Theorem evlseu 19516

Description: For a given interpretation of the variables G and of the scalars F, this extends to a homomorphic interpretation of the polynomial ring in exactly one way. (Contributed by Stefan O'Rear, 9-Mar-2015.)

Hypotheses

Ref	Expression
evlseu.p	|- P = ( I mPoly R )
evlseu.c	|- C = ( Base ` S )
evlseu.a	|- A = (algSc ` P )
evlseu.v	|- V = ( I mVar R )
evlseu.i	|- ( ph -> I e. _V )
evlseu.r	|- ( ph -> R e. CRing )
evlseu.s	|- ( ph -> S e. CRing )
evlseu.f	|- ( ph -> F e. ( R RingHom S ) )
evlseu.g	|- ( ph -> G : I --> C )

Assertion

Ref	Expression
evlseu	|- ( ph -> E! m e. ( P RingHom S ) ( ( m o. A ) = F /\ ( m o. V ) = G ) )

Distinct variable groups:   A, m   m, F   m, G   m, I   P, m   ph, m   S, m   m, V

Allowed substitution hints:   C( m)   R( m)

Proof of Theorem evlseu

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

Step	Hyp	Ref	Expression
1		evlseu.p	. . . 4 |- P = ( I mPoly R )
2		eqid 2622	. . . 4 |- ( Base ` P ) = ( Base ` P )
3		evlseu.c	. . . 4 |- C = ( Base ` S )
4		eqid 2622	. . . 4 |- ( Base ` R ) = ( Base ` R )
5		eqid 2622	. . . 4 |- { z e. ( NN0 ^m I ) | ( `' z " NN ) e. Fin } = { z e. ( NN0 ^m I ) | ( `' z " NN ) e. Fin }
6		eqid 2622	. . . 4 |- (mulGrp ` S ) = (mulGrp ` S )
7		eqid 2622	. . . 4 |- (.g ` (mulGrp ` S ) ) = (.g ` (mulGrp ` S ) )
8		eqid 2622	. . . 4 |- ( .r ` S ) = ( .r ` S )
9		evlseu.v	. . . 4 |- V = ( I mVar R )
10		eqid 2622	. . . 4 |- ( x e. ( Base ` P ) |-> ( S gsumg ( y e. { z e. ( NN0 ^m I ) | ( `' z " NN ) e. Fin } |-> ( ( F ` ( x ` y ) ) ( .r ` S ) ( (mulGrp ` S ) gsumg ( y oF (.g ` (mulGrp ` S ) ) G ) ) ) ) ) ) = ( x e. ( Base ` P ) |-> ( S gsumg ( y e. { z e. ( NN0 ^m I ) | ( `' z " NN ) e. Fin } |-> ( ( F ` ( x ` y ) ) ( .r ` S ) ( (mulGrp ` S ) gsumg ( y oF (.g ` (mulGrp ` S ) ) G ) ) ) ) ) )
11		evlseu.i	. . . 4 |- ( ph -> I e. _V )
12		evlseu.r	. . . 4 |- ( ph -> R e. CRing )
13		evlseu.s	. . . 4 |- ( ph -> S e. CRing )
14		evlseu.f	. . . 4 |- ( ph -> F e. ( R RingHom S ) )
15		evlseu.g	. . . 4 |- ( ph -> G : I --> C )
16		evlseu.a	. . . 4 |- A = (algSc ` P )
17	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16	evlslem1 19515	. . 3 |- ( ph -> ( ( x e. ( Base ` P ) |-> ( S gsumg ( y e. { z e. ( NN0 ^m I ) | ( `' z " NN ) e. Fin } |-> ( ( F ` ( x ` y ) ) ( .r ` S ) ( (mulGrp ` S ) gsumg ( y oF (.g ` (mulGrp ` S ) ) G ) ) ) ) ) ) e. ( P RingHom S ) /\ ( ( x e. ( Base ` P ) |-> ( S gsumg ( y e. { z e. ( NN0 ^m I ) | ( `' z " NN ) e. Fin } |-> ( ( F ` ( x ` y ) ) ( .r ` S ) ( (mulGrp ` S ) gsumg ( y oF (.g ` (mulGrp ` S ) ) G ) ) ) ) ) ) o. A ) = F /\ ( ( x e. ( Base ` P ) |-> ( S gsumg ( y e. { z e. ( NN0 ^m I ) | ( `' z " NN ) e. Fin } |-> ( ( F ` ( x ` y ) ) ( .r ` S ) ( (mulGrp ` S ) gsumg ( y oF (.g ` (mulGrp ` S ) ) G ) ) ) ) ) ) o. V ) = G ) )
