Step | Hyp | Ref
| Expression |
1 | | simp1 1061 |
. . 3
⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) → 𝑀 ∈ LMod) |
2 | | linc1.s |
. . . . . . . . . 10
⊢ 𝑆 = (Scalar‘𝑀) |
3 | 2 | lmodring 18871 |
. . . . . . . . 9
⊢ (𝑀 ∈ LMod → 𝑆 ∈ Ring) |
4 | 2 | eqcomi 2631 |
. . . . . . . . . . . 12
⊢
(Scalar‘𝑀) =
𝑆 |
5 | 4 | fveq2i 6194 |
. . . . . . . . . . 11
⊢
(Base‘(Scalar‘𝑀)) = (Base‘𝑆) |
6 | | linc1.1 |
. . . . . . . . . . 11
⊢ 1 =
(1r‘𝑆) |
7 | 5, 6 | ringidcl 18568 |
. . . . . . . . . 10
⊢ (𝑆 ∈ Ring → 1 ∈
(Base‘(Scalar‘𝑀))) |
8 | | linc1.0 |
. . . . . . . . . . 11
⊢ 0 =
(0g‘𝑆) |
9 | 5, 8 | ring0cl 18569 |
. . . . . . . . . 10
⊢ (𝑆 ∈ Ring → 0 ∈
(Base‘(Scalar‘𝑀))) |
10 | 7, 9 | jca 554 |
. . . . . . . . 9
⊢ (𝑆 ∈ Ring → ( 1 ∈
(Base‘(Scalar‘𝑀)) ∧ 0 ∈
(Base‘(Scalar‘𝑀)))) |
11 | 3, 10 | syl 17 |
. . . . . . . 8
⊢ (𝑀 ∈ LMod → ( 1 ∈
(Base‘(Scalar‘𝑀)) ∧ 0 ∈
(Base‘(Scalar‘𝑀)))) |
12 | 11 | 3ad2ant1 1082 |
. . . . . . 7
⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) → ( 1 ∈
(Base‘(Scalar‘𝑀)) ∧ 0 ∈
(Base‘(Scalar‘𝑀)))) |
13 | 12 | adantr 481 |
. . . . . 6
⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) ∧ 𝑥 ∈ 𝑉) → ( 1 ∈
(Base‘(Scalar‘𝑀)) ∧ 0 ∈
(Base‘(Scalar‘𝑀)))) |
14 | | ifcl 4130 |
. . . . . 6
⊢ (( 1 ∈
(Base‘(Scalar‘𝑀)) ∧ 0 ∈
(Base‘(Scalar‘𝑀))) → if(𝑥 = 𝑋, 1 , 0 ) ∈
(Base‘(Scalar‘𝑀))) |
15 | 13, 14 | syl 17 |
. . . . 5
⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) ∧ 𝑥 ∈ 𝑉) → if(𝑥 = 𝑋, 1 , 0 ) ∈
(Base‘(Scalar‘𝑀))) |
16 | | linc1.f |
. . . . 5
⊢ 𝐹 = (𝑥 ∈ 𝑉 ↦ if(𝑥 = 𝑋, 1 , 0 )) |
17 | 15, 16 | fmptd 6385 |
. . . 4
⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) → 𝐹:𝑉⟶(Base‘(Scalar‘𝑀))) |
18 | | fvex 6201 |
. . . . 5
⊢
(Base‘(Scalar‘𝑀)) ∈ V |
19 | | simp2 1062 |
. . . . 5
⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) → 𝑉 ∈ 𝒫 𝐵) |
20 | | elmapg 7870 |
. . . . 5
⊢
(((Base‘(Scalar‘𝑀)) ∈ V ∧ 𝑉 ∈ 𝒫 𝐵) → (𝐹 ∈ ((Base‘(Scalar‘𝑀)) ↑𝑚
𝑉) ↔ 𝐹:𝑉⟶(Base‘(Scalar‘𝑀)))) |
21 | 18, 19, 20 | sylancr 695 |
. . . 4
⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) → (𝐹 ∈ ((Base‘(Scalar‘𝑀)) ↑𝑚
𝑉) ↔ 𝐹:𝑉⟶(Base‘(Scalar‘𝑀)))) |
22 | 17, 21 | mpbird 247 |
