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 18693 |
. . . . 5
⊢ (𝑎 ∈ LMod → 𝑎 ∈ Grp) |
7 | 6 | ssriv 3572 |
. . . 4
⊢ LMod
⊆ Grp |
8 | | fss 5969 |
. . . 4
⊢ ((𝑅:𝐼⟶LMod ∧ LMod ⊆ Grp) →
𝑅:𝐼⟶Grp) |
9 | 5, 7, 8 | sylancl 693 |
. . 3
⊢ (𝜑 → 𝑅:𝐼⟶Grp) |
10 | 1, 2, 3, 4, 9 | dsmmsubg 19906 |
. 2
⊢ (𝜑 → 𝐻 ∈ (SubGrp‘𝑃)) |
11 | | dsmmlss.k |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → (Scalar‘(𝑅‘𝑥)) = 𝑆) |
12 | 1, 4, 3, 5, 11 | prdslmodd 18790 |
. . . . . 6
⊢ (𝜑 → 𝑃 ∈ LMod) |
13 | 12 | adantr 480 |
. . . . 5
⊢ ((𝜑 ∧ (𝑎 ∈ (Base‘(Scalar‘𝑃)) ∧ 𝑏 ∈ 𝐻)) → 𝑃 ∈ LMod) |
14 | | simprl 790 |
. . . . 5
⊢ ((𝜑 ∧ (𝑎 ∈ (Base‘(Scalar‘𝑃)) ∧ 𝑏 ∈ 𝐻)) → 𝑎 ∈ (Base‘(Scalar‘𝑃))) |
15 | | simprr 792 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝑎 ∈ (Base‘(Scalar‘𝑃)) ∧ 𝑏 ∈ 𝐻)) → 𝑏 ∈ 𝐻) |
16 | | eqid 2610 |
. . . . . . . . 9
⊢ (𝑆 ⊕m 𝑅) = (𝑆 ⊕m 𝑅) |
17 | | eqid 2610 |
. . . . . . . . 9
⊢
(Base‘𝑃) =
(Base‘𝑃) |
18 | | ffn 5958 |
. . . . . . . . . 10
⊢ (𝑅:𝐼⟶LMod → 𝑅 Fn 𝐼) |
19 | 5, 18 | syl 17 |
. . . . . . . . 9
⊢ (𝜑 → 𝑅 Fn 𝐼) |
20 | 1, 16, 17, 2, 3, 19 | dsmmelbas 19902 |
. . . . . . . 8
⊢ (𝜑 → (𝑏 ∈ 𝐻 ↔ (𝑏 ∈ (Base‘𝑃) ∧ {𝑥 ∈ 𝐼 ∣ (𝑏‘𝑥) ≠ (0g‘(𝑅‘𝑥))} ∈ Fin))) |
21 | 20 | adantr 480 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝑎 ∈ (Base‘(Scalar‘𝑃)) ∧ 𝑏 ∈ 𝐻)) → (𝑏 ∈ 𝐻 ↔ (𝑏 ∈ (Base‘𝑃) ∧ {𝑥 ∈ 𝐼 ∣ (𝑏‘𝑥) ≠ (0g‘(𝑅‘𝑥))} ∈ Fin))) |
22 | 15, 21 | mpbid 221 |
. . . . . 6
⊢ ((𝜑 ∧ (𝑎 ∈ (Base‘(Scalar‘𝑃)) ∧ 𝑏 ∈ 𝐻)) → (𝑏 ∈ (Base‘𝑃) ∧ {𝑥 ∈ 𝐼 ∣ (𝑏‘𝑥) ≠ (0g‘(𝑅‘𝑥))} ∈ Fin)) |
23 | 22 | simpld 474 |
. . . . 5
⊢ ((𝜑 ∧ (𝑎 ∈ (Base‘(Scalar‘𝑃)) ∧ 𝑏 ∈ 𝐻)) → 𝑏 ∈ (Base‘𝑃)) |
24 | | eqid 2610 |
. . . . . 6
⊢
(Scalar‘𝑃) =
(Scalar‘𝑃) |
25 | | eqid 2610 |
. . . . . 6
⊢ (
·𝑠 ‘𝑃) = ( ·𝑠
‘𝑃) |
26 | | eqid 2610 |
. . . . . 6
⊢
(Base‘(Scalar‘𝑃)) = (Base‘(Scalar‘𝑃)) |
27 | 17, 24, 25, 26 | lmodvscl 18703 |
. . . . 5
⊢ ((𝑃 ∈ LMod ∧ 𝑎 ∈
(Base‘(Scalar‘𝑃)) ∧ 𝑏 ∈ (Base‘𝑃)) → (𝑎( ·𝑠
‘𝑃)𝑏) ∈ (Base‘𝑃)) |
28 | 13, 14, 23, 27 | syl3anc 1318 |
. . . 4
