| Step | Hyp | Ref | Expression | 
|---|
| 1 |  | dsmmlss.p | . . 3
⊢ 𝑃 = (𝑆Xs𝑅) | 
| 2 |  | dsmmlss.h | . . 3
⊢ 𝐻 = (Base‘(𝑆 ⊕m 𝑅)) | 
| 3 |  | dsmmlss.i | . . 3
⊢ (𝜑 → 𝐼 ∈ 𝑊) | 
| 4 |  | dsmmlss.s | . . 3
⊢ (𝜑 → 𝑆 ∈ Ring) | 
| 5 |  | dsmmlss.r | . . . 4
⊢ (𝜑 → 𝑅:𝐼⟶LMod) | 
| 6 |  | lmodgrp 20865 | . . . . 5
⊢ (𝑎 ∈ LMod → 𝑎 ∈ Grp) | 
| 7 | 6 | ssriv 3987 | . . . 4
⊢ LMod
⊆ Grp | 
| 8 |  | fss 6752 | . . . 4
⊢ ((𝑅:𝐼⟶LMod ∧ LMod ⊆ Grp) →
𝑅:𝐼⟶Grp) | 
| 9 | 5, 7, 8 | sylancl 586 | . . 3
⊢ (𝜑 → 𝑅:𝐼⟶Grp) | 
| 10 | 1, 2, 3, 4, 9 | dsmmsubg 21763 | . 2
⊢ (𝜑 → 𝐻 ∈ (SubGrp‘𝑃)) | 
| 11 |  | dsmmlss.k | . . . . . . 7
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → (Scalar‘(𝑅‘𝑥)) = 𝑆) | 
| 12 | 1, 4, 3, 5, 11 | prdslmodd 20967 | . . . . . 6
⊢ (𝜑 → 𝑃 ∈ LMod) | 
| 13 | 12 | adantr 480 | . . . . 5
⊢ ((𝜑 ∧ (𝑎 ∈ (Base‘(Scalar‘𝑃)) ∧ 𝑏 ∈ 𝐻)) → 𝑃 ∈ LMod) | 
| 14 |  | simprl 771 | . . . . 5
⊢ ((𝜑 ∧ (𝑎 ∈ (Base‘(Scalar‘𝑃)) ∧ 𝑏 ∈ 𝐻)) → 𝑎 ∈ (Base‘(Scalar‘𝑃))) | 
| 15 |  | simprr 773 | . . . . . . 7
⊢ ((𝜑 ∧ (𝑎 ∈ (Base‘(Scalar‘𝑃)) ∧ 𝑏 ∈ 𝐻)) → 𝑏 ∈ 𝐻) | 
| 16 |  | eqid 2737 | . . . . . . . . 9
⊢ (𝑆 ⊕m 𝑅) = (𝑆 ⊕m 𝑅) | 
| 17 |  | eqid 2737 | . . . . . . . . 9
⊢
(Base‘𝑃) =
(Base‘𝑃) | 
| 18 | 5 | ffnd 6737 | . . . . . . . . 9
⊢ (𝜑 → 𝑅 Fn 𝐼) | 
| 19 | 1, 16, 17, 2, 3, 18 | dsmmelbas 21759 | . . . . . . . 8
⊢ (𝜑 → (𝑏 ∈ 𝐻 ↔ (𝑏 ∈ (Base‘𝑃) ∧ {𝑥 ∈ 𝐼 ∣ (𝑏‘𝑥) ≠ (0g‘(𝑅‘𝑥))} ∈ Fin))) | 
| 20 | 19 | adantr 480 | . . . . . . 7
⊢ ((𝜑 ∧ (𝑎 ∈ (Base‘(Scalar‘𝑃)) ∧ 𝑏 ∈ 𝐻)) → (𝑏 ∈ 𝐻 ↔ (𝑏 ∈ (Base‘𝑃) ∧ {𝑥 ∈ 𝐼 ∣ (𝑏‘𝑥) ≠ (0g‘(𝑅‘𝑥))} ∈ Fin))) | 
| 21 | 15, 20 | mpbid 232 | . . . . . 6
