Theorem binom 7040

Description: The binomial theorem: (A + B)^N is the sum from k = 0 to N of (N C. k) x. ((A^k) x. (B^(N - k)). Theorem 15-2.8 of [Gleason] p. 296. This final piece of the proof combines the 0 < N case of binomlem6 7039 with the N = 0 case.

Hypotheses

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

Assertion

Ref	Expression
binom	|- (N e. NN0 -> ((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 binom

Step	Hyp	Ref	Expression
1		elnn0 6062	. 2 |- (N e. NN0 <-> (N e. NN \/ N = 0))
2		binomlem.1	. . . 4 |- A e. CC
3		binomlem.2	. . . 4 |- B e. CC
4	2, 3	binomlem6 7039	. . 3 |- (N e. NN -> ((A + B)^N) = sum_k e. (0...N)((N C. k) x. ((A^(N - k)) x. (B^k))))
5		ax1cn 5256	. . . . . 6 |- 1 e. CC
6		0z 6107	. . . . . 6 |- 0 e. ZZ
7		opreq2 3966	. . . . . . . . . 10 |- (k = 0 -> (0 C. k) = (0 C. 0))
8		0nn0 6074	. . . . . . . . . . 11 |- 0 e. NN0
9		bcn0t 6931	. . . . . . . . . . 11 |- (0 e. NN0 -> (0 C. 0) = 1)
10	8, 9	ax-mp 7	. . . . . . . . . 10 |- (0 C. 0) = 1
11	7, 10	syl6eq 1522	. . . . . . . . 9 |- (k = 0 -> (0 C. k) = 1)
12		opreq2 3966	. . . . . . . . . . . . . 14 |- (k = 0 -> (0 - k) = (0 - 0))
13		0cn 5315	. . . . . . . . . . . . . . 15 |- 0 e. CC
14	13	subid 5378	. . . . . . . . . . . . . 14 |- (0 - 0) = 0
15	12, 14	syl6eq 1522	. . . . . . . . . . . . 13 |- (k = 0 -> (0 - k) = 0)
16	15	opreq2d 3973	. . . . . . . . . . . 12 |- (k = 0 -> (A^(0 - k)) = (A^0))
17		exp0t 6521	. . . . . . . . . . . . 13 |- (A e. CC -> (A^0) = 1)
18	2, 17	ax-mp 7	. . . . . . . . . . . 12 |- (A^0) = 1
19	16, 18	syl6eq 1522	. . . . . . . . . . 11 |- (k = 0 -> (A^(0 - k)) = 1)
20		opreq2 3966	. . . . . . . . . . . 12 |- (k = 0 -> (B^k) = (B^0))
21		exp0t 6521	. . . . . . . . . . . . 13 |- (B e. CC -> (B^0) = 1)
22	3, 21	ax-mp 7	. . . . . . . . . . . 12 |- (B^0) = 1
23	20, 22	syl6eq 1522	. . . . . . . . . . 11 |- (k = 0 -> (B^k) = 1)
24	19, 23	opreq12d 3975	. . . . . . . . . 10 |- (k = 0 -> ((A^(0 - k)) x. (B^k)) = (1 x. 1))
25	5	mulid1 5319	. . . . . . . . . 10 |- (1 x. 1) = 1
26	24, 25	syl6eq 1522	. . . . . . . . 9 |- (k = 0 -> ((A^(0 - k)) x. (B^k)) = 1)
27	11, 26	opreq12d 3975	. . . . . . . 8 |- (k = 0 -> ((0 C. k) x. ((A^(0 - k)) x. (B^k))) = (1 x. 1))
28	27, 25	syl6eq 1522	. . . . . . 7 |- (k = 0 -> ((0 C. k) x. ((A^(0 - k)) x. (B^k))) = 1)
29	28	fsum1 6973	. . . . . 6 |- ((1 e. CC /\ 0 e. ZZ) -> sum_k e. (0...0)((0 C. k) x. ((A^(0 - k)) x. (B^k))) = 1)
30	5, 6, 29	mp2an 696	. . . . 5 |- sum_k e. (0...0)((0 C. k) x. ((A^(0 - k)) x. (B^k))) = 1
31	30	eqcomi 1478	. . . 4 |- 1 = sum_k e. (0...0)((0 C. k) x. ((A^(0 - k)) x. (B^k)))
32		opreq2 3966	. . . . 5 |- (N = 0 -> ((A + B)^N) = ((A + B)^0))
33	2, 3	addcl 5307	. . . . . 6 |- (A + B) e. CC
34		exp0t 6521	. . . . . 6 |- ((A + B) e. CC -> ((A + B)^0) = 1)
35	33, 34	ax-mp 7	. . . . 5 |- ((A + B)^0) = 1
36	32, 35	syl6eq 1522	. . . 4 |- (N = 0 -> ((A + B)^N) = 1)
37		opreq2 3966	. . . . 5 |- (N = 0 -> (0...N) = (0...0))
38		opreq1 3965	. . . . . . 7 |- (N = 0 -> (N C. k) = (0 C. k))
39		opreq1 3965	. . . . . . . . 9 |- (N = 0 -> (N - k) = (0 - k))
40	39	opreq2d 3973	. . . . . . . 8 |- (N = 0 -> (A^(N - k)) = (A^(0 - k)))
41	40	opreq1d 3972	. . . . . . 7 |- (N = 0 -> ((A^(N - k)) x. (B^k)) = ((A^(0 - k)) x. (B^k)))
42	38, 41	opreq12d 3975	. . . . . 6 |- (N = 0 -> ((N C. k) x. ((A^(N - k)) x. (B^k))) = ((0 C. k) x. ((A^(0 - k)) x. (B^k))))
43	42	adantr 389	. . . . 5 |- ((N = 0 /\ k e. (0...0)) -> ((N C. k) x. ((A^(N - k)) x. (B^k))) = ((0 C. k) x. ((A^(0 - k)) x. (B^k))))
44	37, 43	sumeq12rdv 6964	. . . 4 |- (N = 0 -> sum_k e. (0...N)((N C. k) x. ((A^(N - k)) x. (B^k))) = sum_k e. (0...0)((0 C. k) x. ((A^(0 - k)) x. (B^k))))
45	31, 36, 44	3eqtr4a 1531	. . 3 |- (N = 0 -> ((A + B)^N) = sum_k e. (0...N)((N C. k) x. ((A^(N - k)) x. (B^k))))
46	4, 45	jaoi 341	. 2 |- ((N e. NN \/ N = 0) -> ((A + B)^N) = sum_k e. (0...N)((N C. k) x. ((A^(N - k)) x. (B^k))))
47	1, 46	sylbi 199	1 |- (N e. NN0 -> ((A + B)^N) = sum_k e. (0...N)((N C. k) x. ((A^(N - k)) x. (B^k))))

