Theorem bpoly2 14788

Description: The Bernoulli polynomials at two. (Contributed by Scott Fenton, 8-Jul-2015.)

Assertion

Ref	Expression
bpoly2	|- ( X e. CC -> ( 2 BernPoly X ) = ( ( ( X ^ 2 ) - X ) + ( 1 / 6 ) ) )

Proof of Theorem bpoly2

Dummy variable k is distinct from all other variables.

Step	Hyp	Ref	Expression
1		2nn0 11309	. . 3 |- 2 e. NN0
2		bpolyval 14780	. . 3 |- ( ( 2 e. NN0 /\ X e. CC ) -> ( 2 BernPoly X ) = ( ( X ^ 2 ) - sum_ k e. ( 0 ... ( 2 - 1 ) ) ( ( 2 _C k ) x. ( ( k BernPoly X ) / ( ( 2 - k ) + 1 ) ) ) ) )
3	1, 2	mpan 706	. 2 |- ( X e. CC -> ( 2 BernPoly X ) = ( ( X ^ 2 ) - sum_ k e. ( 0 ... ( 2 - 1 ) ) ( ( 2 _C k ) x. ( ( k BernPoly X ) / ( ( 2 - k ) + 1 ) ) ) ) )
4		2m1e1 11135	. . . . . . 7 |- ( 2 - 1 ) = 1
5		0p1e1 11132	. . . . . . 7 |- ( 0 + 1 ) = 1
6	4, 5	eqtr4i 2647	. . . . . 6 |- ( 2 - 1 ) = ( 0 + 1 )
7	6	oveq2i 6661	. . . . 5 |- ( 0 ... ( 2 - 1 ) ) = ( 0 ... ( 0 + 1 ) )
8	7	sumeq1i 14428	. . . 4 |- sum_ k e. ( 0 ... ( 2 - 1 ) ) ( ( 2 _C k ) x. ( ( k BernPoly X ) / ( ( 2 - k ) + 1 ) ) ) = sum_ k e. ( 0 ... ( 0 + 1 ) ) ( ( 2 _C k ) x. ( ( k BernPoly X ) / ( ( 2 - k ) + 1 ) ) )
9		0nn0 11307	. . . . . . . . 9 |- 0 e. NN0
10		nn0uz 11722	. . . . . . . . 9 |- NN0 = ( ZZ>= ` 0 )
11	9, 10	eleqtri 2699	. . . . . . . 8 |- 0 e. ( ZZ>= ` 0 )
12	11	a1i 11	. . . . . . 7 |- ( X e. CC -> 0 e. ( ZZ>= ` 0 ) )
13		0z 11388	. . . . . . . . . . 11 |- 0 e. ZZ
14		fzpr 12396	. . . . . . . . . . 11 |- ( 0 e. ZZ -> ( 0 ... ( 0 + 1 ) ) = { 0 , ( 0 + 1 ) } )
15	13, 14	ax-mp 5	. . . . . . . . . 10 |- ( 0 ... ( 0 + 1 ) ) = { 0 , ( 0 + 1 ) }
16	15	eleq2i 2693	. . . . . . . . 9 |- ( k e. ( 0 ... ( 0 + 1 ) ) <-> k e. { 0 , ( 0 + 1 ) } )
17		vex 3203	. . . . . . . . . 10 |- k e. _V
18	17	elpr 4198	. . . . . . . . 9 |- ( k e. { 0 , ( 0 + 1 ) } <-> ( k = 0 \/ k = ( 0 + 1 ) ) )
19	16, 18	bitri 264	. . . . . . . 8 |- ( k e. ( 0 ... ( 0 + 1 ) ) <-> ( k = 0 \/ k = ( 0 + 1 ) ) )
20		oveq2 6658	. . . . . . . . . . . . 13 |- ( k = 0 -> ( 2 _C k ) = ( 2 _C 0 ) )
21		bcn0 13097	. . . . . . . . . . . . . 14 |- ( 2 e. NN0 -> ( 2 _C 0 ) = 1 )
22	1, 21	ax-mp 5	. . . . . . . . . . . . 13 |- ( 2 _C 0 ) = 1
23	20, 22	syl6eq 2672	. . . . . . . . . . . 12 |- ( k = 0 -> ( 2 _C k ) = 1 )
24		oveq1 6657	. . . . . . . . . . . . 13 |- ( k = 0 -> ( k BernPoly X ) = ( 0 BernPoly X ) )
25		oveq2 6658	. . . . . . . . . . . . . . 15 |- ( k = 0 -> ( 2 - k ) = ( 2 - 0 ) )
