Step | Hyp | Ref
| Expression |
1 | | ply1rem.g |
. . . 4
⊢ 𝐺 = (𝑋 − (𝐴‘𝑁)) |
2 | | ply1rem.1 |
. . . . . . . 8
⊢ (𝜑 → 𝑅 ∈ NzRing) |
3 | | nzrring 20299 |
. . . . . . . 8
⊢ (𝑅 ∈ NzRing → 𝑅 ∈ Ring) |
4 | 2, 3 | syl 17 |
. . . . . . 7
⊢ (𝜑 → 𝑅 ∈ Ring) |
5 | | ply1rem.p |
. . . . . . . 8
⊢ 𝑃 = (Poly1‘𝑅) |
6 | 5 | ply1ring 21169 |
. . . . . . 7
⊢ (𝑅 ∈ Ring → 𝑃 ∈ Ring) |
7 | 4, 6 | syl 17 |
. . . . . 6
⊢ (𝜑 → 𝑃 ∈ Ring) |
8 | | ringgrp 19567 |
. . . . . 6
⊢ (𝑃 ∈ Ring → 𝑃 ∈ Grp) |
9 | 7, 8 | syl 17 |
. . . . 5
⊢ (𝜑 → 𝑃 ∈ Grp) |
10 | | ply1rem.x |
. . . . . . 7
⊢ 𝑋 = (var1‘𝑅) |
11 | | ply1rem.b |
. . . . . . 7
⊢ 𝐵 = (Base‘𝑃) |
12 | 10, 5, 11 | vr1cl 21138 |
. . . . . 6
⊢ (𝑅 ∈ Ring → 𝑋 ∈ 𝐵) |
13 | 4, 12 | syl 17 |
. . . . 5
⊢ (𝜑 → 𝑋 ∈ 𝐵) |
14 | | ply1rem.a |
. . . . . . . 8
⊢ 𝐴 = (algSc‘𝑃) |
15 | | ply1rem.k |
. . . . . . . 8
⊢ 𝐾 = (Base‘𝑅) |
16 | 5, 14, 15, 11 | ply1sclf 21206 |
. . . . . . 7
⊢ (𝑅 ∈ Ring → 𝐴:𝐾⟶𝐵) |
17 | 4, 16 | syl 17 |
. . . . . 6
⊢ (𝜑 → 𝐴:𝐾⟶𝐵) |
18 | | ply1rem.3 |
. . . . . 6
⊢ (𝜑 → 𝑁 ∈ 𝐾) |
19 | 17, 18 | ffvelrnd 6905 |
. . . . 5
⊢ (𝜑 → (𝐴‘𝑁) ∈ 𝐵) |
20 | | ply1rem.m |
. . . . . 6
⊢ − =
(-g‘𝑃) |
21 | 11, 20 | grpsubcl 18443 |
. . . . 5
⊢ ((𝑃 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ (𝐴‘𝑁) ∈ 𝐵) → (𝑋 − (𝐴‘𝑁)) ∈ 𝐵) |
22 | 9, 13, 19, 21 | syl3anc 1373 |
. . . 4
⊢ (𝜑 → (𝑋 − (𝐴‘𝑁)) ∈ 𝐵) |
23 | 1, 22 | eqeltrid 2842 |
. . 3
⊢ (𝜑 → 𝐺 ∈ 𝐵) |
24 | 1 | fveq2i 6720 |
. . . . . . 7
⊢ (𝐷‘𝐺) = (𝐷‘(𝑋 − (𝐴‘𝑁))) |
25 | | ply1rem.d |
. . . . . . . 8
⊢ 𝐷 = ( deg1
‘𝑅) |
26 | 25, 5, 11 | deg1xrcl 24980 |
. . . . . . . . . . 11
⊢ ((𝐴‘𝑁) ∈ 𝐵 → (𝐷‘(𝐴‘𝑁)) ∈
ℝ*) |
27 | 19, 26 | syl 17 |
. . . . . . . . . 10
⊢ (𝜑 → (𝐷‘(𝐴‘𝑁)) ∈
ℝ*) |
28 | | 0xr 10880 |
. . . . . . . . . . 11
⊢ 0 ∈
ℝ* |
