Step | Hyp | Ref
| Expression |
1 | | frlmsslsp.y |
. . . . 5
⊢ 𝑌 = (𝑅 freeLMod 𝐼) |
2 | 1 | frlmlmod 20823 |
. . . 4
⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉) → 𝑌 ∈ LMod) |
3 | 2 | 3adant3 1124 |
. . 3
⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) → 𝑌 ∈ LMod) |
4 | | eqid 2821 |
. . . 4
⊢
(LSubSp‘𝑌) =
(LSubSp‘𝑌) |
5 | | frlmsslsp.b |
. . . 4
⊢ 𝐵 = (Base‘𝑌) |
6 | | frlmsslsp.z |
. . . 4
⊢ 0 =
(0g‘𝑅) |
7 | | frlmsslsp.c |
. . . 4
⊢ 𝐶 = {𝑥 ∈ 𝐵 ∣ (𝑥 supp 0 ) ⊆ 𝐽} |
8 | 1, 4, 5, 6, 7 | frlmsslss2 20849 |
. . 3
⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) → 𝐶 ∈ (LSubSp‘𝑌)) |
9 | | frlmsslsp.u |
. . . . . . . . . 10
⊢ 𝑈 = (𝑅 unitVec 𝐼) |
10 | 9, 1, 5 | uvcff 20865 |
. . . . . . . . 9
⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉) → 𝑈:𝐼⟶𝐵) |
11 | 10 | 3adant3 1124 |
. . . . . . . 8
⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) → 𝑈:𝐼⟶𝐵) |
12 | 11 | adantr 481 |
. . . . . . 7
⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ 𝑦 ∈ 𝐽) → 𝑈:𝐼⟶𝐵) |
13 | | simp3 1130 |
. . . . . . . 8
⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) → 𝐽 ⊆ 𝐼) |
14 | 13 | sselda 3966 |
. . . . . . 7
⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ 𝑦 ∈ 𝐽) → 𝑦 ∈ 𝐼) |
15 | 12, 14 | ffvelrnd 6845 |
. . . . . 6
⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ 𝑦 ∈ 𝐽) → (𝑈‘𝑦) ∈ 𝐵) |
16 | | simpl2 1184 |
. . . . . . . 8
⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ 𝑦 ∈ 𝐽) → 𝐼 ∈ 𝑉) |
17 | | eqid 2821 |
. . . . . . . . 9
⊢
(Base‘𝑅) =
(Base‘𝑅) |
18 | 1, 17, 5 | frlmbasf 20834 |
. . . . . . . 8
⊢ ((𝐼 ∈ 𝑉 ∧ (𝑈‘𝑦) ∈ 𝐵) → (𝑈‘𝑦):𝐼⟶(Base‘𝑅)) |
19 | 16, 15, 18 | syl2anc 584 |
. . . . . . 7
⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ 𝑦 ∈ 𝐽) → (𝑈‘𝑦):𝐼⟶(Base‘𝑅)) |
20 | | simpll1 1204 |
. . . . . . . 8
⊢ ((((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ 𝑦 ∈ 𝐽) ∧ 𝑥 ∈ (𝐼 ∖ 𝐽)) → 𝑅 ∈ Ring) |
21 | | simpll2 1205 |
. . . . . . . 8
⊢ ((((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ 𝑦 ∈ 𝐽) ∧ 𝑥 ∈ (𝐼 ∖ 𝐽)) → 𝐼 ∈ 𝑉) |
22 | 14 | adantr 481 |
. . . . . . . 8
⊢ ((((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ 𝑦 ∈ 𝐽) ∧ 𝑥 ∈ (𝐼 ∖ 𝐽)) → 𝑦 ∈ 𝐼) |
23 | | eldifi 4102 |
. . . . . . . . 9
⊢ (𝑥 ∈ (𝐼 ∖ 𝐽) → 𝑥 ∈ 𝐼) |
24 | 23 | adantl 482 |
. . . . . . . 8
⊢ ((((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ 𝑦 ∈ 𝐽) ∧ 𝑥 ∈ (𝐼 ∖ 𝐽)) → 𝑥 ∈ 𝐼) |
25 | | disjdif 4419 |
. . . . . . . . . 10
⊢ (𝐽 ∩ (𝐼 ∖ 𝐽)) = ∅ |
26 | | disjne 4402 |
. . . . . . . . . 10
⊢ (((𝐽 ∩ (𝐼 ∖ 𝐽)) = ∅ ∧ 𝑦 ∈ 𝐽 ∧ 𝑥 ∈ (𝐼 ∖ 𝐽)) → 𝑦 ≠ 𝑥) |
27 | 25, 26 | mp3an1 1439 |
. . . . . . . . 9
⊢ ((𝑦 ∈ 𝐽 ∧ 𝑥 ∈ (𝐼 ∖ 𝐽)) → 𝑦 ≠ 𝑥) |
28 | 27 | adantll 710 |
. . . . . . . 8
⊢ ((((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ 𝑦 ∈ 𝐽) ∧ 𝑥 ∈ (𝐼 ∖ 𝐽)) → 𝑦 ≠ 𝑥) |
29 | 9, 20, 21, 22, 24, 28, 6 | uvcvv0 20864 |
. . . . . . 7
⊢ ((((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ 𝑦 ∈ 𝐽) ∧ 𝑥 ∈ (𝐼 ∖ 𝐽)) → ((𝑈‘𝑦)‘𝑥) = 0 ) |
30 | 19, 29 | suppss 7851 |
. . . . . 6
⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ 𝑦 ∈ 𝐽) → ((𝑈‘𝑦) supp 0 ) ⊆ 𝐽) |
31 | | oveq1 7152 |
. . . . . . . 8
⊢ (𝑥 = (𝑈‘𝑦) → (𝑥 supp 0 ) = ((𝑈‘𝑦) supp 0 )) |
32 | 31 | sseq1d 3997 |
. . . . . . 7
⊢ (𝑥 = (𝑈‘𝑦) → ((𝑥 supp 0 ) ⊆ 𝐽 ↔ ((𝑈‘𝑦) supp 0 ) ⊆ 𝐽)) |
33 | 32, 7 | elrab2 3682 |
. . . . . 6
⊢ ((𝑈‘𝑦) ∈ 𝐶 ↔ ((𝑈‘𝑦) ∈ 𝐵 ∧ ((𝑈‘𝑦) supp 0 ) ⊆ 𝐽)) |
34 | 15, 30, 33 | sylanbrc 583 |
. . . . 5
⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ 𝑦 ∈ 𝐽) → (𝑈‘𝑦) ∈ 𝐶) |
35 | 34 | ralrimiva 3182 |
. . . 4
⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) → ∀𝑦 ∈ 𝐽 (𝑈‘𝑦) ∈ 𝐶) |
36 | 11 | ffund 6512 |
. . . . 5
⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) → Fun 𝑈) |
37 | 11 | fdmd 6517 |
. . . . . 6
⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) → dom 𝑈 = 𝐼) |
38 | 13, 37 | sseqtrrd 4007 |
. . . . 5
⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) → 𝐽 ⊆ dom 𝑈) |
39 | | funimass4 6724 |
. . . . 5
⊢ ((Fun
𝑈 ∧ 𝐽 ⊆ dom 𝑈) → ((𝑈 “ 𝐽) ⊆ 𝐶 ↔ ∀𝑦 ∈ 𝐽 (𝑈‘𝑦) ∈ 𝐶)) |
40 | 36, 38, 39 | syl2anc 584 |
. . . 4
⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) → ((𝑈 “ 𝐽) ⊆ 𝐶 ↔ ∀𝑦 ∈ 𝐽 (𝑈‘𝑦) ∈ 𝐶)) |
41 | 35, 40 | mpbird 258 |
. . 3
⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) → (𝑈 “ 𝐽) ⊆ 𝐶) |
42 | | frlmsslsp.k |
. . . 4
⊢ 𝐾 = (LSpan‘𝑌) |
43 | 4, 42 | lspssp 19691 |
. . 3
⊢ ((𝑌 ∈ LMod ∧ 𝐶 ∈ (LSubSp‘𝑌) ∧ (𝑈 “ 𝐽) ⊆ 𝐶) → (𝐾‘(𝑈 “ 𝐽)) ⊆ 𝐶) |
44 | 3, 8, 41, 43 | syl3anc 1363 |
. 2
⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) → (𝐾‘(𝑈 “ 𝐽)) ⊆ 𝐶) |
45 | | simpl1 1183 |
. . . 4
⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ 𝑦 ∈ 𝐶) → 𝑅 ∈ Ring) |
46 | | simpl2 1184 |
. . . 4
⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ 𝑦 ∈ 𝐶) → 𝐼 ∈ 𝑉) |
47 | 7 | ssrab3 4056 |
. . . . . 6
⊢ 𝐶 ⊆ 𝐵 |
48 | 47 | a1i 11 |
. . . . 5
⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) → 𝐶 ⊆ 𝐵) |
49 | 48 | sselda 3966 |
. . . 4
⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ 𝑦 ∈ 𝐶) → 𝑦 ∈ 𝐵) |
50 | | eqid 2821 |
. . . . 5
⊢ (
·𝑠 ‘𝑌) = ( ·𝑠
‘𝑌) |
51 | 9, 1, 5, 50 | uvcresum 20867 |
. . . 4
⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝑦 ∈ 𝐵) → 𝑦 = (𝑌 Σg (𝑦 ∘f (
·𝑠 ‘𝑌)𝑈))) |
52 | 45, 46, 49, 51 | syl3anc 1363 |
. . 3
⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ 𝑦 ∈ 𝐶) → 𝑦 = (𝑌 Σg (𝑦 ∘f (
·𝑠 ‘𝑌)𝑈))) |
53 | | eqid 2821 |
. . . 4
⊢
(0g‘𝑌) = (0g‘𝑌) |
54 | | lmodabl 19612 |
. . . . . 6
⊢ (𝑌 ∈ LMod → 𝑌 ∈ Abel) |
55 | 3, 54 | syl 17 |
. . . . 5
⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) → 𝑌 ∈ Abel) |
56 | 55 | adantr 481 |
. . . 4
⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ 𝑦 ∈ 𝐶) → 𝑌 ∈ Abel) |
57 | | imassrn 5934 |
. . . . . . . 8
⊢ (𝑈 “ 𝐽) ⊆ ran 𝑈 |
58 | 11 | frnd 6515 |
. . . . . . . 8
⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) → ran 𝑈 ⊆ 𝐵) |
59 | 57, 58 | sstrid 3977 |
. . . . . . 7
⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) → (𝑈 “ 𝐽) ⊆ 𝐵) |
60 | 5, 4, 42 | lspcl 19679 |
. . . . . . 7
