Theorem evlsval2 19520

Description: Characterizing properties of the polynomial evaluation map function. (Contributed by Stefan O'Rear, 12-Mar-2015.) (Revised by AV, 18-Sep-2021.)

Hypotheses

Ref	Expression
evlsval.q	|- Q = ( ( I evalSub S ) ` R )
evlsval.w	|- W = ( I mPoly U )
evlsval.v	|- V = ( I mVar U )
evlsval.u	|- U = ( S ↾s R )
evlsval.t	|- T = ( S ^s ( B ^m I ) )
evlsval.b	|- B = ( Base ` S )
evlsval.a	|- A = (algSc ` W )
evlsval.x	|- X = ( x e. R |-> ( ( B ^m I ) X. { x } ) )
evlsval.y	|- Y = ( x e. I |-> ( g e. ( B ^m I ) |-> ( g ` x ) ) )

Assertion

Ref	Expression
evlsval2	|- ( ( I e. Z /\ S e. CRing /\ R e. (SubRing ` S ) ) -> ( Q e. ( W RingHom T ) /\ ( ( Q o. A ) = X /\ ( Q o. V ) = Y ) ) )

Distinct variable groups:   g, I, x   x, R   S, g, x   B, g, x   R, g   x, T   g, Z, x

Allowed substitution hints:   A( x, g)   Q( x, g)   T( g)   U( x, g)   V( x, g)   W( x, g)   X( x, g)   Y( x, g)

Proof of Theorem evlsval2

Dummy variable m is distinct from all other variables.