. . 3
⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) → 𝐹 ∈ ((Base‘(Scalar‘𝑀)) ↑𝑚
𝑉)) |
23 | | linc1.b |
. . . . . . 7
⊢ 𝐵 = (Base‘𝑀) |
24 | 23 | pweqi 4162 |
. . . . . 6
⊢ 𝒫
𝐵 = 𝒫
(Base‘𝑀) |
25 | 24 | eleq2i 2693 |
. . . . 5
⊢ (𝑉 ∈ 𝒫 𝐵 ↔ 𝑉 ∈ 𝒫 (Base‘𝑀)) |
26 | 25 | biimpi 206 |
. . . 4
⊢ (𝑉 ∈ 𝒫 𝐵 → 𝑉 ∈ 𝒫 (Base‘𝑀)) |
27 | 26 | 3ad2ant2 1083 |
. . 3
⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) → 𝑉 ∈ 𝒫 (Base‘𝑀)) |
28 | | lincval 42198 |
. . 3
⊢ ((𝑀 ∈ LMod ∧ 𝐹 ∈
((Base‘(Scalar‘𝑀)) ↑𝑚 𝑉) ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) → (𝐹( linC ‘𝑀)𝑉) = (𝑀 Σg (𝑦 ∈ 𝑉 ↦ ((𝐹‘𝑦)( ·𝑠
‘𝑀)𝑦)))) |
29 | 1, 22, 27, 28 | syl3anc 1326 |
. 2
⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) → (𝐹( linC ‘𝑀)𝑉) = (𝑀 Σg (𝑦 ∈ 𝑉 ↦ ((𝐹‘𝑦)( ·𝑠
‘𝑀)𝑦)))) |
30 | | eqid 2622 |
. . 3
⊢
(0g‘𝑀) = (0g‘𝑀) |
31 | | lmodgrp 18870 |
. . . . 5
⊢ (𝑀 ∈ LMod → 𝑀 ∈ Grp) |
32 | | grpmnd 17429 |
. . . . 5
⊢ (𝑀 ∈ Grp → 𝑀 ∈ Mnd) |
33 | 31, 32 | syl 17 |
. . . 4
⊢ (𝑀 ∈ LMod → 𝑀 ∈ Mnd) |
34 | 33 | 3ad2ant1 1082 |
. . 3
⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) → 𝑀 ∈ Mnd) |
35 | | simp3 1063 |
. . 3
⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) → 𝑋 ∈ 𝑉) |
36 | 1 | adantr 481 |
. . . . 5
⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) ∧ 𝑦 ∈ 𝑉) → 𝑀 ∈ LMod) |
37 | | simpr 477 |
. . . . . . 7
⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) ∧ 𝑦 ∈ 𝑉) → 𝑦 ∈ 𝑉) |
38 | | eqid 2622 |
. . . . . . . . . . 11
⊢
(Base‘𝑆) =
(Base‘𝑆) |
39 | 2, 38, 6 | lmod1cl 18890 |
. . . . . . . . . 10
⊢ (𝑀 ∈ LMod → 1 ∈
(Base‘𝑆)) |
40 | 39 | 3ad2ant1 1082 |
. . . . . . . . 9
⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) → 1 ∈ (Base‘𝑆)) |
41 | 40 | adantr 481 |
. . . . . . . 8
⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) ∧ 𝑦 ∈ 𝑉) → 1 ∈ (Base‘𝑆)) |
42 | 2, 38, 8 | lmod0cl 18889 |
. . . . . . . . . 10
⊢ (𝑀 ∈ LMod → 0 ∈
(Base‘𝑆)) |
43 | 42 | 3ad2ant1 1082 |
. . . . . . . . 9
⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) → 0 ∈ (Base‘𝑆)) |
44 | 43 | adantr 481 |
. . . . . . . 8
⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) ∧ 𝑦 ∈ 𝑉) → 0 ∈ (Base‘𝑆)) |
