Theorem binom 12016

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 part of the proof sets up the induction and does the base case, with the bulk of the work (the induction step) in binomlem 12015. This is Metamath 100 proof #44. (Contributed by NM, 7-Dec-2005.) (Proof shortened by Mario Carneiro, 24-Apr-2014.)

Assertion

Ref	Expression
binom	|- ( ( A e. CC /\ B e. CC /\ 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

Dummy variables n x are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		oveq2 6018	. . . . . 6 |- ( x = 0 -> ( ( A + B ) ^ x ) = ( ( A + B ) ^ 0 ) )
2		oveq2 6018	. . . . . . 7 |- ( x = 0 -> ( 0 ... x ) = ( 0 ... 0 ) )
3		oveq1 6017	. . . . . . . . 9 |- ( x = 0 -> ( x _C k ) = ( 0 _C k ) )
4		oveq1 6017	. . . . . . . . . . 11 |- ( x = 0 -> ( x - k ) = ( 0 - k ) )
5	4	oveq2d 6026	. . . . . . . . . 10 |- ( x = 0 -> ( A ^ ( x - k ) ) = ( A ^ ( 0 - k ) ) )
6	5	oveq1d 6025	. . . . . . . . 9 |- ( x = 0 -> ( ( A ^ ( x - k ) ) x. ( B ^ k ) ) = ( ( A ^ ( 0 - k ) ) x. ( B ^ k ) ) )
7	3, 6	oveq12d 6028	. . . . . . . 8 |- ( x = 0 -> ( ( x _C k ) x. ( ( A ^ ( x - k ) ) x. ( B ^ k ) ) ) = ( ( 0 _C k ) x. ( ( A ^ ( 0 - k ) ) x. ( B ^ k ) ) ) )
8	7	adantr 276	. . . . . . 7 |- ( ( x = 0 /\ k e. ( 0 ... x ) ) -> ( ( x _C k ) x. ( ( A ^ ( x - k ) ) x. ( B ^ k ) ) ) = ( ( 0 _C k ) x. ( ( A ^ ( 0 - k ) ) x. ( B ^ k ) ) ) )
9	2, 8	sumeq12dv 11904	. . . . . 6 |- ( x = 0 -> sum_ k e. ( 0 ... x ) ( ( x _C k ) x. ( ( A ^ ( x - k ) ) x. ( B ^ k ) ) ) = sum_ k e. ( 0 ... 0 ) ( ( 0 _C k ) x. ( ( A ^ ( 0 - k ) ) x. ( B ^ k ) ) ) )
10	1, 9	eqeq12d 2244	. . . . 5 |- ( x = 0 -> ( ( ( A + B ) ^ x ) = sum_ k e. ( 0 ... x ) ( ( x _C k ) x. ( ( A ^ ( x - k ) ) x. ( B ^ k ) ) ) <-> ( ( A + B ) ^ 0 ) = sum_ k e. ( 0 ... 0 ) ( ( 0 _C k ) x. ( ( A ^ ( 0 - k ) ) x. ( B ^ k ) ) ) ) )
11	10	imbi2d 230	. . . 4 |- ( x = 0 -> ( ( ( A e. CC /\ B e. CC ) -> ( ( A + B ) ^ x ) = sum_ k e. ( 0 ... x ) ( ( x _C k ) x. ( ( A ^ ( x - k ) ) x. ( B ^ k ) ) ) ) <-> ( ( A e. CC /\ B e. CC ) -> ( ( A + B ) ^ 0 ) = sum_ k e. ( 0 ... 0 ) ( ( 0 _C k ) x. ( ( A ^ ( 0 - k ) ) x. ( B ^ k ) ) ) ) ) )
12		oveq2 6018	. . . . . 6 |- ( x = n -> ( ( A + B ) ^ x ) = ( ( A + B ) ^ n ) )