18		coeq1 5279	. . . . . . 7 |- ( m = ( x e. ( Base ` P ) |-> ( S gsumg ( y e. { z e. ( NN0 ^m I ) | ( `' z " NN ) e. Fin } |-> ( ( F ` ( x ` y ) ) ( .r ` S ) ( (mulGrp ` S ) gsumg ( y oF (.g ` (mulGrp ` S ) ) G ) ) ) ) ) ) -> ( m o. A ) = ( ( x e. ( Base ` P ) |-> ( S gsumg ( y e. { z e. ( NN0 ^m I ) | ( `' z " NN ) e. Fin } |-> ( ( F ` ( x ` y ) ) ( .r ` S ) ( (mulGrp ` S ) gsumg ( y oF (.g ` (mulGrp ` S ) ) G ) ) ) ) ) ) o. A ) )
19	18	eqeq1d 2624	. . . . . 6 |- ( m = ( x e. ( Base ` P ) |-> ( S gsumg ( y e. { z e. ( NN0 ^m I ) | ( `' z " NN ) e. Fin } |-> ( ( F ` ( x ` y ) ) ( .r ` S ) ( (mulGrp ` S ) gsumg ( y oF (.g ` (mulGrp ` S ) ) G ) ) ) ) ) ) -> ( ( m o. A ) = F <-> ( ( x e. ( Base ` P ) |-> ( S gsumg ( y e. { z e. ( NN0 ^m I ) | ( `' z " NN ) e. Fin } |-> ( ( F ` ( x ` y ) ) ( .r ` S ) ( (mulGrp ` S ) gsumg ( y oF (.g ` (mulGrp ` S ) ) G ) ) ) ) ) ) o. A ) = F ) )
20		coeq1 5279	. . . . . . 7 |- ( m = ( x e. ( Base ` P ) |-> ( S gsumg ( y e. { z e. ( NN0 ^m I ) | ( `' z " NN ) e. Fin } |-> ( ( F ` ( x ` y ) ) ( .r ` S ) ( (mulGrp ` S ) gsumg ( y oF (.g ` (mulGrp ` S ) ) G ) ) ) ) ) ) -> ( m o. V ) = ( ( x e. ( Base ` P ) |-> ( S gsumg ( y e. { z e. ( NN0 ^m I ) | ( `' z " NN ) e. Fin } |-> ( ( F ` ( x ` y ) ) ( .r ` S ) ( (mulGrp ` S ) gsumg ( y oF (.g ` (mulGrp ` S ) ) G ) ) ) ) ) ) o. V ) )
21	20	eqeq1d 2624	. . . . . 6 |- ( m = ( x e. ( Base ` P ) |-> ( S gsumg ( y e. { z e. ( NN0 ^m I ) | ( `' z " NN ) e. Fin } |-> ( ( F ` ( x ` y ) ) ( .r ` S ) ( (mulGrp ` S ) gsumg ( y oF (.g ` (mulGrp ` S ) ) G ) ) ) ) ) ) -> ( ( m o. V ) = G <-> ( ( x e. ( Base ` P ) |-> ( S gsumg ( y e. { z e. ( NN0 ^m I ) | ( `' z " NN ) e. Fin } |-> ( ( F ` ( x ` y ) ) ( .r ` S ) ( (mulGrp ` S ) gsumg ( y oF (.g ` (mulGrp ` S ) ) G ) ) ) ) ) ) o. V ) = G ) )
22	19, 21	anbi12d 747	. . . . 5 |- ( m = ( x e. ( Base ` P ) |-> ( S gsumg ( y e. { z e. ( NN0 ^m I ) | ( `' z " NN ) e. Fin } |-> ( ( F ` ( x ` y ) ) ( .r ` S ) ( (mulGrp ` S ) gsumg ( y oF (.g ` (mulGrp ` S ) ) G ) ) ) ) ) ) -> ( ( ( m o. A ) = F /\ ( m o. V ) = G ) <-> ( ( ( x e. ( Base ` P ) |-> ( S gsumg ( y e. { z e. ( NN0 ^m I ) | ( `' z " NN ) e. Fin } |-> ( ( F ` ( x ` y ) ) ( .r ` S ) ( (mulGrp ` S ) gsumg ( y oF (.g ` (mulGrp ` S ) ) G ) ) ) ) ) ) o. A ) = F /\ ( ( x e. ( Base ` P ) |-> ( S gsumg ( y e. { z e. ( NN0 ^m I ) | ( `' z " NN ) e. Fin } |-> ( ( F ` ( x ` y ) ) ( .r ` S ) ( (mulGrp ` S ) gsumg ( y oF (.g ` (mulGrp ` S ) ) G ) ) ) ) ) ) o. V ) = G ) ) )
