Theorem coemulc 24011

Description: The coefficient function is linear under scalar multiplication. (Contributed by Mario Carneiro, 24-Jul-2014.)

Assertion

Ref	Expression
coemulc	|- ( ( A e. CC /\ F e. (Poly ` S ) ) -> (coeff ` ( ( CC X. { A } ) oF x. F ) ) = ( ( NN0 X. { A } ) oF x. (coeff ` F ) ) )

Proof of Theorem coemulc

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

Step	Hyp	Ref	Expression
1		ssid 3624	. . . . 5 |- CC C_ CC
2		plyconst 23962	. . . . 5 |- ( ( CC C_ CC /\ A e. CC ) -> ( CC X. { A } ) e. (Poly ` CC ) )
3	1, 2	mpan 706	. . . 4 |- ( A e. CC -> ( CC X. { A } ) e. (Poly ` CC ) )
4		plyssc 23956	. . . . 5 |- (Poly ` S ) C_ (Poly ` CC )
5	4	sseli 3599	. . . 4 |- ( F e. (Poly ` S ) -> F e. (Poly ` CC ) )
6		plymulcl 23977	. . . 4 |- ( ( ( CC X. { A } ) e. (Poly ` CC ) /\ F e. (Poly ` CC ) ) -> ( ( CC X. { A } ) oF x. F ) e. (Poly ` CC ) )
7	3, 5, 6	syl2an 494	. . 3 |- ( ( A e. CC /\ F e. (Poly ` S ) ) -> ( ( CC X. { A } ) oF x. F ) e. (Poly ` CC ) )
8		eqid 2622	. . . 4 |- (coeff ` ( ( CC X. { A } ) oF x. F ) ) = (coeff ` ( ( CC X. { A } ) oF x. F ) )
9	8	coef3 23988	. . 3 |- ( ( ( CC X. { A } ) oF x. F ) e. (Poly ` CC ) -> (coeff ` ( ( CC X. { A } ) oF x. F ) ) : NN0 --> CC )
10		ffn 6045	. . 3 |- ( (coeff ` ( ( CC X. { A } ) oF x. F ) ) : NN0 --> CC -> (coeff ` ( ( CC X. { A } ) oF x. F ) ) Fn NN0 )
11	7, 9, 10	3syl 18	. 2 |- ( ( A e. CC /\ F e. (Poly ` S ) ) -> (coeff ` ( ( CC X. { A } ) oF x. F ) ) Fn NN0 )
12		fconstg 6092	. . . . 5 |- ( A e. CC -> ( NN0 X. { A } ) : NN0 --> { A } )
13	12	adantr 481	. . . 4 |- ( ( A e. CC /\ F e. (Poly ` S ) ) -> ( NN0 X. { A } ) : NN0 --> { A } )
14		ffn 6045	. . . 4 |- ( ( NN0 X. { A } ) : NN0 --> { A } -> ( NN0 X. { A } ) Fn NN0 )
15	13, 14	syl 17	. . 3 |- ( ( A e. CC /\ F e. (Poly ` S ) ) -> ( NN0 X. { A } ) Fn NN0 )
16		eqid 2622	. . . . . 6 |- (coeff ` F ) = (coeff ` F )
17	16	coef3 23988	. . . . 5 |- ( F e. (Poly ` S ) -> (coeff ` F ) : NN0 --> CC )
18	17	adantl 482	. . . 4 |- ( ( A e. CC /\ F e. (Poly ` S ) ) -> (coeff ` F ) : NN0 --> CC )
19		ffn 6045	. . . 4 |- ( (coeff ` F ) : NN0 --> CC -> (coeff ` F ) Fn NN0 )
20	18, 19	syl 17	. . 3 |- ( ( A e. CC /\ F e. (Poly ` S ) ) -> (coeff ` F ) Fn NN0 )
21		nn0ex 11298	. . . 4 |- NN0 e. _V
22	21	a1i 11	. . 3 |- ( ( A e. CC /\ F e. (Poly ` S ) ) -> NN0 e. _V )
23		inidm 3822	. . 3 |- ( NN0 i^i NN0 ) = NN0