13		oveq2 6018	. . . . . . 7 |- ( x = n -> ( 0 ... x ) = ( 0 ... n ) )
14		oveq1 6017	. . . . . . . . 9 |- ( x = n -> ( x _C k ) = ( n _C k ) )
15		oveq1 6017	. . . . . . . . . . 11 |- ( x = n -> ( x - k ) = ( n - k ) )
16	15	oveq2d 6026	. . . . . . . . . 10 |- ( x = n -> ( A ^ ( x - k ) ) = ( A ^ ( n - k ) ) )
17	16	oveq1d 6025	. . . . . . . . 9 |- ( x = n -> ( ( A ^ ( x - k ) ) x. ( B ^ k ) ) = ( ( A ^ ( n - k ) ) x. ( B ^ k ) ) )
18	14, 17	oveq12d 6028	. . . . . . . 8 |- ( x = n -> ( ( x _C k ) x. ( ( A ^ ( x - k ) ) x. ( B ^ k ) ) ) = ( ( n _C k ) x. ( ( A ^ ( n - k ) ) x. ( B ^ k ) ) ) )
19	18	adantr 276	. . . . . . 7 |- ( ( x = n /\ k e. ( 0 ... x ) ) -> ( ( x _C k ) x. ( ( A ^ ( x - k ) ) x. ( B ^ k ) ) ) = ( ( n _C k ) x. ( ( A ^ ( n - k ) ) x. ( B ^ k ) ) ) )
20	13, 19	sumeq12dv 11904	. . . . . 6 |- ( 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 ) ) ) )
21	12, 20	eqeq12d 2244	. . . . 5 |- ( 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 ) ) ) ) )
22	21	imbi2d 230	. . . 4 |- ( x = n -> ( ( ( A e. CC /\ B e. CC ) -> ( ( A + B ) ^ x ) = sum_ k e. ( 0 ... x ) ( ( x _C k ) x. ( ( A ^ ( x - k ) ) x. ( B ^ k ) ) ) ) <-> ( ( A e. CC /\ B e. CC ) -> ( ( A + B ) ^ n ) = sum_ k e. ( 0 ... n ) ( ( n _C k ) x. ( ( A ^ ( n - k ) ) x. ( B ^ k ) ) ) ) ) )
23		oveq2 6018	. . . . . 6 |- ( x = ( n + 1 ) -> ( ( A + B ) ^ x ) = ( ( A + B ) ^ ( n + 1 ) ) )
24		oveq2 6018	. . . . . . 7 |- ( x = ( n + 1 ) -> ( 0 ... x ) = ( 0 ... ( n + 1 ) ) )
25		oveq1 6017	. . . . . . . . 9 |- ( x = ( n + 1 ) -> ( x _C k ) = ( ( n + 1 ) _C k ) )
26		oveq1 6017	. . . . . . . . . . 11 |- ( x = ( n + 1 ) -> ( x - k ) = ( ( n + 1 ) - k ) )
27	26	oveq2d 6026	. . . . . . . . . 10 |- ( x = ( n + 1 ) -> ( A ^ ( x - k ) ) = ( A ^ ( ( n + 1 ) - k ) ) )
28	27	oveq1d 6025	. . . . . . . . 9 |- ( x = ( n + 1 ) -> ( ( A ^ ( x - k ) ) x. ( B ^ k ) ) = ( ( A ^ ( ( n + 1 ) - k ) ) x. ( B ^ k ) ) )
29	25, 28	oveq12d 6028	. . . . . . . 8 |- ( x = ( n + 1 ) -> ( ( x _C k ) x. ( ( A ^ ( x - k ) ) x. ( B ^ k ) ) ) = ( ( ( n + 1 ) _C k ) x. ( ( A ^ ( ( n + 1 ) - k ) ) x. ( B ^ k ) ) ) )
30	29	adantr 276	. . . . . . 7 |- ( ( x = ( n + 1 ) /\ k e. ( 0 ... x ) ) -> ( ( x _C k ) x. ( ( A ^ ( x - k ) ) x. ( B ^ k ) ) ) = ( ( ( n + 1 ) _C k ) x. ( ( A ^ ( ( n + 1 ) - k ) ) x. ( B ^ k ) ) ) )
