Theorem evls1val 19685

Description: Value of the univariate polynomial evaluation map. (Contributed by AV, 10-Sep-2019.)

Hypotheses

Ref	Expression
evls1fval.q	|- Q = ( S evalSub1 R )
evls1fval.e	|- E = ( 1o evalSub S )
evls1fval.b	|- B = ( Base ` S )
evls1val.m	|- M = ( 1o mPoly ( S ↾s R ) )
evls1val.k	|- K = ( Base ` M )

Assertion

Ref	Expression
evls1val	|- ( ( S e. CRing /\ R e. (SubRing ` S ) /\ A e. K ) -> ( Q ` A ) = ( ( ( E ` R ) ` A ) o. ( y e. B |-> ( 1o X. { y } ) ) ) )

Distinct variable group:   y, B

Allowed substitution hints:   A( y)   Q( y)   R( y)   S( y)   E( y)   K( y)   M( y)

Proof of Theorem evls1val

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		evls1fval.b	. . . . . . . 8 |- B = ( Base ` S )
2	1	subrgss 18781	. . . . . . 7 |- ( R e. (SubRing ` S ) -> R C_ B )
3	2	adantl 482	. . . . . 6 |- ( ( S e. CRing /\ R e. (SubRing ` S ) ) -> R C_ B )
4		elpwg 4166	. . . . . . 7 |- ( R e. (SubRing ` S ) -> ( R e. ~P B <-> R C_ B ) )
5	4	adantl 482	. . . . . 6 |- ( ( S e. CRing /\ R e. (SubRing ` S ) ) -> ( R e. ~P B <-> R C_ B ) )
6	3, 5	mpbird 247	. . . . 5 |- ( ( S e. CRing /\ R e. (SubRing ` S ) ) -> R e. ~P B )
7		evls1fval.q	. . . . . 6 |- Q = ( S evalSub1 R )
8		evls1fval.e	. . . . . 6 |- E = ( 1o evalSub S )
9	7, 8, 1	evls1fval 19684	. . . . 5 |- ( ( S e. CRing /\ R e. ~P B ) -> Q = ( ( x e. ( B ^m ( B ^m 1o ) ) |-> ( x o. ( y e. B |-> ( 1o X. { y } ) ) ) ) o. ( E ` R ) ) )
10	6, 9	syldan 487	. . . 4 |- ( ( S e. CRing /\ R e. (SubRing ` S ) ) -> Q = ( ( x e. ( B ^m ( B ^m 1o ) ) |-> ( x o. ( y e. B |-> ( 1o X. { y } ) ) ) ) o. ( E ` R ) ) )
11	10	fveq1d 6193	. . 3 |- ( ( S e. CRing /\ R e. (SubRing ` S ) ) -> ( Q ` A ) = ( ( ( x e. ( B ^m ( B ^m 1o ) ) |-> ( x o. ( y e. B |-> ( 1o X. { y } ) ) ) ) o. ( E ` R ) ) ` A ) )
12	11	3adant3 1081	. 2 |- ( ( S e. CRing /\ R e. (SubRing ` S ) /\ A e. K ) -> ( Q ` A ) = ( ( ( x e. ( B ^m ( B ^m 1o ) ) |-> ( x o. ( y e. B |-> ( 1o X. { y } ) ) ) ) o. ( E ` R ) ) ` A ) )
13		1on 7567	. . . . . 6 |- 1o e. On
14	13	a1i 11	. . . . 5 |- ( ( S e. CRing /\ R e. (SubRing ` S ) /\ A e. K ) -> 1o e. On )
15		simp1 1061	. . . . 5 |- ( ( S e. CRing /\ R e. (SubRing ` S ) /\ A e. K ) -> S e. CRing )
16		simp2 1062	. . . . 5 |- ( ( S e. CRing /\ R e. (SubRing ` S ) /\ A e. K ) -> R e. (SubRing ` S ) )
17	8	fveq1i 6192	. . . . . 6 |- ( E ` R ) = ( ( 1o evalSub S ) ` R )
18		evls1val.m	. . . . . 6 |- M = ( 1o mPoly ( S ↾s R ) )
19		eqid 2622	. . . . . 6 |- ( S ↾s R ) = ( S ↾s R )
20		eqid 2622	. . . . . 6 |- ( S ^s ( B ^m 1o ) ) = ( S ^s ( B ^m 1o ) )
21	17, 18, 19, 20, 1	evlsrhm 19521	. . . . 5 |- ( ( 1o e. On /\ S e. CRing /\ R e. (SubRing ` S ) ) -> ( E ` R ) e. ( M RingHom ( S ^s ( B ^m 1o ) ) ) )
22	14, 15, 16, 21	syl3anc 1326	. . . 4 |- ( ( S e. CRing /\ R e. (SubRing ` S ) /\ A e. K ) -> ( E ` R ) e. ( M RingHom ( S ^s ( B ^m 1o ) ) ) )
23		evls1val.k	. . . . 5 |- K = ( Base ` M )
24		eqid 2622	. . . . 5 |- ( Base ` ( S ^s ( B ^m 1o ) ) ) = ( Base ` ( S ^s ( B ^m 1o ) ) )