24	15, 20, 22, 22, 23	offn 6908	. 2 |- ( ( A e. CC /\ F e. (Poly ` S ) ) -> ( ( NN0 X. { A } ) oF x. (coeff ` F ) ) Fn NN0 )
25	3	ad2antrr 762	. . . . . 6 |- ( ( ( A e. CC /\ F e. (Poly ` S ) ) /\ n e. NN0 ) -> ( CC X. { A } ) e. (Poly ` CC ) )
26		eqid 2622	. . . . . . 7 |- (coeff ` ( CC X. { A } ) ) = (coeff ` ( CC X. { A } ) )
27	26	coefv0 24004	. . . . . 6 |- ( ( CC X. { A } ) e. (Poly ` CC ) -> ( ( CC X. { A } ) ` 0 ) = ( (coeff ` ( CC X. { A } ) ) ` 0 ) )
28	25, 27	syl 17	. . . . 5 |- ( ( ( A e. CC /\ F e. (Poly ` S ) ) /\ n e. NN0 ) -> ( ( CC X. { A } ) ` 0 ) = ( (coeff ` ( CC X. { A } ) ) ` 0 ) )
29		simpll 790	. . . . . 6 |- ( ( ( A e. CC /\ F e. (Poly ` S ) ) /\ n e. NN0 ) -> A e. CC )
30		0cn 10032	. . . . . 6 |- 0 e. CC
31		fvconst2g 6467	. . . . . 6 |- ( ( A e. CC /\ 0 e. CC ) -> ( ( CC X. { A } ) ` 0 ) = A )
32	29, 30, 31	sylancl 694	. . . . 5 |- ( ( ( A e. CC /\ F e. (Poly ` S ) ) /\ n e. NN0 ) -> ( ( CC X. { A } ) ` 0 ) = A )
33	28, 32	eqtr3d 2658	. . . 4 |- ( ( ( A e. CC /\ F e. (Poly ` S ) ) /\ n e. NN0 ) -> ( (coeff ` ( CC X. { A } ) ) ` 0 ) = A )
34		simpr 477	. . . . . . 7 |- ( ( ( A e. CC /\ F e. (Poly ` S ) ) /\ n e. NN0 ) -> n e. NN0 )
35	34	nn0cnd 11353	. . . . . 6 |- ( ( ( A e. CC /\ F e. (Poly ` S ) ) /\ n e. NN0 ) -> n e. CC )
36	35	subid1d 10381	. . . . 5 |- ( ( ( A e. CC /\ F e. (Poly ` S ) ) /\ n e. NN0 ) -> ( n - 0 ) = n )
37	36	fveq2d 6195	. . . 4 |- ( ( ( A e. CC /\ F e. (Poly ` S ) ) /\ n e. NN0 ) -> ( (coeff ` F ) ` ( n - 0 ) ) = ( (coeff ` F ) ` n ) )
38	33, 37	oveq12d 6668	. . 3 |- ( ( ( A e. CC /\ F e. (Poly ` S ) ) /\ n e. NN0 ) -> ( ( (coeff ` ( CC X. { A } ) ) ` 0 ) x. ( (coeff ` F ) ` ( n - 0 ) ) ) = ( A x. ( (coeff ` F ) ` n ) ) )
39	5	ad2antlr 763	. . . . 5 |- ( ( ( A e. CC /\ F e. (Poly ` S ) ) /\ n e. NN0 ) -> F e. (Poly ` CC ) )
40	26, 16	coemul 24008	. . . . 5 |- ( ( ( CC X. { A } ) e. (Poly ` CC ) /\ F e. (Poly ` CC ) /\ n e. NN0 ) -> ( (coeff ` ( ( CC X. { A } ) oF x. F ) ) ` n ) = sum_ k e. ( 0 ... n ) ( ( (coeff ` ( CC X. { A } ) ) ` k ) x. ( (coeff ` F ) ` ( n - k ) ) ) )
41	25, 39, 34, 40	syl3anc 1326	. . . 4 |- ( ( ( A e. CC /\ F e. (Poly ` S ) ) /\ n e. NN0 ) -> ( (coeff ` ( ( CC X. { A } ) oF x. F ) ) ` n ) = sum_ k e. ( 0 ... n ) ( ( (coeff ` ( CC X. { A } ) ) ` k ) x. ( (coeff ` F ) ` ( n - k ) ) ) )
42		nn0uz 11722	. . . . . . 7 |- NN0 = ( ZZ>= ` 0 )
43	34, 42	syl6eleq 2711	. . . . . 6 |- ( ( ( A e. CC /\ F e. (Poly ` S ) ) /\ n e. NN0 ) -> n e. ( ZZ>= ` 0 ) )
