Step |
Hyp |
Ref |
Expression |
1 |
|
evlslem1.p |
⊢ 𝑃 = ( 𝐼 mPoly 𝑅 ) |
2 |
|
evlslem1.b |
⊢ 𝐵 = ( Base ‘ 𝑃 ) |
3 |
|
evlslem1.c |
⊢ 𝐶 = ( Base ‘ 𝑆 ) |
4 |
|
evlslem1.d |
⊢ 𝐷 = { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } |
5 |
|
evlslem1.t |
⊢ 𝑇 = ( mulGrp ‘ 𝑆 ) |
6 |
|
evlslem1.x |
⊢ ↑ = ( .g ‘ 𝑇 ) |
7 |
|
evlslem1.m |
⊢ · = ( .r ‘ 𝑆 ) |
8 |
|
evlslem1.v |
⊢ 𝑉 = ( 𝐼 mVar 𝑅 ) |
9 |
|
evlslem1.e |
⊢ 𝐸 = ( 𝑝 ∈ 𝐵 ↦ ( 𝑆 Σg ( 𝑏 ∈ 𝐷 ↦ ( ( 𝐹 ‘ ( 𝑝 ‘ 𝑏 ) ) · ( 𝑇 Σg ( 𝑏 ∘f ↑ 𝐺 ) ) ) ) ) ) |
10 |
|
evlslem1.i |
⊢ ( 𝜑 → 𝐼 ∈ 𝑊 ) |
11 |
|
evlslem1.r |
⊢ ( 𝜑 → 𝑅 ∈ CRing ) |
12 |
|
evlslem1.s |
⊢ ( 𝜑 → 𝑆 ∈ CRing ) |
13 |
|
evlslem1.f |
⊢ ( 𝜑 → 𝐹 ∈ ( 𝑅 RingHom 𝑆 ) ) |
14 |
|
evlslem1.g |
⊢ ( 𝜑 → 𝐺 : 𝐼 ⟶ 𝐶 ) |
15 |
|
evlslem1.a |
⊢ 𝐴 = ( algSc ‘ 𝑃 ) |
16 |
|
eqid |
⊢ ( 1r ‘ 𝑃 ) = ( 1r ‘ 𝑃 ) |
17 |
|
eqid |
⊢ ( 1r ‘ 𝑆 ) = ( 1r ‘ 𝑆 ) |
18 |
|
eqid |
⊢ ( .r ‘ 𝑃 ) = ( .r ‘ 𝑃 ) |
19 |
11
|
crngringd |
⊢ ( 𝜑 → 𝑅 ∈ Ring ) |
20 |
1 10 19
|
mplringd |
⊢ ( 𝜑 → 𝑃 ∈ Ring ) |
21 |
12
|
crngringd |
⊢ ( 𝜑 → 𝑆 ∈ Ring ) |
22 |
|
2fveq3 |
⊢ ( 𝑥 = ( 1r ‘ 𝑅 ) → ( 𝐸 ‘ ( 𝐴 ‘ 𝑥 ) ) = ( 𝐸 ‘ ( 𝐴 ‘ ( 1r ‘ 𝑅 ) ) ) ) |
23 |
|
fveq2 |
⊢ ( 𝑥 = ( 1r ‘ 𝑅 ) → ( 𝐹 ‘ 𝑥 ) = ( 𝐹 ‘ ( 1r ‘ 𝑅 ) ) ) |
24 |
22 23
|
eqeq12d |
⊢ ( 𝑥 = ( 1r ‘ 𝑅 ) → ( ( 𝐸 ‘ ( 𝐴 ‘ 𝑥 ) ) = ( 𝐹 ‘ 𝑥 ) ↔ ( 𝐸 ‘ ( 𝐴 ‘ ( 1r ‘ 𝑅 ) ) ) = ( 𝐹 ‘ ( 1r ‘ 𝑅 ) ) ) ) |
25 |
|
eqid |
⊢ ( 0g ‘ 𝑅 ) = ( 0g ‘ 𝑅 ) |
26 |
|
eqid |
⊢ ( Base ‘ 𝑅 ) = ( Base ‘ 𝑅 ) |
27 |
10
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝑅 ) ) → 𝐼 ∈ 𝑊 ) |
28 |
19
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝑅 ) ) → 𝑅 ∈ Ring ) |
29 |
|
simpr |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝑅 ) ) → 𝑥 ∈ ( Base ‘ 𝑅 ) ) |
30 |
1 4 25 26 15 27 28 29
|
mplascl |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝑅 ) ) → ( 𝐴 ‘ 𝑥 ) = ( 𝑦 ∈ 𝐷 ↦ if ( 𝑦 = ( 𝐼 × { 0 } ) , 𝑥 , ( 0g ‘ 𝑅 ) ) ) ) |
31 |
30
|
fveq2d |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝑅 ) ) → ( 𝐸 ‘ ( 𝐴 ‘ 𝑥 ) ) = ( 𝐸 ‘ ( 𝑦 ∈ 𝐷 ↦ if ( 𝑦 = ( 𝐼 × { 0 } ) , 𝑥 , ( 0g ‘ 𝑅 ) ) ) ) ) |
32 |
11
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝑅 ) ) → 𝑅 ∈ CRing ) |
33 |
12
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝑅 ) ) → 𝑆 ∈ CRing ) |
34 |
13
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝑅 ) ) → 𝐹 ∈ ( 𝑅 RingHom 𝑆 ) ) |
35 |
14
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝑅 ) ) → 𝐺 : 𝐼 ⟶ 𝐶 ) |
36 |
4
|
psrbag0 |
⊢ ( 𝐼 ∈ 𝑊 → ( 𝐼 × { 0 } ) ∈ 𝐷 ) |
37 |
10 36
|
syl |
⊢ ( 𝜑 → ( 𝐼 × { 0 } ) ∈ 𝐷 ) |
38 |
37
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝑅 ) ) → ( 𝐼 × { 0 } ) ∈ 𝐷 ) |
39 |
1 2 3 26 4 5 6 7 8 9 27 32 33 34 35 25 38 29
|
evlslem3 |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝑅 ) ) → ( 𝐸 ‘ ( 𝑦 ∈ 𝐷 ↦ if ( 𝑦 = ( 𝐼 × { 0 } ) , 𝑥 , ( 0g ‘ 𝑅 ) ) ) ) = ( ( 𝐹 ‘ 𝑥 ) · ( 𝑇 Σg ( ( 𝐼 × { 0 } ) ∘f ↑ 𝐺 ) ) ) ) |
40 |
|
0zd |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐼 ) → 0 ∈ ℤ ) |
41 |
|
fvexd |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐼 ) → ( 𝐺 ‘ 𝑥 ) ∈ V ) |
42 |
|
fconstmpt |
⊢ ( 𝐼 × { 0 } ) = ( 𝑥 ∈ 𝐼 ↦ 0 ) |
43 |
42
|
a1i |
⊢ ( 𝜑 → ( 𝐼 × { 0 } ) = ( 𝑥 ∈ 𝐼 ↦ 0 ) ) |
44 |
14
|
feqmptd |
⊢ ( 𝜑 → 𝐺 = ( 𝑥 ∈ 𝐼 ↦ ( 𝐺 ‘ 𝑥 ) ) ) |
45 |
10 40 41 43 44
|
offval2 |
⊢ ( 𝜑 → ( ( 𝐼 × { 0 } ) ∘f ↑ 𝐺 ) = ( 𝑥 ∈ 𝐼 ↦ ( 0 ↑ ( 𝐺 ‘ 𝑥 ) ) ) ) |
46 |
14
|
ffvelcdmda |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐼 ) → ( 𝐺 ‘ 𝑥 ) ∈ 𝐶 ) |
47 |
5 3
|
mgpbas |
⊢ 𝐶 = ( Base ‘ 𝑇 ) |
48 |
5 17
|
ringidval |
⊢ ( 1r ‘ 𝑆 ) = ( 0g ‘ 𝑇 ) |
49 |
47 48 6
|
mulg0 |
⊢ ( ( 𝐺 ‘ 𝑥 ) ∈ 𝐶 → ( 0 ↑ ( 𝐺 ‘ 𝑥 ) ) = ( 1r ‘ 𝑆 ) ) |
50 |
46 49
|
syl |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐼 ) → ( 0 ↑ ( 𝐺 ‘ 𝑥 ) ) = ( 1r ‘ 𝑆 ) ) |
51 |
50
|
mpteq2dva |
⊢ ( 𝜑 → ( 𝑥 ∈ 𝐼 ↦ ( 0 ↑ ( 𝐺 ‘ 𝑥 ) ) ) = ( 𝑥 ∈ 𝐼 ↦ ( 1r ‘ 𝑆 ) ) ) |
52 |
45 51
|
eqtrd |
⊢ ( 𝜑 → ( ( 𝐼 × { 0 } ) ∘f ↑ 𝐺 ) = ( 𝑥 ∈ 𝐼 ↦ ( 1r ‘ 𝑆 ) ) ) |
53 |
52
|
oveq2d |
⊢ ( 𝜑 → ( 𝑇 Σg ( ( 𝐼 × { 0 } ) ∘f ↑ 𝐺 ) ) = ( 𝑇 Σg ( 𝑥 ∈ 𝐼 ↦ ( 1r ‘ 𝑆 ) ) ) ) |
54 |
5
|
crngmgp |
⊢ ( 𝑆 ∈ CRing → 𝑇 ∈ CMnd ) |
55 |
12 54
|
syl |
⊢ ( 𝜑 → 𝑇 ∈ CMnd ) |
56 |
55
|
cmnmndd |
⊢ ( 𝜑 → 𝑇 ∈ Mnd ) |
57 |
48
|
gsumz |
⊢ ( ( 𝑇 ∈ Mnd ∧ 𝐼 ∈ 𝑊 ) → ( 𝑇 Σg ( 𝑥 ∈ 𝐼 ↦ ( 1r ‘ 𝑆 ) ) ) = ( 1r ‘ 𝑆 ) ) |
58 |
56 10 57
|
syl2anc |
⊢ ( 𝜑 → ( 𝑇 Σg ( 𝑥 ∈ 𝐼 ↦ ( 1r ‘ 𝑆 ) ) ) = ( 1r ‘ 𝑆 ) ) |
59 |
53 58
|
eqtrd |
⊢ ( 𝜑 → ( 𝑇 Σg ( ( 𝐼 × { 0 } ) ∘f ↑ 𝐺 ) ) = ( 1r ‘ 𝑆 ) ) |
60 |
59
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝑅 ) ) → ( 𝑇 Σg ( ( 𝐼 × { 0 } ) ∘f ↑ 𝐺 ) ) = ( 1r ‘ 𝑆 ) ) |
61 |
60
|
oveq2d |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝑅 ) ) → ( ( 𝐹 ‘ 𝑥 ) · ( 𝑇 Σg ( ( 𝐼 × { 0 } ) ∘f ↑ 𝐺 ) ) ) = ( ( 𝐹 ‘ 𝑥 ) · ( 1r ‘ 𝑆 ) ) ) |
62 |
26 3
|
rhmf |
⊢ ( 𝐹 ∈ ( 𝑅 RingHom 𝑆 ) → 𝐹 : ( Base ‘ 𝑅 ) ⟶ 𝐶 ) |
63 |
13 62
|
syl |
⊢ ( 𝜑 → 𝐹 : ( Base ‘ 𝑅 ) ⟶ 𝐶 ) |
64 |
63
|
ffvelcdmda |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝑅 ) ) → ( 𝐹 ‘ 𝑥 ) ∈ 𝐶 ) |
65 |
3 7 17
|
ringridm |
⊢ ( ( 𝑆 ∈ Ring ∧ ( 𝐹 ‘ 𝑥 ) ∈ 𝐶 ) → ( ( 𝐹 ‘ 𝑥 ) · ( 1r ‘ 𝑆 ) ) = ( 𝐹 ‘ 𝑥 ) ) |
66 |
21 64 65
|
syl2an2r |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝑅 ) ) → ( ( 𝐹 ‘ 𝑥 ) · ( 1r ‘ 𝑆 ) ) = ( 𝐹 ‘ 𝑥 ) ) |
67 |
61 66
|
eqtrd |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝑅 ) ) → ( ( 𝐹 ‘ 𝑥 ) · ( 𝑇 Σg ( ( 𝐼 × { 0 } ) ∘f ↑ 𝐺 ) ) ) = ( 𝐹 ‘ 𝑥 ) ) |
68 |
31 39 67
|
3eqtrd |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝑅 ) ) → ( 𝐸 ‘ ( 𝐴 ‘ 𝑥 ) ) = ( 𝐹 ‘ 𝑥 ) ) |
69 |
68
|
ralrimiva |
⊢ ( 𝜑 → ∀ 𝑥 ∈ ( Base ‘ 𝑅 ) ( 𝐸 ‘ ( 𝐴 ‘ 𝑥 ) ) = ( 𝐹 ‘ 𝑥 ) ) |
70 |
|
eqid |
⊢ ( 1r ‘ 𝑅 ) = ( 1r ‘ 𝑅 ) |
71 |
26 70
|
ringidcl |