⊢ ((𝑌 ∈ LMod ∧ (𝑈 “ 𝐽) ⊆ 𝐵) → (𝐾‘(𝑈 “ 𝐽)) ∈ (LSubSp‘𝑌)) |
61 | 3, 59, 60 | syl2anc 584 |
. . . . . 6
⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) → (𝐾‘(𝑈 “ 𝐽)) ∈ (LSubSp‘𝑌)) |
62 | 4 | lsssubg 19660 |
. . . . . 6
⊢ ((𝑌 ∈ LMod ∧ (𝐾‘(𝑈 “ 𝐽)) ∈ (LSubSp‘𝑌)) → (𝐾‘(𝑈 “ 𝐽)) ∈ (SubGrp‘𝑌)) |
63 | 3, 61, 62 | syl2anc 584 |
. . . . 5
⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) → (𝐾‘(𝑈 “ 𝐽)) ∈ (SubGrp‘𝑌)) |
64 | 63 | adantr 481 |
. . . 4
⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ 𝑦 ∈ 𝐶) → (𝐾‘(𝑈 “ 𝐽)) ∈ (SubGrp‘𝑌)) |
65 | 1, 17, 5 | frlmbasf 20834 |
. . . . . . . . 9
⊢ ((𝐼 ∈ 𝑉 ∧ 𝑦 ∈ 𝐵) → 𝑦:𝐼⟶(Base‘𝑅)) |
66 | 65 | 3ad2antl2 1178 |
. . . . . . . 8
⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ 𝑦 ∈ 𝐵) → 𝑦:𝐼⟶(Base‘𝑅)) |
67 | 66 | ffnd 6509 |
. . . . . . 7
⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ 𝑦 ∈ 𝐵) → 𝑦 Fn 𝐼) |
68 | 11 | ffnd 6509 |
. . . . . . . 8
⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) → 𝑈 Fn 𝐼) |
69 | 68 | adantr 481 |
. . . . . . 7
⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ 𝑦 ∈ 𝐵) → 𝑈 Fn 𝐼) |
70 | | simpl2 1184 |
. . . . . . 7
⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ 𝑦 ∈ 𝐵) → 𝐼 ∈ 𝑉) |
71 | | inidm 4194 |
. . . . . . 7
⊢ (𝐼 ∩ 𝐼) = 𝐼 |
72 | 67, 69, 70, 70, 71 | offn 7409 |
. . . . . 6
⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ 𝑦 ∈ 𝐵) → (𝑦 ∘f (
·𝑠 ‘𝑌)𝑈) Fn 𝐼) |
73 | 49, 72 | syldan 591 |
. . . . 5
⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ 𝑦 ∈ 𝐶) → (𝑦 ∘f (
·𝑠 ‘𝑌)𝑈) Fn 𝐼) |
74 | 49, 67 | syldan 591 |
. . . . . . . . . 10
⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ 𝑦 ∈ 𝐶) → 𝑦 Fn 𝐼) |
75 | 74 | adantrr 713 |
. . . . . . . . 9
⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐼)) → 𝑦 Fn 𝐼) |
76 | 68 | adantr 481 |
. . . . . . . . 9
⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐼)) → 𝑈 Fn 𝐼) |
77 | | simpl2 1184 |
. . . . . . . . 9
⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐼)) → 𝐼 ∈ 𝑉) |
78 | | simprr 769 |
. . . . . . . . 9
⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐼)) → 𝑧 ∈ 𝐼) |
79 | | fnfvof 7412 |
. . . . . . . . 9
⊢ (((𝑦 Fn 𝐼 ∧ 𝑈 Fn 𝐼) ∧ (𝐼 ∈ 𝑉 ∧ 𝑧 ∈ 𝐼)) → ((𝑦 ∘f (
·𝑠 ‘𝑌)𝑈)‘𝑧) = ((𝑦‘𝑧)( ·𝑠
‘𝑌)(𝑈‘𝑧))) |
80 | 75, 76, 77, 78, 79 | syl22anc 834 |
. . . . . . . 8
⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐼)) → ((𝑦 ∘f (
·𝑠 ‘𝑌)𝑈)‘𝑧) = ((𝑦‘𝑧)( ·𝑠
‘𝑌)(𝑈‘𝑧))) |
81 | 3 | adantr 481 |
. . . . . . . . . . . 12
⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐽)) → 𝑌 ∈ LMod) |
82 | 61 | adantr 481 |
. . . . . . . . . . . 12
⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐽)) → (𝐾‘(𝑈 “ 𝐽)) ∈ (LSubSp‘𝑌)) |
83 | 47 | sseli 3962 |
. . . . . . . . . . . . . . . 16
⊢ (𝑦 ∈ 𝐶 → 𝑦 ∈ 𝐵) |
84 | 83, 66 | sylan2 592 |
. . . . . . . . . . . . . . 15
⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ 𝑦 ∈ 𝐶) → 𝑦:𝐼⟶(Base‘𝑅)) |
85 | 84 | adantrr 713 |
. . . . . . . . . . . . . 14
⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐽)) → 𝑦:𝐼⟶(Base‘𝑅)) |
