> (0...x) = (0...N))
33		opreq1 4026	. . . . . 6 |- (x = N -> (x C. k) = (N C. k))
34		opreq1 4026	. . . . . . . 8 |- (x = N -> (x - k) = (N - k))
35	34	opreq2d 4034	. . . . . . 7 |- (x = N -> (A^(x - k)) = (A^(N - k)))
36	35	opreq1d 4033	. . . . . 6 |- (x = N -> ((A^(x - k)) x. (B^k)) = ((A^(N - k)) x. (B^k)))
37	33, 36	opreq12d 4036	. . . . 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 389	. . . 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 7199	. . 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 1532	. 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 5474	. . . 4 |- (A + B) e. CC
44		exp1 6768	. . . 4 |- ((A + B) e. CC -> ((A + B)^1) = (A + B))
45	43, 44	ax-mp 7	. . 3 |- ((A + B)^1) = (A + B)
46		0z 6314	. . . . . 6 |- 0 e. ZZ
47		uzid 6554	. . . . . 6 |- (0 e. ZZ -> 0 e. (ZZ>=` 0))
48	46, 47	ax-mp 7	. . . . 5 |- 0 e. (ZZ>=` 0)
49		oprex 4041	. . . . . 6 |- ((1 C. k) x. ((A^(1 - k)) x. (B^k))) e. V
50	42	elisseti 1864	. . . . . 6 |- B e. V
51		ax1cn 5423	. . . . . . . . 9 |- 1 e. CC
52	51	addid2i 5485	. . . . . . . 8 |- (0 + 1) = 1
53	52	eqeq2i 1528	. . . . . . 7 |- (k = (0 + 1) <-> k = 1)
54		opreq2 4027	. . . . . . . . . 10 |- (k = 1 -> (1 C. k) = (1 C. 1))
55		1nn0 6282	. . . . . . . . . . 11 |- 1 e. NN0
56		bcnn 7167	. . . . . . . . . . 11 |- (1 e. NN0 -> (1 C. 1) = 1)
57	55, 56	ax-mp 7	. . . . . . . . . 10 |- (1 C. 1) = 1
58	54, 57	syl6eq 1566	. . . . . . . . 9 |- (k = 1 -> (1 C. k) = 1)
59		opreq2 4027	. . . . . . . . . . . . . 14 |- (k = 1 -> (1 - k) = (1 - 1))
60	51	subidi 5545	. . . . . . . . . . . . . 14 |- (1 - 1) = 0
61	59, 60	syl6eq 1566	. . . . . . . . . . . . 13 |- (k = 1 -> (1 - k) = 0)
62	61	opreq2d 4034	. . . . . . . . . . . 12 |- (k = 1 -> (A^(1 - k)) = (A^0))
63		exp0 6766	. . . . . . . . . . . . 13 |- (A e. CC -> (A^0) = 1)
64	41, 63	ax-mp 7	. . . . . . . . . . . 12 |- (A^0) = 1
65	62, 64	syl6eq 1566	. . . . . . . . . . 11 |- (k = 1 -> (A^(1 - k)) = 1)
66		opreq2 4027	. . . . . . . . . . . 12 |- (k = 1 -> (B^k) = (B^1))
67		exp1 6768	. . . . . . . . . . . . 13 |- (B e. CC -> (B^1) = B)
68	42, 67	ax-mp 7	. . . . . . . . . . . 12 |- (B^1) = B
69	66, 68	syl6eq 1566	. . . . . . . . . . 11 |- (k = 1 -> (B^k) = B)
70	65, 69	opreq12d 4036	. . . . . . . . . 10 |- (k = 1 -> ((A^(1 - k)) x. (B^k)) = (1 x. B))
71	42	mulid2i 5487	. . . . . . . . . 10 |- (1 x. B) = B
72	70, 71	syl6eq 1566	. . . . . . . . 9 |- (k = 1 -> ((A^(1 - k)) x. (B^k)) = B)
73	58, 72	opreq12d 4036	. . . . . . . 8 |- (k = 1 -> ((1 C. k) x. ((A^(1 - k)) x. (B^k))) = (1 x. B))
74	73, 71	syl6eq 1566	. . . . . . 7 |- (k = 1 -> ((1 C. k) x. ((A^(1 - k)) x. (B^k))) = B)
75	53, 74	sylbi 197	. . . . . 6 |- (k = (0 + 1) -> ((1 C. k) x. ((A^(1 - k)) x. (B^k))) = B)
