Step |
Hyp |
Ref |
Expression |
1 |
|
rmodislmod.v |
⊢ 𝑉 = ( Base ‘ 𝑅 ) |
2 |
|
rmodislmod.a |
⊢ + = ( +g ‘ 𝑅 ) |
3 |
|
rmodislmod.s |
⊢ · = ( ·𝑠 ‘ 𝑅 ) |
4 |
|
rmodislmod.f |
⊢ 𝐹 = ( Scalar ‘ 𝑅 ) |
5 |
|
rmodislmod.k |
⊢ 𝐾 = ( Base ‘ 𝐹 ) |
6 |
|
rmodislmod.p |
⊢ ⨣ = ( +g ‘ 𝐹 ) |
7 |
|
rmodislmod.t |
⊢ × = ( .r ‘ 𝐹 ) |
8 |
|
rmodislmod.u |
⊢ 1 = ( 1r ‘ 𝐹 ) |
9 |
|
rmodislmod.r |
⊢ ( 𝑅 ∈ Grp ∧ 𝐹 ∈ Ring ∧ ∀ 𝑞 ∈ 𝐾 ∀ 𝑟 ∈ 𝐾 ∀ 𝑥 ∈ 𝑉 ∀ 𝑤 ∈ 𝑉 ( ( ( 𝑤 · 𝑟 ) ∈ 𝑉 ∧ ( ( 𝑤 + 𝑥 ) · 𝑟 ) = ( ( 𝑤 · 𝑟 ) + ( 𝑥 · 𝑟 ) ) ∧ ( 𝑤 · ( 𝑞 ⨣ 𝑟 ) ) = ( ( 𝑤 · 𝑞 ) + ( 𝑤 · 𝑟 ) ) ) ∧ ( ( 𝑤 · ( 𝑞 × 𝑟 ) ) = ( ( 𝑤 · 𝑞 ) · 𝑟 ) ∧ ( 𝑤 · 1 ) = 𝑤 ) ) ) |
10 |
|
rmodislmod.m |
⊢ ∗ = ( 𝑠 ∈ 𝐾 , 𝑣 ∈ 𝑉 ↦ ( 𝑣 · 𝑠 ) ) |
11 |
|
rmodislmod.l |
⊢ 𝐿 = ( 𝑅 sSet 〈 ( ·𝑠 ‘ ndx ) , ∗ 〉 ) |
12 |
|
baseid |
⊢ Base = Slot ( Base ‘ ndx ) |
13 |
|
df-base |
⊢ Base = Slot 1 |
14 |
|
1nn |
⊢ 1 ∈ ℕ |
15 |
13 14
|
ndxarg |
⊢ ( Base ‘ ndx ) = 1 |
16 |
|
1re |
⊢ 1 ∈ ℝ |
17 |
|
1lt6 |
⊢ 1 < 6 |
18 |
16 17
|
ltneii |
⊢ 1 ≠ 6 |
19 |
|
vscandx |
⊢ ( ·𝑠 ‘ ndx ) = 6 |
20 |
18 19
|
neeqtrri |
⊢ 1 ≠ ( ·𝑠 ‘ ndx ) |
21 |
15 20
|
eqnetri |
⊢ ( Base ‘ ndx ) ≠ ( ·𝑠 ‘ ndx ) |
22 |
12 21
|
setsnid |
⊢ ( Base ‘ 𝑅 ) = ( Base ‘ ( 𝑅 sSet 〈 ( ·𝑠 ‘ ndx ) , ∗ 〉 ) ) |
23 |
1 22
|
eqtri |
⊢ 𝑉 = ( Base ‘ ( 𝑅 sSet 〈 ( ·𝑠 ‘ ndx ) , ∗ 〉 ) ) |
24 |
11
|
eqcomi |
⊢ ( 𝑅 sSet 〈 ( ·𝑠 ‘ ndx ) , ∗ 〉 ) = 𝐿 |
25 |
24
|
fveq2i |
⊢ ( Base ‘ ( 𝑅 sSet 〈 ( ·𝑠 ‘ ndx ) , ∗ 〉 ) ) = ( Base ‘ 𝐿 ) |
26 |
23 25
|
eqtri |
⊢ 𝑉 = ( Base ‘ 𝐿 ) |
27 |
26
|
a1i |
⊢ ( 𝐹 ∈ CRing → 𝑉 = ( Base ‘ 𝐿 ) ) |
28 |
|
plusgid |
⊢ +g = Slot ( +g ‘ ndx ) |
29 |
|
plusgndx |
⊢ ( +g ‘ ndx ) = 2 |
30 |
|
2re |
⊢ 2 ∈ ℝ |
31 |
|
2lt6 |
⊢ 2 < 6 |
32 |
30 31
|
ltneii |
⊢ 2 ≠ 6 |
33 |
32 19
|
neeqtrri |
⊢ 2 ≠ ( ·𝑠 ‘ ndx ) |
34 |
29 33
|
eqnetri |
⊢ ( +g ‘ ndx ) ≠ ( ·𝑠 ‘ ndx ) |
35 |
28 34
|
setsnid |
⊢ ( +g ‘ 𝑅 ) = ( +g ‘ ( 𝑅 sSet 〈 ( ·𝑠 ‘ ndx ) , ∗ 〉 ) ) |
36 |
11
|
fveq2i |
⊢ ( +g ‘ 𝐿 ) = ( +g ‘ ( 𝑅 sSet 〈 ( ·𝑠 ‘ ndx ) , ∗ 〉 ) ) |
37 |
35 2 36
|
3eqtr4i |
⊢ + = ( +g ‘ 𝐿 ) |
38 |
37
|
a1i |
⊢ ( 𝐹 ∈ CRing → + = ( +g ‘ 𝐿 ) ) |
39 |
|
scaid |
⊢ Scalar = Slot ( Scalar ‘ ndx ) |
40 |
|
scandx |
⊢ ( Scalar ‘ ndx ) = 5 |
41 |
|
5re |
⊢ 5 ∈ ℝ |
42 |
|
5lt6 |
⊢ 5 < 6 |
43 |
41 42
|
ltneii |
⊢ 5 ≠ 6 |
44 |
43 19
|
neeqtrri |
⊢ 5 ≠ ( ·𝑠 ‘ ndx ) |
45 |
40 44
|
eqnetri |
⊢ ( Scalar ‘ ndx ) ≠ ( ·𝑠 ‘ ndx ) |
46 |
39 45
|
setsnid |
⊢ ( Scalar ‘ 𝑅 ) = ( Scalar ‘ ( 𝑅 sSet 〈 ( ·𝑠 ‘ ndx ) , ∗ 〉 ) ) |
47 |
11
|
fveq2i |
⊢ ( Scalar ‘ 𝐿 ) = ( Scalar ‘ ( 𝑅 sSet 〈 ( ·𝑠 ‘ ndx ) , ∗ 〉 ) ) |
48 |
46 4 47
|
3eqtr4i |
⊢ 𝐹 = ( Scalar ‘ 𝐿 ) |
49 |
48
|
a1i |
⊢ ( 𝐹 ∈ CRing → 𝐹 = ( Scalar ‘ 𝐿 ) ) |
50 |
9
|
simp1i |
⊢ 𝑅 ∈ Grp |
51 |
5
|
fvexi |
⊢ 𝐾 ∈ V |
52 |
1
|
fvexi |
⊢ 𝑉 ∈ V |
53 |
10
|
mpoexg |
⊢ ( ( 𝐾 ∈ V ∧ 𝑉 ∈ V ) → ∗ ∈ V ) |
54 |
51 52 53
|
mp2an |
⊢ ∗ ∈ V |
55 |
|
vscaid |
⊢ ·𝑠 = Slot ( ·𝑠 ‘ ndx ) |
56 |
55
|
setsid |
⊢ ( ( 𝑅 ∈ Grp ∧ ∗ ∈ V ) → ∗ = ( ·𝑠 ‘ ( 𝑅 sSet 〈 ( ·𝑠 ‘ ndx ) , ∗ 〉 ) ) ) |
57 |
50 54 56
|
mp2an |
⊢ ∗ = ( ·𝑠 ‘ ( 𝑅 sSet 〈 ( ·𝑠 ‘ ndx ) , ∗ 〉 ) ) |
58 |
24
|
fveq2i |
⊢ ( ·𝑠 ‘ ( 𝑅 sSet 〈 ( ·𝑠 ‘ ndx ) , ∗ 〉 ) ) = ( ·𝑠 ‘ 𝐿 ) |
59 |
57 58
|
eqtri |
⊢ ∗ = ( ·𝑠 ‘ 𝐿 ) |
60 |
59
|
a1i |
⊢ ( 𝐹 ∈ CRing → ∗ = ( ·𝑠 ‘ 𝐿 ) ) |