86 | 13 | sselda 3966 |
. . . . . . . . . . . . . . 15
⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ 𝑧 ∈ 𝐽) → 𝑧 ∈ 𝐼) |
87 | 86 | adantrl 712 |
. . . . . . . . . . . . . 14
⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐽)) → 𝑧 ∈ 𝐼) |
88 | 85, 87 | ffvelrnd 6845 |
. . . . . . . . . . . . 13
⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐽)) → (𝑦‘𝑧) ∈ (Base‘𝑅)) |
89 | 1 | frlmsca 20827 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉) → 𝑅 = (Scalar‘𝑌)) |
90 | 89 | 3adant3 1124 |
. . . . . . . . . . . . . . 15
⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) → 𝑅 = (Scalar‘𝑌)) |
91 | 90 | fveq2d 6668 |
. . . . . . . . . . . . . 14
⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) → (Base‘𝑅) = (Base‘(Scalar‘𝑌))) |
92 | 91 | adantr 481 |
. . . . . . . . . . . . 13
⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐽)) → (Base‘𝑅) = (Base‘(Scalar‘𝑌))) |
93 | 88, 92 | eleqtrd 2915 |
. . . . . . . . . . . 12
⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐽)) → (𝑦‘𝑧) ∈ (Base‘(Scalar‘𝑌))) |
94 | 5, 42 | lspssid 19688 |
. . . . . . . . . . . . . . 15
⊢ ((𝑌 ∈ LMod ∧ (𝑈 “ 𝐽) ⊆ 𝐵) → (𝑈 “ 𝐽) ⊆ (𝐾‘(𝑈 “ 𝐽))) |
95 | 3, 59, 94 | syl2anc 584 |
. . . . . . . . . . . . . 14
⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) → (𝑈 “ 𝐽) ⊆ (𝐾‘(𝑈 “ 𝐽))) |
96 | 95 | adantr 481 |
. . . . . . . . . . . . 13
⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐽)) → (𝑈 “ 𝐽) ⊆ (𝐾‘(𝑈 “ 𝐽))) |
97 | | funfvima2 6985 |
. . . . . . . . . . . . . . . 16
⊢ ((Fun
𝑈 ∧ 𝐽 ⊆ dom 𝑈) → (𝑧 ∈ 𝐽 → (𝑈‘𝑧) ∈ (𝑈 “ 𝐽))) |
98 | 36, 38, 97 | syl2anc 584 |
. . . . . . . . . . . . . . 15
⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) → (𝑧 ∈ 𝐽 → (𝑈‘𝑧) ∈ (𝑈 “ 𝐽))) |
99 | 98 | imp 407 |
. . . . . . . . . . . . . 14
⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ 𝑧 ∈ 𝐽) → (𝑈‘𝑧) ∈ (𝑈 “ 𝐽)) |
100 | 99 | adantrl 712 |
. . . . . . . . . . . . 13
⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐽)) → (𝑈‘𝑧) ∈ (𝑈 “ 𝐽)) |
101 | 96, 100 | sseldd 3967 |
. . . . . . . . . . . 12
⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐽)) → (𝑈‘𝑧) ∈ (𝐾‘(𝑈 “ 𝐽))) |
102 | | eqid 2821 |
. . . . . . . . . . . . 13
⊢
(Scalar‘𝑌) =
(Scalar‘𝑌) |
103 | | eqid 2821 |
. . . . . . . . . . . . 13
⊢
(Base‘(Scalar‘𝑌)) = (Base‘(Scalar‘𝑌)) |
104 | 102, 50, 103, 4 | lssvscl 19658 |
. . . . . . . . . . . 12
⊢ (((𝑌 ∈ LMod ∧ (𝐾‘(𝑈 “ 𝐽)) ∈ (LSubSp‘𝑌)) ∧ ((𝑦‘𝑧) ∈ (Base‘(Scalar‘𝑌)) ∧ (𝑈‘𝑧) ∈ (𝐾‘(𝑈 “ 𝐽)))) → ((𝑦‘𝑧)( ·𝑠
‘𝑌)(𝑈‘𝑧)) ∈ (𝐾‘(𝑈 “ 𝐽))) |
105 | 81, 82, 93, 101, 104 | syl22anc 834 |
. . . . . . . . . . 11
⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐽)) → ((𝑦‘𝑧)( ·𝑠
‘𝑌)(𝑈‘𝑧)) ∈ (𝐾‘(𝑈 “ 𝐽))) |
106 | 105 | anassrs 468 |
. . . . . . . . . 10
⊢ ((((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ 𝑦 ∈ 𝐶) ∧ 𝑧 ∈ 𝐽) → ((𝑦‘𝑧)( ·𝑠
‘𝑌)(𝑈‘𝑧)) ∈ (𝐾‘(𝑈 “ 𝐽))) |
107 | 106 | adantlrr 717 |
. . . . . . . . 9
⊢ ((((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐼)) ∧ 𝑧 ∈ 𝐽) → ((𝑦‘𝑧)( ·𝑠
