Theorem coe1termlem 24014

Description: The coefficient function of a monomial. (Contributed by Mario Carneiro, 26-Jul-2014.) (Revised by Mario Carneiro, 23-Aug-2014.)

Hypothesis

Ref	Expression
coe1term.1	|- F = ( z e. CC |-> ( A x. ( z ^ N ) ) )

Assertion

Ref	Expression
coe1termlem	|- ( ( A e. CC /\ N e. NN0 ) -> ( (coeff ` F ) = ( n e. NN0 |-> if ( n = N , A , 0 ) ) /\ ( A =/= 0 -> (deg ` F ) = N ) ) )

Distinct variable groups:   z, n, A   n, N, z

Allowed substitution hints:   F( z, n)

Proof of Theorem coe1termlem

Dummy variable k is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ssid 3624	. . . 4 |- CC C_ CC
2		coe1term.1	. . . . 5 |- F = ( z e. CC |-> ( A x. ( z ^ N ) ) )
3	2	ply1term 23960	. . . 4 |- ( ( CC C_ CC /\ A e. CC /\ N e. NN0 ) -> F e. (Poly ` CC ) )
4	1, 3	mp3an1 1411	. . 3 |- ( ( A e. CC /\ N e. NN0 ) -> F e. (Poly ` CC ) )
5		simpr 477	. . 3 |- ( ( A e. CC /\ N e. NN0 ) -> N e. NN0 )
6		simpl 473	. . . . . 6 |- ( ( A e. CC /\ N e. NN0 ) -> A e. CC )
7		0cn 10032	. . . . . 6 |- 0 e. CC
8		ifcl 4130	. . . . . 6 |- ( ( A e. CC /\ 0 e. CC ) -> if ( n = N , A , 0 ) e. CC )
9	6, 7, 8	sylancl 694	. . . . 5 |- ( ( A e. CC /\ N e. NN0 ) -> if ( n = N , A , 0 ) e. CC )
10	9	adantr 481	. . . 4 |- ( ( ( A e. CC /\ N e. NN0 ) /\ n e. NN0 ) -> if ( n = N , A , 0 ) e. CC )
11		eqid 2622	. . . 4 |- ( n e. NN0 |-> if ( n = N , A , 0 ) ) = ( n e. NN0 |-> if ( n = N , A , 0 ) )
12	10, 11	fmptd 6385	. . 3 |- ( ( A e. CC /\ N e. NN0 ) -> ( n e. NN0 |-> if ( n = N , A , 0 ) ) : NN0 --> CC )
13		simpr 477	. . . . . . . 8 |- ( ( ( A e. CC /\ N e. NN0 ) /\ k e. NN0 ) -> k e. NN0 )
14		ifcl 4130	. . . . . . . . . 10 |- ( ( A e. CC /\ 0 e. CC ) -> if ( k = N , A , 0 ) e. CC )
15	6, 7, 14	sylancl 694	. . . . . . . . 9 |- ( ( A e. CC /\ N e. NN0 ) -> if ( k = N , A , 0 ) e. CC )
16	15	adantr 481	. . . . . . . 8 |- ( ( ( A e. CC /\ N e. NN0 ) /\ k e. NN0 ) -> if ( k = N , A , 0 ) e. CC )
17		eqeq1 2626	. . . . . . . . . 10 |- ( n = k -> ( n = N <-> k = N ) )
18	17	ifbid 4108	. . . . . . . . 9 |- ( n = k -> if ( n = N , A , 0 ) = if ( k = N , A , 0 ) )
19	18, 11	fvmptg 6280	. . . . . . . 8 |- ( ( k e. NN0 /\ if ( k = N , A , 0 ) e. CC ) -> ( ( n e. NN0 |-> if ( n = N , A , 0 ) ) ` k ) = if ( k = N , A , 0 ) )
20	13, 16, 19	syl2anc 693	. . . . . . 7 |- ( ( ( A e. CC /\ N e. NN0 ) /\ k e. NN0 ) -> ( ( n e. NN0 |-> if ( n = N , A , 0 ) ) ` k ) = if ( k = N , A , 0 ) )