⊢ ( 𝑅 ∈ Ring → ( 1r ‘ 𝑅 ) ∈ ( Base ‘ 𝑅 ) ) |
72 |
19 71
|
syl |
⊢ ( 𝜑 → ( 1r ‘ 𝑅 ) ∈ ( Base ‘ 𝑅 ) ) |
73 |
24 69 72
|
rspcdva |
⊢ ( 𝜑 → ( 𝐸 ‘ ( 𝐴 ‘ ( 1r ‘ 𝑅 ) ) ) = ( 𝐹 ‘ ( 1r ‘ 𝑅 ) ) ) |
74 |
1
|
mplassa |
⊢ ( ( 𝐼 ∈ 𝑊 ∧ 𝑅 ∈ CRing ) → 𝑃 ∈ AssAlg ) |
75 |
10 11 74
|
syl2anc |
⊢ ( 𝜑 → 𝑃 ∈ AssAlg ) |
76 |
|
eqid |
⊢ ( Scalar ‘ 𝑃 ) = ( Scalar ‘ 𝑃 ) |
77 |
15 76
|
asclrhm |
⊢ ( 𝑃 ∈ AssAlg → 𝐴 ∈ ( ( Scalar ‘ 𝑃 ) RingHom 𝑃 ) ) |
78 |
75 77
|
syl |
⊢ ( 𝜑 → 𝐴 ∈ ( ( Scalar ‘ 𝑃 ) RingHom 𝑃 ) ) |
79 |
1 10 11
|
mplsca |
⊢ ( 𝜑 → 𝑅 = ( Scalar ‘ 𝑃 ) ) |
80 |
79
|
oveq1d |
⊢ ( 𝜑 → ( 𝑅 RingHom 𝑃 ) = ( ( Scalar ‘ 𝑃 ) RingHom 𝑃 ) ) |
81 |
78 80
|
eleqtrrd |
⊢ ( 𝜑 → 𝐴 ∈ ( 𝑅 RingHom 𝑃 ) ) |
82 |
70 16
|
rhm1 |
⊢ ( 𝐴 ∈ ( 𝑅 RingHom 𝑃 ) → ( 𝐴 ‘ ( 1r ‘ 𝑅 ) ) = ( 1r ‘ 𝑃 ) ) |
83 |
81 82
|
syl |
⊢ ( 𝜑 → ( 𝐴 ‘ ( 1r ‘ 𝑅 ) ) = ( 1r ‘ 𝑃 ) ) |
84 |
83
|
fveq2d |
⊢ ( 𝜑 → ( 𝐸 ‘ ( 𝐴 ‘ ( 1r ‘ 𝑅 ) ) ) = ( 𝐸 ‘ ( 1r ‘ 𝑃 ) ) ) |
85 |
70 17
|
rhm1 |
⊢ ( 𝐹 ∈ ( 𝑅 RingHom 𝑆 ) → ( 𝐹 ‘ ( 1r ‘ 𝑅 ) ) = ( 1r ‘ 𝑆 ) ) |
86 |
13 85
|
syl |
⊢ ( 𝜑 → ( 𝐹 ‘ ( 1r ‘ 𝑅 ) ) = ( 1r ‘ 𝑆 ) ) |
87 |
73 84 86
|
3eqtr3d |
⊢ ( 𝜑 → ( 𝐸 ‘ ( 1r ‘ 𝑃 ) ) = ( 1r ‘ 𝑆 ) ) |
88 |
|
eqid |
⊢ ( +g ‘ 𝑃 ) = ( +g ‘ 𝑃 ) |
89 |
|
eqid |
⊢ ( +g ‘ 𝑆 ) = ( +g ‘ 𝑆 ) |
90 |
20
|
ringgrpd |
⊢ ( 𝜑 → 𝑃 ∈ Grp ) |
91 |
21
|
ringgrpd |
⊢ ( 𝜑 → 𝑆 ∈ Grp ) |
92 |
|
eqid |
⊢ ( 0g ‘ 𝑆 ) = ( 0g ‘ 𝑆 ) |
93 |
|
ringcmn |
⊢ ( 𝑆 ∈ Ring → 𝑆 ∈ CMnd ) |
94 |
21 93
|
syl |
⊢ ( 𝜑 → 𝑆 ∈ CMnd ) |
95 |
94
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑝 ∈ 𝐵 ) → 𝑆 ∈ CMnd ) |
96 |
|
ovex |
⊢ ( ℕ0 ↑m 𝐼 ) ∈ V |
97 |
4 96
|
rabex2 |
⊢ 𝐷 ∈ V |
98 |
97
|
a1i |
⊢ ( ( 𝜑 ∧ 𝑝 ∈ 𝐵 ) → 𝐷 ∈ V ) |
99 |
10
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑝 ∈ 𝐵 ) → 𝐼 ∈ 𝑊 ) |
100 |
11
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑝 ∈ 𝐵 ) → 𝑅 ∈ CRing ) |
101 |
12
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑝 ∈ 𝐵 ) → 𝑆 ∈ CRing ) |
102 |
13
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑝 ∈ 𝐵 ) → 𝐹 ∈ ( 𝑅 RingHom 𝑆 ) ) |
103 |
14
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑝 ∈ 𝐵 ) → 𝐺 : 𝐼 ⟶ 𝐶 ) |
104 |
|
simpr |
⊢ ( ( 𝜑 ∧ 𝑝 ∈ 𝐵 ) → 𝑝 ∈ 𝐵 ) |
105 |
1 2 3 4 5 6 7 8 9 99 100 101 102 103 104
|
evlslem6 |
⊢ ( ( 𝜑 ∧ 𝑝 ∈ 𝐵 ) → ( ( 𝑏 ∈ 𝐷 ↦ ( ( 𝐹 ‘ ( 𝑝 ‘ 𝑏 ) ) · ( 𝑇 Σg ( 𝑏 ∘f ↑ 𝐺 ) ) ) ) : 𝐷 ⟶ 𝐶 ∧ ( 𝑏 ∈ 𝐷 ↦ ( ( 𝐹 ‘ ( 𝑝 ‘ 𝑏 ) ) · ( 𝑇 Σg ( 𝑏 ∘f ↑ 𝐺 ) ) ) ) finSupp ( 0g ‘ 𝑆 ) ) ) |
106 |
105
|
simpld |
⊢ ( ( 𝜑 ∧ 𝑝 ∈ 𝐵 ) → ( 𝑏 ∈ 𝐷 ↦ ( ( 𝐹 ‘ ( 𝑝 ‘ 𝑏 ) ) · ( 𝑇 Σg ( 𝑏 ∘f ↑ 𝐺 ) ) ) ) : 𝐷 ⟶ 𝐶 ) |
107 |
105
|
simprd |
⊢ ( ( 𝜑 ∧ 𝑝 ∈ 𝐵 ) → ( 𝑏 ∈ 𝐷 ↦ ( ( 𝐹 ‘ ( 𝑝 ‘ 𝑏 ) ) · ( 𝑇 Σg ( 𝑏 ∘f ↑ 𝐺 ) ) ) ) finSupp ( 0g ‘ 𝑆 ) ) |
108 |
3 92 95 98 106 107
|
gsumcl |
⊢ ( ( 𝜑 ∧ 𝑝 ∈ 𝐵 ) → ( 𝑆 Σg ( 𝑏 ∈ 𝐷 ↦ ( ( 𝐹 ‘ ( 𝑝 ‘ 𝑏 ) ) · ( 𝑇 Σg ( 𝑏 ∘f ↑ 𝐺 ) ) ) ) ) ∈ 𝐶 ) |
109 |
108 9
|
fmptd |
⊢ ( 𝜑 → 𝐸 : 𝐵 ⟶ 𝐶 ) |
110 |
|
eqid |
⊢ ( +g ‘ 𝑅 ) = ( +g ‘ 𝑅 ) |
111 |
|
simplrl |
⊢ ( ( ( 𝜑 ∧ ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) ) ∧ 𝑏 ∈ 𝐷 ) → 𝑥 ∈ 𝐵 ) |
112 |
|
simplrr |
⊢ ( ( ( 𝜑 ∧ ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) ) ∧ 𝑏 ∈ 𝐷 ) → 𝑦 ∈ 𝐵 ) |
113 |
1 2 110 88 111 112
|
mpladd |
⊢ ( ( ( 𝜑 ∧ ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) ) ∧ 𝑏 ∈ 𝐷 ) → ( 𝑥 ( +g ‘ 𝑃 ) 𝑦 ) = ( 𝑥 ∘f ( +g ‘ 𝑅 ) 𝑦 ) ) |
114 |
113
|
fveq1d |
⊢ ( ( ( 𝜑 ∧ ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) ) ∧ 𝑏 ∈ 𝐷 ) → ( ( 𝑥 ( +g ‘ 𝑃 ) 𝑦 ) ‘ 𝑏 ) = ( ( 𝑥 ∘f ( +g ‘ 𝑅 ) 𝑦 ) ‘ 𝑏 ) ) |
115 |
|
simprl |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) ) → 𝑥 ∈ 𝐵 ) |
116 |
1 26 2 4 115
|
mplelf |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) ) → 𝑥 : 𝐷 ⟶ ( Base ‘ 𝑅 ) ) |
117 |
116
|
ffnd |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) ) → 𝑥 Fn 𝐷 ) |
118 |
117
|
adantr |
⊢ ( ( ( 𝜑 ∧ ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) ) ∧ 𝑏 ∈ 𝐷 ) → 𝑥 Fn 𝐷 ) |
119 |
|
simprr |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) ) → 𝑦 ∈ 𝐵 ) |
120 |
1 26 2 4 119
|
mplelf |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) ) → 𝑦 : 𝐷 ⟶ ( Base ‘ 𝑅 ) ) |
121 |
120
|
ffnd |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) ) → 𝑦 Fn 𝐷 ) |
122 |
121
|
adantr |
⊢ ( ( ( 𝜑 ∧ ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) ) ∧ 𝑏 ∈ 𝐷 ) → 𝑦 Fn 𝐷 ) |
123 |
97
|
a1i |
⊢ ( ( ( 𝜑 ∧ ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) ) ∧ 𝑏 ∈ 𝐷 ) → 𝐷 ∈ V ) |
124 |
|
simpr |
⊢ ( ( ( 𝜑 ∧ ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) ) ∧ 𝑏 ∈ 𝐷 ) → 𝑏 ∈ 𝐷 ) |
125 |
|
fnfvof |
⊢ ( ( ( 𝑥 Fn 𝐷 ∧ 𝑦 Fn 𝐷 ) ∧ ( 𝐷 ∈ V ∧ 𝑏 ∈ 𝐷 ) ) → ( ( 𝑥 ∘f ( +g ‘ 𝑅 ) 𝑦 ) ‘ 𝑏 ) = ( ( 𝑥 ‘ 𝑏 ) ( +g ‘ 𝑅 ) ( 𝑦 ‘ 𝑏 ) ) ) |
126 |
118 122 123 124 125
|
syl22anc |
⊢ ( ( ( 𝜑 ∧ ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) ) ∧ 𝑏 ∈ 𝐷 ) → ( ( 𝑥 ∘f ( +g ‘ 𝑅 ) 𝑦 ) ‘ 𝑏 ) = ( ( 𝑥 ‘ 𝑏 ) ( +g ‘ 𝑅 ) ( 𝑦 ‘ 𝑏 ) ) ) |
127 |
114 126
|
eqtrd |
⊢ ( ( ( 𝜑 ∧ ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) ) ∧ 𝑏 ∈ 𝐷 ) → ( ( 𝑥 ( +g ‘ 𝑃 ) 𝑦 ) ‘ 𝑏 ) = ( ( 𝑥 ‘ 𝑏 ) ( +g ‘ 𝑅 ) ( 𝑦 ‘ 𝑏 ) ) ) |
128 |
127
|
fveq2d |
⊢ ( ( ( 𝜑 ∧ ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) ) ∧ 𝑏 ∈ 𝐷 ) → ( 𝐹 ‘ ( ( 𝑥 ( +g ‘ 𝑃 ) 𝑦 ) ‘ 𝑏 ) ) = ( 𝐹 ‘ ( ( 𝑥 ‘ 𝑏 ) ( +g ‘ 𝑅 ) ( 𝑦 ‘ 𝑏 ) ) ) ) |
129 |
|
rhmghm |
⊢ ( 𝐹 ∈ ( 𝑅 RingHom 𝑆 ) → 𝐹 ∈ ( 𝑅 GrpHom 𝑆 ) ) |
130 |
13 129
|
syl |
⊢ ( 𝜑 → 𝐹 ∈ ( 𝑅 GrpHom 𝑆 ) ) |
131 |
130
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) ) ∧ 𝑏 ∈ 𝐷 ) → 𝐹 ∈ ( 𝑅 GrpHom 𝑆 ) ) |
132 |
116
|
ffvelcdmda |
⊢ ( ( ( 𝜑 ∧ ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) ) ∧ 𝑏 ∈ 𝐷 ) → ( 𝑥 ‘ 𝑏 ) ∈ ( Base ‘ 𝑅 ) ) |
133 |
120
|
ffvelcdmda |
⊢ ( ( ( 𝜑 ∧ ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) ) ∧ 𝑏 ∈ 𝐷 ) → ( 𝑦 ‘ 𝑏 ) ∈ ( Base ‘ 𝑅 ) ) |
