Step | Hyp | Ref
| Expression |
1 | | simp2 1055 |
. . . 4
⊢ ((𝑆 ∈ 𝒫 𝐵 ∧ 𝑀 ∈ LMod ∧ 𝑋 ∈ 𝑆) → 𝑀 ∈ LMod) |
2 | 1 | 3ad2ant1 1075 |
. . 3
⊢ (((𝑆 ∈ 𝒫 𝐵 ∧ 𝑀 ∈ LMod ∧ 𝑋 ∈ 𝑆) ∧ (𝐹 ∈ (𝐸 ↑𝑚 𝑆) ∧ (𝐹‘𝑋) ∈ 𝑈 ∧ 𝐹 finSupp 0 ) ∧ (𝐹( linC ‘𝑀)𝑆) = 𝑍) → 𝑀 ∈ LMod) |
3 | | simp1 1054 |
. . . . . . . 8
⊢ (((𝑆 ∈ 𝒫 𝐵 ∧ 𝑀 ∈ LMod ∧ 𝑋 ∈ 𝑆) ∧ (𝐹 ∈ (𝐸 ↑𝑚 𝑆) ∧ (𝐹‘𝑋) ∈ 𝑈 ∧ 𝐹 finSupp 0 ) ∧ (𝐹( linC ‘𝑀)𝑆) = 𝑍) → (𝑆 ∈ 𝒫 𝐵 ∧ 𝑀 ∈ LMod ∧ 𝑋 ∈ 𝑆)) |
4 | | 3simpa 1051 |
. . . . . . . . 9
⊢ ((𝐹 ∈ (𝐸 ↑𝑚 𝑆) ∧ (𝐹‘𝑋) ∈ 𝑈 ∧ 𝐹 finSupp 0 ) → (𝐹 ∈ (𝐸 ↑𝑚 𝑆) ∧ (𝐹‘𝑋) ∈ 𝑈)) |
5 | 4 | 3ad2ant2 1076 |
. . . . . . . 8
⊢ (((𝑆 ∈ 𝒫 𝐵 ∧ 𝑀 ∈ LMod ∧ 𝑋 ∈ 𝑆) ∧ (𝐹 ∈ (𝐸 ↑𝑚 𝑆) ∧ (𝐹‘𝑋) ∈ 𝑈 ∧ 𝐹 finSupp 0 ) ∧ (𝐹( linC ‘𝑀)𝑆) = 𝑍) → (𝐹 ∈ (𝐸 ↑𝑚 𝑆) ∧ (𝐹‘𝑋) ∈ 𝑈)) |
6 | 3, 5 | jca 553 |
. . . . . . 7
⊢ (((𝑆 ∈ 𝒫 𝐵 ∧ 𝑀 ∈ LMod ∧ 𝑋 ∈ 𝑆) ∧ (𝐹 ∈ (𝐸 ↑𝑚 𝑆) ∧ (𝐹‘𝑋) ∈ 𝑈 ∧ 𝐹 finSupp 0 ) ∧ (𝐹( linC ‘𝑀)𝑆) = 𝑍) → ((𝑆 ∈ 𝒫 𝐵 ∧ 𝑀 ∈ LMod ∧ 𝑋 ∈ 𝑆) ∧ (𝐹 ∈ (𝐸 ↑𝑚 𝑆) ∧ (𝐹‘𝑋) ∈ 𝑈))) |
7 | | eldifi 3694 |
. . . . . . 7
⊢ (𝑠 ∈ (𝑆 ∖ {𝑋}) → 𝑠 ∈ 𝑆) |
8 | | lincresunit.b |
. . . . . . . 8
⊢ 𝐵 = (Base‘𝑀) |
9 | | lincresunit.r |
. . . . . . . 8
⊢ 𝑅 = (Scalar‘𝑀) |
10 | | lincresunit.e |
. . . . . . . 8
⊢ 𝐸 = (Base‘𝑅) |
11 | | lincresunit.u |
. . . . . . . 8
⊢ 𝑈 = (Unit‘𝑅) |
12 | | lincresunit.0 |
. . . . . . . 8
⊢ 0 =
(0g‘𝑅) |
13 | | lincresunit.z |
. . . . . . . 8
⊢ 𝑍 = (0g‘𝑀) |
14 | | lincresunit.n |
. . . . . . . 8
⊢ 𝑁 = (invg‘𝑅) |
15 | | lincresunit.i |
. . . . . . . 8
⊢ 𝐼 = (invr‘𝑅) |
16 | | lincresunit.t |
. . . . . . . 8
⊢ · =
(.r‘𝑅) |
17 | | lincresunit.g |
. . . . . . . 8
⊢ 𝐺 = (𝑠 ∈ (𝑆 ∖ {𝑋}) ↦ ((𝐼‘(𝑁‘(𝐹‘𝑋))) · (𝐹‘𝑠))) |
18 | 8, 9, 10, 11, 12, 13, 14, 15, 16, 17 | lincresunitlem2 42059 |
. . . . . . 7
⊢ ((((𝑆 ∈ 𝒫 𝐵 ∧ 𝑀 ∈ LMod ∧ 𝑋 ∈ 𝑆) ∧ (𝐹 ∈ (𝐸 ↑𝑚 𝑆) ∧ (𝐹‘𝑋) ∈ 𝑈)) ∧ 𝑠 ∈ 𝑆) → ((𝐼‘(𝑁‘(𝐹‘𝑋))) · (𝐹‘𝑠)) ∈ 𝐸) |
19 | 6, 7, 18 | syl2an 493 |
. . . . . 6
⊢ ((((𝑆 ∈ 𝒫 𝐵 ∧ 𝑀 ∈ LMod ∧ 𝑋 ∈ 𝑆) ∧ (𝐹 ∈ (𝐸 ↑𝑚 𝑆) ∧ (𝐹‘𝑋) ∈ 𝑈 ∧ 𝐹 finSupp 0 ) ∧ (𝐹( linC ‘𝑀)𝑆) = 𝑍) ∧ 𝑠 ∈ (𝑆 ∖ {𝑋})) → ((𝐼‘(𝑁‘(𝐹‘𝑋))) · (𝐹‘𝑠)) ∈ 𝐸) |
20 | 9 | fveq2i 6106 |
. . . . . . 7
⊢
(Base‘𝑅) =
(Base‘(Scalar‘𝑀)) |
21 | 10, 20 | eqtri 2632 |
. . . . . 6
⊢ 𝐸 =
(Base‘(Scalar‘𝑀)) |
22 | 19, 21 | syl6eleq 2698 |
. . . . 5
⊢ ((((𝑆 ∈ 𝒫 𝐵 ∧ 𝑀 ∈ LMod ∧ 𝑋 ∈ 𝑆) ∧ (𝐹 ∈ (𝐸 ↑𝑚 𝑆) ∧ (𝐹‘𝑋) ∈ 𝑈 ∧ 𝐹 finSupp 0 ) ∧ (𝐹( linC ‘𝑀)𝑆) = 𝑍) ∧ 𝑠 ∈ (𝑆 ∖ {𝑋})) → ((𝐼‘(𝑁‘(𝐹‘𝑋))) · (𝐹‘𝑠)) ∈ (Base‘(Scalar‘𝑀))) |
23 | 22, 17 | fmptd 6292 |
. . . 4
⊢ (((𝑆 ∈ 𝒫 𝐵 ∧ 𝑀 ∈ LMod ∧ 𝑋 ∈ 𝑆) ∧ (𝐹 ∈ (𝐸 ↑𝑚 𝑆) ∧ (𝐹‘𝑋) ∈ 𝑈 ∧ 𝐹 finSupp 0 ) ∧ (𝐹( linC ‘𝑀)𝑆) = 𝑍) → 𝐺:(𝑆 ∖ {𝑋})⟶(Base‘(Scalar‘𝑀))) |
24 | | fvex 6113 |
. . . . 5
⊢
(Base‘(Scalar‘𝑀)) ∈ V |
25 | | difexg 4735 |
. . . . . . 7
⊢ (𝑆 ∈ 𝒫 𝐵 → (𝑆 ∖ {𝑋}) ∈ V) |