⊢ ((𝜑 ∧ (𝑎 ∈ (Base‘(Scalar‘𝑃)) ∧ 𝑏 ∈ 𝐻)) → (𝑎( ·𝑠
‘𝑃)𝑏) ∈ (Base‘𝑃)) |
29 | 22 | simprd 478 |
. . . . 5
⊢ ((𝜑 ∧ (𝑎 ∈ (Base‘(Scalar‘𝑃)) ∧ 𝑏 ∈ 𝐻)) → {𝑥 ∈ 𝐼 ∣ (𝑏‘𝑥) ≠ (0g‘(𝑅‘𝑥))} ∈ Fin) |
30 | | eqid 2610 |
. . . . . . . . . . 11
⊢
(Base‘𝑆) =
(Base‘𝑆) |
31 | 4 | ad2antrr 758 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ (𝑎 ∈ (Base‘(Scalar‘𝑃)) ∧ 𝑏 ∈ 𝐻)) ∧ 𝑥 ∈ 𝐼) → 𝑆 ∈ Ring) |
32 | 3 | ad2antrr 758 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ (𝑎 ∈ (Base‘(Scalar‘𝑃)) ∧ 𝑏 ∈ 𝐻)) ∧ 𝑥 ∈ 𝐼) → 𝐼 ∈ 𝑊) |
33 | 19 | ad2antrr 758 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ (𝑎 ∈ (Base‘(Scalar‘𝑃)) ∧ 𝑏 ∈ 𝐻)) ∧ 𝑥 ∈ 𝐼) → 𝑅 Fn 𝐼) |
34 | | fex 6394 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝑅:𝐼⟶LMod ∧ 𝐼 ∈ 𝑊) → 𝑅 ∈ V) |
35 | 5, 3, 34 | syl2anc 691 |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 → 𝑅 ∈ V) |
36 | 1, 4, 35 | prdssca 15939 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → 𝑆 = (Scalar‘𝑃)) |
37 | 36 | fveq2d 6107 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → (Base‘𝑆) =
(Base‘(Scalar‘𝑃))) |
38 | 37 | eleq2d 2673 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → (𝑎 ∈ (Base‘𝑆) ↔ 𝑎 ∈ (Base‘(Scalar‘𝑃)))) |
39 | 38 | biimpar 501 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑎 ∈ (Base‘(Scalar‘𝑃))) → 𝑎 ∈ (Base‘𝑆)) |
40 | 39 | adantrr 749 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ (𝑎 ∈ (Base‘(Scalar‘𝑃)) ∧ 𝑏 ∈ 𝐻)) → 𝑎 ∈ (Base‘𝑆)) |
41 | 40 | adantr 480 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ (𝑎 ∈ (Base‘(Scalar‘𝑃)) ∧ 𝑏 ∈ 𝐻)) ∧ 𝑥 ∈ 𝐼) → 𝑎 ∈ (Base‘𝑆)) |
42 | 23 | adantr 480 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ (𝑎 ∈ (Base‘(Scalar‘𝑃)) ∧ 𝑏 ∈ 𝐻)) ∧ 𝑥 ∈ 𝐼) → 𝑏 ∈ (Base‘𝑃)) |
43 | | simpr 476 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ (𝑎 ∈ (Base‘(Scalar‘𝑃)) ∧ 𝑏 ∈ 𝐻)) ∧ 𝑥 ∈ 𝐼) → 𝑥 ∈ 𝐼) |
44 | 1, 17, 25, 30, 31, 32, 33, 41, 42, 43 | prdsvscafval 15963 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ (𝑎 ∈ (Base‘(Scalar‘𝑃)) ∧ 𝑏 ∈ 𝐻)) ∧ 𝑥 ∈ 𝐼) → ((𝑎( ·𝑠
