Step | Hyp | Ref
| Expression |
1 | | frlmsslsp.y |
. . . . 5
⊢ 𝑌 = (𝑅 freeLMod 𝐼) |
2 | 1 | frlmlmod 20457 |
. . . 4
⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉) → 𝑌 ∈ LMod) |
3 | 2 | 3adant3 1168 |
. . 3
⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) → 𝑌 ∈ LMod) |
4 | | eqid 2826 |
. . . 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 20482 |
. . 3
⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) → 𝐶 ∈ (LSubSp‘𝑌)) |
9 | | frlmsslsp.u |
. . . . . . . . . 10
⊢ 𝑈 = (𝑅 unitVec 𝐼) |
10 | 9, 1, 5 | uvcff 20498 |
. . . . . . . . 9
⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉) → 𝑈:𝐼⟶𝐵) |
11 | 10 | 3adant3 1168 |
. . . . . . . 8
⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) → 𝑈:𝐼⟶𝐵) |
12 | 11 | adantr 474 |
. . . . . . 7
⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ 𝑦 ∈ 𝐽) → 𝑈:𝐼⟶𝐵) |
13 | | simp3 1174 |
. . . . . . . 8
⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) → 𝐽 ⊆ 𝐼) |
14 | 13 | sselda 3828 |
. . . . . . 7
⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ 𝑦 ∈ 𝐽) → 𝑦 ∈ 𝐼) |
15 | 12, 14 | ffvelrnd 6610 |
. . . . . 6
⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ 𝑦 ∈ 𝐽) → (𝑈‘𝑦) ∈ 𝐵) |
16 | | simpl2 1250 |
. . . . . . . 8
⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ 𝑦 ∈ 𝐽) → 𝐼 ∈ 𝑉) |
17 | | eqid 2826 |
. . . . . . . . 9
⊢
(Base‘𝑅) =
(Base‘𝑅) |
18 | 1, 17, 5 | frlmbasf 20468 |
. . . . . . . 8
⊢ ((𝐼 ∈ 𝑉 ∧ (𝑈‘𝑦) ∈ 𝐵) → (𝑈‘𝑦):𝐼⟶(Base‘𝑅)) |
19 | 16, 15, 18 | syl2anc 581 |
. . . . . . 7
⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ 𝑦 ∈ 𝐽) → (𝑈‘𝑦):𝐼⟶(Base‘𝑅)) |
20 | | simpll1 1275 |
. . . . . . . 8
⊢ ((((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ 𝑦 ∈ 𝐽) ∧ 𝑥 ∈ (𝐼 ∖ 𝐽)) → 𝑅 ∈ Ring) |
21 | | simpll2 1277 |
. . . . . . . 8
⊢ ((((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ 𝑦 ∈ 𝐽) ∧ 𝑥 ∈ (𝐼 ∖ 𝐽)) → 𝐼 ∈ 𝑉) |
22 | 14 | adantr 474 |
. . . . . . . 8
⊢ ((((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ 𝑦 ∈ 𝐽) ∧ 𝑥 ∈ (𝐼 ∖ 𝐽)) → 𝑦 ∈ 𝐼) |
23 | | eldifi 3960 |
. . . . . . . . 9
⊢ (𝑥 ∈ (𝐼 ∖ 𝐽) → 𝑥 ∈ 𝐼) |
24 | 23 | adantl 475 |
. . . . . . . 8
⊢ ((((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ 𝑦 ∈ 𝐽) ∧ 𝑥 ∈ (𝐼 ∖ 𝐽)) → 𝑥 ∈ 𝐼) |
25 | | disjdif 4264 |
. . . . . . . . . 10
⊢ (𝐽 ∩ (𝐼 ∖ 𝐽)) = ∅ |
26 | | disjne 4247 |
. . . . . . . . . 10
⊢ (((𝐽 ∩ (𝐼 ∖ 𝐽)) = ∅ ∧ 𝑦 ∈ 𝐽 ∧ 𝑥 ∈ (𝐼 ∖ 𝐽)) → 𝑦 ≠ 𝑥) |
27 | 25, 26 | mp3an1 1578 |
. . . . . . . . 9
⊢ ((𝑦 ∈ 𝐽 ∧ 𝑥 ∈ (𝐼 ∖ 𝐽)) → 𝑦 ≠ 𝑥) |
28 | 27 | adantll 707 |
. . . . . . . 8
⊢ ((((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ 𝑦 ∈ 𝐽) ∧ 𝑥 ∈ (𝐼 ∖ 𝐽)) → 𝑦 ≠ 𝑥) |
29 | 9, 20, 21, 22, 24, 28, 6 | uvcvv0 20497 |
. . . . . . 7
⊢ ((((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ 𝑦 ∈ 𝐽) ∧ 𝑥 ∈ (𝐼 ∖ 𝐽)) → ((𝑈‘𝑦)‘𝑥) = 0 ) |
30 | 19, 29 | suppss 7591 |
. . . . . 6
⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ 𝑦 ∈ 𝐽) → ((𝑈‘𝑦) supp 0 ) ⊆ 𝐽) |
31 | | oveq1 6913 |
. . . . . . . 8