29 | 28 | a1i 11 |
. . . . . . . . . 10
⊢ (𝜑 → 0 ∈
ℝ*) |
30 | | 1re 10833 |
. . . . . . . . . . 11
⊢ 1 ∈
ℝ |
31 | | rexr 10879 |
. . . . . . . . . . 11
⊢ (1 ∈
ℝ → 1 ∈ ℝ*) |
32 | 30, 31 | mp1i 13 |
. . . . . . . . . 10
⊢ (𝜑 → 1 ∈
ℝ*) |
33 | 25, 5, 15, 14 | deg1sclle 25010 |
. . . . . . . . . . 11
⊢ ((𝑅 ∈ Ring ∧ 𝑁 ∈ 𝐾) → (𝐷‘(𝐴‘𝑁)) ≤ 0) |
34 | 4, 18, 33 | syl2anc 587 |
. . . . . . . . . 10
⊢ (𝜑 → (𝐷‘(𝐴‘𝑁)) ≤ 0) |
35 | | 0lt1 11354 |
. . . . . . . . . . 11
⊢ 0 <
1 |
36 | 35 | a1i 11 |
. . . . . . . . . 10
⊢ (𝜑 → 0 < 1) |
37 | 27, 29, 32, 34, 36 | xrlelttrd 12750 |
. . . . . . . . 9
⊢ (𝜑 → (𝐷‘(𝐴‘𝑁)) < 1) |
38 | | eqid 2737 |
. . . . . . . . . . . . . 14
⊢
(mulGrp‘𝑃) =
(mulGrp‘𝑃) |
39 | 38, 11 | mgpbas 19510 |
. . . . . . . . . . . . 13
⊢ 𝐵 =
(Base‘(mulGrp‘𝑃)) |
40 | | eqid 2737 |
. . . . . . . . . . . . 13
⊢
(.g‘(mulGrp‘𝑃)) =
(.g‘(mulGrp‘𝑃)) |
41 | 39, 40 | mulg1 18499 |
. . . . . . . . . . . 12
⊢ (𝑋 ∈ 𝐵 →
(1(.g‘(mulGrp‘𝑃))𝑋) = 𝑋) |
42 | 13, 41 | syl 17 |
. . . . . . . . . . 11
⊢ (𝜑 →
(1(.g‘(mulGrp‘𝑃))𝑋) = 𝑋) |
43 | 42 | fveq2d 6721 |
. . . . . . . . . 10
⊢ (𝜑 → (𝐷‘(1(.g‘(mulGrp‘𝑃))𝑋)) = (𝐷‘𝑋)) |
44 | | 1nn0 12106 |
. . . . . . . . . . 11
⊢ 1 ∈
ℕ0 |
45 | 25, 5, 10, 38, 40 | deg1pw 25018 |
. . . . . . . . . . 11
⊢ ((𝑅 ∈ NzRing ∧ 1 ∈
ℕ0) → (𝐷‘(1(.g‘(mulGrp‘𝑃))𝑋)) = 1) |
46 | 2, 44, 45 | sylancl 589 |
. . . . . . . . . 10
⊢ (𝜑 → (𝐷‘(1(.g‘(mulGrp‘𝑃))𝑋)) = 1) |
47 | 43, 46 | eqtr3d 2779 |
. . . . . . . . 9
⊢ (𝜑 → (𝐷‘𝑋) = 1) |
48 | 37, 47 | breqtrrd 5081 |
. . . . . . . 8
⊢ (𝜑 → (𝐷‘(𝐴‘𝑁)) < (𝐷‘𝑋)) |
49 | 5, 25, 4, 11, 20, 13, 19, 48 | deg1sub 25006 |
. . . . . . 7
⊢ (𝜑 → (𝐷‘(𝑋 − (𝐴‘𝑁))) = (𝐷‘𝑋)) |
50 | 24, 49 | syl5eq 2790 |
. . . . . 6
⊢ (𝜑 → (𝐷‘𝐺) = (𝐷‘𝑋)) |
51 | 50, 47 | eqtrd 2777 |
. . . . 5