‘𝑃)𝑏)‘𝑥) = (𝑎( ·𝑠
‘(𝑅‘𝑥))(𝑏‘𝑥))) |
45 | 44 | adantrr 749 |
. . . . . . . . 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))) |
46 | 5 | ffvelrnda 6267 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → (𝑅‘𝑥) ∈ LMod) |
47 | 46 | adantlr 747 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ (𝑎 ∈ (Base‘(Scalar‘𝑃)) ∧ 𝑏 ∈ 𝐻)) ∧ 𝑥 ∈ 𝐼) → (𝑅‘𝑥) ∈ LMod) |
48 | | simplrl 796 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ (𝑎 ∈ (Base‘(Scalar‘𝑃)) ∧ 𝑏 ∈ 𝐻)) ∧ 𝑥 ∈ 𝐼) → 𝑎 ∈ (Base‘(Scalar‘𝑃))) |
49 | 36 | adantr 480 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → 𝑆 = (Scalar‘𝑃)) |
50 | 11, 49 | eqtrd 2644 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → (Scalar‘(𝑅‘𝑥)) = (Scalar‘𝑃)) |
51 | 50 | fveq2d 6107 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → (Base‘(Scalar‘(𝑅‘𝑥))) = (Base‘(Scalar‘𝑃))) |
52 | 51 | adantlr 747 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ (𝑎 ∈ (Base‘(Scalar‘𝑃)) ∧ 𝑏 ∈ 𝐻)) ∧ 𝑥 ∈ 𝐼) → (Base‘(Scalar‘(𝑅‘𝑥))) = (Base‘(Scalar‘𝑃))) |
53 | 48, 52 | eleqtrrd 2691 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ (𝑎 ∈ (Base‘(Scalar‘𝑃)) ∧ 𝑏 ∈ 𝐻)) ∧ 𝑥 ∈ 𝐼) → 𝑎 ∈ (Base‘(Scalar‘(𝑅‘𝑥)))) |
54 | | eqid 2610 |
. . . . . . . . . . . . 13
⊢
(Scalar‘(𝑅‘𝑥)) = (Scalar‘(𝑅‘𝑥)) |
55 | | eqid 2610 |
. . . . . . . . . . . . 13
⊢ (
·𝑠 ‘(𝑅‘𝑥)) = ( ·𝑠
‘(𝑅‘𝑥)) |
56 | | eqid 2610 |
. . . . . . . . . . . . 13
⊢
(Base‘(Scalar‘(𝑅‘𝑥))) = (Base‘(Scalar‘(𝑅‘𝑥))) |
57 | | eqid 2610 |
. . . . . . . . . . . . 13
⊢
(0g‘(𝑅‘𝑥)) = (0g‘(𝑅‘𝑥)) |
58 | 54, 55, 56, 57 | lmodvs0 18720 |
. . . . . . . . . . . 12
⊢ (((𝑅‘𝑥) ∈ LMod ∧ 𝑎 ∈ (Base‘(Scalar‘(𝑅‘𝑥)))) → (𝑎( ·𝑠
‘(𝑅‘𝑥))(0g‘(𝑅‘𝑥))) = (0g‘(𝑅‘𝑥))) |
59 | 47, 53, 58 | syl2anc 691 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ (𝑎 ∈ (Base‘(Scalar‘𝑃)) ∧ 𝑏 ∈ 𝐻)) ∧ 𝑥 ∈ 𝐼) → (𝑎( ·𝑠
‘(𝑅‘𝑥))(0g‘(𝑅‘𝑥))) = (0g‘(𝑅‘𝑥))) |
60 | | oveq2 6557 |
. . . . . . . . . . . 12
⊢ ((𝑏‘𝑥) = (0g‘(𝑅‘𝑥)) → (𝑎( ·𝑠
‘(𝑅‘𝑥))(𝑏‘𝑥)) = (𝑎( ·𝑠
‘(𝑅‘𝑥))(0g‘(𝑅‘𝑥)))) |
61 | 60 | eqeq1d 2612 |