26 | 25 | 3ad2ant1 1075 |
. . . . . 6
⊢ ((𝑆 ∈ 𝒫 𝐵 ∧ 𝑀 ∈ LMod ∧ 𝑋 ∈ 𝑆) → (𝑆 ∖ {𝑋}) ∈ V) |
27 | 26 | 3ad2ant1 1075 |
. . . . 5
⊢ (((𝑆 ∈ 𝒫 𝐵 ∧ 𝑀 ∈ LMod ∧ 𝑋 ∈ 𝑆) ∧ (𝐹 ∈ (𝐸 ↑𝑚 𝑆) ∧ (𝐹‘𝑋) ∈ 𝑈 ∧ 𝐹 finSupp 0 ) ∧ (𝐹( linC ‘𝑀)𝑆) = 𝑍) → (𝑆 ∖ {𝑋}) ∈ V) |
28 | | elmapg 7757 |
. . . . 5
⊢
(((Base‘(Scalar‘𝑀)) ∈ V ∧ (𝑆 ∖ {𝑋}) ∈ V) → (𝐺 ∈ ((Base‘(Scalar‘𝑀)) ↑𝑚
(𝑆 ∖ {𝑋})) ↔ 𝐺:(𝑆 ∖ {𝑋})⟶(Base‘(Scalar‘𝑀)))) |
29 | 24, 27, 28 | sylancr 694 |
. . . 4
⊢ (((𝑆 ∈ 𝒫 𝐵 ∧ 𝑀 ∈ LMod ∧ 𝑋 ∈ 𝑆) ∧ (𝐹 ∈ (𝐸 ↑𝑚 𝑆) ∧ (𝐹‘𝑋) ∈ 𝑈 ∧ 𝐹 finSupp 0 ) ∧ (𝐹( linC ‘𝑀)𝑆) = 𝑍) → (𝐺 ∈ ((Base‘(Scalar‘𝑀)) ↑𝑚
(𝑆 ∖ {𝑋})) ↔ 𝐺:(𝑆 ∖ {𝑋})⟶(Base‘(Scalar‘𝑀)))) |
30 | 23, 29 | mpbird 246 |
. . 3
⊢ (((𝑆 ∈ 𝒫 𝐵 ∧ 𝑀 ∈ LMod ∧ 𝑋 ∈ 𝑆) ∧ (𝐹 ∈ (𝐸 ↑𝑚 𝑆) ∧ (𝐹‘𝑋) ∈ 𝑈 ∧ 𝐹 finSupp 0 ) ∧ (𝐹( linC ‘𝑀)𝑆) = 𝑍) → 𝐺 ∈ ((Base‘(Scalar‘𝑀)) ↑𝑚
(𝑆 ∖ {𝑋}))) |
31 | | elpwi 4117 |
. . . . . . . . . . 11
⊢ (𝑆 ∈ 𝒫
(Base‘𝑀) → 𝑆 ⊆ (Base‘𝑀)) |
32 | | ssdifss 3703 |
. . . . . . . . . . . 12
⊢ (𝑆 ⊆ (Base‘𝑀) → (𝑆 ∖ {𝑋}) ⊆ (Base‘𝑀)) |
33 | 32 | a1i 11 |
. . . . . . . . . . 11
⊢ (𝑋 ∈ 𝑆 → (𝑆 ⊆ (Base‘𝑀) → (𝑆 ∖ {𝑋}) ⊆ (Base‘𝑀))) |
34 | 31, 33 | syl5com 31 |
. . . . . . . . . 10
⊢ (𝑆 ∈ 𝒫
(Base‘𝑀) →
(𝑋 ∈ 𝑆 → (𝑆 ∖ {𝑋}) ⊆ (Base‘𝑀))) |
35 | 34 | impcom 445 |
. . . . . . . . 9
⊢ ((𝑋 ∈ 𝑆 ∧ 𝑆 ∈ 𝒫 (Base‘𝑀)) → (𝑆 ∖ {𝑋}) ⊆ (Base‘𝑀)) |
36 | | difexg 4735 |
. . . . . . . . . . 11
⊢ (𝑆 ∈ 𝒫
(Base‘𝑀) →
(𝑆 ∖ {𝑋}) ∈ V) |
37 | 36 | adantl 481 |
. . . . . . . . . 10
⊢ ((𝑋 ∈ 𝑆 ∧ 𝑆 ∈ 𝒫 (Base‘𝑀)) → (𝑆 ∖ {𝑋}) ∈ V) |
38 | | elpwg 4116 |
. . . . . . . . . 10
⊢ ((𝑆 ∖ {𝑋}) ∈ V → ((𝑆 ∖ {𝑋}) ∈ 𝒫 (Base‘𝑀) ↔ (𝑆 ∖ {𝑋}) ⊆ (Base‘𝑀))) |
39 | 37, 38 | syl 17 |
. . . . . . . . 9
⊢ ((𝑋 ∈ 𝑆 ∧ 𝑆 ∈ 𝒫 (Base‘𝑀)) → ((𝑆 ∖ {𝑋}) ∈ 𝒫 (Base‘𝑀) ↔ (𝑆 ∖ {𝑋}) ⊆ (Base‘𝑀))) |
40 | 35, 39 | mpbird 246 |
. . . . . . . 8
⊢ ((𝑋 ∈ 𝑆 ∧ 𝑆 ∈ 𝒫 (Base‘𝑀)) → (𝑆 ∖ {𝑋}) ∈ 𝒫 (Base‘𝑀)) |
41 | 40 | expcom 450 |
. . . . . . 7
⊢ (𝑆 ∈ 𝒫
(Base‘𝑀) →
(𝑋 ∈ 𝑆 → (𝑆 ∖ {𝑋}) ∈ 𝒫 (Base‘𝑀))) |
42 | 8 | pweqi 4112 |
. . . . . . 7
⊢ 𝒫
𝐵 = 𝒫
(Base‘𝑀) |
43 | 41, 42 | eleq2s 2706 |
. . . . . 6
⊢ (𝑆 ∈ 𝒫 𝐵 → (𝑋 ∈ 𝑆 → (𝑆 ∖ {𝑋}) ∈ 𝒫 (Base‘𝑀))) |
44 | 43 | imp 444 |
. . . . 5
⊢ ((𝑆 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑆) → (𝑆 ∖ {𝑋}) ∈ 𝒫 (Base‘𝑀)) |
45 | 44 | 3adant2 1073 |
. . . 4
⊢ ((𝑆 ∈ 𝒫 𝐵 ∧ 𝑀 ∈ LMod ∧ 𝑋 ∈ 𝑆) → (𝑆 ∖ {𝑋}) ∈ 𝒫 (Base‘𝑀)) |
46 | 45 | 3ad2ant1 1075 |
. . 3
⊢ (((𝑆 ∈ 𝒫 𝐵 ∧ 𝑀 ∈ LMod ∧ 𝑋 ∈ 𝑆) ∧ (𝐹 ∈ (𝐸 ↑𝑚 𝑆) ∧ (𝐹‘𝑋) ∈ 𝑈 ∧ 𝐹 finSupp 0 ) ∧ (𝐹( linC ‘𝑀)𝑆) = 𝑍) → (𝑆 ∖ {𝑋}) ∈ 𝒫 (Base‘𝑀)) |
47 | | lincval 41992 |
. . 3
⊢ ((𝑀 ∈ LMod ∧ 𝐺 ∈
((Base‘(Scalar‘𝑀)) ↑𝑚 (𝑆 ∖ {𝑋})) ∧ (𝑆 ∖ {𝑋}) ∈ 𝒫 (Base‘𝑀)) → (𝐺( linC ‘𝑀)(𝑆 ∖ {𝑋})) = (𝑀 Σg (𝑠 ∈ (𝑆 ∖ {𝑋}) ↦ ((𝐺‘𝑠)( ·𝑠
‘𝑀)𝑠)))) |
48 | 2, 30, 46, 47 | syl3anc 1318 |
. 2
