Step |
Hyp |
Ref |
Expression |
1 |
|
imaslmod.u |
⊢ ( 𝜑 → 𝑁 = ( 𝐹 “s 𝑀 ) ) |
2 |
|
imaslmod.v |
⊢ 𝑉 = ( Base ‘ 𝑀 ) |
3 |
|
imaslmod.k |
⊢ 𝑆 = ( Base ‘ ( Scalar ‘ 𝑀 ) ) |
4 |
|
imaslmod.p |
⊢ + = ( +g ‘ 𝑀 ) |
5 |
|
imaslmod.t |
⊢ · = ( ·𝑠 ‘ 𝑀 ) |
6 |
|
imaslmod.o |
⊢ 0 = ( 0g ‘ 𝑀 ) |
7 |
|
imaslmod.f |
⊢ ( 𝜑 → 𝐹 : 𝑉 –onto→ 𝐵 ) |
8 |
|
imaslmod.e1 |
⊢ ( ( 𝜑 ∧ ( 𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉 ) ∧ ( 𝑝 ∈ 𝑉 ∧ 𝑞 ∈ 𝑉 ) ) → ( ( ( 𝐹 ‘ 𝑎 ) = ( 𝐹 ‘ 𝑝 ) ∧ ( 𝐹 ‘ 𝑏 ) = ( 𝐹 ‘ 𝑞 ) ) → ( 𝐹 ‘ ( 𝑎 + 𝑏 ) ) = ( 𝐹 ‘ ( 𝑝 + 𝑞 ) ) ) ) |
9 |
|
imaslmod.e2 |
⊢ ( ( 𝜑 ∧ ( 𝑘 ∈ 𝑆 ∧ 𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉 ) ) → ( ( 𝐹 ‘ 𝑎 ) = ( 𝐹 ‘ 𝑏 ) → ( 𝐹 ‘ ( 𝑘 · 𝑎 ) ) = ( 𝐹 ‘ ( 𝑘 · 𝑏 ) ) ) ) |
10 |
|
imaslmod.l |
⊢ ( 𝜑 → 𝑀 ∈ LMod ) |
11 |
2
|
a1i |
⊢ ( 𝜑 → 𝑉 = ( Base ‘ 𝑀 ) ) |
12 |
1 11 7 10
|
imasbas |
⊢ ( 𝜑 → 𝐵 = ( Base ‘ 𝑁 ) ) |
13 |
|
eqidd |
⊢ ( 𝜑 → ( +g ‘ 𝑁 ) = ( +g ‘ 𝑁 ) ) |
14 |
|
eqid |
⊢ ( Scalar ‘ 𝑀 ) = ( Scalar ‘ 𝑀 ) |
15 |
1 11 7 10 14
|
imassca |
⊢ ( 𝜑 → ( Scalar ‘ 𝑀 ) = ( Scalar ‘ 𝑁 ) ) |
16 |
|
eqidd |
⊢ ( 𝜑 → ( ·𝑠 ‘ 𝑁 ) = ( ·𝑠 ‘ 𝑁 ) ) |
17 |
3
|
a1i |
⊢ ( 𝜑 → 𝑆 = ( Base ‘ ( Scalar ‘ 𝑀 ) ) ) |
18 |
|
eqidd |
⊢ ( 𝜑 → ( +g ‘ ( Scalar ‘ 𝑀 ) ) = ( +g ‘ ( Scalar ‘ 𝑀 ) ) ) |
19 |
|
eqidd |
⊢ ( 𝜑 → ( .r ‘ ( Scalar ‘ 𝑀 ) ) = ( .r ‘ ( Scalar ‘ 𝑀 ) ) ) |
20 |
|
eqidd |
⊢ ( 𝜑 → ( 1r ‘ ( Scalar ‘ 𝑀 ) ) = ( 1r ‘ ( Scalar ‘ 𝑀 ) ) ) |
21 |
14
|
lmodring |
⊢ ( 𝑀 ∈ LMod → ( Scalar ‘ 𝑀 ) ∈ Ring ) |
22 |
10 21
|
syl |
⊢ ( 𝜑 → ( Scalar ‘ 𝑀 ) ∈ Ring ) |
23 |
4
|
a1i |
⊢ ( 𝜑 → + = ( +g ‘ 𝑀 ) ) |
24 |
|
lmodgrp |
⊢ ( 𝑀 ∈ LMod → 𝑀 ∈ Grp ) |
25 |
10 24
|
syl |
⊢ ( 𝜑 → 𝑀 ∈ Grp ) |
26 |
1 11 23 7 8 25 6
|
imasgrp |
⊢ ( 𝜑 → ( 𝑁 ∈ Grp ∧ ( 𝐹 ‘ 0 ) = ( 0g ‘ 𝑁 ) ) ) |
27 |
26
|
simpld |
⊢ ( 𝜑 → 𝑁 ∈ Grp ) |
28 |
|
eqid |
⊢ ( ·𝑠 ‘ 𝑁 ) = ( ·𝑠 ‘ 𝑁 ) |
29 |
10
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑘 ∈ 𝑆 ∧ 𝑏 ∈ 𝑉 ) ) → 𝑀 ∈ LMod ) |
30 |
|
simprl |
⊢ ( ( 𝜑 ∧ ( 𝑘 ∈ 𝑆 ∧ 𝑏 ∈ 𝑉 ) ) → 𝑘 ∈ 𝑆 ) |
31 |
|
simprr |
⊢ ( ( 𝜑 ∧ ( 𝑘 ∈ 𝑆 ∧ 𝑏 ∈ 𝑉 ) ) → 𝑏 ∈ 𝑉 ) |
32 |
2 14 5 3
|
lmodvscl |
⊢ ( ( 𝑀 ∈ LMod ∧ 𝑘 ∈ 𝑆 ∧ 𝑏 ∈ 𝑉 ) → ( 𝑘 · 𝑏 ) ∈ 𝑉 ) |
33 |
29 30 31 32
|
syl3anc |
⊢ ( ( 𝜑 ∧ ( 𝑘 ∈ 𝑆 ∧ 𝑏 ∈ 𝑉 ) ) → ( 𝑘 · 𝑏 ) ∈ 𝑉 ) |
34 |
1 11 7 10 14 3 5 28 9 33
|
imasvscaf |
⊢ ( 𝜑 → ( ·𝑠 ‘ 𝑁 ) : ( 𝑆 × 𝐵 ) ⟶ 𝐵 ) |
35 |
34
|
fovcld |
⊢ ( ( 𝜑 ∧ 𝑢 ∈ 𝑆 ∧ 𝑣 ∈ 𝐵 ) → ( 𝑢 ( ·𝑠 ‘ 𝑁 ) 𝑣 ) ∈ 𝐵 ) |
36 |
|
simp-5l |
⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑢 ∈ 𝑆 ∧ 𝑣 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ) ) ∧ 𝑧 ∈ 𝑉 ) ∧ ( 𝐹 ‘ 𝑧 ) = 𝑤 ) ∧ 𝑦 ∈ 𝑉 ) ∧ ( 𝐹 ‘ 𝑦 ) = 𝑣 ) → 𝜑 ) |
37 |
|
simpllr |
⊢ ( ( ( ( 𝜑 ∧ ( 𝑢 ∈ 𝑆 ∧ 𝑣 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ) ) ∧ 𝑧 ∈ 𝑉 ) ∧ ( 𝐹 ‘ 𝑧 ) = 𝑤 ) → ( 𝑢 ∈ 𝑆 ∧ 𝑣 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ) ) |
38 |
37
|
simp1d |
⊢ ( ( ( ( 𝜑 ∧ ( 𝑢 ∈ 𝑆 ∧ 𝑣 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ) ) ∧ 𝑧 ∈ 𝑉 ) ∧ ( 𝐹 ‘ 𝑧 ) = 𝑤 ) → 𝑢 ∈ 𝑆 ) |
