Theorem etransclem4 40455

Description: F expressed as a finite product. (Contributed by Glauco Siliprandi, 5-Apr-2020.)

Hypotheses

Ref	Expression
etransclem4.a	|- ( ph -> A C_ CC )
etransclem4.p	|- ( ph -> P e. NN )
etransclem4.M	|- ( ph -> M e. NN0 )
etransclem4.f	|- F = ( x e. A |-> ( ( x ^ ( P - 1 ) ) x. prod_ j e. ( 1 ... M ) ( ( x - j ) ^ P ) ) )
etransclem4.h	|- H = ( j e. ( 0 ... M ) |-> ( x e. A |-> ( ( x - j ) ^ if ( j = 0 , ( P - 1 ) , P ) ) ) )
etransclem4.e	|- E = ( x e. A |-> prod_ j e. ( 0 ... M ) ( ( H ` j ) ` x ) )

Assertion

Ref	Expression
etransclem4	|- ( ph -> F = E )

Distinct variable groups:   A, j, x   j, M   P, j   ph, j, x

Allowed substitution hints:   P( x)   E( x, j)   F( x, j)   H( x, j)   M( x)

Proof of Theorem etransclem4

Step	Hyp	Ref	Expression
1		simpr 477	. . . . . . . 8 |- ( ( ph /\ j e. ( 0 ... M ) ) -> j e. ( 0 ... M ) )
2		etransclem4.a	. . . . . . . . . 10 |- ( ph -> A C_ CC )
3		cnex 10017	. . . . . . . . . . 11 |- CC e. _V
4	3	ssex 4802	. . . . . . . . . 10 |- ( A C_ CC -> A e. _V )
5		mptexg 6484	. . . . . . . . . 10 |- ( A e. _V -> ( x e. A |-> ( ( x - j ) ^ if ( j = 0 , ( P - 1 ) , P ) ) ) e. _V )
6	2, 4, 5	3syl 18	. . . . . . . . 9 |- ( ph -> ( x e. A |-> ( ( x - j ) ^ if ( j = 0 , ( P - 1 ) , P ) ) ) e. _V )
7	6	adantr 481	. . . . . . . 8 |- ( ( ph /\ j e. ( 0 ... M ) ) -> ( x e. A |-> ( ( x - j ) ^ if ( j = 0 , ( P - 1 ) , P ) ) ) e. _V )
8		etransclem4.h	. . . . . . . . 9 |- H = ( j e. ( 0 ... M ) |-> ( x e. A |-> ( ( x - j ) ^ if ( j = 0 , ( P - 1 ) , P ) ) ) )
9	8	fvmpt2 6291	. . . . . . . 8 |- ( ( j e. ( 0 ... M ) /\ ( x e. A |-> ( ( x - j ) ^ if ( j = 0 , ( P - 1 ) , P ) ) ) e. _V ) -> ( H ` j ) = ( x e. A |-> ( ( x - j ) ^ if ( j = 0 , ( P - 1 ) , P ) ) ) )
10	1, 7, 9	syl2anc 693	. . . . . . 7 |- ( ( ph /\ j e. ( 0 ... M ) ) -> ( H ` j ) = ( x e. A |-> ( ( x - j ) ^ if ( j = 0 , ( P - 1 ) , P ) ) ) )
11		ovexd 6680	. . . . . . 7 |- ( ( ( ph /\ j e. ( 0 ... M ) ) /\ x e. A ) -> ( ( x - j ) ^ if ( j = 0 , ( P - 1 ) , P ) ) e. _V )
12	10, 11	fvmpt2d 6293	. . . . . 6 |- ( ( ( ph /\ j e. ( 0 ... M ) ) /\ x e. A ) -> ( ( H ` j ) ` x ) = ( ( x - j ) ^ if ( j = 0 , ( P - 1 ) , P ) ) )
13	12	an32s 846	. . . . 5 |- ( ( ( ph /\ x e. A ) /\ j e. ( 0 ... M ) ) -> ( ( H ` j ) ` x ) = ( ( x - j ) ^ if ( j = 0 , ( P - 1 ) , P ) ) )
14	13	prodeq2dv 14653	. . . 4 |- ( ( ph /\ x e. A ) -> prod_ j e. ( 0 ... M ) ( ( H ` j ) ` x ) = prod_ j e. ( 0 ... M ) ( ( x - j ) ^ if ( j = 0 , ( P - 1 ) , P ) ) )
15		etransclem4.M	. . . . . . 7 |- ( ph -> M e. NN0 )