⊢ (((𝑆 ∈ 𝒫 𝐵 ∧ 𝑀 ∈ LMod ∧ 𝑋 ∈ 𝑆) ∧ (𝐹 ∈ (𝐸 ↑𝑚 𝑆) ∧ (𝐹‘𝑋) ∈ 𝑈 ∧ 𝐹 finSupp 0 ) ∧ (𝐹( linC ‘𝑀)𝑆) = 𝑍) → (𝐺( linC ‘𝑀)(𝑆 ∖ {𝑋})) = (𝑀 Σg (𝑠 ∈ (𝑆 ∖ {𝑋}) ↦ ((𝐺‘𝑠)( ·𝑠
‘𝑀)𝑠)))) |
49 | | simp1 1054 |
. . . . . . . 8
⊢ ((𝑆 ∈ 𝒫 𝐵 ∧ 𝑀 ∈ LMod ∧ 𝑋 ∈ 𝑆) → 𝑆 ∈ 𝒫 𝐵) |
50 | | simp3 1056 |
. . . . . . . 8
⊢ ((𝑆 ∈ 𝒫 𝐵 ∧ 𝑀 ∈ LMod ∧ 𝑋 ∈ 𝑆) → 𝑋 ∈ 𝑆) |
51 | 1, 49, 50 | 3jca 1235 |
. . . . . . 7
⊢ ((𝑆 ∈ 𝒫 𝐵 ∧ 𝑀 ∈ LMod ∧ 𝑋 ∈ 𝑆) → (𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑆)) |
52 | 51 | adantr 480 |
. . . . . 6
⊢ (((𝑆 ∈ 𝒫 𝐵 ∧ 𝑀 ∈ LMod ∧ 𝑋 ∈ 𝑆) ∧ (𝐹 ∈ (𝐸 ↑𝑚 𝑆) ∧ (𝐹‘𝑋) ∈ 𝑈 ∧ 𝐹 finSupp 0 )) → (𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑆)) |
53 | | 3simpb 1052 |
. . . . . . 7
⊢ ((𝐹 ∈ (𝐸 ↑𝑚 𝑆) ∧ (𝐹‘𝑋) ∈ 𝑈 ∧ 𝐹 finSupp 0 ) → (𝐹 ∈ (𝐸 ↑𝑚 𝑆) ∧ 𝐹 finSupp 0 )) |
54 | 53 | adantl 481 |
. . . . . 6
⊢ (((𝑆 ∈ 𝒫 𝐵 ∧ 𝑀 ∈ LMod ∧ 𝑋 ∈ 𝑆) ∧ (𝐹 ∈ (𝐸 ↑𝑚 𝑆) ∧ (𝐹‘𝑋) ∈ 𝑈 ∧ 𝐹 finSupp 0 )) → (𝐹 ∈ (𝐸 ↑𝑚 𝑆) ∧ 𝐹 finSupp 0 )) |
55 | | eqidd 2611 |
. . . . . 6
⊢ (((𝑆 ∈ 𝒫 𝐵 ∧ 𝑀 ∈ LMod ∧ 𝑋 ∈ 𝑆) ∧ (𝐹 ∈ (𝐸 ↑𝑚 𝑆) ∧ (𝐹‘𝑋) ∈ 𝑈 ∧ 𝐹 finSupp 0 )) → (𝐹 ↾ (𝑆 ∖ {𝑋})) = (𝐹 ↾ (𝑆 ∖ {𝑋}))) |
56 | | eqid 2610 |
. . . . . . 7
⊢ (
·𝑠 ‘𝑀) = ( ·𝑠
‘𝑀) |
57 | | eqid 2610 |
. . . . . . 7
⊢
(+g‘𝑀) = (+g‘𝑀) |
58 | 8, 9, 10, 56, 57, 12 | lincdifsn 42007 |
. . . . . 6
⊢ (((𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑆) ∧ (𝐹 ∈ (𝐸 ↑𝑚 𝑆) ∧ 𝐹 finSupp 0 ) ∧ (𝐹 ↾ (𝑆 ∖ {𝑋})) = (𝐹 ↾ (𝑆 ∖ {𝑋}))) → (𝐹( linC ‘𝑀)𝑆) = (((𝐹 ↾ (𝑆 ∖ {𝑋}))( linC ‘𝑀)(𝑆 ∖ {𝑋}))(+g‘𝑀)((𝐹‘𝑋)( ·𝑠
‘𝑀)𝑋))) |
59 | 52, 54, 55, 58 | syl3anc 1318 |
. . . . 5
⊢ (((𝑆 ∈ 𝒫 𝐵 ∧ 𝑀 ∈ LMod ∧ 𝑋 ∈ 𝑆) ∧ (𝐹 ∈ (𝐸 ↑𝑚 𝑆) ∧ (𝐹‘𝑋) ∈ 𝑈 ∧ 𝐹 finSupp 0 )) → (𝐹( linC ‘𝑀)𝑆) = (((𝐹 ↾ (𝑆 ∖ {𝑋}))( linC ‘𝑀)(𝑆 ∖ {𝑋}))(+g‘𝑀)((𝐹‘𝑋)( ·𝑠
‘𝑀)𝑋))) |
60 | 59 | eqeq1d 2612 |
. . . 4
⊢ (((𝑆 ∈ 𝒫 𝐵 ∧ 𝑀 ∈ LMod ∧ 𝑋 ∈ 𝑆) ∧ (𝐹 ∈ (𝐸 ↑𝑚 𝑆) ∧ (𝐹‘𝑋) ∈ 𝑈 ∧ 𝐹 finSupp 0 )) → ((𝐹( linC ‘𝑀)𝑆) = 𝑍 ↔ (((𝐹 ↾ (𝑆 ∖ {𝑋}))( linC ‘𝑀)(𝑆 ∖ {𝑋}))(+g‘𝑀)((𝐹‘𝑋)( ·𝑠
‘𝑀)𝑋)) = 𝑍)) |
61 | | fveq2 6103 |
. . . . . . . . . . . . 13
⊢ (𝑠 = 𝑧 → (𝐺‘𝑠) = (𝐺‘𝑧)) |
62 | | id 22 |
. . . . . . . . . . . . 13
⊢ (𝑠 = 𝑧 → 𝑠 = 𝑧) |
63 | 61, 62 | oveq12d 6567 |
. . . . . . . . . . . 12
⊢ (𝑠 = 𝑧 → ((𝐺‘𝑠)( ·𝑠
‘𝑀)𝑠) = ((𝐺‘𝑧)( ·𝑠
‘𝑀)𝑧)) |
64 | 63 | cbvmptv 4678 |
. . . . . . . . . . 11
⊢ (𝑠 ∈ (𝑆 ∖ {𝑋}) ↦ ((𝐺‘𝑠)( ·𝑠
‘𝑀)𝑠)) = (𝑧 ∈ (𝑆 ∖ {𝑋}) ↦ ((𝐺‘𝑧)( ·𝑠
‘𝑀)𝑧)) |
65 | 64 | a1i 11 |
. . . . . . . . . 10
⊢ (((𝑆 ∈ 𝒫 𝐵 ∧ 𝑀 ∈ LMod ∧ 𝑋 ∈ 𝑆) ∧ (𝐹 ∈ (𝐸 ↑𝑚 𝑆) ∧ (𝐹‘𝑋) ∈ 𝑈 ∧ 𝐹 finSupp 0 )) → (𝑠 ∈ (𝑆 ∖ {𝑋}) ↦ ((𝐺‘𝑠)( ·𝑠
‘𝑀)𝑠)) = (𝑧 ∈ (𝑆 ∖ {𝑋}) ↦ ((𝐺‘𝑧)( ·𝑠
‘𝑀)𝑧))) |
66 | 65 | oveq2d 6565 |
. . . . . . . . 9
⊢ (((𝑆 ∈ 𝒫 𝐵 ∧ 𝑀 ∈ LMod ∧ 𝑋 ∈ 𝑆) ∧ (𝐹 ∈ (𝐸 ↑𝑚 𝑆) ∧ (𝐹‘𝑋) ∈ 𝑈 ∧ 𝐹 finSupp 0 )) → (𝑀 Σg
(𝑠 ∈ (𝑆 ∖ {𝑋}) ↦ ((𝐺‘𝑠)( ·𝑠