39 |
38
|
ad2antrr |
⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑢 ∈ 𝑆 ∧ 𝑣 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ) ) ∧ 𝑧 ∈ 𝑉 ) ∧ ( 𝐹 ‘ 𝑧 ) = 𝑤 ) ∧ 𝑦 ∈ 𝑉 ) ∧ ( 𝐹 ‘ 𝑦 ) = 𝑣 ) → 𝑢 ∈ 𝑆 ) |
40 |
36 25
|
syl |
⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑢 ∈ 𝑆 ∧ 𝑣 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ) ) ∧ 𝑧 ∈ 𝑉 ) ∧ ( 𝐹 ‘ 𝑧 ) = 𝑤 ) ∧ 𝑦 ∈ 𝑉 ) ∧ ( 𝐹 ‘ 𝑦 ) = 𝑣 ) → 𝑀 ∈ Grp ) |
41 |
|
simplr |
⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑢 ∈ 𝑆 ∧ 𝑣 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ) ) ∧ 𝑧 ∈ 𝑉 ) ∧ ( 𝐹 ‘ 𝑧 ) = 𝑤 ) ∧ 𝑦 ∈ 𝑉 ) ∧ ( 𝐹 ‘ 𝑦 ) = 𝑣 ) → 𝑦 ∈ 𝑉 ) |
42 |
|
simp-4r |
⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑢 ∈ 𝑆 ∧ 𝑣 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ) ) ∧ 𝑧 ∈ 𝑉 ) ∧ ( 𝐹 ‘ 𝑧 ) = 𝑤 ) ∧ 𝑦 ∈ 𝑉 ) ∧ ( 𝐹 ‘ 𝑦 ) = 𝑣 ) → 𝑧 ∈ 𝑉 ) |
43 |
2 4
|
grpcl |
⊢ ( ( 𝑀 ∈ Grp ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉 ) → ( 𝑦 + 𝑧 ) ∈ 𝑉 ) |
44 |
40 41 42 43
|
syl3anc |
⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑢 ∈ 𝑆 ∧ 𝑣 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ) ) ∧ 𝑧 ∈ 𝑉 ) ∧ ( 𝐹 ‘ 𝑧 ) = 𝑤 ) ∧ 𝑦 ∈ 𝑉 ) ∧ ( 𝐹 ‘ 𝑦 ) = 𝑣 ) → ( 𝑦 + 𝑧 ) ∈ 𝑉 ) |
45 |
1 11 7 10 14 3 5 28 9
|
imasvscaval |
⊢ ( ( 𝜑 ∧ 𝑢 ∈ 𝑆 ∧ ( 𝑦 + 𝑧 ) ∈ 𝑉 ) → ( 𝑢 ( ·𝑠 ‘ 𝑁 ) ( 𝐹 ‘ ( 𝑦 + 𝑧 ) ) ) = ( 𝐹 ‘ ( 𝑢 · ( 𝑦 + 𝑧 ) ) ) ) |
46 |
36 39 44 45
|
syl3anc |
⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑢 ∈ 𝑆 ∧ 𝑣 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ) ) ∧ 𝑧 ∈ 𝑉 ) ∧ ( 𝐹 ‘ 𝑧 ) = 𝑤 ) ∧ 𝑦 ∈ 𝑉 ) ∧ ( 𝐹 ‘ 𝑦 ) = 𝑣 ) → ( 𝑢 ( ·𝑠 ‘ 𝑁 ) ( 𝐹 ‘ ( 𝑦 + 𝑧 ) ) ) = ( 𝐹 ‘ ( 𝑢 · ( 𝑦 + 𝑧 ) ) ) ) |
47 |
|
eqid |
⊢ ( +g ‘ 𝑁 ) = ( +g ‘ 𝑁 ) |
48 |
7 8 1 11 10 4 47
|
imasaddval |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉 ) → ( ( 𝐹 ‘ 𝑦 ) ( +g ‘ 𝑁 ) ( 𝐹 ‘ 𝑧 ) ) = ( 𝐹 ‘ ( 𝑦 + 𝑧 ) ) ) |
49 |
36 41 42 48
|
syl3anc |
⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑢 ∈ 𝑆 ∧ 𝑣 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ) ) ∧ 𝑧 ∈ 𝑉 ) ∧ ( 𝐹 ‘ 𝑧 ) = 𝑤 ) ∧ 𝑦 ∈ 𝑉 ) ∧ ( 𝐹 ‘ 𝑦 ) = 𝑣 ) → ( ( 𝐹 ‘ 𝑦 ) ( +g ‘ 𝑁 ) ( 𝐹 ‘ 𝑧 ) ) = ( 𝐹 ‘ ( 𝑦 + 𝑧 ) ) ) |
50 |
|
simpr |
⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑢 ∈ 𝑆 ∧ 𝑣 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ) ) ∧ 𝑧 ∈ 𝑉 ) ∧ ( 𝐹 ‘ 𝑧 ) = 𝑤 ) ∧ 𝑦 ∈ 𝑉 ) ∧ ( 𝐹 ‘ 𝑦 ) = 𝑣 ) → ( 𝐹 ‘ 𝑦 ) = 𝑣 ) |
51 |
|
simpllr |
⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑢 ∈ 𝑆 ∧ 𝑣 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ) ) ∧ 𝑧 ∈ 𝑉 ) ∧ ( 𝐹 ‘ 𝑧 ) = 𝑤 ) ∧ 𝑦 ∈ 𝑉 ) ∧ ( 𝐹 ‘ 𝑦 ) = 𝑣 ) → ( 𝐹 ‘ 𝑧 ) = 𝑤 ) |
52 |
50 51
|
oveq12d |
⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑢 ∈ 𝑆 ∧ 𝑣 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ) ) ∧ 𝑧 ∈ 𝑉 ) ∧ ( 𝐹 ‘ 𝑧 ) = 𝑤 ) ∧ 𝑦 ∈ 𝑉 ) ∧ ( 𝐹 ‘ 𝑦 ) = 𝑣 ) → ( ( 𝐹 ‘ 𝑦 ) ( +g ‘ 𝑁 ) ( 𝐹 ‘ 𝑧 ) ) = ( 𝑣 ( +g ‘ 𝑁 ) 𝑤 ) ) |
53 |
49 52
|
eqtr3d |
⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑢 ∈ 𝑆 ∧ 𝑣 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ) ) ∧ 𝑧 ∈ 𝑉 ) ∧ ( 𝐹 ‘ 𝑧 ) = 𝑤 ) ∧ 𝑦 ∈ 𝑉 ) ∧ ( 𝐹 ‘ 𝑦 ) = 𝑣 ) → ( 𝐹 ‘ ( 𝑦 + 𝑧 ) ) = ( 𝑣 ( +g ‘ 𝑁 ) 𝑤 ) ) |
54 |
53
|
oveq2d |
⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑢 ∈ 𝑆 ∧ 𝑣 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ) ) ∧ 𝑧 ∈ 𝑉 ) ∧ ( 𝐹 ‘ 𝑧 ) = 𝑤 ) ∧ 𝑦 ∈ 𝑉 ) ∧ ( 𝐹 ‘ 𝑦 ) = 𝑣 ) → ( 𝑢 ( ·𝑠 ‘ 𝑁 ) ( 𝐹 ‘ ( 𝑦 + 𝑧 ) ) ) = ( 𝑢 ( ·𝑠 ‘ 𝑁 ) ( 𝑣 ( +g ‘ 𝑁 ) 𝑤 ) ) ) |
55 |
36 10
|
syl |
⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑢 ∈ 𝑆 ∧ 𝑣 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ) ) ∧ 𝑧 ∈ 𝑉 ) ∧ ( 𝐹 ‘ 𝑧 ) = 𝑤 ) ∧ 𝑦 ∈ 𝑉 ) ∧ ( 𝐹 ‘ 𝑦 ) = 𝑣 ) → 𝑀 ∈ LMod ) |
56 |
2 4 14 5 3
|
lmodvsdi |
⊢ ( ( 𝑀 ∈ LMod ∧ ( 𝑢 ∈ 𝑆 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉 ) ) → ( 𝑢 · ( 𝑦 + 𝑧 ) ) = ( ( 𝑢 · 𝑦 ) + ( 𝑢 · 𝑧 ) ) ) |
57 |
55 39 41 42 56
|
syl13anc |
⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑢 ∈ 𝑆 ∧ 𝑣 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ) ) ∧ 𝑧 ∈ 𝑉 ) ∧ ( 𝐹 ‘ 𝑧 ) = 𝑤 ) ∧ 𝑦 ∈ 𝑉 ) ∧ ( 𝐹 ‘ 𝑦 ) = 𝑣 ) → ( 𝑢 · ( 𝑦 + 𝑧 ) ) = ( ( 𝑢 · 𝑦 ) + ( 𝑢 · 𝑧 ) ) ) |
58 |
57
|
fveq2d |
⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑢 ∈ 𝑆 ∧ 𝑣 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ) ) ∧ 𝑧 ∈ 𝑉 ) ∧ ( 𝐹 ‘ 𝑧 ) = 𝑤 ) ∧ 𝑦 ∈ 𝑉 ) ∧ ( 𝐹 ‘ 𝑦 ) = 𝑣 ) → ( 𝐹 ‘ ( 𝑢 · ( 𝑦 + 𝑧 ) ) ) = ( 𝐹 ‘ ( ( 𝑢 · 𝑦 ) + ( 𝑢 · 𝑧 ) ) ) ) |
59 |
46 54 58
|
3eqtr3d |
⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑢 ∈ 𝑆 ∧ 𝑣 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ) ) ∧ 𝑧 ∈ 𝑉 ) ∧ ( 𝐹 ‘ 𝑧 ) = 𝑤 ) ∧ 𝑦 ∈ 𝑉 ) ∧ ( 𝐹 ‘ 𝑦 ) = 𝑣 ) → ( 𝑢 ( ·𝑠 ‘ 𝑁 ) ( 𝑣 ( +g ‘ 𝑁 ) 𝑤 ) ) = ( 𝐹 ‘ ( ( 𝑢 · 𝑦 ) + ( 𝑢 · 𝑧 ) ) ) ) |
60 |
2 14 5 3
|
lmodvscl |
⊢ ( ( 𝑀 ∈ LMod ∧ 𝑢 ∈ 𝑆 ∧ 𝑦 ∈ 𝑉 ) → ( 𝑢 · 𝑦 ) ∈ 𝑉 ) |
61 |
55 39 41 60
|
syl3anc |
⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑢 ∈ 𝑆 ∧ 𝑣 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ) ) ∧ 𝑧 ∈ 𝑉 ) ∧ ( 𝐹 ‘ 𝑧 ) = 𝑤 ) ∧ 𝑦 ∈ 𝑉 ) ∧ ( 𝐹 ‘ 𝑦 ) = 𝑣 ) → ( 𝑢 · 𝑦 ) ∈ 𝑉 ) |
62 |
2 14 5 3
|
lmodvscl |
⊢ ( ( 𝑀 ∈ LMod ∧ 𝑢 ∈ 𝑆 ∧ 𝑧 ∈ 𝑉 ) → ( 𝑢 · 𝑧 ) ∈ 𝑉 ) |
63 |
55 39 42 62
|
syl3anc |
⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑢 ∈ 𝑆 ∧ 𝑣 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ) ) ∧ 𝑧 ∈ 𝑉 ) ∧ ( 𝐹 ‘ 𝑧 ) = 𝑤 ) ∧ 𝑦 ∈ 𝑉 ) ∧ ( 𝐹 ‘ 𝑦 ) = 𝑣 ) → ( 𝑢 · 𝑧 ) ∈ 𝑉 ) |
64 |
7 8 1 11 10 4 47
|
imasaddval |
⊢ ( ( 𝜑 ∧ ( 𝑢 · 𝑦 ) ∈ 𝑉 ∧ ( 𝑢 · 𝑧 ) ∈ 𝑉 ) → ( ( 𝐹 ‘ ( 𝑢 · 𝑦 ) ) ( +g ‘ 𝑁 ) ( 𝐹 ‘ ( 𝑢 · 𝑧 ) ) ) = ( 𝐹 ‘ ( ( 𝑢 · 𝑦 ) + ( 𝑢 · 𝑧 ) ) ) ) |
65 |
36 61 63 64
|
syl3anc |
⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑢 ∈ 𝑆 ∧ 𝑣 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ) ) ∧ 𝑧 ∈ 𝑉 ) ∧ ( 𝐹 ‘ 𝑧 ) = 𝑤 ) ∧ 𝑦 ∈ 𝑉 ) ∧ ( 𝐹 ‘ 𝑦 ) = 𝑣 ) → ( ( 𝐹 ‘ ( 𝑢 · 𝑦 ) ) ( +g ‘ 𝑁 ) ( 𝐹 ‘ ( 𝑢 · 𝑧 ) ) ) = ( 𝐹 ‘ ( ( 𝑢 · 𝑦 ) + ( 𝑢 · 𝑧 ) ) ) ) |
66 |
1 11 7 10 14 3 5 28 9
|
imasvscaval |
⊢ ( ( 𝜑 ∧ 𝑢 ∈ 𝑆 ∧ 𝑦 ∈ 𝑉 ) → ( 𝑢 ( ·𝑠 ‘ 𝑁 ) ( 𝐹 ‘ 𝑦 ) ) = ( 𝐹 ‘ ( 𝑢 · 𝑦 ) ) ) |
67 |
36 39 41 66
|
syl3anc |
⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑢 ∈ 𝑆 ∧ 𝑣 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ) ) ∧ 𝑧 ∈ 𝑉 ) ∧ ( 𝐹 ‘ 𝑧 ) = 𝑤 ) ∧ 𝑦 ∈ 𝑉 ) ∧ ( 𝐹 ‘ 𝑦 ) = 𝑣 ) → ( 𝑢 ( ·𝑠 ‘ 𝑁 ) ( 𝐹 ‘ 𝑦 ) ) = ( 𝐹 ‘ ( 𝑢 · 𝑦 ) ) ) |
68 |
50
|
oveq2d |
⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑢 ∈ 𝑆 ∧ 𝑣 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ) ) ∧ 𝑧 ∈ 𝑉 ) ∧ ( 𝐹 ‘ 𝑧 ) = 𝑤 ) ∧ 𝑦 ∈ 𝑉 ) ∧ ( 𝐹 ‘ 𝑦 ) = 𝑣 ) → ( 𝑢 ( ·𝑠 ‘ 𝑁 ) ( 𝐹 ‘ 𝑦 ) ) = ( 𝑢 ( ·𝑠 ‘ 𝑁 ) 𝑣 ) ) |
69 |
67 68
|
eqtr3d |
⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑢 ∈ 𝑆 ∧ 𝑣 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ) ) ∧ 𝑧 ∈ 𝑉 ) ∧ ( 𝐹 ‘ 𝑧 ) = 𝑤 ) ∧ 𝑦 ∈ 𝑉 ) ∧ ( 𝐹 ‘ 𝑦 ) = 𝑣 ) → ( 𝐹 ‘ ( 𝑢 · 𝑦 ) ) = ( 𝑢 ( ·𝑠 ‘ 𝑁 ) 𝑣 ) ) |
70 |
1 11 7 10 14 3 5 28 9
|
imasvscaval |
⊢ ( ( 𝜑 ∧ 𝑢 ∈ 𝑆 ∧ 𝑧 ∈ 𝑉 ) → ( 𝑢 ( ·𝑠 ‘ 𝑁 ) ( 𝐹 ‘ 𝑧 ) ) = ( 𝐹 ‘ ( 𝑢 · 𝑧 ) ) ) |
71 |
36 39 42 70
|
syl3anc |
⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑢 ∈ 𝑆 ∧ 𝑣 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ) ) ∧ 𝑧 ∈ 𝑉 ) ∧ ( 𝐹 ‘ 𝑧 ) = 𝑤 ) ∧ 𝑦 ∈ 𝑉 ) ∧ ( 𝐹 ‘ 𝑦 ) = 𝑣 ) → ( 𝑢 ( ·𝑠 ‘ 𝑁 ) ( 𝐹 ‘ 𝑧 ) ) = ( 𝐹 ‘ ( 𝑢 · 𝑧 ) ) ) |
72 |
51
|
oveq2d |
⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑢 ∈ 𝑆 ∧ 𝑣 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ) ) ∧ 𝑧 ∈ 𝑉 ) ∧ ( 𝐹 ‘ 𝑧 ) = 𝑤 ) ∧ 𝑦 ∈ 𝑉 ) ∧ ( 𝐹 ‘ 𝑦 ) = 𝑣 ) → ( 𝑢 ( ·𝑠 ‘ 𝑁 ) ( 𝐹 ‘ 𝑧 ) ) = ( 𝑢 ( ·𝑠 ‘ 𝑁 ) 𝑤 ) ) |
73 |
71 72
|
eqtr3d |
⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑢 ∈ 𝑆 ∧ 𝑣 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ) ) ∧ 𝑧 ∈ 𝑉 ) ∧ ( 𝐹 ‘ 𝑧 ) = 𝑤 ) ∧ 𝑦 ∈ 𝑉 ) ∧ ( 𝐹 ‘ 𝑦 ) = 𝑣 ) → ( 𝐹 ‘ ( 𝑢 · 𝑧 ) ) = ( 𝑢 ( ·𝑠 ‘ 𝑁 ) 𝑤 ) ) |
74 |
69 73
|
oveq12d |
⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑢 ∈ 𝑆 ∧ 𝑣 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ) ) ∧ 𝑧 ∈ 𝑉 ) ∧ ( 𝐹 ‘ 𝑧 ) = 𝑤 ) ∧ 𝑦 ∈ 𝑉 ) ∧ ( 𝐹 ‘ 𝑦 ) = 𝑣 ) → ( ( 𝐹 ‘ ( 𝑢 · 𝑦 ) ) ( +g ‘ 𝑁 ) ( 𝐹 ‘ ( 𝑢 · 𝑧 ) ) ) = ( ( 𝑢 ( ·𝑠 ‘ 𝑁 ) 𝑣 ) ( +g ‘ 𝑁 ) ( 𝑢 ( ·𝑠 ‘ 𝑁 ) 𝑤 ) ) ) |
75 |
59 65 74
|
3eqtr2d |
⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑢 ∈ 𝑆 ∧ 𝑣 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ) ) ∧ 𝑧 ∈ 𝑉 ) ∧ ( 𝐹 ‘ 𝑧 ) = 𝑤 ) ∧ 𝑦 ∈ 𝑉 ) ∧ ( 𝐹 ‘ 𝑦 ) = 𝑣 ) → ( 𝑢 ( ·𝑠 ‘ 𝑁 ) ( 𝑣 ( +g ‘ 𝑁 ) 𝑤 ) ) = ( ( 𝑢 ( ·𝑠 ‘ 𝑁 ) 𝑣 ) ( +g ‘ 𝑁 ) ( 𝑢 ( ·𝑠 ‘ 𝑁 ) 𝑤 ) ) ) |
76 |
|
simplll |
⊢ ( ( ( ( 𝜑 ∧ ( 𝑢 ∈ 𝑆 ∧ 𝑣 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ) ) ∧ 𝑧 ∈ 𝑉 ) ∧ ( 𝐹 ‘ 𝑧 ) = 𝑤 ) → 𝜑 ) |
77 |
37
|
simp2d |
⊢ ( ( ( ( 𝜑 ∧ ( 𝑢 ∈ 𝑆 ∧ 𝑣 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ) ) ∧ 𝑧 ∈ 𝑉 ) ∧ ( 𝐹 ‘ 𝑧 ) = 𝑤 ) → 𝑣 ∈ 𝐵 ) |
78 |
|
fofn |
⊢ ( 𝐹 : 𝑉 –onto→ 𝐵 → 𝐹 Fn 𝑉 ) |
79 |
7 78
|
syl |
⊢ ( 𝜑 → 𝐹 Fn 𝑉 ) |
80 |
|
simpr |
⊢ ( ( 𝜑 ∧ 𝑣 ∈ 𝐵 ) → 𝑣 ∈ 𝐵 ) |
81 |
|
forn |
⊢ ( 𝐹 : 𝑉 –onto→ 𝐵 → ran 𝐹 = 𝐵 ) |
82 |
7 81
|
syl |
⊢ ( 𝜑 → ran 𝐹 = 𝐵 ) |
83 |
82
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑣 ∈ 𝐵 ) → ran 𝐹 = 𝐵 ) |
84 |
80 83
|
eleqtrrd |
⊢ ( ( 𝜑 ∧ 𝑣 ∈ 𝐵 ) → 𝑣 ∈ ran 𝐹 ) |