134 |
26 110 89
|
ghmlin |
⊢ ( ( 𝐹 ∈ ( 𝑅 GrpHom 𝑆 ) ∧ ( 𝑥 ‘ 𝑏 ) ∈ ( Base ‘ 𝑅 ) ∧ ( 𝑦 ‘ 𝑏 ) ∈ ( Base ‘ 𝑅 ) ) → ( 𝐹 ‘ ( ( 𝑥 ‘ 𝑏 ) ( +g ‘ 𝑅 ) ( 𝑦 ‘ 𝑏 ) ) ) = ( ( 𝐹 ‘ ( 𝑥 ‘ 𝑏 ) ) ( +g ‘ 𝑆 ) ( 𝐹 ‘ ( 𝑦 ‘ 𝑏 ) ) ) ) |
135 |
131 132 133 134
|
syl3anc |
⊢ ( ( ( 𝜑 ∧ ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) ) ∧ 𝑏 ∈ 𝐷 ) → ( 𝐹 ‘ ( ( 𝑥 ‘ 𝑏 ) ( +g ‘ 𝑅 ) ( 𝑦 ‘ 𝑏 ) ) ) = ( ( 𝐹 ‘ ( 𝑥 ‘ 𝑏 ) ) ( +g ‘ 𝑆 ) ( 𝐹 ‘ ( 𝑦 ‘ 𝑏 ) ) ) ) |
136 |
128 135
|
eqtrd |
⊢ ( ( ( 𝜑 ∧ ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) ) ∧ 𝑏 ∈ 𝐷 ) → ( 𝐹 ‘ ( ( 𝑥 ( +g ‘ 𝑃 ) 𝑦 ) ‘ 𝑏 ) ) = ( ( 𝐹 ‘ ( 𝑥 ‘ 𝑏 ) ) ( +g ‘ 𝑆 ) ( 𝐹 ‘ ( 𝑦 ‘ 𝑏 ) ) ) ) |
137 |
136
|
oveq1d |
⊢ ( ( ( 𝜑 ∧ ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) ) ∧ 𝑏 ∈ 𝐷 ) → ( ( 𝐹 ‘ ( ( 𝑥 ( +g ‘ 𝑃 ) 𝑦 ) ‘ 𝑏 ) ) · ( 𝑇 Σg ( 𝑏 ∘f ↑ 𝐺 ) ) ) = ( ( ( 𝐹 ‘ ( 𝑥 ‘ 𝑏 ) ) ( +g ‘ 𝑆 ) ( 𝐹 ‘ ( 𝑦 ‘ 𝑏 ) ) ) · ( 𝑇 Σg ( 𝑏 ∘f ↑ 𝐺 ) ) ) ) |
138 |
21
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) ) ∧ 𝑏 ∈ 𝐷 ) → 𝑆 ∈ Ring ) |
139 |
63
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) ) ∧ 𝑏 ∈ 𝐷 ) → 𝐹 : ( Base ‘ 𝑅 ) ⟶ 𝐶 ) |
140 |
139 132
|
ffvelcdmd |
⊢ ( ( ( 𝜑 ∧ ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) ) ∧ 𝑏 ∈ 𝐷 ) → ( 𝐹 ‘ ( 𝑥 ‘ 𝑏 ) ) ∈ 𝐶 ) |
141 |
139 133
|
ffvelcdmd |
⊢ ( ( ( 𝜑 ∧ ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) ) ∧ 𝑏 ∈ 𝐷 ) → ( 𝐹 ‘ ( 𝑦 ‘ 𝑏 ) ) ∈ 𝐶 ) |
142 |
55
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) ) ∧ 𝑏 ∈ 𝐷 ) → 𝑇 ∈ CMnd ) |
143 |
14
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) ) ∧ 𝑏 ∈ 𝐷 ) → 𝐺 : 𝐼 ⟶ 𝐶 ) |
144 |
4 47 6 142 124 143
|
psrbagev2 |
⊢ ( ( ( 𝜑 ∧ ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) ) ∧ 𝑏 ∈ 𝐷 ) → ( 𝑇 Σg ( 𝑏 ∘f ↑ 𝐺 ) ) ∈ 𝐶 ) |
145 |
3 89 7
|
ringdir |
⊢ ( ( 𝑆 ∈ Ring ∧ ( ( 𝐹 ‘ ( 𝑥 ‘ 𝑏 ) ) ∈ 𝐶 ∧ ( 𝐹 ‘ ( 𝑦 ‘ 𝑏 ) ) ∈ 𝐶 ∧ ( 𝑇 Σg ( 𝑏 ∘f ↑ 𝐺 ) ) ∈ 𝐶 ) ) → ( ( ( 𝐹 ‘ ( 𝑥 ‘ 𝑏 ) ) ( +g ‘ 𝑆 ) ( 𝐹 ‘ ( 𝑦 ‘ 𝑏 ) ) ) · ( 𝑇 Σg ( 𝑏 ∘f ↑ 𝐺 ) ) ) = ( ( ( 𝐹 ‘ ( 𝑥 ‘ 𝑏 ) ) · ( 𝑇 Σg ( 𝑏 ∘f ↑ 𝐺 ) ) ) ( +g ‘ 𝑆 ) ( ( 𝐹 ‘ ( 𝑦 ‘ 𝑏 ) ) · ( 𝑇 Σg ( 𝑏 ∘f ↑ 𝐺 ) ) ) ) ) |
146 |
138 140 141 144 145
|
syl13anc |
⊢ ( ( ( 𝜑 ∧ ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) ) ∧ 𝑏 ∈ 𝐷 ) → ( ( ( 𝐹 ‘ ( 𝑥 ‘ 𝑏 ) ) ( +g ‘ 𝑆 ) ( 𝐹 ‘ ( 𝑦 ‘ 𝑏 ) ) ) · ( 𝑇 Σg ( 𝑏 ∘f ↑ 𝐺 ) ) ) = ( ( ( 𝐹 ‘ ( 𝑥 ‘ 𝑏 ) ) · ( 𝑇 Σg ( 𝑏 ∘f ↑ 𝐺 ) ) ) ( +g ‘ 𝑆 ) ( ( 𝐹 ‘ ( 𝑦 ‘ 𝑏 ) ) · ( 𝑇 Σg ( 𝑏 ∘f ↑ 𝐺 ) ) ) ) ) |
147 |
137 146
|
eqtrd |
⊢ ( ( ( 𝜑 ∧ ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) ) ∧ 𝑏 ∈ 𝐷 ) → ( ( 𝐹 ‘ ( ( 𝑥 ( +g ‘ 𝑃 ) 𝑦 ) ‘ 𝑏 ) ) · ( 𝑇 Σg ( 𝑏 ∘f ↑ 𝐺 ) ) ) = ( ( ( 𝐹 ‘ ( 𝑥 ‘ 𝑏 ) ) · ( 𝑇 Σg ( 𝑏 ∘f ↑ 𝐺 ) ) ) ( +g ‘ 𝑆 ) ( ( 𝐹 ‘ ( 𝑦 ‘ 𝑏 ) ) · ( 𝑇 Σg ( 𝑏 ∘f ↑ 𝐺 ) ) ) ) ) |
148 |
147
|
mpteq2dva |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) ) → ( 𝑏 ∈ 𝐷 ↦ ( ( 𝐹 ‘ ( ( 𝑥 ( +g ‘ 𝑃 ) 𝑦 ) ‘ 𝑏 ) ) · ( 𝑇 Σg ( 𝑏 ∘f ↑ 𝐺 ) ) ) ) = ( 𝑏 ∈ 𝐷 ↦ ( ( ( 𝐹 ‘ ( 𝑥 ‘ 𝑏 ) ) · ( 𝑇 Σg ( 𝑏 ∘f ↑ 𝐺 ) ) ) ( +g ‘ 𝑆 ) ( ( 𝐹 ‘ ( 𝑦 ‘ 𝑏 ) ) · ( 𝑇 Σg ( 𝑏 ∘f ↑ 𝐺 ) ) ) ) ) ) |
149 |
97
|
a1i |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) ) → 𝐷 ∈ V ) |
150 |
|
ovexd |
⊢ ( ( ( 𝜑 ∧ ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) ) ∧ 𝑏 ∈ 𝐷 ) → ( ( 𝐹 ‘ ( 𝑥 ‘ 𝑏 ) ) · ( 𝑇 Σg ( 𝑏 ∘f ↑ 𝐺 ) ) ) ∈ V ) |
151 |
|
ovexd |
⊢ ( ( ( 𝜑 ∧ ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) ) ∧ 𝑏 ∈ 𝐷 ) → ( ( 𝐹 ‘ ( 𝑦 ‘ 𝑏 ) ) · ( 𝑇 Σg ( 𝑏 ∘f ↑ 𝐺 ) ) ) ∈ V ) |
152 |
|
eqidd |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) ) → ( 𝑏 ∈ 𝐷 ↦ ( ( 𝐹 ‘ ( 𝑥 ‘ 𝑏 ) ) · ( 𝑇 Σg ( 𝑏 ∘f ↑ 𝐺 ) ) ) ) = ( 𝑏 ∈ 𝐷 ↦ ( ( 𝐹 ‘ ( 𝑥 ‘ 𝑏 ) ) · ( 𝑇 Σg ( 𝑏 ∘f ↑ 𝐺 ) ) ) ) ) |
153 |
|
eqidd |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) ) → ( 𝑏 ∈ 𝐷 ↦ ( ( 𝐹 ‘ ( 𝑦 ‘ 𝑏 ) ) · ( 𝑇 Σg ( 𝑏 ∘f ↑ 𝐺 ) ) ) ) = ( 𝑏 ∈ 𝐷 ↦ ( ( 𝐹 ‘ ( 𝑦 ‘ 𝑏 ) ) · ( 𝑇 Σg ( 𝑏 ∘f ↑ 𝐺 ) ) ) ) ) |
154 |
149 150 151 152 153
|
offval2 |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) ) → ( ( 𝑏 ∈ 𝐷 ↦ ( ( 𝐹 ‘ ( 𝑥 ‘ 𝑏 ) ) · ( 𝑇 Σg ( 𝑏 ∘f ↑ 𝐺 ) ) ) ) ∘f ( +g ‘ 𝑆 ) ( 𝑏 ∈ 𝐷 ↦ ( ( 𝐹 ‘ ( 𝑦 ‘ 𝑏 ) ) · ( 𝑇 Σg ( 𝑏 ∘f ↑ 𝐺 ) ) ) ) ) = ( 𝑏 ∈ 𝐷 ↦ ( ( ( 𝐹 ‘ ( 𝑥 ‘ 𝑏 ) ) · ( 𝑇 Σg ( 𝑏 ∘f ↑ 𝐺 ) ) ) ( +g ‘ 𝑆 ) ( ( 𝐹 ‘ ( 𝑦 ‘ 𝑏 ) ) · ( 𝑇 Σg ( 𝑏 ∘f ↑ 𝐺 ) ) ) ) ) ) |
155 |
148 154
|
eqtr4d |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) ) → ( 𝑏 ∈ 𝐷 ↦ ( ( 𝐹 ‘ ( ( 𝑥 ( +g ‘ 𝑃 ) 𝑦 ) ‘ 𝑏 ) ) · ( 𝑇 Σg ( 𝑏 ∘f ↑ 𝐺 ) ) ) ) = ( ( 𝑏 ∈ 𝐷 ↦ ( ( 𝐹 ‘ ( 𝑥 ‘ 𝑏 ) ) · ( 𝑇 Σg ( 𝑏 ∘f ↑ 𝐺 ) ) ) ) ∘f ( +g ‘ 𝑆 ) ( 𝑏 ∈ 𝐷 ↦ ( ( 𝐹 ‘ ( 𝑦 ‘ 𝑏 ) ) · ( 𝑇 Σg ( 𝑏 ∘f ↑ 𝐺 ) ) ) ) ) ) |
156 |
155
|
oveq2d |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) ) → ( 𝑆 Σg ( 𝑏 ∈ 𝐷 ↦ ( ( 𝐹 ‘ ( ( 𝑥 ( +g ‘ 𝑃 ) 𝑦 ) ‘ 𝑏 ) ) · ( 𝑇 Σg ( 𝑏 ∘f ↑ 𝐺 ) ) ) ) ) = ( 𝑆 Σg ( ( 𝑏 ∈ 𝐷 ↦ ( ( 𝐹 ‘ ( 𝑥 ‘ 𝑏 ) ) · ( 𝑇 Σg ( 𝑏 ∘f ↑ 𝐺 ) ) ) ) ∘f ( +g ‘ 𝑆 ) ( 𝑏 ∈ 𝐷 ↦ ( ( 𝐹 ‘ ( 𝑦 ‘ 𝑏 ) ) · ( 𝑇 Σg ( 𝑏 ∘f ↑ 𝐺 ) ) ) ) ) ) ) |
157 |
94
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) ) → 𝑆 ∈ CMnd ) |
158 |
10
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) ) → 𝐼 ∈ 𝑊 ) |
159 |
11
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) ) → 𝑅 ∈ CRing ) |
160 |
12
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) ) → 𝑆 ∈ CRing ) |
161 |
13
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) ) → 𝐹 ∈ ( 𝑅 RingHom 𝑆 ) ) |
162 |
14
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) ) → 𝐺 : 𝐼 ⟶ 𝐶 ) |
163 |
1 2 3 4 5 6 7 8 9 158 159 160 161 162 115
|
evlslem6 |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) ) → ( ( 𝑏 ∈ 𝐷 ↦ ( ( 𝐹 ‘ ( 𝑥 ‘ 𝑏 ) ) · ( 𝑇 Σg ( 𝑏 ∘f ↑ 𝐺 ) ) ) ) : 𝐷 ⟶ 𝐶 ∧ ( 𝑏 ∈ 𝐷 ↦ ( ( 𝐹 ‘ ( 𝑥 ‘ 𝑏 ) ) · ( 𝑇 Σg ( 𝑏 ∘f ↑ 𝐺 ) ) ) ) finSupp ( 0g ‘ 𝑆 ) ) ) |
164 |
163
|