⊢ (𝑥 = (𝑈‘𝑦) → (𝑥 supp 0 ) = ((𝑈‘𝑦) supp 0 )) |
32 | 31 | sseq1d 3858 |
. . . . . . 7
⊢ (𝑥 = (𝑈‘𝑦) → ((𝑥 supp 0 ) ⊆ 𝐽 ↔ ((𝑈‘𝑦) supp 0 ) ⊆ 𝐽)) |
33 | 32, 7 | elrab2 3590 |
. . . . . 6
⊢ ((𝑈‘𝑦) ∈ 𝐶 ↔ ((𝑈‘𝑦) ∈ 𝐵 ∧ ((𝑈‘𝑦) supp 0 ) ⊆ 𝐽)) |
34 | 15, 30, 33 | sylanbrc 580 |
. . . . 5
⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ 𝑦 ∈ 𝐽) → (𝑈‘𝑦) ∈ 𝐶) |
35 | 34 | ralrimiva 3176 |
. . . 4
⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) → ∀𝑦 ∈ 𝐽 (𝑈‘𝑦) ∈ 𝐶) |
36 | 11 | ffnd 6280 |
. . . . . 6
⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) → 𝑈 Fn 𝐼) |
37 | | fnfun 6222 |
. . . . . 6
⊢ (𝑈 Fn 𝐼 → Fun 𝑈) |
38 | 36, 37 | syl 17 |
. . . . 5
⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) → Fun 𝑈) |
39 | | fndm 6224 |
. . . . . . 7
⊢ (𝑈 Fn 𝐼 → dom 𝑈 = 𝐼) |
40 | 36, 39 | syl 17 |
. . . . . 6
⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) → dom 𝑈 = 𝐼) |
41 | 13, 40 | sseqtr4d 3868 |
. . . . 5
⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) → 𝐽 ⊆ dom 𝑈) |
42 | | funimass4 6495 |
. . . . 5
⊢ ((Fun
𝑈 ∧ 𝐽 ⊆ dom 𝑈) → ((𝑈 “ 𝐽) ⊆ 𝐶 ↔ ∀𝑦 ∈ 𝐽 (𝑈‘𝑦) ∈ 𝐶)) |
43 | 38, 41, 42 | syl2anc 581 |
. . . 4
⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) → ((𝑈 “ 𝐽) ⊆ 𝐶 ↔ ∀𝑦 ∈ 𝐽 (𝑈‘𝑦) ∈ 𝐶)) |
44 | 35, 43 | mpbird 249 |
. . 3
⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) → (𝑈 “ 𝐽) ⊆ 𝐶) |
45 | | frlmsslsp.k |
. . . 4
⊢ 𝐾 = (LSpan‘𝑌) |
46 | 4, 45 | lspssp 19348 |
. . 3
⊢ ((𝑌 ∈ LMod ∧ 𝐶 ∈ (LSubSp‘𝑌) ∧ (𝑈 “ 𝐽) ⊆ 𝐶) → (𝐾‘(𝑈 “ 𝐽)) ⊆ 𝐶) |
47 | 3, 8, 44, 46 | syl3anc 1496 |
. 2
⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) → (𝐾‘(𝑈 “ 𝐽)) ⊆ 𝐶) |
48 | | simpl1 1248 |
. . . 4
⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ 𝑦 ∈ 𝐶) → 𝑅 ∈ Ring) |
49 | | simpl2 1250 |
. . . 4
⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ 𝑦 ∈ 𝐶) → 𝐼 ∈ 𝑉) |
50 | | ssrab2 3913 |
. . . . . . 7
⊢ {𝑥 ∈ 𝐵 ∣ (𝑥 supp 0 ) ⊆ 𝐽} ⊆ 𝐵 |
51 | 7, 50 | eqsstri 3861 |
. . . . . 6
⊢ 𝐶 ⊆ 𝐵 |
52 | 51 | a1i 11 |
. . . . 5
⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) → 𝐶 ⊆ 𝐵) |
53 | 52 | sselda 3828 |
. . . 4
⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ 𝑦 ∈ 𝐶) → 𝑦 ∈ 𝐵) |
54 | | eqid 2826 |
. . . . 5
⊢ (
·𝑠 ‘𝑌) = ( ·𝑠
‘𝑌) |
55 | 9, 1, 5, 54 | uvcresum 20500 |
. . . 4
⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝑦 ∈ 𝐵) → 𝑦 = (𝑌 Σg (𝑦 ∘𝑓 (
·𝑠 ‘𝑌)𝑈))) |
56 | 48, 49, 53, 55 | syl3anc 1496 |
. . 3
⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ 𝑦 ∈ 𝐶) → 𝑦 = (𝑌 Σg (𝑦 ∘𝑓 (
·𝑠 ‘𝑌)𝑈))) |
57 | | eqid 2826 |
. . . 4
⊢
(0g‘𝑌) = (0g‘𝑌) |
58 | | lmodabl 19267 |
. . . . . 6
⊢ (𝑌 ∈ LMod → 𝑌 ∈ Abel) |
59 | 3, 58 | syl 17 |
. . . . 5
⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) → 𝑌 ∈ Abel) |
60 | 59 | adantr 474 |
. . . 4
⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ 𝑦 ∈ 𝐶) → 𝑌 ∈ Abel) |
61 | | imassrn 5719 |
. . . . . . . 8
⊢ (𝑈 “ 𝐽) ⊆ ran 𝑈 |
62 | 11 | frnd 6286 |
. . . . . . . 8
⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) → ran 𝑈 ⊆ 𝐵) |
63 | 61, 62 | syl5ss 3839 |
. . . . . . 7
⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) → (𝑈 “ 𝐽) ⊆ 𝐵) |
64 | 5, 4, 45 | lspcl 19336 |
. . . . . . 7
⊢ ((𝑌 ∈ LMod ∧ (𝑈 “ 𝐽) ⊆ 𝐵) → (𝐾‘(𝑈 “ 𝐽)) ∈ (LSubSp‘𝑌)) |
65 | 3, 63, 64 | syl2anc 581 |
. . . . . 6
⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) → (𝐾‘(𝑈 “ 𝐽)) ∈ (LSubSp‘𝑌)) |
66 | 4 | lsssubg 19317 |
. . . . . 6
⊢ ((𝑌 ∈ LMod ∧ (𝐾‘(𝑈 “ 𝐽)) ∈ (LSubSp‘𝑌)) → (𝐾‘(𝑈 “ 𝐽)) ∈ (SubGrp‘𝑌)) |
67 | 3, 65, 66 | syl2anc 581 |
. . . . 5
⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) → (𝐾‘(𝑈 “ 𝐽)) ∈ (SubGrp‘𝑌)) |
68 | 67 | adantr 474 |
. . . 4
⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ 𝑦 ∈ 𝐶) → (𝐾‘(𝑈 “ 𝐽)) ∈ (SubGrp‘𝑌)) |
69 | 1, 17, 5 | frlmbasf 20468 |
. . . . . . . . 9
⊢ ((𝐼 ∈ 𝑉 ∧ 𝑦 ∈ 𝐵) → 𝑦:𝐼⟶(Base‘𝑅)) |
70 | 69 | 3ad2antl2 1243 |
. . . . . . . 8
⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ 𝑦 ∈ 𝐵) → 𝑦:𝐼⟶(Base‘𝑅)) |
71 | 70 | ffnd 6280 |
. . . . . . 7
⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ 𝑦 ∈ 𝐵) → 𝑦 Fn 𝐼) |
72 | 36 | adantr 474 |
. . . . . . 7
⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ 𝑦 ∈ 𝐵) → 𝑈 Fn 𝐼) |
73 | | simpl2 1250 |
. . . . . . 7
⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ 𝑦 ∈ 𝐵) → 𝐼 ∈ 𝑉) |
74 | | inidm 4048 |
. . . . . . 7
⊢ (𝐼 ∩ 𝐼) = 𝐼 |
75 | 71, 72, 73, 73, 74 | offn 7169 |
. . . . . 6
⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ 𝑦 ∈ 𝐵) → (𝑦 ∘𝑓 (
·𝑠 ‘𝑌)𝑈) Fn 𝐼) |
76 | 53, 75 | syldan 587 |
. . . . 5
⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ 𝑦 ∈ 𝐶) → (𝑦 ∘𝑓 (
·𝑠 ‘𝑌)𝑈) Fn 𝐼) |
77 | 53, 71 | syldan 587 |
. . . . . . . . . 10
⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ 𝑦 ∈ 𝐶) → 𝑦 Fn 𝐼) |
78 | 77 | adantrr 710 |
. . . . . . . . 9
⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐼)) → 𝑦 Fn 𝐼) |
79 | 36 | adantr 474 |
. . . . . . . . 9
⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐼)) → 𝑈 Fn 𝐼) |
80 | | simpl2 1250 |
. . . . . . . . 9
⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐼)) → 𝐼 ∈ 𝑉) |
81 | | simprr 791 |
. . . . . . . . 9
⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐼)) → 𝑧 ∈ 𝐼) |
82 | | fnfvof 7172 |
. . . . . . . . 9
⊢ (((𝑦 Fn 𝐼 ∧ 𝑈 Fn 𝐼) ∧ (𝐼 ∈ 𝑉 ∧ 𝑧 ∈ 𝐼)) → ((𝑦 ∘𝑓 (
·𝑠 ‘𝑌)𝑈)‘𝑧) = ((𝑦‘𝑧)( ·𝑠
‘𝑌)(𝑈‘𝑧))) |
83 | 78, 79, 80, 81, 82 | syl22anc 874 |
. . . . . . . 8
⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐼)) → ((𝑦 ∘𝑓 (
·𝑠 ‘𝑌)𝑈)‘𝑧) = ((𝑦‘𝑧)( ·𝑠
‘𝑌)(𝑈‘𝑧))) |
84 | 3 | adantr 474 |
. . . . . . . . . . . 12
⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐽)) → 𝑌 ∈ LMod) |
85 | 65 | adantr 474 |
. . . . . . . . . . . 12
⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐽)) → (𝐾‘(𝑈 “ 𝐽)) ∈ (LSubSp‘𝑌)) |
86 | 51 | sseli 3824 |
. . . . . . . . . . . . . . . 16
⊢ (𝑦 ∈ 𝐶 → 𝑦 ∈ 𝐵) |
87 | 86, 70 | sylan2 588 |
. . . . . . . . . . . . . . 15
⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ 𝑦 ∈ 𝐶) → 𝑦:𝐼⟶(Base‘𝑅)) |
88 | 87 | adantrr 710 |
. . . . . . . . . . . . . 14
⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐽)) → 𝑦:𝐼⟶(Base‘𝑅)) |
89 | 13 | sselda 3828 |
. . . . . . . . . . . . . . 15
⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ 𝑧 ∈ 𝐽) → 𝑧 ∈ 𝐼) |
90 | 89 | adantrl 709 |
. . . . . . . . . . . . . 14
⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐽)) → 𝑧 ∈ 𝐼) |
91 | 88, 90 | ffvelrnd 6610 |
. . . . . . . . . . . . 13
⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐽)) → (𝑦‘𝑧) ∈ (Base‘𝑅)) |
92 | 1 | frlmsca 20461 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉) → 𝑅 = (Scalar‘𝑌)) |
93 | 92 | 3adant3 1168 |
. . . . . . . . . . . . . . 15
⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) → 𝑅 = (Scalar‘𝑌)) |
94 | 93 | fveq2d 6438 |
. . . . . . . . . . . . . 14
⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) → (Base‘𝑅) = (Base‘(Scalar‘𝑌))) |
95 | 94 | adantr 474 |
. . . . . . . . . . . . 13
⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐽)) → (Base‘𝑅) = (Base‘(Scalar‘𝑌))) |
96 | 91, 95 | eleqtrd 2909 |
. . . . . . . . . . . 12
⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐽)) → (𝑦‘𝑧) ∈ (Base‘(Scalar‘𝑌))) |
97 | 5, 45 | lspssid 19345 |
. . . . . . . . . . . . . . 15
⊢ ((𝑌 ∈ LMod ∧ (𝑈 “ 𝐽) ⊆ 𝐵) → (𝑈 “ 𝐽) ⊆ (𝐾‘(𝑈 “ 𝐽))) |
98 | 3, 63, 97 | syl2anc 581 |
. . . . . . . . . . . . . 14
⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) → (𝑈 “ 𝐽) ⊆ (𝐾‘(𝑈 “ 𝐽))) |
99 | 98 | adantr 474 |
. . . . . . . . . . . . 13
⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐽)) → (𝑈 “ 𝐽) ⊆ (𝐾‘(𝑈 “ 𝐽))) |
100 | | funfvima2 6750 |
. . . . . . . . . . . . . . . 16
⊢ ((Fun
𝑈 ∧ 𝐽 ⊆ dom 𝑈) → (𝑧 ∈ 𝐽 → (𝑈‘𝑧) ∈ (𝑈 “ 𝐽))) |
101 | 38, 41, 100 | syl2anc 581 |
. . . . . . . . . . . . . . 15
⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) → (𝑧 ∈ 𝐽 → (𝑈‘𝑧) ∈ (𝑈 “ 𝐽))) |
102 | 101 | imp 397 |
. . . . . . . . . . . . . 14
⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ 𝑧 ∈ 𝐽) → (𝑈‘𝑧) ∈ (𝑈 “ 𝐽)) |
103 | 102 | adantrl 709 |
. . . . . . . . . . . . 13
⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐽)) → (𝑈‘𝑧) ∈ (𝑈 “ 𝐽)) |
104 | 99, 103 | sseldd 3829 |
. . . . . . . . . . . 12
⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐽)) → (𝑈‘𝑧) ∈ (𝐾‘(𝑈 “ 𝐽))) |
105 | | eqid 2826 |
. . . . . . . . . . . . 13
⊢
(Scalar‘𝑌) =
(Scalar‘𝑌) |
106 | | eqid 2826 |
. . . . . . . . . . . . 13
⊢
(Base‘(Scalar‘𝑌)) = (Base‘(Scalar‘𝑌)) |
107 | 105, 54, 106, 4 | lssvscl 19315 |
. . . . . . . . . . . 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))) |
108 | 84, 85, 96, 104, 107 | syl22anc 874 |
. . . . . . . . . . 11
⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐽)) → ((𝑦‘𝑧)( ·𝑠
‘𝑌)(𝑈‘𝑧)) ∈ (𝐾‘(𝑈 “ 𝐽))) |
109 | 108 | anassrs 461 |
. . . . . . . . . 10
⊢ ((((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ 𝑦 ∈ 𝐶) ∧ 𝑧 ∈ 𝐽) → ((𝑦‘𝑧)( ·𝑠
‘𝑌)(𝑈‘𝑧)) ∈ (𝐾‘(𝑈 “ 𝐽))) |
