Theorem evls1rhm 18923

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 18021	. . . . 5 |- ( R e. (SubRing ` S ) -> R C_ B )
3	2	adantl 468	. . . 4 |- ( ( S e. CRing /\ R e. (SubRing ` S ) ) -> R C_ B )
4		elpwg 3961	. . . . 5 |- ( R e. (SubRing ` S ) -> ( R e. ~P B <-> R C_ B ) )
5	4	adantl 468	. . . 4 |- ( ( S e. CRing /\ R e. (SubRing ` S ) ) -> ( R e. ~P B <-> R C_ B ) )
6	3, 5	mpbird 236	. . 3 |- ( ( S e. CRing /\ R e. (SubRing ` S ) ) -> R e. ~P B )
7		evls1rhm.q	. . . 4 |- Q = ( S evalSub1 R )
8		eqid 2453	. . . 4 |- ( 1o evalSub S ) = ( 1o evalSub S )
9	7, 8, 1	evls1fval 18920	. . 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 473	. 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 2453	. . . . 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 18922	. . . 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 467	. . 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 7194	. . . . 5 |- 1o e. On
16		eqid 2453	. . . . . 6 |- ( ( 1o evalSub S ) ` R ) = ( ( 1o evalSub S ) ` R )
17		eqid 2453	. . . . . 6 |- ( 1o mPoly U ) = ( 1o mPoly U )
18		evls1rhm.u	. . . . . 6 |- U = ( S ↾s R )
19		eqid 2453	. . . . . 6 |- ( S ^s ( B ^m 1o ) ) = ( S ^s ( B ^m 1o ) )
20	16, 17, 18, 19, 1	evlsrhm 18756	. . . . 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 1353	. . . 4 |- ( ( S e. CRing /\ R e. (SubRing ` S ) ) -> ( ( 1o evalSub S ) ` R ) e. ( ( 1o mPoly U ) RingHom ( S ^s ( B ^m 1o ) ) ) )
22		eqidd 2454	. . . . 5 |- ( ( S e. CRing /\ R e. (SubRing ` S ) ) -> ( Base ` W ) = ( Base ` W ) )
23		eqidd 2454	. . . . 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 2453	. . . . . . 7 |- (PwSer1 ` U ) = (PwSer1 ` U )
26		eqid 2453	. . . . . . 7 |- ( Base ` W ) = ( Base ` W )
27	24, 25, 26	ply1bas 18800	. . . . . 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 2453	. . . . . . . 8 |- ( +g ` W ) = ( +g ` W )
30	24, 17, 29	ply1plusg 18830	. . . . . . 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 6319	. . . . 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 2454	. . . . 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 2453	. . . . . . . 8 |- ( .r ` W ) = ( .r ` W )
35	24, 17, 34	ply1mulr 18832	. . . . . . 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 6319	. . . . 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 2454	. . . . 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 18055	. . . 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 2534	. . 3 |- ( ( S e. CRing /\ R e. (SubRing ` S ) ) -> ( ( 1o evalSub S ) ` R ) e. ( W RingHom ( S ^s ( B ^m 1o ) ) ) )
41		rhmco 17977	. . 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 667	. 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 2531	1 |- ( ( S e. CRing /\ R e. (SubRing ` S ) ) -> Q e. ( W RingHom T ) )

Syntax hints:   -> wi 4   <-> wb 188   /\ wa 371   = wceq 1446   e. wcel 1889   C_ wss 3406   ~Pcpw 3953   {csn 3970   |-> cmpt 4464   X. cxp 4835   o. ccom 4841   Oncon0 5426   ` cfv 5585  (class class class)co 6295   1oc1o 7180   ^m cmap 7477   Basecbs 15133   ↾_s cress 15134   +g cplusg 15202   .rcmulr 15203   ^_s cpws 15357   CRingccrg 17793   RingHom crh 17952  SubRingcsubrg 18016   mPoly cmpl 18589   evalSub ces 18739  PwSer₁cps1 18780  Poly₁cpl1 18782   evalSub₁ ces1 18914