21	20	neeq1d 2853	. . . . . 6 |- ( ( ( A e. CC /\ N e. NN0 ) /\ k e. NN0 ) -> ( ( ( n e. NN0 |-> if ( n = N , A , 0 ) ) ` k ) =/= 0 <-> if ( k = N , A , 0 ) =/= 0 ) )
22		nn0re 11301	. . . . . . . . 9 |- ( N e. NN0 -> N e. RR )
23	22	leidd 10594	. . . . . . . 8 |- ( N e. NN0 -> N <_ N )
24	23	ad2antlr 763	. . . . . . 7 |- ( ( ( A e. CC /\ N e. NN0 ) /\ k e. NN0 ) -> N <_ N )
25		iffalse 4095	. . . . . . . . 9 |- ( -. k = N -> if ( k = N , A , 0 ) = 0 )
26	25	necon1ai 2821	. . . . . . . 8 |- ( if ( k = N , A , 0 ) =/= 0 -> k = N )
27	26	breq1d 4663	. . . . . . 7 |- ( if ( k = N , A , 0 ) =/= 0 -> ( k <_ N <-> N <_ N ) )
28	24, 27	syl5ibrcom 237	. . . . . 6 |- ( ( ( A e. CC /\ N e. NN0 ) /\ k e. NN0 ) -> ( if ( k = N , A , 0 ) =/= 0 -> k <_ N ) )
29	21, 28	sylbid 230	. . . . 5 |- ( ( ( A e. CC /\ N e. NN0 ) /\ k e. NN0 ) -> ( ( ( n e. NN0 |-> if ( n = N , A , 0 ) ) ` k ) =/= 0 -> k <_ N ) )
30	29	ralrimiva 2966	. . . 4 |- ( ( A e. CC /\ N e. NN0 ) -> A. k e. NN0 ( ( ( n e. NN0 |-> if ( n = N , A , 0 ) ) ` k ) =/= 0 -> k <_ N ) )
31		plyco0 23948	. . . . 5 |- ( ( N e. NN0 /\ ( n e. NN0 |-> if ( n = N , A , 0 ) ) : NN0 --> CC ) -> ( ( ( n e. NN0 |-> if ( n = N , A , 0 ) ) " ( ZZ>= ` ( N + 1 ) ) ) = { 0 } <-> A. k e. NN0 ( ( ( n e. NN0 |-> if ( n = N , A , 0 ) ) ` k ) =/= 0 -> k <_ N ) ) )
32	5, 12, 31	syl2anc 693	. . . 4 |- ( ( A e. CC /\ N e. NN0 ) -> ( ( ( n e. NN0 |-> if ( n = N , A , 0 ) ) " ( ZZ>= ` ( N + 1 ) ) ) = { 0 } <-> A. k e. NN0 ( ( ( n e. NN0 |-> if ( n = N , A , 0 ) ) ` k ) =/= 0 -> k <_ N ) ) )
33	30, 32	mpbird 247	. . 3 |- ( ( A e. CC /\ N e. NN0 ) -> ( ( n e. NN0 |-> if ( n = N , A , 0 ) ) " ( ZZ>= ` ( N + 1 ) ) ) = { 0 } )
34	2	ply1termlem 23959	. . . 4 |- ( ( A e. CC /\ N e. NN0 ) -> F = ( z e. CC |-> sum_ k e. ( 0 ... N ) ( if ( k = N , A , 0 ) x. ( z ^ k ) ) ) )
35		elfznn0 12433	. . . . . . 7 |- ( k e. ( 0 ... N ) -> k e. NN0 )
36	20	oveq1d 6665	. . . . . . 7 |- ( ( ( A e. CC /\ N e. NN0 ) /\ k e. NN0 ) -> ( ( ( n e. NN0 |-> if ( n = N , A , 0 ) ) ` k ) x. ( z ^ k ) ) = ( if ( k = N , A , 0 ) x. ( z ^ k ) ) )
37	35, 36	sylan2 491	. . . . . 6 |- ( ( ( A e. CC /\ N e. NN0 ) /\ k e. ( 0 ... N ) ) -> ( ( ( n e. NN0 |-> if ( n = N , A , 0 ) ) ` k ) x. ( z ^ k ) ) = ( if ( k = N , A , 0 ) x. ( z ^ k ) ) )
38	37	sumeq2dv 14433	. . . . 5 |- ( ( A e. CC /\ N e. NN0 ) -> sum_ k e. ( 0 ... N ) ( ( ( n e. NN0 |-> if ( n = N , A , 0 ) ) ` k ) x. ( z ^ k ) ) = sum_ k e. ( 0 ... N ) ( if ( k = N , A , 0 ) x. ( z ^ k ) ) )
