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 19262 |
. . . . 5
⊢ (𝑎 ∈ LMod → 𝑎 ∈ Grp) |
7 | 6 | ssriv 3825 |
. . . 4
⊢ LMod
⊆ Grp |
8 | | fss 6304 |
. . . 4
⊢ ((𝑅:𝐼⟶LMod ∧ LMod ⊆ Grp) →
𝑅:𝐼⟶Grp) |
9 | 5, 7, 8 | sylancl 580 |
. . 3
⊢ (𝜑 → 𝑅:𝐼⟶Grp) |
10 | 1, 2, 3, 4, 9 | dsmmsubg 20486 |
. 2
⊢ (𝜑 → 𝐻 ∈ (SubGrp‘𝑃)) |
11 | | dsmmlss.k |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → (Scalar‘(𝑅‘𝑥)) = 𝑆) |
12 | 1, 4, 3, 5, 11 | prdslmodd 19364 |
. . . . . 6
⊢ (𝜑 → 𝑃 ∈ LMod) |
13 | 12 | adantr 474 |
. . . . 5
⊢ ((𝜑 ∧ (𝑎 ∈ (Base‘(Scalar‘𝑃)) ∧ 𝑏 ∈ 𝐻)) → 𝑃 ∈ LMod) |
14 | | simprl 761 |
. . . . 5
⊢ ((𝜑 ∧ (𝑎 ∈ (Base‘(Scalar‘𝑃)) ∧ 𝑏 ∈ 𝐻)) → 𝑎 ∈ (Base‘(Scalar‘𝑃))) |
15 | | simprr 763 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝑎 ∈ (Base‘(Scalar‘𝑃)) ∧ 𝑏 ∈ 𝐻)) → 𝑏 ∈ 𝐻) |
16 | | eqid 2778 |
. . . . . . . . 9
⊢ (𝑆 ⊕m 𝑅) = (𝑆 ⊕m 𝑅) |
17 | | eqid 2778 |
. . . . . . . . 9
⊢
(Base‘𝑃) =
(Base‘𝑃) |
18 | 5 | ffnd 6292 |
. . . . . . . . 9
⊢ (𝜑 → 𝑅 Fn 𝐼) |
19 | 1, 16, 17, 2, 3, 18 | dsmmelbas 20482 |
. . . . . . . 8
⊢ (𝜑 → (𝑏 ∈ 𝐻 ↔ (𝑏 ∈ (Base‘𝑃) ∧ {𝑥 ∈ 𝐼 ∣ (𝑏‘𝑥) ≠ (0g‘(𝑅‘𝑥))} ∈ Fin))) |
20 | 19 | adantr 474 |
. . . . . . 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 224 |
. . . . . 6
⊢ ((𝜑 ∧ (𝑎 ∈ (Base‘(Scalar‘𝑃)) ∧ 𝑏 ∈ 𝐻)) → (𝑏 ∈ (Base‘𝑃) ∧ {𝑥 ∈ 𝐼 ∣ (𝑏‘𝑥) ≠ (0g‘(𝑅‘𝑥))} ∈ Fin)) |
22 | 21 | simpld 490 |
. . . . 5
⊢ ((𝜑 ∧ (𝑎 ∈ (Base‘(Scalar‘𝑃)) ∧ 𝑏 ∈ 𝐻)) → 𝑏 ∈ (Base‘𝑃)) |
23 | | eqid 2778 |
. . . . . 6
⊢
(Scalar‘𝑃) =
(Scalar‘𝑃) |
24 | | eqid 2778 |
. . . . . 6
⊢ (
·𝑠 ‘𝑃) = ( ·𝑠
‘𝑃) |
25 | | eqid 2778 |
. . . . . 6
⊢
(Base‘(Scalar‘𝑃)) = (Base‘(Scalar‘𝑃)) |
26 | 17, 23, 24, 25 | lmodvscl 19272 |
. . . . 5
⊢ ((𝑃 ∈ LMod ∧ 𝑎 ∈
(Base‘(Scalar‘𝑃)) ∧ 𝑏 ∈ (Base‘𝑃)) → (𝑎( ·𝑠
‘𝑃)𝑏) ∈ (Base‘𝑃)) |
27 | 13, 14, 22, 26 | syl3anc 1439 |
. . . 4
⊢ ((𝜑 ∧ (𝑎 ∈ (Base‘(Scalar‘𝑃)) ∧ 𝑏 ∈ 𝐻)) → (𝑎( ·𝑠
‘𝑃)𝑏) ∈ (Base‘𝑃)) |
28 | 21 | simprd 491 |
. . . . 5
⊢ ((𝜑 ∧ (𝑎 ∈ (Base‘(Scalar‘𝑃)) ∧ 𝑏 ∈ 𝐻)) → {𝑥 ∈ 𝐼 ∣ (𝑏‘𝑥) ≠ (0g‘(𝑅‘𝑥))} ∈ Fin) |