45 | 41, 44 | ifcld 4131 |
. . . . . . 7
⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) ∧ 𝑦 ∈ 𝑉) → if(𝑦 = 𝑋, 1 , 0 ) ∈ (Base‘𝑆)) |
46 | | eqeq1 2626 |
. . . . . . . . 9
⊢ (𝑥 = 𝑦 → (𝑥 = 𝑋 ↔ 𝑦 = 𝑋)) |
47 | 46 | ifbid 4108 |
. . . . . . . 8
⊢ (𝑥 = 𝑦 → if(𝑥 = 𝑋, 1 , 0 ) = if(𝑦 = 𝑋, 1 , 0 )) |
48 | 47, 16 | fvmptg 6280 |
. . . . . . 7
⊢ ((𝑦 ∈ 𝑉 ∧ if(𝑦 = 𝑋, 1 , 0 ) ∈ (Base‘𝑆)) → (𝐹‘𝑦) = if(𝑦 = 𝑋, 1 , 0 )) |
49 | 37, 45, 48 | syl2anc 693 |
. . . . . 6
⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) ∧ 𝑦 ∈ 𝑉) → (𝐹‘𝑦) = if(𝑦 = 𝑋, 1 , 0 )) |
50 | 49, 45 | eqeltrd 2701 |
. . . . 5
⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) ∧ 𝑦 ∈ 𝑉) → (𝐹‘𝑦) ∈ (Base‘𝑆)) |
51 | | elelpwi 4171 |
. . . . . . . 8
⊢ ((𝑦 ∈ 𝑉 ∧ 𝑉 ∈ 𝒫 𝐵) → 𝑦 ∈ 𝐵) |
52 | 51 | expcom 451 |
. . . . . . 7
⊢ (𝑉 ∈ 𝒫 𝐵 → (𝑦 ∈ 𝑉 → 𝑦 ∈ 𝐵)) |
53 | 52 | 3ad2ant2 1083 |
. . . . . 6
⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) → (𝑦 ∈ 𝑉 → 𝑦 ∈ 𝐵)) |
54 | 53 | imp 445 |
. . . . 5
⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) ∧ 𝑦 ∈ 𝑉) → 𝑦 ∈ 𝐵) |
55 | | eqid 2622 |
. . . . . 6
⊢ (
·𝑠 ‘𝑀) = ( ·𝑠
‘𝑀) |
56 | 23, 2, 55, 38 | lmodvscl 18880 |
. . . . 5
⊢ ((𝑀 ∈ LMod ∧ (𝐹‘𝑦) ∈ (Base‘𝑆) ∧ 𝑦 ∈ 𝐵) → ((𝐹‘𝑦)( ·𝑠
‘𝑀)𝑦) ∈ 𝐵) |
57 | 36, 50, 54, 56 | syl3anc 1326 |
. . . 4
⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) ∧ 𝑦 ∈ 𝑉) → ((𝐹‘𝑦)( ·𝑠
‘𝑀)𝑦) ∈ 𝐵) |
58 | | eqid 2622 |
. . . 4
⊢ (𝑦 ∈ 𝑉 ↦ ((𝐹‘𝑦)( ·𝑠
‘𝑀)𝑦)) = (𝑦 ∈ 𝑉 ↦ ((𝐹‘𝑦)( ·𝑠
‘𝑀)𝑦)) |
59 | 57, 58 | fmptd 6385 |
. . 3
⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) → (𝑦 ∈ 𝑉 ↦ ((𝐹‘𝑦)( ·𝑠
‘𝑀)𝑦)):𝑉⟶𝐵) |
60 | | fveq2 6191 |
. . . . . . 7
⊢ (𝑦 = 𝑣 → (𝐹‘𝑦) = (𝐹‘𝑣)) |
61 | | id 22 |
. . . . . . 7
⊢ (𝑦 = 𝑣 → 𝑦 = 𝑣) |
62 | 60, 61 | oveq12d 6668 |
. . . . . 6
⊢ (𝑦 = 𝑣 → ((𝐹‘𝑦)( ·𝑠
‘𝑀)𝑦) = ((𝐹‘𝑣)( ·𝑠
‘𝑀)𝑣)) |
63 | 62 | cbvmptv 4750 |
. . . . 5
⊢ (𝑦 ∈ 𝑉 ↦ ((𝐹‘𝑦)( ·𝑠
‘𝑀)𝑦)) = (𝑣 ∈ 𝑉 ↦ ((𝐹‘𝑣)( ·𝑠
‘𝑀)𝑣)) |
64 | | fvexd 6203 |
. . . . 5
⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) → (0g‘𝑀) ∈ V) |
65 | | ovexd 6680 |
. . . . 5
⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) ∧ 𝑣 ∈ 𝑉) → ((𝐹‘𝑣)( ·𝑠
‘𝑀)𝑣) ∈ V) |
