Theorem evls1rhm 18486

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 17557	. . . . 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 4023	. . . . 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 2457	. . . 4 |- ( 1o evalSub S ) = ( 1o evalSub S )
9	7, 8, 1	evls1fval 18483	. . 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 2457	. . . . 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 18485	. . . 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 7155	. . . . 5 |- 1o e. On
16		eqid 2457	. . . . . 6 |- ( ( 1o evalSub S ) ` R ) = ( ( 1o evalSub S ) ` R )
17		eqid 2457	. . . . . 6 |- ( 1o mPoly U ) = ( 1o mPoly U )
18		evls1rhm.u	. . . . . 6 |- U = ( S ↾s R )
19		eqid 2457	. . . . . 6 |- ( S ^s ( B ^m 1o ) ) = ( S ^s ( B ^m 1o ) )
20	16, 17, 18, 19, 1	evlsrhm 18317	. . . . 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 1311	. . . 4 |- ( ( S e. CRing /\ R e. (SubRing ` S ) ) -> ( ( 1o evalSub S ) ` R ) e. ( ( 1o mPoly U ) RingHom ( S ^s ( B ^m 1o ) ) ) )
22		eqidd 2458	. . . . 5 |- ( ( S e. CRing /\ R e. (SubRing ` S ) ) -> ( Base ` W ) = ( Base ` W ) )
23		eqidd 2458	. . . . 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 2457	. . . . . . 7 |- (PwSer1 ` U ) = (PwSer1 ` U )
26		eqid 2457	. . . . . . 7 |- ( Base ` W ) = ( Base ` W )
27	24, 25, 26	ply1bas 18361	. . . . . 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 2457	. . . . . . . 8 |- ( +g ` W ) = ( +g ` W )
30	24, 17, 29	ply1plusg 18393	. . . . . . 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	oveqdr 6320	. . . . 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 2458	. . . . 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 2457	. . . . . . . 8 |- ( .r ` W ) = ( .r ` W )
35	24, 17, 34	ply1mulr 18395	. . . . . . 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	oveqdr 6320	. . . . 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 2458	. . . . 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 17591	. . . 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 2548	. . 3 |- ( ( S e. CRing /\ R e. (SubRing ` S ) ) -> ( ( 1o evalSub S ) ` R ) e. ( W RingHom ( S ^s ( B ^m 1o ) ) ) )
41		rhmco 17513	. . 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 2545	1 |- ( ( S e. CRing /\ R e. (SubRing ` S ) ) -> Q e. ( W RingHom T ) )

Syntax hints:   -> wi 4   <-> wb 184   /\ wa 369   = wceq 1395   e. wcel 1819   C_ wss 3471   ~Pcpw 4015   {csn 4032   |-> cmpt 4515   Oncon0 4887   X. cxp 5006   o. ccom 5012   ` cfv 5594  (class class class)co 6296   1oc1o 7141   ^m cmap 7438   Basecbs 14644   ↾_s cress 14645   +g cplusg 14712   .rcmulr 14713   ^_s cpws 14864   CRingccrg 17326   RingHom crh 17488  SubRingcsubrg 17552   mPoly cmpl 18129   evalSub ces 18296  PwSer₁cps1 18341  Poly₁cpl1 18343   evalSub₁ ces1 18477

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-inf2 8075  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-int 4289  df-iun 4334  df-iin 4335  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-se 4848  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-isom 5603  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-of 6539  df-ofr 6540  df-om 6700  df-1st 6799  df-2nd 6800  df-supp 6918  df-recs 7060  df-rdg 7094  df-1o 7148  df-2o 7149  df-oadd 7152  df-er 7329  df-map 7440  df-pm 7441  df-ixp 7489  df-en 7536  df-dom 7537  df-sdom 7538  df-fin 7539  df-fsupp 7848  df-sup 7919  df-oi 7953  df-card 8337  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-nn 10557  df-2 10615  df-3 10616  df-4 10617  df-5 10618  df-6 10619  df-7 10620  df-8 10621  df-9 10622  df-10 10623  df-n0 10817  df-z 10886  df-dec 11001  df-uz 11107  df-fz 11698  df-fzo 11822  df-seq 12111  df-hash 12409  df-struct 14646  df-ndx 14647  df-slot 14648  df-base 14649  df-sets 14650  df-ress 14651  df-plusg 14725  df-mulr 14726  df-sca 14728  df-vsca 14729  df-ip 14730  df-tset 14731  df-ple 14732  df-ds 14734  df-hom 14736  df-cco 14737  df-0g 14859  df-gsum 14860  df-prds 14865  df-pws 14867  df-mre 15003  df-mrc 15004  df-acs 15006  df-mgm 15999  df-sgrp 16038  df-mnd 16048  df-mhm 16093  df-submnd 16094  df-grp 16184  df-minusg 16185  df-sbg 16186  df-mulg 16187  df-subg 16325  df-ghm 16392  df-cntz 16482  df-cmn 16927  df-abl 16928  df-mgp 17269  df-ur 17281  df-srg 17285  df-ring 17327  df-cring 17328  df-rnghom 17491  df-subrg 17554  df-lmod 17641  df-lss 17706  df-lsp 17745  df-assa 18088  df-asp 18089  df-ascl 18090  df-psr 18132  df-mvr 18133  df-mpl 18134  df-opsr 18136  df-evls 18298  df-psr1 18346  df-ply1 18348  df-evls1 18479

This theorem is referenced by:  evls1gsumadd  18488  evls1gsummul  18489  evls1varpw  18490