Metamath Proof Explorer

Theorem evlslem3

Description: Lemma for evlseu . Polynomial evaluation of a scaled monomial. (Contributed by Stefan O'Rear, 8-Mar-2015) (Revised by AV, 11-Apr-2024)

Ref Expression
Hypotheses evlslem3.p 
|- P = ( I mPoly R )
evlslem3.b 
|- B = ( Base ` P )
evlslem3.c 
|- C = ( Base ` S )
evlslem3.k 
|- K = ( Base ` R )
evlslem3.d 
|- D = { h e. ( NN0 ^m I ) | ( `' h " NN ) e. Fin }
evlslem3.t 
|- T = ( mulGrp ` S )
evlslem3.x 
|- .^ = ( .g ` T )
evlslem3.m 
|- .x. = ( .r ` S )
evlslem3.v 
|- V = ( I mVar R )
evlslem3.e 
|- E = ( p e. B |-> ( S gsum ( b e. D |-> ( ( F ` ( p ` b ) ) .x. ( T gsum ( b oF .^ G ) ) ) ) ) )
evlslem3.i 
|- ( ph -> I e. W )
evlslem3.r 
|- ( ph -> R e. CRing )
evlslem3.s 
|- ( ph -> S e. CRing )
evlslem3.f 
|- ( ph -> F e. ( R RingHom S ) )
evlslem3.g 
|- ( ph -> G : I --> C )
evlslem3.z 
|- .0. = ( 0g ` R )
evlslem3.a 
|- ( ph -> A e. D )
evlslem3.q 
|- ( ph -> H e. K )
Assertion evlslem3 
|- ( ph -> ( E ` ( x e. D |-> if ( x = A , H , .0. ) ) ) = ( ( F ` H ) .x. ( T gsum ( A oF .^ G ) ) ) )

Proof

Step Hyp Ref Expression
1 evlslem3.p 
 |-  P = ( I mPoly R )
2 evlslem3.b 
 |-  B = ( Base ` P )
3 evlslem3.c 
 |-  C = ( Base ` S )
4 evlslem3.k 
 |-  K = ( Base ` R )
5 evlslem3.d 
 |-  D = { h e. ( NN0 ^m I ) | ( `' h " NN ) e. Fin }
6 evlslem3.t 
 |-  T = ( mulGrp ` S )
7 evlslem3.x 
 |-  .^ = ( .g ` T )
8 evlslem3.m 
 |-  .x. = ( .r ` S )
9 evlslem3.v 
 |-  V = ( I mVar R )
10 evlslem3.e 
 |-  E = ( p e. B |-> ( S gsum ( b e. D |-> ( ( F ` ( p ` b ) ) .x. ( T gsum ( b oF .^ G ) ) ) ) ) )
11 evlslem3.i 
 |-  ( ph -> I e. W )
12 evlslem3.r 
 |-  ( ph -> R e. CRing )
13 evlslem3.s 
 |-  ( ph -> S e. CRing )
14 evlslem3.f 
 |-  ( ph -> F e. ( R RingHom S ) )
15 evlslem3.g 
 |-  ( ph -> G : I --> C )
16 evlslem3.z 
 |-  .0. = ( 0g ` R )
17 evlslem3.a 
 |-  ( ph -> A e. D )
18 evlslem3.q 
 |-  ( ph -> H e. K )
19 crngring 
 |-  ( R e. CRing -> R e. Ring )
20 12 19 syl 
 |-  ( ph -> R e. Ring )
21 1 5 16 4 11 20 2 18 17 mplmon2cl 
 |-  ( ph -> ( x e. D |-> if ( x = A , H , .0. ) ) e. B )
22 fveq1 
 |-  ( p = ( x e. D |-> if ( x = A , H , .0. ) ) -> ( p ` b ) = ( ( x e. D |-> if ( x = A , H , .0. ) ) ` b ) )
23 22 fveq2d 
 |-  ( p = ( x e. D |-> if ( x = A , H , .0. ) ) -> ( F ` ( p ` b ) ) = ( F ` ( ( x e. D |-> if ( x = A , H , .0. ) ) ` b ) ) )