. . . . . . . . . . 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)))) |
62 | 59, 61 | syl5ibrcom 236 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ (𝑎 ∈ (Base‘(Scalar‘𝑃)) ∧ 𝑏 ∈ 𝐻)) ∧ 𝑥 ∈ 𝐼) → ((𝑏‘𝑥) = (0g‘(𝑅‘𝑥)) → (𝑎( ·𝑠
‘(𝑅‘𝑥))(𝑏‘𝑥)) = (0g‘(𝑅‘𝑥)))) |
63 | 62 | impr 647 |
. . . . . . . . 9
⊢ (((𝜑 ∧ (𝑎 ∈ (Base‘(Scalar‘𝑃)) ∧ 𝑏 ∈ 𝐻)) ∧ (𝑥 ∈ 𝐼 ∧ (𝑏‘𝑥) = (0g‘(𝑅‘𝑥)))) → (𝑎( ·𝑠
‘(𝑅‘𝑥))(𝑏‘𝑥)) = (0g‘(𝑅‘𝑥))) |
64 | 45, 63 | eqtrd 2644 |
. . . . . . . 8
⊢ (((𝜑 ∧ (𝑎 ∈ (Base‘(Scalar‘𝑃)) ∧ 𝑏 ∈ 𝐻)) ∧ (𝑥 ∈ 𝐼 ∧ (𝑏‘𝑥) = (0g‘(𝑅‘𝑥)))) → ((𝑎( ·𝑠
‘𝑃)𝑏)‘𝑥) = (0g‘(𝑅‘𝑥))) |
65 | 64 | expr 641 |
. . . . . . 7
⊢ (((𝜑 ∧ (𝑎 ∈ (Base‘(Scalar‘𝑃)) ∧ 𝑏 ∈ 𝐻)) ∧ 𝑥 ∈ 𝐼) → ((𝑏‘𝑥) = (0g‘(𝑅‘𝑥)) → ((𝑎( ·𝑠
‘𝑃)𝑏)‘𝑥) = (0g‘(𝑅‘𝑥)))) |
66 | 65 | necon3d 2803 |
. . . . . 6
⊢ (((𝜑 ∧ (𝑎 ∈ (Base‘(Scalar‘𝑃)) ∧ 𝑏 ∈ 𝐻)) ∧ 𝑥 ∈ 𝐼) → (((𝑎( ·𝑠
‘𝑃)𝑏)‘𝑥) ≠ (0g‘(𝑅‘𝑥)) → (𝑏‘𝑥) ≠ (0g‘(𝑅‘𝑥)))) |
67 | 66 | ss2rabdv 3646 |
. . . . 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))}) |
68 | | ssfi 8065 |
. . . . 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) |
69 | 29, 67, 68 | syl2anc 691 |
. . . 4
⊢ ((𝜑 ∧ (𝑎 ∈ (Base‘(Scalar‘𝑃)) ∧ 𝑏 ∈ 𝐻)) → {𝑥 ∈ 𝐼 ∣ ((𝑎( ·𝑠
‘𝑃)𝑏)‘𝑥) ≠ (0g‘(𝑅‘𝑥))} ∈ Fin) |
70 | 1, 16, 17, 2, 3, 19 | dsmmelbas 19902 |
. . . . 5
⊢ (𝜑 → ((𝑎( ·𝑠
‘𝑃)𝑏) ∈ 𝐻 ↔ ((𝑎( ·𝑠
‘𝑃)𝑏) ∈ (Base‘𝑃) ∧ {𝑥 ∈ 𝐼 ∣ ((𝑎( ·𝑠
‘𝑃)𝑏)‘𝑥) ≠ (0g‘(𝑅‘𝑥))} ∈ Fin))) |
71 | 70 | 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))) |
72 | 28, 69, 71 | mpbir2and 959 |
. . 3
⊢ ((𝜑 ∧ (𝑎 ∈ (Base‘(Scalar‘𝑃)) ∧ 𝑏 ∈ 𝐻)) → (𝑎( ·𝑠
‘𝑃)𝑏) ∈ 𝐻) |
73 | 72 | ralrimivva 2954 |
. 2
⊢ (𝜑 → ∀𝑎 ∈ (Base‘(Scalar‘𝑃))∀𝑏 ∈ 𝐻 (𝑎( ·𝑠
‘𝑃)𝑏) ∈ 𝐻) |
74 | | dsmmlss.u |
. . . 4
⊢ 𝑈 = (LSubSp‘𝑃) |
75 | 24, 26, 17, 25, 74 | islss4 18783 |
. . 3
⊢ (𝑃 ∈ LMod → (𝐻 ∈ 𝑈 ↔ (𝐻 ∈ (SubGrp‘𝑃) ∧ ∀𝑎 ∈ (Base‘(Scalar‘𝑃))∀𝑏 ∈ 𝐻 (𝑎( ·𝑠
‘𝑃)𝑏) ∈ 𝐻))) |
76 | 12, 75 | syl 17 |
. 2
⊢ (𝜑 → (𝐻 ∈ 𝑈 ↔ (𝐻 ∈ (SubGrp‘𝑃) ∧ ∀𝑎 ∈ (Base‘(Scalar‘𝑃))∀𝑏 ∈ 𝐻 (𝑎( ·𝑠
‘𝑃)𝑏) ∈ 𝐻))) |
77 | 10, 73, 76 | mpbir2and 959 |
1
⊢ (𝜑 → 𝐻 ∈ 𝑈) |