76	49, 50, 75	fsump1i 7209	. . . . 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 4027	. . . . . . . . . 10 |- (k = 0 -> (1 C. k) = (1 C. 0))
79		bcn0 7166	. . . . . . . . . . 11 |- (1 e. NN0 -> (1 C. 0) = 1)
80	55, 79	ax-mp 7	. . . . . . . . . 10 |- (1 C. 0) = 1
81	78, 80	syl6eq 1566	. . . . . . . . 9 |- (k = 0 -> (1 C. k) = 1)
82		opreq2 4027	. . . . . . . . . . . . . 14 |- (k = 0 -> (1 - k) = (1 - 0))
83	51	subid1i 5546	. . . . . . . . . . . . . 14 |- (1 - 0) = 1
84	82, 83	syl6eq 1566	. . . . . . . . . . . . 13 |- (k = 0 -> (1 - k) = 1)
85	84	opreq2d 4034	. . . . . . . . . . . 12 |- (k = 0 -> (A^(1 - k)) = (A^1))
86		exp1 6768	. . . . . . . . . . . . 13 |- (A e. CC -> (A^1) = A)
87	41, 86	ax-mp 7	. . . . . . . . . . . 12 |- (A^1) = A
88	85, 87	syl6eq 1566	. . . . . . . . . . 11 |- (k = 0 -> (A^(1 - k)) = A)
89		opreq2 4027	. . . . . . . . . . . 12 |- (k = 0 -> (B^k) = (B^0))
90		exp0 6766	. . . . . . . . . . . . 13 |- (B e. CC -> (B^0) = 1)
91	42, 90	ax-mp 7	. . . . . . . . . . . 12 |- (B^0) = 1
92	89, 91	syl6eq 1566	. . . . . . . . . . 11 |- (k = 0 -> (B^k) = 1)
93	88, 92	opreq12d 4036	. . . . . . . . . 10 |- (k = 0 -> ((A^(1 - k)) x. (B^k)) = (A x. 1))
94	41	mulid1i 5486	. . . . . . . . . 10 |- (A x. 1) = A
95	93, 94	syl6eq 1566	. . . . . . . . 9 |- (k = 0 -> ((A^(1 - k)) x. (B^k)) = A)
96	81, 95	opreq12d 4036	. . . . . . . 8 |- (k = 0 -> ((1 C. k) x. ((A^(1 - k)) x. (B^k))) = (1 x. A))
97	41	mulid2i 5487	. . . . . . . 8 |- (1 x. A) = A
98	96, 97	syl6eq 1566	. . . . . . 7 |- (k = 0 -> ((1 C. k) x. ((A^(1 - k)) x. (B^k))) = A)
99	98	fsum1i 7208	. . . . . 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 701	. . . . 5 |- sum_k e. (0...0)((1 C. k) x. ((A^(1 - k)) x. (B^k))) = A
101	100	opreq1i 4029	. . . 4 |- (sum_k e. (0...0)((1 C. k) x. ((A^(1 - k)) x. (B^k))) + B) = (A + B)
102	77, 101	eqtr2i 1539	. . 3 |- (A + B) = sum_k e. (0...(0 + 1))((1 C. k) x. ((A^(1 - k)) x. (B^k)))
103	52	opreq2i 4030	. . . 4 |- (0...(0 + 1)) = (0...1)
104	103	sumeq1i 7190	. . 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 1542	. 2 |- ((A + B)^1) = sum_k e. (0...1)((1 C. k) x. ((A^(1 - k)) x. (B^k)))
106		expp1 6769	. . . . . . 7 |- (((A + B) e. CC /\ j e. NN0) -> ((A + B)^(j + 1)) = (((A + B)^j) x. (A + B)))
107		nnnn0 6274	. . . . . . 7 |- (j e. NN -> j e. NN0)
108	106, 107	sylan2 453	. . . . . 6 |- (((A + B) e. CC /\ j e. NN) -> ((A + B)^(j + 1)) = (((A + B)^j) x. (A + B)))
109	43, 108	mpan 699	. . . . 5 |- (j e. NN -> ((A + B)^(j + 1)) = (((A + B)^j) x. (A + B)))
110	109	adantr 389	. . . 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)))
111		pm3.27 321	. . . . 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))))
112	111	opreq1d 4033	. . . 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)))
113	41, 42	binomlem5 7273	. . . . 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))))