39	38	mpteq2dv 4745	. . . 4 |- ( ( A e. CC /\ N e. NN0 ) -> ( z e. CC |-> sum_ k e. ( 0 ... N ) ( ( ( n e. NN0 |-> if ( n = N , A , 0 ) ) ` k ) x. ( z ^ k ) ) ) = ( z e. CC |-> sum_ k e. ( 0 ... N ) ( if ( k = N , A , 0 ) x. ( z ^ k ) ) ) )
40	34, 39	eqtr4d 2659	. . 3 |- ( ( A e. CC /\ N e. NN0 ) -> F = ( z e. CC |-> sum_ k e. ( 0 ... N ) ( ( ( n e. NN0 |-> if ( n = N , A , 0 ) ) ` k ) x. ( z ^ k ) ) ) )
41	4, 5, 12, 33, 40	coeeq 23983	. 2 |- ( ( A e. CC /\ N e. NN0 ) -> (coeff ` F ) = ( n e. NN0 |-> if ( n = N , A , 0 ) ) )
42	4	adantr 481	. . . 4 |- ( ( ( A e. CC /\ N e. NN0 ) /\ A =/= 0 ) -> F e. (Poly ` CC ) )
43	5	adantr 481	. . . 4 |- ( ( ( A e. CC /\ N e. NN0 ) /\ A =/= 0 ) -> N e. NN0 )
44	12	adantr 481	. . . 4 |- ( ( ( A e. CC /\ N e. NN0 ) /\ A =/= 0 ) -> ( n e. NN0 |-> if ( n = N , A , 0 ) ) : NN0 --> CC )
45	33	adantr 481	. . . 4 |- ( ( ( A e. CC /\ N e. NN0 ) /\ A =/= 0 ) -> ( ( n e. NN0 |-> if ( n = N , A , 0 ) ) " ( ZZ>= ` ( N + 1 ) ) ) = { 0 } )
46	40	adantr 481	. . . 4 |- ( ( ( A e. CC /\ N e. NN0 ) /\ A =/= 0 ) -> F = ( z e. CC |-> sum_ k e. ( 0 ... N ) ( ( ( n e. NN0 |-> if ( n = N , A , 0 ) ) ` k ) x. ( z ^ k ) ) ) )
47		iftrue 4092	. . . . . . . 8 |- ( n = N -> if ( n = N , A , 0 ) = A )
48	47, 11	fvmptg 6280	. . . . . . 7 |- ( ( N e. NN0 /\ A e. CC ) -> ( ( n e. NN0 |-> if ( n = N , A , 0 ) ) ` N ) = A )
49	48	ancoms 469	. . . . . 6 |- ( ( A e. CC /\ N e. NN0 ) -> ( ( n e. NN0 |-> if ( n = N , A , 0 ) ) ` N ) = A )
50	49	neeq1d 2853	. . . . 5 |- ( ( A e. CC /\ N e. NN0 ) -> ( ( ( n e. NN0 |-> if ( n = N , A , 0 ) ) ` N ) =/= 0 <-> A =/= 0 ) )
51	50	biimpar 502	. . . 4 |- ( ( ( A e. CC /\ N e. NN0 ) /\ A =/= 0 ) -> ( ( n e. NN0 |-> if ( n = N , A , 0 ) ) ` N ) =/= 0 )
52	42, 43, 44, 45, 46, 51	dgreq 24000	. . 3 |- ( ( ( A e. CC /\ N e. NN0 ) /\ A =/= 0 ) -> (deg ` F ) = N )
53	52	ex 450	. 2 |- ( ( A e. CC /\ N e. NN0 ) -> ( A =/= 0 -> (deg ` F ) = N ) )
54	41, 53	jca 554	1 |- ( ( A e. CC /\ N e. NN0 ) -> ( (coeff ` F ) = ( n e. NN0 |-> if ( n = N , A , 0 ) ) /\ ( A =/= 0 -> (deg ` F ) = N ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   A.wral 2912   C_ wss 3574   ifcif 4086   {csn 4177   class class class wbr 4653   |-> cmpt 4729   "cima 5117   -->wf 5884   ` cfv 5888  (class class class)co 6650   CCcc 9934   0cc0 9936   1c1 9937   + caddc 9939   x. cmul 9941   <_ cle 10075   NN0cn0 11292   ZZ>=cuz 11687   ...cfz 12326   ^cexp 12860   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:  coe1term  24015  dgr1term  24016