| Step | Hyp | Ref | Expression | 
|---|
| 1 |  | simpl 482 | . . . 4
⊢ ((𝑠 = 𝑆 ∧ 𝑚 = 𝑀) → 𝑠 = 𝑆) | 
| 2 |  | fveq2 6906 | . . . . . . 7
⊢ (𝑚 = 𝑀 → (Base‘𝑚) = (Base‘𝑀)) | 
| 3 |  | islininds.b | . . . . . . 7
⊢ 𝐵 = (Base‘𝑀) | 
| 4 | 2, 3 | eqtr4di 2795 | . . . . . 6
⊢ (𝑚 = 𝑀 → (Base‘𝑚) = 𝐵) | 
| 5 | 4 | adantl 481 | . . . . 5
⊢ ((𝑠 = 𝑆 ∧ 𝑚 = 𝑀) → (Base‘𝑚) = 𝐵) | 
| 6 | 5 | pweqd 4617 | . . . 4
⊢ ((𝑠 = 𝑆 ∧ 𝑚 = 𝑀) → 𝒫 (Base‘𝑚) = 𝒫 𝐵) | 
| 7 | 1, 6 | eleq12d 2835 | . . 3
⊢ ((𝑠 = 𝑆 ∧ 𝑚 = 𝑀) → (𝑠 ∈ 𝒫 (Base‘𝑚) ↔ 𝑆 ∈ 𝒫 𝐵)) | 
| 8 |  | fveq2 6906 | . . . . . . . . 9
⊢ (𝑚 = 𝑀 → (Scalar‘𝑚) = (Scalar‘𝑀)) | 
| 9 |  | islininds.r | . . . . . . . . 9
⊢ 𝑅 = (Scalar‘𝑀) | 
| 10 | 8, 9 | eqtr4di 2795 | . . . . . . . 8
⊢ (𝑚 = 𝑀 → (Scalar‘𝑚) = 𝑅) | 
| 11 | 10 | fveq2d 6910 | . . . . . . 7
⊢ (𝑚 = 𝑀 → (Base‘(Scalar‘𝑚)) = (Base‘𝑅)) | 
| 12 |  | islininds.e | . . . . . . 7
⊢ 𝐸 = (Base‘𝑅) | 
| 13 | 11, 12 | eqtr4di 2795 | . . . . . 6
⊢ (𝑚 = 𝑀 → (Base‘(Scalar‘𝑚)) = 𝐸) | 
| 14 | 13 | adantl 481 | . . . . 5
⊢ ((𝑠 = 𝑆 ∧ 𝑚 = 𝑀) → (Base‘(Scalar‘𝑚)) = 𝐸) | 
| 15 | 14, 1 | oveq12d 7449 | . . . 4
⊢ ((𝑠 = 𝑆 ∧ 𝑚 = 𝑀) → ((Base‘(Scalar‘𝑚)) ↑m 𝑠) = (𝐸 ↑m 𝑆)) | 
| 16 | 8 | adantl 481 | . . . . . . . . . 10
⊢ ((𝑠 = 𝑆 ∧ 𝑚 = 𝑀) → (Scalar‘𝑚) = (Scalar‘𝑀)) | 
| 17 | 16, 9 | eqtr4di 2795 | . . . . . . . . 9
⊢ ((𝑠 = 𝑆 ∧ 𝑚 = 𝑀) → (Scalar‘𝑚) = 𝑅) | 
| 18 | 17 | fveq2d 6910 | . . . . . . . 8
⊢ ((𝑠 = 𝑆 ∧ 𝑚 = 𝑀) →
(0g‘(Scalar‘𝑚)) = (0g‘𝑅)) | 
| 19 |  | islininds.0 | . . . . . . . 8
⊢  0 =
(0g‘𝑅) | 
| 20 | 18, 19 | eqtr4di 2795 | . . . . . . 7
⊢ ((𝑠 = 𝑆 ∧ 𝑚 = 𝑀) →
(0g‘(Scalar‘𝑚)) = 0 ) | 
| 21 | 20 | breq2d 5155 | . . . . . 6
⊢ ((𝑠 = 𝑆 ∧ 𝑚 = 𝑀) → (𝑓 finSupp
(0g‘(Scalar‘𝑚)) ↔ 𝑓 finSupp 0 )) | 
| 22 |  | fveq2 6906 | . . . . . . . . 9
⊢ (𝑚 = 𝑀 → ( linC ‘𝑚) = ( linC ‘𝑀)) | 
| 23 | 22 | adantl 481 | . . . . . . . 8
⊢ ((𝑠 = 𝑆 ∧ 𝑚 = 𝑀) → ( linC ‘𝑚) = ( linC ‘𝑀)) | 
| 24 |  | eqidd 2738 | . . . . . . . 8
⊢ ((𝑠 = 𝑆 ∧ 𝑚 = 𝑀) → 𝑓 = 𝑓) | 
| 25 | 23, 24, 1 | oveq123d 7452 | . . . . . . 7
⊢ ((𝑠 = 𝑆 ∧ 𝑚 = 𝑀) → (𝑓( linC ‘𝑚)𝑠) = (𝑓( linC ‘𝑀)𝑆)) | 
| 26 |  | fveq2 6906 | . . . . . . . . 9
⊢ (𝑚 = 𝑀 → (0g‘𝑚) = (0g‘𝑀)) | 
| 27 | 26 | adantl 481 | . . . . . . . 8
⊢ ((𝑠 = 𝑆 ∧ 𝑚 = 𝑀) → (0g‘𝑚) = (0g‘𝑀)) | 
| 28 |  | islininds.z | . . . . . . . 8
⊢ 𝑍 = (0g‘𝑀) | 
| 29 | 27, 28 | eqtr4di 2795 | . . . . . . 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 632 | . . . . 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 6910 | . . . . . . . . 9
⊢ (𝑚 = 𝑀 →
(0g‘(Scalar‘𝑚)) = (0g‘𝑅)) | 
| 33 | 32, 19 | eqtr4di 2795 | . . . . . . . 8
⊢ (𝑚 = 𝑀 →
(0g‘(Scalar‘𝑚)) = 0 ) | 
| 34 | 33 | adantl 481 | . . . . . . 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 3346 | . . . . 5
⊢ ((𝑠 = 𝑆 ∧ 𝑚 = 𝑀) → (∀𝑥 ∈ 𝑠 (𝑓‘𝑥) = (0g‘(Scalar‘𝑚)) ↔ ∀𝑥 ∈ 𝑆 (𝑓‘𝑥) = 0 )) | 
| 37 | 31, 36 | imbi12d 344 | . . . 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 3346 | . . 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 632 | . 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 48359 | . 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 5539 | 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 )))) |