23	22	rspcev 3309	. . . 4 |- ( ( ( x e. ( Base ` P ) |-> ( S gsumg ( y e. { z e. ( NN0 ^m I ) | ( `' z " NN ) e. Fin } |-> ( ( F ` ( x ` y ) ) ( .r ` S ) ( (mulGrp ` S ) gsumg ( y oF (.g ` (mulGrp ` S ) ) G ) ) ) ) ) ) e. ( P RingHom S ) /\ ( ( ( x e. ( Base ` P ) |-> ( S gsumg ( y e. { z e. ( NN0 ^m I ) | ( `' z " NN ) e. Fin } |-> ( ( F ` ( x ` y ) ) ( .r ` S ) ( (mulGrp ` S ) gsumg ( y oF (.g ` (mulGrp ` S ) ) G ) ) ) ) ) ) o. A ) = F /\ ( ( x e. ( Base ` P ) |-> ( S gsumg ( y e. { z e. ( NN0 ^m I ) | ( `' z " NN ) e. Fin } |-> ( ( F ` ( x ` y ) ) ( .r ` S ) ( (mulGrp ` S ) gsumg ( y oF (.g ` (mulGrp ` S ) ) G ) ) ) ) ) ) o. V ) = G ) ) -> E. m e. ( P RingHom S ) ( ( m o. A ) = F /\ ( m o. V ) = G ) )
24	23	3impb 1260	. . 3 |- ( ( ( x e. ( Base ` P ) |-> ( S gsumg ( y e. { z e. ( NN0 ^m I ) | ( `' z " NN ) e. Fin } |-> ( ( F ` ( x ` y ) ) ( .r ` S ) ( (mulGrp ` S ) gsumg ( y oF (.g ` (mulGrp ` S ) ) G ) ) ) ) ) ) e. ( P RingHom S ) /\ ( ( x e. ( Base ` P ) |-> ( S gsumg ( y e. { z e. ( NN0 ^m I ) | ( `' z " NN ) e. Fin } |-> ( ( F ` ( x ` y ) ) ( .r ` S ) ( (mulGrp ` S ) gsumg ( y oF (.g ` (mulGrp ` S ) ) G ) ) ) ) ) ) o. A ) = F /\ ( ( x e. ( Base ` P ) |-> ( S gsumg ( y e. { z e. ( NN0 ^m I ) | ( `' z " NN ) e. Fin } |-> ( ( F ` ( x ` y ) ) ( .r ` S ) ( (mulGrp ` S ) gsumg ( y oF (.g ` (mulGrp ` S ) ) G ) ) ) ) ) ) o. V ) = G ) -> E. m e. ( P RingHom S ) ( ( m o. A ) = F /\ ( m o. V ) = G ) )
25	17, 24	syl 17	. 2 |- ( ph -> E. m e. ( P RingHom S ) ( ( m o. A ) = F /\ ( m o. V ) = G ) )
26		crngring 18558	. . . . . . . . . . 11 |- ( R e. CRing -> R e. Ring )
27	12, 26	syl 17	. . . . . . . . . 10 |- ( ph -> R e. Ring )
28		eqid 2622	. . . . . . . . . . 11 |- (Scalar ` P ) = (Scalar ` P )
29	1	mplring 19452	. . . . . . . . . . 11 |- ( ( I e. _V /\ R e. Ring ) -> P e. Ring )
30	1	mpllmod 19451	. . . . . . . . . . 11 |- ( ( I e. _V /\ R e. Ring ) -> P e. LMod )
31		eqid 2622	. . . . . . . . . . 11 |- ( Base ` (Scalar ` P ) ) = ( Base ` (Scalar ` P ) )
32	16, 28, 29, 30, 31, 2	asclf 19337	. . . . . . . . . 10 |- ( ( I e. _V /\ R e. Ring ) -> A : ( Base ` (Scalar ` P ) ) --> ( Base ` P ) )
33	11, 27, 32	syl2anc 693	. . . . . . . . 9 |- ( ph -> A : ( Base ` (Scalar ` P ) ) --> ( Base ` P ) )
34		ffun 6048	. . . . . . . . 9 |- ( A : ( Base ` (Scalar ` P ) ) --> ( Base ` P ) -> Fun A )
35	33, 34	syl 17	. . . . . . . 8 |- ( ph -> Fun A )
36		funcoeqres 6167	. . . . . . . 8 |- ( ( Fun A /\ ( m o. A ) = F ) -> ( m |` ran A ) = ( F o. `' A ) )
37	35, 36	sylan 488	. . . . . . 7 |- ( ( ph /\ ( m o. A ) = F ) -> ( m |` ran A ) = ( F o. `' A ) )
38	1, 9, 2, 11, 27	mvrf2 19492	. . . . . . . . 9 |- ( ph -> V : I --> ( Base ` P ) )
39		ffun 6048	. . . . . . . . 9 |- ( V : I --> ( Base ` P ) -> Fun V )
40	38, 39	syl 17	. . . . . . . 8 |- ( ph -> Fun V )
41		funcoeqres 6167	. . . . . . . 8 |- ( ( Fun V /\ ( m o. V ) = G ) -> ( m |` ran V ) = ( G o. `' V ) )
42	40, 41	sylan 488	. . . . . . 7 |- ( ( ph /\ ( m o. V ) = G ) -> ( m |` ran V ) = ( G o. `' V ) )
