Theorem evls1rhm 17868

Description: Polynomial evaluation is a homomorphism (into the product ring). (Contributed by AV, 11-Sep-2019.)

Hypotheses

Ref	Expression
evls1rhm.q	|- Q = ( S evalSub1 R )
evls1rhm.b	|- B = ( Base ` S )
evls1rhm.t	|- T = ( S ^s B )
evls1rhm.u	|- U = ( S ↾s R )
evls1rhm.w	|- W = (Poly1 ` U )

Assertion

Ref	Expression
evls1rhm	|- ( ( S e. CRing /\ R e. (SubRing ` S ) ) -> Q e. ( W RingHom T ) )

Proof of Theorem evls1rhm

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		evls1rhm.b	. . . . . 6 |- B = ( Base ` S )
2	1	subrgss 16974	. . . . 5 |- ( R e. (SubRing ` S ) -> R C_ B )
3	2	adantl 466	. . . 4 |- ( ( S e. CRing /\ R e. (SubRing ` S ) ) -> R C_ B )
4		elpwg 3968	. . . . 5 |- ( R e. (SubRing ` S ) -> ( R e. ~P B <-> R C_ B ) )
5	4	adantl 466	. . . 4 |- ( ( S e. CRing /\ R e. (SubRing ` S ) ) -> ( R e. ~P B <-> R C_ B ) )
6	3, 5	mpbird 232	. . 3 |- ( ( S e. CRing /\ R e. (SubRing ` S ) ) -> R e. ~P B )
7		evls1rhm.q	. . . 4 |- Q = ( S evalSub1 R )
8		eqid 2451	. . . 4 |- ( 1o evalSub S ) = ( 1o evalSub S )
9	7, 8, 1	evls1fval 17865	. . 3 |- ( ( S e. CRing /\ R e. ~P B ) -> Q = ( ( x e. ( B ^m ( B ^m 1o ) ) |-> ( x o. ( y e. B |-> ( 1o X. { y } ) ) ) ) o. ( ( 1o evalSub S ) ` R ) ) )
10	6, 9	syldan 470	. 2 |- ( ( S e. CRing /\ R e. (SubRing ` S ) ) -> Q = ( ( x e. ( B ^m ( B ^m 1o ) ) |-> ( x o. ( y e. B |-> ( 1o X. { y } ) ) ) ) o. ( ( 1o evalSub S ) ` R ) ) )
11		evls1rhm.t	. . . . 5 |- T = ( S ^s B )
12		eqid 2451	. . . . 5 |- ( x e. ( B ^m ( B ^m 1o ) ) |-> ( x o. ( y e. B |-> ( 1o X. { y } ) ) ) ) = ( x e. ( B ^m ( B ^m 1o ) ) |-> ( x o. ( y e. B |-> ( 1o X. { y } ) ) ) )
13	1, 11, 12	evls1rhmlem 17867	. . . 4 |- ( S e. CRing -> ( x e. ( B ^m ( B ^m 1o ) ) |-> ( x o. ( y e. B |-> ( 1o X. { y } ) ) ) ) e. ( ( S ^s ( B ^m 1o ) ) RingHom T ) )
14	13	adantr 465	. . 3 |- ( ( S e. CRing /\ R e. (SubRing ` S ) ) -> ( x e. ( B ^m ( B ^m 1o ) ) |-> ( x o. ( y e. B |-> ( 1o X. { y } ) ) ) ) e. ( ( S ^s ( B ^m 1o ) ) RingHom T ) )
15		1on 7029	. . . . 5 |- 1o e. On
16		eqid 2451	. . . . . 6 |- ( ( 1o evalSub S ) ` R ) = ( ( 1o evalSub S ) ` R )
17		eqid 2451	. . . . . 6 |- ( 1o mPoly U ) = ( 1o mPoly U )
18		evls1rhm.u	. . . . . 6 |- U = ( S ↾s R )
19		eqid 2451	. . . . . 6 |- ( S ^s ( B ^m 1o ) ) = ( S ^s ( B ^m 1o ) )
20	16, 17, 18, 19, 1	evlsrhm 17716	. . . . 5 |- ( ( 1o e. On /\ S e. CRing /\ R e. (SubRing ` S ) ) -> ( ( 1o evalSub S ) ` R ) e. ( ( 1o mPoly U ) RingHom ( S ^s ( B ^m 1o ) ) ) )
21	15, 20	mp3an1 1302	. . . 4 |- ( ( S e. CRing /\ R e. (SubRing ` S ) ) -> ( ( 1o evalSub S ) ` R ) e. ( ( 1o mPoly U ) RingHom ( S ^s ( B ^m 1o ) ) ) )
22		eqidd 2452	. . . . 5 |- ( ( S e. CRing /\ R e. (SubRing ` S ) ) -> ( Base ` W ) = ( Base ` W ) )