24 23 oveq1d 
 |-  ( p = ( x e. D |-> if ( x = A , H , .0. ) ) -> ( ( F ` ( p ` b ) ) .x. ( T gsum ( b oF .^ G ) ) ) = ( ( F ` ( ( x e. D |-> if ( x = A , H , .0. ) ) ` b ) ) .x. ( T gsum ( b oF .^ G ) ) ) )
25 24 mpteq2dv 
 |-  ( p = ( x e. D |-> if ( x = A , H , .0. ) ) -> ( b e. D |-> ( ( F ` ( p ` b ) ) .x. ( T gsum ( b oF .^ G ) ) ) ) = ( b e. D |-> ( ( F ` ( ( x e. D |-> if ( x = A , H , .0. ) ) ` b ) ) .x. ( T gsum ( b oF .^ G ) ) ) ) )
26 25 oveq2d 
 |-  ( p = ( x e. D |-> if ( x = A , H , .0. ) ) -> ( S gsum ( b e. D |-> ( ( F ` ( p ` b ) ) .x. ( T gsum ( b oF .^ G ) ) ) ) ) = ( S gsum ( b e. D |-> ( ( F ` ( ( x e. D |-> if ( x = A , H , .0. ) ) ` b ) ) .x. ( T gsum ( b oF .^ G ) ) ) ) ) )
27 ovex 
 |-  ( S gsum ( b e. D |-> ( ( F ` ( ( x e. D |-> if ( x = A , H , .0. ) ) ` b ) ) .x. ( T gsum ( b oF .^ G ) ) ) ) ) e. _V
28 26 10 27 fvmpt 
 |-  ( ( x e. D |-> if ( x = A , H , .0. ) ) e. B -> ( E ` ( x e. D |-> if ( x = A , H , .0. ) ) ) = ( S gsum ( b e. D |-> ( ( F ` ( ( x e. D |-> if ( x = A , H , .0. ) ) ` b ) ) .x. ( T gsum ( b oF .^ G ) ) ) ) ) )
29 21 28 syl 
 |-  ( ph -> ( E ` ( x e. D |-> if ( x = A , H , .0. ) ) ) = ( S gsum ( b e. D |-> ( ( F ` ( ( x e. D |-> if ( x = A , H , .0. ) ) ` b ) ) .x. ( T gsum ( b oF .^ G ) ) ) ) ) )
30 eqid 
 |-  ( x e. D |-> if ( x = A , H , .0. ) ) = ( x e. D |-> if ( x = A , H , .0. ) )
31 eqeq1 
 |-  ( x = b -> ( x = A <-> b = A ) )
32 31 ifbid 
 |-  ( x = b -> if ( x = A , H , .0. ) = if ( b = A , H , .0. ) )
33 simpr 
 |-  ( ( ph /\ b e. D ) -> b e. D )
34 16 fvexi 
 |-  .0. e. _V
35 34 a1i 
 |-  ( ph -> .0. e. _V )
36 18 35 ifexd 
 |-  ( ph -> if ( b = A , H , .0. ) e. _V )
37 36 adantr 
 |-  ( ( ph /\ b e. D ) -> if ( b = A , H , .0. ) e. _V )
38 30 32 33 37 fvmptd3 
 |-  ( ( ph /\ b e. D ) -> ( ( x e. D |-> if ( x = A , H , .0. ) ) ` b ) = if ( b = A , H , .0. ) )
39 38 fveq2d 
 |-  ( ( ph /\ b e. D ) -> ( F ` ( ( x e. D |-> if ( x = A , H , .0. ) ) ` b ) ) = ( F ` if ( b = A , H , .0. ) ) )
40 39 oveq1d 
 |-  ( ( ph /\ b e. D ) -> ( ( F ` ( ( x e. D |-> if ( x = A , H , .0. ) ) ` b ) ) .x. ( T gsum ( b oF .^ G ) ) ) = ( ( F ` if ( b = A , H , .0. ) ) .x. ( T gsum ( b oF .^ G ) ) ) )
41 40 mpteq2dva 
