Theorem binom 11447

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 11446. 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 5861	. . . . . 6 |- ( x = 0 -> ( ( A + B ) ^ x ) = ( ( A + B ) ^ 0 ) )
2		oveq2 5861	. . . . . . 7 |- ( x = 0 -> ( 0 ... x ) = ( 0 ... 0 ) )
3		oveq1 5860	. . . . . . . . 9 |- ( x = 0 -> ( x _C k ) = ( 0 _C k ) )
4		oveq1 5860	. . . . . . . . . . 11 |- ( x = 0 -> ( x - k ) = ( 0 - k ) )
5	4	oveq2d 5869	. . . . . . . . . 10 |- ( x = 0 -> ( A ^ ( x - k ) ) = ( A ^ ( 0 - k ) ) )
6	5	oveq1d 5868	. . . . . . . . 9 |- ( x = 0 -> ( ( A ^ ( x - k ) ) x. ( B ^ k ) ) = ( ( A ^ ( 0 - k ) ) x. ( B ^ k ) ) )
7	3, 6	oveq12d 5871	. . . . . . . 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 274	. . . . . . 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 11335	. . . . . 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 2185	. . . . 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 229	. . . 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 5861	. . . . . 6 |- ( x = n -> ( ( A + B ) ^ x ) = ( ( A + B ) ^ n ) )
13		oveq2 5861	. . . . . . 7 |- ( x = n -> ( 0 ... x ) = ( 0 ... n ) )
14		oveq1 5860	. . . . . . . . 9 |- ( x = n -> ( x _C k ) = ( n _C k ) )
15		oveq1 5860	. . . . . . . . . . 11 |- ( x = n -> ( x - k ) = ( n - k ) )
16	15	oveq2d 5869	. . . . . . . . . 10 |- ( x = n -> ( A ^ ( x - k ) ) = ( A ^ ( n - k ) ) )
17	16	oveq1d 5868	. . . . . . . . 9 |- ( x = n -> ( ( A ^ ( x - k ) ) x. ( B ^ k ) ) = ( ( A ^ ( n - k ) ) x. ( B ^ k ) ) )
18	14, 17	oveq12d 5871	. . . . . . . 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 274	. . . . . . 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 11335	. . . . . 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 2185	. . . . 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 229	. . . 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 5861	. . . . . 6 |- ( x = ( n + 1 ) -> ( ( A + B ) ^ x ) = ( ( A + B ) ^ ( n + 1 ) ) )
24		oveq2 5861	. . . . . . 7 |- ( x = ( n + 1 ) -> ( 0 ... x ) = ( 0 ... ( n + 1 ) ) )
25		oveq1 5860	. . . . . . . . 9 |- ( x = ( n + 1 ) -> ( x _C k ) = ( ( n + 1 ) _C k ) )
26		oveq1 5860	. . . . . . . . . . 11 |- ( x = ( n + 1 ) -> ( x - k ) = ( ( n + 1 ) - k ) )
27	26	oveq2d 5869	. . . . . . . . . 10 |- ( x = ( n + 1 ) -> ( A ^ ( x - k ) ) = ( A ^ ( ( n + 1 ) - k ) ) )
28	27	oveq1d 5868	. . . . . . . . 9 |- ( x = ( n + 1 ) -> ( ( A ^ ( x - k ) ) x. ( B ^ k ) ) = ( ( A ^ ( ( n + 1 ) - k ) ) x. ( B ^ k ) ) )
29	25, 28	oveq12d 5871	. . . . . . . 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 274	. . . . . . 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 11335	. . . . . 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 2185	. . . . 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 229	. . . 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 5861	. . . . . 6 |- ( x = N -> ( ( A + B ) ^ x ) = ( ( A + B ) ^ N ) )
