Theorem etransclem1 40452

Description: H is a function. (Contributed by Glauco Siliprandi, 5-Apr-2020.)

Hypotheses

Ref	Expression
etransclem1.x	|- ( ph -> X C_ CC )
etransclem1.p	|- ( ph -> P e. NN )
etransclem1.h	|- H = ( j e. ( 0 ... M ) |-> ( x e. X |-> ( ( x - j ) ^ if ( j = 0 , ( P - 1 ) , P ) ) ) )
etransclem1.j	|- ( ph -> J e. ( 0 ... M ) )

Assertion

Ref	Expression
etransclem1	|- ( ph -> ( H ` J ) : X --> CC )

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

Allowed substitution hints:   ph( j)   P( x)   H( x, j)   J( j)   M( x)

Proof of Theorem etransclem1

Dummy variable n is distinct from all other variables.

Step	Hyp	Ref	Expression
1		etransclem1.x	. . . . . 6 |- ( ph -> X C_ CC )
2	1	sselda 3603	. . . . 5 |- ( ( ph /\ x e. X ) -> x e. CC )
3		etransclem1.j	. . . . . . . 8 |- ( ph -> J e. ( 0 ... M ) )
4	3	elfzelzd 39530	. . . . . . 7 |- ( ph -> J e. ZZ )
5	4	zcnd 11483	. . . . . 6 |- ( ph -> J e. CC )
6	5	adantr 481	. . . . 5 |- ( ( ph /\ x e. X ) -> J e. CC )
7	2, 6	subcld 10392	. . . 4 |- ( ( ph /\ x e. X ) -> ( x - J ) e. CC )
8		etransclem1.p	. . . . . . 7 |- ( ph -> P e. NN )
9		nnm1nn0 11334	. . . . . . 7 |- ( P e. NN -> ( P - 1 ) e. NN0 )
10	8, 9	syl 17	. . . . . 6 |- ( ph -> ( P - 1 ) e. NN0 )
11	8	nnnn0d 11351	. . . . . 6 |- ( ph -> P e. NN0 )
12	10, 11	ifcld 4131	. . . . 5 |- ( ph -> if ( J = 0 , ( P - 1 ) , P ) e. NN0 )
13	12	adantr 481	. . . 4 |- ( ( ph /\ x e. X ) -> if ( J = 0 , ( P - 1 ) , P ) e. NN0 )
14	7, 13	expcld 13008	. . 3 |- ( ( ph /\ x e. X ) -> ( ( x - J ) ^ if ( J = 0 , ( P - 1 ) , P ) ) e. CC )
15		eqid 2622	. . 3 |- ( x e. X |-> ( ( x - J ) ^ if ( J = 0 , ( P - 1 ) , P ) ) ) = ( x e. X |-> ( ( x - J ) ^ if ( J = 0 , ( P - 1 ) , P ) ) )
16	14, 15	fmptd 6385	. 2 |- ( ph -> ( x e. X |-> ( ( x - J ) ^ if ( J = 0 , ( P - 1 ) , P ) ) ) : X --> CC )
17		etransclem1.h	. . . . . 6 |- H = ( j e. ( 0 ... M ) |-> ( x e. X |-> ( ( x - j ) ^ if ( j = 0 , ( P - 1 ) , P ) ) ) )
18		oveq2 6658	. . . . . . . . 9 |- ( j = n -> ( x - j ) = ( x - n ) )
19		eqeq1 2626	. . . . . . . . . 10 |- ( j = n -> ( j = 0 <-> n = 0 ) )
20	19	ifbid 4108	. . . . . . . . 9 |- ( j = n -> if ( j = 0 , ( P - 1 ) , P ) = if ( n = 0 , ( P - 1 ) , P ) )
21	18, 20	oveq12d 6668	. . . . . . . 8 |- ( j = n -> ( ( x - j ) ^ if ( j = 0 , ( P - 1 ) , P ) ) = ( ( x - n ) ^ if ( n = 0 , ( P - 1 ) , P ) ) )
22	21	mpteq2dv 4745	. . . . . . 7 |- ( j = n -> ( x e. X |-> ( ( x - j ) ^ if ( j = 0 , ( P - 1 ) , P ) ) ) = ( x e. X |-> ( ( x - n ) ^ if ( n = 0 , ( P - 1 ) , P ) ) ) )