 |-  ( ph -> ( b e. D |-> ( ( F ` ( ( x e. D |-> if ( x = A , H , .0. ) ) ` b ) ) .x. ( T gsum ( b oF .^ G ) ) ) ) = ( b e. D |-> ( ( F ` if ( b = A , H , .0. ) ) .x. ( T gsum ( b oF .^ G ) ) ) ) )
42 41 oveq2d 
 |-  ( ph -> ( S gsum ( b e. D |-> ( ( F ` ( ( x e. D |-> if ( x = A , H , .0. ) ) ` b ) ) .x. ( T gsum ( b oF .^ G ) ) ) ) ) = ( S gsum ( b e. D |-> ( ( F ` if ( b = A , H , .0. ) ) .x. ( T gsum ( b oF .^ G ) ) ) ) ) )
43 eqid 
 |-  ( 0g ` S ) = ( 0g ` S )
44 crngring 
 |-  ( S e. CRing -> S e. Ring )
45 13 44 syl 
 |-  ( ph -> S e. Ring )
46 ringmnd 
 |-  ( S e. Ring -> S e. Mnd )
47 45 46 syl 
 |-  ( ph -> S e. Mnd )
48 ovex 
 |-  ( NN0 ^m I ) e. _V
49 5 48 rabex2 
 |-  D e. _V
50 49 a1i 
 |-  ( ph -> D e. _V )
51 45 adantr 
 |-  ( ( ph /\ b e. D ) -> S e. Ring )
52 4 3 rhmf 
 |-  ( F e. ( R RingHom S ) -> F : K --> C )
53 14 52 syl 
 |-  ( ph -> F : K --> C )
54 4 16 ring0cl 
 |-  ( R e. Ring -> .0. e. K )
55 20 54 syl 
 |-  ( ph -> .0. e. K )
56 18 55 ifcld 
 |-  ( ph -> if ( b = A , H , .0. ) e. K )
57 53 56 ffvelcdmd 
 |-  ( ph -> ( F ` if ( b = A , H , .0. ) ) e. C )
58 57 adantr 
 |-  ( ( ph /\ b e. D ) -> ( F ` if ( b = A , H , .0. ) ) e. C )
59 6 3 mgpbas 
 |-  C = ( Base ` T )
60 eqid 
 |-  ( 0g ` T ) = ( 0g ` T )
61 6 crngmgp 
 |-  ( S e. CRing -> T e. CMnd )
62 13 61 syl 
 |-  ( ph -> T e. CMnd )
63 62 adantr 
 |-  ( ( ph /\ b e. D ) -> T e. CMnd )
64 11 adantr 
 |-  ( ( ph /\ b e. D ) -> I e. W )
65 cmnmnd 
 |-  ( T e. CMnd -> T e. Mnd )
66 62 65 syl 
 |-  ( ph -> T e. Mnd )
67 66 ad2antrr 
 |-  ( ( ( ph /\ b e. D ) /\ ( y e. NN0 /\ z e. C ) ) -> T e. Mnd )
68 simprl 
 |-  ( ( ( ph /\ b e. D ) /\ ( y e. NN0 /\ z e. C ) ) -> y e. NN0 )
69 simprr 
 |-  ( ( ( ph /\ b e. D ) /\ ( y e. NN0 /\ z e. C ) ) -> z e. C )
70 59 7 67 68 69 mulgnn0cld 
 |-  ( ( ( ph /\ b e. D ) /\ ( y e. NN0 /\ z e. C ) ) -> ( y .^ z ) e. C )
71 5 psrbagf 
 |-  ( b e. D -> b : I --> NN0 )
72 71 adantl 
 |-  ( ( ph /\ b e. D ) -> b : I --> NN0 )
73 15 adantr 
 |-  ( ( ph /\ b e. D ) -> G : I --> C )
74 inidm 
 |-  ( I i^i I ) = I
75 70 72 73 64 64 74 off 
 |-  ( ( ph /\ b e. D ) -> ( b oF .^ G ) : I --> C )
76 ovex 
 |-  ( b oF .^ G ) e. _V
77 76 a1i 
 |-  ( ( ph /\ b e. D ) -> ( b oF .^ G ) e. _V )
78 75 ffund 
 |-  ( ( ph /\ b e. D ) -> Fun ( b oF .^ G ) )
79 fvexd 
 |-  ( ( ph /\ b e. D ) -> ( 0g ` T ) e. _V )
80 5 psrbag 