35		oveq2 5861	. . . . . . 7 |- ( x = N -> ( 0 ... x ) = ( 0 ... N ) )
36		oveq1 5860	. . . . . . . . 9 |- ( x = N -> ( x _C k ) = ( N _C k ) )
37		oveq1 5860	. . . . . . . . . . 11 |- ( x = N -> ( x - k ) = ( N - k ) )
38	37	oveq2d 5869	. . . . . . . . . 10 |- ( x = N -> ( A ^ ( x - k ) ) = ( A ^ ( N - k ) ) )
39	38	oveq1d 5868	. . . . . . . . 9 |- ( x = N -> ( ( A ^ ( x - k ) ) x. ( B ^ k ) ) = ( ( A ^ ( N - k ) ) x. ( B ^ k ) ) )
40	36, 39	oveq12d 5871	. . . . . . . 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 274	. . . . . . 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 11335	. . . . . 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 2185	. . . . 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 229	. . . 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 10480	. . . . . . . . 9 |- ( A e. CC -> ( A ^ 0 ) = 1 )
46		exp0 10480	. . . . . . . . 9 |- ( B e. CC -> ( B ^ 0 ) = 1 )
47	45, 46	oveqan12d 5872	. . . . . . . 8 |- ( ( A e. CC /\ B e. CC ) -> ( ( A ^ 0 ) x. ( B ^ 0 ) ) = ( 1 x. 1 ) )
48		1t1e1 9030	. . . . . . . 8 |- ( 1 x. 1 ) = 1
49	47, 48	eqtrdi 2219	. . . . . . 7 |- ( ( A e. CC /\ B e. CC ) -> ( ( A ^ 0 ) x. ( B ^ 0 ) ) = 1 )
50	49	oveq2d 5869	. . . . . 6 |- ( ( A e. CC /\ B e. CC ) -> ( 1 x. ( ( A ^ 0 ) x. ( B ^ 0 ) ) ) = ( 1 x. 1 ) )
51	50, 48	eqtrdi 2219	. . . . 5 |- ( ( A e. CC /\ B e. CC ) -> ( 1 x. ( ( A ^ 0 ) x. ( B ^ 0 ) ) ) = 1 )
52		0z 9223	. . . . . 6 |- 0 e. ZZ
53		ax-1cn 7867	. . . . . . 7 |- 1 e. CC
54	51, 53	eqeltrdi 2261	. . . . . 6 |- ( ( A e. CC /\ B e. CC ) -> ( 1 x. ( ( A ^ 0 ) x. ( B ^ 0 ) ) ) e. CC )
55		oveq2 5861	. . . . . . . . 9 |- ( k = 0 -> ( 0 _C k ) = ( 0 _C 0 ) )
56		0nn0 9150	. . . . . . . . . 10 |- 0 e. NN0
57		bcn0 10689	. . . . . . . . . 10 |- ( 0 e. NN0 -> ( 0 _C 0 ) = 1 )
58	56, 57	ax-mp 5	. . . . . . . . 9 |- ( 0 _C 0 ) = 1
59	55, 58	eqtrdi 2219	. . . . . . . 8 |- ( k = 0 -> ( 0 _C k ) = 1 )
60		oveq2 5861	. . . . . . . . . . 11 |- ( k = 0 -> ( 0 - k ) = ( 0 - 0 ) )
61		0m0e0 8990	. . . . . . . . . . 11 |- ( 0 - 0 ) = 0
62	60, 61	eqtrdi 2219	. . . . . . . . . 10 |- ( k = 0 -> ( 0 - k ) = 0 )
63	62	oveq2d 5869	. . . . . . . . 9 |- ( k = 0 -> ( A ^ ( 0 - k ) ) = ( A ^ 0 ) )
64		oveq2 5861	. . . . . . . . 9 |- ( k = 0 -> ( B ^ k ) = ( B ^ 0 ) )
65	63, 64	oveq12d 5871	. . . . . . . 8 |- ( k = 0 -> ( ( A ^ ( 0 - k ) ) x. ( B ^ k ) ) = ( ( A ^ 0 ) x. ( B ^ 0 ) ) )
66	59, 65	oveq12d 5871	. . . . . . 7 |- ( k = 0 -> ( ( 0 _C k ) x. ( ( A ^ ( 0 - k ) ) x. ( B ^ k ) ) ) = ( 1 x. ( ( A ^ 0 ) x. ( B ^ 0 ) ) ) )
67	66	fsum1 11375	. . . . . 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 412	. . . . 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 7899	. . . . . 6 |- ( ( A e. CC /\ B e. CC ) -> ( A + B ) e. CC )
