Theorem mvrfval 15210

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	2	elexd 2835	. . 3 |- ( ph -> I e. _V )
4		mvrfval.r	. . . 4 |- ( ph -> R e. Y )
5	4	elexd 2835	. . 3 |- ( ph -> R e. _V )
6	2	mptexd 5947	. . 3 |- ( ph -> ( x e. I |-> ( f e. D |-> if ( f = ( y e. I |-> if ( y = x , 1 , 0 ) ) , .1. , .0. ) ) ) e. _V )
7		simpl 109	. . . . 5 |- ( ( i = I /\ r = R ) -> i = I )
8	7	oveq2d 6104	. . . . . . . 8 |- ( ( i = I /\ r = R ) -> ( NN0 ^m i ) = ( NN0 ^m I ) )
9	8	rabeqdv 2815	. . . . . . 7 |- ( ( i = I /\ r = R ) -> { h e. ( NN0 ^m i ) | ( `' h " NN ) e. Fin } = { h e. ( NN0 ^m I ) | ( `' h " NN ) e. Fin } )
10		mvrfval.d	. . . . . . 7 |- D = { h e. ( NN0 ^m I ) | ( `' h " NN ) e. Fin }
11	9, 10	eqtr4di 2289	. . . . . 6 |- ( ( i = I /\ r = R ) -> { h e. ( NN0 ^m i ) | ( `' h " NN ) e. Fin } = D )
12		mpteq1 4216	. . . . . . . . 9 |- ( i = I -> ( y e. i |-> if ( y = x , 1 , 0 ) ) = ( y e. I |-> if ( y = x , 1 , 0 ) ) )
13	12	adantr 276	. . . . . . . 8 |- ( ( i = I /\ r = R ) -> ( y e. i |-> if ( y = x , 1 , 0 ) ) = ( y e. I |-> if ( y = x , 1 , 0 ) ) )
14	13	eqeq2d 2250	. . . . . . 7 |- ( ( i = I /\ r = R ) -> ( f = ( y e. i |-> if ( y = x , 1 , 0 ) ) <-> f = ( y e. I |-> if ( y = x , 1 , 0 ) ) ) )
15		simpr 110	. . . . . . . . 9 |- ( ( i = I /\ r = R ) -> r = R )
16	15	fveq2d 5702	. . . . . . . 8 |- ( ( i = I /\ r = R ) -> ( 1r ` r ) = ( 1r ` R ) )
17		mvrfval.o	. . . . . . . 8 |- .1. = ( 1r ` R )
18	16, 17	eqtr4di 2289	. . . . . . 7 |- ( ( i = I /\ r = R ) -> ( 1r ` r ) = .1. )
19	15	fveq2d 5702	. . . . . . . 8 |- ( ( i = I /\ r = R ) -> ( 0g ` r ) = ( 0g ` R ) )
20		mvrfval.z	. . . . . . . 8 |- .0. = ( 0g ` R )
21	19, 20	eqtr4di 2289	. . . . . . 7 |- ( ( i = I /\ r = R ) -> ( 0g ` r ) = .0. )
22	14, 18, 21	ifbieq12d 3667	. . . . . 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. ) )
23	11, 22	mpteq12dv 4214	. . . . 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. ) ) )
24	7, 23	mpteq12dv 4214	. . . 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. ) ) ) )
25		df-mvr 15159	. . . 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 ) ) ) ) )
26	24, 25	ovmpoga 6221	. . 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. ) ) ) )
27	3, 5, 6, 26	syl3anc 1278	. 2 |- ( ph -> ( I mVar R ) = ( x e. I |-> ( f e. D |-> if ( f = ( y e. I |-> if ( y = x , 1 , 0 ) ) , .1. , .0. ) ) ) )
28	1, 27	eqtrid 2283	1 |- ( ph -> V = ( x e. I |-> ( f e. D |-> if ( f = ( y e. I |-> if ( y = x , 1 , 0 ) ) , .1. , .0. ) ) ) )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   = wceq 1402   e. wcel 2209   {crab 2532   _Vcvv 2821   ifcif 3638   |-> cmpt 4193   `'ccnv 4775   "cima 4779   ` cfv 5380  (class class class)co 6088   ^m cmap 6926   Fincfn 7027   0cc0 8186   1c1 8187   NNcn 9314   NN0cn0 9575   0gc0g 13670   1rcur 14370   mVar cmvr 15156

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4248  ax-sep 4251  ax-pow 4313  ax-pr 4348  ax-setind 4686

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-if 3639  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-iun 4014  df-br 4132  df-opab 4194  df-mpt 4195  df-id 4440  df-xp 4782  df-rel 4783  df-cnv 4784  df-co 4785  df-dm 4786  df-rn 4787  df-res 4788  df-ima 4789  df-iota 5340  df-fun 5382  df-fn 5383  df-f 5384  df-f1 5385  df-fo 5386  df-f1o 5387  df-fv 5388  df-ov 6091  df-oprab 6092  df-mpo 6093  df-mvr 15159

This theorem is used by:  mvrval  15211  mvrffi  15214