Theorem mvrval 19421

Description: Value of the generating elements of the power series structure. (Contributed by Mario Carneiro, 7-Jan-2015.)

Hypotheses

Ref	Expression
mvrfval.v	|- V = ( I mVar R )
mvrfval.d	|- D = { h e. ( NN0 ^m I ) | ( `' h " NN ) e. Fin }
mvrfval.z	|- .0. = ( 0g ` R )
mvrfval.o	|- .1. = ( 1r ` R )
mvrfval.i	|- ( ph -> I e. W )
mvrfval.r	|- ( ph -> R e. Y )
mvrval.x	|- ( ph -> X e. I )

Assertion

Ref	Expression
mvrval	|- ( ph -> ( V ` X ) = ( f e. D |-> if ( f = ( y e. I |-> if ( y = X , 1 , 0 ) ) , .1. , .0. ) ) )

Distinct variable groups:   .0. , f   .1. , f   y, f, D   y, W   f, h, I, y   R, f   f, X, h, y

Allowed substitution hints:   ph( y, f, h)   D( h)   R( y, h)   .1. ( y, h)   V( y, f, h)   W( f, h)   Y( y, f, h)   .0. ( y, h)

Proof of Theorem mvrval

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		mvrfval.v	. . . 4 |- V = ( I mVar R )
2		mvrfval.d	. . . 4 |- D = { h e. ( NN0 ^m I ) | ( `' h " NN ) e. Fin }
3		mvrfval.z	. . . 4 |- .0. = ( 0g ` R )
4		mvrfval.o	. . . 4 |- .1. = ( 1r ` R )
5		mvrfval.i	. . . 4 |- ( ph -> I e. W )
6		mvrfval.r	. . . 4 |- ( ph -> R e. Y )
7	1, 2, 3, 4, 5, 6	mvrfval 19420	. . 3 |- ( ph -> V = ( x e. I |-> ( f e. D |-> if ( f = ( y e. I |-> if ( y = x , 1 , 0 ) ) , .1. , .0. ) ) ) )
8	7	fveq1d 6193	. 2 |- ( ph -> ( V ` X ) = ( ( x e. I |-> ( f e. D |-> if ( f = ( y e. I |-> if ( y = x , 1 , 0 ) ) , .1. , .0. ) ) ) ` X ) )
9		mvrval.x	. . 3 |- ( ph -> X e. I )
10		eqeq2 2633	. . . . . . . . 9 |- ( x = X -> ( y = x <-> y = X ) )
11	10	ifbid 4108	. . . . . . . 8 |- ( x = X -> if ( y = x , 1 , 0 ) = if ( y = X , 1 , 0 ) )
12	11	mpteq2dv 4745	. . . . . . 7 |- ( x = X -> ( y e. I |-> if ( y = x , 1 , 0 ) ) = ( y e. I |-> if ( y = X , 1 , 0 ) ) )
13	12	eqeq2d 2632	. . . . . 6 |- ( x = X -> ( f = ( y e. I |-> if ( y = x , 1 , 0 ) ) <-> f = ( y e. I |-> if ( y = X , 1 , 0 ) ) ) )
14	13	ifbid 4108	. . . . 5 |- ( x = X -> if ( f = ( y e. I |-> if ( y = x , 1 , 0 ) ) , .1. , .0. ) = if ( f = ( y e. I |-> if ( y = X , 1 , 0 ) ) , .1. , .0. ) )
15	14	mpteq2dv 4745	. . . 4 |- ( x = X -> ( f e. D |-> if ( f = ( y e. I |-> if ( y = x , 1 , 0 ) ) , .1. , .0. ) ) = ( f e. D |-> if ( f = ( y e. I |-> if ( y = X , 1 , 0 ) ) , .1. , .0. ) ) )
16		eqid 2622	. . . 4 |- ( x e. I |-> ( f e. D |-> if ( f = ( y e. I |-> if ( y = x , 1 , 0 ) ) , .1. , .0. ) ) ) = ( x e. I |-> ( f e. D |-> if ( f = ( y e. I |-> if ( y = x , 1 , 0 ) ) , .1. , .0. ) ) )
17		ovex 6678	. . . . . 6 |- ( NN0 ^m I ) e. _V
18	2, 17	rabex2 4815	. . . . 5 |- D e. _V
19	18	mptex 6486	. . . 4 |- ( f e. D |-> if ( f = ( y e. I |-> if ( y = X , 1 , 0 ) ) , .1. , .0. ) ) e. _V
20	15, 16, 19	fvmpt 6282	. . 3 |- ( X e. I -> ( ( x e. I |-> ( f e. D |-> if ( f = ( y e. I |-> if ( y = x , 1 , 0 ) ) , .1. , .0. ) ) ) ` X ) = ( f e. D |-> if ( f = ( y e. I |-> if ( y = X , 1 , 0 ) ) , .1. , .0. ) ) )
21	9, 20	syl 17	. 2 |- ( ph -> ( ( x e. I |-> ( f e. D |-> if ( f = ( y e. I |-> if ( y = x , 1 , 0 ) ) , .1. , .0. ) ) ) ` X ) = ( f e. D |-> if ( f = ( y e. I |-> if ( y = X , 1 , 0 ) ) , .1. , .0. ) ) )
22	8, 21	eqtrd 2656	1 |- ( ph -> ( V ` X ) = ( f e. D |-> if ( f = ( y e. I |-> if ( y = X , 1 , 0 ) ) , .1. , .0. ) ) )

Syntax hints:   -> wi 4   = wceq 1483   e. wcel 1990   {crab 2916   ifcif 4086   |-> cmpt 4729   `'ccnv 5113   "cima 5117   ` cfv 5888  (class class class)co 6650   ^m cmap 7857   Fincfn 7955   0cc0 9936   1c1 9937   NNcn 11020   NN0cn0 11292   0gc0g 16100   1rcur 18501   mVar cmvr 19352

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-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-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  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-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-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  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-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-mvr 19357

This theorem is referenced by:  mvrval2  19422  mplcoe3  19466  evlslem1  19515