31	24, 30	sumeq12dv 11904	. . . . . 6 |- ( x = ( n + 1 ) -> sum_ k e. ( 0 ... x ) ( ( x _C k ) x. ( ( A ^ ( x - k ) ) x. ( B ^ k ) ) ) = sum_ k e. ( 0 ... ( n + 1 ) ) ( ( ( n + 1 ) _C k ) x. ( ( A ^ ( ( n + 1 ) - k ) ) x. ( B ^ k ) ) ) )
32	23, 31	eqeq12d 2244	. . . . 5 |- ( x = ( n + 1 ) -> ( ( ( A + B ) ^ x ) = sum_ k e. ( 0 ... x ) ( ( x _C k ) x. ( ( A ^ ( x - k ) ) x. ( B ^ k ) ) ) <-> ( ( A + B ) ^ ( n + 1 ) ) = sum_ k e. ( 0 ... ( n + 1 ) ) ( ( ( n + 1 ) _C k ) x. ( ( A ^ ( ( n + 1 ) - k ) ) x. ( B ^ k ) ) ) ) )
33	32	imbi2d 230	. . . 4 |- ( x = ( n + 1 ) -> ( ( ( A e. CC /\ B e. CC ) -> ( ( A + B ) ^ x ) = sum_ k e. ( 0 ... x ) ( ( x _C k ) x. ( ( A ^ ( x - k ) ) x. ( B ^ k ) ) ) ) <-> ( ( A e. CC /\ B e. CC ) -> ( ( A + B ) ^ ( n + 1 ) ) = sum_ k e. ( 0 ... ( n + 1 ) ) ( ( ( n + 1 ) _C k ) x. ( ( A ^ ( ( n + 1 ) - k ) ) x. ( B ^ k ) ) ) ) ) )
34		oveq2 6018	. . . . . 6 |- ( x = N -> ( ( A + B ) ^ x ) = ( ( A + B ) ^ N ) )
35		oveq2 6018	. . . . . . 7 |- ( x = N -> ( 0 ... x ) = ( 0 ... N ) )
36		oveq1 6017	. . . . . . . . 9 |- ( x = N -> ( x _C k ) = ( N _C k ) )
37		oveq1 6017	. . . . . . . . . . 11 |- ( x = N -> ( x - k ) = ( N - k ) )
38	37	oveq2d 6026	. . . . . . . . . 10 |- ( x = N -> ( A ^ ( x - k ) ) = ( A ^ ( N - k ) ) )
39	38	oveq1d 6025	. . . . . . . . 9 |- ( x = N -> ( ( A ^ ( x - k ) ) x. ( B ^ k ) ) = ( ( A ^ ( N - k ) ) x. ( B ^ k ) ) )
40	36, 39	oveq12d 6028	. . . . . . . 8 |- ( x = N -> ( ( x _C k ) x. ( ( A ^ ( x - k ) ) x. ( B ^ k ) ) ) = ( ( N _C k ) x. ( ( A ^ ( N - k ) ) x. ( B ^ k ) ) ) )
41	40	adantr 276	. . . . . . 7 |- ( ( x = N /\ k e. ( 0 ... x ) ) -> ( ( x _C k ) x. ( ( A ^ ( x - k ) ) x. ( B ^ k ) ) ) = ( ( N _C k ) x. ( ( A ^ ( N - k ) ) x. ( B ^ k ) ) ) )
42	35, 41	sumeq12dv 11904	. . . . . 6 |- ( 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 ) ) ) )
43	34, 42	eqeq12d 2244	. . . . 5 |- ( 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 ) ) ) ) )
44	43	imbi2d 230	. . . 4 |- ( x = N -> ( ( ( A e. CC /\ B e. CC ) -> ( ( A + B ) ^ x ) = sum_ k e. ( 0 ... x ) ( ( x _C k ) x. ( ( A ^ ( x - k ) ) x. ( B ^ k ) ) ) ) <-> ( ( A e. CC /\ B e. CC ) -> ( ( A + B ) ^ N ) = sum_ k e. ( 0 ... N ) ( ( N _C k ) x. ( ( A ^ ( N - k ) ) x. ( B ^ k ) ) ) ) ) )