25	23, 24	rhmf 18726	. . . 4 |- ( ( E ` R ) e. ( M RingHom ( S ^s ( B ^m 1o ) ) ) -> ( E ` R ) : K --> ( Base ` ( S ^s ( B ^m 1o ) ) ) )
26	22, 25	syl 17	. . 3 |- ( ( S e. CRing /\ R e. (SubRing ` S ) /\ A e. K ) -> ( E ` R ) : K --> ( Base ` ( S ^s ( B ^m 1o ) ) ) )
27		simp3 1063	. . 3 |- ( ( S e. CRing /\ R e. (SubRing ` S ) /\ A e. K ) -> A e. K )
28		fvco3 6275	. . 3 |- ( ( ( E ` R ) : K --> ( Base ` ( S ^s ( B ^m 1o ) ) ) /\ A e. K ) -> ( ( ( x e. ( B ^m ( B ^m 1o ) ) |-> ( x o. ( y e. B |-> ( 1o X. { y } ) ) ) ) o. ( E ` R ) ) ` A ) = ( ( x e. ( B ^m ( B ^m 1o ) ) |-> ( x o. ( y e. B |-> ( 1o X. { y } ) ) ) ) ` ( ( E ` R ) ` A ) ) )
29	26, 27, 28	syl2anc 693	. 2 |- ( ( S e. CRing /\ R e. (SubRing ` S ) /\ A e. K ) -> ( ( ( x e. ( B ^m ( B ^m 1o ) ) |-> ( x o. ( y e. B |-> ( 1o X. { y } ) ) ) ) o. ( E ` R ) ) ` A ) = ( ( x e. ( B ^m ( B ^m 1o ) ) |-> ( x o. ( y e. B |-> ( 1o X. { y } ) ) ) ) ` ( ( E ` R ) ` A ) ) )
30	26, 27	ffvelrnd 6360	. . . 4 |- ( ( S e. CRing /\ R e. (SubRing ` S ) /\ A e. K ) -> ( ( E ` R ) ` A ) e. ( Base ` ( S ^s ( B ^m 1o ) ) ) )
31		ovex 6678	. . . . 5 |- ( B ^m 1o ) e. _V
32	20, 1	pwsbas 16147	. . . . 5 |- ( ( S e. CRing /\ ( B ^m 1o ) e. _V ) -> ( B ^m ( B ^m 1o ) ) = ( Base ` ( S ^s ( B ^m 1o ) ) ) )
33	15, 31, 32	sylancl 694	. . . 4 |- ( ( S e. CRing /\ R e. (SubRing ` S ) /\ A e. K ) -> ( B ^m ( B ^m 1o ) ) = ( Base ` ( S ^s ( B ^m 1o ) ) ) )
34	30, 33	eleqtrrd 2704	. . 3 |- ( ( S e. CRing /\ R e. (SubRing ` S ) /\ A e. K ) -> ( ( E ` R ) ` A ) e. ( B ^m ( B ^m 1o ) ) )
35		coeq1 5279	. . . 4 |- ( x = ( ( E ` R ) ` A ) -> ( x o. ( y e. B |-> ( 1o X. { y } ) ) ) = ( ( ( E ` R ) ` A ) o. ( y e. B |-> ( 1o X. { y } ) ) ) )
36		eqid 2622	. . . 4 |- ( 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 } ) ) ) )
37		fvex 6201	. . . . 5 |- ( ( E ` R ) ` A ) e. _V
38		fvex 6201	. . . . . . 7 |- ( Base ` S ) e. _V
39	1, 38	eqeltri 2697	. . . . . 6 |- B e. _V
40	39	mptex 6486	. . . . 5 |- ( y e. B |-> ( 1o X. { y } ) ) e. _V
41	37, 40	coex 7118	. . . 4 |- ( ( ( E ` R ) ` A ) o. ( y e. B |-> ( 1o X. { y } ) ) ) e. _V
42	35, 36, 41	fvmpt 6282	. . 3 |- ( ( ( E ` R ) ` A ) e. ( B ^m ( B ^m 1o ) ) -> ( ( x e. ( B ^m ( B ^m 1o ) ) |-> ( x o. ( y e. B |-> ( 1o X. { y } ) ) ) ) ` ( ( E ` R ) ` A ) ) = ( ( ( E ` R ) ` A ) o. ( y e. B |-> ( 1o X. { y } ) ) ) )
43	34, 42	syl 17	. 2 |- ( ( S e. CRing /\ R e. (SubRing ` S ) /\ A e. K ) -> ( ( x e. ( B ^m ( B ^m 1o ) ) |-> ( x o. ( y e. B |-> ( 1o X. { y } ) ) ) ) ` ( ( E ` R ) ` A ) ) = ( ( ( E ` R ) ` A ) o. ( y e. B |-> ( 1o X. { y } ) ) ) )
44	12, 29, 43	3eqtrd 2660	1 |- ( ( S e. CRing /\ R e. (SubRing ` S ) /\ A e. K ) -> ( Q ` A ) = ( ( ( E ` R ) ` A ) o. ( y e. B |-> ( 1o X. { y } ) ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   _Vcvv 3200   C_ wss 3574   ~Pcpw 4158   {csn 4177   |-> cmpt 4729   X. cxp 5112   o. ccom 5118   Oncon0 5723   -->wf 5884   ` cfv 5888  (class class class)co 6650   1oc1o 7553   ^m cmap 7857   Basecbs 15857   ↾_s cress 15858   ^_s cpws 16107   CRingccrg 18548   RingHom crh 18712  SubRingcsubrg 18776   mPoly cmpl 19353   evalSub ces 19504   evalSub₁ ces1 19678