⊢ (𝜑 → (𝐷‘𝐺) = 1) |
52 | 51, 44 | eqeltrdi 2846 |
. . . 4
⊢ (𝜑 → (𝐷‘𝐺) ∈
ℕ0) |
53 | | eqid 2737 |
. . . . . 6
⊢
(0g‘𝑃) = (0g‘𝑃) |
54 | 25, 5, 53, 11 | deg1nn0clb 24988 |
. . . . 5
⊢ ((𝑅 ∈ Ring ∧ 𝐺 ∈ 𝐵) → (𝐺 ≠ (0g‘𝑃) ↔ (𝐷‘𝐺) ∈
ℕ0)) |
55 | 4, 23, 54 | syl2anc 587 |
. . . 4
⊢ (𝜑 → (𝐺 ≠ (0g‘𝑃) ↔ (𝐷‘𝐺) ∈
ℕ0)) |
56 | 52, 55 | mpbird 260 |
. . 3
⊢ (𝜑 → 𝐺 ≠ (0g‘𝑃)) |
57 | 51 | fveq2d 6721 |
. . . 4
⊢ (𝜑 →
((coe1‘𝐺)‘(𝐷‘𝐺)) = ((coe1‘𝐺)‘1)) |
58 | 1 | fveq2i 6720 |
. . . . . 6
⊢
(coe1‘𝐺) = (coe1‘(𝑋 − (𝐴‘𝑁))) |
59 | 58 | fveq1i 6718 |
. . . . 5
⊢
((coe1‘𝐺)‘1) = ((coe1‘(𝑋 − (𝐴‘𝑁)))‘1) |
60 | 44 | a1i 11 |
. . . . . 6
⊢ (𝜑 → 1 ∈
ℕ0) |
61 | | eqid 2737 |
. . . . . . 7
⊢
(-g‘𝑅) = (-g‘𝑅) |
62 | 5, 11, 20, 61 | coe1subfv 21187 |
. . . . . 6
⊢ (((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵 ∧ (𝐴‘𝑁) ∈ 𝐵) ∧ 1 ∈ ℕ0) →
((coe1‘(𝑋
−
(𝐴‘𝑁)))‘1) =
(((coe1‘𝑋)‘1)(-g‘𝑅)((coe1‘(𝐴‘𝑁))‘1))) |
63 | 4, 13, 19, 60, 62 | syl31anc 1375 |
. . . . 5
⊢ (𝜑 →
((coe1‘(𝑋
−
(𝐴‘𝑁)))‘1) =
(((coe1‘𝑋)‘1)(-g‘𝑅)((coe1‘(𝐴‘𝑁))‘1))) |
64 | 59, 63 | syl5eq 2790 |
. . . 4
⊢ (𝜑 →
((coe1‘𝐺)‘1) = (((coe1‘𝑋)‘1)(-g‘𝑅)((coe1‘(𝐴‘𝑁))‘1))) |
65 | 42 | oveq2d 7229 |
. . . . . . . . . 10
⊢ (𝜑 →
((1r‘𝑅)(
·𝑠 ‘𝑃)(1(.g‘(mulGrp‘𝑃))𝑋)) = ((1r‘𝑅)( ·𝑠
‘𝑃)𝑋)) |
66 | 5 | ply1sca 21174 |
. . . . . . . . . . . . 13
⊢ (𝑅 ∈ NzRing → 𝑅 = (Scalar‘𝑃)) |
67 | 2, 66 | syl 17 |
. . . . . . . . . . . 12
⊢ (𝜑 → 𝑅 = (Scalar‘𝑃)) |
68 | 67 | fveq2d 6721 |
. . . . . . . . . . 11
⊢ (𝜑 → (1r‘𝑅) =
(1r‘(Scalar‘𝑃))) |
69 | 68 | oveq1d 7228 |
. . . . . . . . . 10
⊢ (𝜑 →
((1r‘𝑅)(
·𝑠 ‘𝑃)𝑋) =
((1r‘(Scalar‘𝑃))( ·𝑠
‘𝑃)𝑋)) |
70 | 5 | ply1lmod 21173 |
. . . . . . . . . . . 12
⊢ (𝑅 ∈ Ring → 𝑃 ∈ LMod) |
71 | 4, 70 | syl 17 |
. . . . . . . . . . 11
⊢ (𝜑 → 𝑃 ∈ LMod) |
72 | | eqid 2737 |