simpld |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) ) → ( 𝑏 ∈ 𝐷 ↦ ( ( 𝐹 ‘ ( 𝑥 ‘ 𝑏 ) ) · ( 𝑇 Σg ( 𝑏 ∘f ↑ 𝐺 ) ) ) ) : 𝐷 ⟶ 𝐶 ) |
165 |
1 2 3 4 5 6 7 8 9 158 159 160 161 162 119
|
evlslem6 |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) ) → ( ( 𝑏 ∈ 𝐷 ↦ ( ( 𝐹 ‘ ( 𝑦 ‘ 𝑏 ) ) · ( 𝑇 Σg ( 𝑏 ∘f ↑ 𝐺 ) ) ) ) : 𝐷 ⟶ 𝐶 ∧ ( 𝑏 ∈ 𝐷 ↦ ( ( 𝐹 ‘ ( 𝑦 ‘ 𝑏 ) ) · ( 𝑇 Σg ( 𝑏 ∘f ↑ 𝐺 ) ) ) ) finSupp ( 0g ‘ 𝑆 ) ) ) |
166 |
165
|
simpld |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) ) → ( 𝑏 ∈ 𝐷 ↦ ( ( 𝐹 ‘ ( 𝑦 ‘ 𝑏 ) ) · ( 𝑇 Σg ( 𝑏 ∘f ↑ 𝐺 ) ) ) ) : 𝐷 ⟶ 𝐶 ) |
167 |
163
|
simprd |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) ) → ( 𝑏 ∈ 𝐷 ↦ ( ( 𝐹 ‘ ( 𝑥 ‘ 𝑏 ) ) · ( 𝑇 Σg ( 𝑏 ∘f ↑ 𝐺 ) ) ) ) finSupp ( 0g ‘ 𝑆 ) ) |
168 |
165
|
simprd |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) ) → ( 𝑏 ∈ 𝐷 ↦ ( ( 𝐹 ‘ ( 𝑦 ‘ 𝑏 ) ) · ( 𝑇 Σg ( 𝑏 ∘f ↑ 𝐺 ) ) ) ) finSupp ( 0g ‘ 𝑆 ) ) |
169 |
3 92 89 157 149 164 166 167 168
|
gsumadd |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) ) → ( 𝑆 Σg ( ( 𝑏 ∈ 𝐷 ↦ ( ( 𝐹 ‘ ( 𝑥 ‘ 𝑏 ) ) · ( 𝑇 Σg ( 𝑏 ∘f ↑ 𝐺 ) ) ) ) ∘f ( +g ‘ 𝑆 ) ( 𝑏 ∈ 𝐷 ↦ ( ( 𝐹 ‘ ( 𝑦 ‘ 𝑏 ) ) · ( 𝑇 Σg ( 𝑏 ∘f ↑ 𝐺 ) ) ) ) ) ) = ( ( 𝑆 Σg ( 𝑏 ∈ 𝐷 ↦ ( ( 𝐹 ‘ ( 𝑥 ‘ 𝑏 ) ) · ( 𝑇 Σg ( 𝑏 ∘f ↑ 𝐺 ) ) ) ) ) ( +g ‘ 𝑆 ) ( 𝑆 Σg ( 𝑏 ∈ 𝐷 ↦ ( ( 𝐹 ‘ ( 𝑦 ‘ 𝑏 ) ) · ( 𝑇 Σg ( 𝑏 ∘f ↑ 𝐺 ) ) ) ) ) ) ) |
170 |
156 169
|
eqtrd |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) ) → ( 𝑆 Σg ( 𝑏 ∈ 𝐷 ↦ ( ( 𝐹 ‘ ( ( 𝑥 ( +g ‘ 𝑃 ) 𝑦 ) ‘ 𝑏 ) ) · ( 𝑇 Σg ( 𝑏 ∘f ↑ 𝐺 ) ) ) ) ) = ( ( 𝑆 Σg ( 𝑏 ∈ 𝐷 ↦ ( ( 𝐹 ‘ ( 𝑥 ‘ 𝑏 ) ) · ( 𝑇 Σg ( 𝑏 ∘f ↑ 𝐺 ) ) ) ) ) ( +g ‘ 𝑆 ) ( 𝑆 Σg ( 𝑏 ∈ 𝐷 ↦ ( ( 𝐹 ‘ ( 𝑦 ‘ 𝑏 ) ) · ( 𝑇 Σg ( 𝑏 ∘f ↑ 𝐺 ) ) ) ) ) ) ) |
171 |
90
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) ) → 𝑃 ∈ Grp ) |
172 |
2 88
|
grpcl |
⊢ ( ( 𝑃 ∈ Grp ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) → ( 𝑥 ( +g ‘ 𝑃 ) 𝑦 ) ∈ 𝐵 ) |
173 |
171 115 119 172
|
syl3anc |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) ) → ( 𝑥 ( +g ‘ 𝑃 ) 𝑦 ) ∈ 𝐵 ) |
174 |
|
fveq1 |
⊢ ( 𝑝 = ( 𝑥 ( +g ‘ 𝑃 ) 𝑦 ) → ( 𝑝 ‘ 𝑏 ) = ( ( 𝑥 ( +g ‘ 𝑃 ) 𝑦 ) ‘ 𝑏 ) ) |
175 |
174
|
fveq2d |
⊢ ( 𝑝 = ( 𝑥 ( +g ‘ 𝑃 ) 𝑦 ) → ( 𝐹 ‘ ( 𝑝 ‘ 𝑏 ) ) = ( 𝐹 ‘ ( ( 𝑥 ( +g ‘ 𝑃 ) 𝑦 ) ‘ 𝑏 ) ) ) |
176 |
175
|
oveq1d |
⊢ ( 𝑝 = ( 𝑥 ( +g ‘ 𝑃 ) 𝑦 ) → ( ( 𝐹 ‘ ( 𝑝 ‘ 𝑏 ) ) · ( 𝑇 Σg ( 𝑏 ∘f ↑ 𝐺 ) ) ) = ( ( 𝐹 ‘ ( ( 𝑥 ( +g ‘ 𝑃 ) 𝑦 ) ‘ 𝑏 ) ) · ( 𝑇 Σg ( 𝑏 ∘f ↑ 𝐺 ) ) ) ) |
177 |
176
|
mpteq2dv |
⊢ ( 𝑝 = ( 𝑥 ( +g ‘ 𝑃 ) 𝑦 ) → ( 𝑏 ∈ 𝐷 ↦ ( ( 𝐹 ‘ ( 𝑝 ‘ 𝑏 ) ) · ( 𝑇 Σg ( 𝑏 ∘f ↑ 𝐺 ) ) ) ) = ( 𝑏 ∈ 𝐷 ↦ ( ( 𝐹 ‘ ( ( 𝑥 ( +g ‘ 𝑃 ) 𝑦 ) ‘ 𝑏 ) ) · ( 𝑇 Σg ( 𝑏 ∘f ↑ 𝐺 ) ) ) ) ) |
178 |
177
|
oveq2d |
⊢ ( 𝑝 = ( 𝑥 ( +g ‘ 𝑃 ) 𝑦 ) → ( 𝑆 Σg ( 𝑏 ∈ 𝐷 ↦ ( ( 𝐹 ‘ ( 𝑝 ‘ 𝑏 ) ) · ( 𝑇 Σg ( 𝑏 ∘f ↑ 𝐺 ) ) ) ) ) = ( 𝑆 Σg ( 𝑏 ∈ 𝐷 ↦ ( ( 𝐹 ‘ ( ( 𝑥 ( +g ‘ 𝑃 ) 𝑦 ) ‘ 𝑏 ) ) · ( 𝑇 Σg ( 𝑏 ∘f ↑ 𝐺 ) ) ) ) ) ) |
179 |
|
ovex |
⊢ ( 𝑆 Σg ( 𝑏 ∈ 𝐷 ↦ ( ( 𝐹 ‘ ( ( 𝑥 ( +g ‘ 𝑃 ) 𝑦 ) ‘ 𝑏 ) ) · ( 𝑇 Σg ( 𝑏 ∘f ↑ 𝐺 ) ) ) ) ) ∈ V |
180 |
178 9 179
|
fvmpt |
⊢ ( ( 𝑥 ( +g ‘ 𝑃 ) 𝑦 ) ∈ 𝐵 → ( 𝐸 ‘ ( 𝑥 ( +g ‘ 𝑃 ) 𝑦 ) ) = ( 𝑆 Σg ( 𝑏 ∈ 𝐷 ↦ ( ( 𝐹 ‘ ( ( 𝑥 ( +g ‘ 𝑃 ) 𝑦 ) ‘ 𝑏 ) ) · ( 𝑇 Σg ( 𝑏 ∘f ↑ 𝐺 ) ) ) ) ) ) |
181 |
173 180
|
syl |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) ) → ( 𝐸 ‘ ( 𝑥 ( +g ‘ 𝑃 ) 𝑦 ) ) = ( 𝑆 Σg ( 𝑏 ∈ 𝐷 ↦ ( ( 𝐹 ‘ ( ( 𝑥 ( +g ‘ 𝑃 ) 𝑦 ) ‘ 𝑏 ) ) · ( 𝑇 Σg ( 𝑏 ∘f ↑ 𝐺 ) ) ) ) ) ) |
182 |
|
fveq1 |
⊢ ( 𝑝 = 𝑥 → ( 𝑝 ‘ 𝑏 ) = ( 𝑥 ‘ 𝑏 ) ) |
183 |
182
|
fveq2d |
⊢ ( 𝑝 = 𝑥 → ( 𝐹 ‘ ( 𝑝 ‘ 𝑏 ) ) = ( 𝐹 ‘ ( 𝑥 ‘ 𝑏 ) ) ) |
184 |
183
|
oveq1d |
⊢ ( 𝑝 = 𝑥 → ( ( 𝐹 ‘ ( 𝑝 ‘ 𝑏 ) ) · ( 𝑇 Σg ( 𝑏 ∘f ↑ 𝐺 ) ) ) = ( ( 𝐹 ‘ ( 𝑥 ‘ 𝑏 ) ) · ( 𝑇 Σg ( 𝑏 ∘f ↑ 𝐺 ) ) ) ) |
185 |
184
|
mpteq2dv |
⊢ ( 𝑝 = 𝑥 → ( 𝑏 ∈ 𝐷 ↦ ( ( 𝐹 ‘ ( 𝑝 ‘ 𝑏 ) ) · ( 𝑇 Σg ( 𝑏 ∘f ↑ 𝐺 ) ) ) ) = ( 𝑏 ∈ 𝐷 ↦ ( ( 𝐹 ‘ ( 𝑥 ‘ 𝑏 ) ) · ( 𝑇 Σg ( 𝑏 ∘f ↑ 𝐺 ) ) ) ) ) |
186 |
185
|
oveq2d |
⊢ ( 𝑝 = 𝑥 → ( 𝑆 Σg ( 𝑏 ∈ 𝐷 ↦ ( ( 𝐹 ‘ ( 𝑝 ‘ 𝑏 ) ) · ( 𝑇 Σg ( 𝑏 ∘f ↑ 𝐺 ) ) ) ) ) = ( 𝑆 Σg ( 𝑏 ∈ 𝐷 ↦ ( ( 𝐹 ‘ ( 𝑥 ‘ 𝑏 ) ) · ( 𝑇 Σg ( 𝑏 ∘f ↑ 𝐺 ) ) ) ) ) ) |
187 |
|
ovex |
⊢ ( 𝑆 Σg ( 𝑏 ∈ 𝐷 ↦ ( ( 𝐹 ‘ ( 𝑥 ‘ 𝑏 ) ) · ( 𝑇 Σg ( 𝑏 ∘f ↑ 𝐺 ) ) ) ) ) ∈ V |
188 |
186 9 187
|
fvmpt |
⊢ ( 𝑥 ∈ 𝐵 → ( 𝐸 ‘ 𝑥 ) = ( 𝑆 Σg ( 𝑏 ∈ 𝐷 ↦ ( ( 𝐹 ‘ ( 𝑥 ‘ 𝑏 ) ) · ( 𝑇 Σg ( 𝑏 ∘f ↑ 𝐺 ) ) ) ) ) ) |
189 |
115 188
|
syl |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) ) → ( 𝐸 ‘ 𝑥 ) = ( 𝑆 Σg ( 𝑏 ∈ 𝐷 ↦ ( ( 𝐹 ‘ ( 𝑥 ‘ 𝑏 ) ) · ( 𝑇 Σg ( 𝑏 ∘f ↑ 𝐺 ) ) ) ) ) ) |
190 |
|
fveq1 |
⊢ ( 𝑝 = 𝑦 → ( 𝑝 ‘ 𝑏 ) = ( 𝑦 ‘ 𝑏 ) ) |
191 |
190
|
fveq2d |
⊢ ( 𝑝 = 𝑦 → ( 𝐹 ‘ ( 𝑝 ‘ 𝑏 ) ) = ( 𝐹 ‘ ( 𝑦 ‘ 𝑏 ) ) ) |
192 |
191
|
oveq1d |
⊢ ( 𝑝 = 𝑦 → ( ( 𝐹 ‘ ( 𝑝 ‘ 𝑏 ) ) · ( 𝑇 Σg ( 𝑏 ∘f ↑ 𝐺 ) ) ) = ( ( 𝐹 ‘ ( 𝑦 ‘ 𝑏 ) ) · ( 𝑇 Σg ( 𝑏 ∘f ↑ 𝐺 ) ) ) ) |
193 |
192
|
mpteq2dv |
⊢ ( 𝑝 = 𝑦 → ( 𝑏 ∈ 𝐷 ↦ ( ( 𝐹 ‘ ( 𝑝 ‘ 𝑏 ) ) · ( 𝑇 Σg ( 𝑏 ∘f ↑ 𝐺 ) ) ) ) = ( 𝑏 ∈ 𝐷 ↦ ( ( 𝐹 ‘ ( 𝑦 ‘ 𝑏 ) ) · ( 𝑇 Σg ( 𝑏 ∘f ↑ 𝐺 ) ) ) ) ) |
194 |
193
|
oveq2d |
⊢ ( 𝑝 = 𝑦 → ( 𝑆 Σg ( 𝑏 ∈ 𝐷 ↦ ( ( 𝐹 ‘ ( 𝑝 ‘ 𝑏 ) ) · ( 𝑇 Σg ( 𝑏 ∘f ↑ 𝐺 ) ) ) ) ) = ( 𝑆 Σg ( 𝑏 ∈ 𝐷 ↦ ( ( 𝐹 ‘ ( 𝑦 ‘ 𝑏 ) ) · ( 𝑇 Σg ( 𝑏 ∘f ↑ 𝐺 ) ) ) ) ) ) |
195 |
|
ovex |
⊢ ( 𝑆 Σg ( 𝑏 ∈ 𝐷 ↦ ( ( 𝐹 ‘ ( 𝑦 ‘ 𝑏 ) ) · ( 𝑇 Σg ( 𝑏 ∘f ↑ 𝐺 ) ) ) ) ) ∈ V |
196 |
194 9 195
|
fvmpt |
⊢ ( 𝑦 ∈ 𝐵 → ( 𝐸 ‘ 𝑦 ) = ( 𝑆 Σg ( 𝑏 ∈ 𝐷 ↦ ( ( 𝐹 ‘ ( 𝑦 ‘ 𝑏 ) ) · ( 𝑇 Σg ( 𝑏 ∘f ↑ 𝐺 ) ) ) ) ) ) |
197 |
196
|
ad2antll |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) ) → ( 𝐸 ‘ 𝑦 ) = ( 𝑆 Σg ( 𝑏 ∈ 𝐷 ↦ ( ( 𝐹 ‘ ( 𝑦 ‘ 𝑏 ) ) · ( 𝑇 Σg ( 𝑏 ∘f ↑ 𝐺 ) ) ) ) ) ) |
198 |
189 197
|
oveq12d |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) ) → ( ( 𝐸 ‘ 𝑥 ) ( +g ‘ 𝑆 ) ( 𝐸 ‘ 𝑦 ) ) = ( ( 𝑆 Σg ( 𝑏 ∈ 𝐷 ↦ ( ( 𝐹 ‘ ( 𝑥 ‘ 𝑏 ) ) · ( 𝑇 Σg ( 𝑏 ∘f ↑ 𝐺 ) ) ) ) ) ( +g ‘ 𝑆 ) ( 𝑆 Σg ( 𝑏 ∈ 𝐷 ↦ ( ( 𝐹 ‘ ( 𝑦 ‘ 𝑏 ) ) · ( 𝑇 Σg ( 𝑏 ∘f ↑ 𝐺 ) ) ) ) ) ) ) |
199 |
170 181 198
|
3eqtr4d |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) ) → ( 𝐸 ‘ ( 𝑥 ( +g ‘ 𝑃 ) 𝑦 ) ) = ( ( 𝐸 ‘ 𝑥 ) ( +g ‘ 𝑆 ) ( 𝐸 ‘ 𝑦 ) ) ) |