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-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  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-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-iin 4523  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-of 6897  df-ofr 6898  df-om 7066  df-1st 7168  df-2nd 7169  df-supp 7296  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-oadd 7564  df-er 7742  df-map 7859  df-pm 7860  df-ixp 7909  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-fsupp 8276  df-sup 8348  df-oi 8415  df-card 8765  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-2 11079  df-3 11080  df-4 11081  df-5 11082  df-6 11083  df-7 11084  df-8 11085  df-9 11086  df-n0 11293  df-z 11378  df-dec 11494  df-uz 11688  df-fz 12327  df-fzo 12466  df-seq 12802  df-hash 13118  df-struct 15859  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-ress 15865  df-plusg 15954  df-mulr 15955  df-sca 15957  df-vsca 15958  df-ip 15959  df-tset 15960  df-ple 15961  df-ds 15964  df-hom 15966  df-cco 15967  df-0g 16102  df-gsum 16103  df-prds 16108  df-pws 16110  df-mre 16246  df-mrc 16247  df-acs 16249  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-mhm 17335  df-submnd 17336  df-grp 17425  df-minusg 17426  df-sbg 17427  df-mulg 17541  df-subg 17591  df-ghm 17658  df-cntz 17750  df-cmn 18195  df-abl 18196  df-mgp 18490  df-ur 18502  df-srg 18506  df-ring 18549  df-cring 18550  df-rnghom 18715  df-subrg 18778  df-lmod 18865  df-lss 18933  df-lsp 18972  df-assa 19312  df-asp 19313  df-ascl 19314  df-psr 19356  df-mvr 19357  df-mpl 19358  df-evls 19506  df-evls1 19680

This theorem is referenced by:  evls1var  19702