Proof of Theorem frlmphllem
Step | Hyp | Ref
| Expression |
1 | | frlmphl.i |
. . . . . . . . 9
⊢ (𝜑 → 𝐼 ∈ 𝑊) |
2 | 1 | 3ad2ant1 1075 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑔 ∈ 𝑉 ∧ ℎ ∈ 𝑉) → 𝐼 ∈ 𝑊) |
3 | | simp2 1055 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑔 ∈ 𝑉 ∧ ℎ ∈ 𝑉) → 𝑔 ∈ 𝑉) |
4 | | frlmphl.y |
. . . . . . . . 9
⊢ 𝑌 = (𝑅 freeLMod 𝐼) |
5 | | frlmphl.b |
. . . . . . . . 9
⊢ 𝐵 = (Base‘𝑅) |
6 | | frlmphl.v |
. . . . . . . . 9
⊢ 𝑉 = (Base‘𝑌) |
7 | 4, 5, 6 | frlmbasmap 19922 |
. . . . . . . 8
⊢ ((𝐼 ∈ 𝑊 ∧ 𝑔 ∈ 𝑉) → 𝑔 ∈ (𝐵 ↑𝑚 𝐼)) |
8 | 2, 3, 7 | syl2anc 691 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑔 ∈ 𝑉 ∧ ℎ ∈ 𝑉) → 𝑔 ∈ (𝐵 ↑𝑚 𝐼)) |
9 | | elmapi 7765 |
. . . . . . 7
⊢ (𝑔 ∈ (𝐵 ↑𝑚 𝐼) → 𝑔:𝐼⟶𝐵) |
10 | 8, 9 | syl 17 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑔 ∈ 𝑉 ∧ ℎ ∈ 𝑉) → 𝑔:𝐼⟶𝐵) |
11 | | ffn 5958 |
. . . . . 6
⊢ (𝑔:𝐼⟶𝐵 → 𝑔 Fn 𝐼) |
12 | 10, 11 | syl 17 |
. . . . 5
⊢ ((𝜑 ∧ 𝑔 ∈ 𝑉 ∧ ℎ ∈ 𝑉) → 𝑔 Fn 𝐼) |
13 | | simp3 1056 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑔 ∈ 𝑉 ∧ ℎ ∈ 𝑉) → ℎ ∈ 𝑉) |
14 | 4, 5, 6 | frlmbasmap 19922 |
. . . . . . . 8
⊢ ((𝐼 ∈ 𝑊 ∧ ℎ ∈ 𝑉) → ℎ ∈ (𝐵 ↑𝑚 𝐼)) |
15 | 2, 13, 14 | syl2anc 691 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑔 ∈ 𝑉 ∧ ℎ ∈ 𝑉) → ℎ ∈ (𝐵 ↑𝑚 𝐼)) |
16 | | elmapi 7765 |
. . . . . . 7
⊢ (ℎ ∈ (𝐵 ↑𝑚 𝐼) → ℎ:𝐼⟶𝐵) |
17 | 15, 16 | syl 17 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑔 ∈ 𝑉 ∧ ℎ ∈ 𝑉) → ℎ:𝐼⟶𝐵) |
18 | | ffn 5958 |
. . . . . 6
⊢ (ℎ:𝐼⟶𝐵 → ℎ Fn 𝐼) |
19 | 17, 18 | syl 17 |
. . . . 5
⊢ ((𝜑 ∧ 𝑔 ∈ 𝑉 ∧ ℎ ∈ 𝑉) → ℎ Fn 𝐼) |
20 | | inidm 3784 |
. . . . 5
⊢ (𝐼 ∩ 𝐼) = 𝐼 |
21 | | eqidd 2611 |
. . . . 5
⊢ (((𝜑 ∧ 𝑔 ∈ 𝑉 ∧ ℎ ∈ 𝑉) ∧ 𝑥 ∈ 𝐼) → (𝑔‘𝑥) = (𝑔‘𝑥)) |
22 | | eqidd 2611 |
. . . . 5