‘𝑀)𝑠))) = (𝑀 Σg (𝑧 ∈ (𝑆 ∖ {𝑋}) ↦ ((𝐺‘𝑧)( ·𝑠
‘𝑀)𝑧)))) |
67 | 66 | oveq2d 6565 |
. . . . . . . 8
⊢ (((𝑆 ∈ 𝒫 𝐵 ∧ 𝑀 ∈ LMod ∧ 𝑋 ∈ 𝑆) ∧ (𝐹 ∈ (𝐸 ↑𝑚 𝑆) ∧ (𝐹‘𝑋) ∈ 𝑈 ∧ 𝐹 finSupp 0 )) → ((𝑁‘(𝐹‘𝑋))( ·𝑠
‘𝑀)(𝑀 Σg (𝑠 ∈ (𝑆 ∖ {𝑋}) ↦ ((𝐺‘𝑠)( ·𝑠
‘𝑀)𝑠)))) = ((𝑁‘(𝐹‘𝑋))( ·𝑠
‘𝑀)(𝑀 Σg (𝑧 ∈ (𝑆 ∖ {𝑋}) ↦ ((𝐺‘𝑧)( ·𝑠
‘𝑀)𝑧))))) |
68 | 8, 9, 10, 11, 12, 13, 14, 15, 16, 17 | lincresunit3lem2 42063 |
. . . . . . . 8
⊢ (((𝑆 ∈ 𝒫 𝐵 ∧ 𝑀 ∈ LMod ∧ 𝑋 ∈ 𝑆) ∧ (𝐹 ∈ (𝐸 ↑𝑚 𝑆) ∧ (𝐹‘𝑋) ∈ 𝑈 ∧ 𝐹 finSupp 0 )) → ((𝑁‘(𝐹‘𝑋))( ·𝑠
‘𝑀)(𝑀 Σg (𝑧 ∈ (𝑆 ∖ {𝑋}) ↦ ((𝐺‘𝑧)( ·𝑠
‘𝑀)𝑧)))) = ((𝐹 ↾ (𝑆 ∖ {𝑋}))( linC ‘𝑀)(𝑆 ∖ {𝑋}))) |
69 | 67, 68 | eqtr2d 2645 |
. . . . . . 7
⊢ (((𝑆 ∈ 𝒫 𝐵 ∧ 𝑀 ∈ LMod ∧ 𝑋 ∈ 𝑆) ∧ (𝐹 ∈ (𝐸 ↑𝑚 𝑆) ∧ (𝐹‘𝑋) ∈ 𝑈 ∧ 𝐹 finSupp 0 )) → ((𝐹 ↾ (𝑆 ∖ {𝑋}))( linC ‘𝑀)(𝑆 ∖ {𝑋})) = ((𝑁‘(𝐹‘𝑋))( ·𝑠
‘𝑀)(𝑀 Σg (𝑠 ∈ (𝑆 ∖ {𝑋}) ↦ ((𝐺‘𝑠)( ·𝑠
‘𝑀)𝑠))))) |
70 | 69 | oveq1d 6564 |
. . . . . 6
⊢ (((𝑆 ∈ 𝒫 𝐵 ∧ 𝑀 ∈ LMod ∧ 𝑋 ∈ 𝑆) ∧ (𝐹 ∈ (𝐸 ↑𝑚 𝑆) ∧ (𝐹‘𝑋) ∈ 𝑈 ∧ 𝐹 finSupp 0 )) → (((𝐹 ↾ (𝑆 ∖ {𝑋}))( linC ‘𝑀)(𝑆 ∖ {𝑋}))(+g‘𝑀)((𝐹‘𝑋)( ·𝑠
‘𝑀)𝑋)) = (((𝑁‘(𝐹‘𝑋))( ·𝑠
‘𝑀)(𝑀 Σg (𝑠 ∈ (𝑆 ∖ {𝑋}) ↦ ((𝐺‘𝑠)( ·𝑠
‘𝑀)𝑠))))(+g‘𝑀)((𝐹‘𝑋)( ·𝑠
‘𝑀)𝑋))) |
71 | 70 | eqeq1d 2612 |
. . . . 5
⊢ (((𝑆 ∈ 𝒫 𝐵 ∧ 𝑀 ∈ LMod ∧ 𝑋 ∈ 𝑆) ∧ (𝐹 ∈ (𝐸 ↑𝑚 𝑆) ∧ (𝐹‘𝑋) ∈ 𝑈 ∧ 𝐹 finSupp 0 )) → ((((𝐹 ↾ (𝑆 ∖ {𝑋}))( linC ‘𝑀)(𝑆 ∖ {𝑋}))(+g‘𝑀)((𝐹‘𝑋)( ·𝑠
‘𝑀)𝑋)) = 𝑍 ↔ (((𝑁‘(𝐹‘𝑋))( ·𝑠
‘𝑀)(𝑀 Σg (𝑠 ∈ (𝑆 ∖ {𝑋}) ↦ ((𝐺‘𝑠)( ·𝑠
‘𝑀)𝑠))))(+g‘𝑀)((𝐹‘𝑋)( ·𝑠
‘𝑀)𝑋)) = 𝑍)) |
72 | | lmodgrp 18693 |
. . . . . . . . 9
⊢ (𝑀 ∈ LMod → 𝑀 ∈ Grp) |
73 | 72 | 3ad2ant2 1076 |
. . . . . . . 8
⊢ ((𝑆 ∈ 𝒫 𝐵 ∧ 𝑀 ∈ LMod ∧ 𝑋 ∈ 𝑆) → 𝑀 ∈ Grp) |
74 | 73 | adantr 480 |
. . . . . . 7
⊢ (((𝑆 ∈ 𝒫 𝐵 ∧ 𝑀 ∈ LMod ∧ 𝑋 ∈ 𝑆) ∧ (𝐹 ∈ (𝐸 ↑𝑚 𝑆) ∧ (𝐹‘𝑋) ∈ 𝑈 ∧ 𝐹 finSupp 0 )) → 𝑀 ∈ Grp) |
75 | 1 | adantr 480 |
. . . . . . . 8
⊢ (((𝑆 ∈ 𝒫 𝐵 ∧ 𝑀 ∈ LMod ∧ 𝑋 ∈ 𝑆) ∧ (𝐹 ∈ (𝐸 ↑𝑚 𝑆) ∧ (𝐹‘𝑋) ∈ 𝑈 ∧ 𝐹 finSupp 0 )) → 𝑀 ∈ LMod) |
76 | | elmapi 7765 |
. . . . . . . . . 10
⊢ (𝐹 ∈ (𝐸 ↑𝑚 𝑆) → 𝐹:𝑆⟶𝐸) |
77 | 76 | 3ad2ant1 1075 |
. . . . . . . . 9
⊢ ((𝐹 ∈ (𝐸 ↑𝑚 𝑆) ∧ (𝐹‘𝑋) ∈ 𝑈 ∧ 𝐹 finSupp 0 ) → 𝐹:𝑆⟶𝐸) |
78 | | ffvelrn 6265 |
. . . . . . . . 9
⊢ ((𝐹:𝑆⟶𝐸 ∧ 𝑋 ∈ 𝑆) → (𝐹‘𝑋) ∈ 𝐸) |
79 | 77, 50, 78 | syl2anr 494 |
. . . . . . . 8
⊢ (((𝑆 ∈ 𝒫 𝐵 ∧ 𝑀 ∈ LMod ∧ 𝑋 ∈ 𝑆) ∧ (𝐹 ∈ (𝐸 ↑𝑚 𝑆) ∧ (𝐹‘𝑋) ∈ 𝑈 ∧ 𝐹 finSupp 0 )) → (𝐹‘𝑋) ∈ 𝐸) |
80 | | elpwi 4117 |
. . . . . . . . . . 11
⊢ (𝑆 ∈ 𝒫 𝐵 → 𝑆 ⊆ 𝐵) |
81 | 80 | sselda 3568 |
. . . . . . . . . 10
⊢ ((𝑆 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑆) → 𝑋 ∈ 𝐵) |
82 | 81 | 3adant2 1073 |
. . . . . . . . 9
⊢ ((𝑆 ∈ 𝒫 𝐵 ∧ 𝑀 ∈ LMod ∧ 𝑋 ∈ 𝑆) → 𝑋 ∈ 𝐵) |
83 | 82 | adantr 480 |
. . . . . . . 8
⊢ (((𝑆 ∈ 𝒫 𝐵 ∧ 𝑀 ∈ LMod ∧ 𝑋 ∈ 𝑆) ∧ (𝐹 ∈ (𝐸 ↑𝑚 𝑆) ∧ (𝐹‘𝑋) ∈ 𝑈 ∧ 𝐹 finSupp 0 )) → 𝑋 ∈ 𝐵) |
84 | 8, 9, 56, 10 | lmodvscl 18703 |
. . . . . . . 8