85 |
|
fvelrnb |
⊢ ( 𝐹 Fn 𝑉 → ( 𝑣 ∈ ran 𝐹 ↔ ∃ 𝑦 ∈ 𝑉 ( 𝐹 ‘ 𝑦 ) = 𝑣 ) ) |
86 |
85
|
biimpa |
⊢ ( ( 𝐹 Fn 𝑉 ∧ 𝑣 ∈ ran 𝐹 ) → ∃ 𝑦 ∈ 𝑉 ( 𝐹 ‘ 𝑦 ) = 𝑣 ) |
87 |
79 84 86
|
syl2an2r |
⊢ ( ( 𝜑 ∧ 𝑣 ∈ 𝐵 ) → ∃ 𝑦 ∈ 𝑉 ( 𝐹 ‘ 𝑦 ) = 𝑣 ) |
88 |
76 77 87
|
syl2anc |
⊢ ( ( ( ( 𝜑 ∧ ( 𝑢 ∈ 𝑆 ∧ 𝑣 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ) ) ∧ 𝑧 ∈ 𝑉 ) ∧ ( 𝐹 ‘ 𝑧 ) = 𝑤 ) → ∃ 𝑦 ∈ 𝑉 ( 𝐹 ‘ 𝑦 ) = 𝑣 ) |
89 |
75 88
|
r19.29a |
⊢ ( ( ( ( 𝜑 ∧ ( 𝑢 ∈ 𝑆 ∧ 𝑣 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ) ) ∧ 𝑧 ∈ 𝑉 ) ∧ ( 𝐹 ‘ 𝑧 ) = 𝑤 ) → ( 𝑢 ( ·𝑠 ‘ 𝑁 ) ( 𝑣 ( +g ‘ 𝑁 ) 𝑤 ) ) = ( ( 𝑢 ( ·𝑠 ‘ 𝑁 ) 𝑣 ) ( +g ‘ 𝑁 ) ( 𝑢 ( ·𝑠 ‘ 𝑁 ) 𝑤 ) ) ) |
90 |
|
simpr |
⊢ ( ( 𝜑 ∧ 𝑤 ∈ 𝐵 ) → 𝑤 ∈ 𝐵 ) |
91 |
82
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑤 ∈ 𝐵 ) → ran 𝐹 = 𝐵 ) |
92 |
90 91
|
eleqtrrd |
⊢ ( ( 𝜑 ∧ 𝑤 ∈ 𝐵 ) → 𝑤 ∈ ran 𝐹 ) |
93 |
|
fvelrnb |
⊢ ( 𝐹 Fn 𝑉 → ( 𝑤 ∈ ran 𝐹 ↔ ∃ 𝑧 ∈ 𝑉 ( 𝐹 ‘ 𝑧 ) = 𝑤 ) ) |
94 |
93
|
biimpa |
⊢ ( ( 𝐹 Fn 𝑉 ∧ 𝑤 ∈ ran 𝐹 ) → ∃ 𝑧 ∈ 𝑉 ( 𝐹 ‘ 𝑧 ) = 𝑤 ) |
95 |
79 92 94
|
syl2an2r |
⊢ ( ( 𝜑 ∧ 𝑤 ∈ 𝐵 ) → ∃ 𝑧 ∈ 𝑉 ( 𝐹 ‘ 𝑧 ) = 𝑤 ) |
96 |
95
|
3ad2antr3 |
⊢ ( ( 𝜑 ∧ ( 𝑢 ∈ 𝑆 ∧ 𝑣 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ) ) → ∃ 𝑧 ∈ 𝑉 ( 𝐹 ‘ 𝑧 ) = 𝑤 ) |
97 |
89 96
|
r19.29a |
⊢ ( ( 𝜑 ∧ ( 𝑢 ∈ 𝑆 ∧ 𝑣 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ) ) → ( 𝑢 ( ·𝑠 ‘ 𝑁 ) ( 𝑣 ( +g ‘ 𝑁 ) 𝑤 ) ) = ( ( 𝑢 ( ·𝑠 ‘ 𝑁 ) 𝑣 ) ( +g ‘ 𝑁 ) ( 𝑢 ( ·𝑠 ‘ 𝑁 ) 𝑤 ) ) ) |
98 |
|
simplll |
⊢ ( ( ( ( 𝜑 ∧ ( 𝑢 ∈ 𝑆 ∧ 𝑣 ∈ 𝑆 ∧ 𝑤 ∈ 𝐵 ) ) ∧ 𝑧 ∈ 𝑉 ) ∧ ( 𝐹 ‘ 𝑧 ) = 𝑤 ) → 𝜑 ) |
99 |
10
|
ad3antrrr |
⊢ ( ( ( ( 𝜑 ∧ ( 𝑢 ∈ 𝑆 ∧ 𝑣 ∈ 𝑆 ∧ 𝑤 ∈ 𝐵 ) ) ∧ 𝑧 ∈ 𝑉 ) ∧ ( 𝐹 ‘ 𝑧 ) = 𝑤 ) → 𝑀 ∈ LMod ) |
100 |
|
simpllr |
⊢ ( ( ( ( 𝜑 ∧ ( 𝑢 ∈ 𝑆 ∧ 𝑣 ∈ 𝑆 ∧ 𝑤 ∈ 𝐵 ) ) ∧ 𝑧 ∈ 𝑉 ) ∧ ( 𝐹 ‘ 𝑧 ) = 𝑤 ) → ( 𝑢 ∈ 𝑆 ∧ 𝑣 ∈ 𝑆 ∧ 𝑤 ∈ 𝐵 ) ) |
101 |
100
|
simp1d |
⊢ ( ( ( ( 𝜑 ∧ ( 𝑢 ∈ 𝑆 ∧ 𝑣 ∈ 𝑆 ∧ 𝑤 ∈ 𝐵 ) ) ∧ 𝑧 ∈ 𝑉 ) ∧ ( 𝐹 ‘ 𝑧 ) = 𝑤 ) → 𝑢 ∈ 𝑆 ) |
102 |
100
|
simp2d |
⊢ ( ( ( ( 𝜑 ∧ ( 𝑢 ∈ 𝑆 ∧ 𝑣 ∈ 𝑆 ∧ 𝑤 ∈ 𝐵 ) ) ∧ 𝑧 ∈ 𝑉 ) ∧ ( 𝐹 ‘ 𝑧 ) = 𝑤 ) → 𝑣 ∈ 𝑆 ) |
103 |
|
eqid |
⊢ ( +g ‘ ( Scalar ‘ 𝑀 ) ) = ( +g ‘ ( Scalar ‘ 𝑀 ) ) |
104 |
14 3 103
|
lmodacl |
⊢ ( ( 𝑀 ∈ LMod ∧ 𝑢 ∈ 𝑆 ∧ 𝑣 ∈ 𝑆 ) → ( 𝑢 ( +g ‘ ( Scalar ‘ 𝑀 ) ) 𝑣 ) ∈ 𝑆 ) |
105 |
99 101 102 104
|
syl3anc |
⊢ ( ( ( ( 𝜑 ∧ ( 𝑢 ∈ 𝑆 ∧ 𝑣 ∈ 𝑆 ∧ 𝑤 ∈ 𝐵 ) ) ∧ 𝑧 ∈ 𝑉 ) ∧ ( 𝐹 ‘ 𝑧 ) = 𝑤 ) → ( 𝑢 ( +g ‘ ( Scalar ‘ 𝑀 ) ) 𝑣 ) ∈ 𝑆 ) |
106 |
|
simplr |
⊢ ( ( ( ( 𝜑 ∧ ( 𝑢 ∈ 𝑆 ∧ 𝑣 ∈ 𝑆 ∧ 𝑤 ∈ 𝐵 ) ) ∧ 𝑧 ∈ 𝑉 ) ∧ ( 𝐹 ‘ 𝑧 ) = 𝑤 ) → 𝑧 ∈ 𝑉 ) |
107 |
1 11 7 10 14 3 5 28 9
|
imasvscaval |
⊢ ( ( 𝜑 ∧ ( 𝑢 ( +g ‘ ( Scalar ‘ 𝑀 ) ) 𝑣 ) ∈ 𝑆 ∧ 𝑧 ∈ 𝑉 ) → ( ( 𝑢 ( +g ‘ ( Scalar ‘ 𝑀 ) ) 𝑣 ) ( ·𝑠 ‘ 𝑁 ) ( 𝐹 ‘ 𝑧 ) ) = ( 𝐹 ‘ ( ( 𝑢 ( +g ‘ ( Scalar ‘ 𝑀 ) ) 𝑣 ) · 𝑧 ) ) ) |
108 |
98 105 106 107
|
syl3anc |
⊢ ( ( ( ( 𝜑 ∧ ( 𝑢 ∈ 𝑆 ∧ 𝑣 ∈ 𝑆 ∧ 𝑤 ∈ 𝐵 ) ) ∧ 𝑧 ∈ 𝑉 ) ∧ ( 𝐹 ‘ 𝑧 ) = 𝑤 ) → ( ( 𝑢 ( +g ‘ ( Scalar ‘ 𝑀 ) ) 𝑣 ) ( ·𝑠 ‘ 𝑁 ) ( 𝐹 ‘ 𝑧 ) ) = ( 𝐹 ‘ ( ( 𝑢 ( +g ‘ ( Scalar ‘ 𝑀 ) ) 𝑣 ) · 𝑧 ) ) ) |
109 |
|
simpr |
⊢ ( ( ( ( 𝜑 ∧ ( 𝑢 ∈ 𝑆 ∧ 𝑣 ∈ 𝑆 ∧ 𝑤 ∈ 𝐵 ) ) ∧ 𝑧 ∈ 𝑉 ) ∧ ( 𝐹 ‘ 𝑧 ) = 𝑤 ) → ( 𝐹 ‘ 𝑧 ) = 𝑤 ) |
110 |
109
|
oveq2d |