110 | 109 | adantlrr 714 |
. . . . . . . . 9
⊢ ((((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐼)) ∧ 𝑧 ∈ 𝐽) → ((𝑦‘𝑧)( ·𝑠
‘𝑌)(𝑈‘𝑧)) ∈ (𝐾‘(𝑈 “ 𝐽))) |
111 | | id 22 |
. . . . . . . . . . . . . . . 16
⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ 𝑦 ∈ 𝐶) → ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ 𝑦 ∈ 𝐶)) |
112 | 111 | adantrr 710 |
. . . . . . . . . . . . . . 15
⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐼)) → ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ 𝑦 ∈ 𝐶)) |
113 | 112 | adantr 474 |
. . . . . . . . . . . . . 14
⊢ ((((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐼)) ∧ ¬ 𝑧 ∈ 𝐽) → ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ 𝑦 ∈ 𝐶)) |
114 | | simplrr 798 |
. . . . . . . . . . . . . . 15
⊢ ((((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐼)) ∧ ¬ 𝑧 ∈ 𝐽) → 𝑧 ∈ 𝐼) |
115 | | simpr 479 |
. . . . . . . . . . . . . . 15
⊢ ((((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐼)) ∧ ¬ 𝑧 ∈ 𝐽) → ¬ 𝑧 ∈ 𝐽) |
116 | 114, 115 | eldifd 3810 |
. . . . . . . . . . . . . 14
⊢ ((((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐼)) ∧ ¬ 𝑧 ∈ 𝐽) → 𝑧 ∈ (𝐼 ∖ 𝐽)) |
117 | | oveq1 6913 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑥 = 𝑦 → (𝑥 supp 0 ) = (𝑦 supp 0 )) |
118 | 117 | sseq1d 3858 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑥 = 𝑦 → ((𝑥 supp 0 ) ⊆ 𝐽 ↔ (𝑦 supp 0 ) ⊆ 𝐽)) |
119 | 118, 7 | elrab2 3590 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑦 ∈ 𝐶 ↔ (𝑦 ∈ 𝐵 ∧ (𝑦 supp 0 ) ⊆ 𝐽)) |
120 | 119 | simprbi 492 |
. . . . . . . . . . . . . . . 16
⊢ (𝑦 ∈ 𝐶 → (𝑦 supp 0 ) ⊆ 𝐽) |
121 | 120 | adantl 475 |
. . . . . . . . . . . . . . 15
⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ 𝑦 ∈ 𝐶) → (𝑦 supp 0 ) ⊆ 𝐽) |
122 | 6 | fvexi 6448 |
. . . . . . . . . . . . . . . 16
⊢ 0 ∈
V |
123 | 122 | a1i 11 |
. . . . . . . . . . . . . . 15
⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ 𝑦 ∈ 𝐶) → 0 ∈ V) |
124 | 87, 121, 49, 123 | suppssr 7592 |
. . . . . . . . . . . . . 14
⊢ ((((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ 𝑦 ∈ 𝐶) ∧ 𝑧 ∈ (𝐼 ∖ 𝐽)) → (𝑦‘𝑧) = 0 ) |
125 | 113, 116,
124 | syl2anc 581 |
. . . . . . . . . . . . 13
⊢ ((((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐼)) ∧ ¬ 𝑧 ∈ 𝐽) → (𝑦‘𝑧) = 0 ) |
126 | 93 | fveq2d 6438 |
. . . . . . . . . . . . . . 15
⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) → (0g‘𝑅) =
(0g‘(Scalar‘𝑌))) |
127 | 6, 126 | syl5eq 2874 |
. . . . . . . . . . . . . 14
⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) → 0 =
(0g‘(Scalar‘𝑌))) |
128 | 127 | ad2antrr 719 |
. . . . . . . . . . . . 13
⊢ ((((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐼)) ∧ ¬ 𝑧 ∈ 𝐽) → 0 =
(0g‘(Scalar‘𝑌))) |
129 | 125, 128 | eqtrd 2862 |
. . . . . . . . . . . 12
⊢ ((((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐼)) ∧ ¬ 𝑧 ∈ 𝐽) → (𝑦‘𝑧) = (0g‘(Scalar‘𝑌))) |
130 | 129 | oveq1d 6921 |
