Theorem mzpcl34 37294

Description: Defining properties 3 and 4 of a polynomially closed function set P: it is closed under pointwise addition and multiplication. (Contributed by Stefan O'Rear, 4-Oct-2014.)

Assertion

Ref	Expression
mzpcl34	|- ( ( P e. (mzPolyCld ` V ) /\ F e. P /\ G e. P ) -> ( ( F oF + G ) e. P /\ ( F oF x. G ) e. P ) )

Proof of Theorem mzpcl34

Dummy variables f g are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simp2 1062	. 2 |- ( ( P e. (mzPolyCld ` V ) /\ F e. P /\ G e. P ) -> F e. P )
2		simp3 1063	. 2 |- ( ( P e. (mzPolyCld ` V ) /\ F e. P /\ G e. P ) -> G e. P )
3		simp1 1061	. . . 4 |- ( ( P e. (mzPolyCld ` V ) /\ F e. P /\ G e. P ) -> P e. (mzPolyCld ` V ) )
4	3	elfvexd 6222	. . . . 5 |- ( ( P e. (mzPolyCld ` V ) /\ F e. P /\ G e. P ) -> V e. _V )
5		elmzpcl 37289	. . . . 5 |- ( V e. _V -> ( P e. (mzPolyCld ` V ) <-> ( P C_ ( ZZ ^m ( ZZ ^m V ) ) /\ ( ( A. f e. ZZ ( ( ZZ ^m V ) X. { f } ) e. P /\ A. f e. V ( g e. ( ZZ ^m V ) |-> ( g ` f ) ) e. P ) /\ A. f e. P A. g e. P ( ( f oF + g ) e. P /\ ( f oF x. g ) e. P ) ) ) ) )
6	4, 5	syl 17	. . . 4 |- ( ( P e. (mzPolyCld ` V ) /\ F e. P /\ G e. P ) -> ( P e. (mzPolyCld ` V ) <-> ( P C_ ( ZZ ^m ( ZZ ^m V ) ) /\ ( ( A. f e. ZZ ( ( ZZ ^m V ) X. { f } ) e. P /\ A. f e. V ( g e. ( ZZ ^m V ) |-> ( g ` f ) ) e. P ) /\ A. f e. P A. g e. P ( ( f oF + g ) e. P /\ ( f oF x. g ) e. P ) ) ) ) )
7	3, 6	mpbid 222	. . 3 |- ( ( P e. (mzPolyCld ` V ) /\ F e. P /\ G e. P ) -> ( P C_ ( ZZ ^m ( ZZ ^m V ) ) /\ ( ( A. f e. ZZ ( ( ZZ ^m V ) X. { f } ) e. P /\ A. f e. V ( g e. ( ZZ ^m V ) |-> ( g ` f ) ) e. P ) /\ A. f e. P A. g e. P ( ( f oF + g ) e. P /\ ( f oF x. g ) e. P ) ) ) )
8	7	simprrd 797	. 2 |- ( ( P e. (mzPolyCld ` V ) /\ F e. P /\ G e. P ) -> A. f e. P A. g e. P ( ( f oF + g ) e. P /\ ( f oF x. g ) e. P ) )
9		oveq1 6657	. . . . 5 |- ( f = F -> ( f oF + g ) = ( F oF + g ) )
10	9	eleq1d 2686	. . . 4 |- ( f = F -> ( ( f oF + g ) e. P <-> ( F oF + g ) e. P ) )
11		oveq1 6657	. . . . 5 |- ( f = F -> ( f oF x. g ) = ( F oF x. g ) )
12	11	eleq1d 2686	. . . 4 |- ( f = F -> ( ( f oF x. g ) e. P <-> ( F oF x. g ) e. P ) )
13	10, 12	anbi12d 747	. . 3 |- ( f = F -> ( ( ( f oF + g ) e. P /\ ( f oF x. g ) e. P ) <-> ( ( F oF + g ) e. P /\ ( F oF x. g ) e. P ) ) )
14		oveq2 6658	. . . . 5 |- ( g = G -> ( F oF + g ) = ( F oF + G ) )
15	14	eleq1d 2686	. . . 4 |- ( g = G -> ( ( F oF + g ) e. P <-> ( F oF + G ) e. P ) )
16		oveq2 6658	. . . . 5 |- ( g = G -> ( F oF x. g ) = ( F oF x. G ) )
17	16	eleq1d 2686	. . . 4 |- ( g = G -> ( ( F oF x. g ) e. P <-> ( F oF x. G ) e. P ) )
18	15, 17	anbi12d 747	. . 3 |- ( g = G -> ( ( ( F oF + g ) e. P /\ ( F oF x. g ) e. P ) <-> ( ( F oF + G ) e. P /\ ( F oF x. G ) e. P ) ) )
19	13, 18	rspc2va 3323	. 2 |- ( ( ( F e. P /\ G e. P ) /\ A. f e. P A. g e. P ( ( f oF + g ) e. P /\ ( f oF x. g ) e. P ) ) -> ( ( F oF + G ) e. P /\ ( F oF x. G ) e. P ) )
20	1, 2, 8, 19	syl21anc 1325	1 |- ( ( P e. (mzPolyCld ` V ) /\ F e. P /\ G e. P ) -> ( ( F oF + G ) e. P /\ ( F oF x. G ) e. P ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   A.wral 2912   _Vcvv 3200   C_ wss 3574   {csn 4177   |-> cmpt 4729   X. cxp 5112   ` cfv 5888  (class class class)co 6650   oFcof 6895   ^m cmap 7857   + caddc 9939   x. cmul 9941   ZZcz 11377  mzPolyCldcmzpcl 37284

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-ne 2795  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-pw 4160  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-mzpcl 37286

This theorem is referenced by:  mzpincl  37297  mzpadd  37301  mzpmul  37302