. . . . . . . . . . . 12
⊢
(Scalar‘𝑃) =
(Scalar‘𝑃) |
73 | | eqid 2737 |
. . . . . . . . . . . 12
⊢ (
·𝑠 ‘𝑃) = ( ·𝑠
‘𝑃) |
74 | | eqid 2737 |
. . . . . . . . . . . 12
⊢
(1r‘(Scalar‘𝑃)) =
(1r‘(Scalar‘𝑃)) |
75 | 11, 72, 73, 74 | lmodvs1 19927 |
. . . . . . . . . . 11
⊢ ((𝑃 ∈ LMod ∧ 𝑋 ∈ 𝐵) →
((1r‘(Scalar‘𝑃))( ·𝑠
‘𝑃)𝑋) = 𝑋) |
76 | 71, 13, 75 | syl2anc 587 |
. . . . . . . . . 10
⊢ (𝜑 →
((1r‘(Scalar‘𝑃))( ·𝑠
‘𝑃)𝑋) = 𝑋) |
77 | 65, 69, 76 | 3eqtrd 2781 |
. . . . . . . . 9
⊢ (𝜑 →
((1r‘𝑅)(
·𝑠 ‘𝑃)(1(.g‘(mulGrp‘𝑃))𝑋)) = 𝑋) |
78 | 77 | fveq2d 6721 |
. . . . . . . 8
⊢ (𝜑 →
(coe1‘((1r‘𝑅)( ·𝑠
‘𝑃)(1(.g‘(mulGrp‘𝑃))𝑋))) = (coe1‘𝑋)) |
79 | 78 | fveq1d 6719 |
. . . . . . 7
⊢ (𝜑 →
((coe1‘((1r‘𝑅)( ·𝑠
‘𝑃)(1(.g‘(mulGrp‘𝑃))𝑋)))‘1) = ((coe1‘𝑋)‘1)) |
80 | | eqid 2737 |
. . . . . . . . . 10
⊢
(1r‘𝑅) = (1r‘𝑅) |
81 | 15, 80 | ringidcl 19586 |
. . . . . . . . 9
⊢ (𝑅 ∈ Ring →
(1r‘𝑅)
∈ 𝐾) |
82 | 4, 81 | syl 17 |
. . . . . . . 8
⊢ (𝜑 → (1r‘𝑅) ∈ 𝐾) |
83 | | ply1rem.z |
. . . . . . . . 9
⊢ 0 =
(0g‘𝑅) |
84 | 83, 15, 5, 10, 73, 38, 40 | coe1tmfv1 21195 |
. . . . . . . 8
⊢ ((𝑅 ∈ Ring ∧
(1r‘𝑅)
∈ 𝐾 ∧ 1 ∈
ℕ0) → ((coe1‘((1r‘𝑅)(
·𝑠 ‘𝑃)(1(.g‘(mulGrp‘𝑃))𝑋)))‘1) = (1r‘𝑅)) |
85 | 4, 82, 60, 84 | syl3anc 1373 |
. . . . . . 7
⊢ (𝜑 →
((coe1‘((1r‘𝑅)( ·𝑠
‘𝑃)(1(.g‘(mulGrp‘𝑃))𝑋)))‘1) = (1r‘𝑅)) |
86 | 79, 85 | eqtr3d 2779 |
. . . . . 6
⊢ (𝜑 →
((coe1‘𝑋)‘1) = (1r‘𝑅)) |
87 | | eqid 2737 |
. . . . . . . . . 10
⊢
(0g‘𝑅) = (0g‘𝑅) |
88 | 5, 14, 15, 87 | coe1scl 21208 |
. . . . . . . . 9
⊢ ((𝑅 ∈ Ring ∧ 𝑁 ∈ 𝐾) → (coe1‘(𝐴‘𝑁)) = (𝑥 ∈ ℕ0 ↦ if(𝑥 = 0, 𝑁, (0g‘𝑅)))) |
89 | 4, 18, 88 | syl2anc 587 |
. . . . . . . 8
⊢ (𝜑 →
(coe1‘(𝐴‘𝑁)) = (𝑥 ∈ ℕ0 ↦ if(𝑥 = 0, 𝑁, (0g‘𝑅)))) |
90 | 89 | fveq1d 6719 |
. . . . . . 7
⊢ (𝜑 →
((coe1‘(𝐴‘𝑁))‘1) = ((𝑥 ∈ ℕ0 ↦ if(𝑥 = 0, 𝑁, (0g‘𝑅)))‘1)) |