45		exp0 10782	. . . . . . . . 9 |- ( A e. CC -> ( A ^ 0 ) = 1 )
46		exp0 10782	. . . . . . . . 9 |- ( B e. CC -> ( B ^ 0 ) = 1 )
47	45, 46	oveqan12d 6029	. . . . . . . 8 |- ( ( A e. CC /\ B e. CC ) -> ( ( A ^ 0 ) x. ( B ^ 0 ) ) = ( 1 x. 1 ) )
48		1t1e1 9279	. . . . . . . 8 |- ( 1 x. 1 ) = 1
49	47, 48	eqtrdi 2278	. . . . . . 7 |- ( ( A e. CC /\ B e. CC ) -> ( ( A ^ 0 ) x. ( B ^ 0 ) ) = 1 )
50	49	oveq2d 6026	. . . . . 6 |- ( ( A e. CC /\ B e. CC ) -> ( 1 x. ( ( A ^ 0 ) x. ( B ^ 0 ) ) ) = ( 1 x. 1 ) )
51	50, 48	eqtrdi 2278	. . . . 5 |- ( ( A e. CC /\ B e. CC ) -> ( 1 x. ( ( A ^ 0 ) x. ( B ^ 0 ) ) ) = 1 )
52		0z 9473	. . . . . 6 |- 0 e. ZZ
53		ax-1cn 8108	. . . . . . 7 |- 1 e. CC
54	51, 53	eqeltrdi 2320	. . . . . 6 |- ( ( A e. CC /\ B e. CC ) -> ( 1 x. ( ( A ^ 0 ) x. ( B ^ 0 ) ) ) e. CC )
55		oveq2 6018	. . . . . . . . 9 |- ( k = 0 -> ( 0 _C k ) = ( 0 _C 0 ) )
56		0nn0 9400	. . . . . . . . . 10 |- 0 e. NN0
57		bcn0 10994	. . . . . . . . . 10 |- ( 0 e. NN0 -> ( 0 _C 0 ) = 1 )
58	56, 57	ax-mp 5	. . . . . . . . 9 |- ( 0 _C 0 ) = 1
59	55, 58	eqtrdi 2278	. . . . . . . 8 |- ( k = 0 -> ( 0 _C k ) = 1 )
60		oveq2 6018	. . . . . . . . . . 11 |- ( k = 0 -> ( 0 - k ) = ( 0 - 0 ) )
61		0m0e0 9238	. . . . . . . . . . 11 |- ( 0 - 0 ) = 0
62	60, 61	eqtrdi 2278	. . . . . . . . . 10 |- ( k = 0 -> ( 0 - k ) = 0 )
63	62	oveq2d 6026	. . . . . . . . 9 |- ( k = 0 -> ( A ^ ( 0 - k ) ) = ( A ^ 0 ) )
64		oveq2 6018	. . . . . . . . 9 |- ( k = 0 -> ( B ^ k ) = ( B ^ 0 ) )
65	63, 64	oveq12d 6028	. . . . . . . 8 |- ( k = 0 -> ( ( A ^ ( 0 - k ) ) x. ( B ^ k ) ) = ( ( A ^ 0 ) x. ( B ^ 0 ) ) )
66	59, 65	oveq12d 6028	. . . . . . 7 |- ( k = 0 -> ( ( 0 _C k ) x. ( ( A ^ ( 0 - k ) ) x. ( B ^ k ) ) ) = ( 1 x. ( ( A ^ 0 ) x. ( B ^ 0 ) ) ) )
67	66	fsum1 11944	. . . . . 6 |- ( ( 0 e. ZZ /\ ( 1 x. ( ( A ^ 0 ) x. ( B ^ 0 ) ) ) e. CC ) -> sum_ k e. ( 0 ... 0 ) ( ( 0 _C k ) x. ( ( A ^ ( 0 - k ) ) x. ( B ^ k ) ) ) = ( 1 x. ( ( A ^ 0 ) x. ( B ^ 0 ) ) ) )
68	52, 54, 67	sylancr 414	. . . . 5 |- ( ( A e. CC /\ B e. CC ) -> sum_ k e. ( 0 ... 0 ) ( ( 0 _C k ) x. ( ( A ^ ( 0 - k ) ) x. ( B ^ k ) ) ) = ( 1 x. ( ( A ^ 0 ) x. ( B ^ 0 ) ) ) )
69		addcl 8140	. . . . . 6 |- ( ( A e. CC /\ B e. CC ) -> ( A + B ) e. CC )