Syntax hints:   -> wi 3   \/ wo 222   = wceq 955   e. wcel 957  (class class class)co 3960  CCcc 5219  0cc0 5221  1c1 5222   + caddc 5224   x. cmul 5226   - cmin 5279  NNcn 5283  NN0cn0 5284  ZZcz 5285  ...cfz 6417  ^cexp 6518   C. cbc 6922  sum_csu 6947

This theorem is referenced by:  binom1p 7041  efaddlem5 7320

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 1210  ax-11o 1218  ax-ext 1459  ax-rep 2690  ax-sep 2700  ax-nul 2707  ax-pow 2739  ax-pr 2776  ax-un 2863  ax-inf2 4612

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 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1586  df-nel 1587  df-ral 1648  df-rex 1649  df-reu 1650  df-rab 1651  df-v 1810  df-sbc 1940  df-csb 2000  df-dif 2047  df-un 2048  df-in 2049  df-ss 2051  df-pss 2053  df-nul 2279  df-if 2360  df-pw 2400  df-sn 2410  df-pr 2411  df-tp 2413  df-op 2414  df-uni 2501  df-int 2531  df-iun 2565  df-br 2617  df-opab 2664  df-tr 2678  df-eprel 2829  df-id 2832  df-po 2837  df-so 2847  df-fr 2914  df-we 2931  df-ord 2948  df-on 2949  df-lim 2950  df-suc 2951  df-om 3129  df-xp 3181  df-rel 3182  df-cnv 3183  df-co 3184  df-dm 3185  df-rn 3186  df-res 3187  df-ima 3188  df-fun 3189  df-fn 3190  df-f 3191  df-f1 3192  df-fo 3193  df-f1o 3194  df-fv 3195  df-rdg 3929  df-opr 3962  df-oprab 3963  df-1st 4076  df-2nd 4077  df-1o 4130  df-oadd 4132  df-omul 4133  df-er 4258  df-ec 4260  df-qs 4263  df-en 4364  df-dom 4365  df-sdom 4366  df-ni 4987  df-pli 4988  df-mi 4989  df-lti 4990  df-plpq 5022  df-mpq 5023  df-enq 5024  df-nq 5025  df-plq 5026  df-mq 5027  df-rq 5028  df-ltq 5029  df-1q 5030  df-np 5073  df-1p 5074  df-plp 5075  df-mp 5076  df-ltp 5077  df-plpr 5151  df-mpr 5152  df-enr 5153  df-nr 5154  df-plr 5155  df-mr 5156  df-ltr 5157  df-0r 5158  df-1r 5159  df-m1r 5160  df-c 5227  df-0 5228  df-1 5229  df-i 5230  df-r 5231  df-plus 5232  df-mul 5233  df-lt 5234  df-sub 5343  df-neg 5345  df-pnf 5474  df-mnf 5475  df-xr 5476  df-ltxr 5477  df-le 5478  df-div 5686  df-n 5887  df-n0 6061  df-z 6097  df-seq1 6263  df-shft 6296  df-uz 6368  df-fz 6418  df-seqz 6483  df-exp 6519  df-fac 6898  df-bc 6923  df-sum 6948

Copyright terms: Public domain