29 | | eqid 2778 |
. . . . . . . . . . 11
⊢
(Base‘𝑆) =
(Base‘𝑆) |
30 | 4 | ad2antrr 716 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ (𝑎 ∈ (Base‘(Scalar‘𝑃)) ∧ 𝑏 ∈ 𝐻)) ∧ 𝑥 ∈ 𝐼) → 𝑆 ∈ Ring) |
31 | 3 | ad2antrr 716 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ (𝑎 ∈ (Base‘(Scalar‘𝑃)) ∧ 𝑏 ∈ 𝐻)) ∧ 𝑥 ∈ 𝐼) → 𝐼 ∈ 𝑊) |
32 | 18 | ad2antrr 716 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ (𝑎 ∈ (Base‘(Scalar‘𝑃)) ∧ 𝑏 ∈ 𝐻)) ∧ 𝑥 ∈ 𝐼) → 𝑅 Fn 𝐼) |
33 | | fex 6761 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝑅:𝐼⟶LMod ∧ 𝐼 ∈ 𝑊) → 𝑅 ∈ V) |
34 | 5, 3, 33 | syl2anc 579 |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 → 𝑅 ∈ V) |
35 | 1, 4, 34 | prdssca 16502 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → 𝑆 = (Scalar‘𝑃)) |
36 | 35 | fveq2d 6450 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → (Base‘𝑆) =
(Base‘(Scalar‘𝑃))) |
37 | 36 | eleq2d 2845 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → (𝑎 ∈ (Base‘𝑆) ↔ 𝑎 ∈ (Base‘(Scalar‘𝑃)))) |
38 | 37 | biimpar 471 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑎 ∈ (Base‘(Scalar‘𝑃))) → 𝑎 ∈ (Base‘𝑆)) |
39 | 38 | adantrr 707 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ (𝑎 ∈ (Base‘(Scalar‘𝑃)) ∧ 𝑏 ∈ 𝐻)) → 𝑎 ∈ (Base‘𝑆)) |
40 | 39 | adantr 474 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ (𝑎 ∈ (Base‘(Scalar‘𝑃)) ∧ 𝑏 ∈ 𝐻)) ∧ 𝑥 ∈ 𝐼) → 𝑎 ∈ (Base‘𝑆)) |
41 | 22 | adantr 474 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ (𝑎 ∈ (Base‘(Scalar‘𝑃)) ∧ 𝑏 ∈ 𝐻)) ∧ 𝑥 ∈ 𝐼) → 𝑏 ∈ (Base‘𝑃)) |
42 | | simpr 479 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ (𝑎 ∈ (Base‘(Scalar‘𝑃)) ∧ 𝑏 ∈ 𝐻)) ∧ 𝑥 ∈ 𝐼) → 𝑥 ∈ 𝐼) |
43 | 1, 17, 24, 29, 30, 31, 32, 40, 41, 42 | prdsvscafval 16526 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ (𝑎 ∈ (Base‘(Scalar‘𝑃)) ∧ 𝑏 ∈ 𝐻)) ∧ 𝑥 ∈ 𝐼) → ((𝑎( ·𝑠
‘𝑃)𝑏)‘𝑥) = (𝑎( ·𝑠
‘(𝑅‘𝑥))(𝑏‘𝑥))) |
44 | 43 | adantrr 707 |
. . . . . . . . 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))) |