26	25	oveq1d 6665	. . . . . . . . . . . . . 14 |- ( k = 0 -> ( ( 2 - k ) + 1 ) = ( ( 2 - 0 ) + 1 ) )
27		2cn 11091	. . . . . . . . . . . . . . . . 17 |- 2 e. CC
28	27	subid1i 10353	. . . . . . . . . . . . . . . 16 |- ( 2 - 0 ) = 2
29	28	oveq1i 6660	. . . . . . . . . . . . . . 15 |- ( ( 2 - 0 ) + 1 ) = ( 2 + 1 )
30		df-3 11080	. . . . . . . . . . . . . . 15 |- 3 = ( 2 + 1 )
31	29, 30	eqtr4i 2647	. . . . . . . . . . . . . 14 |- ( ( 2 - 0 ) + 1 ) = 3
32	26, 31	syl6eq 2672	. . . . . . . . . . . . 13 |- ( k = 0 -> ( ( 2 - k ) + 1 ) = 3 )
33	24, 32	oveq12d 6668	. . . . . . . . . . . 12 |- ( k = 0 -> ( ( k BernPoly X ) / ( ( 2 - k ) + 1 ) ) = ( ( 0 BernPoly X ) / 3 ) )
34	23, 33	oveq12d 6668	. . . . . . . . . . 11 |- ( k = 0 -> ( ( 2 _C k ) x. ( ( k BernPoly X ) / ( ( 2 - k ) + 1 ) ) ) = ( 1 x. ( ( 0 BernPoly X ) / 3 ) ) )
35		bpoly0 14781	. . . . . . . . . . . . . 14 |- ( X e. CC -> ( 0 BernPoly X ) = 1 )
36	35	oveq1d 6665	. . . . . . . . . . . . 13 |- ( X e. CC -> ( ( 0 BernPoly X ) / 3 ) = ( 1 / 3 ) )
37	36	oveq2d 6666	. . . . . . . . . . . 12 |- ( X e. CC -> ( 1 x. ( ( 0 BernPoly X ) / 3 ) ) = ( 1 x. ( 1 / 3 ) ) )
38		3cn 11095	. . . . . . . . . . . . . 14 |- 3 e. CC
39		3ne0 11115	. . . . . . . . . . . . . 14 |- 3 =/= 0
40	38, 39	reccli 10755	. . . . . . . . . . . . 13 |- ( 1 / 3 ) e. CC
41	40	mulid2i 10043	. . . . . . . . . . . 12 |- ( 1 x. ( 1 / 3 ) ) = ( 1 / 3 )
42	37, 41	syl6eq 2672	. . . . . . . . . . 11 |- ( X e. CC -> ( 1 x. ( ( 0 BernPoly X ) / 3 ) ) = ( 1 / 3 ) )
43	34, 42	sylan9eqr 2678	. . . . . . . . . 10 |- ( ( X e. CC /\ k = 0 ) -> ( ( 2 _C k ) x. ( ( k BernPoly X ) / ( ( 2 - k ) + 1 ) ) ) = ( 1 / 3 ) )
44	43, 40	syl6eqel 2709	. . . . . . . . 9 |- ( ( X e. CC /\ k = 0 ) -> ( ( 2 _C k ) x. ( ( k BernPoly X ) / ( ( 2 - k ) + 1 ) ) ) e. CC )
45	5	eqeq2i 2634	. . . . . . . . . . . 12 |- ( k = ( 0 + 1 ) <-> k = 1 )
46		oveq2 6658	. . . . . . . . . . . . . 14 |- ( k = 1 -> ( 2 _C k ) = ( 2 _C 1 ) )
47		bcn1 13100	. . . . . . . . . . . . . . 15 |- ( 2 e. NN0 -> ( 2 _C 1 ) = 2 )
48	1, 47	ax-mp 5	. . . . . . . . . . . . . 14 |- ( 2 _C 1 ) = 2
49	46, 48	syl6eq 2672	. . . . . . . . . . . . 13 |- ( k = 1 -> ( 2 _C k ) = 2 )
50		oveq1 6657	. . . . . . . . . . . . . 14 |- ( k = 1 -> ( k BernPoly X ) = ( 1 BernPoly X ) )
51		oveq2 6658	. . . . . . . . . . . . . . . 16 |- ( k = 1 -> ( 2 - k ) = ( 2 - 1 ) )
52	51	oveq1d 6665	. . . . . . . . . . . . . . 15 |- ( k = 1 -> ( ( 2 - k ) + 1 ) = ( ( 2 - 1 ) + 1 ) )