43	37, 42	anim12dan 882	. . . . . 6 |- ( ( ph /\ ( ( m o. A ) = F /\ ( m o. V ) = G ) ) -> ( ( m |` ran A ) = ( F o. `' A ) /\ ( m |` ran V ) = ( G o. `' V ) ) )
44	43	ex 450	. . . . 5 |- ( ph -> ( ( ( m o. A ) = F /\ ( m o. V ) = G ) -> ( ( m |` ran A ) = ( F o. `' A ) /\ ( m |` ran V ) = ( G o. `' V ) ) ) )
45		resundi 5410	. . . . . 6 |- ( m |` ( ran A u. ran V ) ) = ( ( m |` ran A ) u. ( m |` ran V ) )
46		uneq12 3762	. . . . . 6 |- ( ( ( m |` ran A ) = ( F o. `' A ) /\ ( m |` ran V ) = ( G o. `' V ) ) -> ( ( m |` ran A ) u. ( m |` ran V ) ) = ( ( F o. `' A ) u. ( G o. `' V ) ) )
47	45, 46	syl5eq 2668	. . . . 5 |- ( ( ( m |` ran A ) = ( F o. `' A ) /\ ( m |` ran V ) = ( G o. `' V ) ) -> ( m |` ( ran A u. ran V ) ) = ( ( F o. `' A ) u. ( G o. `' V ) ) )
48	44, 47	syl6 35	. . . 4 |- ( ph -> ( ( ( m o. A ) = F /\ ( m o. V ) = G ) -> ( m |` ( ran A u. ran V ) ) = ( ( F o. `' A ) u. ( G o. `' V ) ) ) )
49	48	ralrimivw 2967	. . 3 |- ( ph -> A. m e. ( P RingHom S ) ( ( ( m o. A ) = F /\ ( m o. V ) = G ) -> ( m |` ( ran A u. ran V ) ) = ( ( F o. `' A ) u. ( G o. `' V ) ) ) )
50		eqtr3 2643	. . . . . 6 |- ( ( ( m |` ( ran A u. ran V ) ) = ( ( F o. `' A ) u. ( G o. `' V ) ) /\ ( n |` ( ran A u. ran V ) ) = ( ( F o. `' A ) u. ( G o. `' V ) ) ) -> ( m |` ( ran A u. ran V ) ) = ( n |` ( ran A u. ran V ) ) )
51		eqid 2622	. . . . . . . . . . . . 13 |- ( I mPwSer R ) = ( I mPwSer R )
52	51, 11, 12	psrassa 19414	. . . . . . . . . . . 12 |- ( ph -> ( I mPwSer R ) e. AssAlg )
53		eqid 2622	. . . . . . . . . . . . . 14 |- ( Base ` ( I mPwSer R ) ) = ( Base ` ( I mPwSer R ) )
54	51, 9, 53, 11, 27	mvrf 19424	. . . . . . . . . . . . 13 |- ( ph -> V : I --> ( Base ` ( I mPwSer R ) ) )
55		frn 6053	. . . . . . . . . . . . 13 |- ( V : I --> ( Base ` ( I mPwSer R ) ) -> ran V C_ ( Base ` ( I mPwSer R ) ) )
56	54, 55	syl 17	. . . . . . . . . . . 12 |- ( ph -> ran V C_ ( Base ` ( I mPwSer R ) ) )
57		eqid 2622	. . . . . . . . . . . . 13 |- (AlgSpan ` ( I mPwSer R ) ) = (AlgSpan ` ( I mPwSer R ) )
58		eqid 2622	. . . . . . . . . . . . 13 |- (algSc ` ( I mPwSer R ) ) = (algSc ` ( I mPwSer R ) )
59		eqid 2622	. . . . . . . . . . . . 13 |- (mrCls ` (SubRing ` ( I mPwSer R ) ) ) = (mrCls ` (SubRing ` ( I mPwSer R ) ) )
60	57, 58, 59, 53	aspval2 19347	. . . . . . . . . . . 12 |- ( ( ( I mPwSer R ) e. AssAlg /\ ran V C_ ( Base ` ( I mPwSer R ) ) ) -> ( (AlgSpan ` ( I mPwSer R ) ) ` ran V ) = ( (mrCls ` (SubRing ` ( I mPwSer R ) ) ) ` ( ran (algSc ` ( I mPwSer R ) ) u. ran V ) ) )
61	52, 56, 60	syl2anc 693	. . . . . . . . . . 11 |- ( ph -> ( (AlgSpan ` ( I mPwSer R ) ) ` ran V ) = ( (mrCls ` (SubRing ` ( I mPwSer R ) ) ) ` ( ran (algSc ` ( I mPwSer R ) ) u. ran V ) ) )
62	1, 51, 9, 57, 11, 12	mplbas2 19470	. . . . . . . . . . 11 |- ( ph -> ( (AlgSpan ` ( I mPwSer R ) ) ` ran V ) = ( Base ` P ) )