16		nn0uz 11722	. . . . . . 7 |- NN0 = ( ZZ>= ` 0 )
17	15, 16	syl6eleq 2711	. . . . . 6 |- ( ph -> M e. ( ZZ>= ` 0 ) )
18	17	adantr 481	. . . . 5 |- ( ( ph /\ x e. A ) -> M e. ( ZZ>= ` 0 ) )
19	2	sselda 3603	. . . . . . . 8 |- ( ( ph /\ x e. A ) -> x e. CC )
20	19	adantr 481	. . . . . . 7 |- ( ( ( ph /\ x e. A ) /\ j e. ( 0 ... M ) ) -> x e. CC )
21		elfzelz 12342	. . . . . . . . 9 |- ( j e. ( 0 ... M ) -> j e. ZZ )
22	21	zcnd 11483	. . . . . . . 8 |- ( j e. ( 0 ... M ) -> j e. CC )
23	22	adantl 482	. . . . . . 7 |- ( ( ( ph /\ x e. A ) /\ j e. ( 0 ... M ) ) -> j e. CC )
24	20, 23	subcld 10392	. . . . . 6 |- ( ( ( ph /\ x e. A ) /\ j e. ( 0 ... M ) ) -> ( x - j ) e. CC )
25		etransclem4.p	. . . . . . . . 9 |- ( ph -> P e. NN )
26		nnm1nn0 11334	. . . . . . . . 9 |- ( P e. NN -> ( P - 1 ) e. NN0 )
27	25, 26	syl 17	. . . . . . . 8 |- ( ph -> ( P - 1 ) e. NN0 )
28	25	nnnn0d 11351	. . . . . . . 8 |- ( ph -> P e. NN0 )
29	27, 28	ifcld 4131	. . . . . . 7 |- ( ph -> if ( j = 0 , ( P - 1 ) , P ) e. NN0 )
30	29	ad2antrr 762	. . . . . 6 |- ( ( ( ph /\ x e. A ) /\ j e. ( 0 ... M ) ) -> if ( j = 0 , ( P - 1 ) , P ) e. NN0 )
31	24, 30	expcld 13008	. . . . 5 |- ( ( ( ph /\ x e. A ) /\ j e. ( 0 ... M ) ) -> ( ( x - j ) ^ if ( j = 0 , ( P - 1 ) , P ) ) e. CC )
32		oveq2 6658	. . . . . 6 |- ( j = 0 -> ( x - j ) = ( x - 0 ) )
33		iftrue 4092	. . . . . 6 |- ( j = 0 -> if ( j = 0 , ( P - 1 ) , P ) = ( P - 1 ) )
34	32, 33	oveq12d 6668	. . . . 5 |- ( j = 0 -> ( ( x - j ) ^ if ( j = 0 , ( P - 1 ) , P ) ) = ( ( x - 0 ) ^ ( P - 1 ) ) )
35	18, 31, 34	fprod1p 14698	. . . 4 |- ( ( ph /\ x e. A ) -> prod_ j e. ( 0 ... M ) ( ( x - j ) ^ if ( j = 0 , ( P - 1 ) , P ) ) = ( ( ( x - 0 ) ^ ( P - 1 ) ) x. prod_ j e. ( ( 0 + 1 ) ... M ) ( ( x - j ) ^ if ( j = 0 , ( P - 1 ) , P ) ) ) )
36	19	subid1d 10381	. . . . . 6 |- ( ( ph /\ x e. A ) -> ( x - 0 ) = x )
37	36	oveq1d 6665	. . . . 5 |- ( ( ph /\ x e. A ) -> ( ( x - 0 ) ^ ( P - 1 ) ) = ( x ^ ( P - 1 ) ) )
38		0p1e1 11132	. . . . . . . . 9 |- ( 0 + 1 ) = 1
39	38	oveq1i 6660	. . . . . . . 8 |- ( ( 0 + 1 ) ... M ) = ( 1 ... M )
40	39	a1i 11	. . . . . . 7 |- ( ph -> ( ( 0 + 1 ) ... M ) = ( 1 ... M ) )
41		0red 10041	. . . . . . . . . . . . 13 |- ( j e. ( 1 ... M ) -> 0 e. RR )
42		1red 10055	. . . . . . . . . . . . 13 |- ( j e. ( 1 ... M ) -> 1 e. RR )
43		elfzelz 12342	. . . . . . . . . . . . . 14 |- ( j e. ( 1 ... M ) -> j e. ZZ )
44	43	zred 11482	. . . . . . . . . . . . 13 |- ( j e. ( 1 ... M ) -> j e. RR )
45		0lt1 10550	. . . . . . . . . . . . . 14 |- 0 < 1