⊢ ((𝜑 ∧ (𝑎 ∈ (Base‘(Scalar‘𝑃)) ∧ 𝑏 ∈ 𝐻)) → (𝑏 ∈ (Base‘𝑃) ∧ {𝑥 ∈ 𝐼 ∣ (𝑏‘𝑥) ≠ (0g‘(𝑅‘𝑥))} ∈ Fin)) | 
| 22 | 21 | simpld 494 | . . . . 5
⊢ ((𝜑 ∧ (𝑎 ∈ (Base‘(Scalar‘𝑃)) ∧ 𝑏 ∈ 𝐻)) → 𝑏 ∈ (Base‘𝑃)) | 
| 23 |  | eqid 2737 | . . . . . 6
⊢
(Scalar‘𝑃) =
(Scalar‘𝑃) | 
| 24 |  | eqid 2737 | . . . . . 6
⊢ (
·𝑠 ‘𝑃) = ( ·𝑠
‘𝑃) | 
| 25 |  | eqid 2737 | . . . . . 6
⊢
(Base‘(Scalar‘𝑃)) = (Base‘(Scalar‘𝑃)) | 
| 26 | 17, 23, 24, 25 | lmodvscl 20876 | . . . . 5
⊢ ((𝑃 ∈ LMod ∧ 𝑎 ∈
(Base‘(Scalar‘𝑃)) ∧ 𝑏 ∈ (Base‘𝑃)) → (𝑎( ·𝑠
‘𝑃)𝑏) ∈ (Base‘𝑃)) | 
| 27 | 13, 14, 22, 26 | syl3anc 1373 | . . . 4
⊢ ((𝜑 ∧ (𝑎 ∈ (Base‘(Scalar‘𝑃)) ∧ 𝑏 ∈ 𝐻)) → (𝑎( ·𝑠
‘𝑃)𝑏) ∈ (Base‘𝑃)) | 
| 28 | 21 | simprd 495 | . . . . 5
⊢ ((𝜑 ∧ (𝑎 ∈ (Base‘(Scalar‘𝑃)) ∧ 𝑏 ∈ 𝐻)) → {𝑥 ∈ 𝐼 ∣ (𝑏‘𝑥) ≠ (0g‘(𝑅‘𝑥))} ∈ Fin) | 
| 29 |  | eqid 2737 | . . . . . . . . . . 11
⊢
(Base‘𝑆) =
(Base‘𝑆) | 
| 30 | 4 | ad2antrr 726 | . . . . . . . . . . 11
⊢ (((𝜑 ∧ (𝑎 ∈ (Base‘(Scalar‘𝑃)) ∧ 𝑏 ∈ 𝐻)) ∧ 𝑥 ∈ 𝐼) → 𝑆 ∈ Ring) | 
| 31 | 3 | ad2antrr 726 | . . . . . . . . . . 11
⊢ (((𝜑 ∧ (𝑎 ∈ (Base‘(Scalar‘𝑃)) ∧ 𝑏 ∈ 𝐻)) ∧ 𝑥 ∈ 𝐼) → 𝐼 ∈ 𝑊) | 
| 32 | 18 | ad2antrr 726 | . . . . . . . . . . 11
⊢ (((𝜑 ∧ (𝑎 ∈ (Base‘(Scalar‘𝑃)) ∧ 𝑏 ∈ 𝐻)) ∧ 𝑥 ∈ 𝐼) → 𝑅 Fn 𝐼) | 
| 33 | 5, 3 | fexd 7247 | . . . . . . . . . . . . . . . . 17
⊢ (𝜑 → 𝑅 ∈ V) | 
| 34 | 1, 4, 33 | prdssca 17501 | . . . . . . . . . . . . . . . 16
⊢ (𝜑 → 𝑆 = (Scalar‘𝑃)) | 
| 35 | 34 | fveq2d 6910 | . . . . . . . . . . . . . . 15
⊢ (𝜑 → (Base‘𝑆) =
(Base‘(Scalar‘𝑃))) | 
| 36 | 35 | eleq2d 2827 | . . . . . . . . . . . . . 14
⊢ (𝜑 → (𝑎 ∈ (Base‘𝑆) ↔ 𝑎 ∈ (Base‘(Scalar‘𝑃)))) | 