63	51, 1, 2, 11, 27	mplsubrg 19440	. . . . . . . . . . . . . . 15 |- ( ph -> ( Base ` P ) e. (SubRing ` ( I mPwSer R ) ) )
64	1, 51, 2	mplval2 19431	. . . . . . . . . . . . . . . 16 |- P = ( ( I mPwSer R ) ↾s ( Base ` P ) )
65	64	subsubrg2 18807	. . . . . . . . . . . . . . 15 |- ( ( Base ` P ) e. (SubRing ` ( I mPwSer R ) ) -> (SubRing ` P ) = ( (SubRing ` ( I mPwSer R ) ) i^i ~P ( Base ` P ) ) )
66	63, 65	syl 17	. . . . . . . . . . . . . 14 |- ( ph -> (SubRing ` P ) = ( (SubRing ` ( I mPwSer R ) ) i^i ~P ( Base ` P ) ) )
67	66	fveq2d 6195	. . . . . . . . . . . . 13 |- ( ph -> (mrCls ` (SubRing ` P ) ) = (mrCls ` ( (SubRing ` ( I mPwSer R ) ) i^i ~P ( Base ` P ) ) ) )
68	58, 64	ressascl 19344	. . . . . . . . . . . . . . . . 17 |- ( ( Base ` P ) e. (SubRing ` ( I mPwSer R ) ) -> (algSc ` ( I mPwSer R ) ) = (algSc ` P ) )
69	63, 68	syl 17	. . . . . . . . . . . . . . . 16 |- ( ph -> (algSc ` ( I mPwSer R ) ) = (algSc ` P ) )
70	69, 16	syl6reqr 2675	. . . . . . . . . . . . . . 15 |- ( ph -> A = (algSc ` ( I mPwSer R ) ) )
71	70	rneqd 5353	. . . . . . . . . . . . . 14 |- ( ph -> ran A = ran (algSc ` ( I mPwSer R ) ) )
72	71	uneq1d 3766	. . . . . . . . . . . . 13 |- ( ph -> ( ran A u. ran V ) = ( ran (algSc ` ( I mPwSer R ) ) u. ran V ) )
73	67, 72	fveq12d 6197	. . . . . . . . . . . 12 |- ( ph -> ( (mrCls ` (SubRing ` P ) ) ` ( ran A u. ran V ) ) = ( (mrCls ` ( (SubRing ` ( I mPwSer R ) ) i^i ~P ( Base ` P ) ) ) ` ( ran (algSc ` ( I mPwSer R ) ) u. ran V ) ) )
74		assaring 19320	. . . . . . . . . . . . . 14 |- ( ( I mPwSer R ) e. AssAlg -> ( I mPwSer R ) e. Ring )
75	53	subrgmre 18804	. . . . . . . . . . . . . 14 |- ( ( I mPwSer R ) e. Ring -> (SubRing ` ( I mPwSer R ) ) e. (Moore ` ( Base ` ( I mPwSer R ) ) ) )
76	52, 74, 75	3syl 18	. . . . . . . . . . . . 13 |- ( ph -> (SubRing ` ( I mPwSer R ) ) e. (Moore ` ( Base ` ( I mPwSer R ) ) ) )
77		frn 6053	. . . . . . . . . . . . . . . 16 |- ( A : ( Base ` (Scalar ` P ) ) --> ( Base ` P ) -> ran A C_ ( Base ` P ) )
78	33, 77	syl 17	. . . . . . . . . . . . . . 15 |- ( ph -> ran A C_ ( Base ` P ) )
79	71, 78	eqsstr3d 3640	. . . . . . . . . . . . . 14 |- ( ph -> ran (algSc ` ( I mPwSer R ) ) C_ ( Base ` P ) )
80		frn 6053	. . . . . . . . . . . . . . 15 |- ( V : I --> ( Base ` P ) -> ran V C_ ( Base ` P ) )
81	38, 80	syl 17	. . . . . . . . . . . . . 14 |- ( ph -> ran V C_ ( Base ` P ) )
82	79, 81	unssd 3789	. . . . . . . . . . . . 13 |- ( ph -> ( ran (algSc ` ( I mPwSer R ) ) u. ran V ) C_ ( Base ` P ) )
83		eqid 2622	. . . . . . . . . . . . . 14 |- (mrCls ` ( (SubRing ` ( I mPwSer R ) ) i^i ~P ( Base ` P ) ) ) = (mrCls ` ( (SubRing ` ( I mPwSer R ) ) i^i ~P ( Base ` P ) ) )