46	45	a1i 11	. . . . . . . . . . . . 13 |- ( j e. ( 1 ... M ) -> 0 < 1 )
47		elfzle1 12344	. . . . . . . . . . . . 13 |- ( j e. ( 1 ... M ) -> 1 <_ j )
48	41, 42, 44, 46, 47	ltletrd 10197	. . . . . . . . . . . 12 |- ( j e. ( 1 ... M ) -> 0 < j )
49	48	gt0ne0d 10592	. . . . . . . . . . 11 |- ( j e. ( 1 ... M ) -> j =/= 0 )
50	49	neneqd 2799	. . . . . . . . . 10 |- ( j e. ( 1 ... M ) -> -. j = 0 )
51	50	iffalsed 4097	. . . . . . . . 9 |- ( j e. ( 1 ... M ) -> if ( j = 0 , ( P - 1 ) , P ) = P )
52	51	oveq2d 6666	. . . . . . . 8 |- ( j e. ( 1 ... M ) -> ( ( x - j ) ^ if ( j = 0 , ( P - 1 ) , P ) ) = ( ( x - j ) ^ P ) )
53	52	adantl 482	. . . . . . 7 |- ( ( ph /\ j e. ( 1 ... M ) ) -> ( ( x - j ) ^ if ( j = 0 , ( P - 1 ) , P ) ) = ( ( x - j ) ^ P ) )
54	40, 53	prodeq12rdv 14657	. . . . . 6 |- ( ph -> prod_ j e. ( ( 0 + 1 ) ... M ) ( ( x - j ) ^ if ( j = 0 , ( P - 1 ) , P ) ) = prod_ j e. ( 1 ... M ) ( ( x - j ) ^ P ) )
55	54	adantr 481	. . . . 5 |- ( ( ph /\ x e. A ) -> prod_ j e. ( ( 0 + 1 ) ... M ) ( ( x - j ) ^ if ( j = 0 , ( P - 1 ) , P ) ) = prod_ j e. ( 1 ... M ) ( ( x - j ) ^ P ) )
56	37, 55	oveq12d 6668	. . . 4 |- ( ( ph /\ x e. A ) -> ( ( ( x - 0 ) ^ ( P - 1 ) ) x. prod_ j e. ( ( 0 + 1 ) ... M ) ( ( x - j ) ^ if ( j = 0 , ( P - 1 ) , P ) ) ) = ( ( x ^ ( P - 1 ) ) x. prod_ j e. ( 1 ... M ) ( ( x - j ) ^ P ) ) )
57	14, 35, 56	3eqtrrd 2661	. . 3 |- ( ( ph /\ x e. A ) -> ( ( x ^ ( P - 1 ) ) x. prod_ j e. ( 1 ... M ) ( ( x - j ) ^ P ) ) = prod_ j e. ( 0 ... M ) ( ( H ` j ) ` x ) )
58	57	mpteq2dva 4744	. 2 |- ( ph -> ( x e. A |-> ( ( x ^ ( P - 1 ) ) x. prod_ j e. ( 1 ... M ) ( ( x - j ) ^ P ) ) ) = ( x e. A |-> prod_ j e. ( 0 ... M ) ( ( H ` j ) ` x ) ) )
59		etransclem4.f	. 2 |- F = ( x e. A |-> ( ( x ^ ( P - 1 ) ) x. prod_ j e. ( 1 ... M ) ( ( x - j ) ^ P ) ) )
60		etransclem4.e	. 2 |- E = ( x e. A |-> prod_ j e. ( 0 ... M ) ( ( H ` j ) ` x ) )
61	58, 59, 60	3eqtr4g 2681	1 |- ( ph -> F = E )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   _Vcvv 3200   C_ wss 3574   ifcif 4086   class class class wbr 4653   |-> cmpt 4729   ` cfv 5888  (class class class)co 6650   CCcc 9934   0cc0 9936   1c1 9937   + caddc 9939   x. cmul 9941   < clt 10074   - cmin 10266   NNcn 11020   NN0cn0 11292   ZZ>=cuz 11687   ...cfz 12326   ^cexp 12860   prod_cprod 14635

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-n0 11293  df-z 11378  df-uz 11688  df-rp 11833  df-fz 12327  df-fzo 12466  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-prod 14636

This theorem is referenced by:  etransclem13  40464  etransclem29  40480