Theorem taylfvallem1 24111

Description: Lemma for taylfval 24113. (Contributed by Mario Carneiro, 30-Dec-2016.)

Hypotheses

Ref	Expression
taylfval.s	|- ( ph -> S e. { RR , CC } )
taylfval.f	|- ( ph -> F : A --> CC )
taylfval.a	|- ( ph -> A C_ S )
taylfval.n	|- ( ph -> ( N e. NN0 \/ N = +oo ) )
taylfval.b	|- ( ( ph /\ k e. ( ( 0 [,] N ) i^i ZZ ) ) -> B e. dom ( ( S Dn F ) ` k ) )

Assertion

Ref	Expression
taylfvallem1	|- ( ( ( ph /\ X e. CC ) /\ k e. ( ( 0 [,] N ) i^i ZZ ) ) -> ( ( ( ( ( S Dn F ) ` k ) ` B ) / ( ! ` k ) ) x. ( ( X - B ) ^ k ) ) e. CC )

Distinct variable groups:   B, k   k, F   ph, k   k, N   S, k   k, X

Allowed substitution hint:   A( k)

Proof of Theorem taylfvallem1

Step	Hyp	Ref	Expression
1		taylfval.s	. . . . . 6 |- ( ph -> S e. { RR , CC } )
2	1	ad2antrr 762	. . . . 5 |- ( ( ( ph /\ X e. CC ) /\ k e. ( ( 0 [,] N ) i^i ZZ ) ) -> S e. { RR , CC } )
3		cnex 10017	. . . . . . . 8 |- CC e. _V
4	3	a1i 11	. . . . . . 7 |- ( ph -> CC e. _V )
5		taylfval.f	. . . . . . 7 |- ( ph -> F : A --> CC )
6		taylfval.a	. . . . . . 7 |- ( ph -> A C_ S )
7		elpm2r 7875	. . . . . . 7 |- ( ( ( CC e. _V /\ S e. { RR , CC } ) /\ ( F : A --> CC /\ A C_ S ) ) -> F e. ( CC ^pm S ) )
8	4, 1, 5, 6, 7	syl22anc 1327	. . . . . 6 |- ( ph -> F e. ( CC ^pm S ) )
9	8	ad2antrr 762	. . . . 5 |- ( ( ( ph /\ X e. CC ) /\ k e. ( ( 0 [,] N ) i^i ZZ ) ) -> F e. ( CC ^pm S ) )
10		inss2 3834	. . . . . . 7 |- ( ( 0 [,] N ) i^i ZZ ) C_ ZZ
11		simpr 477	. . . . . . 7 |- ( ( ( ph /\ X e. CC ) /\ k e. ( ( 0 [,] N ) i^i ZZ ) ) -> k e. ( ( 0 [,] N ) i^i ZZ ) )
12	10, 11	sseldi 3601	. . . . . 6 |- ( ( ( ph /\ X e. CC ) /\ k e. ( ( 0 [,] N ) i^i ZZ ) ) -> k e. ZZ )
13		inss1 3833	. . . . . . . . 9 |- ( ( 0 [,] N ) i^i ZZ ) C_ ( 0 [,] N )
14	13, 11	sseldi 3601	. . . . . . . 8 |- ( ( ( ph /\ X e. CC ) /\ k e. ( ( 0 [,] N ) i^i ZZ ) ) -> k e. ( 0 [,] N ) )
15		0xr 10086	. . . . . . . . 9 |- 0 e. RR*
16		taylfval.n	. . . . . . . . . . 11 |- ( ph -> ( N e. NN0 \/ N = +oo ) )
17		nn0re 11301	. . . . . . . . . . . . 13 |- ( N e. NN0 -> N e. RR )
18	17	rexrd 10089	. . . . . . . . . . . 12 |- ( N e. NN0 -> N e. RR* )
19		id 22	. . . . . . . . . . . . 13 |- ( N = +oo -> N = +oo )
20		pnfxr 10092	. . . . . . . . . . . . 13 |- +oo e. RR*
21	19, 20	syl6eqel 2709	. . . . . . . . . . . 12 |- ( N = +oo -> N e. RR* )
22	18, 21	jaoi 394	. . . . . . . . . . 11 |- ( ( N e. NN0 \/ N = +oo ) -> N e. RR* )
23	16, 22	syl 17	. . . . . . . . . 10 |- ( ph -> N e. RR* )
24	23	ad2antrr 762	. . . . . . . . 9 |- ( ( ( ph /\ X e. CC ) /\ k e. ( ( 0 [,] N ) i^i ZZ ) ) -> N e. RR* )
25		elicc1 12219	. . . . . . . . 9 |- ( ( 0 e. RR* /\ N e. RR* ) -> ( k e. ( 0 [,] N ) <-> ( k e. RR* /\ 0 <_ k /\ k <_ N ) ) )
26	15, 24, 25	sylancr 695	. . . . . . . 8 |- ( ( ( ph /\ X e. CC ) /\ k e. ( ( 0 [,] N ) i^i ZZ ) ) -> ( k e. ( 0 [,] N ) <-> ( k e. RR* /\ 0 <_ k /\ k <_ N ) ) )
27	14, 26	mpbid 222	. . . . . . 7 |- ( ( ( ph /\ X e. CC ) /\ k e. ( ( 0 [,] N ) i^i ZZ ) ) -> ( k e. RR* /\ 0 <_ k /\ k <_ N ) )
