Theorem etransclem13 40464

Description: F applied to Y. (Contributed by Glauco Siliprandi, 5-Apr-2020.)

Hypotheses

Ref	Expression
etransclem13.x	|- ( ph -> X C_ CC )
etransclem13.p	|- ( ph -> P e. NN )
etransclem13.m	|- ( ph -> M e. NN0 )
etransclem13.f	|- F = ( x e. X |-> ( ( x ^ ( P - 1 ) ) x. prod_ j e. ( 1 ... M ) ( ( x - j ) ^ P ) ) )
etransclem13.y	|- ( ph -> Y e. X )

Assertion

Ref	Expression
etransclem13	|- ( ph -> ( F ` Y ) = prod_ j e. ( 0 ... M ) ( ( Y - j ) ^ if ( j = 0 , ( P - 1 ) , P ) ) )

Distinct variable groups:   j, M, x   P, j, x   j, X, x   j, Y, x   ph, j, x

Allowed substitution hints:   F( x, j)

Proof of Theorem etransclem13

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		etransclem13.x	. . 3 |- ( ph -> X C_ CC )
2		etransclem13.p	. . 3 |- ( ph -> P e. NN )
3		etransclem13.m	. . 3 |- ( ph -> M e. NN0 )
4		etransclem13.f	. . 3 |- F = ( x e. X |-> ( ( x ^ ( P - 1 ) ) x. prod_ j e. ( 1 ... M ) ( ( x - j ) ^ P ) ) )
5		eqid 2622	. . 3 |- ( j e. ( 0 ... M ) |-> ( x e. X |-> ( ( x - j ) ^ if ( j = 0 , ( P - 1 ) , P ) ) ) ) = ( j e. ( 0 ... M ) |-> ( x e. X |-> ( ( x - j ) ^ if ( j = 0 , ( P - 1 ) , P ) ) ) )
6		eqid 2622	. . 3 |- ( x e. X |-> prod_ j e. ( 0 ... M ) ( ( ( j e. ( 0 ... M ) |-> ( x e. X |-> ( ( x - j ) ^ if ( j = 0 , ( P - 1 ) , P ) ) ) ) ` j ) ` x ) ) = ( x e. X |-> prod_ j e. ( 0 ... M ) ( ( ( j e. ( 0 ... M ) |-> ( x e. X |-> ( ( x - j ) ^ if ( j = 0 , ( P - 1 ) , P ) ) ) ) ` j ) ` x ) )
7	1, 2, 3, 4, 5, 6	etransclem4 40455	. 2 |- ( ph -> F = ( x e. X |-> prod_ j e. ( 0 ... M ) ( ( ( j e. ( 0 ... M ) |-> ( x e. X |-> ( ( x - j ) ^ if ( j = 0 , ( P - 1 ) , P ) ) ) ) ` j ) ` x ) ) )
8		simpr 477	. . . . . 6 |- ( ( ph /\ j e. ( 0 ... M ) ) -> j e. ( 0 ... M ) )
9		cnex 10017	. . . . . . . . 9 |- CC e. _V
10	9	ssex 4802	. . . . . . . 8 |- ( X C_ CC -> X e. _V )
11		mptexg 6484	. . . . . . . 8 |- ( X e. _V -> ( y e. X |-> ( ( y - j ) ^ if ( j = 0 , ( P - 1 ) , P ) ) ) e. _V )
12	1, 10, 11	3syl 18	. . . . . . 7 |- ( ph -> ( y e. X |-> ( ( y - j ) ^ if ( j = 0 , ( P - 1 ) , P ) ) ) e. _V )
13	12	adantr 481	. . . . . 6 |- ( ( ph /\ j e. ( 0 ... M ) ) -> ( y e. X |-> ( ( y - j ) ^ if ( j = 0 , ( P - 1 ) , P ) ) ) e. _V )
14		oveq1 6657	. . . . . . . . . 10 |- ( x = y -> ( x - j ) = ( y - j ) )
15	14	oveq1d 6665	. . . . . . . . 9 |- ( x = y -> ( ( x - j ) ^ if ( j = 0 , ( P - 1 ) , P ) ) = ( ( y - j ) ^ if ( j = 0 , ( P - 1 ) , P ) ) )
16	15	cbvmptv 4750	. . . . . . . 8 |- ( x e. X |-> ( ( x - j ) ^ if ( j = 0 , ( P - 1 ) , P ) ) ) = ( y e. X |-> ( ( y - j ) ^ if ( j = 0 , ( P - 1 ) , P ) ) )
17	16	mpteq2i 4741	. . . . . . 7 |- ( j e. ( 0 ... M ) |-> ( x e. X |-> ( ( x - j ) ^ if ( j = 0 , ( P - 1 ) , P ) ) ) ) = ( j e. ( 0 ... M ) |-> ( y e. X |-> ( ( y - j ) ^ if ( j = 0 , ( P - 1 ) , P ) ) ) )
18	17	fvmpt2 6291	. . . . . 6 |- ( ( j e. ( 0 ... M ) /\ ( y e. X |-> ( ( y - j ) ^ if ( j = 0 , ( P - 1 ) , P ) ) ) e. _V ) -> ( ( j e. ( 0 ... M ) |-> ( x e. X |-> ( ( x - j ) ^ if ( j = 0 , ( P - 1 ) , P ) ) ) ) ` j ) = ( y e. X |-> ( ( y - j ) ^ if ( j = 0 , ( P - 1 ) , P ) ) ) )