. . . . . . . . . . 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))) |
131 | 3 | ad2antrr 719 |
. . . . . . . . . . . 12
⊢ ((((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐼)) ∧ ¬ 𝑧 ∈ 𝐽) → 𝑌 ∈ LMod) |
132 | 11 | ffvelrnda 6609 |
. . . . . . . . . . . . . 14
⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ 𝑧 ∈ 𝐼) → (𝑈‘𝑧) ∈ 𝐵) |
133 | 132 | adantrl 709 |
. . . . . . . . . . . . 13
⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐼)) → (𝑈‘𝑧) ∈ 𝐵) |
134 | 133 | adantr 474 |
. . . . . . . . . . . 12
⊢ ((((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐼)) ∧ ¬ 𝑧 ∈ 𝐽) → (𝑈‘𝑧) ∈ 𝐵) |
135 | | eqid 2826 |
. . . . . . . . . . . . 13
⊢
(0g‘(Scalar‘𝑌)) =
(0g‘(Scalar‘𝑌)) |
136 | 5, 105, 54, 135, 57 | lmod0vs 19253 |
. . . . . . . . . . . 12
⊢ ((𝑌 ∈ LMod ∧ (𝑈‘𝑧) ∈ 𝐵) →
((0g‘(Scalar‘𝑌))( ·𝑠
‘𝑌)(𝑈‘𝑧)) = (0g‘𝑌)) |
137 | 131, 134,
136 | syl2anc 581 |
. . . . . . . . . . 11
⊢ ((((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐼)) ∧ ¬ 𝑧 ∈ 𝐽) →
((0g‘(Scalar‘𝑌))( ·𝑠
‘𝑌)(𝑈‘𝑧)) = (0g‘𝑌)) |
138 | 130, 137 | eqtrd 2862 |
. . . . . . . . . 10
⊢ ((((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐼)) ∧ ¬ 𝑧 ∈ 𝐽) → ((𝑦‘𝑧)( ·𝑠
‘𝑌)(𝑈‘𝑧)) = (0g‘𝑌)) |
139 | 65 | ad2antrr 719 |
. . . . . . . . . . 11
⊢ ((((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐼)) ∧ ¬ 𝑧 ∈ 𝐽) → (𝐾‘(𝑈 “ 𝐽)) ∈ (LSubSp‘𝑌)) |
140 | 57, 4 | lss0cl 19304 |
. . . . . . . . . . 11
⊢ ((𝑌 ∈ LMod ∧ (𝐾‘(𝑈 “ 𝐽)) ∈ (LSubSp‘𝑌)) → (0g‘𝑌) ∈ (𝐾‘(𝑈 “ 𝐽))) |
141 | 131, 139,
140 | syl2anc 581 |
. . . . . . . . . 10
⊢ ((((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐼)) ∧ ¬ 𝑧 ∈ 𝐽) → (0g‘𝑌) ∈ (𝐾‘(𝑈 “ 𝐽))) |
142 | 138, 141 | eqeltrd 2907 |
. . . . . . . . 9
⊢ ((((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐼)) ∧ ¬ 𝑧 ∈ 𝐽) → ((𝑦‘𝑧)( ·𝑠
‘𝑌)(𝑈‘𝑧)) ∈ (𝐾‘(𝑈 “ 𝐽))) |
143 | 110, 142 | pm2.61dan 849 |
. . . . . . . 8
⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐼)) → ((𝑦‘𝑧)( ·𝑠
‘𝑌)(𝑈‘𝑧)) ∈ (𝐾‘(𝑈 “ 𝐽))) |
144 | 83, 143 | eqeltrd 2907 |
. . . . . . 7
⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐼)) → ((𝑦 ∘𝑓 (
·𝑠 ‘𝑌)𝑈)‘𝑧) ∈ (𝐾‘(𝑈 “ 𝐽))) |
145 | 144 | expr 450 |
. . . . . 6
⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ 𝑦 ∈ 𝐶) → (𝑧 ∈ 𝐼 → ((𝑦 ∘𝑓 (
·𝑠 ‘𝑌)𝑈)‘𝑧) ∈ (𝐾‘(𝑈 “ 𝐽)))) |
146 | 145 | ralrimiv 3175 |
. . . . 5
⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ 𝑦 ∈ 𝐶) → ∀𝑧 ∈ 𝐼 ((𝑦 ∘𝑓 (
·𝑠 ‘𝑌)𝑈)‘𝑧) ∈ (𝐾‘(𝑈 “ 𝐽))) |
147 | | ffnfv 6638 |
. . . . 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)))) |
148 | 76, 146, 147 | sylanbrc 580 |
. . . 4
⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ 𝑦 ∈ 𝐶) → (𝑦 ∘𝑓 (
·𝑠 ‘𝑌)𝑈):𝐼⟶(𝐾‘(𝑈 “ 𝐽))) |
149 | 1, 6, 5 | frlmbasfsupp 20466 |
. . . . . . . 8
⊢ ((𝐼 ∈ 𝑉 ∧ 𝑦 ∈ 𝐵) → 𝑦 finSupp 0 ) |
150 | 149 | fsuppimpd 8552 |
. . . . . . 7
⊢ ((𝐼 ∈ 𝑉 ∧ 𝑦 ∈ 𝐵) → (𝑦 supp 0 ) ∈
Fin) |
151 | 49, 53, 150 | syl2anc 581 |
. . . . . 6
⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ 𝑦 ∈ 𝐶) → (𝑦 supp 0 ) ∈
Fin) |