45 | 5 | ffvelrnda 6623 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → (𝑅‘𝑥) ∈ LMod) |
46 | 45 | adantlr 705 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ (𝑎 ∈ (Base‘(Scalar‘𝑃)) ∧ 𝑏 ∈ 𝐻)) ∧ 𝑥 ∈ 𝐼) → (𝑅‘𝑥) ∈ LMod) |
47 | | simplrl 767 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ (𝑎 ∈ (Base‘(Scalar‘𝑃)) ∧ 𝑏 ∈ 𝐻)) ∧ 𝑥 ∈ 𝐼) → 𝑎 ∈ (Base‘(Scalar‘𝑃))) |
48 | 35 | adantr 474 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → 𝑆 = (Scalar‘𝑃)) |
49 | 11, 48 | eqtrd 2814 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → (Scalar‘(𝑅‘𝑥)) = (Scalar‘𝑃)) |
50 | 49 | fveq2d 6450 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → (Base‘(Scalar‘(𝑅‘𝑥))) = (Base‘(Scalar‘𝑃))) |
51 | 50 | adantlr 705 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ (𝑎 ∈ (Base‘(Scalar‘𝑃)) ∧ 𝑏 ∈ 𝐻)) ∧ 𝑥 ∈ 𝐼) → (Base‘(Scalar‘(𝑅‘𝑥))) = (Base‘(Scalar‘𝑃))) |
52 | 47, 51 | eleqtrrd 2862 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ (𝑎 ∈ (Base‘(Scalar‘𝑃)) ∧ 𝑏 ∈ 𝐻)) ∧ 𝑥 ∈ 𝐼) → 𝑎 ∈ (Base‘(Scalar‘(𝑅‘𝑥)))) |
53 | | eqid 2778 |
. . . . . . . . . . . . 13
⊢
(Scalar‘(𝑅‘𝑥)) = (Scalar‘(𝑅‘𝑥)) |
54 | | eqid 2778 |
. . . . . . . . . . . . 13
⊢ (
·𝑠 ‘(𝑅‘𝑥)) = ( ·𝑠
‘(𝑅‘𝑥)) |
55 | | eqid 2778 |
. . . . . . . . . . . . 13
⊢
(Base‘(Scalar‘(𝑅‘𝑥))) = (Base‘(Scalar‘(𝑅‘𝑥))) |
56 | | eqid 2778 |
. . . . . . . . . . . . 13
⊢
(0g‘(𝑅‘𝑥)) = (0g‘(𝑅‘𝑥)) |
57 | 53, 54, 55, 56 | lmodvs0 19289 |
. . . . . . . . . . . 12
⊢ (((𝑅‘𝑥) ∈ LMod ∧ 𝑎 ∈ (Base‘(Scalar‘(𝑅‘𝑥)))) → (𝑎( ·𝑠
‘(𝑅‘𝑥))(0g‘(𝑅‘𝑥))) = (0g‘(𝑅‘𝑥))) |
58 | 46, 52, 57 | syl2anc 579 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ (𝑎 ∈ (Base‘(Scalar‘𝑃)) ∧ 𝑏 ∈ 𝐻)) ∧ 𝑥 ∈ 𝐼) → (𝑎( ·𝑠
‘(𝑅‘𝑥))(0g‘(𝑅‘𝑥))) = (0g‘(𝑅‘𝑥))) |
59 | | oveq2 6930 |
. . . . . . . . . . . 12
⊢ ((𝑏‘𝑥) = (0g‘(𝑅‘𝑥)) → (𝑎( ·𝑠
‘(𝑅‘𝑥))(𝑏‘𝑥)) = (𝑎( ·𝑠
‘(𝑅‘𝑥))(0g‘(𝑅‘𝑥)))) |
60 | 59 | eqeq1d 2780 |
. . . . . . . . . . 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)))) |