28	27	simp2d 1074	. . . . . 6 |- ( ( ( ph /\ X e. CC ) /\ k e. ( ( 0 [,] N ) i^i ZZ ) ) -> 0 <_ k )
29		elnn0z 11390	. . . . . 6 |- ( k e. NN0 <-> ( k e. ZZ /\ 0 <_ k ) )
30	12, 28, 29	sylanbrc 698	. . . . 5 |- ( ( ( ph /\ X e. CC ) /\ k e. ( ( 0 [,] N ) i^i ZZ ) ) -> k e. NN0 )
31		dvnf 23690	. . . . 5 |- ( ( S e. { RR , CC } /\ F e. ( CC ^pm S ) /\ k e. NN0 ) -> ( ( S Dn F ) ` k ) : dom ( ( S Dn F ) ` k ) --> CC )
32	2, 9, 30, 31	syl3anc 1326	. . . 4 |- ( ( ( ph /\ X e. CC ) /\ k e. ( ( 0 [,] N ) i^i ZZ ) ) -> ( ( S Dn F ) ` k ) : dom ( ( S Dn F ) ` k ) --> CC )
33		taylfval.b	. . . . 5 |- ( ( ph /\ k e. ( ( 0 [,] N ) i^i ZZ ) ) -> B e. dom ( ( S Dn F ) ` k ) )
34	33	adantlr 751	. . . 4 |- ( ( ( ph /\ X e. CC ) /\ k e. ( ( 0 [,] N ) i^i ZZ ) ) -> B e. dom ( ( S Dn F ) ` k ) )
35	32, 34	ffvelrnd 6360	. . 3 |- ( ( ( ph /\ X e. CC ) /\ k e. ( ( 0 [,] N ) i^i ZZ ) ) -> ( ( ( S Dn F ) ` k ) ` B ) e. CC )
36		faccl 13070	. . . . 5 |- ( k e. NN0 -> ( ! ` k ) e. NN )
37	30, 36	syl 17	. . . 4 |- ( ( ( ph /\ X e. CC ) /\ k e. ( ( 0 [,] N ) i^i ZZ ) ) -> ( ! ` k ) e. NN )
38	37	nncnd 11036	. . 3 |- ( ( ( ph /\ X e. CC ) /\ k e. ( ( 0 [,] N ) i^i ZZ ) ) -> ( ! ` k ) e. CC )
39	37	nnne0d 11065	. . 3 |- ( ( ( ph /\ X e. CC ) /\ k e. ( ( 0 [,] N ) i^i ZZ ) ) -> ( ! ` k ) =/= 0 )
40	35, 38, 39	divcld 10801	. 2 |- ( ( ( ph /\ X e. CC ) /\ k e. ( ( 0 [,] N ) i^i ZZ ) ) -> ( ( ( ( S Dn F ) ` k ) ` B ) / ( ! ` k ) ) e. CC )
41		simplr 792	. . . 4 |- ( ( ( ph /\ X e. CC ) /\ k e. ( ( 0 [,] N ) i^i ZZ ) ) -> X e. CC )
42		dvnbss 23691	. . . . . . . 8 |- ( ( S e. { RR , CC } /\ F e. ( CC ^pm S ) /\ k e. NN0 ) -> dom ( ( S Dn F ) ` k ) C_ dom F )
43	2, 9, 30, 42	syl3anc 1326	. . . . . . 7 |- ( ( ( ph /\ X e. CC ) /\ k e. ( ( 0 [,] N ) i^i ZZ ) ) -> dom ( ( S Dn F ) ` k ) C_ dom F )
44	5	ad2antrr 762	. . . . . . . 8 |- ( ( ( ph /\ X e. CC ) /\ k e. ( ( 0 [,] N ) i^i ZZ ) ) -> F : A --> CC )
45		fdm 6051	. . . . . . . 8 |- ( F : A --> CC -> dom F = A )
46	44, 45	syl 17	. . . . . . 7 |- ( ( ( ph /\ X e. CC ) /\ k e. ( ( 0 [,] N ) i^i ZZ ) ) -> dom F = A )
47	43, 46	sseqtrd 3641	. . . . . 6 |- ( ( ( ph /\ X e. CC ) /\ k e. ( ( 0 [,] N ) i^i ZZ ) ) -> dom ( ( S Dn F ) ` k ) C_ A )
48		recnprss 23668	. . . . . . . . 9 |- ( S e. { RR , CC } -> S C_ CC )
49	1, 48	syl 17	. . . . . . . 8 |- ( ph -> S C_ CC )
50	6, 49	sstrd 3613	. . . . . . 7 |- ( ph -> A C_ CC )
51	50	ad2antrr 762	. . . . . 6 |- ( ( ( ph /\ X e. CC ) /\ k e. ( ( 0 [,] N ) i^i ZZ ) ) -> A C_ CC )
52	47, 51	sstrd 3613	. . . . 5 |- ( ( ( ph /\ X e. CC ) /\ k e. ( ( 0 [,] N ) i^i ZZ ) ) -> dom ( ( S Dn F ) ` k ) C_ CC )
53	52, 34	sseldd 3604	. . . 4 |- ( ( ( ph /\ X e. CC ) /\ k e. ( ( 0 [,] N ) i^i ZZ ) ) -> B e. CC )
54	41, 53	subcld 10392	. . 3 |- ( ( ( ph /\ X e. CC ) /\ k e. ( ( 0 [,] N ) i^i ZZ ) ) -> ( X - B ) e. CC )
55	54, 30	expcld 13008	. 2 |- ( ( ( ph /\ X e. CC ) /\ k e. ( ( 0 [,] N ) i^i ZZ ) ) -> ( ( X - B ) ^ k ) e. CC )
56	40, 55	mulcld 10060	1 |- ( ( ( ph /\ X e. CC ) /\ k e. ( ( 0 [,] N ) i^i ZZ ) ) -> ( ( ( ( ( S Dn F ) ` k ) ` B ) / ( ! ` k ) ) x. ( ( X - B ) ^ k ) ) e. CC )