152 | | dffn2 6281 |
. . . . . . . . 9
⊢ ((𝑦 ∘𝑓 (
·𝑠 ‘𝑌)𝑈) Fn 𝐼 ↔ (𝑦 ∘𝑓 (
·𝑠 ‘𝑌)𝑈):𝐼⟶V) |
153 | 75, 152 | sylib 210 |
. . . . . . . 8
⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ 𝑦 ∈ 𝐵) → (𝑦 ∘𝑓 (
·𝑠 ‘𝑌)𝑈):𝐼⟶V) |
154 | 71 | adantr 474 |
. . . . . . . . . 10
⊢ ((((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ 𝑦 ∈ 𝐵) ∧ 𝑥 ∈ (𝐼 ∖ (𝑦 supp 0 ))) → 𝑦 Fn 𝐼) |
155 | 36 | ad2antrr 719 |
. . . . . . . . . 10
⊢ ((((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ 𝑦 ∈ 𝐵) ∧ 𝑥 ∈ (𝐼 ∖ (𝑦 supp 0 ))) → 𝑈 Fn 𝐼) |
156 | | simpll2 1277 |
. . . . . . . . . 10
⊢ ((((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ 𝑦 ∈ 𝐵) ∧ 𝑥 ∈ (𝐼 ∖ (𝑦 supp 0 ))) → 𝐼 ∈ 𝑉) |
157 | | eldifi 3960 |
. . . . . . . . . . 11
⊢ (𝑥 ∈ (𝐼 ∖ (𝑦 supp 0 )) → 𝑥 ∈ 𝐼) |
158 | 157 | adantl 475 |
. . . . . . . . . 10
⊢ ((((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ 𝑦 ∈ 𝐵) ∧ 𝑥 ∈ (𝐼 ∖ (𝑦 supp 0 ))) → 𝑥 ∈ 𝐼) |
159 | | fnfvof 7172 |
. . . . . . . . . 10
⊢ (((𝑦 Fn 𝐼 ∧ 𝑈 Fn 𝐼) ∧ (𝐼 ∈ 𝑉 ∧ 𝑥 ∈ 𝐼)) → ((𝑦 ∘𝑓 (
·𝑠 ‘𝑌)𝑈)‘𝑥) = ((𝑦‘𝑥)( ·𝑠
‘𝑌)(𝑈‘𝑥))) |
160 | 154, 155,
156, 158, 159 | syl22anc 874 |
. . . . . . . . 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))) |
161 | | ssidd 3850 |
. . . . . . . . . . . 12
⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ 𝑦 ∈ 𝐵) → (𝑦 supp 0 ) ⊆ (𝑦 supp 0 )) |
162 | 122 | a1i 11 |
. . . . . . . . . . . 12
⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ 𝑦 ∈ 𝐵) → 0 ∈ V) |
163 | 70, 161, 73, 162 | suppssr 7592 |
. . . . . . . . . . 11
⊢ ((((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ 𝑦 ∈ 𝐵) ∧ 𝑥 ∈ (𝐼 ∖ (𝑦 supp 0 ))) → (𝑦‘𝑥) = 0 ) |
164 | 127 | ad2antrr 719 |
. . . . . . . . . . 11
⊢ ((((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ 𝑦 ∈ 𝐵) ∧ 𝑥 ∈ (𝐼 ∖ (𝑦 supp 0 ))) → 0 =
(0g‘(Scalar‘𝑌))) |
165 | 163, 164 | eqtrd 2862 |
. . . . . . . . . 10
⊢ ((((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ 𝑦 ∈ 𝐵) ∧ 𝑥 ∈ (𝐼 ∖ (𝑦 supp 0 ))) → (𝑦‘𝑥) = (0g‘(Scalar‘𝑌))) |
166 | 165 | oveq1d 6921 |
. . . . . . . . 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))) |
167 | 3 | ad2antrr 719 |
. . . . . . . . . 10
⊢ ((((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ 𝑦 ∈ 𝐵) ∧ 𝑥 ∈ (𝐼 ∖ (𝑦 supp 0 ))) → 𝑌 ∈ LMod) |
168 | 11 | adantr 474 |
. . . . . . . . . . 11
⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ 𝑦 ∈ 𝐵) → 𝑈:𝐼⟶𝐵) |
169 | | ffvelrn 6607 |
. . . . . . . . . . 11
⊢ ((𝑈:𝐼⟶𝐵 ∧ 𝑥 ∈ 𝐼) → (𝑈‘𝑥) ∈ 𝐵) |
170 | 168, 157,
169 | syl2an 591 |
. . . . . . . . . 10
⊢ ((((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ 𝑦 ∈ 𝐵) ∧ 𝑥 ∈ (𝐼 ∖ (𝑦 supp 0 ))) → (𝑈‘𝑥) ∈ 𝐵) |
171 | 5, 105, 54, 135, 57 | lmod0vs 19253 |
. . . . . . . . . 10
⊢ ((𝑌 ∈ LMod ∧ (𝑈‘𝑥) ∈ 𝐵) →
((0g‘(Scalar‘𝑌))( ·𝑠
‘𝑌)(𝑈‘𝑥)) = (0g‘𝑌)) |
172 | 167, 170,
171 | syl2anc 581 |
. . . . . . . . 9
⊢ ((((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ 𝑦 ∈ 𝐵) ∧ 𝑥 ∈ (𝐼 ∖ (𝑦 supp 0 ))) →
((0g‘(Scalar‘𝑌))( ·𝑠
‘𝑌)(𝑈‘𝑥)) = (0g‘𝑌)) |
173 | 160, 166,
172 | 3eqtrd 2866 |