61 |
5
|
a1i |
⊢ ( 𝐹 ∈ CRing → 𝐾 = ( Base ‘ 𝐹 ) ) |
62 |
6
|
a1i |
⊢ ( 𝐹 ∈ CRing → ⨣ = ( +g ‘ 𝐹 ) ) |
63 |
7
|
a1i |
⊢ ( 𝐹 ∈ CRing → × = ( .r ‘ 𝐹 ) ) |
64 |
8
|
a1i |
⊢ ( 𝐹 ∈ CRing → 1 = ( 1r ‘ 𝐹 ) ) |
65 |
|
crngring |
⊢ ( 𝐹 ∈ CRing → 𝐹 ∈ Ring ) |
66 |
1
|
eqcomi |
⊢ ( Base ‘ 𝑅 ) = 𝑉 |
67 |
66 26
|
eqtri |
⊢ ( Base ‘ 𝑅 ) = ( Base ‘ 𝐿 ) |
68 |
35 36
|
eqtr4i |
⊢ ( +g ‘ 𝑅 ) = ( +g ‘ 𝐿 ) |
69 |
67 68
|
grpprop |
⊢ ( 𝑅 ∈ Grp ↔ 𝐿 ∈ Grp ) |
70 |
50 69
|
mpbi |
⊢ 𝐿 ∈ Grp |
71 |
70
|
a1i |
⊢ ( 𝐹 ∈ CRing → 𝐿 ∈ Grp ) |
72 |
10
|
a1i |
⊢ ( ( 𝐹 ∈ CRing ∧ 𝑎 ∈ 𝐾 ∧ 𝑏 ∈ 𝑉 ) → ∗ = ( 𝑠 ∈ 𝐾 , 𝑣 ∈ 𝑉 ↦ ( 𝑣 · 𝑠 ) ) ) |
73 |
|
oveq12 |
⊢ ( ( 𝑣 = 𝑏 ∧ 𝑠 = 𝑎 ) → ( 𝑣 · 𝑠 ) = ( 𝑏 · 𝑎 ) ) |
74 |
73
|
ancoms |
⊢ ( ( 𝑠 = 𝑎 ∧ 𝑣 = 𝑏 ) → ( 𝑣 · 𝑠 ) = ( 𝑏 · 𝑎 ) ) |
75 |
74
|
adantl |
⊢ ( ( ( 𝐹 ∈ CRing ∧ 𝑎 ∈ 𝐾 ∧ 𝑏 ∈ 𝑉 ) ∧ ( 𝑠 = 𝑎 ∧ 𝑣 = 𝑏 ) ) → ( 𝑣 · 𝑠 ) = ( 𝑏 · 𝑎 ) ) |
76 |
|
simp2 |
⊢ ( ( 𝐹 ∈ CRing ∧ 𝑎 ∈ 𝐾 ∧ 𝑏 ∈ 𝑉 ) → 𝑎 ∈ 𝐾 ) |
77 |
|
simp3 |
⊢ ( ( 𝐹 ∈ CRing ∧ 𝑎 ∈ 𝐾 ∧ 𝑏 ∈ 𝑉 ) → 𝑏 ∈ 𝑉 ) |
78 |
|
ovexd |
⊢ ( ( 𝐹 ∈ CRing ∧ 𝑎 ∈ 𝐾 ∧ 𝑏 ∈ 𝑉 ) → ( 𝑏 · 𝑎 ) ∈ V ) |
79 |
72 75 76 77 78
|
ovmpod |
⊢ ( ( 𝐹 ∈ CRing ∧ 𝑎 ∈ 𝐾 ∧ 𝑏 ∈ 𝑉 ) → ( 𝑎 ∗ 𝑏 ) = ( 𝑏 · 𝑎 ) ) |
80 |
|
simpl1 |
⊢ ( ( ( ( 𝑤 · 𝑟 ) ∈ 𝑉 ∧ ( ( 𝑤 + 𝑥 ) · 𝑟 ) = ( ( 𝑤 · 𝑟 ) + ( 𝑥 · 𝑟 ) ) ∧ ( 𝑤 · ( 𝑞 ⨣ 𝑟 ) ) = ( ( 𝑤 · 𝑞 ) + ( 𝑤 · 𝑟 ) ) ) ∧ ( ( 𝑤 · ( 𝑞 × 𝑟 ) ) = ( ( 𝑤 · 𝑞 ) · 𝑟 ) ∧ ( 𝑤 · 1 ) = 𝑤 ) ) → ( 𝑤 · 𝑟 ) ∈ 𝑉 ) |
81 |
80
|
2ralimi |
⊢ ( ∀ 𝑥 ∈ 𝑉 ∀ 𝑤 ∈ 𝑉 ( ( ( 𝑤 · 𝑟 ) ∈ 𝑉 ∧ ( ( 𝑤 + 𝑥 ) · 𝑟 ) = ( ( 𝑤 · 𝑟 ) + ( 𝑥 · 𝑟 ) ) ∧ ( 𝑤 · ( 𝑞 ⨣ 𝑟 ) ) = ( ( 𝑤 · 𝑞 ) + ( 𝑤 · 𝑟 ) ) ) ∧ ( ( 𝑤 · ( 𝑞 × 𝑟 ) ) = ( ( 𝑤 · 𝑞 ) · 𝑟 ) ∧ ( 𝑤 · 1 ) = 𝑤 ) ) → ∀ 𝑥 ∈ 𝑉 ∀ 𝑤 ∈ 𝑉 ( 𝑤 · 𝑟 ) ∈ 𝑉 ) |
82 |
81
|
2ralimi |
⊢ ( ∀ 𝑞 ∈ 𝐾 ∀ 𝑟 ∈ 𝐾 ∀ 𝑥 ∈ 𝑉 ∀ 𝑤 ∈ 𝑉 ( ( ( 𝑤 · 𝑟 ) ∈ 𝑉 ∧ ( ( 𝑤 + 𝑥 ) · 𝑟 ) = ( ( 𝑤 · 𝑟 ) + ( 𝑥 · 𝑟 ) ) ∧ ( 𝑤 · ( 𝑞 ⨣ 𝑟 ) ) = ( ( 𝑤 · 𝑞 ) + ( 𝑤 · 𝑟 ) ) ) ∧ ( ( 𝑤 · ( 𝑞 × 𝑟 ) ) = ( ( 𝑤 · 𝑞 ) · 𝑟 ) ∧ ( 𝑤 · 1 ) = 𝑤 ) ) → ∀ 𝑞 ∈ 𝐾 ∀ 𝑟 ∈ 𝐾 ∀ 𝑥 ∈ 𝑉 ∀ 𝑤 ∈ 𝑉 ( 𝑤 · 𝑟 ) ∈ 𝑉 ) |
83 |
|
ringgrp |
⊢ ( 𝐹 ∈ Ring → 𝐹 ∈ Grp ) |
84 |
5
|
grpbn0 |
⊢ ( 𝐹 ∈ Grp → 𝐾 ≠ ∅ ) |
85 |
83 84
|
syl |
⊢ ( 𝐹 ∈ Ring → 𝐾 ≠ ∅ ) |
86 |
85
|
3ad2ant2 |
⊢ ( ( 𝑅 ∈ Grp ∧ 𝐹 ∈ Ring ∧ ∀ 𝑞 ∈ 𝐾 ∀ 𝑟 ∈ 𝐾 ∀ 𝑥 ∈ 𝑉 ∀ 𝑤 ∈ 𝑉 ( ( ( 𝑤 · 𝑟 ) ∈ 𝑉 ∧ ( ( 𝑤 + 𝑥 ) · 𝑟 ) = ( ( 𝑤 · 𝑟 ) + ( 𝑥 · 𝑟 ) ) ∧ ( 𝑤 · ( 𝑞 ⨣ 𝑟 ) ) = ( ( 𝑤 · 𝑞 ) + ( 𝑤 · 𝑟 ) ) ) ∧ ( ( 𝑤 · ( 𝑞 × 𝑟 ) ) = ( ( 𝑤 · 𝑞 ) · 𝑟 ) ∧ ( 𝑤 · 1 ) = 𝑤 ) ) ) → 𝐾 ≠ ∅ ) |
87 |
9 86
|
ax-mp |
⊢ 𝐾 ≠ ∅ |
88 |
|
rspn0 |
⊢ ( 𝐾 ≠ ∅ → ( ∀ 𝑞 ∈ 𝐾 ∀ 𝑟 ∈ 𝐾 ∀ 𝑥 ∈ 𝑉 ∀ 𝑤 ∈ 𝑉 ( 𝑤 · 𝑟 ) ∈ 𝑉 → ∀ 𝑟 ∈ 𝐾 ∀ 𝑥 ∈ 𝑉 ∀ 𝑤 ∈ 𝑉 ( 𝑤 · 𝑟 ) ∈ 𝑉 ) ) |
89 |
87 88
|
ax-mp |
⊢ ( ∀ 𝑞 ∈ 𝐾 ∀ 𝑟 ∈ 𝐾 ∀ 𝑥 ∈ 𝑉 ∀ 𝑤 ∈ 𝑉 ( 𝑤 · 𝑟 ) ∈ 𝑉 → ∀ 𝑟 ∈ 𝐾 ∀ 𝑥 ∈ 𝑉 ∀ 𝑤 ∈ 𝑉 ( 𝑤 · 𝑟 ) ∈ 𝑉 ) |
90 |
|
ralcom |
⊢ ( ∀ 𝑟 ∈ 𝐾 ∀ 𝑥 ∈ 𝑉 ∀ 𝑤 ∈ 𝑉 ( 𝑤 · 𝑟 ) ∈ 𝑉 ↔ ∀ 𝑥 ∈ 𝑉 ∀ 𝑟 ∈ 𝐾 ∀ 𝑤 ∈ 𝑉 ( 𝑤 · 𝑟 ) ∈ 𝑉 ) |
91 |
1
|
grpbn0 |
⊢ ( 𝑅 ∈ Grp → 𝑉 ≠ ∅ ) |
92 |
91
|
3ad2ant1 |
⊢ ( ( 𝑅 ∈ Grp ∧ 𝐹 ∈ Ring ∧ ∀ 𝑞 ∈ 𝐾 ∀ 𝑟 ∈ 𝐾 ∀ 𝑥 ∈ 𝑉 ∀ 𝑤 ∈ 𝑉 ( ( ( 𝑤 · 𝑟 ) ∈ 𝑉 ∧ ( ( 𝑤 + 𝑥 ) · 𝑟 ) = ( ( 𝑤 · 𝑟 ) + ( 𝑥 · 𝑟 ) ) ∧ ( 𝑤 · ( 𝑞 ⨣ 𝑟 ) ) = ( ( 𝑤 · 𝑞 ) + ( 𝑤 · 𝑟 ) ) ) ∧ ( ( 𝑤 · ( 𝑞 × 𝑟 ) ) = ( ( 𝑤 · 𝑞 ) · 𝑟 ) ∧ ( 𝑤 · 1 ) = 𝑤 ) ) ) → 𝑉 ≠ ∅ ) |
93 |
9 92
|
ax-mp |
⊢ 𝑉 ≠ ∅ |
94 |
|
rspn0 |
⊢ ( 𝑉 ≠ ∅ → ( ∀ 𝑥 ∈ 𝑉 ∀ 𝑟 ∈ 𝐾 ∀ 𝑤 ∈ 𝑉 ( 𝑤 · 𝑟 ) ∈ 𝑉 → ∀ 𝑟 ∈ 𝐾 ∀ 𝑤 ∈ 𝑉 ( 𝑤 · 𝑟 ) ∈ 𝑉 ) ) |