‘𝑌)(𝑈‘𝑧)) ∈ (𝐾‘(𝑈 “ 𝐽))) |
108 | | id 22 |
. . . . . . . . . . . . . . . 16
⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ 𝑦 ∈ 𝐶) → ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ 𝑦 ∈ 𝐶)) |
109 | 108 | adantrr 713 |
. . . . . . . . . . . . . . 15
⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐼)) → ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ 𝑦 ∈ 𝐶)) |
110 | 109 | adantr 481 |
. . . . . . . . . . . . . 14
⊢ ((((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐼)) ∧ ¬ 𝑧 ∈ 𝐽) → ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ 𝑦 ∈ 𝐶)) |
111 | | simplrr 774 |
. . . . . . . . . . . . . . 15
⊢ ((((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐼)) ∧ ¬ 𝑧 ∈ 𝐽) → 𝑧 ∈ 𝐼) |
112 | | simpr 485 |
. . . . . . . . . . . . . . 15
⊢ ((((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐼)) ∧ ¬ 𝑧 ∈ 𝐽) → ¬ 𝑧 ∈ 𝐽) |
113 | 111, 112 | eldifd 3946 |
. . . . . . . . . . . . . 14
⊢ ((((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐼)) ∧ ¬ 𝑧 ∈ 𝐽) → 𝑧 ∈ (𝐼 ∖ 𝐽)) |
114 | | oveq1 7152 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑥 = 𝑦 → (𝑥 supp 0 ) = (𝑦 supp 0 )) |
115 | 114 | sseq1d 3997 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑥 = 𝑦 → ((𝑥 supp 0 ) ⊆ 𝐽 ↔ (𝑦 supp 0 ) ⊆ 𝐽)) |
116 | 115, 7 | elrab2 3682 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑦 ∈ 𝐶 ↔ (𝑦 ∈ 𝐵 ∧ (𝑦 supp 0 ) ⊆ 𝐽)) |
117 | 116 | simprbi 497 |
. . . . . . . . . . . . . . . 16
⊢ (𝑦 ∈ 𝐶 → (𝑦 supp 0 ) ⊆ 𝐽) |
118 | 117 | adantl 482 |
. . . . . . . . . . . . . . 15
⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ 𝑦 ∈ 𝐶) → (𝑦 supp 0 ) ⊆ 𝐽) |
119 | 6 | fvexi 6678 |
. . . . . . . . . . . . . . . 16
⊢ 0 ∈
V |
120 | 119 | a1i 11 |
. . . . . . . . . . . . . . 15
⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ 𝑦 ∈ 𝐶) → 0 ∈ V) |
121 | 84, 118, 46, 120 | suppssr 7852 |
. . . . . . . . . . . . . 14
⊢ ((((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ 𝑦 ∈ 𝐶) ∧ 𝑧 ∈ (𝐼 ∖ 𝐽)) → (𝑦‘𝑧) = 0 ) |
122 | 110, 113,
121 | syl2anc 584 |
. . . . . . . . . . . . 13
⊢ ((((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐼)) ∧ ¬ 𝑧 ∈ 𝐽) → (𝑦‘𝑧) = 0 ) |
123 | 90 | fveq2d 6668 |
. . . . . . . . . . . . . . 15
⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) → (0g‘𝑅) =
(0g‘(Scalar‘𝑌))) |
124 | 6, 123 | syl5eq 2868 |
. . . . . . . . . . . . . 14
⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) → 0 =
(0g‘(Scalar‘𝑌))) |
125 | 124 | ad2antrr 722 |
. . . . . . . . . . . . 13
⊢ ((((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐼)) ∧ ¬ 𝑧 ∈ 𝐽) → 0 =
(0g‘(Scalar‘𝑌))) |
126 | 122, 125 | eqtrd 2856 |
. . . . . . . . . . . 12
⊢ ((((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐼)) ∧ ¬ 𝑧 ∈ 𝐽) → (𝑦‘𝑧) = (0g‘(Scalar‘𝑌))) |
127 | 126 | oveq1d 7160 |
. . . . . . . . . . 11
⊢ ((((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐼)) ∧ ¬ 𝑧 ∈ 𝐽) → ((𝑦‘𝑧)( ·𝑠
‘𝑌)(𝑈‘𝑧)) =
((0g‘(Scalar‘𝑌))( ·𝑠
‘𝑌)(𝑈‘𝑧))) |
128 | 3 | ad2antrr 722 |