. . . . . . . 8
⊢ ((((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ 𝑦 ∈ 𝐵) ∧ 𝑥 ∈ (𝐼 ∖ (𝑦 supp 0 ))) → ((𝑦 ∘𝑓 (
·𝑠 ‘𝑌)𝑈)‘𝑥) = (0g‘𝑌)) |
174 | 153, 173 | suppss 7591 |
. . . . . . 7
⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ 𝑦 ∈ 𝐵) → ((𝑦 ∘𝑓 (
·𝑠 ‘𝑌)𝑈) supp (0g‘𝑌)) ⊆ (𝑦 supp 0 )) |
175 | 53, 174 | syldan 587 |
. . . . . 6
⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ 𝑦 ∈ 𝐶) → ((𝑦 ∘𝑓 (
·𝑠 ‘𝑌)𝑈) supp (0g‘𝑌)) ⊆ (𝑦 supp 0 )) |
176 | | ssfi 8450 |
. . . . . 6
⊢ (((𝑦 supp 0 ) ∈ Fin ∧ ((𝑦 ∘𝑓 (
·𝑠 ‘𝑌)𝑈) supp (0g‘𝑌)) ⊆ (𝑦 supp 0 )) → ((𝑦 ∘𝑓 (
·𝑠 ‘𝑌)𝑈) supp (0g‘𝑌)) ∈ Fin) |
177 | 151, 175,
176 | syl2anc 581 |
. . . . 5
⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ 𝑦 ∈ 𝐶) → ((𝑦 ∘𝑓 (
·𝑠 ‘𝑌)𝑈) supp (0g‘𝑌)) ∈ Fin) |
178 | | simp2 1173 |
. . . . . . . . . 10
⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) → 𝐼 ∈ 𝑉) |
179 | 1, 17, 5 | frlmbasmap 20467 |
. . . . . . . . . 10
⊢ ((𝐼 ∈ 𝑉 ∧ 𝑦 ∈ 𝐵) → 𝑦 ∈ ((Base‘𝑅) ↑𝑚 𝐼)) |
180 | 178, 86, 179 | syl2an 591 |
. . . . . . . . 9
⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ 𝑦 ∈ 𝐶) → 𝑦 ∈ ((Base‘𝑅) ↑𝑚 𝐼)) |
181 | | elmapfn 8146 |
. . . . . . . . 9
⊢ (𝑦 ∈ ((Base‘𝑅) ↑𝑚
𝐼) → 𝑦 Fn 𝐼) |
182 | 180, 181 | syl 17 |
. . . . . . . 8
⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ 𝑦 ∈ 𝐶) → 𝑦 Fn 𝐼) |
183 | 11 | adantr 474 |
. . . . . . . . 9
⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ 𝑦 ∈ 𝐶) → 𝑈:𝐼⟶𝐵) |
184 | 183 | ffnd 6280 |
. . . . . . . 8
⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ 𝑦 ∈ 𝐶) → 𝑈 Fn 𝐼) |
185 | 182, 184,
49, 49, 74 | offn 7169 |
. . . . . . 7
⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ 𝑦 ∈ 𝐶) → (𝑦 ∘𝑓 (
·𝑠 ‘𝑌)𝑈) Fn 𝐼) |
186 | | fnfun 6222 |
. . . . . . 7
⊢ ((𝑦 ∘𝑓 (
·𝑠 ‘𝑌)𝑈) Fn 𝐼 → Fun (𝑦 ∘𝑓 (
·𝑠 ‘𝑌)𝑈)) |
187 | 185, 186 | syl 17 |
. . . . . 6
⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ 𝑦 ∈ 𝐶) → Fun (𝑦 ∘𝑓 (
·𝑠 ‘𝑌)𝑈)) |
188 | | ovexd 6940 |
. . . . . 6
⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ 𝑦 ∈ 𝐶) → (𝑦 ∘𝑓 (
·𝑠 ‘𝑌)𝑈) ∈ V) |
189 | | fvex 6447 |
. . . . . . 7
⊢
(0g‘𝑌) ∈ V |
190 | 189 | a1i 11 |
. . . . . 6
⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ 𝑦 ∈ 𝐶) → (0g‘𝑌) ∈ V) |
191 | | funisfsupp 8550 |
. . . . . 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)) |
192 | 187, 188,
190, 191 | syl3anc 1496 |
. . . . 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)) |
193 | 177, 192 | mpbird 249 |
. . . 4
⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ 𝑦 ∈ 𝐶) → (𝑦 ∘𝑓 (
·𝑠 ‘𝑌)𝑈) finSupp (0g‘𝑌)) |
194 | 57, 60, 49, 68, 148, 193 | gsumsubgcl 18674 |
. . 3
⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ 𝑦 ∈ 𝐶) → (𝑌 Σg (𝑦 ∘𝑓 (
·𝑠 ‘𝑌)𝑈)) ∈ (𝐾‘(𝑈 “ 𝐽))) |
195 | 56, 194 | eqeltrd 2907 |
. 2
⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ 𝑦 ∈ 𝐶) → 𝑦 ∈ (𝐾‘(𝑈 “ 𝐽))) |
196 | 47, 195 | eqelssd 3848 |
1
⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) → (𝐾‘(𝑈 “ 𝐽)) = 𝐶) |