Step	Hyp	Ref	Expression
1		evlsval.q	. . . 4 |- Q = ( ( I evalSub S ) ` R )
2		evlsval.w	. . . 4 |- W = ( I mPoly U )
3		evlsval.v	. . . 4 |- V = ( I mVar U )
4		evlsval.u	. . . 4 |- U = ( S ↾s R )
5		evlsval.t	. . . 4 |- T = ( S ^s ( B ^m I ) )
6		evlsval.b	. . . 4 |- B = ( Base ` S )
7		evlsval.a	. . . 4 |- A = (algSc ` W )
8		evlsval.x	. . . 4 |- X = ( x e. R |-> ( ( B ^m I ) X. { x } ) )
9		evlsval.y	. . . 4 |- Y = ( x e. I |-> ( g e. ( B ^m I ) |-> ( g ` x ) ) )
10	1, 2, 3, 4, 5, 6, 7, 8, 9	evlsval 19519	. . 3 |- ( ( I e. Z /\ S e. CRing /\ R e. (SubRing ` S ) ) -> Q = ( iota_ m e. ( W RingHom T ) ( ( m o. A ) = X /\ ( m o. V ) = Y ) ) )
11		eqid 2622	. . . . 5 |- ( Base ` T ) = ( Base ` T )
12		elex 3212	. . . . . 6 |- ( I e. Z -> I e. _V )
13	12	3ad2ant1 1082	. . . . 5 |- ( ( I e. Z /\ S e. CRing /\ R e. (SubRing ` S ) ) -> I e. _V )
14	4	subrgcrng 18784	. . . . . 6 |- ( ( S e. CRing /\ R e. (SubRing ` S ) ) -> U e. CRing )
15	14	3adant1 1079	. . . . 5 |- ( ( I e. Z /\ S e. CRing /\ R e. (SubRing ` S ) ) -> U e. CRing )
16		simp2 1062	. . . . . 6 |- ( ( I e. Z /\ S e. CRing /\ R e. (SubRing ` S ) ) -> S e. CRing )
17		ovex 6678	. . . . . 6 |- ( B ^m I ) e. _V
18	5	pwscrng 18617	. . . . . 6 |- ( ( S e. CRing /\ ( B ^m I ) e. _V ) -> T e. CRing )
19	16, 17, 18	sylancl 694	. . . . 5 |- ( ( I e. Z /\ S e. CRing /\ R e. (SubRing ` S ) ) -> T e. CRing )
20	6	subrgss 18781	. . . . . . . . 9 |- ( R e. (SubRing ` S ) -> R C_ B )
21	20	3ad2ant3 1084	. . . . . . . 8 |- ( ( I e. Z /\ S e. CRing /\ R e. (SubRing ` S ) ) -> R C_ B )
22	21	resmptd 5452	. . . . . . 7 |- ( ( I e. Z /\ S e. CRing /\ R e. (SubRing ` S ) ) -> ( ( x e. B |-> ( ( B ^m I ) X. { x } ) ) |` R ) = ( x e. R |-> ( ( B ^m I ) X. { x } ) ) )
23	22, 8	syl6eqr 2674	. . . . . 6 |- ( ( I e. Z /\ S e. CRing /\ R e. (SubRing ` S ) ) -> ( ( x e. B |-> ( ( B ^m I ) X. { x } ) ) |` R ) = X )
24		crngring 18558	. . . . . . . . 9 |- ( S e. CRing -> S e. Ring )
25	24	3ad2ant2 1083	. . . . . . . 8 |- ( ( I e. Z /\ S e. CRing /\ R e. (SubRing ` S ) ) -> S e. Ring )
26		eqid 2622	. . . . . . . . 9 |- ( x e. B |-> ( ( B ^m I ) X. { x } ) ) = ( x e. B |-> ( ( B ^m I ) X. { x } ) )
27	5, 6, 26	pwsdiagrhm 18813	. . . . . . . 8 |- ( ( S e. Ring /\ ( B ^m I ) e. _V ) -> ( x e. B |-> ( ( B ^m I ) X. { x } ) ) e. ( S RingHom T ) )
28	25, 17, 27	sylancl 694	. . . . . . 7 |- ( ( I e. Z /\ S e. CRing /\ R e. (SubRing ` S ) ) -> ( x e. B |-> ( ( B ^m I ) X. { x } ) ) e. ( S RingHom T ) )
29		simp3 1063	. . . . . . 7 |- ( ( I e. Z /\ S e. CRing /\ R e. (SubRing ` S ) ) -> R e. (SubRing ` S ) )
30	4	resrhm 18809	. . . . . . 7 |- ( ( ( x e. B |-> ( ( B ^m I ) X. { x } ) ) e. ( S RingHom T ) /\ R e. (SubRing ` S ) ) -> ( ( x e. B |-> ( ( B ^m I ) X. { x } ) ) |` R ) e. ( U RingHom T ) )
31	28, 29, 30	syl2anc 693	. . . . . 6 |- ( ( I e. Z /\ S e. CRing /\ R e. (SubRing ` S ) ) -> ( ( x e. B |-> ( ( B ^m I ) X. { x } ) ) |` R ) e. ( U RingHom T ) )
32	23, 31	eqeltrrd 2702	. . . . 5 |- ( ( I e. Z /\ S e. CRing /\ R e. (SubRing ` S ) ) -> X e. ( U RingHom T ) )
33		fvex 6201	. . . . . . . . . . . 12 |- ( Base ` S ) e. _V
34	6, 33	eqeltri 2697	. . . . . . . . . . 11 |- B e. _V
35		simpl1 1064	. . . . . . . . . . 11 |- ( ( ( I e. Z /\ S e. CRing /\ R e. (SubRing ` S ) ) /\ x e. I ) -> I e. Z )
36		elmapg 7870	. . . . . . . . . . 11 |- ( ( B e. _V /\ I e. Z ) -> ( g e. ( B ^m I ) <-> g : I --> B ) )
37	34, 35, 36	sylancr 695	. . . . . . . . . 10 |- ( ( ( I e. Z /\ S e. CRing /\ R e. (SubRing ` S ) ) /\ x e. I ) -> ( g e. ( B ^m I ) <-> g : I --> B ) )