95 |
93 94
|
ax-mp |
⊢ ( ∀ 𝑥 ∈ 𝑉 ∀ 𝑟 ∈ 𝐾 ∀ 𝑤 ∈ 𝑉 ( 𝑤 · 𝑟 ) ∈ 𝑉 → ∀ 𝑟 ∈ 𝐾 ∀ 𝑤 ∈ 𝑉 ( 𝑤 · 𝑟 ) ∈ 𝑉 ) |
96 |
|
oveq2 |
⊢ ( 𝑟 = 𝑎 → ( 𝑤 · 𝑟 ) = ( 𝑤 · 𝑎 ) ) |
97 |
96
|
eleq1d |
⊢ ( 𝑟 = 𝑎 → ( ( 𝑤 · 𝑟 ) ∈ 𝑉 ↔ ( 𝑤 · 𝑎 ) ∈ 𝑉 ) ) |
98 |
|
oveq1 |
⊢ ( 𝑤 = 𝑏 → ( 𝑤 · 𝑎 ) = ( 𝑏 · 𝑎 ) ) |
99 |
98
|
eleq1d |
⊢ ( 𝑤 = 𝑏 → ( ( 𝑤 · 𝑎 ) ∈ 𝑉 ↔ ( 𝑏 · 𝑎 ) ∈ 𝑉 ) ) |
100 |
97 99
|
rspc2v |
⊢ ( ( 𝑎 ∈ 𝐾 ∧ 𝑏 ∈ 𝑉 ) → ( ∀ 𝑟 ∈ 𝐾 ∀ 𝑤 ∈ 𝑉 ( 𝑤 · 𝑟 ) ∈ 𝑉 → ( 𝑏 · 𝑎 ) ∈ 𝑉 ) ) |
101 |
100
|
3adant1 |
⊢ ( ( 𝐹 ∈ CRing ∧ 𝑎 ∈ 𝐾 ∧ 𝑏 ∈ 𝑉 ) → ( ∀ 𝑟 ∈ 𝐾 ∀ 𝑤 ∈ 𝑉 ( 𝑤 · 𝑟 ) ∈ 𝑉 → ( 𝑏 · 𝑎 ) ∈ 𝑉 ) ) |
102 |
95 101
|
syl5com |
⊢ ( ∀ 𝑥 ∈ 𝑉 ∀ 𝑟 ∈ 𝐾 ∀ 𝑤 ∈ 𝑉 ( 𝑤 · 𝑟 ) ∈ 𝑉 → ( ( 𝐹 ∈ CRing ∧ 𝑎 ∈ 𝐾 ∧ 𝑏 ∈ 𝑉 ) → ( 𝑏 · 𝑎 ) ∈ 𝑉 ) ) |
103 |
90 102
|
sylbi |
⊢ ( ∀ 𝑟 ∈ 𝐾 ∀ 𝑥 ∈ 𝑉 ∀ 𝑤 ∈ 𝑉 ( 𝑤 · 𝑟 ) ∈ 𝑉 → ( ( 𝐹 ∈ CRing ∧ 𝑎 ∈ 𝐾 ∧ 𝑏 ∈ 𝑉 ) → ( 𝑏 · 𝑎 ) ∈ 𝑉 ) ) |
104 |
82 89 103
|
3syl |
⊢ ( ∀ 𝑞 ∈ 𝐾 ∀ 𝑟 ∈ 𝐾 ∀ 𝑥 ∈ 𝑉 ∀ 𝑤 ∈ 𝑉 ( ( ( 𝑤 · 𝑟 ) ∈ 𝑉 ∧ ( ( 𝑤 + 𝑥 ) · 𝑟 ) = ( ( 𝑤 · 𝑟 ) + ( 𝑥 · 𝑟 ) ) ∧ ( 𝑤 · ( 𝑞 ⨣ 𝑟 ) ) = ( ( 𝑤 · 𝑞 ) + ( 𝑤 · 𝑟 ) ) ) ∧ ( ( 𝑤 · ( 𝑞 × 𝑟 ) ) = ( ( 𝑤 · 𝑞 ) · 𝑟 ) ∧ ( 𝑤 · 1 ) = 𝑤 ) ) → ( ( 𝐹 ∈ CRing ∧ 𝑎 ∈ 𝐾 ∧ 𝑏 ∈ 𝑉 ) → ( 𝑏 · 𝑎 ) ∈ 𝑉 ) ) |
105 |
104
|
3ad2ant3 |
⊢ ( ( 𝑅 ∈ Grp ∧ 𝐹 ∈ Ring ∧ ∀ 𝑞 ∈ 𝐾 ∀ 𝑟 ∈ 𝐾 ∀ 𝑥 ∈ 𝑉 ∀ 𝑤 ∈ 𝑉 ( ( ( 𝑤 · 𝑟 ) ∈ 𝑉 ∧ ( ( 𝑤 + 𝑥 ) · 𝑟 ) = ( ( 𝑤 · 𝑟 ) + ( 𝑥 · 𝑟 ) ) ∧ ( 𝑤 · ( 𝑞 ⨣ 𝑟 ) ) = ( ( 𝑤 · 𝑞 ) + ( 𝑤 · 𝑟 ) ) ) ∧ ( ( 𝑤 · ( 𝑞 × 𝑟 ) ) = ( ( 𝑤 · 𝑞 ) · 𝑟 ) ∧ ( 𝑤 · 1 ) = 𝑤 ) ) ) → ( ( 𝐹 ∈ CRing ∧ 𝑎 ∈ 𝐾 ∧ 𝑏 ∈ 𝑉 ) → ( 𝑏 · 𝑎 ) ∈ 𝑉 ) ) |
106 |
9 105
|
ax-mp |
⊢ ( ( 𝐹 ∈ CRing ∧ 𝑎 ∈ 𝐾 ∧ 𝑏 ∈ 𝑉 ) → ( 𝑏 · 𝑎 ) ∈ 𝑉 ) |
107 |
79 106
|
eqeltrd |
⊢ ( ( 𝐹 ∈ CRing ∧ 𝑎 ∈ 𝐾 ∧ 𝑏 ∈ 𝑉 ) → ( 𝑎 ∗ 𝑏 ) ∈ 𝑉 ) |
108 |
10
|
a1i |
⊢ ( ( 𝑎 ∈ 𝐾 ∧ 𝑏 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉 ) → ∗ = ( 𝑠 ∈ 𝐾 , 𝑣 ∈ 𝑉 ↦ ( 𝑣 · 𝑠 ) ) ) |
109 |
|
oveq12 |
⊢ ( ( 𝑣 = ( 𝑏 + 𝑐 ) ∧ 𝑠 = 𝑎 ) → ( 𝑣 · 𝑠 ) = ( ( 𝑏 + 𝑐 ) · 𝑎 ) ) |
110 |
109
|
ancoms |
⊢ ( ( 𝑠 = 𝑎 ∧ 𝑣 = ( 𝑏 + 𝑐 ) ) → ( 𝑣 · 𝑠 ) = ( ( 𝑏 + 𝑐 ) · 𝑎 ) ) |
111 |
110
|
adantl |
⊢ ( ( ( 𝑎 ∈ 𝐾 ∧ 𝑏 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉 ) ∧ ( 𝑠 = 𝑎 ∧ 𝑣 = ( 𝑏 + 𝑐 ) ) ) → ( 𝑣 · 𝑠 ) = ( ( 𝑏 + 𝑐 ) · 𝑎 ) ) |
112 |
|
simp1 |
⊢ ( ( 𝑎 ∈ 𝐾 ∧ 𝑏 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉 ) → 𝑎 ∈ 𝐾 ) |
113 |
1 2
|
grpcl |
⊢ ( ( 𝑅 ∈ Grp ∧ 𝑏 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉 ) → ( 𝑏 + 𝑐 ) ∈ 𝑉 ) |
114 |
50 113
|
mp3an1 |
⊢ ( ( 𝑏 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉 ) → ( 𝑏 + 𝑐 ) ∈ 𝑉 ) |
115 |
114
|
3adant1 |
⊢ ( ( 𝑎 ∈ 𝐾 ∧ 𝑏 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉 ) → ( 𝑏 + 𝑐 ) ∈ 𝑉 ) |
116 |
|
ovexd |
⊢ ( ( 𝑎 ∈ 𝐾 ∧ 𝑏 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉 ) → ( ( 𝑏 + 𝑐 ) · 𝑎 ) ∈ V ) |
117 |
108 111 112 115 116
|
ovmpod |
⊢ ( ( 𝑎 ∈ 𝐾 ∧ 𝑏 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉 ) → ( 𝑎 ∗ ( 𝑏 + 𝑐 ) ) = ( ( 𝑏 + 𝑐 ) · 𝑎 ) ) |
118 |
|
simpl2 |
⊢ ( ( ( ( 𝑤 · 𝑟 ) ∈ 𝑉 ∧ ( ( 𝑤 + 𝑥 ) · 𝑟 ) = ( ( 𝑤 · 𝑟 ) + ( 𝑥 · 𝑟 ) ) ∧ ( 𝑤 · ( 𝑞 ⨣ 𝑟 ) ) = ( ( 𝑤 · 𝑞 ) + ( 𝑤 · 𝑟 ) ) ) ∧ ( ( 𝑤 · ( 𝑞 × 𝑟 ) ) = ( ( 𝑤 · 𝑞 ) · 𝑟 ) ∧ ( 𝑤 · 1 ) = 𝑤 ) ) → ( ( 𝑤 + 𝑥 ) · 𝑟 ) = ( ( 𝑤 · 𝑟 ) + ( 𝑥 · 𝑟 ) ) ) |
119 |
118
|
2ralimi |