44		fzss2 12381	. . . . . 6 |- ( n e. ( ZZ>= ` 0 ) -> ( 0 ... 0 ) C_ ( 0 ... n ) )
45	43, 44	syl 17	. . . . 5 |- ( ( ( A e. CC /\ F e. (Poly ` S ) ) /\ n e. NN0 ) -> ( 0 ... 0 ) C_ ( 0 ... n ) )
46		elfz1eq 12352	. . . . . . . 8 |- ( k e. ( 0 ... 0 ) -> k = 0 )
47	46	adantl 482	. . . . . . 7 |- ( ( ( ( A e. CC /\ F e. (Poly ` S ) ) /\ n e. NN0 ) /\ k e. ( 0 ... 0 ) ) -> k = 0 )
48		fveq2 6191	. . . . . . . 8 |- ( k = 0 -> ( (coeff ` ( CC X. { A } ) ) ` k ) = ( (coeff ` ( CC X. { A } ) ) ` 0 ) )
49		oveq2 6658	. . . . . . . . 9 |- ( k = 0 -> ( n - k ) = ( n - 0 ) )
50	49	fveq2d 6195	. . . . . . . 8 |- ( k = 0 -> ( (coeff ` F ) ` ( n - k ) ) = ( (coeff ` F ) ` ( n - 0 ) ) )
51	48, 50	oveq12d 6668	. . . . . . 7 |- ( k = 0 -> ( ( (coeff ` ( CC X. { A } ) ) ` k ) x. ( (coeff ` F ) ` ( n - k ) ) ) = ( ( (coeff ` ( CC X. { A } ) ) ` 0 ) x. ( (coeff ` F ) ` ( n - 0 ) ) ) )
52	47, 51	syl 17	. . . . . 6 |- ( ( ( ( A e. CC /\ F e. (Poly ` S ) ) /\ n e. NN0 ) /\ k e. ( 0 ... 0 ) ) -> ( ( (coeff ` ( CC X. { A } ) ) ` k ) x. ( (coeff ` F ) ` ( n - k ) ) ) = ( ( (coeff ` ( CC X. { A } ) ) ` 0 ) x. ( (coeff ` F ) ` ( n - 0 ) ) ) )
53	18	ffvelrnda 6359	. . . . . . . . 9 |- ( ( ( A e. CC /\ F e. (Poly ` S ) ) /\ n e. NN0 ) -> ( (coeff ` F ) ` n ) e. CC )
54	29, 53	mulcld 10060	. . . . . . . 8 |- ( ( ( A e. CC /\ F e. (Poly ` S ) ) /\ n e. NN0 ) -> ( A x. ( (coeff ` F ) ` n ) ) e. CC )
55	38, 54	eqeltrd 2701	. . . . . . 7 |- ( ( ( A e. CC /\ F e. (Poly ` S ) ) /\ n e. NN0 ) -> ( ( (coeff ` ( CC X. { A } ) ) ` 0 ) x. ( (coeff ` F ) ` ( n - 0 ) ) ) e. CC )
56	55	adantr 481	. . . . . 6 |- ( ( ( ( A e. CC /\ F e. (Poly ` S ) ) /\ n e. NN0 ) /\ k e. ( 0 ... 0 ) ) -> ( ( (coeff ` ( CC X. { A } ) ) ` 0 ) x. ( (coeff ` F ) ` ( n - 0 ) ) ) e. CC )
57	52, 56	eqeltrd 2701	. . . . 5 |- ( ( ( ( A e. CC /\ F e. (Poly ` S ) ) /\ n e. NN0 ) /\ k e. ( 0 ... 0 ) ) -> ( ( (coeff ` ( CC X. { A } ) ) ` k ) x. ( (coeff ` F ) ` ( n - k ) ) ) e. CC )
58		eldifn 3733	. . . . . . . . 9 |- ( k e. ( ( 0 ... n ) \ ( 0 ... 0 ) ) -> -. k e. ( 0 ... 0 ) )
59	58	adantl 482	. . . . . . . 8 |- ( ( ( ( A e. CC /\ F e. (Poly ` S ) ) /\ n e. NN0 ) /\ k e. ( ( 0 ... n ) \ ( 0 ... 0 ) ) ) -> -. k e. ( 0 ... 0 ) )
60		eldifi 3732	. . . . . . . . . . . . 13 |- ( k e. ( ( 0 ... n ) \ ( 0 ... 0 ) ) -> k e. ( 0 ... n ) )