. . . . . . . . . . . 12
⊢ ((((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐼)) ∧ ¬ 𝑧 ∈ 𝐽) → 𝑌 ∈ LMod) |
129 | 11 | ffvelrnda 6844 |
. . . . . . . . . . . . . 14
⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ 𝑧 ∈ 𝐼) → (𝑈‘𝑧) ∈ 𝐵) |
130 | 129 | adantrl 712 |
. . . . . . . . . . . . 13
⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐼)) → (𝑈‘𝑧) ∈ 𝐵) |
131 | 130 | adantr 481 |
. . . . . . . . . . . 12
⊢ ((((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐼)) ∧ ¬ 𝑧 ∈ 𝐽) → (𝑈‘𝑧) ∈ 𝐵) |
132 | | eqid 2821 |
. . . . . . . . . . . . 13
⊢
(0g‘(Scalar‘𝑌)) =
(0g‘(Scalar‘𝑌)) |
133 | 5, 102, 50, 132, 53 | lmod0vs 19598 |
. . . . . . . . . . . 12
⊢ ((𝑌 ∈ LMod ∧ (𝑈‘𝑧) ∈ 𝐵) →
((0g‘(Scalar‘𝑌))( ·𝑠
‘𝑌)(𝑈‘𝑧)) = (0g‘𝑌)) |
134 | 128, 131,
133 | syl2anc 584 |
. . . . . . . . . . 11
⊢ ((((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐼)) ∧ ¬ 𝑧 ∈ 𝐽) →
((0g‘(Scalar‘𝑌))( ·𝑠
‘𝑌)(𝑈‘𝑧)) = (0g‘𝑌)) |
135 | 127, 134 | eqtrd 2856 |
. . . . . . . . . 10
⊢ ((((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐼)) ∧ ¬ 𝑧 ∈ 𝐽) → ((𝑦‘𝑧)( ·𝑠
‘𝑌)(𝑈‘𝑧)) = (0g‘𝑌)) |
136 | 61 | ad2antrr 722 |
. . . . . . . . . . 11
⊢ ((((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐼)) ∧ ¬ 𝑧 ∈ 𝐽) → (𝐾‘(𝑈 “ 𝐽)) ∈ (LSubSp‘𝑌)) |
137 | 53, 4 | lss0cl 19649 |
. . . . . . . . . . 11
⊢ ((𝑌 ∈ LMod ∧ (𝐾‘(𝑈 “ 𝐽)) ∈ (LSubSp‘𝑌)) → (0g‘𝑌) ∈ (𝐾‘(𝑈 “ 𝐽))) |
138 | 128, 136,
137 | syl2anc 584 |
. . . . . . . . . 10
⊢ ((((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐼)) ∧ ¬ 𝑧 ∈ 𝐽) → (0g‘𝑌) ∈ (𝐾‘(𝑈 “ 𝐽))) |
139 | 135, 138 | eqeltrd 2913 |
. . . . . . . . 9
⊢ ((((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐼)) ∧ ¬ 𝑧 ∈ 𝐽) → ((𝑦‘𝑧)( ·𝑠
‘𝑌)(𝑈‘𝑧)) ∈ (𝐾‘(𝑈 “ 𝐽))) |
140 | 107, 139 | pm2.61dan 809 |
. . . . . . . 8
⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐼)) → ((𝑦‘𝑧)( ·𝑠
‘𝑌)(𝑈‘𝑧)) ∈ (𝐾‘(𝑈 “ 𝐽))) |
141 | 80, 140 | eqeltrd 2913 |
. . . . . . 7
⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐼)) → ((𝑦 ∘f (
·𝑠 ‘𝑌)𝑈)‘𝑧) ∈ (𝐾‘(𝑈 “ 𝐽))) |
142 | 141 | expr 457 |
. . . . . 6
⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ 𝑦 ∈ 𝐶) → (𝑧 ∈ 𝐼 → ((𝑦 ∘f (
·𝑠 ‘𝑌)𝑈)‘𝑧) ∈ (𝐾‘(𝑈 “ 𝐽)))) |
143 | 142 | ralrimiv 3181 |
. . . . 5
⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ 𝑦 ∈ 𝐶) → ∀𝑧 ∈ 𝐼 ((𝑦 ∘f (
·𝑠 ‘𝑌)𝑈)‘𝑧) ∈ (𝐾‘(𝑈 “ 𝐽))) |
144 | | ffnfv 6875 |
. . . . 5
⊢ ((𝑦 ∘f (
·𝑠 ‘𝑌)𝑈):𝐼⟶(𝐾‘(𝑈 “ 𝐽)) ↔ ((𝑦 ∘f (
·𝑠 ‘𝑌)𝑈) Fn 𝐼 ∧ ∀𝑧 ∈ 𝐼 ((𝑦 ∘f (
·𝑠 ‘𝑌)𝑈)‘𝑧) ∈ (𝐾‘(𝑈 “ 𝐽)))) |
145 | 73, 143, 144 | sylanbrc 583 |
. . . 4
⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ 𝑦 ∈ 𝐶) → (𝑦 ∘f (
·𝑠 ‘𝑌)𝑈):𝐼⟶(𝐾‘(𝑈 “ 𝐽))) |
146 | 1, 6, 5 | frlmbasfsupp 20832 |
. . . . . . . 8
⊢ ((𝐼 ∈ 𝑉 ∧ 𝑦 ∈ 𝐵) → 𝑦 finSupp 0 ) |
147 | 146 | fsuppimpd 8829 |
. . . . . . 7
⊢ ((𝐼 ∈ 𝑉 ∧ 𝑦 ∈ 𝐵) → (𝑦 supp 0 ) ∈
Fin) |
148 | 46, 49, 147 | syl2anc 584 |
. . . . . 6
⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ 𝑦 ∈ 𝐶) → (𝑦 supp 0 ) ∈
Fin) |
149 | | dffn2 6510 |
. . . . . . . . 9
⊢ ((𝑦 ∘f (
·𝑠 ‘𝑌)𝑈) Fn 𝐼 ↔ (𝑦 ∘f (