| 37 | 36 | biimpar 477 | . . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑎 ∈ (Base‘(Scalar‘𝑃))) → 𝑎 ∈ (Base‘𝑆)) | 
| 38 | 37 | adantrr 717 | . . . . . . . . . . . 12
⊢ ((𝜑 ∧ (𝑎 ∈ (Base‘(Scalar‘𝑃)) ∧ 𝑏 ∈ 𝐻)) → 𝑎 ∈ (Base‘𝑆)) | 
| 39 | 38 | adantr 480 | . . . . . . . . . . 11
⊢ (((𝜑 ∧ (𝑎 ∈ (Base‘(Scalar‘𝑃)) ∧ 𝑏 ∈ 𝐻)) ∧ 𝑥 ∈ 𝐼) → 𝑎 ∈ (Base‘𝑆)) | 
| 40 | 22 | adantr 480 | . . . . . . . . . . 11
⊢ (((𝜑 ∧ (𝑎 ∈ (Base‘(Scalar‘𝑃)) ∧ 𝑏 ∈ 𝐻)) ∧ 𝑥 ∈ 𝐼) → 𝑏 ∈ (Base‘𝑃)) | 
| 41 |  | simpr 484 | . . . . . . . . . . 11
⊢ (((𝜑 ∧ (𝑎 ∈ (Base‘(Scalar‘𝑃)) ∧ 𝑏 ∈ 𝐻)) ∧ 𝑥 ∈ 𝐼) → 𝑥 ∈ 𝐼) | 
| 42 | 1, 17, 24, 29, 30, 31, 32, 39, 40, 41 | prdsvscafval 17525 | . . . . . . . . . 10
⊢ (((𝜑 ∧ (𝑎 ∈ (Base‘(Scalar‘𝑃)) ∧ 𝑏 ∈ 𝐻)) ∧ 𝑥 ∈ 𝐼) → ((𝑎( ·𝑠
‘𝑃)𝑏)‘𝑥) = (𝑎( ·𝑠
‘(𝑅‘𝑥))(𝑏‘𝑥))) | 
| 43 | 42 | adantrr 717 | . . . . . . . . 9
⊢ (((𝜑 ∧ (𝑎 ∈ (Base‘(Scalar‘𝑃)) ∧ 𝑏 ∈ 𝐻)) ∧ (𝑥 ∈ 𝐼 ∧ (𝑏‘𝑥) = (0g‘(𝑅‘𝑥)))) → ((𝑎( ·𝑠
‘𝑃)𝑏)‘𝑥) = (𝑎( ·𝑠
‘(𝑅‘𝑥))(𝑏‘𝑥))) | 
| 44 | 5 | ffvelcdmda 7104 | . . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → (𝑅‘𝑥) ∈ LMod) | 
| 45 | 44 | adantlr 715 | . . . . . . . . . . . 12
⊢ (((𝜑 ∧ (𝑎 ∈ (Base‘(Scalar‘𝑃)) ∧ 𝑏 ∈ 𝐻)) ∧ 𝑥 ∈ 𝐼) → (𝑅‘𝑥) ∈ LMod) | 
| 46 |  | simplrl 777 | . . . . . . . . . . . . 13
⊢ (((𝜑 ∧ (𝑎 ∈ (Base‘(Scalar‘𝑃)) ∧ 𝑏 ∈ 𝐻)) ∧ 𝑥 ∈ 𝐼) → 𝑎 ∈ (Base‘(Scalar‘𝑃))) | 
| 47 | 34 | adantr 480 | . . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → 𝑆 = (Scalar‘𝑃)) | 
| 48 | 11, 47 | eqtrd 2777 | . . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → (Scalar‘(𝑅‘𝑥)) = (Scalar‘𝑃)) | 
| 49 | 48 | fveq2d 6910 | . . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → (Base‘(Scalar‘(𝑅‘𝑥))) = (Base‘(Scalar‘𝑃))) | 