91 | | ax-1ne0 10798 |
. . . . . . . . . . 11
⊢ 1 ≠
0 |
92 | | neeq1 3003 |
. . . . . . . . . . 11
⊢ (𝑥 = 1 → (𝑥 ≠ 0 ↔ 1 ≠ 0)) |
93 | 91, 92 | mpbiri 261 |
. . . . . . . . . 10
⊢ (𝑥 = 1 → 𝑥 ≠ 0) |
94 | | ifnefalse 4451 |
. . . . . . . . . 10
⊢ (𝑥 ≠ 0 → if(𝑥 = 0, 𝑁, (0g‘𝑅)) = (0g‘𝑅)) |
95 | 93, 94 | syl 17 |
. . . . . . . . 9
⊢ (𝑥 = 1 → if(𝑥 = 0, 𝑁, (0g‘𝑅)) = (0g‘𝑅)) |
96 | | eqid 2737 |
. . . . . . . . 9
⊢ (𝑥 ∈ ℕ0
↦ if(𝑥 = 0, 𝑁, (0g‘𝑅))) = (𝑥 ∈ ℕ0 ↦ if(𝑥 = 0, 𝑁, (0g‘𝑅))) |
97 | | fvex 6730 |
. . . . . . . . 9
⊢
(0g‘𝑅) ∈ V |
98 | 95, 96, 97 | fvmpt 6818 |
. . . . . . . 8
⊢ (1 ∈
ℕ0 → ((𝑥 ∈ ℕ0 ↦ if(𝑥 = 0, 𝑁, (0g‘𝑅)))‘1) = (0g‘𝑅)) |
99 | 44, 98 | ax-mp 5 |
. . . . . . 7
⊢ ((𝑥 ∈ ℕ0
↦ if(𝑥 = 0, 𝑁, (0g‘𝑅)))‘1) =
(0g‘𝑅) |
100 | 90, 99 | eqtrdi 2794 |
. . . . . 6
⊢ (𝜑 →
((coe1‘(𝐴‘𝑁))‘1) = (0g‘𝑅)) |
101 | 86, 100 | oveq12d 7231 |
. . . . 5
⊢ (𝜑 →
(((coe1‘𝑋)‘1)(-g‘𝑅)((coe1‘(𝐴‘𝑁))‘1)) = ((1r‘𝑅)(-g‘𝑅)(0g‘𝑅))) |
102 | | ringgrp 19567 |
. . . . . . 7
⊢ (𝑅 ∈ Ring → 𝑅 ∈ Grp) |
103 | 4, 102 | syl 17 |
. . . . . 6
⊢ (𝜑 → 𝑅 ∈ Grp) |
104 | 15, 87, 61 | grpsubid1 18448 |
. . . . . 6
⊢ ((𝑅 ∈ Grp ∧
(1r‘𝑅)
∈ 𝐾) →
((1r‘𝑅)(-g‘𝑅)(0g‘𝑅)) = (1r‘𝑅)) |
105 | 103, 82, 104 | syl2anc 587 |
. . . . 5
⊢ (𝜑 →
((1r‘𝑅)(-g‘𝑅)(0g‘𝑅)) = (1r‘𝑅)) |
106 | 101, 105 | eqtrd 2777 |
. . . 4
⊢ (𝜑 →
(((coe1‘𝑋)‘1)(-g‘𝑅)((coe1‘(𝐴‘𝑁))‘1)) = (1r‘𝑅)) |
107 | 57, 64, 106 | 3eqtrd 2781 |
. . 3
⊢ (𝜑 →
((coe1‘𝐺)‘(𝐷‘𝐺)) = (1r‘𝑅)) |
108 | | ply1rem.u |
. . . 4
⊢ 𝑈 =
(Monic1p‘𝑅) |
109 | 5, 11, 53, 25, 108, 80 | ismon1p 25040 |
. . 3
⊢ (𝐺 ∈ 𝑈 ↔ (𝐺 ∈ 𝐵 ∧ 𝐺 ≠ (0g‘𝑃) ∧ ((coe1‘𝐺)‘(𝐷‘𝐺)) = (1r‘𝑅))) |
110 | 23, 56, 107, 109 | syl3anbrc 1345 |
. 2
⊢ (𝜑 → 𝐺 ∈ 𝑈) |
111 | 1 | fveq2i 6720 |
. . . . . . . . . 10
⊢ (𝑂‘𝐺) = (𝑂‘(𝑋 − (𝐴‘𝑁))) |
112 | 111 | fveq1i 6718 |