84	59, 83	submrc 16288	. . . . . . . . . . . . 13 |- ( ( (SubRing ` ( I mPwSer R ) ) e. (Moore ` ( Base ` ( I mPwSer R ) ) ) /\ ( Base ` P ) e. (SubRing ` ( I mPwSer R ) ) /\ ( ran (algSc ` ( I mPwSer R ) ) u. ran V ) C_ ( Base ` P ) ) -> ( (mrCls ` ( (SubRing ` ( I mPwSer R ) ) i^i ~P ( Base ` P ) ) ) ` ( ran (algSc ` ( I mPwSer R ) ) u. ran V ) ) = ( (mrCls ` (SubRing ` ( I mPwSer R ) ) ) ` ( ran (algSc ` ( I mPwSer R ) ) u. ran V ) ) )
85	76, 63, 82, 84	syl3anc 1326	. . . . . . . . . . . 12 |- ( ph -> ( (mrCls ` ( (SubRing ` ( I mPwSer R ) ) i^i ~P ( Base ` P ) ) ) ` ( ran (algSc ` ( I mPwSer R ) ) u. ran V ) ) = ( (mrCls ` (SubRing ` ( I mPwSer R ) ) ) ` ( ran (algSc ` ( I mPwSer R ) ) u. ran V ) ) )
86	73, 85	eqtr2d 2657	. . . . . . . . . . 11 |- ( ph -> ( (mrCls ` (SubRing ` ( I mPwSer R ) ) ) ` ( ran (algSc ` ( I mPwSer R ) ) u. ran V ) ) = ( (mrCls ` (SubRing ` P ) ) ` ( ran A u. ran V ) ) )
87	61, 62, 86	3eqtr3d 2664	. . . . . . . . . 10 |- ( ph -> ( Base ` P ) = ( (mrCls ` (SubRing ` P ) ) ` ( ran A u. ran V ) ) )
88	87	ad2antrr 762	. . . . . . . . 9 |- ( ( ( ph /\ ( m e. ( P RingHom S ) /\ n e. ( P RingHom S ) ) ) /\ ( ran A u. ran V ) C_ dom ( m i^i n ) ) -> ( Base ` P ) = ( (mrCls ` (SubRing ` P ) ) ` ( ran A u. ran V ) ) )
89	11, 27, 29	syl2anc 693	. . . . . . . . . . . 12 |- ( ph -> P e. Ring )
90	2	subrgmre 18804	. . . . . . . . . . . 12 |- ( P e. Ring -> (SubRing ` P ) e. (Moore ` ( Base ` P ) ) )
91	89, 90	syl 17	. . . . . . . . . . 11 |- ( ph -> (SubRing ` P ) e. (Moore ` ( Base ` P ) ) )
92	91	ad2antrr 762	. . . . . . . . . 10 |- ( ( ( ph /\ ( m e. ( P RingHom S ) /\ n e. ( P RingHom S ) ) ) /\ ( ran A u. ran V ) C_ dom ( m i^i n ) ) -> (SubRing ` P ) e. (Moore ` ( Base ` P ) ) )
93		simpr 477	. . . . . . . . . 10 |- ( ( ( ph /\ ( m e. ( P RingHom S ) /\ n e. ( P RingHom S ) ) ) /\ ( ran A u. ran V ) C_ dom ( m i^i n ) ) -> ( ran A u. ran V ) C_ dom ( m i^i n ) )
94		rhmeql 18810	. . . . . . . . . . 11 |- ( ( m e. ( P RingHom S ) /\ n e. ( P RingHom S ) ) -> dom ( m i^i n ) e. (SubRing ` P ) )
95	94	ad2antlr 763	. . . . . . . . . 10 |- ( ( ( ph /\ ( m e. ( P RingHom S ) /\ n e. ( P RingHom S ) ) ) /\ ( ran A u. ran V ) C_ dom ( m i^i n ) ) -> dom ( m i^i n ) e. (SubRing ` P ) )
96		eqid 2622	. . . . . . . . . . 11 |- (mrCls ` (SubRing ` P ) ) = (mrCls ` (SubRing ` P ) )
97	96	mrcsscl 16280	. . . . . . . . . 10 |- ( ( (SubRing ` P ) e. (Moore ` ( Base ` P ) ) /\ ( ran A u. ran V ) C_ dom ( m i^i n ) /\ dom ( m i^i n ) e. (SubRing ` P ) ) -> ( (mrCls ` (SubRing ` P ) ) ` ( ran A u. ran V ) ) C_ dom ( m i^i n ) )
98	92, 93, 95, 97	syl3anc 1326	. . . . . . . . 9 |- ( ( ( ph /\ ( m e. ( P RingHom S ) /\ n e. ( P RingHom S ) ) ) /\ ( ran A u. ran V ) C_ dom ( m i^i n ) ) -> ( (mrCls ` (SubRing ` P ) ) ` ( ran A u. ran V ) ) C_ dom ( m i^i n ) )
