Step | Hyp | Ref
| Expression |
1 | | simpl 476 |
. . 3
⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵) → 𝑀 ∈ LMod) |
2 | | linc0scn0.s |
. . . . . . . . 9
⊢ 𝑆 = (Scalar‘𝑀) |
3 | 2 | lmodring 19226 |
. . . . . . . 8
⊢ (𝑀 ∈ LMod → 𝑆 ∈ Ring) |
4 | 2 | eqcomi 2833 |
. . . . . . . . . . 11
⊢
(Scalar‘𝑀) =
𝑆 |
5 | 4 | fveq2i 6435 |
. . . . . . . . . 10
⊢
(Base‘(Scalar‘𝑀)) = (Base‘𝑆) |
6 | | linc0scn0.1 |
. . . . . . . . . 10
⊢ 1 =
(1r‘𝑆) |
7 | 5, 6 | ringidcl 18921 |
. . . . . . . . 9
⊢ (𝑆 ∈ Ring → 1 ∈
(Base‘(Scalar‘𝑀))) |
8 | | linc0scn0.0 |
. . . . . . . . . 10
⊢ 0 =
(0g‘𝑆) |
9 | 5, 8 | ring0cl 18922 |
. . . . . . . . 9
⊢ (𝑆 ∈ Ring → 0 ∈
(Base‘(Scalar‘𝑀))) |
10 | 7, 9 | jca 509 |
. . . . . . . 8
⊢ (𝑆 ∈ Ring → ( 1 ∈
(Base‘(Scalar‘𝑀)) ∧ 0 ∈
(Base‘(Scalar‘𝑀)))) |
11 | 3, 10 | syl 17 |
. . . . . . 7
⊢ (𝑀 ∈ LMod → ( 1 ∈
(Base‘(Scalar‘𝑀)) ∧ 0 ∈
(Base‘(Scalar‘𝑀)))) |
12 | 11 | ad2antrr 719 |
. . . . . 6
⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵) ∧ 𝑥 ∈ 𝑉) → ( 1 ∈
(Base‘(Scalar‘𝑀)) ∧ 0 ∈
(Base‘(Scalar‘𝑀)))) |
13 | | ifcl 4349 |
. . . . . 6
⊢ (( 1 ∈
(Base‘(Scalar‘𝑀)) ∧ 0 ∈
(Base‘(Scalar‘𝑀))) → if(𝑥 = 𝑍, 1 , 0 ) ∈
(Base‘(Scalar‘𝑀))) |
14 | 12, 13 | syl 17 |
. . . . 5
⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵) ∧ 𝑥 ∈ 𝑉) → if(𝑥 = 𝑍, 1 , 0 ) ∈
(Base‘(Scalar‘𝑀))) |
15 | | linc0scn0.f |
. . . . 5
⊢ 𝐹 = (𝑥 ∈ 𝑉 ↦ if(𝑥 = 𝑍, 1 , 0 )) |
16 | 14, 15 | fmptd 6632 |
. . . 4
⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵) → 𝐹:𝑉⟶(Base‘(Scalar‘𝑀))) |
17 | | fvex 6445 |
. . . . . 6
⊢
(Base‘(Scalar‘𝑀)) ∈ V |
18 | 17 | a1i 11 |
. . . . 5
⊢ (𝑀 ∈ LMod →
(Base‘(Scalar‘𝑀)) ∈ V) |
19 | | elmapg 8134 |
. . . . 5
⊢
(((Base‘(Scalar‘𝑀)) ∈ V ∧ 𝑉 ∈ 𝒫 𝐵) → (𝐹 ∈ ((Base‘(Scalar‘𝑀)) ↑𝑚
𝑉) ↔ 𝐹:𝑉⟶(Base‘(Scalar‘𝑀)))) |
20 | 18, 19 | sylan 577 |
. . . 4
⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵) → (𝐹 ∈ ((Base‘(Scalar‘𝑀)) ↑𝑚
𝑉) ↔ 𝐹:𝑉⟶(Base‘(Scalar‘𝑀)))) |
21 | 16, 20 | mpbird 249 |
. . 3
⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵) → 𝐹 ∈ ((Base‘(Scalar‘𝑀)) ↑𝑚
𝑉)) |
22 | | linc0scn0.b |
. . . . . . 7
⊢ 𝐵 = (Base‘𝑀) |
23 | 22 | pweqi 4381 |
. . . . . 6
⊢ 𝒫
𝐵 = 𝒫
(Base‘𝑀) |
24 | 23 | eleq2i 2897 |
. . . . 5
⊢ (𝑉 ∈ 𝒫 𝐵 ↔ 𝑉 ∈ 𝒫 (Base‘𝑀)) |
25 | 24 | biimpi 208 |
. . . 4
⊢ (𝑉 ∈ 𝒫 𝐵 → 𝑉 ∈ 𝒫 (Base‘𝑀)) |
26 | 25 | adantl 475 |
. . 3
⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵) → 𝑉 ∈ 𝒫 (Base‘𝑀)) |