⊢ ((𝑀 ∈ LMod ∧ (𝐹‘𝑋) ∈ 𝐸 ∧ 𝑋 ∈ 𝐵) → ((𝐹‘𝑋)( ·𝑠
‘𝑀)𝑋) ∈ 𝐵) |
85 | 75, 79, 83, 84 | syl3anc 1318 |
. . . . . . 7
⊢ (((𝑆 ∈ 𝒫 𝐵 ∧ 𝑀 ∈ LMod ∧ 𝑋 ∈ 𝑆) ∧ (𝐹 ∈ (𝐸 ↑𝑚 𝑆) ∧ (𝐹‘𝑋) ∈ 𝑈 ∧ 𝐹 finSupp 0 )) → ((𝐹‘𝑋)( ·𝑠
‘𝑀)𝑋) ∈ 𝐵) |
86 | 9 | lmodfgrp 18695 |
. . . . . . . . . . 11
⊢ (𝑀 ∈ LMod → 𝑅 ∈ Grp) |
87 | 86 | 3ad2ant2 1076 |
. . . . . . . . . 10
⊢ ((𝑆 ∈ 𝒫 𝐵 ∧ 𝑀 ∈ LMod ∧ 𝑋 ∈ 𝑆) → 𝑅 ∈ Grp) |
88 | 87 | adantr 480 |
. . . . . . . . 9
⊢ (((𝑆 ∈ 𝒫 𝐵 ∧ 𝑀 ∈ LMod ∧ 𝑋 ∈ 𝑆) ∧ (𝐹 ∈ (𝐸 ↑𝑚 𝑆) ∧ (𝐹‘𝑋) ∈ 𝑈 ∧ 𝐹 finSupp 0 )) → 𝑅 ∈ Grp) |
89 | 10, 14 | grpinvcl 17290 |
. . . . . . . . 9
⊢ ((𝑅 ∈ Grp ∧ (𝐹‘𝑋) ∈ 𝐸) → (𝑁‘(𝐹‘𝑋)) ∈ 𝐸) |
90 | 88, 79, 89 | syl2anc 691 |
. . . . . . . 8
⊢ (((𝑆 ∈ 𝒫 𝐵 ∧ 𝑀 ∈ LMod ∧ 𝑋 ∈ 𝑆) ∧ (𝐹 ∈ (𝐸 ↑𝑚 𝑆) ∧ (𝐹‘𝑋) ∈ 𝑈 ∧ 𝐹 finSupp 0 )) → (𝑁‘(𝐹‘𝑋)) ∈ 𝐸) |
91 | | lmodcmn 18734 |
. . . . . . . . . . 11
⊢ (𝑀 ∈ LMod → 𝑀 ∈ CMnd) |
92 | 91 | 3ad2ant2 1076 |
. . . . . . . . . 10
⊢ ((𝑆 ∈ 𝒫 𝐵 ∧ 𝑀 ∈ LMod ∧ 𝑋 ∈ 𝑆) → 𝑀 ∈ CMnd) |
93 | 92 | adantr 480 |
. . . . . . . . 9
⊢ (((𝑆 ∈ 𝒫 𝐵 ∧ 𝑀 ∈ LMod ∧ 𝑋 ∈ 𝑆) ∧ (𝐹 ∈ (𝐸 ↑𝑚 𝑆) ∧ (𝐹‘𝑋) ∈ 𝑈 ∧ 𝐹 finSupp 0 )) → 𝑀 ∈ CMnd) |
94 | 26 | adantr 480 |
. . . . . . . . 9
⊢ (((𝑆 ∈ 𝒫 𝐵 ∧ 𝑀 ∈ LMod ∧ 𝑋 ∈ 𝑆) ∧ (𝐹 ∈ (𝐸 ↑𝑚 𝑆) ∧ (𝐹‘𝑋) ∈ 𝑈 ∧ 𝐹 finSupp 0 )) → (𝑆 ∖ {𝑋}) ∈ V) |
95 | | simpll2 1094 |
. . . . . . . . . . 11
⊢ ((((𝑆 ∈ 𝒫 𝐵 ∧ 𝑀 ∈ LMod ∧ 𝑋 ∈ 𝑆) ∧ (𝐹 ∈ (𝐸 ↑𝑚 𝑆) ∧ (𝐹‘𝑋) ∈ 𝑈 ∧ 𝐹 finSupp 0 )) ∧ 𝑠 ∈ (𝑆 ∖ {𝑋})) → 𝑀 ∈ LMod) |
96 | 8, 9, 10, 11, 12, 13, 14, 15, 16, 17 | lincresunit1 42060 |
. . . . . . . . . . . . . 14
⊢ (((𝑆 ∈ 𝒫 𝐵 ∧ 𝑀 ∈ LMod ∧ 𝑋 ∈ 𝑆) ∧ (𝐹 ∈ (𝐸 ↑𝑚 𝑆) ∧ (𝐹‘𝑋) ∈ 𝑈)) → 𝐺 ∈ (𝐸 ↑𝑚 (𝑆 ∖ {𝑋}))) |
97 | 96 | 3adantr3 1215 |
. . . . . . . . . . . . 13
⊢ (((𝑆 ∈ 𝒫 𝐵 ∧ 𝑀 ∈ LMod ∧ 𝑋 ∈ 𝑆) ∧ (𝐹 ∈ (𝐸 ↑𝑚 𝑆) ∧ (𝐹‘𝑋) ∈ 𝑈 ∧ 𝐹 finSupp 0 )) → 𝐺 ∈ (𝐸 ↑𝑚 (𝑆 ∖ {𝑋}))) |
98 | | elmapi 7765 |
. . . . . . . . . . . . 13
⊢ (𝐺 ∈ (𝐸 ↑𝑚 (𝑆 ∖ {𝑋})) → 𝐺:(𝑆 ∖ {𝑋})⟶𝐸) |
99 | 97, 98 | syl 17 |
. . . . . . . . . . . 12
⊢ (((𝑆 ∈ 𝒫 𝐵 ∧ 𝑀 ∈ LMod ∧ 𝑋 ∈ 𝑆) ∧ (𝐹 ∈ (𝐸 ↑𝑚 𝑆) ∧ (𝐹‘𝑋) ∈ 𝑈 ∧ 𝐹 finSupp 0 )) → 𝐺:(𝑆 ∖ {𝑋})⟶𝐸) |
100 | 99 | ffvelrnda 6267 |
. . . . . . . . . . 11
⊢ ((((𝑆 ∈ 𝒫 𝐵 ∧ 𝑀 ∈ LMod ∧ 𝑋 ∈ 𝑆) ∧ (𝐹 ∈ (𝐸 ↑𝑚 𝑆) ∧ (𝐹‘𝑋) ∈ 𝑈 ∧ 𝐹 finSupp 0 )) ∧ 𝑠 ∈ (𝑆 ∖ {𝑋})) → (𝐺‘𝑠) ∈ 𝐸) |
101 | | ssel2 3563 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝑆 ⊆ 𝐵 ∧ 𝑠 ∈ 𝑆) → 𝑠 ∈ 𝐵) |
102 | 101 | expcom 450 |
. . . . . . . . . . . . . . . 16
⊢ (𝑠 ∈ 𝑆 → (𝑆 ⊆ 𝐵 → 𝑠 ∈ 𝐵)) |
103 | 7, 102 | syl 17 |
. . . . . . . . . . . . . . 15
⊢ (𝑠 ∈ (𝑆 ∖ {𝑋}) → (𝑆 ⊆ 𝐵 → 𝑠 ∈ 𝐵)) |
104 | 80, 103 | syl5com 31 |
. . . . . . . . . . . . . 14
⊢ (𝑆 ∈ 𝒫 𝐵 → (𝑠 ∈ (𝑆 ∖ {𝑋}) → 𝑠 ∈ 𝐵)) |
105 | 104 | 3ad2ant1 1075 |
. . . . . . . . . . . . 13
⊢ ((𝑆 ∈ 𝒫 𝐵 ∧ 𝑀 ∈ LMod ∧ 𝑋 ∈ 𝑆) → (𝑠 ∈ (𝑆 ∖ {𝑋}) → 𝑠 ∈ 𝐵)) |