⊢ ( ( ( ( 𝜑 ∧ ( 𝑢 ∈ 𝑆 ∧ 𝑣 ∈ 𝑆 ∧ 𝑤 ∈ 𝐵 ) ) ∧ 𝑧 ∈ 𝑉 ) ∧ ( 𝐹 ‘ 𝑧 ) = 𝑤 ) → ( ( 𝑢 ( +g ‘ ( Scalar ‘ 𝑀 ) ) 𝑣 ) ( ·𝑠 ‘ 𝑁 ) ( 𝐹 ‘ 𝑧 ) ) = ( ( 𝑢 ( +g ‘ ( Scalar ‘ 𝑀 ) ) 𝑣 ) ( ·𝑠 ‘ 𝑁 ) 𝑤 ) ) |
111 |
2 4 14 5 3 103
|
lmodvsdir |
⊢ ( ( 𝑀 ∈ LMod ∧ ( 𝑢 ∈ 𝑆 ∧ 𝑣 ∈ 𝑆 ∧ 𝑧 ∈ 𝑉 ) ) → ( ( 𝑢 ( +g ‘ ( Scalar ‘ 𝑀 ) ) 𝑣 ) · 𝑧 ) = ( ( 𝑢 · 𝑧 ) + ( 𝑣 · 𝑧 ) ) ) |
112 |
99 101 102 106 111
|
syl13anc |
⊢ ( ( ( ( 𝜑 ∧ ( 𝑢 ∈ 𝑆 ∧ 𝑣 ∈ 𝑆 ∧ 𝑤 ∈ 𝐵 ) ) ∧ 𝑧 ∈ 𝑉 ) ∧ ( 𝐹 ‘ 𝑧 ) = 𝑤 ) → ( ( 𝑢 ( +g ‘ ( Scalar ‘ 𝑀 ) ) 𝑣 ) · 𝑧 ) = ( ( 𝑢 · 𝑧 ) + ( 𝑣 · 𝑧 ) ) ) |
113 |
112
|
fveq2d |
⊢ ( ( ( ( 𝜑 ∧ ( 𝑢 ∈ 𝑆 ∧ 𝑣 ∈ 𝑆 ∧ 𝑤 ∈ 𝐵 ) ) ∧ 𝑧 ∈ 𝑉 ) ∧ ( 𝐹 ‘ 𝑧 ) = 𝑤 ) → ( 𝐹 ‘ ( ( 𝑢 ( +g ‘ ( Scalar ‘ 𝑀 ) ) 𝑣 ) · 𝑧 ) ) = ( 𝐹 ‘ ( ( 𝑢 · 𝑧 ) + ( 𝑣 · 𝑧 ) ) ) ) |
114 |
99 101 106 62
|
syl3anc |
⊢ ( ( ( ( 𝜑 ∧ ( 𝑢 ∈ 𝑆 ∧ 𝑣 ∈ 𝑆 ∧ 𝑤 ∈ 𝐵 ) ) ∧ 𝑧 ∈ 𝑉 ) ∧ ( 𝐹 ‘ 𝑧 ) = 𝑤 ) → ( 𝑢 · 𝑧 ) ∈ 𝑉 ) |
115 |
2 14 5 3
|
lmodvscl |
⊢ ( ( 𝑀 ∈ LMod ∧ 𝑣 ∈ 𝑆 ∧ 𝑧 ∈ 𝑉 ) → ( 𝑣 · 𝑧 ) ∈ 𝑉 ) |
116 |
99 102 106 115
|
syl3anc |
⊢ ( ( ( ( 𝜑 ∧ ( 𝑢 ∈ 𝑆 ∧ 𝑣 ∈ 𝑆 ∧ 𝑤 ∈ 𝐵 ) ) ∧ 𝑧 ∈ 𝑉 ) ∧ ( 𝐹 ‘ 𝑧 ) = 𝑤 ) → ( 𝑣 · 𝑧 ) ∈ 𝑉 ) |
117 |
7 8 1 11 10 4 47
|
imasaddval |
⊢ ( ( 𝜑 ∧ ( 𝑢 · 𝑧 ) ∈ 𝑉 ∧ ( 𝑣 · 𝑧 ) ∈ 𝑉 ) → ( ( 𝐹 ‘ ( 𝑢 · 𝑧 ) ) ( +g ‘ 𝑁 ) ( 𝐹 ‘ ( 𝑣 · 𝑧 ) ) ) = ( 𝐹 ‘ ( ( 𝑢 · 𝑧 ) + ( 𝑣 · 𝑧 ) ) ) ) |
118 |
98 114 116 117
|
syl3anc |
⊢ ( ( ( ( 𝜑 ∧ ( 𝑢 ∈ 𝑆 ∧ 𝑣 ∈ 𝑆 ∧ 𝑤 ∈ 𝐵 ) ) ∧ 𝑧 ∈ 𝑉 ) ∧ ( 𝐹 ‘ 𝑧 ) = 𝑤 ) → ( ( 𝐹 ‘ ( 𝑢 · 𝑧 ) ) ( +g ‘ 𝑁 ) ( 𝐹 ‘ ( 𝑣 · 𝑧 ) ) ) = ( 𝐹 ‘ ( ( 𝑢 · 𝑧 ) + ( 𝑣 · 𝑧 ) ) ) ) |
119 |
98 101 106 70
|
syl3anc |
⊢ ( ( ( ( 𝜑 ∧ ( 𝑢 ∈ 𝑆 ∧ 𝑣 ∈ 𝑆 ∧ 𝑤 ∈ 𝐵 ) ) ∧ 𝑧 ∈ 𝑉 ) ∧ ( 𝐹 ‘ 𝑧 ) = 𝑤 ) → ( 𝑢 ( ·𝑠 ‘ 𝑁 ) ( 𝐹 ‘ 𝑧 ) ) = ( 𝐹 ‘ ( 𝑢 · 𝑧 ) ) ) |
120 |
109
|
oveq2d |
⊢ ( ( ( ( 𝜑 ∧ ( 𝑢 ∈ 𝑆 ∧ 𝑣 ∈ 𝑆 ∧ 𝑤 ∈ 𝐵 ) ) ∧ 𝑧 ∈ 𝑉 ) ∧ ( 𝐹 ‘ 𝑧 ) = 𝑤 ) → ( 𝑢 ( ·𝑠 ‘ 𝑁 ) ( 𝐹 ‘ 𝑧 ) ) = ( 𝑢 ( ·𝑠 ‘ 𝑁 ) 𝑤 ) ) |
121 |
119 120
|
eqtr3d |
⊢ ( ( ( ( 𝜑 ∧ ( 𝑢 ∈ 𝑆 ∧ 𝑣 ∈ 𝑆 ∧ 𝑤 ∈ 𝐵 ) ) ∧ 𝑧 ∈ 𝑉 ) ∧ ( 𝐹 ‘ 𝑧 ) = 𝑤 ) → ( 𝐹 ‘ ( 𝑢 · 𝑧 ) ) = ( 𝑢 ( ·𝑠 ‘ 𝑁 ) 𝑤 ) ) |
122 |
1 11 7 10 14 3 5 28 9
|
imasvscaval |
⊢ ( ( 𝜑 ∧ 𝑣 ∈ 𝑆 ∧ 𝑧 ∈ 𝑉 ) → ( 𝑣 ( ·𝑠 ‘ 𝑁 ) ( 𝐹 ‘ 𝑧 ) ) = ( 𝐹 ‘ ( 𝑣 · 𝑧 ) ) ) |
123 |
98 102 106 122
|
syl3anc |
⊢ ( ( ( ( 𝜑 ∧ ( 𝑢 ∈ 𝑆 ∧ 𝑣 ∈ 𝑆 ∧ 𝑤 ∈ 𝐵 ) ) ∧ 𝑧 ∈ 𝑉 ) ∧ ( 𝐹 ‘ 𝑧 ) = 𝑤 ) → ( 𝑣 ( ·𝑠 ‘ 𝑁 ) ( 𝐹 ‘ 𝑧 ) ) = ( 𝐹 ‘ ( 𝑣 · 𝑧 ) ) ) |
124 |
109
|
oveq2d |
⊢ ( ( ( ( 𝜑 ∧ ( 𝑢 ∈ 𝑆 ∧ 𝑣 ∈ 𝑆 ∧ 𝑤 ∈ 𝐵 ) ) ∧ 𝑧 ∈ 𝑉 ) ∧ ( 𝐹 ‘ 𝑧 ) = 𝑤 ) → ( 𝑣 ( ·𝑠 ‘ 𝑁 ) ( 𝐹 ‘ 𝑧 ) ) = ( 𝑣 ( ·𝑠 ‘ 𝑁 ) 𝑤 ) ) |
125 |
123 124
|
eqtr3d |
⊢ ( ( ( ( 𝜑 ∧ ( 𝑢 ∈ 𝑆 ∧ 𝑣 ∈ 𝑆 ∧ 𝑤 ∈ 𝐵 ) ) ∧ 𝑧 ∈ 𝑉 ) ∧ ( 𝐹 ‘ 𝑧 ) = 𝑤 ) → ( 𝐹 ‘ ( 𝑣 · 𝑧 ) ) = ( 𝑣 ( ·𝑠 ‘ 𝑁 ) 𝑤 ) ) |
126 |
121 125
|
oveq12d |
⊢ ( ( ( ( 𝜑 ∧ ( 𝑢 ∈ 𝑆 ∧ 𝑣 ∈ 𝑆 ∧ 𝑤 ∈ 𝐵 ) ) ∧ 𝑧 ∈ 𝑉 ) ∧ ( 𝐹 ‘ 𝑧 ) = 𝑤 ) → ( ( 𝐹 ‘ ( 𝑢 · 𝑧 ) ) ( +g ‘ 𝑁 ) ( 𝐹 ‘ ( 𝑣 · 𝑧 ) ) ) = ( ( 𝑢 ( ·𝑠 ‘ 𝑁 ) 𝑤 ) ( +g ‘ 𝑁 ) ( 𝑣 ( ·𝑠 ‘ 𝑁 ) 𝑤 ) ) ) |
127 |
113 118 126
|
3eqtr2d |
⊢ ( ( ( ( 𝜑 ∧ ( 𝑢 ∈ 𝑆 ∧ 𝑣 ∈ 𝑆 ∧ 𝑤 ∈ 𝐵 ) ) ∧ 𝑧 ∈ 𝑉 ) ∧ ( 𝐹 ‘ 𝑧 ) = 𝑤 ) → ( 𝐹 ‘ ( ( 𝑢 ( +g ‘ ( Scalar ‘ 𝑀 ) ) 𝑣 ) · 𝑧 ) ) = ( ( 𝑢 ( ·𝑠 ‘ 𝑁 ) 𝑤 ) ( +g ‘ 𝑁 ) ( 𝑣 ( ·𝑠 ‘ 𝑁 ) 𝑤 ) ) ) |
128 |
108 110 127
|
3eqtr3d |