 |-  ( I e. W -> ( b e. D <-> ( b : I --> NN0 /\ ( `' b " NN ) e. Fin ) ) )
81 11 80 syl 
 |-  ( ph -> ( b e. D <-> ( b : I --> NN0 /\ ( `' b " NN ) e. Fin ) ) )
82 81 simplbda 
 |-  ( ( ph /\ b e. D ) -> ( `' b " NN ) e. Fin )
83 72 ffnd 
 |-  ( ( ph /\ b e. D ) -> b Fn I )
84 83 adantr 
 |-  ( ( ( ph /\ b e. D ) /\ y e. ( I \ ( `' b " NN ) ) ) -> b Fn I )
85 15 ffnd 
 |-  ( ph -> G Fn I )
86 85 ad2antrr 
 |-  ( ( ( ph /\ b e. D ) /\ y e. ( I \ ( `' b " NN ) ) ) -> G Fn I )
87 11 ad2antrr 
 |-  ( ( ( ph /\ b e. D ) /\ y e. ( I \ ( `' b " NN ) ) ) -> I e. W )
88 eldifi 
 |-  ( y e. ( I \ ( `' b " NN ) ) -> y e. I )
89 88 adantl 
 |-  ( ( ( ph /\ b e. D ) /\ y e. ( I \ ( `' b " NN ) ) ) -> y e. I )
90 fnfvof 
 |-  ( ( ( b Fn I /\ G Fn I ) /\ ( I e. W /\ y e. I ) ) -> ( ( b oF .^ G ) ` y ) = ( ( b ` y ) .^ ( G ` y ) ) )
91 84 86 87 89 90 syl22anc 
 |-  ( ( ( ph /\ b e. D ) /\ y e. ( I \ ( `' b " NN ) ) ) -> ( ( b oF .^ G ) ` y ) = ( ( b ` y ) .^ ( G ` y ) ) )
92 ffvelcdm 
 |-  ( ( b : I --> NN0 /\ y e. I ) -> ( b ` y ) e. NN0 )
93 72 88 92 syl2an 
 |-  ( ( ( ph /\ b e. D ) /\ y e. ( I \ ( `' b " NN ) ) ) -> ( b ` y ) e. NN0 )
94 elnn0 
 |-  ( ( b ` y ) e. NN0 <-> ( ( b ` y ) e. NN \/ ( b ` y ) = 0 ) )
95 93 94 sylib 
 |-  ( ( ( ph /\ b e. D ) /\ y e. ( I \ ( `' b " NN ) ) ) -> ( ( b ` y ) e. NN \/ ( b ` y ) = 0 ) )
96 eldifn 
 |-  ( y e. ( I \ ( `' b " NN ) ) -> -. y e. ( `' b " NN ) )
97 96 adantl 
 |-  ( ( ( ph /\ b e. D ) /\ y e. ( I \ ( `' b " NN ) ) ) -> -. y e. ( `' b " NN ) )
98 83 ad2antrr 
 |-  ( ( ( ( ph /\ b e. D ) /\ y e. ( I \ ( `' b " NN ) ) ) /\ ( b ` y ) e. NN ) -> b Fn I )
99 88 ad2antlr 
 |-  ( ( ( ( ph /\ b e. D ) /\ y e. ( I \ ( `' b " NN ) ) ) /\ ( b ` y ) e. NN ) -> y e. I )
100 simpr 
 |-  ( ( ( ( ph /\ b e. D ) /\ y e. ( I \ ( `' b " NN ) ) ) /\ ( b ` y ) e. NN ) -> ( b ` y ) e. NN )
101 98 99 100 elpreimad 
 |-  ( ( ( ( ph /\ b e. D ) /\ y e. ( I \ ( `' b " NN ) ) ) /\ ( b ` y ) e. NN ) -> y e. ( `' b " NN ) )
102 97 101 mtand 
 |-  ( ( ( ph /\ b e. D ) /\ y e. ( I \ ( `' b " NN ) ) ) -> -. ( b ` y ) e. NN )
103 95 102 orcnd 
 |-  ( ( ( ph /\ b e. D ) /\ y e. ( I \ ( `' b " NN ) ) ) -> ( b ` y ) = 0 )
104 103 oveq1d 
 |-  ( ( ( ph /\ b e. D ) /\ y e. ( I \ ( `' b " NN ) ) ) -> ( ( b ` y ) .^ ( G ` y ) ) = ( 0 .^ ( G ` y ) ) )
105 ffvelcdm 
 |-  ( ( G : I --> C /\ y e. I ) -> ( G ` y ) e. C )