106 | 105 | adantr 480 |
. . . . . . . . . . . 12
⊢ (((𝑆 ∈ 𝒫 𝐵 ∧ 𝑀 ∈ LMod ∧ 𝑋 ∈ 𝑆) ∧ (𝐹 ∈ (𝐸 ↑𝑚 𝑆) ∧ (𝐹‘𝑋) ∈ 𝑈 ∧ 𝐹 finSupp 0 )) → (𝑠 ∈ (𝑆 ∖ {𝑋}) → 𝑠 ∈ 𝐵)) |
107 | 106 | imp 444 |
. . . . . . . . . . 11
⊢ ((((𝑆 ∈ 𝒫 𝐵 ∧ 𝑀 ∈ LMod ∧ 𝑋 ∈ 𝑆) ∧ (𝐹 ∈ (𝐸 ↑𝑚 𝑆) ∧ (𝐹‘𝑋) ∈ 𝑈 ∧ 𝐹 finSupp 0 )) ∧ 𝑠 ∈ (𝑆 ∖ {𝑋})) → 𝑠 ∈ 𝐵) |
108 | 8, 9, 56, 10 | lmodvscl 18703 |
. . . . . . . . . . 11
⊢ ((𝑀 ∈ LMod ∧ (𝐺‘𝑠) ∈ 𝐸 ∧ 𝑠 ∈ 𝐵) → ((𝐺‘𝑠)( ·𝑠
‘𝑀)𝑠) ∈ 𝐵) |
109 | 95, 100, 107, 108 | syl3anc 1318 |
. . . . . . . . . 10
⊢ ((((𝑆 ∈ 𝒫 𝐵 ∧ 𝑀 ∈ LMod ∧ 𝑋 ∈ 𝑆) ∧ (𝐹 ∈ (𝐸 ↑𝑚 𝑆) ∧ (𝐹‘𝑋) ∈ 𝑈 ∧ 𝐹 finSupp 0 )) ∧ 𝑠 ∈ (𝑆 ∖ {𝑋})) → ((𝐺‘𝑠)( ·𝑠
‘𝑀)𝑠) ∈ 𝐵) |
110 | | eqid 2610 |
. . . . . . . . . 10
⊢ (𝑠 ∈ (𝑆 ∖ {𝑋}) ↦ ((𝐺‘𝑠)( ·𝑠
‘𝑀)𝑠)) = (𝑠 ∈ (𝑆 ∖ {𝑋}) ↦ ((𝐺‘𝑠)( ·𝑠
‘𝑀)𝑠)) |
111 | 109, 110 | fmptd 6292 |
. . . . . . . . 9
⊢ (((𝑆 ∈ 𝒫 𝐵 ∧ 𝑀 ∈ LMod ∧ 𝑋 ∈ 𝑆) ∧ (𝐹 ∈ (𝐸 ↑𝑚 𝑆) ∧ (𝐹‘𝑋) ∈ 𝑈 ∧ 𝐹 finSupp 0 )) → (𝑠 ∈ (𝑆 ∖ {𝑋}) ↦ ((𝐺‘𝑠)( ·𝑠
‘𝑀)𝑠)):(𝑆 ∖ {𝑋})⟶𝐵) |
112 | | ssdifss 3703 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑆 ⊆ 𝐵 → (𝑆 ∖ {𝑋}) ⊆ 𝐵) |
113 | 80, 112 | syl 17 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑆 ∈ 𝒫 𝐵 → (𝑆 ∖ {𝑋}) ⊆ 𝐵) |
114 | 113 | adantr 480 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑆 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑆) → (𝑆 ∖ {𝑋}) ⊆ 𝐵) |
115 | 114, 8 | syl6sseq 3614 |
. . . . . . . . . . . . . . 15
⊢ ((𝑆 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑆) → (𝑆 ∖ {𝑋}) ⊆ (Base‘𝑀)) |
116 | 25 | adantr 480 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑆 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑆) → (𝑆 ∖ {𝑋}) ∈ V) |
117 | 116, 38 | syl 17 |
. . . . . . . . . . . . . . 15
⊢ ((𝑆 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑆) → ((𝑆 ∖ {𝑋}) ∈ 𝒫 (Base‘𝑀) ↔ (𝑆 ∖ {𝑋}) ⊆ (Base‘𝑀))) |
118 | 115, 117 | mpbird 246 |
. . . . . . . . . . . . . 14
⊢ ((𝑆 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑆) → (𝑆 ∖ {𝑋}) ∈ 𝒫 (Base‘𝑀)) |
119 | 118 | 3adant2 1073 |
. . . . . . . . . . . . 13
⊢ ((𝑆 ∈ 𝒫 𝐵 ∧ 𝑀 ∈ LMod ∧ 𝑋 ∈ 𝑆) → (𝑆 ∖ {𝑋}) ∈ 𝒫 (Base‘𝑀)) |
120 | 1, 119 | jca 553 |
. . . . . . . . . . . 12
⊢ ((𝑆 ∈ 𝒫 𝐵 ∧ 𝑀 ∈ LMod ∧ 𝑋 ∈ 𝑆) → (𝑀 ∈ LMod ∧ (𝑆 ∖ {𝑋}) ∈ 𝒫 (Base‘𝑀))) |
121 | 120 | adantr 480 |
. . . . . . . . . . 11
⊢ (((𝑆 ∈ 𝒫 𝐵 ∧ 𝑀 ∈ LMod ∧ 𝑋 ∈ 𝑆) ∧ (𝐹 ∈ (𝐸 ↑𝑚 𝑆) ∧ (𝐹‘𝑋) ∈ 𝑈 ∧ 𝐹 finSupp 0 )) → (𝑀 ∈ LMod ∧ (𝑆 ∖ {𝑋}) ∈ 𝒫 (Base‘𝑀))) |
122 | 8, 9, 10, 11, 12, 13, 14, 15, 16, 17 | lincresunit2 42061 |
. . . . . . . . . . . 12
⊢ (((𝑆 ∈ 𝒫 𝐵 ∧ 𝑀 ∈ LMod ∧ 𝑋 ∈ 𝑆) ∧ (𝐹 ∈ (𝐸 ↑𝑚 𝑆) ∧ (𝐹‘𝑋) ∈ 𝑈 ∧ 𝐹 finSupp 0 )) → 𝐺 finSupp 0 ) |
123 | 122, 12 | syl6breq 4624 |
. . . . . . . . . . 11