| 50 | 49 | adantlr 715 | . . . . . . . . . . . . 13
⊢ (((𝜑 ∧ (𝑎 ∈ (Base‘(Scalar‘𝑃)) ∧ 𝑏 ∈ 𝐻)) ∧ 𝑥 ∈ 𝐼) → (Base‘(Scalar‘(𝑅‘𝑥))) = (Base‘(Scalar‘𝑃))) | 
| 51 | 46, 50 | eleqtrrd 2844 | . . . . . . . . . . . 12
⊢ (((𝜑 ∧ (𝑎 ∈ (Base‘(Scalar‘𝑃)) ∧ 𝑏 ∈ 𝐻)) ∧ 𝑥 ∈ 𝐼) → 𝑎 ∈ (Base‘(Scalar‘(𝑅‘𝑥)))) | 
| 52 |  | eqid 2737 | . . . . . . . . . . . . 13
⊢
(Scalar‘(𝑅‘𝑥)) = (Scalar‘(𝑅‘𝑥)) | 
| 53 |  | eqid 2737 | . . . . . . . . . . . . 13
⊢ (
·𝑠 ‘(𝑅‘𝑥)) = ( ·𝑠
‘(𝑅‘𝑥)) | 
| 54 |  | eqid 2737 | . . . . . . . . . . . . 13
⊢
(Base‘(Scalar‘(𝑅‘𝑥))) = (Base‘(Scalar‘(𝑅‘𝑥))) | 
| 55 |  | eqid 2737 | . . . . . . . . . . . . 13
⊢
(0g‘(𝑅‘𝑥)) = (0g‘(𝑅‘𝑥)) | 
| 56 | 52, 53, 54, 55 | lmodvs0 20894 | . . . . . . . . . . . 12
⊢ (((𝑅‘𝑥) ∈ LMod ∧ 𝑎 ∈ (Base‘(Scalar‘(𝑅‘𝑥)))) → (𝑎( ·𝑠
‘(𝑅‘𝑥))(0g‘(𝑅‘𝑥))) = (0g‘(𝑅‘𝑥))) | 
| 57 | 45, 51, 56 | syl2anc 584 | . . . . . . . . . . 11
⊢ (((𝜑 ∧ (𝑎 ∈ (Base‘(Scalar‘𝑃)) ∧ 𝑏 ∈ 𝐻)) ∧ 𝑥 ∈ 𝐼) → (𝑎( ·𝑠
‘(𝑅‘𝑥))(0g‘(𝑅‘𝑥))) = (0g‘(𝑅‘𝑥))) | 
| 58 |  | oveq2 7439 | . . . . . . . . . . . 12
⊢ ((𝑏‘𝑥) = (0g‘(𝑅‘𝑥)) → (𝑎( ·𝑠
‘(𝑅‘𝑥))(𝑏‘𝑥)) = (𝑎( ·𝑠
‘(𝑅‘𝑥))(0g‘(𝑅‘𝑥)))) | 
| 59 | 58 | eqeq1d 2739 | . . . . . . . . . . 11
⊢ ((𝑏‘𝑥) = (0g‘(𝑅‘𝑥)) → ((𝑎( ·𝑠
‘(𝑅‘𝑥))(𝑏‘𝑥)) = (0g‘(𝑅‘𝑥)) ↔ (𝑎( ·𝑠
‘(𝑅‘𝑥))(0g‘(𝑅‘𝑥))) = (0g‘(𝑅‘𝑥)))) | 
| 60 | 57, 59 | syl5ibrcom 247 | . . . . . . . . . 10
⊢ (((𝜑 ∧ (𝑎 ∈ (Base‘(Scalar‘𝑃)) ∧ 𝑏 ∈ 𝐻)) ∧ 𝑥 ∈ 𝐼) → ((𝑏‘𝑥) = (0g‘(𝑅‘𝑥)) → (𝑎( ·𝑠
‘(𝑅‘𝑥))(𝑏‘𝑥)) = (0g‘(𝑅‘𝑥)))) | 
| 61 | 60 | impr 454 | . . . . . . . . 9
⊢ (((𝜑 ∧ (𝑎 ∈ (Base‘(Scalar‘𝑃)) ∧ 𝑏 ∈ 𝐻)) ∧ (𝑥 ∈ 𝐼 ∧ (𝑏‘𝑥) = (0g‘(𝑅‘𝑥)))) → (𝑎( ·𝑠
‘(𝑅‘𝑥))(𝑏‘𝑥)) = (0g‘(𝑅‘𝑥))) | 
| 62 | 43, 61 | eqtrd 2777 | . . . . . . . 8