27 | | lincval 43044 |
. . 3
⊢ ((𝑀 ∈ LMod ∧ 𝐹 ∈
((Base‘(Scalar‘𝑀)) ↑𝑚 𝑉) ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) → (𝐹( linC ‘𝑀)𝑉) = (𝑀 Σg (𝑣 ∈ 𝑉 ↦ ((𝐹‘𝑣)( ·𝑠
‘𝑀)𝑣)))) |
28 | 1, 21, 26, 27 | syl3anc 1496 |
. 2
⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵) → (𝐹( linC ‘𝑀)𝑉) = (𝑀 Σg (𝑣 ∈ 𝑉 ↦ ((𝐹‘𝑣)( ·𝑠
‘𝑀)𝑣)))) |
29 | | simpr 479 |
. . . . . . 7
⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵) ∧ 𝑣 ∈ 𝑉) → 𝑣 ∈ 𝑉) |
30 | 6 | fvexi 6446 |
. . . . . . . 8
⊢ 1 ∈
V |
31 | 8 | fvexi 6446 |
. . . . . . . 8
⊢ 0 ∈
V |
32 | 30, 31 | ifex 4353 |
. . . . . . 7
⊢ if(𝑣 = 𝑍, 1 , 0 ) ∈
V |
33 | | eqeq1 2828 |
. . . . . . . . 9
⊢ (𝑥 = 𝑣 → (𝑥 = 𝑍 ↔ 𝑣 = 𝑍)) |
34 | 33 | ifbid 4327 |
. . . . . . . 8
⊢ (𝑥 = 𝑣 → if(𝑥 = 𝑍, 1 , 0 ) = if(𝑣 = 𝑍, 1 , 0 )) |
35 | 34, 15 | fvmptg 6526 |
. . . . . . 7
⊢ ((𝑣 ∈ 𝑉 ∧ if(𝑣 = 𝑍, 1 , 0 ) ∈ V) → (𝐹‘𝑣) = if(𝑣 = 𝑍, 1 , 0 )) |
36 | 29, 32, 35 | sylancl 582 |
. . . . . 6
⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵) ∧ 𝑣 ∈ 𝑉) → (𝐹‘𝑣) = if(𝑣 = 𝑍, 1 , 0 )) |
37 | 36 | oveq1d 6919 |
. . . . 5
⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵) ∧ 𝑣 ∈ 𝑉) → ((𝐹‘𝑣)( ·𝑠
‘𝑀)𝑣) = (if(𝑣 = 𝑍, 1 , 0 )(
·𝑠 ‘𝑀)𝑣)) |
38 | | ovif 6996 |
. . . . . 6
⊢ (if(𝑣 = 𝑍, 1 , 0 )(
·𝑠 ‘𝑀)𝑣) = if(𝑣 = 𝑍, ( 1 (
·𝑠 ‘𝑀)𝑣), ( 0 (
·𝑠 ‘𝑀)𝑣)) |
39 | 38 | a1i 11 |
. . . . 5
⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵) ∧ 𝑣 ∈ 𝑉) → (if(𝑣 = 𝑍, 1 , 0 )(
·𝑠 ‘𝑀)𝑣) = if(𝑣 = 𝑍, ( 1 (
·𝑠 ‘𝑀)𝑣), ( 0 (
·𝑠 ‘𝑀)𝑣))) |
40 | | oveq2 6912 |
. . . . . . . 8
⊢ (𝑣 = 𝑍 → ( 1 (
·𝑠 ‘𝑀)𝑣) = ( 1 (
·𝑠 ‘𝑀)𝑍)) |
41 | 40 | adantl 475 |
. . . . . . 7
⊢ ((((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵) ∧ 𝑣 ∈ 𝑉) ∧ 𝑣 = 𝑍) → ( 1 (
·𝑠 ‘𝑀)𝑣) = ( 1 (
·𝑠 ‘𝑀)𝑍)) |
42 | | eqid 2824 |
. . . . . . . . . . . 12
⊢
(Base‘𝑆) =
(Base‘𝑆) |
43 | 2, 42, 6 | lmod1cl 19245 |
. . . . . . . . . . 11
⊢ (𝑀 ∈ LMod → 1 ∈
(Base‘𝑆)) |
44 | 43 | ancli 546 |
. . . . . . . . . 10
⊢ (𝑀 ∈ LMod → (𝑀 ∈ LMod ∧ 1 ∈
(Base‘𝑆))) |
45 | 44 | adantr 474 |
. . . . . . . . 9
⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵) → (𝑀 ∈ LMod ∧ 1 ∈ (Base‘𝑆))) |
46 | 45 | ad2antrr 719 |
. . . . . . . 8
⊢ ((((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵) ∧ 𝑣 ∈ 𝑉) ∧ 𝑣 = 𝑍) → (𝑀 ∈ LMod ∧ 1 ∈ (Base‘𝑆))) |
47 | | eqid 2824 |
. . . . . . . . 9
⊢ (
·𝑠 ‘𝑀) = ( ·𝑠
‘𝑀) |
48 | | linc0scn0.z |
. . . . . . . . 9