61		elfznn0 12433	. . . . . . . . . . . . 13 |- ( k e. ( 0 ... n ) -> k e. NN0 )
62	60, 61	syl 17	. . . . . . . . . . . 12 |- ( k e. ( ( 0 ... n ) \ ( 0 ... 0 ) ) -> k e. NN0 )
63		eqid 2622	. . . . . . . . . . . . . 14 |- (deg ` ( CC X. { A } ) ) = (deg ` ( CC X. { A } ) )
64	26, 63	dgrub 23990	. . . . . . . . . . . . 13 |- ( ( ( CC X. { A } ) e. (Poly ` CC ) /\ k e. NN0 /\ ( (coeff ` ( CC X. { A } ) ) ` k ) =/= 0 ) -> k <_ (deg ` ( CC X. { A } ) ) )
65	64	3expia 1267	. . . . . . . . . . . 12 |- ( ( ( CC X. { A } ) e. (Poly ` CC ) /\ k e. NN0 ) -> ( ( (coeff ` ( CC X. { A } ) ) ` k ) =/= 0 -> k <_ (deg ` ( CC X. { A } ) ) ) )
66	25, 62, 65	syl2an 494	. . . . . . . . . . 11 |- ( ( ( ( A e. CC /\ F e. (Poly ` S ) ) /\ n e. NN0 ) /\ k e. ( ( 0 ... n ) \ ( 0 ... 0 ) ) ) -> ( ( (coeff ` ( CC X. { A } ) ) ` k ) =/= 0 -> k <_ (deg ` ( CC X. { A } ) ) ) )
67		0dgr 24001	. . . . . . . . . . . . . 14 |- ( A e. CC -> (deg ` ( CC X. { A } ) ) = 0 )
68	67	ad3antrrr 766	. . . . . . . . . . . . 13 |- ( ( ( ( A e. CC /\ F e. (Poly ` S ) ) /\ n e. NN0 ) /\ k e. ( ( 0 ... n ) \ ( 0 ... 0 ) ) ) -> (deg ` ( CC X. { A } ) ) = 0 )
69	68	breq2d 4665	. . . . . . . . . . . 12 |- ( ( ( ( A e. CC /\ F e. (Poly ` S ) ) /\ n e. NN0 ) /\ k e. ( ( 0 ... n ) \ ( 0 ... 0 ) ) ) -> ( k <_ (deg ` ( CC X. { A } ) ) <-> k <_ 0 ) )
70	62	adantl 482	. . . . . . . . . . . . 13 |- ( ( ( ( A e. CC /\ F e. (Poly ` S ) ) /\ n e. NN0 ) /\ k e. ( ( 0 ... n ) \ ( 0 ... 0 ) ) ) -> k e. NN0 )
71		nn0le0eq0 11321	. . . . . . . . . . . . 13 |- ( k e. NN0 -> ( k <_ 0 <-> k = 0 ) )
72	70, 71	syl 17	. . . . . . . . . . . 12 |- ( ( ( ( A e. CC /\ F e. (Poly ` S ) ) /\ n e. NN0 ) /\ k e. ( ( 0 ... n ) \ ( 0 ... 0 ) ) ) -> ( k <_ 0 <-> k = 0 ) )
73	69, 72	bitrd 268	. . . . . . . . . . 11 |- ( ( ( ( A e. CC /\ F e. (Poly ` S ) ) /\ n e. NN0 ) /\ k e. ( ( 0 ... n ) \ ( 0 ... 0 ) ) ) -> ( k <_ (deg ` ( CC X. { A } ) ) <-> k = 0 ) )
74	66, 73	sylibd 229	. . . . . . . . . 10 |- ( ( ( ( A e. CC /\ F e. (Poly ` S ) ) /\ n e. NN0 ) /\ k e. ( ( 0 ... n ) \ ( 0 ... 0 ) ) ) -> ( ( (coeff ` ( CC X. { A } ) ) ` k ) =/= 0 -> k = 0 ) )
75		id 22	. . . . . . . . . . 11 |- ( k = 0 -> k = 0 )
76		0z 11388	. . . . . . . . . . . 12 |- 0 e. ZZ
77		elfz3 12351	. . . . . . . . . . . 12 |- ( 0 e. ZZ -> 0 e. ( 0 ... 0 ) )
78	76, 77	ax-mp 5	. . . . . . . . . . 11 |- 0 e. ( 0 ... 0 )
79	75, 78	syl6eqel 2709	. . . . . . . . . 10 |- ( k = 0 -> k e. ( 0 ... 0 ) )