99	88, 98	eqsstrd 3639	. . . . . . . 8 |- ( ( ( ph /\ ( m e. ( P RingHom S ) /\ n e. ( P RingHom S ) ) ) /\ ( ran A u. ran V ) C_ dom ( m i^i n ) ) -> ( Base ` P ) C_ dom ( m i^i n ) )
100	99	ex 450	. . . . . . 7 |- ( ( ph /\ ( m e. ( P RingHom S ) /\ n e. ( P RingHom S ) ) ) -> ( ( ran A u. ran V ) C_ dom ( m i^i n ) -> ( Base ` P ) C_ dom ( m i^i n ) ) )
101		simprl 794	. . . . . . . . 9 |- ( ( ph /\ ( m e. ( P RingHom S ) /\ n e. ( P RingHom S ) ) ) -> m e. ( P RingHom S ) )
102	2, 3	rhmf 18726	. . . . . . . . 9 |- ( m e. ( P RingHom S ) -> m : ( Base ` P ) --> C )
103		ffn 6045	. . . . . . . . 9 |- ( m : ( Base ` P ) --> C -> m Fn ( Base ` P ) )
104	101, 102, 103	3syl 18	. . . . . . . 8 |- ( ( ph /\ ( m e. ( P RingHom S ) /\ n e. ( P RingHom S ) ) ) -> m Fn ( Base ` P ) )
105		simprr 796	. . . . . . . . 9 |- ( ( ph /\ ( m e. ( P RingHom S ) /\ n e. ( P RingHom S ) ) ) -> n e. ( P RingHom S ) )
106	2, 3	rhmf 18726	. . . . . . . . 9 |- ( n e. ( P RingHom S ) -> n : ( Base ` P ) --> C )
107		ffn 6045	. . . . . . . . 9 |- ( n : ( Base ` P ) --> C -> n Fn ( Base ` P ) )
108	105, 106, 107	3syl 18	. . . . . . . 8 |- ( ( ph /\ ( m e. ( P RingHom S ) /\ n e. ( P RingHom S ) ) ) -> n Fn ( Base ` P ) )
109	78	adantr 481	. . . . . . . . 9 |- ( ( ph /\ ( m e. ( P RingHom S ) /\ n e. ( P RingHom S ) ) ) -> ran A C_ ( Base ` P ) )
110	81	adantr 481	. . . . . . . . 9 |- ( ( ph /\ ( m e. ( P RingHom S ) /\ n e. ( P RingHom S ) ) ) -> ran V C_ ( Base ` P ) )
111	109, 110	unssd 3789	. . . . . . . 8 |- ( ( ph /\ ( m e. ( P RingHom S ) /\ n e. ( P RingHom S ) ) ) -> ( ran A u. ran V ) C_ ( Base ` P ) )
112		fnreseql 6327	. . . . . . . 8 |- ( ( m Fn ( Base ` P ) /\ n Fn ( Base ` P ) /\ ( ran A u. ran V ) C_ ( Base ` P ) ) -> ( ( m |` ( ran A u. ran V ) ) = ( n |` ( ran A u. ran V ) ) <-> ( ran A u. ran V ) C_ dom ( m i^i n ) ) )
113	104, 108, 111, 112	syl3anc 1326	. . . . . . 7 |- ( ( ph /\ ( m e. ( P RingHom S ) /\ n e. ( P RingHom S ) ) ) -> ( ( m |` ( ran A u. ran V ) ) = ( n |` ( ran A u. ran V ) ) <-> ( ran A u. ran V ) C_ dom ( m i^i n ) ) )
114		fneqeql2 6326	. . . . . . . 8 |- ( ( m Fn ( Base ` P ) /\ n Fn ( Base ` P ) ) -> ( m = n <-> ( Base ` P ) C_ dom ( m i^i n ) ) )
115	104, 108, 114	syl2anc 693	. . . . . . 7 |- ( ( ph /\ ( m e. ( P RingHom S ) /\ n e. ( P RingHom S ) ) ) -> ( m = n <-> ( Base ` P ) C_ dom ( m i^i n ) ) )
116	100, 113, 115	3imtr4d 283	. . . . . 6 |- ( ( ph /\ ( m e. ( P RingHom S ) /\ n e. ( P RingHom S ) ) ) -> ( ( m |` ( ran A u. ran V ) ) = ( n |` ( ran A u. ran V ) ) -> m = n ) )
117	50, 116	syl5 34	. . . . 5 |- ( ( ph /\ ( m e. ( P RingHom S ) /\ n e. ( P RingHom S ) ) ) -> ( ( ( m |` ( ran A u. ran V ) ) = ( ( F o. `' A ) u. ( G o. `' V ) ) /\ ( n |` ( ran A u. ran V ) ) = ( ( F o. `' A ) u. ( G o. `' V ) ) ) -> m = n ) )
