Theorem ply1val 19564

Description: The value of the set of univariate polynomials. (Contributed by Mario Carneiro, 9-Feb-2015.)

Hypotheses

Ref	Expression
ply1val.1	|- P = (Poly1 ` R )
ply1val.2	|- S = (PwSer1 ` R )

Assertion

Ref	Expression
ply1val	|- P = ( S ↾s ( Base ` ( 1o mPoly R ) ) )

Proof of Theorem ply1val

Dummy variable r is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ply1val.1	. 2 |- P = (Poly1 ` R )
2		fveq2 6191	. . . . . 6 |- ( r = R -> (PwSer1 ` r ) = (PwSer1 ` R ) )
3		ply1val.2	. . . . . 6 |- S = (PwSer1 ` R )
4	2, 3	syl6eqr 2674	. . . . 5 |- ( r = R -> (PwSer1 ` r ) = S )
5		oveq2 6658	. . . . . 6 |- ( r = R -> ( 1o mPoly r ) = ( 1o mPoly R ) )
6	5	fveq2d 6195	. . . . 5 |- ( r = R -> ( Base ` ( 1o mPoly r ) ) = ( Base ` ( 1o mPoly R ) ) )
7	4, 6	oveq12d 6668	. . . 4 |- ( r = R -> ( (PwSer1 ` r ) ↾s ( Base ` ( 1o mPoly r ) ) ) = ( S ↾s ( Base ` ( 1o mPoly R ) ) ) )
8		df-ply1 19552	. . . 4 |- Poly1 = ( r e. _V |-> ( (PwSer1 ` r ) ↾s ( Base ` ( 1o mPoly r ) ) ) )
9		ovex 6678	. . . 4 |- ( S ↾s ( Base ` ( 1o mPoly R ) ) ) e. _V
10	7, 8, 9	fvmpt 6282	. . 3 |- ( R e. _V -> (Poly1 ` R ) = ( S ↾s ( Base ` ( 1o mPoly R ) ) ) )
11		fvprc 6185	. . . . 5 |- ( -. R e. _V -> (Poly1 ` R ) = (/) )
12		ress0 15934	. . . . 5 |- ( (/) ↾s ( Base ` ( 1o mPoly R ) ) ) = (/)
13	11, 12	syl6eqr 2674	. . . 4 |- ( -. R e. _V -> (Poly1 ` R ) = ( (/) ↾s ( Base ` ( 1o mPoly R ) ) ) )
14		fvprc 6185	. . . . . 6 |- ( -. R e. _V -> (PwSer1 ` R ) = (/) )
15	3, 14	syl5eq 2668	. . . . 5 |- ( -. R e. _V -> S = (/) )
16	15	oveq1d 6665	. . . 4 |- ( -. R e. _V -> ( S ↾s ( Base ` ( 1o mPoly R ) ) ) = ( (/) ↾s ( Base ` ( 1o mPoly R ) ) ) )
17	13, 16	eqtr4d 2659	. . 3 |- ( -. R e. _V -> (Poly1 ` R ) = ( S ↾s ( Base ` ( 1o mPoly R ) ) ) )
18	10, 17	pm2.61i 176	. 2 |- (Poly1 ` R ) = ( S ↾s ( Base ` ( 1o mPoly R ) ) )
19	1, 18	eqtri 2644	1 |- P = ( S ↾s ( Base ` ( 1o mPoly R ) ) )

Syntax hints:   -. wn 3   = wceq 1483   e. wcel 1990   _Vcvv 3200   (/)c0 3915   ` cfv 5888  (class class class)co 6650   1oc1o 7553   Basecbs 15857   ↾_s cress 15858   mPoly cmpl 19353  PwSer₁cps1 19545  Poly₁cpl1 19547

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-slot 15861  df-base 15863  df-ress 15865  df-ply1 19552

This theorem is referenced by:  ply1bas  19565  ply1crng  19568  ply1assa  19569  ply1bascl  19573  ply1plusg  19595  ply1vsca  19596  ply1mulr  19597  ply1ring  19618  ply1lmod  19622  ply1sca  19623