Step | Hyp | Ref
| Expression |
1 | | simpl 486 |
. . . 4
⊢ ((𝑠 = 𝑆 ∧ 𝑚 = 𝑀) → 𝑠 = 𝑆) |
2 | | fveq2 6717 |
. . . . . . 7
⊢ (𝑚 = 𝑀 → (Base‘𝑚) = (Base‘𝑀)) |
3 | | islininds.b |
. . . . . . 7
⊢ 𝐵 = (Base‘𝑀) |
4 | 2, 3 | eqtr4di 2796 |
. . . . . 6
⊢ (𝑚 = 𝑀 → (Base‘𝑚) = 𝐵) |
5 | 4 | adantl 485 |
. . . . 5
⊢ ((𝑠 = 𝑆 ∧ 𝑚 = 𝑀) → (Base‘𝑚) = 𝐵) |
6 | 5 | pweqd 4532 |
. . . 4
⊢ ((𝑠 = 𝑆 ∧ 𝑚 = 𝑀) → 𝒫 (Base‘𝑚) = 𝒫 𝐵) |
7 | 1, 6 | eleq12d 2832 |
. . 3
⊢ ((𝑠 = 𝑆 ∧ 𝑚 = 𝑀) → (𝑠 ∈ 𝒫 (Base‘𝑚) ↔ 𝑆 ∈ 𝒫 𝐵)) |
8 | | fveq2 6717 |
. . . . . . . . 9
⊢ (𝑚 = 𝑀 → (Scalar‘𝑚) = (Scalar‘𝑀)) |
9 | | islininds.r |
. . . . . . . . 9
⊢ 𝑅 = (Scalar‘𝑀) |
10 | 8, 9 | eqtr4di 2796 |
. . . . . . . 8
⊢ (𝑚 = 𝑀 → (Scalar‘𝑚) = 𝑅) |
11 | 10 | fveq2d 6721 |
. . . . . . 7
⊢ (𝑚 = 𝑀 → (Base‘(Scalar‘𝑚)) = (Base‘𝑅)) |
12 | | islininds.e |
. . . . . . 7
⊢ 𝐸 = (Base‘𝑅) |
13 | 11, 12 | eqtr4di 2796 |
. . . . . 6
⊢ (𝑚 = 𝑀 → (Base‘(Scalar‘𝑚)) = 𝐸) |
14 | 13 | adantl 485 |
. . . . 5
⊢ ((𝑠 = 𝑆 ∧ 𝑚 = 𝑀) → (Base‘(Scalar‘𝑚)) = 𝐸) |
15 | 14, 1 | oveq12d 7231 |
. . . 4
⊢ ((𝑠 = 𝑆 ∧ 𝑚 = 𝑀) → ((Base‘(Scalar‘𝑚)) ↑m 𝑠) = (𝐸 ↑m 𝑆)) |
16 | 8 | adantl 485 |
. . . . . . . . . 10
⊢ ((𝑠 = 𝑆 ∧ 𝑚 = 𝑀) → (Scalar‘𝑚) = (Scalar‘𝑀)) |
17 | 16, 9 | eqtr4di 2796 |
. . . . . . . . 9
⊢ ((𝑠 = 𝑆 ∧ 𝑚 = 𝑀) → (Scalar‘𝑚) = 𝑅) |
18 | 17 | fveq2d 6721 |
. . . . . . . 8
⊢ ((𝑠 = 𝑆 ∧ 𝑚 = 𝑀) →
(0g‘(Scalar‘𝑚)) = (0g‘𝑅)) |
19 | | islininds.0 |
. . . . . . . 8
⊢ 0 =
(0g‘𝑅) |
20 | 18, 19 | eqtr4di 2796 |
. . . . . . 7
⊢ ((𝑠 = 𝑆 ∧ 𝑚 = 𝑀) →
(0g‘(Scalar‘𝑚)) = 0 ) |
21 | 20 | breq2d 5065 |
. . . . . 6
⊢ ((𝑠 = 𝑆 ∧ 𝑚 = 𝑀) → (𝑓 finSupp
(0g‘(Scalar‘𝑚)) ↔ 𝑓 finSupp 0 )) |
22 | | fveq2 6717 |
. . . . . . . . 9
⊢ (𝑚 = 𝑀 → ( linC ‘𝑚) = ( linC ‘𝑀)) |
23 | 22 | adantl 485 |
. . . . . . . 8
⊢ ((𝑠 = 𝑆 ∧ 𝑚 = 𝑀) → ( linC ‘𝑚) = ( linC ‘𝑀)) |
24 | | eqidd 2738 |
. . . . . . . 8
⊢ ((𝑠 = 𝑆 ∧ 𝑚 = 𝑀) → 𝑓 = 𝑓) |
25 | 23, 24, 1 | oveq123d 7234 |
. . . . . . 7
⊢ ((𝑠 = 𝑆 ∧ 𝑚 = 𝑀) → (𝑓( linC ‘𝑚)𝑠) = (𝑓( linC ‘𝑀)𝑆)) |
26 | | fveq2 6717 |
. . . . . . . . 9
⊢ (𝑚 = 𝑀 → (0g‘𝑚) = (0g‘𝑀)) |
27 | 26 | adantl 485 |
. . . . . . . 8
⊢ ((𝑠 = 𝑆 ∧ 𝑚 = 𝑀) → (0g‘𝑚) = (0g‘𝑀)) |
28 | | islininds.z |