53		ax-1cn 9994	. . . . . . . . . . . . . . . 16 |- 1 e. CC
54		npcan 10290	. . . . . . . . . . . . . . . 16 |- ( ( 2 e. CC /\ 1 e. CC ) -> ( ( 2 - 1 ) + 1 ) = 2 )
55	27, 53, 54	mp2an 708	. . . . . . . . . . . . . . 15 |- ( ( 2 - 1 ) + 1 ) = 2
56	52, 55	syl6eq 2672	. . . . . . . . . . . . . 14 |- ( k = 1 -> ( ( 2 - k ) + 1 ) = 2 )
57	50, 56	oveq12d 6668	. . . . . . . . . . . . 13 |- ( k = 1 -> ( ( k BernPoly X ) / ( ( 2 - k ) + 1 ) ) = ( ( 1 BernPoly X ) / 2 ) )
58	49, 57	oveq12d 6668	. . . . . . . . . . . 12 |- ( k = 1 -> ( ( 2 _C k ) x. ( ( k BernPoly X ) / ( ( 2 - k ) + 1 ) ) ) = ( 2 x. ( ( 1 BernPoly X ) / 2 ) ) )
59	45, 58	sylbi 207	. . . . . . . . . . 11 |- ( k = ( 0 + 1 ) -> ( ( 2 _C k ) x. ( ( k BernPoly X ) / ( ( 2 - k ) + 1 ) ) ) = ( 2 x. ( ( 1 BernPoly X ) / 2 ) ) )
60		bpoly1 14782	. . . . . . . . . . . . . 14 |- ( X e. CC -> ( 1 BernPoly X ) = ( X - ( 1 / 2 ) ) )
61	60	oveq1d 6665	. . . . . . . . . . . . 13 |- ( X e. CC -> ( ( 1 BernPoly X ) / 2 ) = ( ( X - ( 1 / 2 ) ) / 2 ) )
62	61	oveq2d 6666	. . . . . . . . . . . 12 |- ( X e. CC -> ( 2 x. ( ( 1 BernPoly X ) / 2 ) ) = ( 2 x. ( ( X - ( 1 / 2 ) ) / 2 ) ) )
63		halfcn 11247	. . . . . . . . . . . . . 14 |- ( 1 / 2 ) e. CC
64		subcl 10280	. . . . . . . . . . . . . 14 |- ( ( X e. CC /\ ( 1 / 2 ) e. CC ) -> ( X - ( 1 / 2 ) ) e. CC )
65	63, 64	mpan2 707	. . . . . . . . . . . . 13 |- ( X e. CC -> ( X - ( 1 / 2 ) ) e. CC )
66		2ne0 11113	. . . . . . . . . . . . . 14 |- 2 =/= 0
67		divcan2 10693	. . . . . . . . . . . . . 14 |- ( ( ( X - ( 1 / 2 ) ) e. CC /\ 2 e. CC /\ 2 =/= 0 ) -> ( 2 x. ( ( X - ( 1 / 2 ) ) / 2 ) ) = ( X - ( 1 / 2 ) ) )
68	27, 66, 67	mp3an23 1416	. . . . . . . . . . . . 13 |- ( ( X - ( 1 / 2 ) ) e. CC -> ( 2 x. ( ( X - ( 1 / 2 ) ) / 2 ) ) = ( X - ( 1 / 2 ) ) )
69	65, 68	syl 17	. . . . . . . . . . . 12 |- ( X e. CC -> ( 2 x. ( ( X - ( 1 / 2 ) ) / 2 ) ) = ( X - ( 1 / 2 ) ) )
70	62, 69	eqtrd 2656	. . . . . . . . . . 11 |- ( X e. CC -> ( 2 x. ( ( 1 BernPoly X ) / 2 ) ) = ( X - ( 1 / 2 ) ) )
71	59, 70	sylan9eqr 2678	. . . . . . . . . 10 |- ( ( X e. CC /\ k = ( 0 + 1 ) ) -> ( ( 2 _C k ) x. ( ( k BernPoly X ) / ( ( 2 - k ) + 1 ) ) ) = ( X - ( 1 / 2 ) ) )
72	65	adantr 481	. . . . . . . . . 10 |- ( ( X e. CC /\ k = ( 0 + 1 ) ) -> ( X - ( 1 / 2 ) ) e. CC )
73	71, 72	eqeltrd 2701	. . . . . . . . 9 |- ( ( X e. CC /\ k = ( 0 + 1 ) ) -> ( ( 2 _C k ) x. ( ( k BernPoly X ) / ( ( 2 - k ) + 1 ) ) ) e. CC )
