Theorem mvrfval 19420

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 )

Assertion

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

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

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

Proof of Theorem mvrfval

Dummy variables i r are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		mvrfval.v	. 2 |- V = ( I mVar R )
2		mvrfval.i	. . . 4 |- ( ph -> I e. W )
3		elex 3212	. . . 4 |- ( I e. W -> I e. _V )
4	2, 3	syl 17	. . 3 |- ( ph -> I e. _V )
5		mvrfval.r	. . . 4 |- ( ph -> R e. Y )
6		elex 3212	. . . 4 |- ( R e. Y -> R e. _V )
7	5, 6	syl 17	. . 3 |- ( ph -> R e. _V )
8		mptexg 6484	. . . 4 |- ( I e. W -> ( x e. I |-> ( f e. D |-> if ( f = ( y e. I |-> if ( y = x , 1 , 0 ) ) , .1. , .0. ) ) ) e. _V )
9	2, 8	syl 17	. . 3 |- ( ph -> ( x e. I |-> ( f e. D |-> if ( f = ( y e. I |-> if ( y = x , 1 , 0 ) ) , .1. , .0. ) ) ) e. _V )
10		simpl 473	. . . . 5 |- ( ( i = I /\ r = R ) -> i = I )
11	10	oveq2d 6666	. . . . . . . 8 |- ( ( i = I /\ r = R ) -> ( NN0 ^m i ) = ( NN0 ^m I ) )
12		rabeq 3192	. . . . . . . 8 |- ( ( NN0 ^m i ) = ( NN0 ^m I ) -> { h e. ( NN0 ^m i ) | ( `' h " NN ) e. Fin } = { h e. ( NN0 ^m I ) | ( `' h " NN ) e. Fin } )
13	11, 12	syl 17	. . . . . . 7 |- ( ( i = I /\ r = R ) -> { h e. ( NN0 ^m i ) | ( `' h " NN ) e. Fin } = { h e. ( NN0 ^m I ) | ( `' h " NN ) e. Fin } )
14		mvrfval.d	. . . . . . 7 |- D = { h e. ( NN0 ^m I ) | ( `' h " NN ) e. Fin }
15	13, 14	syl6eqr 2674	. . . . . 6 |- ( ( i = I /\ r = R ) -> { h e. ( NN0 ^m i ) | ( `' h " NN ) e. Fin } = D )
16		mpteq1 4737	. . . . . . . . 9 |- ( i = I -> ( y e. i |-> if ( y = x , 1 , 0 ) ) = ( y e. I |-> if ( y = x , 1 , 0 ) ) )
17	16	adantr 481	. . . . . . . 8 |- ( ( i = I /\ r = R ) -> ( y e. i |-> if ( y = x , 1 , 0 ) ) = ( y e. I |-> if ( y = x , 1 , 0 ) ) )
18	17	eqeq2d 2632	. . . . . . 7 |- ( ( i = I /\ r = R ) -> ( f = ( y e. i |-> if ( y = x , 1 , 0 ) ) <-> f = ( y e. I |-> if ( y = x , 1 , 0 ) ) ) )
19		simpr 477	. . . . . . . . 9 |- ( ( i = I /\ r = R ) -> r = R )
20	19	fveq2d 6195	. . . . . . . 8 |- ( ( i = I /\ r = R ) -> ( 1r ` r ) = ( 1r ` R ) )
21		mvrfval.o	. . . . . . . 8 |- .1. = ( 1r ` R )
22	20, 21	syl6eqr 2674	. . . . . . 7 |- ( ( i = I /\ r = R ) -> ( 1r ` r ) = .1. )
23	19	fveq2d 6195	. . . . . . . 8 |- ( ( i = I /\ r = R ) -> ( 0g ` r ) = ( 0g ` R ) )
24		mvrfval.z	. . . . . . . 8 |- .0. = ( 0g ` R )
25	23, 24	syl6eqr 2674	. . . . . . 7 |- ( ( i = I /\ r = R ) -> ( 0g ` r ) = .0. )
26	18, 22, 25	ifbieq12d 4113	. . . . . 6 |- ( ( i = I /\ r = R ) -> if ( f = ( y e. i |-> if ( y = x , 1 , 0 ) ) , ( 1r ` r ) , ( 0g ` r ) ) = if ( f = ( y e. I |-> if ( y = x , 1 , 0 ) ) , .1. , .0. ) )
27	15, 26	mpteq12dv 4733	. . . . 5 |- ( ( i = I /\ r = R ) -> ( f e. { h e. ( NN0 ^m i ) | ( `' h " NN ) e. Fin } |-> if ( f = ( y e. i |-> if ( y = x , 1 , 0 ) ) , ( 1r ` r ) , ( 0g ` r ) ) ) = ( f e. D |-> if ( f = ( y e. I |-> if ( y = x , 1 , 0 ) ) , .1. , .0. ) ) )
28	10, 27	mpteq12dv 4733	. . . 4 |- ( ( i = I /\ r = R ) -> ( x e. i |-> ( f e. { h e. ( NN0 ^m i ) | ( `' h " NN ) e. Fin } |-> if ( f = ( y e. i |-> if ( y = x , 1 , 0 ) ) , ( 1r ` r ) , ( 0g ` r ) ) ) ) = ( x e. I |-> ( f e. D |-> if ( f = ( y e. I |-> if ( y = x , 1 , 0 ) ) , .1. , .0. ) ) ) )
29		df-mvr 19357	. . . 4 |- mVar = ( i e. _V , r e. _V |-> ( x e. i |-> ( f e. { h e. ( NN0 ^m i ) | ( `' h " NN ) e. Fin } |-> if ( f = ( y e. i |-> if ( y = x , 1 , 0 ) ) , ( 1r ` r ) , ( 0g ` r ) ) ) ) )
30	28, 29	ovmpt2ga 6790	. . 3 |- ( ( I e. _V /\ R e. _V /\ ( x e. I |-> ( f e. D |-> if ( f = ( y e. I |-> if ( y = x , 1 , 0 ) ) , .1. , .0. ) ) ) e. _V ) -> ( I mVar R ) = ( x e. I |-> ( f e. D |-> if ( f = ( y e. I |-> if ( y = x , 1 , 0 ) ) , .1. , .0. ) ) ) )
31	4, 7, 9, 30	syl3anc 1326	. 2 |- ( ph -> ( I mVar R ) = ( x e. I |-> ( f e. D |-> if ( f = ( y e. I |-> if ( y = x , 1 , 0 ) ) , .1. , .0. ) ) ) )
32	1, 31	syl5eq 2668	1 |- ( ph -> V = ( x e. I |-> ( f e. D |-> if ( f = ( y e. I |-> if ( y = x , 1 , 0 ) ) , .1. , .0. ) ) ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   {crab 2916   _Vcvv 3200   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:  mvrval  19421  mvrf  19424  subrgmvr  19461