80	74, 79	syl6 35	. . . . . . . . 9 |- ( ( ( ( A e. CC /\ F e. (Poly ` S ) ) /\ n e. NN0 ) /\ k e. ( ( 0 ... n ) \ ( 0 ... 0 ) ) ) -> ( ( (coeff ` ( CC X. { A } ) ) ` k ) =/= 0 -> k e. ( 0 ... 0 ) ) )
81	80	necon1bd 2812	. . . . . . . 8 |- ( ( ( ( A e. CC /\ F e. (Poly ` S ) ) /\ n e. NN0 ) /\ k e. ( ( 0 ... n ) \ ( 0 ... 0 ) ) ) -> ( -. k e. ( 0 ... 0 ) -> ( (coeff ` ( CC X. { A } ) ) ` k ) = 0 ) )
82	59, 81	mpd 15	. . . . . . 7 |- ( ( ( ( A e. CC /\ F e. (Poly ` S ) ) /\ n e. NN0 ) /\ k e. ( ( 0 ... n ) \ ( 0 ... 0 ) ) ) -> ( (coeff ` ( CC X. { A } ) ) ` k ) = 0 )
83	82	oveq1d 6665	. . . . . 6 |- ( ( ( ( A e. CC /\ F e. (Poly ` S ) ) /\ n e. NN0 ) /\ k e. ( ( 0 ... n ) \ ( 0 ... 0 ) ) ) -> ( ( (coeff ` ( CC X. { A } ) ) ` k ) x. ( (coeff ` F ) ` ( n - k ) ) ) = ( 0 x. ( (coeff ` F ) ` ( n - k ) ) ) )
84	18	adantr 481	. . . . . . . 8 |- ( ( ( A e. CC /\ F e. (Poly ` S ) ) /\ n e. NN0 ) -> (coeff ` F ) : NN0 --> CC )
85		fznn0sub 12373	. . . . . . . . 9 |- ( k e. ( 0 ... n ) -> ( n - k ) e. NN0 )
86	60, 85	syl 17	. . . . . . . 8 |- ( k e. ( ( 0 ... n ) \ ( 0 ... 0 ) ) -> ( n - k ) e. NN0 )
87		ffvelrn 6357	. . . . . . . 8 |- ( ( (coeff ` F ) : NN0 --> CC /\ ( n - k ) e. NN0 ) -> ( (coeff ` F ) ` ( n - k ) ) e. CC )
88	84, 86, 87	syl2an 494	. . . . . . 7 |- ( ( ( ( A e. CC /\ F e. (Poly ` S ) ) /\ n e. NN0 ) /\ k e. ( ( 0 ... n ) \ ( 0 ... 0 ) ) ) -> ( (coeff ` F ) ` ( n - k ) ) e. CC )
89	88	mul02d 10234	. . . . . 6 |- ( ( ( ( A e. CC /\ F e. (Poly ` S ) ) /\ n e. NN0 ) /\ k e. ( ( 0 ... n ) \ ( 0 ... 0 ) ) ) -> ( 0 x. ( (coeff ` F ) ` ( n - k ) ) ) = 0 )
90	83, 89	eqtrd 2656	. . . . 5 |- ( ( ( ( A e. CC /\ F e. (Poly ` S ) ) /\ n e. NN0 ) /\ k e. ( ( 0 ... n ) \ ( 0 ... 0 ) ) ) -> ( ( (coeff ` ( CC X. { A } ) ) ` k ) x. ( (coeff ` F ) ` ( n - k ) ) ) = 0 )
91		fzfid 12772	. . . . 5 |- ( ( ( A e. CC /\ F e. (Poly ` S ) ) /\ n e. NN0 ) -> ( 0 ... n ) e. Fin )
92	45, 57, 90, 91	fsumss 14456	. . . 4 |- ( ( ( A e. CC /\ F e. (Poly ` S ) ) /\ n e. NN0 ) -> sum_ k e. ( 0 ... 0 ) ( ( (coeff ` ( CC X. { A } ) ) ` k ) x. ( (coeff ` F ) ` ( n - k ) ) ) = sum_ k e. ( 0 ... n ) ( ( (coeff ` ( CC X. { A } ) ) ` k ) x. ( (coeff ` F ) ` ( n - k ) ) ) )
93	51	fsum1 14476	. . . . 5 |- ( ( 0 e. ZZ /\ ( ( (coeff ` ( CC X. { A } ) ) ` 0 ) x. ( (coeff ` F ) ` ( n - 0 ) ) ) e. CC ) -> sum_ k e. ( 0 ... 0 ) ( ( (coeff ` ( CC X. { A } ) ) ` k ) x. ( (coeff ` F ) ` ( n - k ) ) ) = ( ( (coeff ` ( CC X. { A } ) ) ` 0 ) x. ( (coeff ` F ) ` ( n - 0 ) ) ) )