200 |
2 3 88 89 90 91 109 199
|
isghmd |
⊢ ( 𝜑 → 𝐸 ∈ ( 𝑃 GrpHom 𝑆 ) ) |
201 |
|
eqid |
⊢ ( mulGrp ‘ 𝑅 ) = ( mulGrp ‘ 𝑅 ) |
202 |
201 5
|
rhmmhm |
⊢ ( 𝐹 ∈ ( 𝑅 RingHom 𝑆 ) → 𝐹 ∈ ( ( mulGrp ‘ 𝑅 ) MndHom 𝑇 ) ) |
203 |
13 202
|
syl |
⊢ ( 𝜑 → 𝐹 ∈ ( ( mulGrp ‘ 𝑅 ) MndHom 𝑇 ) ) |
204 |
203
|
adantr |
⊢ ( ( 𝜑 ∧ ( ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) ∧ ( 𝑧 ∈ 𝐷 ∧ 𝑤 ∈ 𝐷 ) ) ) → 𝐹 ∈ ( ( mulGrp ‘ 𝑅 ) MndHom 𝑇 ) ) |
205 |
|
simprll |
⊢ ( ( 𝜑 ∧ ( ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) ∧ ( 𝑧 ∈ 𝐷 ∧ 𝑤 ∈ 𝐷 ) ) ) → 𝑥 ∈ 𝐵 ) |
206 |
1 26 2 4 205
|
mplelf |
⊢ ( ( 𝜑 ∧ ( ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) ∧ ( 𝑧 ∈ 𝐷 ∧ 𝑤 ∈ 𝐷 ) ) ) → 𝑥 : 𝐷 ⟶ ( Base ‘ 𝑅 ) ) |
207 |
|
simprrl |
⊢ ( ( 𝜑 ∧ ( ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) ∧ ( 𝑧 ∈ 𝐷 ∧ 𝑤 ∈ 𝐷 ) ) ) → 𝑧 ∈ 𝐷 ) |
208 |
206 207
|
ffvelcdmd |
⊢ ( ( 𝜑 ∧ ( ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) ∧ ( 𝑧 ∈ 𝐷 ∧ 𝑤 ∈ 𝐷 ) ) ) → ( 𝑥 ‘ 𝑧 ) ∈ ( Base ‘ 𝑅 ) ) |
209 |
|
simprlr |
⊢ ( ( 𝜑 ∧ ( ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) ∧ ( 𝑧 ∈ 𝐷 ∧ 𝑤 ∈ 𝐷 ) ) ) → 𝑦 ∈ 𝐵 ) |
210 |
1 26 2 4 209
|
mplelf |
⊢ ( ( 𝜑 ∧ ( ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) ∧ ( 𝑧 ∈ 𝐷 ∧ 𝑤 ∈ 𝐷 ) ) ) → 𝑦 : 𝐷 ⟶ ( Base ‘ 𝑅 ) ) |
211 |
|
simprrr |
⊢ ( ( 𝜑 ∧ ( ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) ∧ ( 𝑧 ∈ 𝐷 ∧ 𝑤 ∈ 𝐷 ) ) ) → 𝑤 ∈ 𝐷 ) |
212 |
210 211
|
ffvelcdmd |
⊢ ( ( 𝜑 ∧ ( ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) ∧ ( 𝑧 ∈ 𝐷 ∧ 𝑤 ∈ 𝐷 ) ) ) → ( 𝑦 ‘ 𝑤 ) ∈ ( Base ‘ 𝑅 ) ) |
213 |
201 26
|
mgpbas |
⊢ ( Base ‘ 𝑅 ) = ( Base ‘ ( mulGrp ‘ 𝑅 ) ) |
214 |
|
eqid |
⊢ ( .r ‘ 𝑅 ) = ( .r ‘ 𝑅 ) |
215 |
201 214
|
mgpplusg |
⊢ ( .r ‘ 𝑅 ) = ( +g ‘ ( mulGrp ‘ 𝑅 ) ) |
216 |
5 7
|
mgpplusg |
⊢ · = ( +g ‘ 𝑇 ) |
217 |
213 215 216
|
mhmlin |
⊢ ( ( 𝐹 ∈ ( ( mulGrp ‘ 𝑅 ) MndHom 𝑇 ) ∧ ( 𝑥 ‘ 𝑧 ) ∈ ( Base ‘ 𝑅 ) ∧ ( 𝑦 ‘ 𝑤 ) ∈ ( Base ‘ 𝑅 ) ) → ( 𝐹 ‘ ( ( 𝑥 ‘ 𝑧 ) ( .r ‘ 𝑅 ) ( 𝑦 ‘ 𝑤 ) ) ) = ( ( 𝐹 ‘ ( 𝑥 ‘ 𝑧 ) ) · ( 𝐹 ‘ ( 𝑦 ‘ 𝑤 ) ) ) ) |
218 |
204 208 212 217
|
syl3anc |
⊢ ( ( 𝜑 ∧ ( ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) ∧ ( 𝑧 ∈ 𝐷 ∧ 𝑤 ∈ 𝐷 ) ) ) → ( 𝐹 ‘ ( ( 𝑥 ‘ 𝑧 ) ( .r ‘ 𝑅 ) ( 𝑦 ‘ 𝑤 ) ) ) = ( ( 𝐹 ‘ ( 𝑥 ‘ 𝑧 ) ) · ( 𝐹 ‘ ( 𝑦 ‘ 𝑤 ) ) ) ) |
219 |
56
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ ( 𝑧 ∈ 𝐷 ∧ 𝑤 ∈ 𝐷 ) ) ∧ 𝑣 ∈ 𝐼 ) → 𝑇 ∈ Mnd ) |
220 |
|
simprl |
⊢ ( ( 𝜑 ∧ ( 𝑧 ∈ 𝐷 ∧ 𝑤 ∈ 𝐷 ) ) → 𝑧 ∈ 𝐷 ) |
221 |
4
|
psrbagf |
⊢ ( 𝑧 ∈ 𝐷 → 𝑧 : 𝐼 ⟶ ℕ0 ) |
222 |
220 221
|
syl |
⊢ ( ( 𝜑 ∧ ( 𝑧 ∈ 𝐷 ∧ 𝑤 ∈ 𝐷 ) ) → 𝑧 : 𝐼 ⟶ ℕ0 ) |
223 |
222
|
ffvelcdmda |
⊢ ( ( ( 𝜑 ∧ ( 𝑧 ∈ 𝐷 ∧ 𝑤 ∈ 𝐷 ) ) ∧ 𝑣 ∈ 𝐼 ) → ( 𝑧 ‘ 𝑣 ) ∈ ℕ0 ) |
224 |
4
|
psrbagf |
⊢ ( 𝑤 ∈ 𝐷 → 𝑤 : 𝐼 ⟶ ℕ0 ) |
225 |
224
|
ad2antll |
⊢ ( ( 𝜑 ∧ ( 𝑧 ∈ 𝐷 ∧ 𝑤 ∈ 𝐷 ) ) → 𝑤 : 𝐼 ⟶ ℕ0 ) |
226 |
225
|
ffvelcdmda |
⊢ ( ( ( 𝜑 ∧ ( 𝑧 ∈ 𝐷 ∧ 𝑤 ∈ 𝐷 ) ) ∧ 𝑣 ∈ 𝐼 ) → ( 𝑤 ‘ 𝑣 ) ∈ ℕ0 ) |
227 |
14
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑧 ∈ 𝐷 ∧ 𝑤 ∈ 𝐷 ) ) → 𝐺 : 𝐼 ⟶ 𝐶 ) |
228 |
227
|
ffvelcdmda |
⊢ ( ( ( 𝜑 ∧ ( 𝑧 ∈ 𝐷 ∧ 𝑤 ∈ 𝐷 ) ) ∧ 𝑣 ∈ 𝐼 ) → ( 𝐺 ‘ 𝑣 ) ∈ 𝐶 ) |
229 |
47 6 216
|
mulgnn0dir |
⊢ ( ( 𝑇 ∈ Mnd ∧ ( ( 𝑧 ‘ 𝑣 ) ∈ ℕ0 ∧ ( 𝑤 ‘ 𝑣 ) ∈ ℕ0 ∧ ( 𝐺 ‘ 𝑣 ) ∈ 𝐶 ) ) → ( ( ( 𝑧 ‘ 𝑣 ) + ( 𝑤 ‘ 𝑣 ) ) ↑ ( 𝐺 ‘ 𝑣 ) ) = ( ( ( 𝑧 ‘ 𝑣 ) ↑ ( 𝐺 ‘ 𝑣 ) ) · ( ( 𝑤 ‘ 𝑣 ) ↑ ( 𝐺 ‘ 𝑣 ) ) ) ) |
230 |
219 223 226 228 229
|
syl13anc |
⊢ ( ( ( 𝜑 ∧ ( 𝑧 ∈ 𝐷 ∧ 𝑤 ∈ 𝐷 ) ) ∧ 𝑣 ∈ 𝐼 ) → ( ( ( 𝑧 ‘ 𝑣 ) + ( 𝑤 ‘ 𝑣 ) ) ↑ ( 𝐺 ‘ 𝑣 ) ) = ( ( ( 𝑧 ‘ 𝑣 ) ↑ ( 𝐺 ‘ 𝑣 ) ) · ( ( 𝑤 ‘ 𝑣 ) ↑ ( 𝐺 ‘ 𝑣 ) ) ) ) |
231 |
230
|
mpteq2dva |
⊢ ( ( 𝜑 ∧ ( 𝑧 ∈ 𝐷 ∧ 𝑤 ∈ 𝐷 ) ) → ( 𝑣 ∈ 𝐼 ↦ ( ( ( 𝑧 ‘ 𝑣 ) + ( 𝑤 ‘ 𝑣 ) ) ↑ ( 𝐺 ‘ 𝑣 ) ) ) = ( 𝑣 ∈ 𝐼 ↦ ( ( ( 𝑧 ‘ 𝑣 ) ↑ ( 𝐺 ‘ 𝑣 ) ) · ( ( 𝑤 ‘ 𝑣 ) ↑ ( 𝐺 ‘ 𝑣 ) ) ) ) ) |
232 |
10
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑧 ∈ 𝐷 ∧ 𝑤 ∈ 𝐷 ) ) → 𝐼 ∈ 𝑊 ) |
233 |
|
ovexd |
⊢ ( ( ( 𝜑 ∧ ( 𝑧 ∈ 𝐷 ∧ 𝑤 ∈ 𝐷 ) ) ∧ 𝑣 ∈ 𝐼 ) → ( ( 𝑧 ‘ 𝑣 ) + ( 𝑤 ‘ 𝑣 ) ) ∈ V ) |
234 |
|
fvexd |
⊢ ( ( ( 𝜑 ∧ ( 𝑧 ∈ 𝐷 ∧ 𝑤 ∈ 𝐷 ) ) ∧ 𝑣 ∈ 𝐼 ) → ( 𝐺 ‘ 𝑣 ) ∈ V ) |
235 |
222
|
ffnd |
⊢ ( ( 𝜑 ∧ ( 𝑧 ∈ 𝐷 ∧ 𝑤 ∈ 𝐷 ) ) → 𝑧 Fn 𝐼 ) |
236 |
225
|
ffnd |
⊢ ( ( 𝜑 ∧ ( 𝑧 ∈ 𝐷 ∧ 𝑤 ∈ 𝐷 ) ) → 𝑤 Fn 𝐼 ) |
237 |
|
inidm |
⊢ ( 𝐼 ∩ 𝐼 ) = 𝐼 |
238 |
|
eqidd |
⊢ ( ( ( 𝜑 ∧ ( 𝑧 ∈ 𝐷 ∧ 𝑤 ∈ 𝐷 ) ) ∧ 𝑣 ∈ 𝐼 ) → ( 𝑧 ‘ 𝑣 ) = ( 𝑧 ‘ 𝑣 ) ) |
239 |
|
eqidd |
⊢ ( ( ( 𝜑 ∧ ( 𝑧 ∈ 𝐷 ∧ 𝑤 ∈ 𝐷 ) ) ∧ 𝑣 ∈ 𝐼 ) → ( 𝑤 ‘ 𝑣 ) = ( 𝑤 ‘ 𝑣 ) ) |
240 |
235 236 232 232 237 238 239
|
offval |
⊢ ( ( 𝜑 ∧ ( 𝑧 ∈ 𝐷 ∧ 𝑤 ∈ 𝐷 ) ) → ( 𝑧 ∘f + 𝑤 ) = ( 𝑣 ∈ 𝐼 ↦ ( ( 𝑧 ‘ 𝑣 ) + ( 𝑤 ‘ 𝑣 ) ) ) ) |
241 |
14
|
feqmptd |
⊢ ( 𝜑 → 𝐺 = ( 𝑣 ∈ 𝐼 ↦ ( 𝐺 ‘ 𝑣 ) ) ) |
242 |
241
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑧 ∈ 𝐷 ∧ 𝑤 ∈ 𝐷 ) ) → 𝐺 = ( 𝑣 ∈ 𝐼 ↦ ( 𝐺 ‘ 𝑣 ) ) ) |
243 |
232 233 234 240 242
|
offval2 |
⊢ ( ( 𝜑 ∧ ( 𝑧 ∈ 𝐷 ∧ 𝑤 ∈ 𝐷 ) ) → ( ( 𝑧 ∘f + 𝑤 ) ∘f ↑ 𝐺 ) = ( 𝑣 ∈ 𝐼 ↦ ( ( ( 𝑧 ‘ 𝑣 ) + ( 𝑤 ‘ 𝑣 ) ) ↑ ( 𝐺 ‘ 𝑣 ) ) ) ) |
244 |
|
ovexd |
⊢ ( ( ( 𝜑 ∧ ( 𝑧 ∈ 𝐷 ∧ 𝑤 ∈ 𝐷 ) ) ∧ 𝑣 ∈ 𝐼 ) → ( ( 𝑧 ‘ 𝑣 ) ↑ ( 𝐺 ‘ 𝑣 ) ) ∈ V ) |
245 |
|
ovexd |
⊢ ( ( ( 𝜑 ∧ ( 𝑧 ∈ 𝐷 ∧ 𝑤 ∈ 𝐷 ) ) ∧ 𝑣 ∈ 𝐼 ) → ( ( 𝑤 ‘ 𝑣 ) ↑ ( 𝐺 ‘ 𝑣 ) ) ∈ V ) |
246 |
14
|
ffnd |
⊢ ( 𝜑 → 𝐺 Fn 𝐼 ) |
247 |
246
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑧 ∈ 𝐷 ∧ 𝑤 ∈ 𝐷 ) ) → 𝐺 Fn 𝐼 ) |
248 |
|
eqidd |
⊢ ( ( ( 𝜑 ∧ ( 𝑧 ∈ 𝐷 ∧ 𝑤 ∈ 𝐷 ) ) ∧ 𝑣 ∈ 𝐼 ) → ( 𝐺 ‘ 𝑣 ) = ( 𝐺 ‘ 𝑣 ) ) |
249 |
235 247 232 232 237 238 248
|
offval |
⊢ ( ( 𝜑 ∧ ( 𝑧 ∈ 𝐷 ∧ 𝑤 ∈ 𝐷 ) ) → ( 𝑧 ∘f ↑ 𝐺 ) = ( 𝑣 ∈ 𝐼 ↦ ( ( 𝑧 ‘ 𝑣 ) ↑ ( 𝐺 ‘ 𝑣 ) ) ) ) |
250 |
236 247 232 232 237 239 248
|
offval |
⊢ ( ( 𝜑 ∧ ( 𝑧 ∈ 𝐷 ∧ 𝑤 ∈ 𝐷 ) ) → ( 𝑤 ∘f ↑ 𝐺 ) = ( 𝑣 ∈ 𝐼 ↦ ( ( 𝑤 ‘ 𝑣 ) ↑ ( 𝐺 ‘ 𝑣 ) ) ) ) |
251 |
232 244 245 249 250
|
offval2 |
⊢ ( ( 𝜑 ∧ ( 𝑧 ∈ 𝐷 ∧ 𝑤 ∈ 𝐷 ) ) → ( ( 𝑧 ∘f ↑ 𝐺 ) ∘f · ( 𝑤 ∘f ↑ 𝐺 ) ) = ( 𝑣 ∈ 𝐼 ↦ ( ( ( 𝑧 ‘ 𝑣 ) ↑ ( 𝐺 ‘ 𝑣 ) ) · ( ( 𝑤 ‘ 𝑣 ) ↑ ( 𝐺 ‘ 𝑣 ) ) ) ) ) |