. . . . . . . . 9
⊢ ((𝑂‘𝐺)‘𝑥) = ((𝑂‘(𝑋 − (𝐴‘𝑁)))‘𝑥) |
113 | | ply1rem.o |
. . . . . . . . . . 11
⊢ 𝑂 = (eval1‘𝑅) |
114 | | ply1rem.2 |
. . . . . . . . . . . 12
⊢ (𝜑 → 𝑅 ∈ CRing) |
115 | 114 | adantr 484 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐾) → 𝑅 ∈ CRing) |
116 | | simpr 488 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐾) → 𝑥 ∈ 𝐾) |
117 | 113, 10, 15, 5, 11, 115, 116 | evl1vard 21253 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐾) → (𝑋 ∈ 𝐵 ∧ ((𝑂‘𝑋)‘𝑥) = 𝑥)) |
118 | 18 | adantr 484 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐾) → 𝑁 ∈ 𝐾) |
119 | 113, 5, 15, 14, 11, 115, 118, 116 | evl1scad 21251 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐾) → ((𝐴‘𝑁) ∈ 𝐵 ∧ ((𝑂‘(𝐴‘𝑁))‘𝑥) = 𝑁)) |
120 | 113, 5, 15, 11, 115, 116, 117, 119, 20, 61 | evl1subd 21258 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐾) → ((𝑋 − (𝐴‘𝑁)) ∈ 𝐵 ∧ ((𝑂‘(𝑋 − (𝐴‘𝑁)))‘𝑥) = (𝑥(-g‘𝑅)𝑁))) |
121 | 120 | simprd 499 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐾) → ((𝑂‘(𝑋 − (𝐴‘𝑁)))‘𝑥) = (𝑥(-g‘𝑅)𝑁)) |
122 | 112, 121 | syl5eq 2790 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐾) → ((𝑂‘𝐺)‘𝑥) = (𝑥(-g‘𝑅)𝑁)) |
123 | 122 | eqeq1d 2739 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐾) → (((𝑂‘𝐺)‘𝑥) = 0 ↔ (𝑥(-g‘𝑅)𝑁) = 0 )) |
124 | 103 | adantr 484 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐾) → 𝑅 ∈ Grp) |
125 | 15, 83, 61 | grpsubeq0 18449 |
. . . . . . . 8
⊢ ((𝑅 ∈ Grp ∧ 𝑥 ∈ 𝐾 ∧ 𝑁 ∈ 𝐾) → ((𝑥(-g‘𝑅)𝑁) = 0 ↔ 𝑥 = 𝑁)) |
126 | 124, 116,
118, 125 | syl3anc 1373 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐾) → ((𝑥(-g‘𝑅)𝑁) = 0 ↔ 𝑥 = 𝑁)) |
127 | 123, 126 | bitrd 282 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐾) → (((𝑂‘𝐺)‘𝑥) = 0 ↔ 𝑥 = 𝑁)) |