61 | 58, 60 | syl5ibrcom 239 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ (𝑎 ∈ (Base‘(Scalar‘𝑃)) ∧ 𝑏 ∈ 𝐻)) ∧ 𝑥 ∈ 𝐼) → ((𝑏‘𝑥) = (0g‘(𝑅‘𝑥)) → (𝑎( ·𝑠
‘(𝑅‘𝑥))(𝑏‘𝑥)) = (0g‘(𝑅‘𝑥)))) |
62 | 61 | impr 448 |
. . . . . . . . 9
⊢ (((𝜑 ∧ (𝑎 ∈ (Base‘(Scalar‘𝑃)) ∧ 𝑏 ∈ 𝐻)) ∧ (𝑥 ∈ 𝐼 ∧ (𝑏‘𝑥) = (0g‘(𝑅‘𝑥)))) → (𝑎( ·𝑠
‘(𝑅‘𝑥))(𝑏‘𝑥)) = (0g‘(𝑅‘𝑥))) |
63 | 44, 62 | eqtrd 2814 |
. . . . . . . 8
⊢ (((𝜑 ∧ (𝑎 ∈ (Base‘(Scalar‘𝑃)) ∧ 𝑏 ∈ 𝐻)) ∧ (𝑥 ∈ 𝐼 ∧ (𝑏‘𝑥) = (0g‘(𝑅‘𝑥)))) → ((𝑎( ·𝑠
‘𝑃)𝑏)‘𝑥) = (0g‘(𝑅‘𝑥))) |
64 | 63 | expr 450 |
. . . . . . 7
⊢ (((𝜑 ∧ (𝑎 ∈ (Base‘(Scalar‘𝑃)) ∧ 𝑏 ∈ 𝐻)) ∧ 𝑥 ∈ 𝐼) → ((𝑏‘𝑥) = (0g‘(𝑅‘𝑥)) → ((𝑎( ·𝑠
‘𝑃)𝑏)‘𝑥) = (0g‘(𝑅‘𝑥)))) |
65 | 64 | necon3d 2990 |
. . . . . 6
⊢ (((𝜑 ∧ (𝑎 ∈ (Base‘(Scalar‘𝑃)) ∧ 𝑏 ∈ 𝐻)) ∧ 𝑥 ∈ 𝐼) → (((𝑎( ·𝑠
‘𝑃)𝑏)‘𝑥) ≠ (0g‘(𝑅‘𝑥)) → (𝑏‘𝑥) ≠ (0g‘(𝑅‘𝑥)))) |
66 | 65 | ss2rabdv 3904 |
. . . . 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))}) |
67 | | ssfi 8468 |
. . . . 5
⊢ (({𝑥 ∈ 𝐼 ∣ (𝑏‘𝑥) ≠ (0g‘(𝑅‘𝑥))} ∈ Fin ∧ {𝑥 ∈ 𝐼 ∣ ((𝑎( ·𝑠
‘𝑃)𝑏)‘𝑥) ≠ (0g‘(𝑅‘𝑥))} ⊆ {𝑥 ∈ 𝐼 ∣ (𝑏‘𝑥) ≠ (0g‘(𝑅‘𝑥))}) → {𝑥 ∈ 𝐼 ∣ ((𝑎( ·𝑠
‘𝑃)𝑏)‘𝑥) ≠ (0g‘(𝑅‘𝑥))} ∈ Fin) |
68 | 28, 66, 67 | syl2anc 579 |
. . . 4
⊢ ((𝜑 ∧ (𝑎 ∈ (Base‘(Scalar‘𝑃)) ∧ 𝑏 ∈ 𝐻)) → {𝑥 ∈ 𝐼 ∣ ((𝑎( ·𝑠
‘𝑃)𝑏)‘𝑥) ≠ (0g‘(𝑅‘𝑥))} ∈ Fin) |
69 | 1, 16, 17, 2, 3, 18 | dsmmelbas 20482 |
. . . . 5
⊢ (𝜑 → ((𝑎( ·𝑠
‘𝑃)𝑏) ∈ 𝐻 ↔ ((𝑎( ·𝑠
‘𝑃)𝑏) ∈ (Base‘𝑃) ∧ {𝑥 ∈ 𝐼 ∣ ((𝑎( ·𝑠
‘𝑃)𝑏)‘𝑥) ≠ (0g‘(𝑅‘𝑥))} ∈ Fin))) |
70 | 69 | adantr 474 |
. . . 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))) |
71 | 27, 68, 70 | mpbir2and 703 |
. . 3
⊢ ((𝜑 ∧ (𝑎 ∈ (Base‘(Scalar‘𝑃)) ∧ 𝑏 ∈ 𝐻)) → (𝑎( ·𝑠
‘𝑃)𝑏) ∈ 𝐻) |
72 | 71 | ralrimivva 3153 |
. 2
⊢ (𝜑 → ∀𝑎 ∈ (Base‘(Scalar‘𝑃))∀𝑏 ∈ 𝐻 (𝑎( ·𝑠
‘𝑃)𝑏) ∈ 𝐻) |
73 | | dsmmlss.u |
. . . 4
⊢ 𝑈 = (LSubSp‘𝑃) |
74 | 23, 25, 17, 24, 73 | islss4 19357 |
. . 3
⊢ (𝑃 ∈ LMod → (𝐻 ∈ 𝑈 ↔ (𝐻 ∈ (SubGrp‘𝑃) ∧ ∀𝑎 ∈ (Base‘(Scalar‘𝑃))∀𝑏 ∈ 𝐻 (𝑎( ·𝑠
‘𝑃)𝑏) ∈ 𝐻))) |
75 | 12, 74 | syl 17 |
. 2
⊢ (𝜑 → (𝐻 ∈ 𝑈 ↔ (𝐻 ∈ (SubGrp‘𝑃) ∧ ∀𝑎 ∈ (Base‘(Scalar‘𝑃))∀𝑏 ∈ 𝐻 (𝑎( ·𝑠
‘𝑃)𝑏) ∈ 𝐻))) |
76 | 10, 72, 75 | mpbir2and 703 |
1
⊢ (𝜑 → 𝐻 ∈ 𝑈) |