23		eqidd 2452	. . . . 5 |- ( ( S e. CRing /\ R e. (SubRing ` S ) ) -> ( Base ` ( S ^s ( B ^m 1o ) ) ) = ( Base ` ( S ^s ( B ^m 1o ) ) ) )
24		evls1rhm.w	. . . . . . 7 |- W = (Poly1 ` U )
25		eqid 2451	. . . . . . 7 |- (PwSer1 ` U ) = (PwSer1 ` U )
26		eqid 2451	. . . . . . 7 |- ( Base ` W ) = ( Base ` W )
27	24, 25, 26	ply1bas 17760	. . . . . 6 |- ( Base ` W ) = ( Base ` ( 1o mPoly U ) )
28	27	a1i 11	. . . . 5 |- ( ( S e. CRing /\ R e. (SubRing ` S ) ) -> ( Base ` W ) = ( Base ` ( 1o mPoly U ) ) )
29		eqid 2451	. . . . . . . 8 |- ( +g ` W ) = ( +g ` W )
30	24, 17, 29	ply1plusg 17788	. . . . . . 7 |- ( +g ` W ) = ( +g ` ( 1o mPoly U ) )
31	30	a1i 11	. . . . . 6 |- ( ( S e. CRing /\ R e. (SubRing ` S ) ) -> ( +g ` W ) = ( +g ` ( 1o mPoly U ) ) )
32	31	proplem3 14733	. . . . 5 |- ( ( ( S e. CRing /\ R e. (SubRing ` S ) ) /\ ( x e. ( Base ` W ) /\ y e. ( Base ` W ) ) ) -> ( x ( +g ` W ) y ) = ( x ( +g ` ( 1o mPoly U ) ) y ) )
33		eqidd 2452	. . . . 5 |- ( ( ( S e. CRing /\ R e. (SubRing ` S ) ) /\ ( x e. ( Base ` ( S ^s ( B ^m 1o ) ) ) /\ y e. ( Base ` ( S ^s ( B ^m 1o ) ) ) ) ) -> ( x ( +g ` ( S ^s ( B ^m 1o ) ) ) y ) = ( x ( +g ` ( S ^s ( B ^m 1o ) ) ) y ) )
34		eqid 2451	. . . . . . . 8 |- ( .r ` W ) = ( .r ` W )
35	24, 17, 34	ply1mulr 17790	. . . . . . 7 |- ( .r ` W ) = ( .r ` ( 1o mPoly U ) )
36	35	a1i 11	. . . . . 6 |- ( ( S e. CRing /\ R e. (SubRing ` S ) ) -> ( .r ` W ) = ( .r ` ( 1o mPoly U ) ) )
37	36	proplem3 14733	. . . . 5 |- ( ( ( S e. CRing /\ R e. (SubRing ` S ) ) /\ ( x e. ( Base ` W ) /\ y e. ( Base ` W ) ) ) -> ( x ( .r ` W ) y ) = ( x ( .r ` ( 1o mPoly U ) ) y ) )
38		eqidd 2452	. . . . 5 |- ( ( ( S e. CRing /\ R e. (SubRing ` S ) ) /\ ( x e. ( Base ` ( S ^s ( B ^m 1o ) ) ) /\ y e. ( Base ` ( S ^s ( B ^m 1o ) ) ) ) ) -> ( x ( .r ` ( S ^s ( B ^m 1o ) ) ) y ) = ( x ( .r ` ( S ^s ( B ^m 1o ) ) ) y ) )
39	22, 23, 28, 23, 32, 33, 37, 38	rhmpropd 17008	. . . 4 |- ( ( S e. CRing /\ R e. (SubRing ` S ) ) -> ( W RingHom ( S ^s ( B ^m 1o ) ) ) = ( ( 1o mPoly U ) RingHom ( S ^s ( B ^m 1o ) ) ) )
40	21, 39	eleqtrrd 2542	. . 3 |- ( ( S e. CRing /\ R e. (SubRing ` S ) ) -> ( ( 1o evalSub S ) ` R ) e. ( W RingHom ( S ^s ( B ^m 1o ) ) ) )
41		rhmco 16933	. . 3 |- ( ( ( x e. ( B ^m ( B ^m 1o ) ) |-> ( x o. ( y e. B |-> ( 1o X. { y } ) ) ) ) e. ( ( S ^s ( B ^m 1o ) ) RingHom T ) /\ ( ( 1o evalSub S ) ` R ) e. ( W RingHom ( S ^s ( B ^m 1o ) ) ) ) -> ( ( x e. ( B ^m ( B ^m 1o ) ) |-> ( x o. ( y e. B |-> ( 1o X. { y } ) ) ) ) o. ( ( 1o evalSub S ) ` R ) ) e. ( W RingHom T ) )
42	14, 40, 41	syl2anc 661	. 2 |- ( ( S e. CRing /\ R e. (SubRing ` S ) ) -> ( ( x e. ( B ^m ( B ^m 1o ) ) |-> ( x o. ( y e. B |-> ( 1o X. { y } ) ) ) ) o. ( ( 1o evalSub S ) ` R ) ) e. ( W RingHom T ) )
43	10, 42	eqeltrd 2539	1 |- ( ( S e. CRing /\ R e. (SubRing ` S ) ) -> Q e. ( W RingHom T ) )