19	8, 13, 18	syl2anc 693	. . . . 5 |- ( ( ph /\ j e. ( 0 ... M ) ) -> ( ( j e. ( 0 ... M ) |-> ( x e. X |-> ( ( x - j ) ^ if ( j = 0 , ( P - 1 ) , P ) ) ) ) ` j ) = ( y e. X |-> ( ( y - j ) ^ if ( j = 0 , ( P - 1 ) , P ) ) ) )
20	19	adantlr 751	. . . 4 |- ( ( ( ph /\ x = Y ) /\ j e. ( 0 ... M ) ) -> ( ( j e. ( 0 ... M ) |-> ( x e. X |-> ( ( x - j ) ^ if ( j = 0 , ( P - 1 ) , P ) ) ) ) ` j ) = ( y e. X |-> ( ( y - j ) ^ if ( j = 0 , ( P - 1 ) , P ) ) ) )
21		simpr 477	. . . . . . . 8 |- ( ( x = Y /\ y = x ) -> y = x )
22		simpl 473	. . . . . . . 8 |- ( ( x = Y /\ y = x ) -> x = Y )
23	21, 22	eqtrd 2656	. . . . . . 7 |- ( ( x = Y /\ y = x ) -> y = Y )
24		oveq1 6657	. . . . . . . 8 |- ( y = Y -> ( y - j ) = ( Y - j ) )
25	24	oveq1d 6665	. . . . . . 7 |- ( y = Y -> ( ( y - j ) ^ if ( j = 0 , ( P - 1 ) , P ) ) = ( ( Y - j ) ^ if ( j = 0 , ( P - 1 ) , P ) ) )
26	23, 25	syl 17	. . . . . 6 |- ( ( x = Y /\ y = x ) -> ( ( y - j ) ^ if ( j = 0 , ( P - 1 ) , P ) ) = ( ( Y - j ) ^ if ( j = 0 , ( P - 1 ) , P ) ) )
27	26	adantll 750	. . . . 5 |- ( ( ( ph /\ x = Y ) /\ y = x ) -> ( ( y - j ) ^ if ( j = 0 , ( P - 1 ) , P ) ) = ( ( Y - j ) ^ if ( j = 0 , ( P - 1 ) , P ) ) )
28	27	adantlr 751	. . . 4 |- ( ( ( ( ph /\ x = Y ) /\ j e. ( 0 ... M ) ) /\ y = x ) -> ( ( y - j ) ^ if ( j = 0 , ( P - 1 ) , P ) ) = ( ( Y - j ) ^ if ( j = 0 , ( P - 1 ) , P ) ) )
29		simpr 477	. . . . . 6 |- ( ( ph /\ x = Y ) -> x = Y )
30		etransclem13.y	. . . . . . 7 |- ( ph -> Y e. X )
31	30	adantr 481	. . . . . 6 |- ( ( ph /\ x = Y ) -> Y e. X )
32	29, 31	eqeltrd 2701	. . . . 5 |- ( ( ph /\ x = Y ) -> x e. X )
33	32	adantr 481	. . . 4 |- ( ( ( ph /\ x = Y ) /\ j e. ( 0 ... M ) ) -> x e. X )
34		ovexd 6680	. . . 4 |- ( ( ( ph /\ x = Y ) /\ j e. ( 0 ... M ) ) -> ( ( Y - j ) ^ if ( j = 0 , ( P - 1 ) , P ) ) e. _V )
35	20, 28, 33, 34	fvmptd 6288	. . 3 |- ( ( ( ph /\ x = Y ) /\ j e. ( 0 ... M ) ) -> ( ( ( j e. ( 0 ... M ) |-> ( x e. X |-> ( ( x - j ) ^ if ( j = 0 , ( P - 1 ) , P ) ) ) ) ` j ) ` x ) = ( ( Y - j ) ^ if ( j = 0 , ( P - 1 ) , P ) ) )
36	35	prodeq2dv 14653	. 2 |- ( ( ph /\ x = Y ) -> prod_ j e. ( 0 ... M ) ( ( ( j e. ( 0 ... M ) |-> ( x e. X |-> ( ( x - j ) ^ if ( j = 0 , ( P - 1 ) , P ) ) ) ) ` j ) ` x ) = prod_ j e. ( 0 ... M ) ( ( Y - j ) ^ if ( j = 0 , ( P - 1 ) , P ) ) )
37		prodex 14637	. . 3 |- prod_ j e. ( 0 ... M ) ( ( Y - j ) ^ if ( j = 0 , ( P - 1 ) , P ) ) e. _V
38	37	a1i 11	. 2 |- ( ph -> prod_ j e. ( 0 ... M ) ( ( Y - j ) ^ if ( j = 0 , ( P - 1 ) , P ) ) e. _V )
39	7, 36, 30, 38	fvmptd 6288	1 |- ( ph -> ( F ` Y ) = prod_ j e. ( 0 ... M ) ( ( Y - j ) ^ if ( j = 0 , ( P - 1 ) , P ) ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   _Vcvv 3200   C_ wss 3574   ifcif 4086   |-> cmpt 4729   ` cfv 5888  (class class class)co 6650   CCcc 9934   0cc0 9936   1c1 9937   x. cmul 9941   - cmin 10266   NNcn 11020   NN0cn0 11292   ...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:  etransclem18  40469  etransclem23  40474  etransclem46  40497