38	37	biimpa 501	. . . . . . . . 9 |- ( ( ( ( I e. Z /\ S e. CRing /\ R e. (SubRing ` S ) ) /\ x e. I ) /\ g e. ( B ^m I ) ) -> g : I --> B )
39		simplr 792	. . . . . . . . 9 |- ( ( ( ( I e. Z /\ S e. CRing /\ R e. (SubRing ` S ) ) /\ x e. I ) /\ g e. ( B ^m I ) ) -> x e. I )
40	38, 39	ffvelrnd 6360	. . . . . . . 8 |- ( ( ( ( I e. Z /\ S e. CRing /\ R e. (SubRing ` S ) ) /\ x e. I ) /\ g e. ( B ^m I ) ) -> ( g ` x ) e. B )
41		eqid 2622	. . . . . . . 8 |- ( g e. ( B ^m I ) |-> ( g ` x ) ) = ( g e. ( B ^m I ) |-> ( g ` x ) )
42	40, 41	fmptd 6385	. . . . . . 7 |- ( ( ( I e. Z /\ S e. CRing /\ R e. (SubRing ` S ) ) /\ x e. I ) -> ( g e. ( B ^m I ) |-> ( g ` x ) ) : ( B ^m I ) --> B )
43		simpl2 1065	. . . . . . . 8 |- ( ( ( I e. Z /\ S e. CRing /\ R e. (SubRing ` S ) ) /\ x e. I ) -> S e. CRing )
44	5, 6, 11	pwselbasb 16148	. . . . . . . 8 |- ( ( S e. CRing /\ ( B ^m I ) e. _V ) -> ( ( g e. ( B ^m I ) |-> ( g ` x ) ) e. ( Base ` T ) <-> ( g e. ( B ^m I ) |-> ( g ` x ) ) : ( B ^m I ) --> B ) )
45	43, 17, 44	sylancl 694	. . . . . . 7 |- ( ( ( I e. Z /\ S e. CRing /\ R e. (SubRing ` S ) ) /\ x e. I ) -> ( ( g e. ( B ^m I ) |-> ( g ` x ) ) e. ( Base ` T ) <-> ( g e. ( B ^m I ) |-> ( g ` x ) ) : ( B ^m I ) --> B ) )
46	42, 45	mpbird 247	. . . . . 6 |- ( ( ( I e. Z /\ S e. CRing /\ R e. (SubRing ` S ) ) /\ x e. I ) -> ( g e. ( B ^m I ) |-> ( g ` x ) ) e. ( Base ` T ) )
47	46, 9	fmptd 6385	. . . . 5 |- ( ( I e. Z /\ S e. CRing /\ R e. (SubRing ` S ) ) -> Y : I --> ( Base ` T ) )
48	2, 11, 7, 3, 13, 15, 19, 32, 47	evlseu 19516	. . . 4 |- ( ( I e. Z /\ S e. CRing /\ R e. (SubRing ` S ) ) -> E! m e. ( W RingHom T ) ( ( m o. A ) = X /\ ( m o. V ) = Y ) )
49		riotacl2 6624	. . . 4 |- ( E! m e. ( W RingHom T ) ( ( m o. A ) = X /\ ( m o. V ) = Y ) -> ( iota_ m e. ( W RingHom T ) ( ( m o. A ) = X /\ ( m o. V ) = Y ) ) e. { m e. ( W RingHom T ) | ( ( m o. A ) = X /\ ( m o. V ) = Y ) } )
50	48, 49	syl 17	. . 3 |- ( ( I e. Z /\ S e. CRing /\ R e. (SubRing ` S ) ) -> ( iota_ m e. ( W RingHom T ) ( ( m o. A ) = X /\ ( m o. V ) = Y ) ) e. { m e. ( W RingHom T ) | ( ( m o. A ) = X /\ ( m o. V ) = Y ) } )
51	10, 50	eqeltrd 2701	. 2 |- ( ( I e. Z /\ S e. CRing /\ R e. (SubRing ` S ) ) -> Q e. { m e. ( W RingHom T ) | ( ( m o. A ) = X /\ ( m o. V ) = Y ) } )
52		coeq1 5279	. . . . 5 |- ( m = Q -> ( m o. A ) = ( Q o. A ) )
53	52	eqeq1d 2624	. . . 4 |- ( m = Q -> ( ( m o. A ) = X <-> ( Q o. A ) = X ) )
54		coeq1 5279	. . . . 5 |- ( m = Q -> ( m o. V ) = ( Q o. V ) )
55	54	eqeq1d 2624	. . . 4 |- ( m = Q -> ( ( m o. V ) = Y <-> ( Q o. V ) = Y ) )
56	53, 55	anbi12d 747	. . 3 |- ( m = Q -> ( ( ( m o. A ) = X /\ ( m o. V ) = Y ) <-> ( ( Q o. A ) = X /\ ( Q o. V ) = Y ) ) )
57	56	elrab 3363	. 2 |- ( Q e. { m e. ( W RingHom T ) | ( ( m o. A ) = X /\ ( m o. V ) = Y ) } <-> ( Q e. ( W RingHom T ) /\ ( ( Q o. A ) = X /\ ( Q o. V ) = Y ) ) )
58	51, 57	sylib 208	1 |- ( ( I e. Z /\ S e. CRing /\ R e. (SubRing ` S ) ) -> ( Q e. ( W RingHom T ) /\ ( ( Q o. A ) = X /\ ( Q o. V ) = Y ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   E!wreu 2914   {crab 2916   _Vcvv 3200   C_ wss 3574   {csn 4177   |-> cmpt 4729   X. cxp 5112   |` cres 5116   o. ccom 5118   -->wf 5884   ` cfv 5888   iota_crio 6610  (class class class)co 6650   ^m cmap 7857   Basecbs 15857   ↾_s cress 15858   ^_s cpws 16107   Ringcrg 18547   CRingccrg 18548   RingHom crh 18712  SubRingcsubrg 18776  algSccascl 19311   mVar cmvr 19352   mPoly cmpl 19353   evalSub ces 19504

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

This theorem is referenced by:  evlsrhm  19521  evlssca  19522  evlsvar  19523