94	76, 55, 93	sylancr 695	. . . 4 |- ( ( ( A e. CC /\ F e. (Poly ` S ) ) /\ n e. NN0 ) -> sum_ k e. ( 0 ... 0 ) ( ( (coeff ` ( CC X. { A } ) ) ` k ) x. ( (coeff ` F ) ` ( n - k ) ) ) = ( ( (coeff ` ( CC X. { A } ) ) ` 0 ) x. ( (coeff ` F ) ` ( n - 0 ) ) ) )
95	41, 92, 94	3eqtr2d 2662	. . 3 |- ( ( ( A e. CC /\ F e. (Poly ` S ) ) /\ n e. NN0 ) -> ( (coeff ` ( ( CC X. { A } ) oF x. F ) ) ` n ) = ( ( (coeff ` ( CC X. { A } ) ) ` 0 ) x. ( (coeff ` F ) ` ( n - 0 ) ) ) )
96		simpl 473	. . . 4 |- ( ( A e. CC /\ F e. (Poly ` S ) ) -> A e. CC )
97		eqidd 2623	. . . 4 |- ( ( ( A e. CC /\ F e. (Poly ` S ) ) /\ n e. NN0 ) -> ( (coeff ` F ) ` n ) = ( (coeff ` F ) ` n ) )
98	22, 96, 20, 97	ofc1 6920	. . 3 |- ( ( ( A e. CC /\ F e. (Poly ` S ) ) /\ n e. NN0 ) -> ( ( ( NN0 X. { A } ) oF x. (coeff ` F ) ) ` n ) = ( A x. ( (coeff ` F ) ` n ) ) )
99	38, 95, 98	3eqtr4d 2666	. 2 |- ( ( ( A e. CC /\ F e. (Poly ` S ) ) /\ n e. NN0 ) -> ( (coeff ` ( ( CC X. { A } ) oF x. F ) ) ` n ) = ( ( ( NN0 X. { A } ) oF x. (coeff ` F ) ) ` n ) )
100	11, 24, 99	eqfnfvd 6314	1 |- ( ( A e. CC /\ F e. (Poly ` S ) ) -> (coeff ` ( ( CC X. { A } ) oF x. F ) ) = ( ( NN0 X. { A } ) oF x. (coeff ` F ) ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   _Vcvv 3200   \ cdif 3571   C_ wss 3574   {csn 4177   class class class wbr 4653   X. cxp 5112   Fn wfn 5883   -->wf 5884   ` cfv 5888  (class class class)co 6650   oFcof 6895   CCcc 9934   0cc0 9936   x. cmul 9941   <_ cle 10075   - cmin 10266   NN0cn0 11292   ZZcz 11377   ZZ>=cuz 11687   ...cfz 12326   sum_csu 14416  Polycply 23940  coeffccoe 23942  degcdgr 23943

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  ax-addf 10015

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-of 6897  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-map 7859  df-pm 7860  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-sup 8348  df-inf 8349  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-n0 11293  df-z 11378  df-uz 11688  df-rp 11833  df-fz 12327  df-fzo 12466  df-fl 12593  df-seq 12802  df-exp 12861  df-hash 13118  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-clim 14219  df-rlim 14220  df-sum 14417  df-0p 23437  df-ply 23944  df-coe 23946  df-dgr 23947

This theorem is referenced by:  coe0  24012  coesub  24013  mpaaeu  37720