118	117	ralrimivva 2971	. . . 4 |- ( ph -> A. m e. ( P RingHom S ) A. n e. ( P RingHom S ) ( ( ( m |` ( ran A u. ran V ) ) = ( ( F o. `' A ) u. ( G o. `' V ) ) /\ ( n |` ( ran A u. ran V ) ) = ( ( F o. `' A ) u. ( G o. `' V ) ) ) -> m = n ) )
119		reseq1 5390	. . . . . 6 |- ( m = n -> ( m |` ( ran A u. ran V ) ) = ( n |` ( ran A u. ran V ) ) )
120	119	eqeq1d 2624	. . . . 5 |- ( m = n -> ( ( m |` ( ran A u. ran V ) ) = ( ( F o. `' A ) u. ( G o. `' V ) ) <-> ( n |` ( ran A u. ran V ) ) = ( ( F o. `' A ) u. ( G o. `' V ) ) ) )
121	120	rmo4 3399	. . . 4 |- ( E* m e. ( P RingHom S ) ( m |` ( ran A u. ran V ) ) = ( ( F o. `' A ) u. ( G o. `' V ) ) <-> A. m e. ( P RingHom S ) A. n e. ( P RingHom S ) ( ( ( m |` ( ran A u. ran V ) ) = ( ( F o. `' A ) u. ( G o. `' V ) ) /\ ( n |` ( ran A u. ran V ) ) = ( ( F o. `' A ) u. ( G o. `' V ) ) ) -> m = n ) )
122	118, 121	sylibr 224	. . 3 |- ( ph -> E* m e. ( P RingHom S ) ( m |` ( ran A u. ran V ) ) = ( ( F o. `' A ) u. ( G o. `' V ) ) )
123		rmoim 3407	. . 3 |- ( A. m e. ( P RingHom S ) ( ( ( m o. A ) = F /\ ( m o. V ) = G ) -> ( m |` ( ran A u. ran V ) ) = ( ( F o. `' A ) u. ( G o. `' V ) ) ) -> ( E* m e. ( P RingHom S ) ( m |` ( ran A u. ran V ) ) = ( ( F o. `' A ) u. ( G o. `' V ) ) -> E* m e. ( P RingHom S ) ( ( m o. A ) = F /\ ( m o. V ) = G ) ) )
124	49, 122, 123	sylc 65	. 2 |- ( ph -> E* m e. ( P RingHom S ) ( ( m o. A ) = F /\ ( m o. V ) = G ) )
125		reu5 3159	. 2 |- ( E! m e. ( P RingHom S ) ( ( m o. A ) = F /\ ( m o. V ) = G ) <-> ( E. m e. ( P RingHom S ) ( ( m o. A ) = F /\ ( m o. V ) = G ) /\ E* m e. ( P RingHom S ) ( ( m o. A ) = F /\ ( m o. V ) = G ) ) )
126	25, 124, 125	sylanbrc 698	1 |- ( ph -> E! m e. ( P RingHom S ) ( ( m o. A ) = F /\ ( m o. V ) = G ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   A.wral 2912   E.wrex 2913   E!wreu 2914   E*wrmo 2915   {crab 2916   _Vcvv 3200   u. cun 3572   i^i cin 3573   C_ wss 3574   ~Pcpw 4158   |-> cmpt 4729   `'ccnv 5113   dom cdm 5114   ran crn 5115   |` cres 5116   "cima 5117   o. ccom 5118   Fun wfun 5882   Fn wfn 5883   -->wf 5884   ` cfv 5888  (class class class)co 6650   oFcof 6895   ^m cmap 7857   Fincfn 7955   NNcn 11020   NN0cn0 11292   Basecbs 15857   .rcmulr 15942  Scalarcsca 15944   gsum_g cgsu 16101  Moorecmre 16242  mrClscmrc 16243  ._gcmg 17540  mulGrpcmgp 18489   Ringcrg 18547   CRingccrg 18548   RingHom crh 18712  SubRingcsubrg 18776  AssAlgcasa 19309  AlgSpancasp 19310  algSccascl 19311   mPwSer cmps 19351   mVar cmvr 19352   mPoly cmpl 19353

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-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-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-tset 15960  df-0g 16102  df-gsum 16103  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

This theorem is referenced by:  evlsval2  19520