⊢ ( ∀ 𝑥 ∈ 𝑉 ∀ 𝑤 ∈ 𝑉 ( ( ( 𝑤 · 𝑟 ) ∈ 𝑉 ∧ ( ( 𝑤 + 𝑥 ) · 𝑟 ) = ( ( 𝑤 · 𝑟 ) + ( 𝑥 · 𝑟 ) ) ∧ ( 𝑤 · ( 𝑞 ⨣ 𝑟 ) ) = ( ( 𝑤 · 𝑞 ) + ( 𝑤 · 𝑟 ) ) ) ∧ ( ( 𝑤 · ( 𝑞 × 𝑟 ) ) = ( ( 𝑤 · 𝑞 ) · 𝑟 ) ∧ ( 𝑤 · 1 ) = 𝑤 ) ) → ∀ 𝑥 ∈ 𝑉 ∀ 𝑤 ∈ 𝑉 ( ( 𝑤 + 𝑥 ) · 𝑟 ) = ( ( 𝑤 · 𝑟 ) + ( 𝑥 · 𝑟 ) ) ) |
120 |
119
|
2ralimi |
⊢ ( ∀ 𝑞 ∈ 𝐾 ∀ 𝑟 ∈ 𝐾 ∀ 𝑥 ∈ 𝑉 ∀ 𝑤 ∈ 𝑉 ( ( ( 𝑤 · 𝑟 ) ∈ 𝑉 ∧ ( ( 𝑤 + 𝑥 ) · 𝑟 ) = ( ( 𝑤 · 𝑟 ) + ( 𝑥 · 𝑟 ) ) ∧ ( 𝑤 · ( 𝑞 ⨣ 𝑟 ) ) = ( ( 𝑤 · 𝑞 ) + ( 𝑤 · 𝑟 ) ) ) ∧ ( ( 𝑤 · ( 𝑞 × 𝑟 ) ) = ( ( 𝑤 · 𝑞 ) · 𝑟 ) ∧ ( 𝑤 · 1 ) = 𝑤 ) ) → ∀ 𝑞 ∈ 𝐾 ∀ 𝑟 ∈ 𝐾 ∀ 𝑥 ∈ 𝑉 ∀ 𝑤 ∈ 𝑉 ( ( 𝑤 + 𝑥 ) · 𝑟 ) = ( ( 𝑤 · 𝑟 ) + ( 𝑥 · 𝑟 ) ) ) |
121 |
|
rspn0 |
⊢ ( 𝐾 ≠ ∅ → ( ∀ 𝑞 ∈ 𝐾 ∀ 𝑟 ∈ 𝐾 ∀ 𝑥 ∈ 𝑉 ∀ 𝑤 ∈ 𝑉 ( ( 𝑤 + 𝑥 ) · 𝑟 ) = ( ( 𝑤 · 𝑟 ) + ( 𝑥 · 𝑟 ) ) → ∀ 𝑟 ∈ 𝐾 ∀ 𝑥 ∈ 𝑉 ∀ 𝑤 ∈ 𝑉 ( ( 𝑤 + 𝑥 ) · 𝑟 ) = ( ( 𝑤 · 𝑟 ) + ( 𝑥 · 𝑟 ) ) ) ) |
122 |
87 121
|
ax-mp |
⊢ ( ∀ 𝑞 ∈ 𝐾 ∀ 𝑟 ∈ 𝐾 ∀ 𝑥 ∈ 𝑉 ∀ 𝑤 ∈ 𝑉 ( ( 𝑤 + 𝑥 ) · 𝑟 ) = ( ( 𝑤 · 𝑟 ) + ( 𝑥 · 𝑟 ) ) → ∀ 𝑟 ∈ 𝐾 ∀ 𝑥 ∈ 𝑉 ∀ 𝑤 ∈ 𝑉 ( ( 𝑤 + 𝑥 ) · 𝑟 ) = ( ( 𝑤 · 𝑟 ) + ( 𝑥 · 𝑟 ) ) ) |
123 |
|
oveq2 |
⊢ ( 𝑟 = 𝑎 → ( ( 𝑤 + 𝑥 ) · 𝑟 ) = ( ( 𝑤 + 𝑥 ) · 𝑎 ) ) |
124 |
|
oveq2 |
⊢ ( 𝑟 = 𝑎 → ( 𝑥 · 𝑟 ) = ( 𝑥 · 𝑎 ) ) |
125 |
96 124
|
oveq12d |
⊢ ( 𝑟 = 𝑎 → ( ( 𝑤 · 𝑟 ) + ( 𝑥 · 𝑟 ) ) = ( ( 𝑤 · 𝑎 ) + ( 𝑥 · 𝑎 ) ) ) |
126 |
123 125
|
eqeq12d |
⊢ ( 𝑟 = 𝑎 → ( ( ( 𝑤 + 𝑥 ) · 𝑟 ) = ( ( 𝑤 · 𝑟 ) + ( 𝑥 · 𝑟 ) ) ↔ ( ( 𝑤 + 𝑥 ) · 𝑎 ) = ( ( 𝑤 · 𝑎 ) + ( 𝑥 · 𝑎 ) ) ) ) |
127 |
|
oveq2 |
⊢ ( 𝑥 = 𝑐 → ( 𝑤 + 𝑥 ) = ( 𝑤 + 𝑐 ) ) |
128 |
127
|
oveq1d |
⊢ ( 𝑥 = 𝑐 → ( ( 𝑤 + 𝑥 ) · 𝑎 ) = ( ( 𝑤 + 𝑐 ) · 𝑎 ) ) |
129 |
|
oveq1 |
⊢ ( 𝑥 = 𝑐 → ( 𝑥 · 𝑎 ) = ( 𝑐 · 𝑎 ) ) |
130 |
129
|
oveq2d |
⊢ ( 𝑥 = 𝑐 → ( ( 𝑤 · 𝑎 ) + ( 𝑥 · 𝑎 ) ) = ( ( 𝑤 · 𝑎 ) + ( 𝑐 · 𝑎 ) ) ) |
131 |
128 130
|
eqeq12d |
⊢ ( 𝑥 = 𝑐 → ( ( ( 𝑤 + 𝑥 ) · 𝑎 ) = ( ( 𝑤 · 𝑎 ) + ( 𝑥 · 𝑎 ) ) ↔ ( ( 𝑤 + 𝑐 ) · 𝑎 ) = ( ( 𝑤 · 𝑎 ) + ( 𝑐 · 𝑎 ) ) ) ) |
132 |
|
oveq1 |
⊢ ( 𝑤 = 𝑏 → ( 𝑤 + 𝑐 ) = ( 𝑏 + 𝑐 ) ) |
133 |
132
|
oveq1d |
⊢ ( 𝑤 = 𝑏 → ( ( 𝑤 + 𝑐 ) · 𝑎 ) = ( ( 𝑏 + 𝑐 ) · 𝑎 ) ) |
134 |
98
|
oveq1d |
⊢ ( 𝑤 = 𝑏 → ( ( 𝑤 · 𝑎 ) + ( 𝑐 · 𝑎 ) ) = ( ( 𝑏 · 𝑎 ) + ( 𝑐 · 𝑎 ) ) ) |
135 |
133 134
|
eqeq12d |
⊢ ( 𝑤 = 𝑏 → ( ( ( 𝑤 + 𝑐 ) · 𝑎 ) = ( ( 𝑤 · 𝑎 ) + ( 𝑐 · 𝑎 ) ) ↔ ( ( 𝑏 + 𝑐 ) · 𝑎 ) = ( ( 𝑏 · 𝑎 ) + ( 𝑐 · 𝑎 ) ) ) ) |
136 |
126 131 135
|
rspc3v |
⊢ ( ( 𝑎 ∈ 𝐾 ∧ 𝑐 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉 ) → ( ∀ 𝑟 ∈ 𝐾 ∀ 𝑥 ∈ 𝑉 ∀ 𝑤 ∈ 𝑉 ( ( 𝑤 + 𝑥 ) · 𝑟 ) = ( ( 𝑤 · 𝑟 ) + ( 𝑥 · 𝑟 ) ) → ( ( 𝑏 + 𝑐 ) · 𝑎 ) = ( ( 𝑏 · 𝑎 ) + ( 𝑐 · 𝑎 ) ) ) ) |
137 |
136
|
3com23 |
⊢ ( ( 𝑎 ∈ 𝐾 ∧ 𝑏 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉 ) → ( ∀ 𝑟 ∈ 𝐾 ∀ 𝑥 ∈ 𝑉 ∀ 𝑤 ∈ 𝑉 ( ( 𝑤 + 𝑥 ) · 𝑟 ) = ( ( 𝑤 · 𝑟 ) + ( 𝑥 · 𝑟 ) ) → ( ( 𝑏 + 𝑐 ) · 𝑎 ) = ( ( 𝑏 · 𝑎 ) + ( 𝑐 · 𝑎 ) ) ) ) |
138 |
122 137
|
syl5com |
⊢ ( ∀ 𝑞 ∈ 𝐾 ∀ 𝑟 ∈ 𝐾 ∀ 𝑥 ∈ 𝑉 ∀ 𝑤 ∈ 𝑉 ( ( 𝑤 + 𝑥 ) · 𝑟 ) = ( ( 𝑤 · 𝑟 ) + ( 𝑥 · 𝑟 ) ) → ( ( 𝑎 ∈ 𝐾 ∧ 𝑏 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉 ) → ( ( 𝑏 + 𝑐 ) · 𝑎 ) = ( ( 𝑏 · 𝑎 ) + ( 𝑐 · 𝑎 ) ) ) ) |
139 |
120 138
|
syl |
⊢ ( ∀ 𝑞 ∈ 𝐾 ∀ 𝑟 ∈ 𝐾 ∀ 𝑥 ∈ 𝑉 ∀ 𝑤 ∈ 𝑉 ( ( ( 𝑤 · 𝑟 ) ∈ 𝑉 ∧ ( ( 𝑤 + 𝑥 ) · 𝑟 ) = ( ( 𝑤 · 𝑟 ) + ( 𝑥 · 𝑟 ) ) ∧ ( 𝑤 · ( 𝑞 ⨣ 𝑟 ) ) = ( ( 𝑤 · 𝑞 ) + ( 𝑤 · 𝑟 ) ) ) ∧ ( ( 𝑤 · ( 𝑞 × 𝑟 ) ) = ( ( 𝑤 · 𝑞 ) · 𝑟 ) ∧ ( 𝑤 · 1 ) = 𝑤 ) ) → ( ( 𝑎 ∈ 𝐾 ∧ 𝑏 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉 ) → ( ( 𝑏 + 𝑐 ) · 𝑎 ) = ( ( 𝑏 · 𝑎 ) + ( 𝑐 · 𝑎 ) ) ) ) |
140 |
139
|
3ad2ant3 |