106 73 88 105 syl2an 
 |-  ( ( ( ph /\ b e. D ) /\ y e. ( I \ ( `' b " NN ) ) ) -> ( G ` y ) e. C )
107 59 60 7 mulg0 
 |-  ( ( G ` y ) e. C -> ( 0 .^ ( G ` y ) ) = ( 0g ` T ) )
108 106 107 syl 
 |-  ( ( ( ph /\ b e. D ) /\ y e. ( I \ ( `' b " NN ) ) ) -> ( 0 .^ ( G ` y ) ) = ( 0g ` T ) )
109 91 104 108 3eqtrd 
 |-  ( ( ( ph /\ b e. D ) /\ y e. ( I \ ( `' b " NN ) ) ) -> ( ( b oF .^ G ) ` y ) = ( 0g ` T ) )
110 75 109 suppss 
 |-  ( ( ph /\ b e. D ) -> ( ( b oF .^ G ) supp ( 0g ` T ) ) C_ ( `' b " NN ) )
111 suppssfifsupp 
 |-  ( ( ( ( b oF .^ G ) e. _V /\ Fun ( b oF .^ G ) /\ ( 0g ` T ) e. _V ) /\ ( ( `' b " NN ) e. Fin /\ ( ( b oF .^ G ) supp ( 0g ` T ) ) C_ ( `' b " NN ) ) ) -> ( b oF .^ G ) finSupp ( 0g ` T ) )
112 77 78 79 82 110 111 syl32anc 
 |-  ( ( ph /\ b e. D ) -> ( b oF .^ G ) finSupp ( 0g ` T ) )
113 59 60 63 64 75 112 gsumcl 
 |-  ( ( ph /\ b e. D ) -> ( T gsum ( b oF .^ G ) ) e. C )
114 3 8 ringcl 
 |-  ( ( S e. Ring /\ ( F ` if ( b = A , H , .0. ) ) e. C /\ ( T gsum ( b oF .^ G ) ) e. C ) -> ( ( F ` if ( b = A , H , .0. ) ) .x. ( T gsum ( b oF .^ G ) ) ) e. C )
115 51 58 113 114 syl3anc 
 |-  ( ( ph /\ b e. D ) -> ( ( F ` if ( b = A , H , .0. ) ) .x. ( T gsum ( b oF .^ G ) ) ) e. C )
116 115 fmpttd 
 |-  ( ph -> ( b e. D |-> ( ( F ` if ( b = A , H , .0. ) ) .x. ( T gsum ( b oF .^ G ) ) ) ) : D --> C )
117 eldifsnneq 
 |-  ( b e. ( D \ { A } ) -> -. b = A )
118 117 iffalsed 
 |-  ( b e. ( D \ { A } ) -> if ( b = A , H , .0. ) = .0. )
119 118 adantl 
 |-  ( ( ph /\ b e. ( D \ { A } ) ) -> if ( b = A , H , .0. ) = .0. )
120 119 fveq2d 
 |-  ( ( ph /\ b e. ( D \ { A } ) ) -> ( F ` if ( b = A , H , .0. ) ) = ( F ` .0. ) )
121 rhmghm 
 |-  ( F e. ( R RingHom S ) -> F e. ( R GrpHom S ) )
122 14 121 syl 
 |-  ( ph -> F e. ( R GrpHom S ) )
123 16 43 ghmid 
 |-  ( F e. ( R GrpHom S ) -> ( F ` .0. ) = ( 0g ` S ) )
124 122 123 syl 
 |-  ( ph -> ( F ` .0. ) = ( 0g ` S ) )
125 124 adantr 
 |-  ( ( ph /\ b e. ( D \ { A } ) ) -> ( F ` .0. ) = ( 0g ` S ) )
126 120 125 eqtrd 
 |-  ( ( ph /\ b e. ( D \ { A } ) ) -> ( F ` if ( b = A , H , .0. ) ) = ( 0g ` S ) )
127 126 oveq1d 
 |-  ( ( ph /\ b e. ( D \ { A } ) ) -> ( ( F ` if ( b = A , H , .0. ) ) .x. ( T gsum ( b oF .^ G ) ) ) = ( ( 0g ` S ) .x. ( T gsum ( b oF .^ G ) ) ) )
128 45 adantr 