252 |
231 243 251
|
3eqtr4d |
⊢ ( ( 𝜑 ∧ ( 𝑧 ∈ 𝐷 ∧ 𝑤 ∈ 𝐷 ) ) → ( ( 𝑧 ∘f + 𝑤 ) ∘f ↑ 𝐺 ) = ( ( 𝑧 ∘f ↑ 𝐺 ) ∘f · ( 𝑤 ∘f ↑ 𝐺 ) ) ) |
253 |
252
|
oveq2d |
⊢ ( ( 𝜑 ∧ ( 𝑧 ∈ 𝐷 ∧ 𝑤 ∈ 𝐷 ) ) → ( 𝑇 Σg ( ( 𝑧 ∘f + 𝑤 ) ∘f ↑ 𝐺 ) ) = ( 𝑇 Σg ( ( 𝑧 ∘f ↑ 𝐺 ) ∘f · ( 𝑤 ∘f ↑ 𝐺 ) ) ) ) |
254 |
55
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑧 ∈ 𝐷 ∧ 𝑤 ∈ 𝐷 ) ) → 𝑇 ∈ CMnd ) |
255 |
4 47 6 48 254 220 227
|
psrbagev1 |
⊢ ( ( 𝜑 ∧ ( 𝑧 ∈ 𝐷 ∧ 𝑤 ∈ 𝐷 ) ) → ( ( 𝑧 ∘f ↑ 𝐺 ) : 𝐼 ⟶ 𝐶 ∧ ( 𝑧 ∘f ↑ 𝐺 ) finSupp ( 1r ‘ 𝑆 ) ) ) |
256 |
255
|
simpld |
⊢ ( ( 𝜑 ∧ ( 𝑧 ∈ 𝐷 ∧ 𝑤 ∈ 𝐷 ) ) → ( 𝑧 ∘f ↑ 𝐺 ) : 𝐼 ⟶ 𝐶 ) |
257 |
|
simprr |
⊢ ( ( 𝜑 ∧ ( 𝑧 ∈ 𝐷 ∧ 𝑤 ∈ 𝐷 ) ) → 𝑤 ∈ 𝐷 ) |
258 |
4 47 6 48 254 257 227
|
psrbagev1 |
⊢ ( ( 𝜑 ∧ ( 𝑧 ∈ 𝐷 ∧ 𝑤 ∈ 𝐷 ) ) → ( ( 𝑤 ∘f ↑ 𝐺 ) : 𝐼 ⟶ 𝐶 ∧ ( 𝑤 ∘f ↑ 𝐺 ) finSupp ( 1r ‘ 𝑆 ) ) ) |
259 |
258
|
simpld |
⊢ ( ( 𝜑 ∧ ( 𝑧 ∈ 𝐷 ∧ 𝑤 ∈ 𝐷 ) ) → ( 𝑤 ∘f ↑ 𝐺 ) : 𝐼 ⟶ 𝐶 ) |
260 |
255
|
simprd |
⊢ ( ( 𝜑 ∧ ( 𝑧 ∈ 𝐷 ∧ 𝑤 ∈ 𝐷 ) ) → ( 𝑧 ∘f ↑ 𝐺 ) finSupp ( 1r ‘ 𝑆 ) ) |
261 |
258
|
simprd |
⊢ ( ( 𝜑 ∧ ( 𝑧 ∈ 𝐷 ∧ 𝑤 ∈ 𝐷 ) ) → ( 𝑤 ∘f ↑ 𝐺 ) finSupp ( 1r ‘ 𝑆 ) ) |
262 |
47 48 216 254 232 256 259 260 261
|
gsumadd |
⊢ ( ( 𝜑 ∧ ( 𝑧 ∈ 𝐷 ∧ 𝑤 ∈ 𝐷 ) ) → ( 𝑇 Σg ( ( 𝑧 ∘f ↑ 𝐺 ) ∘f · ( 𝑤 ∘f ↑ 𝐺 ) ) ) = ( ( 𝑇 Σg ( 𝑧 ∘f ↑ 𝐺 ) ) · ( 𝑇 Σg ( 𝑤 ∘f ↑ 𝐺 ) ) ) ) |
263 |
253 262
|
eqtrd |
⊢ ( ( 𝜑 ∧ ( 𝑧 ∈ 𝐷 ∧ 𝑤 ∈ 𝐷 ) ) → ( 𝑇 Σg ( ( 𝑧 ∘f + 𝑤 ) ∘f ↑ 𝐺 ) ) = ( ( 𝑇 Σg ( 𝑧 ∘f ↑ 𝐺 ) ) · ( 𝑇 Σg ( 𝑤 ∘f ↑ 𝐺 ) ) ) ) |
264 |
263
|
adantrl |
⊢ ( ( 𝜑 ∧ ( ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) ∧ ( 𝑧 ∈ 𝐷 ∧ 𝑤 ∈ 𝐷 ) ) ) → ( 𝑇 Σg ( ( 𝑧 ∘f + 𝑤 ) ∘f ↑ 𝐺 ) ) = ( ( 𝑇 Σg ( 𝑧 ∘f ↑ 𝐺 ) ) · ( 𝑇 Σg ( 𝑤 ∘f ↑ 𝐺 ) ) ) ) |
265 |
218 264
|
oveq12d |
⊢ ( ( 𝜑 ∧ ( ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) ∧ ( 𝑧 ∈ 𝐷 ∧ 𝑤 ∈ 𝐷 ) ) ) → ( ( 𝐹 ‘ ( ( 𝑥 ‘ 𝑧 ) ( .r ‘ 𝑅 ) ( 𝑦 ‘ 𝑤 ) ) ) · ( 𝑇 Σg ( ( 𝑧 ∘f + 𝑤 ) ∘f ↑ 𝐺 ) ) ) = ( ( ( 𝐹 ‘ ( 𝑥 ‘ 𝑧 ) ) · ( 𝐹 ‘ ( 𝑦 ‘ 𝑤 ) ) ) · ( ( 𝑇 Σg ( 𝑧 ∘f ↑ 𝐺 ) ) · ( 𝑇 Σg ( 𝑤 ∘f ↑ 𝐺 ) ) ) ) ) |
266 |
55
|
adantr |
⊢ ( ( 𝜑 ∧ ( ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) ∧ ( 𝑧 ∈ 𝐷 ∧ 𝑤 ∈ 𝐷 ) ) ) → 𝑇 ∈ CMnd ) |
267 |
63
|
adantr |
⊢ ( ( 𝜑 ∧ ( ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) ∧ ( 𝑧 ∈ 𝐷 ∧ 𝑤 ∈ 𝐷 ) ) ) → 𝐹 : ( Base ‘ 𝑅 ) ⟶ 𝐶 ) |
268 |
267 208
|
ffvelcdmd |
⊢ ( ( 𝜑 ∧ ( ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) ∧ ( 𝑧 ∈ 𝐷 ∧ 𝑤 ∈ 𝐷 ) ) ) → ( 𝐹 ‘ ( 𝑥 ‘ 𝑧 ) ) ∈ 𝐶 ) |
269 |
267 212
|
ffvelcdmd |
⊢ ( ( 𝜑 ∧ ( ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) ∧ ( 𝑧 ∈ 𝐷 ∧ 𝑤 ∈ 𝐷 ) ) ) → ( 𝐹 ‘ ( 𝑦 ‘ 𝑤 ) ) ∈ 𝐶 ) |
270 |
4 47 6 254 220 227
|
psrbagev2 |
⊢ ( ( 𝜑 ∧ ( 𝑧 ∈ 𝐷 ∧ 𝑤 ∈ 𝐷 ) ) → ( 𝑇 Σg ( 𝑧 ∘f ↑ 𝐺 ) ) ∈ 𝐶 ) |
271 |
270
|
adantrl |
⊢ ( ( 𝜑 ∧ ( ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) ∧ ( 𝑧 ∈ 𝐷 ∧ 𝑤 ∈ 𝐷 ) ) ) → ( 𝑇 Σg ( 𝑧 ∘f ↑ 𝐺 ) ) ∈ 𝐶 ) |
272 |
4 47 6 254 257 227
|
psrbagev2 |
⊢ ( ( 𝜑 ∧ ( 𝑧 ∈ 𝐷 ∧ 𝑤 ∈ 𝐷 ) ) → ( 𝑇 Σg ( 𝑤 ∘f ↑ 𝐺 ) ) ∈ 𝐶 ) |
273 |
272
|
adantrl |
⊢ ( ( 𝜑 ∧ ( ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) ∧ ( 𝑧 ∈ 𝐷 ∧ 𝑤 ∈ 𝐷 ) ) ) → ( 𝑇 Σg ( 𝑤 ∘f ↑ 𝐺 ) ) ∈ 𝐶 ) |
274 |
47 216
|
cmn4 |
⊢ ( ( 𝑇 ∈ CMnd ∧ ( ( 𝐹 ‘ ( 𝑥 ‘ 𝑧 ) ) ∈ 𝐶 ∧ ( 𝐹 ‘ ( 𝑦 ‘ 𝑤 ) ) ∈ 𝐶 ) ∧ ( ( 𝑇 Σg ( 𝑧 ∘f ↑ 𝐺 ) ) ∈ 𝐶 ∧ ( 𝑇 Σg ( 𝑤 ∘f ↑ 𝐺 ) ) ∈ 𝐶 ) ) → ( ( ( 𝐹 ‘ ( 𝑥 ‘ 𝑧 ) ) · ( 𝐹 ‘ ( 𝑦 ‘ 𝑤 ) ) ) · ( ( 𝑇 Σg ( 𝑧 ∘f ↑ 𝐺 ) ) · ( 𝑇 Σg ( 𝑤 ∘f ↑ 𝐺 ) ) ) ) = ( ( ( 𝐹 ‘ ( 𝑥 ‘ 𝑧 ) ) · ( 𝑇 Σg ( 𝑧 ∘f ↑ 𝐺 ) ) ) · ( ( 𝐹 ‘ ( 𝑦 ‘ 𝑤 ) ) · ( 𝑇 Σg ( 𝑤 ∘f ↑ 𝐺 ) ) ) ) ) |
275 |
266 268 269 271 273 274
|
syl122anc |
⊢ ( ( 𝜑 ∧ ( ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) ∧ ( 𝑧 ∈ 𝐷 ∧ 𝑤 ∈ 𝐷 ) ) ) → ( ( ( 𝐹 ‘ ( 𝑥 ‘ 𝑧 ) ) · ( 𝐹 ‘ ( 𝑦 ‘ 𝑤 ) ) ) · ( ( 𝑇 Σg ( 𝑧 ∘f ↑ 𝐺 ) ) · ( 𝑇 Σg ( 𝑤 ∘f ↑ 𝐺 ) ) ) ) = ( ( ( 𝐹 ‘ ( 𝑥 ‘ 𝑧 ) ) · ( 𝑇 Σg ( 𝑧 ∘f ↑ 𝐺 ) ) ) · ( ( 𝐹 ‘ ( 𝑦 ‘ 𝑤 ) ) · ( 𝑇 Σg ( 𝑤 ∘f ↑ 𝐺 ) ) ) ) ) |
276 |
265 275
|
eqtrd |
⊢ ( ( 𝜑 ∧ ( ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) ∧ ( 𝑧 ∈ 𝐷 ∧ 𝑤 ∈ 𝐷 ) ) ) → ( ( 𝐹 ‘ ( ( 𝑥 ‘ 𝑧 ) ( .r ‘ 𝑅 ) ( 𝑦 ‘ 𝑤 ) ) ) · ( 𝑇 Σg ( ( 𝑧 ∘f + 𝑤 ) ∘f ↑ 𝐺 ) ) ) = ( ( ( 𝐹 ‘ ( 𝑥 ‘ 𝑧 ) ) · ( 𝑇 Σg ( 𝑧 ∘f ↑ 𝐺 ) ) ) · ( ( 𝐹 ‘ ( 𝑦 ‘ 𝑤 ) ) · ( 𝑇 Σg ( 𝑤 ∘f ↑ 𝐺 ) ) ) ) ) |
277 |
10
|
adantr |
⊢ ( ( 𝜑 ∧ ( ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) ∧ ( 𝑧 ∈ 𝐷 ∧ 𝑤 ∈ 𝐷 ) ) ) → 𝐼 ∈ 𝑊 ) |
278 |
11
|
adantr |
⊢ ( ( 𝜑 ∧ ( ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) ∧ ( 𝑧 ∈ 𝐷 ∧ 𝑤 ∈ 𝐷 ) ) ) → 𝑅 ∈ CRing ) |
279 |
12
|
adantr |
⊢ ( ( 𝜑 ∧ ( ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) ∧ ( 𝑧 ∈ 𝐷 ∧ 𝑤 ∈ 𝐷 ) ) ) → 𝑆 ∈ CRing ) |
280 |
13
|
adantr |
⊢ ( ( 𝜑 ∧ ( ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) ∧ ( 𝑧 ∈ 𝐷 ∧ 𝑤 ∈ 𝐷 ) ) ) → 𝐹 ∈ ( 𝑅 RingHom 𝑆 ) ) |
281 |
14
|
adantr |
⊢ ( ( 𝜑 ∧ ( ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) ∧ ( 𝑧 ∈ 𝐷 ∧ 𝑤 ∈ 𝐷 ) ) ) → 𝐺 : 𝐼 ⟶ 𝐶 ) |
282 |
4
|
psrbagaddcl |
⊢ ( ( 𝑧 ∈ 𝐷 ∧ 𝑤 ∈ 𝐷 ) → ( 𝑧 ∘f + 𝑤 ) ∈ 𝐷 ) |
283 |
282
|
ad2antll |
⊢ ( ( 𝜑 ∧ ( ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) ∧ ( 𝑧 ∈ 𝐷 ∧ 𝑤 ∈ 𝐷 ) ) ) → ( 𝑧 ∘f + 𝑤 ) ∈ 𝐷 ) |
284 |
19
|
adantr |
⊢ ( ( 𝜑 ∧ ( ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) ∧ ( 𝑧 ∈ 𝐷 ∧ 𝑤 ∈ 𝐷 ) ) ) → 𝑅 ∈ Ring ) |
285 |
26 214
|
ringcl |
⊢ ( ( 𝑅 ∈ Ring ∧ ( 𝑥 ‘ 𝑧 ) ∈ ( Base ‘ 𝑅 ) ∧ ( 𝑦 ‘ 𝑤 ) ∈ ( Base ‘ 𝑅 ) ) → ( ( 𝑥 ‘ 𝑧 ) ( .r ‘ 𝑅 ) ( 𝑦 ‘ 𝑤 ) ) ∈ ( Base ‘ 𝑅 ) ) |
286 |
284 208 212 285
|
syl3anc |
⊢ ( ( 𝜑 ∧ ( ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) ∧ ( 𝑧 ∈ 𝐷 ∧ 𝑤 ∈ 𝐷 ) ) ) → ( ( 𝑥 ‘ 𝑧 ) ( .r ‘ 𝑅 ) ( 𝑦 ‘ 𝑤 ) ) ∈ ( Base ‘ 𝑅 ) ) |
287 |
1 2 3 26 4 5 6 7 8 9 277 278 279 280 281 25 283 286
|
evlslem3 |
⊢ ( ( 𝜑 ∧ ( ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) ∧ ( 𝑧 ∈ 𝐷 ∧ 𝑤 ∈ 𝐷 ) ) ) → ( 𝐸 ‘ ( 𝑣 ∈ 𝐷 ↦ if ( 𝑣 = ( 𝑧 ∘f + 𝑤 ) , ( ( 𝑥 ‘ 𝑧 ) ( .r ‘ 𝑅 ) ( 𝑦 ‘ 𝑤 ) ) , ( 0g ‘ 𝑅 ) ) ) ) = ( ( 𝐹 ‘ ( ( 𝑥 ‘ 𝑧 ) ( .r ‘ 𝑅 ) ( 𝑦 ‘ 𝑤 ) ) ) · ( 𝑇 Σg ( ( 𝑧 ∘f + 𝑤 ) ∘f ↑ 𝐺 ) ) ) ) |