⊢ ( ( 𝑅 ∈ Grp ∧ 𝐹 ∈ Ring ∧ ∀ 𝑞 ∈ 𝐾 ∀ 𝑟 ∈ 𝐾 ∀ 𝑥 ∈ 𝑉 ∀ 𝑤 ∈ 𝑉 ( ( ( 𝑤 · 𝑟 ) ∈ 𝑉 ∧ ( ( 𝑤 + 𝑥 ) · 𝑟 ) = ( ( 𝑤 · 𝑟 ) + ( 𝑥 · 𝑟 ) ) ∧ ( 𝑤 · ( 𝑞 ⨣ 𝑟 ) ) = ( ( 𝑤 · 𝑞 ) + ( 𝑤 · 𝑟 ) ) ) ∧ ( ( 𝑤 · ( 𝑞 × 𝑟 ) ) = ( ( 𝑤 · 𝑞 ) · 𝑟 ) ∧ ( 𝑤 · 1 ) = 𝑤 ) ) ) → ( ( 𝑎 ∈ 𝐾 ∧ 𝑏 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉 ) → ( ( 𝑏 + 𝑐 ) · 𝑎 ) = ( ( 𝑏 · 𝑎 ) + ( 𝑐 · 𝑎 ) ) ) ) |
141 |
9 140
|
ax-mp |
⊢ ( ( 𝑎 ∈ 𝐾 ∧ 𝑏 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉 ) → ( ( 𝑏 + 𝑐 ) · 𝑎 ) = ( ( 𝑏 · 𝑎 ) + ( 𝑐 · 𝑎 ) ) ) |
142 |
117 141
|
eqtrd |
⊢ ( ( 𝑎 ∈ 𝐾 ∧ 𝑏 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉 ) → ( 𝑎 ∗ ( 𝑏 + 𝑐 ) ) = ( ( 𝑏 · 𝑎 ) + ( 𝑐 · 𝑎 ) ) ) |
143 |
142
|
adantl |
⊢ ( ( 𝐹 ∈ CRing ∧ ( 𝑎 ∈ 𝐾 ∧ 𝑏 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉 ) ) → ( 𝑎 ∗ ( 𝑏 + 𝑐 ) ) = ( ( 𝑏 · 𝑎 ) + ( 𝑐 · 𝑎 ) ) ) |
144 |
74
|
adantl |
⊢ ( ( ( 𝑎 ∈ 𝐾 ∧ 𝑏 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉 ) ∧ ( 𝑠 = 𝑎 ∧ 𝑣 = 𝑏 ) ) → ( 𝑣 · 𝑠 ) = ( 𝑏 · 𝑎 ) ) |
145 |
|
simp2 |
⊢ ( ( 𝑎 ∈ 𝐾 ∧ 𝑏 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉 ) → 𝑏 ∈ 𝑉 ) |
146 |
|
ovexd |
⊢ ( ( 𝑎 ∈ 𝐾 ∧ 𝑏 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉 ) → ( 𝑏 · 𝑎 ) ∈ V ) |
147 |
108 144 112 145 146
|
ovmpod |
⊢ ( ( 𝑎 ∈ 𝐾 ∧ 𝑏 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉 ) → ( 𝑎 ∗ 𝑏 ) = ( 𝑏 · 𝑎 ) ) |
148 |
|
oveq12 |
⊢ ( ( 𝑣 = 𝑐 ∧ 𝑠 = 𝑎 ) → ( 𝑣 · 𝑠 ) = ( 𝑐 · 𝑎 ) ) |
149 |
148
|
ancoms |
⊢ ( ( 𝑠 = 𝑎 ∧ 𝑣 = 𝑐 ) → ( 𝑣 · 𝑠 ) = ( 𝑐 · 𝑎 ) ) |
150 |
149
|
adantl |
⊢ ( ( ( 𝑎 ∈ 𝐾 ∧ 𝑏 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉 ) ∧ ( 𝑠 = 𝑎 ∧ 𝑣 = 𝑐 ) ) → ( 𝑣 · 𝑠 ) = ( 𝑐 · 𝑎 ) ) |
151 |
|
simp3 |
⊢ ( ( 𝑎 ∈ 𝐾 ∧ 𝑏 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉 ) → 𝑐 ∈ 𝑉 ) |
152 |
|
ovexd |
⊢ ( ( 𝑎 ∈ 𝐾 ∧ 𝑏 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉 ) → ( 𝑐 · 𝑎 ) ∈ V ) |
153 |
108 150 112 151 152
|
ovmpod |
⊢ ( ( 𝑎 ∈ 𝐾 ∧ 𝑏 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉 ) → ( 𝑎 ∗ 𝑐 ) = ( 𝑐 · 𝑎 ) ) |
154 |
147 153
|
oveq12d |
⊢ ( ( 𝑎 ∈ 𝐾 ∧ 𝑏 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉 ) → ( ( 𝑎 ∗ 𝑏 ) + ( 𝑎 ∗ 𝑐 ) ) = ( ( 𝑏 · 𝑎 ) + ( 𝑐 · 𝑎 ) ) ) |
155 |
154
|
adantl |
⊢ ( ( 𝐹 ∈ CRing ∧ ( 𝑎 ∈ 𝐾 ∧ 𝑏 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉 ) ) → ( ( 𝑎 ∗ 𝑏 ) + ( 𝑎 ∗ 𝑐 ) ) = ( ( 𝑏 · 𝑎 ) + ( 𝑐 · 𝑎 ) ) ) |
156 |
143 155
|
eqtr4d |
⊢ ( ( 𝐹 ∈ CRing ∧ ( 𝑎 ∈ 𝐾 ∧ 𝑏 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉 ) ) → ( 𝑎 ∗ ( 𝑏 + 𝑐 ) ) = ( ( 𝑎 ∗ 𝑏 ) + ( 𝑎 ∗ 𝑐 ) ) ) |
157 |
|
simpl3 |
⊢ ( ( ( ( 𝑤 · 𝑟 ) ∈ 𝑉 ∧ ( ( 𝑤 + 𝑥 ) · 𝑟 ) = ( ( 𝑤 · 𝑟 ) + ( 𝑥 · 𝑟 ) ) ∧ ( 𝑤 · ( 𝑞 ⨣ 𝑟 ) ) = ( ( 𝑤 · 𝑞 ) + ( 𝑤 · 𝑟 ) ) ) ∧ ( ( 𝑤 · ( 𝑞 × 𝑟 ) ) = ( ( 𝑤 · 𝑞 ) · 𝑟 ) ∧ ( 𝑤 · 1 ) = 𝑤 ) ) → ( 𝑤 · ( 𝑞 ⨣ 𝑟 ) ) = ( ( 𝑤 · 𝑞 ) + ( 𝑤 · 𝑟 ) ) ) |
158 |
157
|
2ralimi |
⊢ ( ∀ 𝑥 ∈ 𝑉 ∀ 𝑤 ∈ 𝑉 ( ( ( 𝑤 · 𝑟 ) ∈ 𝑉 ∧ ( ( 𝑤 + 𝑥 ) · 𝑟 ) = ( ( 𝑤 · 𝑟 ) + ( 𝑥 · 𝑟 ) ) ∧ ( 𝑤 · ( 𝑞 ⨣ 𝑟 ) ) = ( ( 𝑤 · 𝑞 ) + ( 𝑤 · 𝑟 ) ) ) ∧ ( ( 𝑤 · ( 𝑞 × 𝑟 ) ) = ( ( 𝑤 · 𝑞 ) · 𝑟 ) ∧ ( 𝑤 · 1 ) = 𝑤 ) ) → ∀ 𝑥 ∈ 𝑉 ∀ 𝑤 ∈ 𝑉 ( 𝑤 · ( 𝑞 ⨣ 𝑟 ) ) = ( ( 𝑤 · 𝑞 ) + ( 𝑤 · 𝑟 ) ) ) |
159 |
158
|
2ralimi |