Syntax hints:   -> wi 4   <-> wb 196   \/ wo 383   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   _Vcvv 3200   i^i cin 3573   C_ wss 3574   {cpr 4179   class class class wbr 4653   dom cdm 5114   -->wf 5884   ` cfv 5888  (class class class)co 6650   ^pm cpm 7858   CCcc 9934   RRcr 9935   0cc0 9936   x. cmul 9941   +oocpnf 10071   RR*cxr 10073   <_ cle 10075   - cmin 10266   / cdiv 10684   NNcn 11020   NN0cn0 11292   ZZcz 11377   [,]cicc 12178   ^cexp 12860   !cfa 13060   Dncdvn 23628

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-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-iin 4523  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-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-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-map 7859  df-pm 7860  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-fi 8317  df-sup 8348  df-inf 8349  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-7 11084  df-8 11085  df-9 11086  df-n0 11293  df-z 11378  df-dec 11494  df-uz 11688  df-q 11789  df-rp 11833  df-xneg 11946  df-xadd 11947  df-xmul 11948  df-icc 12182  df-fz 12327  df-seq 12802  df-exp 12861  df-fac 13061  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-struct 15859  df-ndx 15860  df-slot 15861  df-base 15863  df-plusg 15954  df-mulr 15955  df-starv 15956  df-tset 15960  df-ple 15961  df-ds 15964  df-unif 15965  df-rest 16083  df-topn 16084  df-topgen 16104  df-psmet 19738  df-xmet 19739  df-met 19740  df-bl 19741  df-mopn 19742  df-fbas 19743  df-fg 19744  df-cnfld 19747  df-top 20699  df-topon 20716  df-topsp 20737  df-bases 20750  df-cld 20823  df-ntr 20824  df-cls 20825  df-nei 20902  df-lp 20940  df-perf 20941  df-cnp 21032  df-haus 21119  df-fil 21650  df-fm 21742  df-flim 21743  df-flf 21744  df-xms 22125  df-ms 22126  df-limc 23630  df-dv 23631  df-dvn 23632

This theorem is referenced by:  taylfvallem  24112  taylf  24115  taylplem2  24118  taylpfval  24119