⊢ (((𝜑 ∧ 𝑔 ∈ 𝑉 ∧ ℎ ∈ 𝑉) ∧ 𝑥 ∈ 𝐼) → (ℎ‘𝑥) = (ℎ‘𝑥)) |
23 | 12, 19, 2, 2, 20, 21, 22 | offval 6802 |
. . . 4
⊢ ((𝜑 ∧ 𝑔 ∈ 𝑉 ∧ ℎ ∈ 𝑉) → (𝑔 ∘𝑓 · ℎ) = (𝑥 ∈ 𝐼 ↦ ((𝑔‘𝑥) · (ℎ‘𝑥)))) |
24 | 23 | oveq1d 6564 |
. . 3
⊢ ((𝜑 ∧ 𝑔 ∈ 𝑉 ∧ ℎ ∈ 𝑉) → ((𝑔 ∘𝑓 · ℎ) supp 0 ) = ((𝑥 ∈ 𝐼 ↦ ((𝑔‘𝑥) · (ℎ‘𝑥))) supp 0 )) |
25 | | ovex 6577 |
. . . . 5
⊢ (𝑔 ∘𝑓
·
ℎ) ∈ V |
26 | 25 | a1i 11 |
. . . 4
⊢ ((𝜑 ∧ 𝑔 ∈ 𝑉 ∧ ℎ ∈ 𝑉) → (𝑔 ∘𝑓 · ℎ) ∈ V) |
27 | | funmpt 5840 |
. . . . . . 7
⊢ Fun
(𝑥 ∈ 𝐼 ↦ ((𝑔‘𝑥) · (ℎ‘𝑥))) |
28 | 27 | a1i 11 |
. . . . . 6
⊢ ((𝑔 ∘𝑓
·
ℎ) = (𝑥 ∈ 𝐼 ↦ ((𝑔‘𝑥) · (ℎ‘𝑥))) → Fun (𝑥 ∈ 𝐼 ↦ ((𝑔‘𝑥) · (ℎ‘𝑥)))) |
29 | | funeq 5823 |
. . . . . 6
⊢ ((𝑔 ∘𝑓
·
ℎ) = (𝑥 ∈ 𝐼 ↦ ((𝑔‘𝑥) · (ℎ‘𝑥))) → (Fun (𝑔 ∘𝑓 · ℎ) ↔ Fun (𝑥 ∈ 𝐼 ↦ ((𝑔‘𝑥) · (ℎ‘𝑥))))) |
30 | 28, 29 | mpbird 246 |
. . . . 5
⊢ ((𝑔 ∘𝑓
·
ℎ) = (𝑥 ∈ 𝐼 ↦ ((𝑔‘𝑥) · (ℎ‘𝑥))) → Fun (𝑔 ∘𝑓 · ℎ)) |
31 | 23, 30 | syl 17 |
. . . 4
⊢ ((𝜑 ∧ 𝑔 ∈ 𝑉 ∧ ℎ ∈ 𝑉) → Fun (𝑔 ∘𝑓 · ℎ)) |
32 | | frlmphl.0 |
. . . . . 6
⊢ 0 =
(0g‘𝑅) |
33 | 4, 32, 6 | frlmbasfsupp 19921 |
. . . . 5
⊢ ((𝐼 ∈ 𝑊 ∧ 𝑔 ∈ 𝑉) → 𝑔 finSupp 0 ) |
34 | 2, 3, 33 | syl2anc 691 |
. . . 4
⊢ ((𝜑 ∧ 𝑔 ∈ 𝑉 ∧ ℎ ∈ 𝑉) → 𝑔 finSupp 0 ) |
35 | | frlmphl.f |
. . . . . . . . . 10
⊢ (𝜑 → 𝑅 ∈ Field) |
36 | | isfld 18579 |
. . . . . . . . . 10
⊢ (𝑅 ∈ Field ↔ (𝑅 ∈ DivRing ∧ 𝑅 ∈ CRing)) |
37 | 35, 36 | sylib 207 |
. . . . . . . . 9
⊢ (𝜑 → (𝑅 ∈ DivRing ∧ 𝑅 ∈ CRing)) |
38 | 37 | simpld 474 |
. . . . . . . 8
⊢ (𝜑 → 𝑅 ∈ DivRing) |
39 | | drngring 18577 |
. . . . . . . 8
⊢ (𝑅 ∈ DivRing → 𝑅 ∈ Ring) |
40 | 38, 39 | syl 17 |
. . . . . . 7
⊢ (𝜑 → 𝑅 ∈ Ring) |
41 | 40 | 3ad2ant1 1075 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑔 ∈ 𝑉 ∧ ℎ ∈ 𝑉) → 𝑅 ∈ Ring) |
42 | 5, 32 | ring0cl 18392 |
. . . . . 6
⊢ (𝑅 ∈ Ring → 0 ∈ 𝐵) |
43 | 41, 42 | syl 17 |
. . . . 5