⊢ (((𝑆 ∈ 𝒫 𝐵 ∧ 𝑀 ∈ LMod ∧ 𝑋 ∈ 𝑆) ∧ (𝐹 ∈ (𝐸 ↑𝑚 𝑆) ∧ (𝐹‘𝑋) ∈ 𝑈 ∧ 𝐹 finSupp 0 )) → 𝐺 finSupp
(0g‘𝑅)) |
124 | 9, 10 | scmfsupp 41953 |
. . . . . . . . . . 11
⊢ (((𝑀 ∈ LMod ∧ (𝑆 ∖ {𝑋}) ∈ 𝒫 (Base‘𝑀)) ∧ 𝐺 ∈ (𝐸 ↑𝑚 (𝑆 ∖ {𝑋})) ∧ 𝐺 finSupp (0g‘𝑅)) → (𝑠 ∈ (𝑆 ∖ {𝑋}) ↦ ((𝐺‘𝑠)( ·𝑠
‘𝑀)𝑠)) finSupp (0g‘𝑀)) |
125 | 121, 97, 123, 124 | syl3anc 1318 |
. . . . . . . . . 10
⊢ (((𝑆 ∈ 𝒫 𝐵 ∧ 𝑀 ∈ LMod ∧ 𝑋 ∈ 𝑆) ∧ (𝐹 ∈ (𝐸 ↑𝑚 𝑆) ∧ (𝐹‘𝑋) ∈ 𝑈 ∧ 𝐹 finSupp 0 )) → (𝑠 ∈ (𝑆 ∖ {𝑋}) ↦ ((𝐺‘𝑠)( ·𝑠
‘𝑀)𝑠)) finSupp (0g‘𝑀)) |
126 | 125, 13 | syl6breqr 4625 |
. . . . . . . . 9
⊢ (((𝑆 ∈ 𝒫 𝐵 ∧ 𝑀 ∈ LMod ∧ 𝑋 ∈ 𝑆) ∧ (𝐹 ∈ (𝐸 ↑𝑚 𝑆) ∧ (𝐹‘𝑋) ∈ 𝑈 ∧ 𝐹 finSupp 0 )) → (𝑠 ∈ (𝑆 ∖ {𝑋}) ↦ ((𝐺‘𝑠)( ·𝑠
‘𝑀)𝑠)) finSupp 𝑍) |
127 | 8, 13, 93, 94, 111, 126 | gsumcl 18139 |
. . . . . . . 8
⊢ (((𝑆 ∈ 𝒫 𝐵 ∧ 𝑀 ∈ LMod ∧ 𝑋 ∈ 𝑆) ∧ (𝐹 ∈ (𝐸 ↑𝑚 𝑆) ∧ (𝐹‘𝑋) ∈ 𝑈 ∧ 𝐹 finSupp 0 )) → (𝑀 Σg
(𝑠 ∈ (𝑆 ∖ {𝑋}) ↦ ((𝐺‘𝑠)( ·𝑠
‘𝑀)𝑠))) ∈ 𝐵) |
128 | 8, 9, 56, 10 | lmodvscl 18703 |
. . . . . . . 8
⊢ ((𝑀 ∈ LMod ∧ (𝑁‘(𝐹‘𝑋)) ∈ 𝐸 ∧ (𝑀 Σg (𝑠 ∈ (𝑆 ∖ {𝑋}) ↦ ((𝐺‘𝑠)( ·𝑠
‘𝑀)𝑠))) ∈ 𝐵) → ((𝑁‘(𝐹‘𝑋))( ·𝑠
‘𝑀)(𝑀 Σg (𝑠 ∈ (𝑆 ∖ {𝑋}) ↦ ((𝐺‘𝑠)( ·𝑠
‘𝑀)𝑠)))) ∈ 𝐵) |
129 | 75, 90, 127, 128 | syl3anc 1318 |
. . . . . . 7
⊢ (((𝑆 ∈ 𝒫 𝐵 ∧ 𝑀 ∈ LMod ∧ 𝑋 ∈ 𝑆) ∧ (𝐹 ∈ (𝐸 ↑𝑚 𝑆) ∧ (𝐹‘𝑋) ∈ 𝑈 ∧ 𝐹 finSupp 0 )) → ((𝑁‘(𝐹‘𝑋))( ·𝑠
‘𝑀)(𝑀 Σg (𝑠 ∈ (𝑆 ∖ {𝑋}) ↦ ((𝐺‘𝑠)( ·𝑠
‘𝑀)𝑠)))) ∈ 𝐵) |
130 | | eqid 2610 |
. . . . . . . 8
⊢
(invg‘𝑀) = (invg‘𝑀) |
131 | 8, 57, 13, 130 | grpinvid2 17294 |
. . . . . . 7
⊢ ((𝑀 ∈ Grp ∧ ((𝐹‘𝑋)( ·𝑠
‘𝑀)𝑋) ∈ 𝐵 ∧ ((𝑁‘(𝐹‘𝑋))( ·𝑠
‘𝑀)(𝑀 Σg (𝑠 ∈ (𝑆 ∖ {𝑋}) ↦ ((𝐺‘𝑠)( ·𝑠
‘𝑀)𝑠)))) ∈ 𝐵) → (((invg‘𝑀)‘((𝐹‘𝑋)( ·𝑠
‘𝑀)𝑋)) = ((𝑁‘(𝐹‘𝑋))( ·𝑠
‘𝑀)(𝑀 Σg (𝑠 ∈ (𝑆 ∖ {𝑋}) ↦ ((𝐺‘𝑠)( ·𝑠
‘𝑀)𝑠)))) ↔ (((𝑁‘(𝐹‘𝑋))( ·𝑠
‘𝑀)(𝑀 Σg (𝑠 ∈ (𝑆 ∖ {𝑋}) ↦ ((𝐺‘𝑠)( ·𝑠
‘𝑀)𝑠))))(+g‘𝑀)((𝐹‘𝑋)( ·𝑠
‘𝑀)𝑋)) = 𝑍)) |
132 | 74, 85, 129, 131 | syl3anc 1318 |
. . . . . 6
⊢ (((𝑆 ∈ 𝒫 𝐵 ∧ 𝑀 ∈ LMod ∧ 𝑋 ∈ 𝑆) ∧ (𝐹 ∈ (𝐸 ↑𝑚 𝑆) ∧ (𝐹‘𝑋) ∈ 𝑈 ∧ 𝐹 finSupp 0 )) →
(((invg‘𝑀)‘((𝐹‘𝑋)( ·𝑠
‘𝑀)𝑋)) = ((𝑁‘(𝐹‘𝑋))( ·𝑠
‘𝑀)(𝑀 Σg (𝑠 ∈ (𝑆 ∖ {𝑋}) ↦ ((𝐺‘𝑠)( ·𝑠
‘𝑀)𝑠)))) ↔ (((𝑁‘(𝐹‘𝑋))( ·𝑠
‘𝑀)(𝑀 Σg (𝑠 ∈ (𝑆 ∖ {𝑋}) ↦ ((𝐺‘𝑠)( ·𝑠
‘𝑀)𝑠))))(+g‘𝑀)((𝐹‘𝑋)( ·𝑠
‘𝑀)𝑋)) = 𝑍)) |
133 | 8, 9, 56, 130, 10, 14, 75, 83, 79 | lmodvsneg 18730 |
. . . . . . . 8
⊢ (((𝑆 ∈ 𝒫 𝐵 ∧ 𝑀 ∈ LMod ∧ 𝑋 ∈ 𝑆) ∧ (𝐹 ∈ (𝐸 ↑𝑚 𝑆) ∧ (𝐹‘𝑋) ∈ 𝑈 ∧ 𝐹 finSupp 0 )) →
((invg‘𝑀)‘((𝐹‘𝑋)( ·𝑠
‘𝑀)𝑋)) = ((𝑁‘(𝐹‘𝑋))( ·𝑠
‘𝑀)𝑋)) |
134 | 133 | eqeq1d 2612 |
. . . . . . 7
⊢ (((𝑆 ∈ 𝒫 𝐵 ∧ 𝑀 ∈ LMod ∧ 𝑋 ∈ 𝑆) ∧ (𝐹 ∈ (𝐸 ↑𝑚 𝑆) ∧ (𝐹‘𝑋) ∈ 𝑈 ∧ 𝐹 finSupp 0 )) →
(((invg‘𝑀)‘((𝐹‘𝑋)( ·𝑠
‘𝑀)𝑋)) = ((𝑁‘(𝐹‘𝑋))( ·𝑠
‘𝑀)(𝑀 Σg (𝑠 ∈ (𝑆 ∖ {𝑋}) ↦ ((𝐺‘𝑠)( ·𝑠
‘𝑀)𝑠)))) ↔ ((𝑁‘(𝐹‘𝑋))( ·𝑠
‘𝑀)𝑋) = ((𝑁‘(𝐹‘𝑋))( ·𝑠
‘𝑀)(𝑀 Σg (𝑠 ∈ (𝑆 ∖ {𝑋}) ↦ ((𝐺‘𝑠)( ·𝑠
‘𝑀)𝑠)))))) |
135 | | simpr2 1061 |
. . . . . . . . 9
⊢ (((𝑆 ∈ 𝒫 𝐵 ∧ 𝑀 ∈ LMod ∧ 𝑋 ∈ 𝑆) ∧ (𝐹 ∈ (𝐸 ↑𝑚 𝑆) ∧ (𝐹‘𝑋) ∈ 𝑈 ∧ 𝐹 finSupp 0 )) → (𝐹‘𝑋) ∈ 𝑈) |