·𝑠 ‘𝑌)𝑈):𝐼⟶V) |
150 | 72, 149 | sylib 219 |
. . . . . . . 8
⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ 𝑦 ∈ 𝐵) → (𝑦 ∘f (
·𝑠 ‘𝑌)𝑈):𝐼⟶V) |
151 | 67 | adantr 481 |
. . . . . . . . . 10
⊢ ((((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ 𝑦 ∈ 𝐵) ∧ 𝑥 ∈ (𝐼 ∖ (𝑦 supp 0 ))) → 𝑦 Fn 𝐼) |
152 | 68 | ad2antrr 722 |
. . . . . . . . . 10
⊢ ((((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ 𝑦 ∈ 𝐵) ∧ 𝑥 ∈ (𝐼 ∖ (𝑦 supp 0 ))) → 𝑈 Fn 𝐼) |
153 | | simpll2 1205 |
. . . . . . . . . 10
⊢ ((((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ 𝑦 ∈ 𝐵) ∧ 𝑥 ∈ (𝐼 ∖ (𝑦 supp 0 ))) → 𝐼 ∈ 𝑉) |
154 | | eldifi 4102 |
. . . . . . . . . . 11
⊢ (𝑥 ∈ (𝐼 ∖ (𝑦 supp 0 )) → 𝑥 ∈ 𝐼) |
155 | 154 | adantl 482 |
. . . . . . . . . 10
⊢ ((((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ 𝑦 ∈ 𝐵) ∧ 𝑥 ∈ (𝐼 ∖ (𝑦 supp 0 ))) → 𝑥 ∈ 𝐼) |
156 | | fnfvof 7412 |
. . . . . . . . . 10
⊢ (((𝑦 Fn 𝐼 ∧ 𝑈 Fn 𝐼) ∧ (𝐼 ∈ 𝑉 ∧ 𝑥 ∈ 𝐼)) → ((𝑦 ∘f (
·𝑠 ‘𝑌)𝑈)‘𝑥) = ((𝑦‘𝑥)( ·𝑠
‘𝑌)(𝑈‘𝑥))) |
157 | 151, 152,
153, 155, 156 | syl22anc 834 |
. . . . . . . . 9
⊢ ((((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ 𝑦 ∈ 𝐵) ∧ 𝑥 ∈ (𝐼 ∖ (𝑦 supp 0 ))) → ((𝑦 ∘f (
·𝑠 ‘𝑌)𝑈)‘𝑥) = ((𝑦‘𝑥)( ·𝑠
‘𝑌)(𝑈‘𝑥))) |
158 | | ssidd 3989 |
. . . . . . . . . . . 12
⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ 𝑦 ∈ 𝐵) → (𝑦 supp 0 ) ⊆ (𝑦 supp 0 )) |
159 | 119 | a1i 11 |
. . . . . . . . . . . 12
⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ 𝑦 ∈ 𝐵) → 0 ∈ V) |
160 | 66, 158, 70, 159 | suppssr 7852 |
. . . . . . . . . . 11
⊢ ((((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ 𝑦 ∈ 𝐵) ∧ 𝑥 ∈ (𝐼 ∖ (𝑦 supp 0 ))) → (𝑦‘𝑥) = 0 ) |
161 | 124 | ad2antrr 722 |
. . . . . . . . . . 11
⊢ ((((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ 𝑦 ∈ 𝐵) ∧ 𝑥 ∈ (𝐼 ∖ (𝑦 supp 0 ))) → 0 =
(0g‘(Scalar‘𝑌))) |
162 | 160, 161 | eqtrd 2856 |
. . . . . . . . . 10
⊢ ((((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ 𝑦 ∈ 𝐵) ∧ 𝑥 ∈ (𝐼 ∖ (𝑦 supp 0 ))) → (𝑦‘𝑥) = (0g‘(Scalar‘𝑌))) |
163 | 162 | oveq1d 7160 |
. . . . . . . . 9
⊢ ((((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ 𝑦 ∈ 𝐵) ∧ 𝑥 ∈ (𝐼 ∖ (𝑦 supp 0 ))) → ((𝑦‘𝑥)( ·𝑠
‘𝑌)(𝑈‘𝑥)) =
((0g‘(Scalar‘𝑌))( ·𝑠
‘𝑌)(𝑈‘𝑥))) |
164 | 3 | ad2antrr 722 |
. . . . . . . . . 10
⊢ ((((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ 𝑦 ∈ 𝐵) ∧ 𝑥 ∈ (𝐼 ∖ (𝑦 supp 0 ))) → 𝑌 ∈ LMod) |
165 | 11 | adantr 481 |
. . . . . . . . . . 11
⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ 𝑦 ∈ 𝐵) → 𝑈:𝐼⟶𝐵) |
166 | | ffvelrn 6842 |
. . . . . . . . . . 11
⊢ ((𝑈:𝐼⟶𝐵 ∧ 𝑥 ∈ 𝐼) → (𝑈‘𝑥) ∈ 𝐵) |
167 | 165, 154,
166 | syl2an 595 |
. . . . . . . . . 10
⊢ ((((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ 𝑦 ∈ 𝐵) ∧ 𝑥 ∈ (𝐼 ∖ (𝑦 supp 0 ))) → (𝑈‘𝑥) ∈ 𝐵) |
168 | 5, 102, 50, 132, 53 | lmod0vs 19598 |
. . . . . . . . . 10
⊢ ((𝑌 ∈ LMod ∧ (𝑈‘𝑥) ∈ 𝐵) →
((0g‘(Scalar‘𝑌))( ·𝑠
‘𝑌)(𝑈‘𝑥)) = (0g‘𝑌)) |
169 | 164, 167,
168 | syl2anc 584 |
. . . . . . . . 9
⊢ ((((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ 𝑦 ∈ 𝐵) ∧ 𝑥 ∈ (𝐼 ∖ (𝑦 supp 0 ))) →