128 | | velsn 4557 |
. . . . . 6
⊢ (𝑥 ∈ {𝑁} ↔ 𝑥 = 𝑁) |
129 | 127, 128 | bitr4di 292 |
. . . . 5
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐾) → (((𝑂‘𝐺)‘𝑥) = 0 ↔ 𝑥 ∈ {𝑁})) |
130 | 129 | pm5.32da 582 |
. . . 4
⊢ (𝜑 → ((𝑥 ∈ 𝐾 ∧ ((𝑂‘𝐺)‘𝑥) = 0 ) ↔ (𝑥 ∈ 𝐾 ∧ 𝑥 ∈ {𝑁}))) |
131 | | eqid 2737 |
. . . . . . 7
⊢ (𝑅 ↑s 𝐾) = (𝑅 ↑s 𝐾) |
132 | | eqid 2737 |
. . . . . . 7
⊢
(Base‘(𝑅
↑s 𝐾)) = (Base‘(𝑅 ↑s 𝐾)) |
133 | 15 | fvexi 6731 |
. . . . . . . 8
⊢ 𝐾 ∈ V |
134 | 133 | a1i 11 |
. . . . . . 7
⊢ (𝜑 → 𝐾 ∈ V) |
135 | 113, 5, 131, 15 | evl1rhm 21248 |
. . . . . . . . . 10
⊢ (𝑅 ∈ CRing → 𝑂 ∈ (𝑃 RingHom (𝑅 ↑s 𝐾))) |
136 | 114, 135 | syl 17 |
. . . . . . . . 9
⊢ (𝜑 → 𝑂 ∈ (𝑃 RingHom (𝑅 ↑s 𝐾))) |
137 | 11, 132 | rhmf 19746 |
. . . . . . . . 9
⊢ (𝑂 ∈ (𝑃 RingHom (𝑅 ↑s 𝐾)) → 𝑂:𝐵⟶(Base‘(𝑅 ↑s 𝐾))) |
138 | 136, 137 | syl 17 |
. . . . . . . 8
⊢ (𝜑 → 𝑂:𝐵⟶(Base‘(𝑅 ↑s 𝐾))) |
139 | 138, 23 | ffvelrnd 6905 |
. . . . . . 7
⊢ (𝜑 → (𝑂‘𝐺) ∈ (Base‘(𝑅 ↑s 𝐾))) |
140 | 131, 15, 132, 2, 134, 139 | pwselbas 16994 |
. . . . . 6
⊢ (𝜑 → (𝑂‘𝐺):𝐾⟶𝐾) |
141 | 140 | ffnd 6546 |
. . . . 5
⊢ (𝜑 → (𝑂‘𝐺) Fn 𝐾) |
142 | | fniniseg 6880 |
. . . . 5
⊢ ((𝑂‘𝐺) Fn 𝐾 → (𝑥 ∈ (◡(𝑂‘𝐺) “ { 0 }) ↔ (𝑥 ∈ 𝐾 ∧ ((𝑂‘𝐺)‘𝑥) = 0 ))) |
143 | 141, 142 | syl 17 |
. . . 4
⊢ (𝜑 → (𝑥 ∈ (◡(𝑂‘𝐺) “ { 0 }) ↔ (𝑥 ∈ 𝐾 ∧ ((𝑂‘𝐺)‘𝑥) = 0 ))) |
144 | 18 | snssd 4722 |
. . . . . 6
⊢ (𝜑 → {𝑁} ⊆ 𝐾) |
145 | 144 | sseld 3900 |
. . . . 5
⊢ (𝜑 → (𝑥 ∈ {𝑁} → 𝑥 ∈ 𝐾)) |
146 | 145 | pm4.71rd 566 |
. . . 4
⊢ (𝜑 → (𝑥 ∈ {𝑁} ↔ (𝑥 ∈ 𝐾 ∧ 𝑥 ∈ {𝑁}))) |
147 | 130, 143,
146 | 3bitr4d 314 |
. . 3
⊢ (𝜑 → (𝑥 ∈ (◡(𝑂‘𝐺) “ { 0 }) ↔ 𝑥 ∈ {𝑁})) |
148 | 147 | eqrdv 2735 |
. 2
⊢ (𝜑 → (◡(𝑂‘𝐺) “ { 0 }) = {𝑁}) |
149 | 110, 51, 148 | 3jca 1130 |
1
⊢ (𝜑 → (𝐺 ∈ 𝑈 ∧ (𝐷‘𝐺) = 1 ∧ (◡(𝑂‘𝐺) “ { 0 }) = {𝑁})) |