Theorem vr1val 19562

Description: The value of the generator of the power series algebra (the X in R [ [ X ] ]). Since all univariate polynomial rings over a fixed base ring R are isomorphic, we don't bother to pass this in as a parameter; internally we are actually using the empty set as this generator and 1o = { (/) } is the index set (but for most purposes this choice should not be visible anyway). (Contributed by Mario Carneiro, 8-Feb-2015.) (Revised by Mario Carneiro, 12-Jun-2015.)

Hypothesis

Ref	Expression
vr1val.1	|- X = (var1 ` R )

Assertion

Ref	Expression
vr1val	|- X = ( ( 1o mVar R ) ` (/) )

Proof of Theorem vr1val

Dummy variables f h i r x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		vr1val.1	. . 3 |- X = (var1 ` R )
2		oveq2 6658	. . . . 5 |- ( r = R -> ( 1o mVar r ) = ( 1o mVar R ) )
3	2	fveq1d 6193	. . . 4 |- ( r = R -> ( ( 1o mVar r ) ` (/) ) = ( ( 1o mVar R ) ` (/) ) )
4		df-vr1 19551	. . . 4 |- var1 = ( r e. _V |-> ( ( 1o mVar r ) ` (/) ) )
5		fvex 6201	. . . 4 |- ( ( 1o mVar R ) ` (/) ) e. _V
6	3, 4, 5	fvmpt 6282	. . 3 |- ( R e. _V -> (var1 ` R ) = ( ( 1o mVar R ) ` (/) ) )
7	1, 6	syl5eq 2668	. 2 |- ( R e. _V -> X = ( ( 1o mVar R ) ` (/) ) )
8		fvprc 6185	. . . 4 |- ( -. R e. _V -> (var1 ` R ) = (/) )
9		0fv 6227	. . . 4 |- ( (/) ` (/) ) = (/)
10	8, 1, 9	3eqtr4g 2681	. . 3 |- ( -. R e. _V -> X = ( (/) ` (/) ) )
11		df-mvr 19357	. . . . . 6 |- 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 ) ) ) ) )
12	11	reldmmpt2 6771	. . . . 5 |- Rel dom mVar
13	12	ovprc2 6685	. . . 4 |- ( -. R e. _V -> ( 1o mVar R ) = (/) )
14	13	fveq1d 6193	. . 3 |- ( -. R e. _V -> ( ( 1o mVar R ) ` (/) ) = ( (/) ` (/) ) )
15	10, 14	eqtr4d 2659	. 2 |- ( -. R e. _V -> X = ( ( 1o mVar R ) ` (/) ) )
16	7, 15	pm2.61i 176	1 |- X = ( ( 1o mVar R ) ` (/) )

Syntax hints:   -. wn 3   = wceq 1483   e. wcel 1990   {crab 2916   _Vcvv 3200   (/)c0 3915   ifcif 4086   |-> cmpt 4729   `'ccnv 5113   "cima 5117   ` cfv 5888  (class class class)co 6650   1oc1o 7553   ^m cmap 7857   Fincfn 7955   0cc0 9936   1c1 9937   NNcn 11020   NN0cn0 11292   0gc0g 16100   1rcur 18501   mVar cmvr 19352  var₁cv1 19546

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-sep 4781  ax-nul 4789  ax-pow 4843  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-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  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-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-iota 5851  df-fun 5890  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-mvr 19357  df-vr1 19551

This theorem is referenced by:  vr1cl2  19563  vr1cl  19587  subrgvr1  19631  subrgvr1cl  19632  coe1tm  19643  ply1coe  19666  evl1var  19700  evls1var  19702