⊢ ( ∀ 𝑞 ∈ 𝐾 ∀ 𝑟 ∈ 𝐾 ∀ 𝑥 ∈ 𝑉 ∀ 𝑤 ∈ 𝑉 ( ( ( 𝑤 · 𝑟 ) ∈ 𝑉 ∧ ( ( 𝑤 + 𝑥 ) · 𝑟 ) = ( ( 𝑤 · 𝑟 ) + ( 𝑥 · 𝑟 ) ) ∧ ( 𝑤 · ( 𝑞 ⨣ 𝑟 ) ) = ( ( 𝑤 · 𝑞 ) + ( 𝑤 · 𝑟 ) ) ) ∧ ( ( 𝑤 · ( 𝑞 × 𝑟 ) ) = ( ( 𝑤 · 𝑞 ) · 𝑟 ) ∧ ( 𝑤 · 1 ) = 𝑤 ) ) → ∀ 𝑞 ∈ 𝐾 ∀ 𝑟 ∈ 𝐾 ∀ 𝑥 ∈ 𝑉 ∀ 𝑤 ∈ 𝑉 ( 𝑤 · ( 𝑞 ⨣ 𝑟 ) ) = ( ( 𝑤 · 𝑞 ) + ( 𝑤 · 𝑟 ) ) ) |
160 |
|
ralrot3 |
⊢ ( ∀ 𝑞 ∈ 𝐾 ∀ 𝑟 ∈ 𝐾 ∀ 𝑥 ∈ 𝑉 ∀ 𝑤 ∈ 𝑉 ( 𝑤 · ( 𝑞 ⨣ 𝑟 ) ) = ( ( 𝑤 · 𝑞 ) + ( 𝑤 · 𝑟 ) ) ↔ ∀ 𝑥 ∈ 𝑉 ∀ 𝑞 ∈ 𝐾 ∀ 𝑟 ∈ 𝐾 ∀ 𝑤 ∈ 𝑉 ( 𝑤 · ( 𝑞 ⨣ 𝑟 ) ) = ( ( 𝑤 · 𝑞 ) + ( 𝑤 · 𝑟 ) ) ) |
161 |
|
rspn0 |
⊢ ( 𝑉 ≠ ∅ → ( ∀ 𝑥 ∈ 𝑉 ∀ 𝑞 ∈ 𝐾 ∀ 𝑟 ∈ 𝐾 ∀ 𝑤 ∈ 𝑉 ( 𝑤 · ( 𝑞 ⨣ 𝑟 ) ) = ( ( 𝑤 · 𝑞 ) + ( 𝑤 · 𝑟 ) ) → ∀ 𝑞 ∈ 𝐾 ∀ 𝑟 ∈ 𝐾 ∀ 𝑤 ∈ 𝑉 ( 𝑤 · ( 𝑞 ⨣ 𝑟 ) ) = ( ( 𝑤 · 𝑞 ) + ( 𝑤 · 𝑟 ) ) ) ) |
162 |
93 161
|
ax-mp |
⊢ ( ∀ 𝑥 ∈ 𝑉 ∀ 𝑞 ∈ 𝐾 ∀ 𝑟 ∈ 𝐾 ∀ 𝑤 ∈ 𝑉 ( 𝑤 · ( 𝑞 ⨣ 𝑟 ) ) = ( ( 𝑤 · 𝑞 ) + ( 𝑤 · 𝑟 ) ) → ∀ 𝑞 ∈ 𝐾 ∀ 𝑟 ∈ 𝐾 ∀ 𝑤 ∈ 𝑉 ( 𝑤 · ( 𝑞 ⨣ 𝑟 ) ) = ( ( 𝑤 · 𝑞 ) + ( 𝑤 · 𝑟 ) ) ) |
163 |
|
oveq1 |
⊢ ( 𝑞 = 𝑎 → ( 𝑞 ⨣ 𝑟 ) = ( 𝑎 ⨣ 𝑟 ) ) |
164 |
163
|
oveq2d |
⊢ ( 𝑞 = 𝑎 → ( 𝑤 · ( 𝑞 ⨣ 𝑟 ) ) = ( 𝑤 · ( 𝑎 ⨣ 𝑟 ) ) ) |
165 |
|
oveq2 |
⊢ ( 𝑞 = 𝑎 → ( 𝑤 · 𝑞 ) = ( 𝑤 · 𝑎 ) ) |
166 |
165
|
oveq1d |
⊢ ( 𝑞 = 𝑎 → ( ( 𝑤 · 𝑞 ) + ( 𝑤 · 𝑟 ) ) = ( ( 𝑤 · 𝑎 ) + ( 𝑤 · 𝑟 ) ) ) |
167 |
164 166
|
eqeq12d |
⊢ ( 𝑞 = 𝑎 → ( ( 𝑤 · ( 𝑞 ⨣ 𝑟 ) ) = ( ( 𝑤 · 𝑞 ) + ( 𝑤 · 𝑟 ) ) ↔ ( 𝑤 · ( 𝑎 ⨣ 𝑟 ) ) = ( ( 𝑤 · 𝑎 ) + ( 𝑤 · 𝑟 ) ) ) ) |
168 |
|
oveq2 |
⊢ ( 𝑟 = 𝑏 → ( 𝑎 ⨣ 𝑟 ) = ( 𝑎 ⨣ 𝑏 ) ) |
169 |
168
|
oveq2d |
⊢ ( 𝑟 = 𝑏 → ( 𝑤 · ( 𝑎 ⨣ 𝑟 ) ) = ( 𝑤 · ( 𝑎 ⨣ 𝑏 ) ) ) |
170 |
|
oveq2 |
⊢ ( 𝑟 = 𝑏 → ( 𝑤 · 𝑟 ) = ( 𝑤 · 𝑏 ) ) |
171 |
170
|
oveq2d |
⊢ ( 𝑟 = 𝑏 → ( ( 𝑤 · 𝑎 ) + ( 𝑤 · 𝑟 ) ) = ( ( 𝑤 · 𝑎 ) + ( 𝑤 · 𝑏 ) ) ) |
172 |
169 171
|
eqeq12d |
⊢ ( 𝑟 = 𝑏 → ( ( 𝑤 · ( 𝑎 ⨣ 𝑟 ) ) = ( ( 𝑤 · 𝑎 ) + ( 𝑤 · 𝑟 ) ) ↔ ( 𝑤 · ( 𝑎 ⨣ 𝑏 ) ) = ( ( 𝑤 · 𝑎 ) + ( 𝑤 · 𝑏 ) ) ) ) |
173 |
|
oveq1 |
⊢ ( 𝑤 = 𝑐 → ( 𝑤 · ( 𝑎 ⨣ 𝑏 ) ) = ( 𝑐 · ( 𝑎 ⨣ 𝑏 ) ) ) |
174 |
|
oveq1 |
⊢ ( 𝑤 = 𝑐 → ( 𝑤 · 𝑎 ) = ( 𝑐 · 𝑎 ) ) |
175 |
|
oveq1 |
⊢ ( 𝑤 = 𝑐 → ( 𝑤 · 𝑏 ) = ( 𝑐 · 𝑏 ) ) |
176 |
174 175
|
oveq12d |
⊢ ( 𝑤 = 𝑐 → ( ( 𝑤 · 𝑎 ) + ( 𝑤 · 𝑏 ) ) = ( ( 𝑐 · 𝑎 ) + ( 𝑐 · 𝑏 ) ) ) |
177 |
173 176
|
eqeq12d |
⊢ ( 𝑤 = 𝑐 → ( ( 𝑤 · ( 𝑎 ⨣ 𝑏 ) ) = ( ( 𝑤 · 𝑎 ) + ( 𝑤 · 𝑏 ) ) ↔ ( 𝑐 · ( 𝑎 ⨣ 𝑏 ) ) = ( ( 𝑐 · 𝑎 ) + ( 𝑐 · 𝑏 ) ) ) ) |
178 |
167 172 177
|
rspc3v |
⊢ ( ( 𝑎 ∈ 𝐾 ∧ 𝑏 ∈ 𝐾 ∧ 𝑐 ∈ 𝑉 ) → ( ∀ 𝑞 ∈ 𝐾 ∀ 𝑟 ∈ 𝐾 ∀ 𝑤 ∈ 𝑉 ( 𝑤 · ( 𝑞 ⨣ 𝑟 ) ) = ( ( 𝑤 · 𝑞 ) + ( 𝑤 · 𝑟 ) ) → ( 𝑐 · ( 𝑎 ⨣ 𝑏 ) ) = ( ( 𝑐 · 𝑎 ) + ( 𝑐 · 𝑏 ) ) ) ) |
179 |
162 178
|
syl5com |
⊢ ( ∀ 𝑥 ∈ 𝑉 ∀ 𝑞 ∈ 𝐾 ∀ 𝑟 ∈ 𝐾 ∀ 𝑤 ∈ 𝑉 ( 𝑤 · ( 𝑞 ⨣ 𝑟 ) ) = ( ( 𝑤 · 𝑞 ) + ( 𝑤 · 𝑟 ) ) → ( ( 𝑎 ∈ 𝐾 ∧ 𝑏 ∈ 𝐾 ∧ 𝑐 ∈ 𝑉 ) → ( 𝑐 · ( 𝑎 ⨣ 𝑏 ) ) = ( ( 𝑐 · 𝑎 ) + ( 𝑐 · 𝑏 ) ) ) ) |
180 |
160 179
|
sylbi |
⊢ ( ∀ 𝑞 ∈ 𝐾 ∀ 𝑟 ∈ 𝐾 ∀ 𝑥 ∈ 𝑉 ∀ 𝑤 ∈ 𝑉 ( 𝑤 · ( 𝑞 ⨣ 𝑟 ) ) = ( ( 𝑤 · 𝑞 ) + ( 𝑤 · 𝑟 ) ) → ( ( 𝑎 ∈ 𝐾 ∧ 𝑏 ∈ 𝐾 ∧ 𝑐 ∈ 𝑉 ) → ( 𝑐 · ( 𝑎 ⨣ 𝑏 ) ) = ( ( 𝑐 · 𝑎 ) + ( 𝑐 · 𝑏 ) ) ) ) |
181 |
159 180
|
syl |
⊢ ( ∀ 𝑞 ∈ 𝐾 ∀ 𝑟 ∈ 𝐾 ∀ 𝑥 ∈ 𝑉 ∀ 𝑤 ∈ 𝑉 ( ( ( 𝑤 · 𝑟 ) ∈ 𝑉 ∧ ( ( 𝑤 + 𝑥 ) · 𝑟 ) = ( ( 𝑤 · 𝑟 ) + ( 𝑥 · 𝑟 ) ) ∧ ( 𝑤 · ( 𝑞 ⨣ 𝑟 ) ) = ( ( 𝑤 · 𝑞 ) + ( 𝑤 · 𝑟 ) ) ) ∧ ( ( 𝑤 · ( 𝑞 × 𝑟 ) ) = ( ( 𝑤 · 𝑞 ) · 𝑟 ) ∧ ( 𝑤 · 1 ) = 𝑤 ) ) → ( ( 𝑎 ∈ 𝐾 ∧ 𝑏 ∈ 𝐾 ∧ 𝑐 ∈ 𝑉 ) → ( 𝑐 · ( 𝑎 ⨣ 𝑏 ) ) = ( ( 𝑐 · 𝑎 ) + ( 𝑐 · 𝑏 ) ) ) ) |
182 |
181
|
3ad2ant3 |
⊢ ( ( 𝑅 ∈ Grp ∧ 𝐹 ∈ Ring ∧ ∀ 𝑞 ∈ 𝐾 ∀ 𝑟 ∈ 𝐾 ∀ 𝑥 ∈ 𝑉 ∀ 𝑤 ∈ 𝑉 ( ( ( 𝑤 · 𝑟 ) ∈ 𝑉 ∧ ( ( 𝑤 + 𝑥 ) · 𝑟 ) = ( ( 𝑤 · 𝑟 ) + ( 𝑥 · 𝑟 ) ) ∧ ( 𝑤 · ( 𝑞 ⨣ 𝑟 ) ) = ( ( 𝑤 · 𝑞 ) + ( 𝑤 · 𝑟 ) ) ) ∧ ( ( 𝑤 · ( 𝑞 × 𝑟 ) ) = ( ( 𝑤 · 𝑞 ) · 𝑟 ) ∧ ( 𝑤 · 1 ) = 𝑤 ) ) ) → ( ( 𝑎 ∈ 𝐾 ∧ 𝑏 ∈ 𝐾 ∧ 𝑐 ∈ 𝑉 ) → ( 𝑐 · ( 𝑎 ⨣ 𝑏 ) ) = ( ( 𝑐 · 𝑎 ) + ( 𝑐 · 𝑏 ) ) ) ) |