⊢ 𝑍 = (0g‘𝑀) |
49 | 2, 47, 42, 48 | lmodvs0 19252 |
. . . . . . . 8
⊢ ((𝑀 ∈ LMod ∧ 1 ∈
(Base‘𝑆)) → (
1 (
·𝑠 ‘𝑀)𝑍) = 𝑍) |
50 | 46, 49 | syl 17 |
. . . . . . 7
⊢ ((((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵) ∧ 𝑣 ∈ 𝑉) ∧ 𝑣 = 𝑍) → ( 1 (
·𝑠 ‘𝑀)𝑍) = 𝑍) |
51 | 41, 50 | eqtrd 2860 |
. . . . . 6
⊢ ((((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵) ∧ 𝑣 ∈ 𝑉) ∧ 𝑣 = 𝑍) → ( 1 (
·𝑠 ‘𝑀)𝑣) = 𝑍) |
52 | 1 | adantr 474 |
. . . . . . . 8
⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵) ∧ 𝑣 ∈ 𝑉) → 𝑀 ∈ LMod) |
53 | | elelpwi 4390 |
. . . . . . . . . . 11
⊢ ((𝑣 ∈ 𝑉 ∧ 𝑉 ∈ 𝒫 𝐵) → 𝑣 ∈ 𝐵) |
54 | 53 | expcom 404 |
. . . . . . . . . 10
⊢ (𝑉 ∈ 𝒫 𝐵 → (𝑣 ∈ 𝑉 → 𝑣 ∈ 𝐵)) |
55 | 54 | adantl 475 |
. . . . . . . . 9
⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵) → (𝑣 ∈ 𝑉 → 𝑣 ∈ 𝐵)) |
56 | 55 | imp 397 |
. . . . . . . 8
⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵) ∧ 𝑣 ∈ 𝑉) → 𝑣 ∈ 𝐵) |
57 | 22, 2, 47, 8, 48 | lmod0vs 19251 |
. . . . . . . 8
⊢ ((𝑀 ∈ LMod ∧ 𝑣 ∈ 𝐵) → ( 0 (
·𝑠 ‘𝑀)𝑣) = 𝑍) |
58 | 52, 56, 57 | syl2anc 581 |
. . . . . . 7
⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵) ∧ 𝑣 ∈ 𝑉) → ( 0 (
·𝑠 ‘𝑀)𝑣) = 𝑍) |
59 | 58 | adantr 474 |
. . . . . 6
⊢ ((((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵) ∧ 𝑣 ∈ 𝑉) ∧ ¬ 𝑣 = 𝑍) → ( 0 (
·𝑠 ‘𝑀)𝑣) = 𝑍) |
60 | 51, 59 | ifeqda 4340 |
. . . . 5
⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵) ∧ 𝑣 ∈ 𝑉) → if(𝑣 = 𝑍, ( 1 (
·𝑠 ‘𝑀)𝑣), ( 0 (
·𝑠 ‘𝑀)𝑣)) = 𝑍) |
61 | 37, 39, 60 | 3eqtrd 2864 |
. . . 4
⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵) ∧ 𝑣 ∈ 𝑉) → ((𝐹‘𝑣)( ·𝑠
‘𝑀)𝑣) = 𝑍) |
62 | 61 | mpteq2dva 4966 |
. . 3
⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵) → (𝑣 ∈ 𝑉 ↦ ((𝐹‘𝑣)( ·𝑠
‘𝑀)𝑣)) = (𝑣 ∈ 𝑉 ↦ 𝑍)) |
63 | 62 | oveq2d 6920 |
. 2
⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵) → (𝑀 Σg (𝑣 ∈ 𝑉 ↦ ((𝐹‘𝑣)( ·𝑠
‘𝑀)𝑣))) = (𝑀 Σg (𝑣 ∈ 𝑉 ↦ 𝑍))) |
64 | | lmodgrp 19225 |
. . . 4
⊢ (𝑀 ∈ LMod → 𝑀 ∈ Grp) |
65 | | grpmnd 17782 |
. . . 4
⊢ (𝑀 ∈ Grp → 𝑀 ∈ Mnd) |
66 | 64, 65 | syl 17 |
. . 3
⊢ (𝑀 ∈ LMod → 𝑀 ∈ Mnd) |
67 | 48 | gsumz 17726 |
. . 3
⊢ ((𝑀 ∈ Mnd ∧ 𝑉 ∈ 𝒫 𝐵) → (𝑀 Σg (𝑣 ∈ 𝑉 ↦ 𝑍)) = 𝑍) |
68 | 66, 67 | sylan 577 |
. 2
⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵) → (𝑀 Σg (𝑣 ∈ 𝑉 ↦ 𝑍)) = 𝑍) |
69 | 28, 63, 68 | 3eqtrd 2864 |
1
⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵) → (𝐹( linC ‘𝑀)𝑉) = 𝑍) |