288 |
1 2 3 26 4 5 6 7 8 9 277 278 279 280 281 25 207 208
|
evlslem3 |
⊢ ( ( 𝜑 ∧ ( ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) ∧ ( 𝑧 ∈ 𝐷 ∧ 𝑤 ∈ 𝐷 ) ) ) → ( 𝐸 ‘ ( 𝑣 ∈ 𝐷 ↦ if ( 𝑣 = 𝑧 , ( 𝑥 ‘ 𝑧 ) , ( 0g ‘ 𝑅 ) ) ) ) = ( ( 𝐹 ‘ ( 𝑥 ‘ 𝑧 ) ) · ( 𝑇 Σg ( 𝑧 ∘f ↑ 𝐺 ) ) ) ) |
289 |
1 2 3 26 4 5 6 7 8 9 277 278 279 280 281 25 211 212
|
evlslem3 |
⊢ ( ( 𝜑 ∧ ( ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) ∧ ( 𝑧 ∈ 𝐷 ∧ 𝑤 ∈ 𝐷 ) ) ) → ( 𝐸 ‘ ( 𝑣 ∈ 𝐷 ↦ if ( 𝑣 = 𝑤 , ( 𝑦 ‘ 𝑤 ) , ( 0g ‘ 𝑅 ) ) ) ) = ( ( 𝐹 ‘ ( 𝑦 ‘ 𝑤 ) ) · ( 𝑇 Σg ( 𝑤 ∘f ↑ 𝐺 ) ) ) ) |
290 |
288 289
|
oveq12d |
⊢ ( ( 𝜑 ∧ ( ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) ∧ ( 𝑧 ∈ 𝐷 ∧ 𝑤 ∈ 𝐷 ) ) ) → ( ( 𝐸 ‘ ( 𝑣 ∈ 𝐷 ↦ if ( 𝑣 = 𝑧 , ( 𝑥 ‘ 𝑧 ) , ( 0g ‘ 𝑅 ) ) ) ) · ( 𝐸 ‘ ( 𝑣 ∈ 𝐷 ↦ if ( 𝑣 = 𝑤 , ( 𝑦 ‘ 𝑤 ) , ( 0g ‘ 𝑅 ) ) ) ) ) = ( ( ( 𝐹 ‘ ( 𝑥 ‘ 𝑧 ) ) · ( 𝑇 Σg ( 𝑧 ∘f ↑ 𝐺 ) ) ) · ( ( 𝐹 ‘ ( 𝑦 ‘ 𝑤 ) ) · ( 𝑇 Σg ( 𝑤 ∘f ↑ 𝐺 ) ) ) ) ) |
291 |
276 287 290
|
3eqtr4d |
⊢ ( ( 𝜑 ∧ ( ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) ∧ ( 𝑧 ∈ 𝐷 ∧ 𝑤 ∈ 𝐷 ) ) ) → ( 𝐸 ‘ ( 𝑣 ∈ 𝐷 ↦ if ( 𝑣 = ( 𝑧 ∘f + 𝑤 ) , ( ( 𝑥 ‘ 𝑧 ) ( .r ‘ 𝑅 ) ( 𝑦 ‘ 𝑤 ) ) , ( 0g ‘ 𝑅 ) ) ) ) = ( ( 𝐸 ‘ ( 𝑣 ∈ 𝐷 ↦ if ( 𝑣 = 𝑧 , ( 𝑥 ‘ 𝑧 ) , ( 0g ‘ 𝑅 ) ) ) ) · ( 𝐸 ‘ ( 𝑣 ∈ 𝐷 ↦ if ( 𝑣 = 𝑤 , ( 𝑦 ‘ 𝑤 ) , ( 0g ‘ 𝑅 ) ) ) ) ) ) |
292 |
1 2 7 25 4 10 11 12 200 291
|
evlslem2 |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) ) → ( 𝐸 ‘ ( 𝑥 ( .r ‘ 𝑃 ) 𝑦 ) ) = ( ( 𝐸 ‘ 𝑥 ) · ( 𝐸 ‘ 𝑦 ) ) ) |
293 |
2 16 17 18 7 20 21 87 292 3 88 89 109 199
|
isrhmd |
⊢ ( 𝜑 → 𝐸 ∈ ( 𝑃 RingHom 𝑆 ) ) |
294 |
|
ovex |
⊢ ( 𝑆 Σg ( 𝑏 ∈ 𝐷 ↦ ( ( 𝐹 ‘ ( 𝑝 ‘ 𝑏 ) ) · ( 𝑇 Σg ( 𝑏 ∘f ↑ 𝐺 ) ) ) ) ) ∈ V |
295 |
294 9
|
fnmpti |
⊢ 𝐸 Fn 𝐵 |
296 |
295
|
a1i |
⊢ ( 𝜑 → 𝐸 Fn 𝐵 ) |
297 |
26 2
|
rhmf |
⊢ ( 𝐴 ∈ ( 𝑅 RingHom 𝑃 ) → 𝐴 : ( Base ‘ 𝑅 ) ⟶ 𝐵 ) |
298 |
81 297
|
syl |
⊢ ( 𝜑 → 𝐴 : ( Base ‘ 𝑅 ) ⟶ 𝐵 ) |
299 |
298
|
ffnd |
⊢ ( 𝜑 → 𝐴 Fn ( Base ‘ 𝑅 ) ) |
300 |
298
|
frnd |
⊢ ( 𝜑 → ran 𝐴 ⊆ 𝐵 ) |
301 |
|
fnco |
⊢ ( ( 𝐸 Fn 𝐵 ∧ 𝐴 Fn ( Base ‘ 𝑅 ) ∧ ran 𝐴 ⊆ 𝐵 ) → ( 𝐸 ∘ 𝐴 ) Fn ( Base ‘ 𝑅 ) ) |
302 |
296 299 300 301
|
syl3anc |
⊢ ( 𝜑 → ( 𝐸 ∘ 𝐴 ) Fn ( Base ‘ 𝑅 ) ) |
303 |
63
|
ffnd |
⊢ ( 𝜑 → 𝐹 Fn ( Base ‘ 𝑅 ) ) |
304 |
|
fvco2 |
⊢ ( ( 𝐴 Fn ( Base ‘ 𝑅 ) ∧ 𝑥 ∈ ( Base ‘ 𝑅 ) ) → ( ( 𝐸 ∘ 𝐴 ) ‘ 𝑥 ) = ( 𝐸 ‘ ( 𝐴 ‘ 𝑥 ) ) ) |
305 |
299 304
|
sylan |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝑅 ) ) → ( ( 𝐸 ∘ 𝐴 ) ‘ 𝑥 ) = ( 𝐸 ‘ ( 𝐴 ‘ 𝑥 ) ) ) |
306 |
305 68
|
eqtrd |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝑅 ) ) → ( ( 𝐸 ∘ 𝐴 ) ‘ 𝑥 ) = ( 𝐹 ‘ 𝑥 ) ) |
307 |
302 303 306
|
eqfnfvd |
⊢ ( 𝜑 → ( 𝐸 ∘ 𝐴 ) = 𝐹 ) |
308 |
1 8 2 10 19
|
mvrf2 |
⊢ ( 𝜑 → 𝑉 : 𝐼 ⟶ 𝐵 ) |
309 |
308
|
ffnd |
⊢ ( 𝜑 → 𝑉 Fn 𝐼 ) |
310 |
308
|
frnd |
⊢ ( 𝜑 → ran 𝑉 ⊆ 𝐵 ) |
311 |
|
fnco |
⊢ ( ( 𝐸 Fn 𝐵 ∧ 𝑉 Fn 𝐼 ∧ ran 𝑉 ⊆ 𝐵 ) → ( 𝐸 ∘ 𝑉 ) Fn 𝐼 ) |
312 |
296 309 310 311
|
syl3anc |
⊢ ( 𝜑 → ( 𝐸 ∘ 𝑉 ) Fn 𝐼 ) |
313 |
|
fvco2 |
⊢ ( ( 𝑉 Fn 𝐼 ∧ 𝑥 ∈ 𝐼 ) → ( ( 𝐸 ∘ 𝑉 ) ‘ 𝑥 ) = ( 𝐸 ‘ ( 𝑉 ‘ 𝑥 ) ) ) |
314 |
309 313
|
sylan |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐼 ) → ( ( 𝐸 ∘ 𝑉 ) ‘ 𝑥 ) = ( 𝐸 ‘ ( 𝑉 ‘ 𝑥 ) ) ) |
315 |
10
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐼 ) → 𝐼 ∈ 𝑊 ) |
316 |
11
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐼 ) → 𝑅 ∈ CRing ) |
317 |
|
simpr |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐼 ) → 𝑥 ∈ 𝐼 ) |
318 |
8 4 25 70 315 316 317
|
mvrval |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐼 ) → ( 𝑉 ‘ 𝑥 ) = ( 𝑦 ∈ 𝐷 ↦ if ( 𝑦 = ( 𝑧 ∈ 𝐼 ↦ if ( 𝑧 = 𝑥 , 1 , 0 ) ) , ( 1r ‘ 𝑅 ) , ( 0g ‘ 𝑅 ) ) ) ) |
319 |
318
|
fveq2d |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐼 ) → ( 𝐸 ‘ ( 𝑉 ‘ 𝑥 ) ) = ( 𝐸 ‘ ( 𝑦 ∈ 𝐷 ↦ if ( 𝑦 = ( 𝑧 ∈ 𝐼 ↦ if ( 𝑧 = 𝑥 , 1 , 0 ) ) , ( 1r ‘ 𝑅 ) , ( 0g ‘ 𝑅 ) ) ) ) ) |
320 |
12
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐼 ) → 𝑆 ∈ CRing ) |
321 |
13
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐼 ) → 𝐹 ∈ ( 𝑅 RingHom 𝑆 ) ) |
322 |
14
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐼 ) → 𝐺 : 𝐼 ⟶ 𝐶 ) |
323 |
4
|
psrbagsn |
⊢ ( 𝐼 ∈ 𝑊 → ( 𝑧 ∈ 𝐼 ↦ if ( 𝑧 = 𝑥 , 1 , 0 ) ) ∈ 𝐷 ) |
324 |
10 323
|
syl |
⊢ ( 𝜑 → ( 𝑧 ∈ 𝐼 ↦ if ( 𝑧 = 𝑥 , 1 , 0 ) ) ∈ 𝐷 ) |
325 |
324
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐼 ) → ( 𝑧 ∈ 𝐼 ↦ if ( 𝑧 = 𝑥 , 1 , 0 ) ) ∈ 𝐷 ) |
326 |
72
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐼 ) → ( 1r ‘ 𝑅 ) ∈ ( Base ‘ 𝑅 ) ) |
327 |
1 2 3 26 4 5 6 7 8 9 315 316 320 321 322 25 325 326
|
evlslem3 |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐼 ) → ( 𝐸 ‘ ( 𝑦 ∈ 𝐷 ↦ if ( 𝑦 = ( 𝑧 ∈ 𝐼 ↦ if ( 𝑧 = 𝑥 , 1 , 0 ) ) , ( 1r ‘ 𝑅 ) , ( 0g ‘ 𝑅 ) ) ) ) = ( ( 𝐹 ‘ ( 1r ‘ 𝑅 ) ) · ( 𝑇 Σg ( ( 𝑧 ∈ 𝐼 ↦ if ( 𝑧 = 𝑥 , 1 , 0 ) ) ∘f ↑ 𝐺 ) ) ) ) |
328 |
86
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐼 ) → ( 𝐹 ‘ ( 1r ‘ 𝑅 ) ) = ( 1r ‘ 𝑆 ) ) |
329 |
|
1nn0 |
⊢ 1 ∈ ℕ0 |
330 |
|
0nn0 |
⊢ 0 ∈ ℕ0 |
331 |
329 330
|
ifcli |
⊢ if ( 𝑧 = 𝑥 , 1 , 0 ) ∈ ℕ0 |
332 |
331
|
a1i |
⊢ ( ( 𝜑 ∧ 𝑧 ∈ 𝐼 ) → if ( 𝑧 = 𝑥 , 1 , 0 ) ∈ ℕ0 ) |
333 |
14
|
ffvelcdmda |
⊢ ( ( 𝜑 ∧ 𝑧 ∈ 𝐼 ) → ( 𝐺 ‘ 𝑧 ) ∈ 𝐶 ) |
334 |
|
eqidd |
⊢ ( 𝜑 → ( 𝑧 ∈ 𝐼 ↦ if ( 𝑧 = 𝑥 , 1 , 0 ) ) = ( 𝑧 ∈ 𝐼 ↦ if ( 𝑧 = 𝑥 , 1 , 0 ) ) ) |
335 |
14
|
feqmptd |
⊢ ( 𝜑 → 𝐺 = ( 𝑧 ∈ 𝐼 ↦ ( 𝐺 ‘ 𝑧 ) ) ) |
336 |
10 332 333 334 335
|
offval2 |
⊢ ( 𝜑 → ( ( 𝑧 ∈ 𝐼 ↦ if ( 𝑧 = 𝑥 , 1 , 0 ) ) ∘f ↑ 𝐺 ) = ( 𝑧 ∈ 𝐼 ↦ ( if ( 𝑧 = 𝑥 , 1 , 0 ) ↑ ( 𝐺 ‘ 𝑧 ) ) ) ) |
337 |
|
oveq1 |
⊢ ( 1 = if ( 𝑧 = 𝑥 , 1 , 0 ) → ( 1 ↑ ( 𝐺 ‘ 𝑧 ) ) = ( if ( 𝑧 = 𝑥 , 1 , 0 ) ↑ ( 𝐺 ‘ 𝑧 ) ) ) |
338 |
337
|
eqeq1d |