66 | 63, 19, 64, 65 | mptsuppd 7318 |
. . . 4
⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) → ((𝑦 ∈ 𝑉 ↦ ((𝐹‘𝑦)( ·𝑠
‘𝑀)𝑦)) supp (0g‘𝑀)) = {𝑣 ∈ 𝑉 ∣ ((𝐹‘𝑣)( ·𝑠
‘𝑀)𝑣) ≠ (0g‘𝑀)}) |
67 | | 2a1 28 |
. . . . . . 7
⊢ (𝑣 = 𝑋 → (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) ∧ 𝑣 ∈ 𝑉) → (((𝐹‘𝑣)( ·𝑠
‘𝑀)𝑣) ≠ (0g‘𝑀) → 𝑣 = 𝑋))) |
68 | | simprr 796 |
. . . . . . . . . . . . . 14
⊢ ((¬
𝑣 = 𝑋 ∧ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) ∧ 𝑣 ∈ 𝑉)) → 𝑣 ∈ 𝑉) |
69 | | fvex 6201 |
. . . . . . . . . . . . . . . 16
⊢
(1r‘𝑆) ∈ V |
70 | 6, 69 | eqeltri 2697 |
. . . . . . . . . . . . . . 15
⊢ 1 ∈
V |
71 | | fvex 6201 |
. . . . . . . . . . . . . . . 16
⊢
(0g‘𝑆) ∈ V |
72 | 8, 71 | eqeltri 2697 |
. . . . . . . . . . . . . . 15
⊢ 0 ∈
V |
73 | 70, 72 | ifex 4156 |
. . . . . . . . . . . . . 14
⊢ if(𝑣 = 𝑋, 1 , 0 ) ∈
V |
74 | | eqeq1 2626 |
. . . . . . . . . . . . . . . 16
⊢ (𝑥 = 𝑣 → (𝑥 = 𝑋 ↔ 𝑣 = 𝑋)) |
75 | 74 | ifbid 4108 |
. . . . . . . . . . . . . . 15
⊢ (𝑥 = 𝑣 → if(𝑥 = 𝑋, 1 , 0 ) = if(𝑣 = 𝑋, 1 , 0 )) |
76 | 75, 16 | fvmptg 6280 |
. . . . . . . . . . . . . 14
⊢ ((𝑣 ∈ 𝑉 ∧ if(𝑣 = 𝑋, 1 , 0 ) ∈ V) → (𝐹‘𝑣) = if(𝑣 = 𝑋, 1 , 0 )) |
77 | 68, 73, 76 | sylancl 694 |
. . . . . . . . . . . . 13
⊢ ((¬
𝑣 = 𝑋 ∧ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) ∧ 𝑣 ∈ 𝑉)) → (𝐹‘𝑣) = if(𝑣 = 𝑋, 1 , 0 )) |
78 | | iffalse 4095 |
. . . . . . . . . . . . . 14
⊢ (¬
𝑣 = 𝑋 → if(𝑣 = 𝑋, 1 , 0 ) = 0 ) |
79 | 78 | adantr 481 |
. . . . . . . . . . . . 13
⊢ ((¬
𝑣 = 𝑋 ∧ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) ∧ 𝑣 ∈ 𝑉)) → if(𝑣 = 𝑋, 1 , 0 ) = 0 ) |
80 | 77, 79 | eqtrd 2656 |
. . . . . . . . . . . 12
⊢ ((¬
𝑣 = 𝑋 ∧ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) ∧ 𝑣 ∈ 𝑉)) → (𝐹‘𝑣) = 0 ) |
81 | 80 | oveq1d 6665 |
. . . . . . . . . . 11
⊢ ((¬
𝑣 = 𝑋 ∧ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) ∧ 𝑣 ∈ 𝑉)) → ((𝐹‘𝑣)( ·𝑠
‘𝑀)𝑣) = ( 0 (
·𝑠 ‘𝑀)𝑣)) |
82 | 1 | adantr 481 |
. . . . . . . . . . . . 13
⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) ∧ 𝑣 ∈ 𝑉) → 𝑀 ∈ LMod) |
83 | 82 | adantl 482 |
. . . . . . . . . . . 12
⊢ ((¬
𝑣 = 𝑋 ∧ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) ∧ 𝑣 ∈ 𝑉)) → 𝑀 ∈ LMod) |