183 |
9 182
|
ax-mp |
⊢ ( ( 𝑎 ∈ 𝐾 ∧ 𝑏 ∈ 𝐾 ∧ 𝑐 ∈ 𝑉 ) → ( 𝑐 · ( 𝑎 ⨣ 𝑏 ) ) = ( ( 𝑐 · 𝑎 ) + ( 𝑐 · 𝑏 ) ) ) |
184 |
10
|
a1i |
⊢ ( ( 𝑎 ∈ 𝐾 ∧ 𝑏 ∈ 𝐾 ∧ 𝑐 ∈ 𝑉 ) → ∗ = ( 𝑠 ∈ 𝐾 , 𝑣 ∈ 𝑉 ↦ ( 𝑣 · 𝑠 ) ) ) |
185 |
|
oveq12 |
⊢ ( ( 𝑣 = 𝑐 ∧ 𝑠 = ( 𝑎 ⨣ 𝑏 ) ) → ( 𝑣 · 𝑠 ) = ( 𝑐 · ( 𝑎 ⨣ 𝑏 ) ) ) |
186 |
185
|
ancoms |
⊢ ( ( 𝑠 = ( 𝑎 ⨣ 𝑏 ) ∧ 𝑣 = 𝑐 ) → ( 𝑣 · 𝑠 ) = ( 𝑐 · ( 𝑎 ⨣ 𝑏 ) ) ) |
187 |
186
|
adantl |
⊢ ( ( ( 𝑎 ∈ 𝐾 ∧ 𝑏 ∈ 𝐾 ∧ 𝑐 ∈ 𝑉 ) ∧ ( 𝑠 = ( 𝑎 ⨣ 𝑏 ) ∧ 𝑣 = 𝑐 ) ) → ( 𝑣 · 𝑠 ) = ( 𝑐 · ( 𝑎 ⨣ 𝑏 ) ) ) |
188 |
5 6
|
grpcl |
⊢ ( ( 𝐹 ∈ Grp ∧ 𝑎 ∈ 𝐾 ∧ 𝑏 ∈ 𝐾 ) → ( 𝑎 ⨣ 𝑏 ) ∈ 𝐾 ) |
189 |
188
|
3expib |
⊢ ( 𝐹 ∈ Grp → ( ( 𝑎 ∈ 𝐾 ∧ 𝑏 ∈ 𝐾 ) → ( 𝑎 ⨣ 𝑏 ) ∈ 𝐾 ) ) |
190 |
83 189
|
syl |
⊢ ( 𝐹 ∈ Ring → ( ( 𝑎 ∈ 𝐾 ∧ 𝑏 ∈ 𝐾 ) → ( 𝑎 ⨣ 𝑏 ) ∈ 𝐾 ) ) |
191 |
190
|
3ad2ant2 |
⊢ ( ( 𝑅 ∈ Grp ∧ 𝐹 ∈ Ring ∧ ∀ 𝑞 ∈ 𝐾 ∀ 𝑟 ∈ 𝐾 ∀ 𝑥 ∈ 𝑉 ∀ 𝑤 ∈ 𝑉 ( ( ( 𝑤 · 𝑟 ) ∈ 𝑉 ∧ ( ( 𝑤 + 𝑥 ) · 𝑟 ) = ( ( 𝑤 · 𝑟 ) + ( 𝑥 · 𝑟 ) ) ∧ ( 𝑤 · ( 𝑞 ⨣ 𝑟 ) ) = ( ( 𝑤 · 𝑞 ) + ( 𝑤 · 𝑟 ) ) ) ∧ ( ( 𝑤 · ( 𝑞 × 𝑟 ) ) = ( ( 𝑤 · 𝑞 ) · 𝑟 ) ∧ ( 𝑤 · 1 ) = 𝑤 ) ) ) → ( ( 𝑎 ∈ 𝐾 ∧ 𝑏 ∈ 𝐾 ) → ( 𝑎 ⨣ 𝑏 ) ∈ 𝐾 ) ) |
192 |
9 191
|
ax-mp |
⊢ ( ( 𝑎 ∈ 𝐾 ∧ 𝑏 ∈ 𝐾 ) → ( 𝑎 ⨣ 𝑏 ) ∈ 𝐾 ) |
193 |
192
|
3adant3 |
⊢ ( ( 𝑎 ∈ 𝐾 ∧ 𝑏 ∈ 𝐾 ∧ 𝑐 ∈ 𝑉 ) → ( 𝑎 ⨣ 𝑏 ) ∈ 𝐾 ) |
194 |
|
simp3 |
⊢ ( ( 𝑎 ∈ 𝐾 ∧ 𝑏 ∈ 𝐾 ∧ 𝑐 ∈ 𝑉 ) → 𝑐 ∈ 𝑉 ) |
195 |
|
ovexd |
⊢ ( ( 𝑎 ∈ 𝐾 ∧ 𝑏 ∈ 𝐾 ∧ 𝑐 ∈ 𝑉 ) → ( 𝑐 · ( 𝑎 ⨣ 𝑏 ) ) ∈ V ) |
196 |
184 187 193 194 195
|
ovmpod |
⊢ ( ( 𝑎 ∈ 𝐾 ∧ 𝑏 ∈ 𝐾 ∧ 𝑐 ∈ 𝑉 ) → ( ( 𝑎 ⨣ 𝑏 ) ∗ 𝑐 ) = ( 𝑐 · ( 𝑎 ⨣ 𝑏 ) ) ) |
197 |
149
|
adantl |
⊢ ( ( ( 𝑎 ∈ 𝐾 ∧ 𝑏 ∈ 𝐾 ∧ 𝑐 ∈ 𝑉 ) ∧ ( 𝑠 = 𝑎 ∧ 𝑣 = 𝑐 ) ) → ( 𝑣 · 𝑠 ) = ( 𝑐 · 𝑎 ) ) |
198 |
|
simp1 |
⊢ ( ( 𝑎 ∈ 𝐾 ∧ 𝑏 ∈ 𝐾 ∧ 𝑐 ∈ 𝑉 ) → 𝑎 ∈ 𝐾 ) |
199 |
|
ovexd |
⊢ ( ( 𝑎 ∈ 𝐾 ∧ 𝑏 ∈ 𝐾 ∧ 𝑐 ∈ 𝑉 ) → ( 𝑐 · 𝑎 ) ∈ V ) |
200 |
184 197 198 194 199
|
ovmpod |
⊢ ( ( 𝑎 ∈ 𝐾 ∧ 𝑏 ∈ 𝐾 ∧ 𝑐 ∈ 𝑉 ) → ( 𝑎 ∗ 𝑐 ) = ( 𝑐 · 𝑎 ) ) |
201 |
|
oveq12 |
⊢ ( ( 𝑣 = 𝑐 ∧ 𝑠 = 𝑏 ) → ( 𝑣 · 𝑠 ) = ( 𝑐 · 𝑏 ) ) |
202 |
201
|
ancoms |
⊢ ( ( 𝑠 = 𝑏 ∧ 𝑣 = 𝑐 ) → ( 𝑣 · 𝑠 ) = ( 𝑐 · 𝑏 ) ) |
203 |
202
|
adantl |
⊢ ( ( ( 𝑎 ∈ 𝐾 ∧ 𝑏 ∈ 𝐾 ∧ 𝑐 ∈ 𝑉 ) ∧ ( 𝑠 = 𝑏 ∧ 𝑣 = 𝑐 ) ) → ( 𝑣 · 𝑠 ) = ( 𝑐 · 𝑏 ) ) |
204 |
|
simp2 |
⊢ ( ( 𝑎 ∈ 𝐾 ∧ 𝑏 ∈ 𝐾 ∧ 𝑐 ∈ 𝑉 ) → 𝑏 ∈ 𝐾 ) |
205 |
|
ovexd |
⊢ ( ( 𝑎 ∈ 𝐾 ∧ 𝑏 ∈ 𝐾 ∧ 𝑐 ∈ 𝑉 ) → ( 𝑐 · 𝑏 ) ∈ V ) |
206 |
184 203 204 194 205
|
ovmpod |
⊢ ( ( 𝑎 ∈ 𝐾 ∧ 𝑏 ∈ 𝐾 ∧ 𝑐 ∈ 𝑉 ) → ( 𝑏 ∗ 𝑐 ) = ( 𝑐 · 𝑏 ) ) |
207 |
200 206
|
oveq12d |
⊢ ( ( 𝑎 ∈ 𝐾 ∧ 𝑏 ∈ 𝐾 ∧ 𝑐 ∈ 𝑉 ) → ( ( 𝑎 ∗ 𝑐 ) + ( 𝑏 ∗ 𝑐 ) ) = ( ( 𝑐 · 𝑎 ) + ( 𝑐 · 𝑏 ) ) ) |
208 |
183 196 207
|
3eqtr4d |
⊢ ( ( 𝑎 ∈ 𝐾 ∧ 𝑏 ∈ 𝐾 ∧ 𝑐 ∈ 𝑉 ) → ( ( 𝑎 ⨣ 𝑏 ) ∗ 𝑐 ) = ( ( 𝑎 ∗ 𝑐 ) + ( 𝑏 ∗ 𝑐 ) ) ) |
209 |
208
|
adantl |
⊢ ( ( 𝐹 ∈ CRing ∧ ( 𝑎 ∈ 𝐾 ∧ 𝑏 ∈ 𝐾 ∧ 𝑐 ∈ 𝑉 ) ) → ( ( 𝑎 ⨣ 𝑏 ) ∗ 𝑐 ) = ( ( 𝑎 ∗ 𝑐 ) + ( 𝑏 ∗ 𝑐 ) ) ) |
210 |
1 2 3 4 5 6 7 8 9 10 11
|
rmodislmodlem |
⊢ ( ( 𝐹 ∈ CRing ∧ ( 𝑎 ∈ 𝐾 ∧ 𝑏 ∈ 𝐾 ∧ 𝑐 ∈ 𝑉 ) ) → ( ( 𝑎 × 𝑏 ) ∗ 𝑐 ) = ( 𝑎 ∗ ( 𝑏 ∗ 𝑐 ) ) ) |
211 |
10
|
a1i |
⊢ ( ( 𝐹 ∈ CRing ∧ 𝑎 ∈ 𝑉 ) → ∗ = ( 𝑠 ∈ 𝐾 , 𝑣 ∈ 𝑉 ↦ ( 𝑣 · 𝑠 ) ) ) |
212 |
|
oveq12 |
⊢ ( ( 𝑣 = 𝑎 ∧ 𝑠 = 1 ) → ( 𝑣 · 𝑠 ) = ( 𝑎 · 1 ) ) |
213 |
212
|
ancoms |
⊢ ( ( 𝑠 = 1 ∧ 𝑣 = 𝑎 ) → ( 𝑣 · 𝑠 ) = ( 𝑎 · 1 ) ) |
214 |
213
|
adantl |
⊢ ( ( ( 𝐹 ∈ CRing ∧ 𝑎 ∈ 𝑉 ) ∧ ( 𝑠 = 1 ∧ 𝑣 = 𝑎 ) ) → ( 𝑣 · 𝑠 ) = ( 𝑎 · 1 ) ) |
215 |
5 8
|
ringidcl |