70	69	exp0d 10906	. . . . 5 |- ( ( A e. CC /\ B e. CC ) -> ( ( A + B ) ^ 0 ) = 1 )
71	51, 68, 70	3eqtr4rd 2273	. . . 4 |- ( ( A e. CC /\ B e. CC ) -> ( ( A + B ) ^ 0 ) = sum_ k e. ( 0 ... 0 ) ( ( 0 _C k ) x. ( ( A ^ ( 0 - k ) ) x. ( B ^ k ) ) ) )
72		simprl 529	. . . . . . 7 |- ( ( n e. NN0 /\ ( A e. CC /\ B e. CC ) ) -> A e. CC )
73		simprr 531	. . . . . . 7 |- ( ( n e. NN0 /\ ( A e. CC /\ B e. CC ) ) -> B e. CC )
74		simpl 109	. . . . . . 7 |- ( ( n e. NN0 /\ ( A e. CC /\ B e. CC ) ) -> n e. NN0 )
75		id 19	. . . . . . 7 |- ( ( ( A + B ) ^ n ) = sum_ k e. ( 0 ... n ) ( ( n _C k ) x. ( ( A ^ ( n - k ) ) x. ( B ^ k ) ) ) -> ( ( A + B ) ^ n ) = sum_ k e. ( 0 ... n ) ( ( n _C k ) x. ( ( A ^ ( n - k ) ) x. ( B ^ k ) ) ) )
76	72, 73, 74, 75	binomlem 12015	. . . . . 6 |- ( ( ( n e. NN0 /\ ( A e. CC /\ B e. CC ) ) /\ ( ( A + B ) ^ n ) = sum_ k e. ( 0 ... n ) ( ( n _C k ) x. ( ( A ^ ( n - k ) ) x. ( B ^ k ) ) ) ) -> ( ( A + B ) ^ ( n + 1 ) ) = sum_ k e. ( 0 ... ( n + 1 ) ) ( ( ( n + 1 ) _C k ) x. ( ( A ^ ( ( n + 1 ) - k ) ) x. ( B ^ k ) ) ) )
77	76	exp31 364	. . . . 5 |- ( n e. NN0 -> ( ( A e. CC /\ B e. CC ) -> ( ( ( A + B ) ^ n ) = sum_ k e. ( 0 ... n ) ( ( n _C k ) x. ( ( A ^ ( n - k ) ) x. ( B ^ k ) ) ) -> ( ( A + B ) ^ ( n + 1 ) ) = sum_ k e. ( 0 ... ( n + 1 ) ) ( ( ( n + 1 ) _C k ) x. ( ( A ^ ( ( n + 1 ) - k ) ) x. ( B ^ k ) ) ) ) ) )
78	77	a2d 26	. . . 4 |- ( n e. NN0 -> ( ( ( A e. CC /\ B e. CC ) -> ( ( A + B ) ^ n ) = sum_ k e. ( 0 ... n ) ( ( n _C k ) x. ( ( A ^ ( n - k ) ) x. ( B ^ k ) ) ) ) -> ( ( A e. CC /\ B e. CC ) -> ( ( A + B ) ^ ( n + 1 ) ) = sum_ k e. ( 0 ... ( n + 1 ) ) ( ( ( n + 1 ) _C k ) x. ( ( A ^ ( ( n + 1 ) - k ) ) x. ( B ^ k ) ) ) ) ) )
79	11, 22, 33, 44, 71, 78	nn0ind 9577	. . 3 |- ( N e. NN0 -> ( ( A e. CC /\ B e. CC ) -> ( ( A + B ) ^ N ) = sum_ k e. ( 0 ... N ) ( ( N _C k ) x. ( ( A ^ ( N - k ) ) x. ( B ^ k ) ) ) ) )
80	79	impcom 125	. 2 |- ( ( ( A e. CC /\ B e. CC ) /\ N e. NN0 ) -> ( ( A + B ) ^ N ) = sum_ k e. ( 0 ... N ) ( ( N _C k ) x. ( ( A ^ ( N - k ) ) x. ( B ^ k ) ) ) )