⊢ (((𝜑 ∧ (𝑎 ∈ (Base‘(Scalar‘𝑃)) ∧ 𝑏 ∈ 𝐻)) ∧ (𝑥 ∈ 𝐼 ∧ (𝑏‘𝑥) = (0g‘(𝑅‘𝑥)))) → ((𝑎( ·𝑠
‘𝑃)𝑏)‘𝑥) = (0g‘(𝑅‘𝑥))) | 
| 63 | 62 | expr 456 | . . . . . . 7
⊢ (((𝜑 ∧ (𝑎 ∈ (Base‘(Scalar‘𝑃)) ∧ 𝑏 ∈ 𝐻)) ∧ 𝑥 ∈ 𝐼) → ((𝑏‘𝑥) = (0g‘(𝑅‘𝑥)) → ((𝑎( ·𝑠
‘𝑃)𝑏)‘𝑥) = (0g‘(𝑅‘𝑥)))) | 
| 64 | 63 | necon3d 2961 | . . . . . 6
⊢ (((𝜑 ∧ (𝑎 ∈ (Base‘(Scalar‘𝑃)) ∧ 𝑏 ∈ 𝐻)) ∧ 𝑥 ∈ 𝐼) → (((𝑎( ·𝑠
‘𝑃)𝑏)‘𝑥) ≠ (0g‘(𝑅‘𝑥)) → (𝑏‘𝑥) ≠ (0g‘(𝑅‘𝑥)))) | 
| 65 | 64 | ss2rabdv 4076 | . . . . 5
⊢ ((𝜑 ∧ (𝑎 ∈ (Base‘(Scalar‘𝑃)) ∧ 𝑏 ∈ 𝐻)) → {𝑥 ∈ 𝐼 ∣ ((𝑎( ·𝑠
‘𝑃)𝑏)‘𝑥) ≠ (0g‘(𝑅‘𝑥))} ⊆ {𝑥 ∈ 𝐼 ∣ (𝑏‘𝑥) ≠ (0g‘(𝑅‘𝑥))}) | 
| 66 | 28, 65 | ssfid 9301 | . . . 4
⊢ ((𝜑 ∧ (𝑎 ∈ (Base‘(Scalar‘𝑃)) ∧ 𝑏 ∈ 𝐻)) → {𝑥 ∈ 𝐼 ∣ ((𝑎( ·𝑠
‘𝑃)𝑏)‘𝑥) ≠ (0g‘(𝑅‘𝑥))} ∈ Fin) | 
| 67 | 1, 16, 17, 2, 3, 18 | dsmmelbas 21759 | . . . . 5
⊢ (𝜑 → ((𝑎( ·𝑠
‘𝑃)𝑏) ∈ 𝐻 ↔ ((𝑎( ·𝑠
‘𝑃)𝑏) ∈ (Base‘𝑃) ∧ {𝑥 ∈ 𝐼 ∣ ((𝑎( ·𝑠
‘𝑃)𝑏)‘𝑥) ≠ (0g‘(𝑅‘𝑥))} ∈ Fin))) | 
| 68 | 67 | adantr 480 | . . . 4
⊢ ((𝜑 ∧ (𝑎 ∈ (Base‘(Scalar‘𝑃)) ∧ 𝑏 ∈ 𝐻)) → ((𝑎( ·𝑠
‘𝑃)𝑏) ∈ 𝐻 ↔ ((𝑎( ·𝑠
‘𝑃)𝑏) ∈ (Base‘𝑃) ∧ {𝑥 ∈ 𝐼 ∣ ((𝑎( ·𝑠
‘𝑃)𝑏)‘𝑥) ≠ (0g‘(𝑅‘𝑥))} ∈ Fin))) | 
| 69 | 27, 66, 68 | mpbir2and 713 | . . 3
⊢ ((𝜑 ∧ (𝑎 ∈ (Base‘(Scalar‘𝑃)) ∧ 𝑏 ∈ 𝐻)) → (𝑎( ·𝑠
‘𝑃)𝑏) ∈ 𝐻) | 
| 70 | 69 | ralrimivva 3202 | . 2
⊢ (𝜑 → ∀𝑎 ∈ (Base‘(Scalar‘𝑃))∀𝑏 ∈ 𝐻 (𝑎( ·𝑠
‘𝑃)𝑏) ∈ 𝐻) | 
| 71 |  | dsmmlss.u | . . . 4
⊢ 𝑈 = (LSubSp‘𝑃) | 
| 72 | 23, 25, 17, 24, 71 | islss4 20960 | . . 3
⊢ (𝑃 ∈ LMod → (𝐻 ∈ 𝑈 ↔ (𝐻 ∈ (SubGrp‘𝑃) ∧ ∀𝑎 ∈ (Base‘(Scalar‘𝑃))∀𝑏 ∈ 𝐻 (𝑎( ·𝑠
‘𝑃)𝑏) ∈ 𝐻))) | 
| 73 | 12, 72 | syl 17 | . 2
⊢ (𝜑 → (𝐻 ∈ 𝑈 ↔ (𝐻 ∈ (SubGrp‘𝑃) ∧ ∀𝑎 ∈ (Base‘(Scalar‘𝑃))∀𝑏 ∈ 𝐻 (𝑎( ·𝑠
‘𝑃)𝑏) ∈ 𝐻))) | 
| 74 | 10, 70, 73 | mpbir2and 713 | 1
⊢ (𝜑 → 𝐻 ∈ 𝑈) |