⊢ ( ( ( ( 𝜑 ∧ ( 𝑢 ∈ 𝑆 ∧ 𝑣 ∈ 𝑆 ∧ 𝑤 ∈ 𝐵 ) ) ∧ 𝑧 ∈ 𝑉 ) ∧ ( 𝐹 ‘ 𝑧 ) = 𝑤 ) → ( ( 𝑢 ( +g ‘ ( Scalar ‘ 𝑀 ) ) 𝑣 ) ( ·𝑠 ‘ 𝑁 ) 𝑤 ) = ( ( 𝑢 ( ·𝑠 ‘ 𝑁 ) 𝑤 ) ( +g ‘ 𝑁 ) ( 𝑣 ( ·𝑠 ‘ 𝑁 ) 𝑤 ) ) ) |
129 |
95
|
3ad2antr3 |
⊢ ( ( 𝜑 ∧ ( 𝑢 ∈ 𝑆 ∧ 𝑣 ∈ 𝑆 ∧ 𝑤 ∈ 𝐵 ) ) → ∃ 𝑧 ∈ 𝑉 ( 𝐹 ‘ 𝑧 ) = 𝑤 ) |
130 |
128 129
|
r19.29a |
⊢ ( ( 𝜑 ∧ ( 𝑢 ∈ 𝑆 ∧ 𝑣 ∈ 𝑆 ∧ 𝑤 ∈ 𝐵 ) ) → ( ( 𝑢 ( +g ‘ ( Scalar ‘ 𝑀 ) ) 𝑣 ) ( ·𝑠 ‘ 𝑁 ) 𝑤 ) = ( ( 𝑢 ( ·𝑠 ‘ 𝑁 ) 𝑤 ) ( +g ‘ 𝑁 ) ( 𝑣 ( ·𝑠 ‘ 𝑁 ) 𝑤 ) ) ) |
131 |
|
eqid |
⊢ ( .r ‘ ( Scalar ‘ 𝑀 ) ) = ( .r ‘ ( Scalar ‘ 𝑀 ) ) |
132 |
14 3 131
|
lmodmcl |
⊢ ( ( 𝑀 ∈ LMod ∧ 𝑢 ∈ 𝑆 ∧ 𝑣 ∈ 𝑆 ) → ( 𝑢 ( .r ‘ ( Scalar ‘ 𝑀 ) ) 𝑣 ) ∈ 𝑆 ) |
133 |
99 101 102 132
|
syl3anc |
⊢ ( ( ( ( 𝜑 ∧ ( 𝑢 ∈ 𝑆 ∧ 𝑣 ∈ 𝑆 ∧ 𝑤 ∈ 𝐵 ) ) ∧ 𝑧 ∈ 𝑉 ) ∧ ( 𝐹 ‘ 𝑧 ) = 𝑤 ) → ( 𝑢 ( .r ‘ ( Scalar ‘ 𝑀 ) ) 𝑣 ) ∈ 𝑆 ) |
134 |
1 11 7 10 14 3 5 28 9
|
imasvscaval |
⊢ ( ( 𝜑 ∧ ( 𝑢 ( .r ‘ ( Scalar ‘ 𝑀 ) ) 𝑣 ) ∈ 𝑆 ∧ 𝑧 ∈ 𝑉 ) → ( ( 𝑢 ( .r ‘ ( Scalar ‘ 𝑀 ) ) 𝑣 ) ( ·𝑠 ‘ 𝑁 ) ( 𝐹 ‘ 𝑧 ) ) = ( 𝐹 ‘ ( ( 𝑢 ( .r ‘ ( Scalar ‘ 𝑀 ) ) 𝑣 ) · 𝑧 ) ) ) |
135 |
98 133 106 134
|
syl3anc |
⊢ ( ( ( ( 𝜑 ∧ ( 𝑢 ∈ 𝑆 ∧ 𝑣 ∈ 𝑆 ∧ 𝑤 ∈ 𝐵 ) ) ∧ 𝑧 ∈ 𝑉 ) ∧ ( 𝐹 ‘ 𝑧 ) = 𝑤 ) → ( ( 𝑢 ( .r ‘ ( Scalar ‘ 𝑀 ) ) 𝑣 ) ( ·𝑠 ‘ 𝑁 ) ( 𝐹 ‘ 𝑧 ) ) = ( 𝐹 ‘ ( ( 𝑢 ( .r ‘ ( Scalar ‘ 𝑀 ) ) 𝑣 ) · 𝑧 ) ) ) |
136 |
109
|
oveq2d |
⊢ ( ( ( ( 𝜑 ∧ ( 𝑢 ∈ 𝑆 ∧ 𝑣 ∈ 𝑆 ∧ 𝑤 ∈ 𝐵 ) ) ∧ 𝑧 ∈ 𝑉 ) ∧ ( 𝐹 ‘ 𝑧 ) = 𝑤 ) → ( ( 𝑢 ( .r ‘ ( Scalar ‘ 𝑀 ) ) 𝑣 ) ( ·𝑠 ‘ 𝑁 ) ( 𝐹 ‘ 𝑧 ) ) = ( ( 𝑢 ( .r ‘ ( Scalar ‘ 𝑀 ) ) 𝑣 ) ( ·𝑠 ‘ 𝑁 ) 𝑤 ) ) |
137 |
1 11 7 10 14 3 5 28 9
|
imasvscaval |
⊢ ( ( 𝜑 ∧ 𝑢 ∈ 𝑆 ∧ ( 𝑣 · 𝑧 ) ∈ 𝑉 ) → ( 𝑢 ( ·𝑠 ‘ 𝑁 ) ( 𝐹 ‘ ( 𝑣 · 𝑧 ) ) ) = ( 𝐹 ‘ ( 𝑢 · ( 𝑣 · 𝑧 ) ) ) ) |
138 |
98 101 116 137
|
syl3anc |
⊢ ( ( ( ( 𝜑 ∧ ( 𝑢 ∈ 𝑆 ∧ 𝑣 ∈ 𝑆 ∧ 𝑤 ∈ 𝐵 ) ) ∧ 𝑧 ∈ 𝑉 ) ∧ ( 𝐹 ‘ 𝑧 ) = 𝑤 ) → ( 𝑢 ( ·𝑠 ‘ 𝑁 ) ( 𝐹 ‘ ( 𝑣 · 𝑧 ) ) ) = ( 𝐹 ‘ ( 𝑢 · ( 𝑣 · 𝑧 ) ) ) ) |
139 |
123
|
oveq2d |
⊢ ( ( ( ( 𝜑 ∧ ( 𝑢 ∈ 𝑆 ∧ 𝑣 ∈ 𝑆 ∧ 𝑤 ∈ 𝐵 ) ) ∧ 𝑧 ∈ 𝑉 ) ∧ ( 𝐹 ‘ 𝑧 ) = 𝑤 ) → ( 𝑢 ( ·𝑠 ‘ 𝑁 ) ( 𝑣 ( ·𝑠 ‘ 𝑁 ) ( 𝐹 ‘ 𝑧 ) ) ) = ( 𝑢 ( ·𝑠 ‘ 𝑁 ) ( 𝐹 ‘ ( 𝑣 · 𝑧 ) ) ) ) |
140 |
2 14 5 3 131
|
lmodvsass |
⊢ ( ( 𝑀 ∈ LMod ∧ ( 𝑢 ∈ 𝑆 ∧ 𝑣 ∈ 𝑆 ∧ 𝑧 ∈ 𝑉 ) ) → ( ( 𝑢 ( .r ‘ ( Scalar ‘ 𝑀 ) ) 𝑣 ) · 𝑧 ) = ( 𝑢 · ( 𝑣 · 𝑧 ) ) ) |
141 |
99 101 102 106 140
|
syl13anc |
⊢ ( ( ( ( 𝜑 ∧ ( 𝑢 ∈ 𝑆 ∧ 𝑣 ∈ 𝑆 ∧ 𝑤 ∈ 𝐵 ) ) ∧ 𝑧 ∈ 𝑉 ) ∧ ( 𝐹 ‘ 𝑧 ) = 𝑤 ) → ( ( 𝑢 ( .r ‘ ( Scalar ‘ 𝑀 ) ) 𝑣 ) · 𝑧 ) = ( 𝑢 · ( 𝑣 · 𝑧 ) ) ) |
142 |
141
|
fveq2d |
⊢ ( ( ( ( 𝜑 ∧ ( 𝑢 ∈ 𝑆 ∧ 𝑣 ∈ 𝑆 ∧ 𝑤 ∈ 𝐵 ) ) ∧ 𝑧 ∈ 𝑉 ) ∧ ( 𝐹 ‘ 𝑧 ) = 𝑤 ) → ( 𝐹 ‘ ( ( 𝑢 ( .r ‘ ( Scalar ‘ 𝑀 ) ) 𝑣 ) · 𝑧 ) ) = ( 𝐹 ‘ ( 𝑢 · ( 𝑣 · 𝑧 ) ) ) ) |
143 |
138 139 142
|
3eqtr4rd |
⊢ ( ( ( ( 𝜑 ∧ ( 𝑢 ∈ 𝑆 ∧ 𝑣 ∈ 𝑆 ∧ 𝑤 ∈ 𝐵 ) ) ∧ 𝑧 ∈ 𝑉 ) ∧ ( 𝐹 ‘ 𝑧 ) = 𝑤 ) → ( 𝐹 ‘ ( ( 𝑢 ( .r ‘ ( Scalar ‘ 𝑀 ) ) 𝑣 ) · 𝑧 ) ) = ( 𝑢 ( ·𝑠 ‘ 𝑁 ) ( 𝑣 ( ·𝑠 ‘ 𝑁 ) ( 𝐹 ‘ 𝑧 ) ) ) ) |
144 |
135 136 143
|
3eqtr3d |
⊢ ( ( ( ( 𝜑 ∧ ( 𝑢 ∈ 𝑆 ∧ 𝑣 ∈ 𝑆 ∧ 𝑤 ∈ 𝐵 ) ) ∧ 𝑧 ∈ 𝑉 ) ∧ ( 𝐹 ‘ 𝑧 ) = 𝑤 ) → ( ( 𝑢 ( .r ‘ ( Scalar ‘ 𝑀 ) ) 𝑣 ) ( ·𝑠 ‘ 𝑁 ) 𝑤 ) = ( 𝑢 ( ·𝑠 ‘ 𝑁 ) ( 𝑣 ( ·𝑠 ‘ 𝑁 ) ( 𝐹 ‘ 𝑧 ) ) ) ) |
145 |
124
|
oveq2d |