84 | | elelpwi 4171 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑣 ∈ 𝑉 ∧ 𝑉 ∈ 𝒫 𝐵) → 𝑣 ∈ 𝐵) |
85 | 84 | expcom 451 |
. . . . . . . . . . . . . . 15
⊢ (𝑉 ∈ 𝒫 𝐵 → (𝑣 ∈ 𝑉 → 𝑣 ∈ 𝐵)) |
86 | 85 | 3ad2ant2 1083 |
. . . . . . . . . . . . . 14
⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) → (𝑣 ∈ 𝑉 → 𝑣 ∈ 𝐵)) |
87 | 86 | imp 445 |
. . . . . . . . . . . . 13
⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) ∧ 𝑣 ∈ 𝑉) → 𝑣 ∈ 𝐵) |
88 | 87 | adantl 482 |
. . . . . . . . . . . 12
⊢ ((¬
𝑣 = 𝑋 ∧ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) ∧ 𝑣 ∈ 𝑉)) → 𝑣 ∈ 𝐵) |
89 | 23, 2, 55, 8, 30 | lmod0vs 18896 |
. . . . . . . . . . . 12
⊢ ((𝑀 ∈ LMod ∧ 𝑣 ∈ 𝐵) → ( 0 (
·𝑠 ‘𝑀)𝑣) = (0g‘𝑀)) |
90 | 83, 88, 89 | syl2anc 693 |
. . . . . . . . . . 11
⊢ ((¬
𝑣 = 𝑋 ∧ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) ∧ 𝑣 ∈ 𝑉)) → ( 0 (
·𝑠 ‘𝑀)𝑣) = (0g‘𝑀)) |
91 | 81, 90 | eqtrd 2656 |
. . . . . . . . . 10
⊢ ((¬
𝑣 = 𝑋 ∧ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) ∧ 𝑣 ∈ 𝑉)) → ((𝐹‘𝑣)( ·𝑠
‘𝑀)𝑣) = (0g‘𝑀)) |
92 | 91 | neeq1d 2853 |
. . . . . . . . 9
⊢ ((¬
𝑣 = 𝑋 ∧ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) ∧ 𝑣 ∈ 𝑉)) → (((𝐹‘𝑣)( ·𝑠
‘𝑀)𝑣) ≠ (0g‘𝑀) ↔ (0g‘𝑀) ≠
(0g‘𝑀))) |
93 | | eqneqall 2805 |
. . . . . . . . . 10
⊢
((0g‘𝑀) = (0g‘𝑀) → ((0g‘𝑀) ≠
(0g‘𝑀)
→ 𝑣 = 𝑋)) |
94 | 30, 93 | ax-mp 5 |
. . . . . . . . 9
⊢
((0g‘𝑀) ≠ (0g‘𝑀) → 𝑣 = 𝑋) |
95 | 92, 94 | syl6bi 243 |
. . . . . . . 8
⊢ ((¬
𝑣 = 𝑋 ∧ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) ∧ 𝑣 ∈ 𝑉)) → (((𝐹‘𝑣)( ·𝑠
‘𝑀)𝑣) ≠ (0g‘𝑀) → 𝑣 = 𝑋)) |
96 | 95 | ex 450 |
. . . . . . 7
⊢ (¬
𝑣 = 𝑋 → (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) ∧ 𝑣 ∈ 𝑉) → (((𝐹‘𝑣)( ·𝑠
‘𝑀)𝑣) ≠ (0g‘𝑀) → 𝑣 = 𝑋))) |
97 | 67, 96 | pm2.61i 176 |
. . . . . 6
⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) ∧ 𝑣 ∈ 𝑉) → (((𝐹‘𝑣)( ·𝑠
‘𝑀)𝑣) ≠ (0g‘𝑀) → 𝑣 = 𝑋)) |
98 | 97 | ralrimiva 2966 |
. . . . 5
⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) → ∀𝑣 ∈ 𝑉 (((𝐹‘𝑣)( ·𝑠
‘𝑀)𝑣) ≠ (0g‘𝑀) → 𝑣 = 𝑋)) |
99 | | rabsssn 4215 |
. . . . 5
⊢ ({𝑣 ∈ 𝑉 ∣ ((𝐹‘𝑣)( ·𝑠
‘𝑀)𝑣) ≠ (0g‘𝑀)} ⊆ {𝑋} ↔ ∀𝑣 ∈ 𝑉 (((𝐹‘𝑣)( ·𝑠
‘𝑀)𝑣) ≠ (0g‘𝑀) → 𝑣 = 𝑋)) |
100 | 98, 99 | sylibr 224 |