 |-  ( ( ph /\ b e. ( D \ { A } ) ) -> S e. Ring )
129 eldifi 
 |-  ( b e. ( D \ { A } ) -> b e. D )
130 129 113 sylan2 
 |-  ( ( ph /\ b e. ( D \ { A } ) ) -> ( T gsum ( b oF .^ G ) ) e. C )
131 3 8 43 ringlz 
 |-  ( ( S e. Ring /\ ( T gsum ( b oF .^ G ) ) e. C ) -> ( ( 0g ` S ) .x. ( T gsum ( b oF .^ G ) ) ) = ( 0g ` S ) )
132 128 130 131 syl2anc 
 |-  ( ( ph /\ b e. ( D \ { A } ) ) -> ( ( 0g ` S ) .x. ( T gsum ( b oF .^ G ) ) ) = ( 0g ` S ) )
133 127 132 eqtrd 
 |-  ( ( ph /\ b e. ( D \ { A } ) ) -> ( ( F ` if ( b = A , H , .0. ) ) .x. ( T gsum ( b oF .^ G ) ) ) = ( 0g ` S ) )
134 133 50 suppss2 
 |-  ( ph -> ( ( b e. D |-> ( ( F ` if ( b = A , H , .0. ) ) .x. ( T gsum ( b oF .^ G ) ) ) ) supp ( 0g ` S ) ) C_ { A } )
135 3 43 47 50 17 116 134 gsumpt 
 |-  ( ph -> ( S gsum ( b e. D |-> ( ( F ` if ( b = A , H , .0. ) ) .x. ( T gsum ( b oF .^ G ) ) ) ) ) = ( ( b e. D |-> ( ( F ` if ( b = A , H , .0. ) ) .x. ( T gsum ( b oF .^ G ) ) ) ) ` A ) )
136 42 135 eqtrd 
 |-  ( ph -> ( S gsum ( b e. D |-> ( ( F ` ( ( x e. D |-> if ( x = A , H , .0. ) ) ` b ) ) .x. ( T gsum ( b oF .^ G ) ) ) ) ) = ( ( b e. D |-> ( ( F ` if ( b = A , H , .0. ) ) .x. ( T gsum ( b oF .^ G ) ) ) ) ` A ) )
137 iftrue 
 |-  ( b = A -> if ( b = A , H , .0. ) = H )
138 137 fveq2d 
 |-  ( b = A -> ( F ` if ( b = A , H , .0. ) ) = ( F ` H ) )
139 oveq1 
 |-  ( b = A -> ( b oF .^ G ) = ( A oF .^ G ) )
140 139 oveq2d 
 |-  ( b = A -> ( T gsum ( b oF .^ G ) ) = ( T gsum ( A oF .^ G ) ) )
141 138 140 oveq12d 
 |-  ( b = A -> ( ( F ` if ( b = A , H , .0. ) ) .x. ( T gsum ( b oF .^ G ) ) ) = ( ( F ` H ) .x. ( T gsum ( A oF .^ G ) ) ) )
142 eqid 
 |-  ( b e. D |-> ( ( F ` if ( b = A , H , .0. ) ) .x. ( T gsum ( b oF .^ G ) ) ) ) = ( b e. D |-> ( ( F ` if ( b = A , H , .0. ) ) .x. ( T gsum ( b oF .^ G ) ) ) )
143 ovex 
 |-  ( ( F ` H ) .x. ( T gsum ( A oF .^ G ) ) ) e. _V
144 141 142 143 fvmpt 
 |-  ( A e. D -> ( ( b e. D |-> ( ( F ` if ( b = A , H , .0. ) ) .x. ( T gsum ( b oF .^ G ) ) ) ) ` A ) = ( ( F ` H ) .x. ( T gsum ( A oF .^ G ) ) ) )
145 17 144 syl 
 |-  ( ph -> ( ( b e. D |-> ( ( F ` if ( b = A , H , .0. ) ) .x. ( T gsum ( b oF .^ G ) ) ) ) ` A ) = ( ( F ` H ) .x. ( T gsum ( A oF .^ G ) ) ) )
146 29 136 145 3eqtrd 
 |-  ( ph -> ( E ` ( x e. D |-> if ( x = A , H , .0. ) ) ) = ( ( F ` H ) .x. ( T gsum ( A oF .^ G ) ) ) )