⊢ ( ( ( ( 𝜑 ∧ ( 𝑢 ∈ 𝑆 ∧ 𝑣 ∈ 𝑆 ∧ 𝑤 ∈ 𝐵 ) ) ∧ 𝑧 ∈ 𝑉 ) ∧ ( 𝐹 ‘ 𝑧 ) = 𝑤 ) → ( 𝑢 ( ·𝑠 ‘ 𝑁 ) ( 𝑣 ( ·𝑠 ‘ 𝑁 ) ( 𝐹 ‘ 𝑧 ) ) ) = ( 𝑢 ( ·𝑠 ‘ 𝑁 ) ( 𝑣 ( ·𝑠 ‘ 𝑁 ) 𝑤 ) ) ) |
146 |
144 145
|
eqtrd |
⊢ ( ( ( ( 𝜑 ∧ ( 𝑢 ∈ 𝑆 ∧ 𝑣 ∈ 𝑆 ∧ 𝑤 ∈ 𝐵 ) ) ∧ 𝑧 ∈ 𝑉 ) ∧ ( 𝐹 ‘ 𝑧 ) = 𝑤 ) → ( ( 𝑢 ( .r ‘ ( Scalar ‘ 𝑀 ) ) 𝑣 ) ( ·𝑠 ‘ 𝑁 ) 𝑤 ) = ( 𝑢 ( ·𝑠 ‘ 𝑁 ) ( 𝑣 ( ·𝑠 ‘ 𝑁 ) 𝑤 ) ) ) |
147 |
146 129
|
r19.29a |
⊢ ( ( 𝜑 ∧ ( 𝑢 ∈ 𝑆 ∧ 𝑣 ∈ 𝑆 ∧ 𝑤 ∈ 𝐵 ) ) → ( ( 𝑢 ( .r ‘ ( Scalar ‘ 𝑀 ) ) 𝑣 ) ( ·𝑠 ‘ 𝑁 ) 𝑤 ) = ( 𝑢 ( ·𝑠 ‘ 𝑁 ) ( 𝑣 ( ·𝑠 ‘ 𝑁 ) 𝑤 ) ) ) |
148 |
|
simplll |
⊢ ( ( ( ( 𝜑 ∧ 𝑢 ∈ 𝐵 ) ∧ 𝑥 ∈ 𝑉 ) ∧ ( 𝐹 ‘ 𝑥 ) = 𝑢 ) → 𝜑 ) |
149 |
|
eqid |
⊢ ( 1r ‘ ( Scalar ‘ 𝑀 ) ) = ( 1r ‘ ( Scalar ‘ 𝑀 ) ) |
150 |
3 149
|
ringidcl |
⊢ ( ( Scalar ‘ 𝑀 ) ∈ Ring → ( 1r ‘ ( Scalar ‘ 𝑀 ) ) ∈ 𝑆 ) |
151 |
22 150
|
syl |
⊢ ( 𝜑 → ( 1r ‘ ( Scalar ‘ 𝑀 ) ) ∈ 𝑆 ) |
152 |
151
|
ad3antrrr |
⊢ ( ( ( ( 𝜑 ∧ 𝑢 ∈ 𝐵 ) ∧ 𝑥 ∈ 𝑉 ) ∧ ( 𝐹 ‘ 𝑥 ) = 𝑢 ) → ( 1r ‘ ( Scalar ‘ 𝑀 ) ) ∈ 𝑆 ) |
153 |
|
simplr |
⊢ ( ( ( ( 𝜑 ∧ 𝑢 ∈ 𝐵 ) ∧ 𝑥 ∈ 𝑉 ) ∧ ( 𝐹 ‘ 𝑥 ) = 𝑢 ) → 𝑥 ∈ 𝑉 ) |
154 |
1 11 7 10 14 3 5 28 9
|
imasvscaval |
⊢ ( ( 𝜑 ∧ ( 1r ‘ ( Scalar ‘ 𝑀 ) ) ∈ 𝑆 ∧ 𝑥 ∈ 𝑉 ) → ( ( 1r ‘ ( Scalar ‘ 𝑀 ) ) ( ·𝑠 ‘ 𝑁 ) ( 𝐹 ‘ 𝑥 ) ) = ( 𝐹 ‘ ( ( 1r ‘ ( Scalar ‘ 𝑀 ) ) · 𝑥 ) ) ) |
155 |
148 152 153 154
|
syl3anc |
⊢ ( ( ( ( 𝜑 ∧ 𝑢 ∈ 𝐵 ) ∧ 𝑥 ∈ 𝑉 ) ∧ ( 𝐹 ‘ 𝑥 ) = 𝑢 ) → ( ( 1r ‘ ( Scalar ‘ 𝑀 ) ) ( ·𝑠 ‘ 𝑁 ) ( 𝐹 ‘ 𝑥 ) ) = ( 𝐹 ‘ ( ( 1r ‘ ( Scalar ‘ 𝑀 ) ) · 𝑥 ) ) ) |
156 |
|
simpr |
⊢ ( ( ( ( 𝜑 ∧ 𝑢 ∈ 𝐵 ) ∧ 𝑥 ∈ 𝑉 ) ∧ ( 𝐹 ‘ 𝑥 ) = 𝑢 ) → ( 𝐹 ‘ 𝑥 ) = 𝑢 ) |
157 |
156
|
oveq2d |
⊢ ( ( ( ( 𝜑 ∧ 𝑢 ∈ 𝐵 ) ∧ 𝑥 ∈ 𝑉 ) ∧ ( 𝐹 ‘ 𝑥 ) = 𝑢 ) → ( ( 1r ‘ ( Scalar ‘ 𝑀 ) ) ( ·𝑠 ‘ 𝑁 ) ( 𝐹 ‘ 𝑥 ) ) = ( ( 1r ‘ ( Scalar ‘ 𝑀 ) ) ( ·𝑠 ‘ 𝑁 ) 𝑢 ) ) |
158 |
10
|
ad3antrrr |
⊢ ( ( ( ( 𝜑 ∧ 𝑢 ∈ 𝐵 ) ∧ 𝑥 ∈ 𝑉 ) ∧ ( 𝐹 ‘ 𝑥 ) = 𝑢 ) → 𝑀 ∈ LMod ) |
159 |
2 14 5 149
|
lmodvs1 |
⊢ ( ( 𝑀 ∈ LMod ∧ 𝑥 ∈ 𝑉 ) → ( ( 1r ‘ ( Scalar ‘ 𝑀 ) ) · 𝑥 ) = 𝑥 ) |
160 |
158 153 159
|
syl2anc |
⊢ ( ( ( ( 𝜑 ∧ 𝑢 ∈ 𝐵 ) ∧ 𝑥 ∈ 𝑉 ) ∧ ( 𝐹 ‘ 𝑥 ) = 𝑢 ) → ( ( 1r ‘ ( Scalar ‘ 𝑀 ) ) · 𝑥 ) = 𝑥 ) |
161 |
160
|
fveq2d |
⊢ ( ( ( ( 𝜑 ∧ 𝑢 ∈ 𝐵 ) ∧ 𝑥 ∈ 𝑉 ) ∧ ( 𝐹 ‘ 𝑥 ) = 𝑢 ) → ( 𝐹 ‘ ( ( 1r ‘ ( Scalar ‘ 𝑀 ) ) · 𝑥 ) ) = ( 𝐹 ‘ 𝑥 ) ) |
162 |
161 156
|
eqtrd |
⊢ ( ( ( ( 𝜑 ∧ 𝑢 ∈ 𝐵 ) ∧ 𝑥 ∈ 𝑉 ) ∧ ( 𝐹 ‘ 𝑥 ) = 𝑢 ) → ( 𝐹 ‘ ( ( 1r ‘ ( Scalar ‘ 𝑀 ) ) · 𝑥 ) ) = 𝑢 ) |
163 |
155 157 162
|
3eqtr3d |
⊢ ( ( ( ( 𝜑 ∧ 𝑢 ∈ 𝐵 ) ∧ 𝑥 ∈ 𝑉 ) ∧ ( 𝐹 ‘ 𝑥 ) = 𝑢 ) → ( ( 1r ‘ ( Scalar ‘ 𝑀 ) ) ( ·𝑠 ‘ 𝑁 ) 𝑢 ) = 𝑢 ) |
164 |
|
simpr |
⊢ ( ( 𝜑 ∧ 𝑢 ∈ 𝐵 ) → 𝑢 ∈ 𝐵 ) |
165 |
82
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑢 ∈ 𝐵 ) → ran 𝐹 = 𝐵 ) |
166 |
164 165
|
eleqtrrd |
⊢ ( ( 𝜑 ∧ 𝑢 ∈ 𝐵 ) → 𝑢 ∈ ran 𝐹 ) |
167 |
|
fvelrnb |
⊢ ( 𝐹 Fn 𝑉 → ( 𝑢 ∈ ran 𝐹 ↔ ∃ 𝑥 ∈ 𝑉 ( 𝐹 ‘ 𝑥 ) = 𝑢 ) ) |
168 |
167
|
biimpa |
⊢ ( ( 𝐹 Fn 𝑉 ∧ 𝑢 ∈ ran 𝐹 ) → ∃ 𝑥 ∈ 𝑉 ( 𝐹 ‘ 𝑥 ) = 𝑢 ) |
169 |
79 166 168
|
syl2an2r |
⊢ ( ( 𝜑 ∧ 𝑢 ∈ 𝐵 ) → ∃ 𝑥 ∈ 𝑉 ( 𝐹 ‘ 𝑥 ) = 𝑢 ) |
170 |
163 169
|
r19.29a |
⊢ ( ( 𝜑 ∧ 𝑢 ∈ 𝐵 ) → ( ( 1r ‘ ( Scalar ‘ 𝑀 ) ) ( ·𝑠 ‘ 𝑁 ) 𝑢 ) = 𝑢 ) |
171 |
12 13 15 16 17 18 19 20 22 27 35 97 130 147 170
|
islmodd |
⊢ ( 𝜑 → 𝑁 ∈ LMod ) |