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1671  ax-4 1684  ax-5 1760  ax-6 1807  ax-7 1853  ax-8 1891  ax-9 1898  ax-10 1917  ax-11 1922  ax-12 1935  ax-13 2093  ax-ext 2433  ax-rep 4518  ax-sep 4528  ax-nul 4537  ax-pow 4584  ax-pr 4642  ax-un 6588  ax-inf2 8151  ax-cnex 9600  ax-resscn 9601  ax-1cn 9602  ax-icn 9603  ax-addcl 9604  ax-addrcl 9605  ax-mulcl 9606  ax-mulrcl 9607  ax-mulcom 9608  ax-addass 9609  ax-mulass 9610  ax-distr 9611  ax-i2m1 9612  ax-1ne0 9613  ax-1rid 9614  ax-rnegex 9615  ax-rrecex 9616  ax-cnre 9617  ax-pre-lttri 9618  ax-pre-lttrn 9619  ax-pre-ltadd 9620  ax-pre-mulgt0 9621

This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 987  df-3an 988  df-tru 1449  df-ex 1666  df-nf 1670  df-sb 1800  df-eu 2305  df-mo 2306  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2583  df-ne 2626  df-nel 2627  df-ral 2744  df-rex 2745  df-reu 2746  df-rmo 2747  df-rab 2748  df-v 3049  df-sbc 3270  df-csb 3366  df-dif 3409  df-un 3411  df-in 3413  df-ss 3420  df-pss 3422  df-nul 3734  df-if 3884  df-pw 3955  df-sn 3971  df-pr 3973  df-tp 3975  df-op 3977  df-uni 4202  df-int 4238  df-iun 4283  df-iin 4284  df-br 4406  df-opab 4465  df-mpt 4466  df-tr 4501  df-eprel 4748  df-id 4752  df-po 4758  df-so 4759  df-fr 4796  df-se 4797  df-we 4798  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-pred 5383  df-ord 5429  df-on 5430  df-lim 5431  df-suc 5432  df-iota 5549  df-fun 5587  df-fn 5588  df-f 5589  df-f1 5590  df-fo 5591  df-f1o 5592  df-fv 5593  df-isom 5594  df-riota 6257  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-of 6536  df-ofr 6537  df-om 6698  df-1st 6798  df-2nd 6799  df-supp 6920  df-wrecs 7033  df-recs 7095  df-rdg 7133  df-1o 7187  df-2o 7188  df-oadd 7191  df-er 7368  df-map 7479  df-pm 7480  df-ixp 7528  df-en 7575  df-dom 7576  df-sdom 7577  df-fin 7578  df-fsupp 7889  df-sup 7961  df-oi 8030  df-card 8378  df-pnf 9682  df-mnf 9683  df-xr 9684  df-ltxr 9685  df-le 9686  df-sub 9867  df-neg 9868  df-nn 10617  df-2 10675  df-3 10676  df-4 10677  df-5 10678  df-6 10679  df-7 10680  df-8 10681  df-9 10682  df-10 10683  df-n0 10877  df-z 10945  df-dec 11059  df-uz 11167  df-fz 11792  df-fzo 11923  df-seq 12221  df-hash 12523  df-struct 15135  df-ndx 15136  df-slot 15137  df-base 15138  df-sets 15139  df-ress 15140  df-plusg 15215  df-mulr 15216  df-sca 15218  df-vsca 15219  df-ip 15220  df-tset 15221  df-ple 15222  df-ds 15224  df-hom 15226  df-cco 15227  df-0g 15352  df-gsum 15353  df-prds 15358  df-pws 15360  df-mre 15504  df-mrc 15505  df-acs 15507  df-mgm 16500  df-sgrp 16539  df-mnd 16549  df-mhm 16594  df-submnd 16595  df-grp 16685  df-minusg 16686  df-sbg 16687  df-mulg 16688  df-subg 16826  df-ghm 16893  df-cntz 16983  df-cmn 17444  df-abl 17445  df-mgp 17736  df-ur 17748  df-srg 17752  df-ring 17794  df-cring 17795  df-rnghom 17955  df-subrg 18018  df-lmod 18105  df-lss 18168  df-lsp 18207  df-assa 18548  df-asp 18549  df-ascl 18550  df-psr 18592  df-mvr 18593  df-mpl 18594  df-opsr 18596  df-evls 18741  df-psr1 18785  df-ply1 18787  df-evls1 18916

This theorem is referenced by:  evls1gsumadd  18925  evls1gsummul  18926  evls1varpw  18927