. . . 4
⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) → {𝑣 ∈ 𝑉 ∣ ((𝐹‘𝑣)( ·𝑠
‘𝑀)𝑣) ≠ (0g‘𝑀)} ⊆ {𝑋}) |
101 | 66, 100 | eqsstrd 3639 |
. . 3
⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) → ((𝑦 ∈ 𝑉 ↦ ((𝐹‘𝑦)( ·𝑠
‘𝑀)𝑦)) supp (0g‘𝑀)) ⊆ {𝑋}) |
102 | 23, 30, 34, 19, 35, 59, 101 | gsumpt 18361 |
. 2
⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) → (𝑀 Σg (𝑦 ∈ 𝑉 ↦ ((𝐹‘𝑦)( ·𝑠
‘𝑀)𝑦))) = ((𝑦 ∈ 𝑉 ↦ ((𝐹‘𝑦)( ·𝑠
‘𝑀)𝑦))‘𝑋)) |
103 | | ovex 6678 |
. . . 4
⊢ ((𝐹‘𝑋)( ·𝑠
‘𝑀)𝑋) ∈ V |
104 | | fveq2 6191 |
. . . . . 6
⊢ (𝑦 = 𝑋 → (𝐹‘𝑦) = (𝐹‘𝑋)) |
105 | | id 22 |
. . . . . 6
⊢ (𝑦 = 𝑋 → 𝑦 = 𝑋) |
106 | 104, 105 | oveq12d 6668 |
. . . . 5
⊢ (𝑦 = 𝑋 → ((𝐹‘𝑦)( ·𝑠
‘𝑀)𝑦) = ((𝐹‘𝑋)( ·𝑠
‘𝑀)𝑋)) |
107 | 106, 58 | fvmptg 6280 |
. . . 4
⊢ ((𝑋 ∈ 𝑉 ∧ ((𝐹‘𝑋)( ·𝑠
‘𝑀)𝑋) ∈ V) → ((𝑦 ∈ 𝑉 ↦ ((𝐹‘𝑦)( ·𝑠
‘𝑀)𝑦))‘𝑋) = ((𝐹‘𝑋)( ·𝑠
‘𝑀)𝑋)) |
108 | 35, 103, 107 | sylancl 694 |
. . 3
⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) → ((𝑦 ∈ 𝑉 ↦ ((𝐹‘𝑦)( ·𝑠
‘𝑀)𝑦))‘𝑋) = ((𝐹‘𝑋)( ·𝑠
‘𝑀)𝑋)) |
109 | | iftrue 4092 |
. . . . . 6
⊢ (𝑥 = 𝑋 → if(𝑥 = 𝑋, 1 , 0 ) = 1 ) |
110 | 109, 16 | fvmptg 6280 |
. . . . 5
⊢ ((𝑋 ∈ 𝑉 ∧ 1 ∈ V) → (𝐹‘𝑋) = 1 ) |
111 | 35, 70, 110 | sylancl 694 |
. . . 4
⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) → (𝐹‘𝑋) = 1 ) |
112 | 111 | oveq1d 6665 |
. . 3
⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) → ((𝐹‘𝑋)( ·𝑠
‘𝑀)𝑋) = ( 1 (
·𝑠 ‘𝑀)𝑋)) |
113 | | elelpwi 4171 |
. . . . . 6
⊢ ((𝑋 ∈ 𝑉 ∧ 𝑉 ∈ 𝒫 𝐵) → 𝑋 ∈ 𝐵) |
114 | 113 | ancoms 469 |
. . . . 5
⊢ ((𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) → 𝑋 ∈ 𝐵) |
115 | 114 | 3adant1 1079 |
. . . 4
⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) → 𝑋 ∈ 𝐵) |
116 | 23, 2, 55, 6 | lmodvs1 18891 |
. . . 4
⊢ ((𝑀 ∈ LMod ∧ 𝑋 ∈ 𝐵) → ( 1 (
·𝑠 ‘𝑀)𝑋) = 𝑋) |
117 | 1, 115, 116 | syl2anc 693 |
. . 3
⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) → ( 1 (
·𝑠 ‘𝑀)𝑋) = 𝑋) |
118 | 108, 112,
117 | 3eqtrd 2660 |
. 2
⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) → ((𝑦 ∈ 𝑉 ↦ ((𝐹‘𝑦)( ·𝑠
‘𝑀)𝑦))‘𝑋) = 𝑋) |
119 | 29, 102, 118 | 3eqtrd 2660 |
1
⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) → (𝐹( linC ‘𝑀)𝑉) = 𝑋) |