Syntax hints:   -> wi 4   <-> wb 184   /\ wa 369   = wceq 1370   e. wcel 1758   C_ wss 3428   ~Pcpw 3960   {csn 3977   |-> cmpt 4450   Oncon0 4819   X. cxp 4938   o. ccom 4944   ` cfv 5518  (class class class)co 6192   1oc1o 7015   ^m cmap 7316   Basecbs 14278   ↾_s cress 14279   +g cplusg 14342   .rcmulr 14343   ^_s cpws 14489   CRingccrg 16754   RingHom crh 16912  SubRingcsubrg 16969   mPoly cmpl 17528   evalSub ces 17695  PwSer₁cps1 17740  Poly₁cpl1 17742   evalSub₁ ces1 17859

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4503  ax-sep 4513  ax-nul 4521  ax-pow 4570  ax-pr 4631  ax-un 6474  ax-inf2 7950  ax-cnex 9441  ax-resscn 9442  ax-1cn 9443  ax-icn 9444  ax-addcl 9445  ax-addrcl 9446  ax-mulcl 9447  ax-mulrcl 9448  ax-mulcom 9449  ax-addass 9450  ax-mulass 9451  ax-distr 9452  ax-i2m1 9453  ax-1ne0 9454  ax-1rid 9455  ax-rnegex 9456  ax-rrecex 9457  ax-cnre 9458  ax-pre-lttri 9459  ax-pre-lttrn 9460  ax-pre-ltadd 9461  ax-pre-mulgt0 9462

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-nel 2647  df-ral 2800  df-rex 2801  df-reu 2802  df-rmo 2803  df-rab 2804  df-v 3072  df-sbc 3287  df-csb 3389  df-dif 3431  df-un 3433  df-in 3435  df-ss 3442  df-pss 3444  df-nul 3738  df-if 3892  df-pw 3962  df-sn 3978  df-pr 3980  df-tp 3982  df-op 3984  df-uni 4192  df-int 4229  df-iun 4273  df-iin 4274  df-br 4393  df-opab 4451  df-mpt 4452  df-tr 4486  df-eprel 4732  df-id 4736  df-po 4741  df-so 4742  df-fr 4779  df-se 4780  df-we 4781  df-ord 4822  df-on 4823  df-lim 4824  df-suc 4825  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-rn 4951  df-res 4952  df-ima 4953  df-iota 5481  df-fun 5520  df-fn 5521  df-f 5522  df-f1 5523  df-fo 5524  df-f1o 5525  df-fv 5526  df-isom 5527  df-riota 6153  df-ov 6195  df-oprab 6196  df-mpt2 6197  df-of 6422  df-ofr 6423  df-om 6579  df-1st 6679  df-2nd 6680  df-supp 6793  df-recs 6934  df-rdg 6968  df-1o 7022  df-2o 7023  df-oadd 7026  df-er 7203  df-map 7318  df-pm 7319  df-ixp 7366  df-en 7413  df-dom 7414  df-sdom 7415  df-fin 7416  df-fsupp 7724  df-sup 7794  df-oi 7827  df-card 8212  df-pnf 9523  df-mnf 9524  df-xr 9525  df-ltxr 9526  df-le 9527  df-sub 9700  df-neg 9701  df-nn 10426  df-2 10483  df-3 10484  df-4 10485  df-5 10486  df-6 10487  df-7 10488  df-8 10489  df-9 10490  df-10 10491  df-n0 10683  df-z 10750  df-dec 10859  df-uz 10965  df-fz 11541  df-fzo 11652  df-seq 11910  df-hash 12207  df-struct 14280  df-ndx 14281  df-slot 14282  df-base 14283  df-sets 14284  df-ress 14285  df-plusg 14355  df-mulr 14356  df-sca 14358  df-vsca 14359  df-ip 14360  df-tset 14361  df-ple 14362  df-ds 14364  df-hom 14366  df-cco 14367  df-0g 14484  df-gsum 14485  df-prds 14490  df-pws 14492  df-mre 14628  df-mrc 14629  df-acs 14631  df-mnd 15519  df-mhm 15568  df-submnd 15569  df-grp 15649  df-minusg 15650  df-sbg 15651  df-mulg 15652  df-subg 15782  df-ghm 15849  df-cntz 15939  df-cmn 16385  df-abl 16386  df-mgp 16699  df-ur 16711  df-srg 16715  df-rng 16755  df-cring 16756  df-rnghom 16914  df-subrg 16971  df-lmod 17058  df-lss 17122  df-lsp 17161  df-assa 17492  df-asp 17493  df-ascl 17494  df-psr 17531  df-mvr 17532  df-mpl 17533  df-opsr 17535  df-evls 17697  df-psr1 17745  df-ply1 17747  df-evls1 17861

This theorem is referenced by:  evls1gsumadd  17870  evls1gsummul  17871  evls1varpw  17872