74	44, 73	jaodan 826	. . . . . . . 8 |- ( ( X e. CC /\ ( k = 0 \/ k = ( 0 + 1 ) ) ) -> ( ( 2 _C k ) x. ( ( k BernPoly X ) / ( ( 2 - k ) + 1 ) ) ) e. CC )
75	19, 74	sylan2b 492	. . . . . . 7 |- ( ( X e. CC /\ k e. ( 0 ... ( 0 + 1 ) ) ) -> ( ( 2 _C k ) x. ( ( k BernPoly X ) / ( ( 2 - k ) + 1 ) ) ) e. CC )
76	12, 75, 59	fsump1 14487	. . . . . 6 |- ( X e. CC -> sum_ k e. ( 0 ... ( 0 + 1 ) ) ( ( 2 _C k ) x. ( ( k BernPoly X ) / ( ( 2 - k ) + 1 ) ) ) = ( sum_ k e. ( 0 ... 0 ) ( ( 2 _C k ) x. ( ( k BernPoly X ) / ( ( 2 - k ) + 1 ) ) ) + ( 2 x. ( ( 1 BernPoly X ) / 2 ) ) ) )
77	42, 40	syl6eqel 2709	. . . . . . . . 9 |- ( X e. CC -> ( 1 x. ( ( 0 BernPoly X ) / 3 ) ) e. CC )
78	34	fsum1 14476	. . . . . . . . 9 |- ( ( 0 e. ZZ /\ ( 1 x. ( ( 0 BernPoly X ) / 3 ) ) e. CC ) -> sum_ k e. ( 0 ... 0 ) ( ( 2 _C k ) x. ( ( k BernPoly X ) / ( ( 2 - k ) + 1 ) ) ) = ( 1 x. ( ( 0 BernPoly X ) / 3 ) ) )
79	13, 77, 78	sylancr 695	. . . . . . . 8 |- ( X e. CC -> sum_ k e. ( 0 ... 0 ) ( ( 2 _C k ) x. ( ( k BernPoly X ) / ( ( 2 - k ) + 1 ) ) ) = ( 1 x. ( ( 0 BernPoly X ) / 3 ) ) )
80	79, 42	eqtrd 2656	. . . . . . 7 |- ( X e. CC -> sum_ k e. ( 0 ... 0 ) ( ( 2 _C k ) x. ( ( k BernPoly X ) / ( ( 2 - k ) + 1 ) ) ) = ( 1 / 3 ) )
81	80, 70	oveq12d 6668	. . . . . 6 |- ( X e. CC -> ( sum_ k e. ( 0 ... 0 ) ( ( 2 _C k ) x. ( ( k BernPoly X ) / ( ( 2 - k ) + 1 ) ) ) + ( 2 x. ( ( 1 BernPoly X ) / 2 ) ) ) = ( ( 1 / 3 ) + ( X - ( 1 / 2 ) ) ) )
82	76, 81	eqtrd 2656	. . . . 5 |- ( X e. CC -> sum_ k e. ( 0 ... ( 0 + 1 ) ) ( ( 2 _C k ) x. ( ( k BernPoly X ) / ( ( 2 - k ) + 1 ) ) ) = ( ( 1 / 3 ) + ( X - ( 1 / 2 ) ) ) )
83		addsub12 10294	. . . . . . 7 |- ( ( ( 1 / 3 ) e. CC /\ X e. CC /\ ( 1 / 2 ) e. CC ) -> ( ( 1 / 3 ) + ( X - ( 1 / 2 ) ) ) = ( X + ( ( 1 / 3 ) - ( 1 / 2 ) ) ) )
84	40, 63, 83	mp3an13 1415	. . . . . 6 |- ( X e. CC -> ( ( 1 / 3 ) + ( X - ( 1 / 2 ) ) ) = ( X + ( ( 1 / 3 ) - ( 1 / 2 ) ) ) )
85	63, 40	negsubdi2i 10367	. . . . . . . 8 |- -u ( ( 1 / 2 ) - ( 1 / 3 ) ) = ( ( 1 / 3 ) - ( 1 / 2 ) )
86		halfthird 11685	. . . . . . . . 9 |- ( ( 1 / 2 ) - ( 1 / 3 ) ) = ( 1 / 6 )
87	86	negeqi 10274	. . . . . . . 8 |- -u ( ( 1 / 2 ) - ( 1 / 3 ) ) = -u ( 1 / 6 )
88	85, 87	eqtr3i 2646	. . . . . . 7 |- ( ( 1 / 3 ) - ( 1 / 2 ) ) = -u ( 1 / 6 )
89	88	oveq2i 6661	. . . . . 6 |- ( X + ( ( 1 / 3 ) - ( 1 / 2 ) ) ) = ( X + -u ( 1 / 6 ) )
90	84, 89	syl6eq 2672	. . . . 5 |- ( X e. CC -> ( ( 1 / 3 ) + ( X - ( 1 / 2 ) ) ) = ( X + -u ( 1 / 6 ) ) )
91		6cn 11102	. . . . . . 7 |- 6 e. CC