((0g‘(Scalar‘𝑌))( ·𝑠
‘𝑌)(𝑈‘𝑥)) = (0g‘𝑌)) |
170 | 157, 163,
169 | 3eqtrd 2860 |
. . . . . . . 8
⊢ ((((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ 𝑦 ∈ 𝐵) ∧ 𝑥 ∈ (𝐼 ∖ (𝑦 supp 0 ))) → ((𝑦 ∘f (
·𝑠 ‘𝑌)𝑈)‘𝑥) = (0g‘𝑌)) |
171 | 150, 170 | suppss 7851 |
. . . . . . 7
⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ 𝑦 ∈ 𝐵) → ((𝑦 ∘f (
·𝑠 ‘𝑌)𝑈) supp (0g‘𝑌)) ⊆ (𝑦 supp 0 )) |
172 | 49, 171 | syldan 591 |
. . . . . 6
⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ 𝑦 ∈ 𝐶) → ((𝑦 ∘f (
·𝑠 ‘𝑌)𝑈) supp (0g‘𝑌)) ⊆ (𝑦 supp 0 )) |
173 | 148, 172 | ssfid 8730 |
. . . . 5
⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ 𝑦 ∈ 𝐶) → ((𝑦 ∘f (
·𝑠 ‘𝑌)𝑈) supp (0g‘𝑌)) ∈ Fin) |
174 | | simp2 1129 |
. . . . . . . . . 10
⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) → 𝐼 ∈ 𝑉) |
175 | 1, 17, 5 | frlmbasmap 20833 |
. . . . . . . . . 10
⊢ ((𝐼 ∈ 𝑉 ∧ 𝑦 ∈ 𝐵) → 𝑦 ∈ ((Base‘𝑅) ↑m 𝐼)) |
176 | 174, 83, 175 | syl2an 595 |
. . . . . . . . 9
⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ 𝑦 ∈ 𝐶) → 𝑦 ∈ ((Base‘𝑅) ↑m 𝐼)) |
177 | | elmapfn 8419 |
. . . . . . . . 9
⊢ (𝑦 ∈ ((Base‘𝑅) ↑m 𝐼) → 𝑦 Fn 𝐼) |
178 | 176, 177 | syl 17 |
. . . . . . . 8
⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ 𝑦 ∈ 𝐶) → 𝑦 Fn 𝐼) |
179 | 11 | adantr 481 |
. . . . . . . . 9
⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ 𝑦 ∈ 𝐶) → 𝑈:𝐼⟶𝐵) |
180 | 179 | ffnd 6509 |
. . . . . . . 8
⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ 𝑦 ∈ 𝐶) → 𝑈 Fn 𝐼) |
181 | 178, 180,
46, 46, 71 | offn 7409 |
. . . . . . 7
⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ 𝑦 ∈ 𝐶) → (𝑦 ∘f (
·𝑠 ‘𝑌)𝑈) Fn 𝐼) |
182 | | fnfun 6447 |
. . . . . . 7
⊢ ((𝑦 ∘f (
·𝑠 ‘𝑌)𝑈) Fn 𝐼 → Fun (𝑦 ∘f (
·𝑠 ‘𝑌)𝑈)) |
183 | 181, 182 | syl 17 |
. . . . . 6
⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ 𝑦 ∈ 𝐶) → Fun (𝑦 ∘f (
·𝑠 ‘𝑌)𝑈)) |
184 | | ovexd 7180 |
. . . . . 6
⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ 𝑦 ∈ 𝐶) → (𝑦 ∘f (
·𝑠 ‘𝑌)𝑈) ∈ V) |
185 | | fvexd 6679 |
. . . . . 6
⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ 𝑦 ∈ 𝐶) → (0g‘𝑌) ∈ V) |
186 | | funisfsupp 8827 |
. . . . . 6
⊢ ((Fun
(𝑦 ∘f (
·𝑠 ‘𝑌)𝑈) ∧ (𝑦 ∘f (
·𝑠 ‘𝑌)𝑈) ∈ V ∧ (0g‘𝑌) ∈ V) → ((𝑦 ∘f (
·𝑠 ‘𝑌)𝑈) finSupp (0g‘𝑌) ↔ ((𝑦 ∘f (
·𝑠 ‘𝑌)𝑈) supp (0g‘𝑌)) ∈ Fin)) |
187 | 183, 184,
185, 186 | syl3anc 1363 |
. . . . 5
⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ 𝑦 ∈ 𝐶) → ((𝑦 ∘f (
·𝑠 ‘𝑌)𝑈) finSupp (0g‘𝑌) ↔ ((𝑦 ∘f (
·𝑠 ‘𝑌)𝑈) supp (0g‘𝑌)) ∈ Fin)) |
188 | 173, 187 | mpbird 258 |
. . . 4
⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ 𝑦 ∈ 𝐶) → (𝑦 ∘f (
·𝑠 ‘𝑌)𝑈) finSupp (0g‘𝑌)) |
189 | 53, 56, 46, 64, 145, 188 | gsumsubgcl 18971 |
. . 3
⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ 𝑦 ∈ 𝐶) → (𝑌 Σg (𝑦 ∘f (
·𝑠 ‘𝑌)𝑈)) ∈ (𝐾‘(𝑈 “ 𝐽))) |
190 | 52, 189 | eqeltrd 2913 |
. 2
⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ 𝑦 ∈ 𝐶) → 𝑦 ∈ (𝐾‘(𝑈 “ 𝐽))) |
191 | 44, 190 | eqelssd 3987 |
1
⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) → (𝐾‘(𝑈 “ 𝐽)) = 𝐶) |