136 | 8, 9, 10, 11, 14, 56 | lincresunit3lem3 42057 |
. . . . . . . . . 10
⊢ (((𝑀 ∈ LMod ∧ 𝑋 ∈ 𝐵 ∧ (𝑀 Σg (𝑠 ∈ (𝑆 ∖ {𝑋}) ↦ ((𝐺‘𝑠)( ·𝑠
‘𝑀)𝑠))) ∈ 𝐵) ∧ (𝐹‘𝑋) ∈ 𝑈) → (((𝑁‘(𝐹‘𝑋))( ·𝑠
‘𝑀)𝑋) = ((𝑁‘(𝐹‘𝑋))( ·𝑠
‘𝑀)(𝑀 Σg (𝑠 ∈ (𝑆 ∖ {𝑋}) ↦ ((𝐺‘𝑠)( ·𝑠
‘𝑀)𝑠)))) ↔ 𝑋 = (𝑀 Σg (𝑠 ∈ (𝑆 ∖ {𝑋}) ↦ ((𝐺‘𝑠)( ·𝑠
‘𝑀)𝑠))))) |
137 | | eqcom 2617 |
. . . . . . . . . 10
⊢ (𝑋 = (𝑀 Σg (𝑠 ∈ (𝑆 ∖ {𝑋}) ↦ ((𝐺‘𝑠)( ·𝑠
‘𝑀)𝑠))) ↔ (𝑀 Σg (𝑠 ∈ (𝑆 ∖ {𝑋}) ↦ ((𝐺‘𝑠)( ·𝑠
‘𝑀)𝑠))) = 𝑋) |
138 | 136, 137 | syl6bb 275 |
. . . . . . . . 9
⊢ (((𝑀 ∈ LMod ∧ 𝑋 ∈ 𝐵 ∧ (𝑀 Σg (𝑠 ∈ (𝑆 ∖ {𝑋}) ↦ ((𝐺‘𝑠)( ·𝑠
‘𝑀)𝑠))) ∈ 𝐵) ∧ (𝐹‘𝑋) ∈ 𝑈) → (((𝑁‘(𝐹‘𝑋))( ·𝑠
‘𝑀)𝑋) = ((𝑁‘(𝐹‘𝑋))( ·𝑠
‘𝑀)(𝑀 Σg (𝑠 ∈ (𝑆 ∖ {𝑋}) ↦ ((𝐺‘𝑠)( ·𝑠
‘𝑀)𝑠)))) ↔ (𝑀 Σg (𝑠 ∈ (𝑆 ∖ {𝑋}) ↦ ((𝐺‘𝑠)( ·𝑠
‘𝑀)𝑠))) = 𝑋)) |
139 | 75, 83, 127, 135, 138 | syl31anc 1321 |
. . . . . . . 8
⊢ (((𝑆 ∈ 𝒫 𝐵 ∧ 𝑀 ∈ LMod ∧ 𝑋 ∈ 𝑆) ∧ (𝐹 ∈ (𝐸 ↑𝑚 𝑆) ∧ (𝐹‘𝑋) ∈ 𝑈 ∧ 𝐹 finSupp 0 )) → (((𝑁‘(𝐹‘𝑋))( ·𝑠
‘𝑀)𝑋) = ((𝑁‘(𝐹‘𝑋))( ·𝑠
‘𝑀)(𝑀 Σg (𝑠 ∈ (𝑆 ∖ {𝑋}) ↦ ((𝐺‘𝑠)( ·𝑠
‘𝑀)𝑠)))) ↔ (𝑀 Σg (𝑠 ∈ (𝑆 ∖ {𝑋}) ↦ ((𝐺‘𝑠)( ·𝑠
‘𝑀)𝑠))) = 𝑋)) |
140 | 139 | biimpd 218 |
. . . . . . 7
⊢ (((𝑆 ∈ 𝒫 𝐵 ∧ 𝑀 ∈ LMod ∧ 𝑋 ∈ 𝑆) ∧ (𝐹 ∈ (𝐸 ↑𝑚 𝑆) ∧ (𝐹‘𝑋) ∈ 𝑈 ∧ 𝐹 finSupp 0 )) → (((𝑁‘(𝐹‘𝑋))( ·𝑠
‘𝑀)𝑋) = ((𝑁‘(𝐹‘𝑋))( ·𝑠
‘𝑀)(𝑀 Σg (𝑠 ∈ (𝑆 ∖ {𝑋}) ↦ ((𝐺‘𝑠)( ·𝑠
‘𝑀)𝑠)))) → (𝑀 Σg (𝑠 ∈ (𝑆 ∖ {𝑋}) ↦ ((𝐺‘𝑠)( ·𝑠
‘𝑀)𝑠))) = 𝑋)) |
141 | 134, 140 | sylbid 229 |
. . . . . 6
⊢ (((𝑆 ∈ 𝒫 𝐵 ∧ 𝑀 ∈ LMod ∧ 𝑋 ∈ 𝑆) ∧ (𝐹 ∈ (𝐸 ↑𝑚 𝑆) ∧ (𝐹‘𝑋) ∈ 𝑈 ∧ 𝐹 finSupp 0 )) →
(((invg‘𝑀)‘((𝐹‘𝑋)( ·𝑠
‘𝑀)𝑋)) = ((𝑁‘(𝐹‘𝑋))( ·𝑠
‘𝑀)(𝑀 Σg (𝑠 ∈ (𝑆 ∖ {𝑋}) ↦ ((𝐺‘𝑠)( ·𝑠
‘𝑀)𝑠)))) → (𝑀 Σg (𝑠 ∈ (𝑆 ∖ {𝑋}) ↦ ((𝐺‘𝑠)( ·𝑠
‘𝑀)𝑠))) = 𝑋)) |
142 | 132, 141 | sylbird 249 |
. . . . 5
⊢ (((𝑆 ∈ 𝒫 𝐵 ∧ 𝑀 ∈ LMod ∧ 𝑋 ∈ 𝑆) ∧ (𝐹 ∈ (𝐸 ↑𝑚 𝑆) ∧ (𝐹‘𝑋) ∈ 𝑈 ∧ 𝐹 finSupp 0 )) → ((((𝑁‘(𝐹‘𝑋))( ·𝑠
‘𝑀)(𝑀 Σg (𝑠 ∈ (𝑆 ∖ {𝑋}) ↦ ((𝐺‘𝑠)( ·𝑠
‘𝑀)𝑠))))(+g‘𝑀)((𝐹‘𝑋)( ·𝑠
‘𝑀)𝑋)) = 𝑍 → (𝑀 Σg (𝑠 ∈ (𝑆 ∖ {𝑋}) ↦ ((𝐺‘𝑠)( ·𝑠
‘𝑀)𝑠))) = 𝑋)) |
143 | 71, 142 | sylbid 229 |
. . . 4
⊢ (((𝑆 ∈ 𝒫 𝐵 ∧ 𝑀 ∈ LMod ∧ 𝑋 ∈ 𝑆) ∧ (𝐹 ∈ (𝐸 ↑𝑚 𝑆) ∧ (𝐹‘𝑋) ∈ 𝑈 ∧ 𝐹 finSupp 0 )) → ((((𝐹 ↾ (𝑆 ∖ {𝑋}))( linC ‘𝑀)(𝑆 ∖ {𝑋}))(+g‘𝑀)((𝐹‘𝑋)( ·𝑠
‘𝑀)𝑋)) = 𝑍 → (𝑀 Σg (𝑠 ∈ (𝑆 ∖ {𝑋}) ↦ ((𝐺‘𝑠)( ·𝑠
‘𝑀)𝑠))) = 𝑋)) |
144 | 60, 143 | sylbid 229 |
. . 3
⊢ (((𝑆 ∈ 𝒫 𝐵 ∧ 𝑀 ∈ LMod ∧ 𝑋 ∈ 𝑆) ∧ (𝐹 ∈ (𝐸 ↑𝑚 𝑆) ∧ (𝐹‘𝑋) ∈ 𝑈 ∧ 𝐹 finSupp 0 )) → ((𝐹( linC ‘𝑀)𝑆) = 𝑍 → (𝑀 Σg (𝑠 ∈ (𝑆 ∖ {𝑋}) ↦ ((𝐺‘𝑠)( ·𝑠
‘𝑀)𝑠))) = 𝑋)) |
145 | 144 | 3impia 1253 |
. 2
⊢ (((𝑆 ∈ 𝒫 𝐵 ∧ 𝑀 ∈ LMod ∧ 𝑋 ∈ 𝑆) ∧ (𝐹 ∈ (𝐸 ↑𝑚 𝑆) ∧ (𝐹‘𝑋) ∈ 𝑈 ∧ 𝐹 finSupp 0 ) ∧ (𝐹( linC ‘𝑀)𝑆) = 𝑍) → (𝑀 Σg (𝑠 ∈ (𝑆 ∖ {𝑋}) ↦ ((𝐺‘𝑠)( ·𝑠
‘𝑀)𝑠))) = 𝑋) |
146 | 48, 145 | eqtrd 2644 |
1
⊢ (((𝑆 ∈ 𝒫 𝐵 ∧ 𝑀 ∈ LMod ∧ 𝑋 ∈ 𝑆) ∧ (𝐹 ∈ (𝐸 ↑𝑚 𝑆) ∧ (𝐹‘𝑋) ∈ 𝑈 ∧ 𝐹 finSupp 0 ) ∧ (𝐹( linC ‘𝑀)𝑆) = 𝑍) → (𝐺( linC ‘𝑀)(𝑆 ∖ {𝑋})) = 𝑋) |