⊢ ((𝜑 ∧ 𝑔 ∈ 𝑉 ∧ ℎ ∈ 𝑉) → 0 ∈ 𝐵) |
44 | | frlmphl.t |
. . . . . . 7
⊢ · =
(.r‘𝑅) |
45 | 5, 44, 32 | ringlz 18410 |
. . . . . 6
⊢ ((𝑅 ∈ Ring ∧ 𝑥 ∈ 𝐵) → ( 0 · 𝑥) = 0 ) |
46 | 41, 45 | sylan 487 |
. . . . 5
⊢ (((𝜑 ∧ 𝑔 ∈ 𝑉 ∧ ℎ ∈ 𝑉) ∧ 𝑥 ∈ 𝐵) → ( 0 · 𝑥) = 0 ) |
47 | 2, 43, 10, 17, 46 | suppofss1d 7219 |
. . . 4
⊢ ((𝜑 ∧ 𝑔 ∈ 𝑉 ∧ ℎ ∈ 𝑉) → ((𝑔 ∘𝑓 · ℎ) supp 0 ) ⊆ (𝑔 supp 0 )) |
48 | | fsuppsssupp 8174 |
. . . . 5
⊢ ((((𝑔 ∘𝑓
·
ℎ) ∈ V ∧ Fun (𝑔 ∘𝑓
·
ℎ)) ∧ (𝑔 finSupp 0 ∧ ((𝑔 ∘𝑓 · ℎ) supp 0 ) ⊆ (𝑔 supp 0 ))) → (𝑔 ∘𝑓
·
ℎ) finSupp 0
) |
49 | 48 | fsuppimpd 8165 |
. . . 4
⊢ ((((𝑔 ∘𝑓
·
ℎ) ∈ V ∧ Fun (𝑔 ∘𝑓
·
ℎ)) ∧ (𝑔 finSupp 0 ∧ ((𝑔 ∘𝑓 · ℎ) supp 0 ) ⊆ (𝑔 supp 0 ))) → ((𝑔 ∘𝑓
·
ℎ) supp 0 ) ∈
Fin) |
50 | 26, 31, 34, 47, 49 | syl22anc 1319 |
. . 3
⊢ ((𝜑 ∧ 𝑔 ∈ 𝑉 ∧ ℎ ∈ 𝑉) → ((𝑔 ∘𝑓 · ℎ) supp 0 ) ∈
Fin) |
51 | 24, 50 | eqeltrrd 2689 |
. 2
⊢ ((𝜑 ∧ 𝑔 ∈ 𝑉 ∧ ℎ ∈ 𝑉) → ((𝑥 ∈ 𝐼 ↦ ((𝑔‘𝑥) · (ℎ‘𝑥))) supp 0 ) ∈
Fin) |
52 | 27 | a1i 11 |
. . 3
⊢ ((𝜑 ∧ 𝑔 ∈ 𝑉 ∧ ℎ ∈ 𝑉) → Fun (𝑥 ∈ 𝐼 ↦ ((𝑔‘𝑥) · (ℎ‘𝑥)))) |
53 | | mptexg 6389 |
. . . 4
⊢ (𝐼 ∈ 𝑊 → (𝑥 ∈ 𝐼 ↦ ((𝑔‘𝑥) · (ℎ‘𝑥))) ∈ V) |
54 | 2, 53 | syl 17 |
. . 3
⊢ ((𝜑 ∧ 𝑔 ∈ 𝑉 ∧ ℎ ∈ 𝑉) → (𝑥 ∈ 𝐼 ↦ ((𝑔‘𝑥) · (ℎ‘𝑥))) ∈ V) |
55 | | fvex 6113 |
. . . . 5
⊢
(0g‘𝑅) ∈ V |
56 | 32, 55 | eqeltri 2684 |
. . . 4
⊢ 0 ∈
V |
57 | 56 | a1i 11 |
. . 3
⊢ ((𝜑 ∧ 𝑔 ∈ 𝑉 ∧ ℎ ∈ 𝑉) → 0 ∈ V) |
58 | | funisfsupp 8163 |
. . 3
⊢ ((Fun
(𝑥 ∈ 𝐼 ↦ ((𝑔‘𝑥) · (ℎ‘𝑥))) ∧ (𝑥 ∈ 𝐼 ↦ ((𝑔‘𝑥) · (ℎ‘𝑥))) ∈ V ∧ 0 ∈ V) → ((𝑥 ∈ 𝐼 ↦ ((𝑔‘𝑥) · (ℎ‘𝑥))) finSupp 0 ↔ ((𝑥 ∈ 𝐼 ↦ ((𝑔‘𝑥) · (ℎ‘𝑥))) supp 0 ) ∈
Fin)) |
59 | 52, 54, 57, 58 | syl3anc 1318 |
. 2
⊢ ((𝜑 ∧ 𝑔 ∈ 𝑉 ∧ ℎ ∈ 𝑉) → ((𝑥 ∈ 𝐼 ↦ ((𝑔‘𝑥) · (ℎ‘𝑥))) finSupp 0 ↔ ((𝑥 ∈ 𝐼 ↦ ((𝑔‘𝑥) · (ℎ‘𝑥))) supp 0 ) ∈
Fin)) |
60 | 51, 59 | mpbird 246 |
1
⊢ ((𝜑 ∧ 𝑔 ∈ 𝑉 ∧ ℎ ∈ 𝑉) → (𝑥 ∈ 𝐼 ↦ ((𝑔‘𝑥) · (ℎ‘𝑥))) finSupp 0 ) |