⊢ ( 1 = if ( 𝑧 = 𝑥 , 1 , 0 ) → ( ( 1 ↑ ( 𝐺 ‘ 𝑧 ) ) = if ( 𝑧 = 𝑥 , ( 𝐺 ‘ 𝑧 ) , ( 1r ‘ 𝑆 ) ) ↔ ( if ( 𝑧 = 𝑥 , 1 , 0 ) ↑ ( 𝐺 ‘ 𝑧 ) ) = if ( 𝑧 = 𝑥 , ( 𝐺 ‘ 𝑧 ) , ( 1r ‘ 𝑆 ) ) ) ) |
339 |
|
oveq1 |
⊢ ( 0 = if ( 𝑧 = 𝑥 , 1 , 0 ) → ( 0 ↑ ( 𝐺 ‘ 𝑧 ) ) = ( if ( 𝑧 = 𝑥 , 1 , 0 ) ↑ ( 𝐺 ‘ 𝑧 ) ) ) |
340 |
339
|
eqeq1d |
⊢ ( 0 = if ( 𝑧 = 𝑥 , 1 , 0 ) → ( ( 0 ↑ ( 𝐺 ‘ 𝑧 ) ) = if ( 𝑧 = 𝑥 , ( 𝐺 ‘ 𝑧 ) , ( 1r ‘ 𝑆 ) ) ↔ ( if ( 𝑧 = 𝑥 , 1 , 0 ) ↑ ( 𝐺 ‘ 𝑧 ) ) = if ( 𝑧 = 𝑥 , ( 𝐺 ‘ 𝑧 ) , ( 1r ‘ 𝑆 ) ) ) ) |
341 |
333
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑧 ∈ 𝐼 ) ∧ 𝑧 = 𝑥 ) → ( 𝐺 ‘ 𝑧 ) ∈ 𝐶 ) |
342 |
47 6
|
mulg1 |
⊢ ( ( 𝐺 ‘ 𝑧 ) ∈ 𝐶 → ( 1 ↑ ( 𝐺 ‘ 𝑧 ) ) = ( 𝐺 ‘ 𝑧 ) ) |
343 |
341 342
|
syl |
⊢ ( ( ( 𝜑 ∧ 𝑧 ∈ 𝐼 ) ∧ 𝑧 = 𝑥 ) → ( 1 ↑ ( 𝐺 ‘ 𝑧 ) ) = ( 𝐺 ‘ 𝑧 ) ) |
344 |
|
iftrue |
⊢ ( 𝑧 = 𝑥 → if ( 𝑧 = 𝑥 , ( 𝐺 ‘ 𝑧 ) , ( 1r ‘ 𝑆 ) ) = ( 𝐺 ‘ 𝑧 ) ) |
345 |
344
|
adantl |
⊢ ( ( ( 𝜑 ∧ 𝑧 ∈ 𝐼 ) ∧ 𝑧 = 𝑥 ) → if ( 𝑧 = 𝑥 , ( 𝐺 ‘ 𝑧 ) , ( 1r ‘ 𝑆 ) ) = ( 𝐺 ‘ 𝑧 ) ) |
346 |
343 345
|
eqtr4d |
⊢ ( ( ( 𝜑 ∧ 𝑧 ∈ 𝐼 ) ∧ 𝑧 = 𝑥 ) → ( 1 ↑ ( 𝐺 ‘ 𝑧 ) ) = if ( 𝑧 = 𝑥 , ( 𝐺 ‘ 𝑧 ) , ( 1r ‘ 𝑆 ) ) ) |
347 |
47 48 6
|
mulg0 |
⊢ ( ( 𝐺 ‘ 𝑧 ) ∈ 𝐶 → ( 0 ↑ ( 𝐺 ‘ 𝑧 ) ) = ( 1r ‘ 𝑆 ) ) |
348 |
333 347
|
syl |
⊢ ( ( 𝜑 ∧ 𝑧 ∈ 𝐼 ) → ( 0 ↑ ( 𝐺 ‘ 𝑧 ) ) = ( 1r ‘ 𝑆 ) ) |
349 |
348
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑧 ∈ 𝐼 ) ∧ ¬ 𝑧 = 𝑥 ) → ( 0 ↑ ( 𝐺 ‘ 𝑧 ) ) = ( 1r ‘ 𝑆 ) ) |
350 |
|
iffalse |
⊢ ( ¬ 𝑧 = 𝑥 → if ( 𝑧 = 𝑥 , ( 𝐺 ‘ 𝑧 ) , ( 1r ‘ 𝑆 ) ) = ( 1r ‘ 𝑆 ) ) |
351 |
350
|
adantl |
⊢ ( ( ( 𝜑 ∧ 𝑧 ∈ 𝐼 ) ∧ ¬ 𝑧 = 𝑥 ) → if ( 𝑧 = 𝑥 , ( 𝐺 ‘ 𝑧 ) , ( 1r ‘ 𝑆 ) ) = ( 1r ‘ 𝑆 ) ) |
352 |
349 351
|
eqtr4d |
⊢ ( ( ( 𝜑 ∧ 𝑧 ∈ 𝐼 ) ∧ ¬ 𝑧 = 𝑥 ) → ( 0 ↑ ( 𝐺 ‘ 𝑧 ) ) = if ( 𝑧 = 𝑥 , ( 𝐺 ‘ 𝑧 ) , ( 1r ‘ 𝑆 ) ) ) |
353 |
338 340 346 352
|
ifbothda |
⊢ ( ( 𝜑 ∧ 𝑧 ∈ 𝐼 ) → ( if ( 𝑧 = 𝑥 , 1 , 0 ) ↑ ( 𝐺 ‘ 𝑧 ) ) = if ( 𝑧 = 𝑥 , ( 𝐺 ‘ 𝑧 ) , ( 1r ‘ 𝑆 ) ) ) |
354 |
353
|
mpteq2dva |
⊢ ( 𝜑 → ( 𝑧 ∈ 𝐼 ↦ ( if ( 𝑧 = 𝑥 , 1 , 0 ) ↑ ( 𝐺 ‘ 𝑧 ) ) ) = ( 𝑧 ∈ 𝐼 ↦ if ( 𝑧 = 𝑥 , ( 𝐺 ‘ 𝑧 ) , ( 1r ‘ 𝑆 ) ) ) ) |
355 |
336 354
|
eqtrd |
⊢ ( 𝜑 → ( ( 𝑧 ∈ 𝐼 ↦ if ( 𝑧 = 𝑥 , 1 , 0 ) ) ∘f ↑ 𝐺 ) = ( 𝑧 ∈ 𝐼 ↦ if ( 𝑧 = 𝑥 , ( 𝐺 ‘ 𝑧 ) , ( 1r ‘ 𝑆 ) ) ) ) |
356 |
355
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐼 ) → ( ( 𝑧 ∈ 𝐼 ↦ if ( 𝑧 = 𝑥 , 1 , 0 ) ) ∘f ↑ 𝐺 ) = ( 𝑧 ∈ 𝐼 ↦ if ( 𝑧 = 𝑥 , ( 𝐺 ‘ 𝑧 ) , ( 1r ‘ 𝑆 ) ) ) ) |
357 |
356
|
oveq2d |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐼 ) → ( 𝑇 Σg ( ( 𝑧 ∈ 𝐼 ↦ if ( 𝑧 = 𝑥 , 1 , 0 ) ) ∘f ↑ 𝐺 ) ) = ( 𝑇 Σg ( 𝑧 ∈ 𝐼 ↦ if ( 𝑧 = 𝑥 , ( 𝐺 ‘ 𝑧 ) , ( 1r ‘ 𝑆 ) ) ) ) ) |
358 |
56
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐼 ) → 𝑇 ∈ Mnd ) |
359 |
333
|
adantlr |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ 𝐼 ) ∧ 𝑧 ∈ 𝐼 ) → ( 𝐺 ‘ 𝑧 ) ∈ 𝐶 ) |
360 |
3 17
|
ringidcl |
⊢ ( 𝑆 ∈ Ring → ( 1r ‘ 𝑆 ) ∈ 𝐶 ) |
361 |
21 360
|
syl |
⊢ ( 𝜑 → ( 1r ‘ 𝑆 ) ∈ 𝐶 ) |
362 |
361
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ 𝐼 ) ∧ 𝑧 ∈ 𝐼 ) → ( 1r ‘ 𝑆 ) ∈ 𝐶 ) |
363 |
359 362
|
ifcld |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ 𝐼 ) ∧ 𝑧 ∈ 𝐼 ) → if ( 𝑧 = 𝑥 , ( 𝐺 ‘ 𝑧 ) , ( 1r ‘ 𝑆 ) ) ∈ 𝐶 ) |
364 |
363
|
fmpttd |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐼 ) → ( 𝑧 ∈ 𝐼 ↦ if ( 𝑧 = 𝑥 , ( 𝐺 ‘ 𝑧 ) , ( 1r ‘ 𝑆 ) ) ) : 𝐼 ⟶ 𝐶 ) |
365 |
|
eldifsnneq |
⊢ ( 𝑧 ∈ ( 𝐼 ∖ { 𝑥 } ) → ¬ 𝑧 = 𝑥 ) |
366 |
365 350
|
syl |
⊢ ( 𝑧 ∈ ( 𝐼 ∖ { 𝑥 } ) → if ( 𝑧 = 𝑥 , ( 𝐺 ‘ 𝑧 ) , ( 1r ‘ 𝑆 ) ) = ( 1r ‘ 𝑆 ) ) |
367 |
366
|
adantl |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ 𝐼 ) ∧ 𝑧 ∈ ( 𝐼 ∖ { 𝑥 } ) ) → if ( 𝑧 = 𝑥 , ( 𝐺 ‘ 𝑧 ) , ( 1r ‘ 𝑆 ) ) = ( 1r ‘ 𝑆 ) ) |
368 |
367 315
|
suppss2 |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐼 ) → ( ( 𝑧 ∈ 𝐼 ↦ if ( 𝑧 = 𝑥 , ( 𝐺 ‘ 𝑧 ) , ( 1r ‘ 𝑆 ) ) ) supp ( 1r ‘ 𝑆 ) ) ⊆ { 𝑥 } ) |
369 |
47 48 358 315 317 364 368
|
gsumpt |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐼 ) → ( 𝑇 Σg ( 𝑧 ∈ 𝐼 ↦ if ( 𝑧 = 𝑥 , ( 𝐺 ‘ 𝑧 ) , ( 1r ‘ 𝑆 ) ) ) ) = ( ( 𝑧 ∈ 𝐼 ↦ if ( 𝑧 = 𝑥 , ( 𝐺 ‘ 𝑧 ) , ( 1r ‘ 𝑆 ) ) ) ‘ 𝑥 ) ) |
370 |
|
fveq2 |
⊢ ( 𝑧 = 𝑥 → ( 𝐺 ‘ 𝑧 ) = ( 𝐺 ‘ 𝑥 ) ) |
371 |
344 370
|
eqtrd |
⊢ ( 𝑧 = 𝑥 → if ( 𝑧 = 𝑥 , ( 𝐺 ‘ 𝑧 ) , ( 1r ‘ 𝑆 ) ) = ( 𝐺 ‘ 𝑥 ) ) |
372 |
|
eqid |
⊢ ( 𝑧 ∈ 𝐼 ↦ if ( 𝑧 = 𝑥 , ( 𝐺 ‘ 𝑧 ) , ( 1r ‘ 𝑆 ) ) ) = ( 𝑧 ∈ 𝐼 ↦ if ( 𝑧 = 𝑥 , ( 𝐺 ‘ 𝑧 ) , ( 1r ‘ 𝑆 ) ) ) |
373 |
|
fvex |
⊢ ( 𝐺 ‘ 𝑥 ) ∈ V |
374 |
371 372 373
|
fvmpt |
⊢ ( 𝑥 ∈ 𝐼 → ( ( 𝑧 ∈ 𝐼 ↦ if ( 𝑧 = 𝑥 , ( 𝐺 ‘ 𝑧 ) , ( 1r ‘ 𝑆 ) ) ) ‘ 𝑥 ) = ( 𝐺 ‘ 𝑥 ) ) |
375 |
374
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐼 ) → ( ( 𝑧 ∈ 𝐼 ↦ if ( 𝑧 = 𝑥 , ( 𝐺 ‘ 𝑧 ) , ( 1r ‘ 𝑆 ) ) ) ‘ 𝑥 ) = ( 𝐺 ‘ 𝑥 ) ) |
376 |
357 369 375
|
3eqtrd |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐼 ) → ( 𝑇 Σg ( ( 𝑧 ∈ 𝐼 ↦ if ( 𝑧 = 𝑥 , 1 , 0 ) ) ∘f ↑ 𝐺 ) ) = ( 𝐺 ‘ 𝑥 ) ) |
377 |
328 376
|
oveq12d |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐼 ) → ( ( 𝐹 ‘ ( 1r ‘ 𝑅 ) ) · ( 𝑇 Σg ( ( 𝑧 ∈ 𝐼 ↦ if ( 𝑧 = 𝑥 , 1 , 0 ) ) ∘f ↑ 𝐺 ) ) ) = ( ( 1r ‘ 𝑆 ) · ( 𝐺 ‘ 𝑥 ) ) ) |
378 |
3 7 17
|
ringlidm |
⊢ ( ( 𝑆 ∈ Ring ∧ ( 𝐺 ‘ 𝑥 ) ∈ 𝐶 ) → ( ( 1r ‘ 𝑆 ) · ( 𝐺 ‘ 𝑥 ) ) = ( 𝐺 ‘ 𝑥 ) ) |
379 |
21 46 378
|
syl2an2r |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐼 ) → ( ( 1r ‘ 𝑆 ) · ( 𝐺 ‘ 𝑥 ) ) = ( 𝐺 ‘ 𝑥 ) ) |
380 |
377 379
|
eqtrd |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐼 ) → ( ( 𝐹 ‘ ( 1r ‘ 𝑅 ) ) · ( 𝑇 Σg ( ( 𝑧 ∈ 𝐼 ↦ if ( 𝑧 = 𝑥 , 1 , 0 ) ) ∘f ↑ 𝐺 ) ) ) = ( 𝐺 ‘ 𝑥 ) ) |
381 |
319 327 380
|
3eqtrd |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐼 ) → ( 𝐸 ‘ ( 𝑉 ‘ 𝑥 ) ) = ( 𝐺 ‘ 𝑥 ) ) |
382 |
314 381
|
eqtrd |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐼 ) → ( ( 𝐸 ∘ 𝑉 ) ‘ 𝑥 ) = ( 𝐺 ‘ 𝑥 ) ) |
383 |
312 246 382
|
eqfnfvd |
⊢ ( 𝜑 → ( 𝐸 ∘ 𝑉 ) = 𝐺 ) |
384 |
293 307 383
|
3jca |
⊢ ( 𝜑 → ( 𝐸 ∈ ( 𝑃 RingHom 𝑆 ) ∧ ( 𝐸 ∘ 𝐴 ) = 𝐹 ∧ ( 𝐸 ∘ 𝑉 ) = 𝐺 ) ) |