23	22	cbvmptv 4750	. . . . . 6 |- ( j e. ( 0 ... M ) |-> ( x e. X |-> ( ( x - j ) ^ if ( j = 0 , ( P - 1 ) , P ) ) ) ) = ( n e. ( 0 ... M ) |-> ( x e. X |-> ( ( x - n ) ^ if ( n = 0 , ( P - 1 ) , P ) ) ) )
24	17, 23	eqtri 2644	. . . . 5 |- H = ( n e. ( 0 ... M ) |-> ( x e. X |-> ( ( x - n ) ^ if ( n = 0 , ( P - 1 ) , P ) ) ) )
25	24	a1i 11	. . . 4 |- ( ph -> H = ( n e. ( 0 ... M ) |-> ( x e. X |-> ( ( x - n ) ^ if ( n = 0 , ( P - 1 ) , P ) ) ) ) )
26		oveq2 6658	. . . . . . 7 |- ( n = J -> ( x - n ) = ( x - J ) )
27		eqeq1 2626	. . . . . . . 8 |- ( n = J -> ( n = 0 <-> J = 0 ) )
28	27	ifbid 4108	. . . . . . 7 |- ( n = J -> if ( n = 0 , ( P - 1 ) , P ) = if ( J = 0 , ( P - 1 ) , P ) )
29	26, 28	oveq12d 6668	. . . . . 6 |- ( n = J -> ( ( x - n ) ^ if ( n = 0 , ( P - 1 ) , P ) ) = ( ( x - J ) ^ if ( J = 0 , ( P - 1 ) , P ) ) )
30	29	mpteq2dv 4745	. . . . 5 |- ( n = J -> ( x e. X |-> ( ( x - n ) ^ if ( n = 0 , ( P - 1 ) , P ) ) ) = ( x e. X |-> ( ( x - J ) ^ if ( J = 0 , ( P - 1 ) , P ) ) ) )
31	30	adantl 482	. . . 4 |- ( ( ph /\ n = J ) -> ( x e. X |-> ( ( x - n ) ^ if ( n = 0 , ( P - 1 ) , P ) ) ) = ( x e. X |-> ( ( x - J ) ^ if ( J = 0 , ( P - 1 ) , P ) ) ) )
32		cnex 10017	. . . . . 6 |- CC e. _V
33	32	ssex 4802	. . . . 5 |- ( X C_ CC -> X e. _V )
34		mptexg 6484	. . . . 5 |- ( X e. _V -> ( x e. X |-> ( ( x - J ) ^ if ( J = 0 , ( P - 1 ) , P ) ) ) e. _V )
35	1, 33, 34	3syl 18	. . . 4 |- ( ph -> ( x e. X |-> ( ( x - J ) ^ if ( J = 0 , ( P - 1 ) , P ) ) ) e. _V )
36	25, 31, 3, 35	fvmptd 6288	. . 3 |- ( ph -> ( H ` J ) = ( x e. X |-> ( ( x - J ) ^ if ( J = 0 , ( P - 1 ) , P ) ) ) )
37	36	feq1d 6030	. 2 |- ( ph -> ( ( H ` J ) : X --> CC <-> ( x e. X |-> ( ( x - J ) ^ if ( J = 0 , ( P - 1 ) , P ) ) ) : X --> CC ) )
38	16, 37	mpbird 247	1 |- ( ph -> ( H ` J ) : X --> CC )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   _Vcvv 3200   C_ wss 3574   ifcif 4086   |-> cmpt 4729   -->wf 5884   ` cfv 5888  (class class class)co 6650   CCcc 9934   0cc0 9936   1c1 9937   - cmin 10266   NNcn 11020   NN0cn0 11292   ...cfz 12326   ^cexp 12860

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-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

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-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-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-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-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-n0 11293  df-z 11378  df-uz 11688  df-fz 12327  df-seq 12802  df-exp 12861

This theorem is referenced by:  etransclem29  40480