. . . . . . . 8
⊢ 𝑍 = (0g‘𝑀) |
29 | 27, 28 | eqtr4di 2796 |
. . . . . . 7
⊢ ((𝑠 = 𝑆 ∧ 𝑚 = 𝑀) → (0g‘𝑚) = 𝑍) |
30 | 25, 29 | eqeq12d 2753 |
. . . . . 6
⊢ ((𝑠 = 𝑆 ∧ 𝑚 = 𝑀) → ((𝑓( linC ‘𝑚)𝑠) = (0g‘𝑚) ↔ (𝑓( linC ‘𝑀)𝑆) = 𝑍)) |
31 | 21, 30 | anbi12d 634 |
. . . . 5
⊢ ((𝑠 = 𝑆 ∧ 𝑚 = 𝑀) → ((𝑓 finSupp
(0g‘(Scalar‘𝑚)) ∧ (𝑓( linC ‘𝑚)𝑠) = (0g‘𝑚)) ↔ (𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍))) |
32 | 10 | fveq2d 6721 |
. . . . . . . . 9
⊢ (𝑚 = 𝑀 →
(0g‘(Scalar‘𝑚)) = (0g‘𝑅)) |
33 | 32, 19 | eqtr4di 2796 |
. . . . . . . 8
⊢ (𝑚 = 𝑀 →
(0g‘(Scalar‘𝑚)) = 0 ) |
34 | 33 | adantl 485 |
. . . . . . 7
⊢ ((𝑠 = 𝑆 ∧ 𝑚 = 𝑀) →
(0g‘(Scalar‘𝑚)) = 0 ) |
35 | 34 | eqeq2d 2748 |
. . . . . 6
⊢ ((𝑠 = 𝑆 ∧ 𝑚 = 𝑀) → ((𝑓‘𝑥) = (0g‘(Scalar‘𝑚)) ↔ (𝑓‘𝑥) = 0 )) |
36 | 1, 35 | raleqbidv 3313 |
. . . . 5
⊢ ((𝑠 = 𝑆 ∧ 𝑚 = 𝑀) → (∀𝑥 ∈ 𝑠 (𝑓‘𝑥) = (0g‘(Scalar‘𝑚)) ↔ ∀𝑥 ∈ 𝑆 (𝑓‘𝑥) = 0 )) |
37 | 31, 36 | imbi12d 348 |
. . . 4
⊢ ((𝑠 = 𝑆 ∧ 𝑚 = 𝑀) → (((𝑓 finSupp
(0g‘(Scalar‘𝑚)) ∧ (𝑓( linC ‘𝑚)𝑠) = (0g‘𝑚)) → ∀𝑥 ∈ 𝑠 (𝑓‘𝑥) = (0g‘(Scalar‘𝑚))) ↔ ((𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍) → ∀𝑥 ∈ 𝑆 (𝑓‘𝑥) = 0 ))) |
38 | 15, 37 | raleqbidv 3313 |
. . 3
⊢ ((𝑠 = 𝑆 ∧ 𝑚 = 𝑀) → (∀𝑓 ∈ ((Base‘(Scalar‘𝑚)) ↑m 𝑠)((𝑓 finSupp
(0g‘(Scalar‘𝑚)) ∧ (𝑓( linC ‘𝑚)𝑠) = (0g‘𝑚)) → ∀𝑥 ∈ 𝑠 (𝑓‘𝑥) = (0g‘(Scalar‘𝑚))) ↔ ∀𝑓 ∈ (𝐸 ↑m 𝑆)((𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍) → ∀𝑥 ∈ 𝑆 (𝑓‘𝑥) = 0 ))) |
39 | 7, 38 | anbi12d 634 |
. 2
⊢ ((𝑠 = 𝑆 ∧ 𝑚 = 𝑀) → ((𝑠 ∈ 𝒫 (Base‘𝑚) ∧ ∀𝑓 ∈
((Base‘(Scalar‘𝑚)) ↑m 𝑠)((𝑓 finSupp
(0g‘(Scalar‘𝑚)) ∧ (𝑓( linC ‘𝑚)𝑠) = (0g‘𝑚)) → ∀𝑥 ∈ 𝑠 (𝑓‘𝑥) = (0g‘(Scalar‘𝑚)))) ↔ (𝑆 ∈ 𝒫 𝐵 ∧ ∀𝑓 ∈ (𝐸 ↑m 𝑆)((𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍) → ∀𝑥 ∈ 𝑆 (𝑓‘𝑥) = 0 )))) |
40 | | df-lininds 45456 |
. 2
⊢ linIndS
= {〈𝑠, 𝑚〉 ∣ (𝑠 ∈ 𝒫
(Base‘𝑚) ∧
∀𝑓 ∈
((Base‘(Scalar‘𝑚)) ↑m 𝑠)((𝑓 finSupp
(0g‘(Scalar‘𝑚)) ∧ (𝑓( linC ‘𝑚)𝑠) = (0g‘𝑚)) → ∀𝑥 ∈ 𝑠 (𝑓‘𝑥) = (0g‘(Scalar‘𝑚))))} |
41 | 39, 40 | brabga 5415 |
1
⊢ ((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ 𝑊) → (𝑆 linIndS 𝑀 ↔ (𝑆 ∈ 𝒫 𝐵 ∧ ∀𝑓 ∈ (𝐸 ↑m 𝑆)((𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍) → ∀𝑥 ∈ 𝑆 (𝑓‘𝑥) = 0 )))) |