70	69	exp0d 10603	. . . . 5 |- ( ( A e. CC /\ B e. CC ) -> ( ( A + B ) ^ 0 ) = 1 )
71	51, 68, 70	3eqtr4rd 2214	. . . 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 526	. . . . . . 7 |- ( ( n e. NN0 /\ ( A e. CC /\ B e. CC ) ) -> A e. CC )
73		simprr 527	. . . . . . 7 |- ( ( n e. NN0 /\ ( A e. CC /\ B e. CC ) ) -> B e. CC )
74		simpl 108	. . . . . . 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 11446	. . . . . 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 362	. . . . 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 9326	. . 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 124	. 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 1189	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 103   /\ w3a 973   = wceq 1348   e. wcel 2141  (class class class)co 5853   CCcc 7772   0cc0 7774   1c1 7775   + caddc 7777   x. cmul 7779   - cmin 8090   NN0cn0 9135   ZZcz 9212   ...cfz 9965   ^cexp 10475   _C cbc 10681   sum_csu 11316

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-coll 4104  ax-sep 4107  ax-nul 4115  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-iinf 4572  ax-cnex 7865  ax-resscn 7866  ax-1cn 7867  ax-1re 7868  ax-icn 7869  ax-addcl 7870  ax-addrcl 7871  ax-mulcl 7872  ax-mulrcl 7873  ax-addcom 7874  ax-mulcom 7875  ax-addass 7876  ax-mulass 7877  ax-distr 7878  ax-i2m1 7879  ax-0lt1 7880  ax-1rid 7881  ax-0id 7882  ax-rnegex 7883  ax-precex 7884  ax-cnre 7885  ax-pre-ltirr 7886  ax-pre-ltwlin 7887  ax-pre-lttrn 7888  ax-pre-apti 7889  ax-pre-ltadd 7890  ax-pre-mulgt0 7891  ax-pre-mulext 7892  ax-arch 7893  ax-caucvg 7894

This theorem depends on definitions:  df-bi 116  df-dc 830  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-nel 2436  df-ral 2453  df-rex 2454  df-reu 2455  df-rmo 2456  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-if 3527  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-int 3832  df-iun 3875  df-br 3990  df-opab 4051  df-mpt 4052  df-tr 4088  df-id 4278  df-po 4281  df-iso 4282  df-iord 4351  df-on 4353  df-ilim 4354  df-suc 4356  df-iom 4575  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-fo 5204  df-f1o 5205  df-fv 5206  df-isom 5207  df-riota 5809  df-ov 5856  df-oprab 5857  df-mpo 5858  df-1st 6119  df-2nd 6120  df-recs 6284  df-irdg 6349  df-frec 6370  df-1o 6395  df-oadd 6399  df-er 6513  df-en 6719  df-dom 6720  df-fin 6721  df-pnf 7956  df-mnf 7957  df-xr 7958  df-ltxr 7959  df-le 7960  df-sub 8092  df-neg 8093  df-reap 8494  df-ap 8501  df-div 8590  df-inn 8879  df-2 8937  df-3 8938  df-4 8939  df-n0 9136  df-z 9213  df-uz 9488  df-q 9579  df-rp 9611  df-fz 9966  df-fzo 10099  df-seqfrec 10402  df-exp 10476  df-fac 10660  df-bc 10682  df-ihash 10710  df-cj 10806  df-re 10807  df-im 10808  df-rsqrt 10962  df-abs 10963  df-clim 11242  df-sumdc 11317

This theorem is referenced by:  binom1p  11448  efaddlem  11637