92		6re 11101	. . . . . . . 8 |- 6 e. RR
93		6pos 11119	. . . . . . . 8 |- 0 < 6
94	92, 93	gt0ne0ii 10564	. . . . . . 7 |- 6 =/= 0
95	91, 94	reccli 10755	. . . . . 6 |- ( 1 / 6 ) e. CC
96		negsub 10329	. . . . . 6 |- ( ( X e. CC /\ ( 1 / 6 ) e. CC ) -> ( X + -u ( 1 / 6 ) ) = ( X - ( 1 / 6 ) ) )
97	95, 96	mpan2 707	. . . . 5 |- ( X e. CC -> ( X + -u ( 1 / 6 ) ) = ( X - ( 1 / 6 ) ) )
98	82, 90, 97	3eqtrd 2660	. . . 4 |- ( X e. CC -> sum_ k e. ( 0 ... ( 0 + 1 ) ) ( ( 2 _C k ) x. ( ( k BernPoly X ) / ( ( 2 - k ) + 1 ) ) ) = ( X - ( 1 / 6 ) ) )
99	8, 98	syl5eq 2668	. . 3 |- ( X e. CC -> sum_ k e. ( 0 ... ( 2 - 1 ) ) ( ( 2 _C k ) x. ( ( k BernPoly X ) / ( ( 2 - k ) + 1 ) ) ) = ( X - ( 1 / 6 ) ) )
100	99	oveq2d 6666	. 2 |- ( X e. CC -> ( ( X ^ 2 ) - sum_ k e. ( 0 ... ( 2 - 1 ) ) ( ( 2 _C k ) x. ( ( k BernPoly X ) / ( ( 2 - k ) + 1 ) ) ) ) = ( ( X ^ 2 ) - ( X - ( 1 / 6 ) ) ) )
101		sqcl 12925	. . 3 |- ( X e. CC -> ( X ^ 2 ) e. CC )
102		subsub 10311	. . . 4 |- ( ( ( X ^ 2 ) e. CC /\ X e. CC /\ ( 1 / 6 ) e. CC ) -> ( ( X ^ 2 ) - ( X - ( 1 / 6 ) ) ) = ( ( ( X ^ 2 ) - X ) + ( 1 / 6 ) ) )
103	95, 102	mp3an3 1413	. . 3 |- ( ( ( X ^ 2 ) e. CC /\ X e. CC ) -> ( ( X ^ 2 ) - ( X - ( 1 / 6 ) ) ) = ( ( ( X ^ 2 ) - X ) + ( 1 / 6 ) ) )
104	101, 103	mpancom 703	. 2 |- ( X e. CC -> ( ( X ^ 2 ) - ( X - ( 1 / 6 ) ) ) = ( ( ( X ^ 2 ) - X ) + ( 1 / 6 ) ) )
105	3, 100, 104	3eqtrd 2660	1 |- ( X e. CC -> ( 2 BernPoly X ) = ( ( ( X ^ 2 ) - X ) + ( 1 / 6 ) ) )

Syntax hints:   -> wi 4   \/ wo 383   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   {cpr 4179   ` cfv 5888  (class class class)co 6650   CCcc 9934   0cc0 9936   1c1 9937   + caddc 9939   x. cmul 9941   - cmin 10266   -ucneg 10267   / cdiv 10684   2c2 11070   3c3 11071   6c6 11074   NN0cn0 11292   ZZcz 11377   ZZ>=cuz 11687   ...cfz 12326   ^cexp 12860   _C cbc 13089   sum_csu 14416   BernPoly cbp 14777

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-fal 1489  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-sup 8348  df-oi 8415  df-card 8765  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-4 11081  df-5 11082  df-6 11083  df-n0 11293  df-z 11378  df-uz 11688  df-rp 11833  df-fz 12327  df-fzo 12466  df-seq 12802  df-exp 12861  df-fac 13061  df-bc 13090  df-hash 13118  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-clim 14219  df-sum 14417  df-bpoly 14778

This theorem is referenced by:  bpoly3  14789  bpoly4  14790