⊢ ( 𝐹 ∈ Ring → 1 ∈ 𝐾 ) |
216 |
65 215
|
syl |
⊢ ( 𝐹 ∈ CRing → 1 ∈ 𝐾 ) |
217 |
216
|
adantr |
⊢ ( ( 𝐹 ∈ CRing ∧ 𝑎 ∈ 𝑉 ) → 1 ∈ 𝐾 ) |
218 |
|
simpr |
⊢ ( ( 𝐹 ∈ CRing ∧ 𝑎 ∈ 𝑉 ) → 𝑎 ∈ 𝑉 ) |
219 |
|
ovexd |
⊢ ( ( 𝐹 ∈ CRing ∧ 𝑎 ∈ 𝑉 ) → ( 𝑎 · 1 ) ∈ V ) |
220 |
211 214 217 218 219
|
ovmpod |
⊢ ( ( 𝐹 ∈ CRing ∧ 𝑎 ∈ 𝑉 ) → ( 1 ∗ 𝑎 ) = ( 𝑎 · 1 ) ) |
221 |
|
simprr |
⊢ ( ( ( ( 𝑤 · 𝑟 ) ∈ 𝑉 ∧ ( ( 𝑤 + 𝑥 ) · 𝑟 ) = ( ( 𝑤 · 𝑟 ) + ( 𝑥 · 𝑟 ) ) ∧ ( 𝑤 · ( 𝑞 ⨣ 𝑟 ) ) = ( ( 𝑤 · 𝑞 ) + ( 𝑤 · 𝑟 ) ) ) ∧ ( ( 𝑤 · ( 𝑞 × 𝑟 ) ) = ( ( 𝑤 · 𝑞 ) · 𝑟 ) ∧ ( 𝑤 · 1 ) = 𝑤 ) ) → ( 𝑤 · 1 ) = 𝑤 ) |
222 |
221
|
2ralimi |
⊢ ( ∀ 𝑥 ∈ 𝑉 ∀ 𝑤 ∈ 𝑉 ( ( ( 𝑤 · 𝑟 ) ∈ 𝑉 ∧ ( ( 𝑤 + 𝑥 ) · 𝑟 ) = ( ( 𝑤 · 𝑟 ) + ( 𝑥 · 𝑟 ) ) ∧ ( 𝑤 · ( 𝑞 ⨣ 𝑟 ) ) = ( ( 𝑤 · 𝑞 ) + ( 𝑤 · 𝑟 ) ) ) ∧ ( ( 𝑤 · ( 𝑞 × 𝑟 ) ) = ( ( 𝑤 · 𝑞 ) · 𝑟 ) ∧ ( 𝑤 · 1 ) = 𝑤 ) ) → ∀ 𝑥 ∈ 𝑉 ∀ 𝑤 ∈ 𝑉 ( 𝑤 · 1 ) = 𝑤 ) |
223 |
222
|
2ralimi |
⊢ ( ∀ 𝑞 ∈ 𝐾 ∀ 𝑟 ∈ 𝐾 ∀ 𝑥 ∈ 𝑉 ∀ 𝑤 ∈ 𝑉 ( ( ( 𝑤 · 𝑟 ) ∈ 𝑉 ∧ ( ( 𝑤 + 𝑥 ) · 𝑟 ) = ( ( 𝑤 · 𝑟 ) + ( 𝑥 · 𝑟 ) ) ∧ ( 𝑤 · ( 𝑞 ⨣ 𝑟 ) ) = ( ( 𝑤 · 𝑞 ) + ( 𝑤 · 𝑟 ) ) ) ∧ ( ( 𝑤 · ( 𝑞 × 𝑟 ) ) = ( ( 𝑤 · 𝑞 ) · 𝑟 ) ∧ ( 𝑤 · 1 ) = 𝑤 ) ) → ∀ 𝑞 ∈ 𝐾 ∀ 𝑟 ∈ 𝐾 ∀ 𝑥 ∈ 𝑉 ∀ 𝑤 ∈ 𝑉 ( 𝑤 · 1 ) = 𝑤 ) |
224 |
|
rspn0 |
⊢ ( 𝐾 ≠ ∅ → ( ∀ 𝑞 ∈ 𝐾 ∀ 𝑟 ∈ 𝐾 ∀ 𝑥 ∈ 𝑉 ∀ 𝑤 ∈ 𝑉 ( 𝑤 · 1 ) = 𝑤 → ∀ 𝑟 ∈ 𝐾 ∀ 𝑥 ∈ 𝑉 ∀ 𝑤 ∈ 𝑉 ( 𝑤 · 1 ) = 𝑤 ) ) |
225 |
87 224
|
ax-mp |
⊢ ( ∀ 𝑞 ∈ 𝐾 ∀ 𝑟 ∈ 𝐾 ∀ 𝑥 ∈ 𝑉 ∀ 𝑤 ∈ 𝑉 ( 𝑤 · 1 ) = 𝑤 → ∀ 𝑟 ∈ 𝐾 ∀ 𝑥 ∈ 𝑉 ∀ 𝑤 ∈ 𝑉 ( 𝑤 · 1 ) = 𝑤 ) |
226 |
|
rspn0 |
⊢ ( 𝐾 ≠ ∅ → ( ∀ 𝑟 ∈ 𝐾 ∀ 𝑥 ∈ 𝑉 ∀ 𝑤 ∈ 𝑉 ( 𝑤 · 1 ) = 𝑤 → ∀ 𝑥 ∈ 𝑉 ∀ 𝑤 ∈ 𝑉 ( 𝑤 · 1 ) = 𝑤 ) ) |
227 |
87 226
|
ax-mp |
⊢ ( ∀ 𝑟 ∈ 𝐾 ∀ 𝑥 ∈ 𝑉 ∀ 𝑤 ∈ 𝑉 ( 𝑤 · 1 ) = 𝑤 → ∀ 𝑥 ∈ 𝑉 ∀ 𝑤 ∈ 𝑉 ( 𝑤 · 1 ) = 𝑤 ) |
228 |
|
rspn0 |
⊢ ( 𝑉 ≠ ∅ → ( ∀ 𝑥 ∈ 𝑉 ∀ 𝑤 ∈ 𝑉 ( 𝑤 · 1 ) = 𝑤 → ∀ 𝑤 ∈ 𝑉 ( 𝑤 · 1 ) = 𝑤 ) ) |
229 |
93 228
|
ax-mp |
⊢ ( ∀ 𝑥 ∈ 𝑉 ∀ 𝑤 ∈ 𝑉 ( 𝑤 · 1 ) = 𝑤 → ∀ 𝑤 ∈ 𝑉 ( 𝑤 · 1 ) = 𝑤 ) |
230 |
|
oveq1 |
⊢ ( 𝑤 = 𝑎 → ( 𝑤 · 1 ) = ( 𝑎 · 1 ) ) |
231 |
|
id |
⊢ ( 𝑤 = 𝑎 → 𝑤 = 𝑎 ) |
232 |
230 231
|
eqeq12d |
⊢ ( 𝑤 = 𝑎 → ( ( 𝑤 · 1 ) = 𝑤 ↔ ( 𝑎 · 1 ) = 𝑎 ) ) |
233 |
232
|
rspcv |
⊢ ( 𝑎 ∈ 𝑉 → ( ∀ 𝑤 ∈ 𝑉 ( 𝑤 · 1 ) = 𝑤 → ( 𝑎 · 1 ) = 𝑎 ) ) |
234 |
233
|
adantl |
⊢ ( ( 𝐹 ∈ CRing ∧ 𝑎 ∈ 𝑉 ) → ( ∀ 𝑤 ∈ 𝑉 ( 𝑤 · 1 ) = 𝑤 → ( 𝑎 · 1 ) = 𝑎 ) ) |
235 |
229 234
|
syl5com |
⊢ ( ∀ 𝑥 ∈ 𝑉 ∀ 𝑤 ∈ 𝑉 ( 𝑤 · 1 ) = 𝑤 → ( ( 𝐹 ∈ CRing ∧ 𝑎 ∈ 𝑉 ) → ( 𝑎 · 1 ) = 𝑎 ) ) |
236 |
227 235
|
syl |
⊢ ( ∀ 𝑟 ∈ 𝐾 ∀ 𝑥 ∈ 𝑉 ∀ 𝑤 ∈ 𝑉 ( 𝑤 · 1 ) = 𝑤 → ( ( 𝐹 ∈ CRing ∧ 𝑎 ∈ 𝑉 ) → ( 𝑎 · 1 ) = 𝑎 ) ) |
237 |
223 225 236
|
3syl |
⊢ ( ∀ 𝑞 ∈ 𝐾 ∀ 𝑟 ∈ 𝐾 ∀ 𝑥 ∈ 𝑉 ∀ 𝑤 ∈ 𝑉 ( ( ( 𝑤 · 𝑟 ) ∈ 𝑉 ∧ ( ( 𝑤 + 𝑥 ) · 𝑟 ) = ( ( 𝑤 · 𝑟 ) + ( 𝑥 · 𝑟 ) ) ∧ ( 𝑤 · ( 𝑞 ⨣ 𝑟 ) ) = ( ( 𝑤 · 𝑞 ) + ( 𝑤 · 𝑟 ) ) ) ∧ ( ( 𝑤 · ( 𝑞 × 𝑟 ) ) = ( ( 𝑤 · 𝑞 ) · 𝑟 ) ∧ ( 𝑤 · 1 ) = 𝑤 ) ) → ( ( 𝐹 ∈ CRing ∧ 𝑎 ∈ 𝑉 ) → ( 𝑎 · 1 ) = 𝑎 ) ) |
238 |
237
|
3ad2ant3 |
⊢ ( ( 𝑅 ∈ Grp ∧ 𝐹 ∈ Ring ∧ ∀ 𝑞 ∈ 𝐾 ∀ 𝑟 ∈ 𝐾 ∀ 𝑥 ∈ 𝑉 ∀ 𝑤 ∈ 𝑉 ( ( ( 𝑤 · 𝑟 ) ∈ 𝑉 ∧ ( ( 𝑤 + 𝑥 ) · 𝑟 ) = ( ( 𝑤 · 𝑟 ) + ( 𝑥 · 𝑟 ) ) ∧ ( 𝑤 · ( 𝑞 ⨣ 𝑟 ) ) = ( ( 𝑤 · 𝑞 ) + ( 𝑤 · 𝑟 ) ) ) ∧ ( ( 𝑤 · ( 𝑞 × 𝑟 ) ) = ( ( 𝑤 · 𝑞 ) · 𝑟 ) ∧ ( 𝑤 · 1 ) = 𝑤 ) ) ) → ( ( 𝐹 ∈ CRing ∧ 𝑎 ∈ 𝑉 ) → ( 𝑎 · 1 ) = 𝑎 ) ) |
239 |
9 238
|
ax-mp |
⊢ ( ( 𝐹 ∈ CRing ∧ 𝑎 ∈ 𝑉 ) → ( 𝑎 · 1 ) = 𝑎 ) |
240 |
220 239
|
eqtrd |
⊢ ( ( 𝐹 ∈ CRing ∧ 𝑎 ∈ 𝑉 ) → ( 1 ∗ 𝑎 ) = 𝑎 ) |
241 |
27 38 49 60 61 62 63 64 65 71 107 156 209 210 240
|
islmodd |
⊢ ( 𝐹 ∈ CRing → 𝐿 ∈ LMod ) |