114	113	adantr 389	. . . 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))))
115	110, 112, 114	3eqtrd 1554	. . 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))))
116	115	ex 371	. 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)))))
117	10, 20, 30, 40, 105, 116	nnind 6082	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 221   = wceq 992   e. wcel 994  ` cfv 3263  (class class class)co 4021  CCcc 5386  0cc0 5388  1c1 5389   + caddc 5391   x. cmul 5393   - cmin 5446  NNcn 5450  NN0cn0 5451  ZZcz 5452  ZZ>=cuz 6544  ...cfz 6595  ^cexp 6763   C. cbc 7159  sum_csu 7182

This theorem is referenced by:  binomi 7275

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 998  ax-gen 999  ax-8 1000  ax-9 1001  ax-10 1002  ax-11 1003  ax-12 1004  ax-13 1005  ax-14 1006  ax-17 1007  ax-4 1009  ax-5o 1011  ax-6o 1014  ax-9o 1159  ax-10o 1177  ax-16 1247  ax-11o 1255  ax-ext 1500  ax-rep 2767  ax-sep 2777  ax-nul 2784  ax-pow 2818  ax-pr 2855  ax-un 3089  ax-inf2 4770

This theorem depends on definitions:  df-bi 145  df-or 222  df-an 223  df-3or 782  df-3an 783  df-ex 1017  df-sb 1209  df-eu 1421  df-mo 1422  df-clab 1506  df-cleq 1511  df-clel 1514  df-ne 1630  df-nel 1631  df-ral 1695  df-rex 1696  df-reu 1697  df-rab 1698  df-v 1858  df-sbc 1987  df-csb 2052  df-dif 2101  df-un 2102  df-in 2103  df-ss 2105  df-pss 2107  df-nul 2333  df-if 2416  df-pw 2459  df-sn 2470  df-pr 2471  df-tp 2473  df-op 2474  df-uni 2570  df-int 2601  df-iun 2635  df-br 2693  df-opab 2741  df-tr 2755  df-eprel 2910  df-id 2913  df-po 2918  df-so 2929  df-fr 2947  df-we 2962  df-ord 2978  df-on 2979  df-lim 2980  df-suc 2981  df-om 3219  df-xp 3265  df-rel 3266  df-cnv 3267  df-co 3268  df-dm 3269  df-rn 3270  df-res 3271  df-ima 3272  df-fun 3273  df-fn 3274  df-f 3275  df-f1 3276  df-fo 3277  df-f1o 3278  df-fv 3279  df-opr 4023  df-oprab 4024  df-1st 4140  df-2nd 4141  df-rdg 4233  df-1o 4269  df-oadd 4271  df-omul 4272  df-er 4401  df-ec 4403  df-qs 4406  df-en 4509  df-dom 4510  df-sdom 4511  df-ni 5154  df-pli 5155  df-mi 5156  df-lti 5157  df-plpq 5189  df-mpq 5190  df-enq 5191  df-nq 5192  df-plq 5193  df-mq 5194  df-rq 5195  df-ltq 5196  df-1q 5197  df-np 5240  df-1p 5241  df-plp 5242  df-mp 5243  df-ltp 5244  df-plpr 5318  df-mpr 5319  df-enr 5320  df-nr 5321  df-plr 5322  df-mr 5323  df-ltr 5324  df-0r 5325  df-1r 5326  df-m1r 5327  df-c 5394  df-0 5395  df-1 5396  df-i 5397  df-r 5398  df-plus 5399  df-mul 5400  df-lt 5401  df-sub 5510  df-neg 5512  df-pnf 5641  df-mnf 5642  df-xr 5643  df-ltxr 5644  df-le 5645  df-div 5855  df-n 6070  df-n0 6268  df-z 6304  df-uz 6545  df-fz 6596  df-seq1 6673  df-shft 6706  df-seqz 6728  df-exp 6764  df-fac 7135  df-bc 7160  df-sum 7183

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