81	80	3impa 1218	1 |- ( ( A e. CC /\ B e. CC /\ N e. NN0 ) -> ( ( A + B ) ^ N ) = sum_ k e. ( 0 ... N ) ( ( N _C k ) x. ( ( A ^ ( N - k ) ) x. ( B ^ k ) ) ) )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 1002   = wceq 1395   e. wcel 2200  (class class class)co 6010   CCcc 8013   0cc0 8015   1c1 8016   + caddc 8018   x. cmul 8020   - cmin 8333   NN0cn0 9385   ZZcz 9462   ...cfz 10221   ^cexp 10777   _C cbc 10986   sum_csu 11885

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4199  ax-sep 4202  ax-nul 4210  ax-pow 4259  ax-pr 4294  ax-un 4525  ax-setind 4630  ax-iinf 4681  ax-cnex 8106  ax-resscn 8107  ax-1cn 8108  ax-1re 8109  ax-icn 8110  ax-addcl 8111  ax-addrcl 8112  ax-mulcl 8113  ax-mulrcl 8114  ax-addcom 8115  ax-mulcom 8116  ax-addass 8117  ax-mulass 8118  ax-distr 8119  ax-i2m1 8120  ax-0lt1 8121  ax-1rid 8122  ax-0id 8123  ax-rnegex 8124  ax-precex 8125  ax-cnre 8126  ax-pre-ltirr 8127  ax-pre-ltwlin 8128  ax-pre-lttrn 8129  ax-pre-apti 8130  ax-pre-ltadd 8131  ax-pre-mulgt0 8132  ax-pre-mulext 8133  ax-arch 8134  ax-caucvg 8135

This theorem depends on definitions:  df-bi 117  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rmo 2516  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-if 3603  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-int 3924  df-iun 3967  df-br 4084  df-opab 4146  df-mpt 4147  df-tr 4183  df-id 4385  df-po 4388  df-iso 4389  df-iord 4458  df-on 4460  df-ilim 4461  df-suc 4463  df-iom 4684  df-xp 4726  df-rel 4727  df-cnv 4728  df-co 4729  df-dm 4730  df-rn 4731  df-res 4732  df-ima 4733  df-iota 5281  df-fun 5323  df-fn 5324  df-f 5325  df-f1 5326  df-fo 5327  df-f1o 5328  df-fv 5329  df-isom 5330  df-riota 5963  df-ov 6013  df-oprab 6014  df-mpo 6015  df-1st 6295  df-2nd 6296  df-recs 6462  df-irdg 6527  df-frec 6548  df-1o 6573  df-oadd 6577  df-er 6693  df-en 6901  df-dom 6902  df-fin 6903  df-pnf 8199  df-mnf 8200  df-xr 8201  df-ltxr 8202  df-le 8203  df-sub 8335  df-neg 8336  df-reap 8738  df-ap 8745  df-div 8836  df-inn 9127  df-2 9185  df-3 9186  df-4 9187  df-n0 9386  df-z 9463  df-uz 9739  df-q 9832  df-rp 9867  df-fz 10222  df-fzo 10356  df-seqfrec 10687  df-exp 10778  df-fac 10965  df-bc 10987  df-ihash 11015  df-cj 11374  df-re 11375  df-im 11376  df-rsqrt 11530  df-abs 11531  df-clim 11811  df-sumdc 11886

This theorem is referenced by:  binom1p  12017  efaddlem  12206