Proof of Theorem frlmphllem
Step | Hyp | Ref
| Expression |
1 | | frlmphl.i |
. . . . . . . . 9
⊢ (𝜑 → 𝐼 ∈ 𝑊) |
2 | 1 | 3ad2ant1 1169 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑔 ∈ 𝑉 ∧ ℎ ∈ 𝑉) → 𝐼 ∈ 𝑊) |
3 | | simp2 1173 |
. . . . . . . 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 20467 |
. . . . . . . 8
⊢ ((𝐼 ∈ 𝑊 ∧ 𝑔 ∈ 𝑉) → 𝑔 ∈ (𝐵 ↑𝑚 𝐼)) |
8 | 2, 3, 7 | syl2anc 581 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑔 ∈ 𝑉 ∧ ℎ ∈ 𝑉) → 𝑔 ∈ (𝐵 ↑𝑚 𝐼)) |
9 | | elmapi 8145 |
. . . . . . 7
⊢ (𝑔 ∈ (𝐵 ↑𝑚 𝐼) → 𝑔:𝐼⟶𝐵) |
10 | 8, 9 | syl 17 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑔 ∈ 𝑉 ∧ ℎ ∈ 𝑉) → 𝑔:𝐼⟶𝐵) |
11 | 10 | ffnd 6280 |
. . . . 5
⊢ ((𝜑 ∧ 𝑔 ∈ 𝑉 ∧ ℎ ∈ 𝑉) → 𝑔 Fn 𝐼) |
12 | | simp3 1174 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑔 ∈ 𝑉 ∧ ℎ ∈ 𝑉) → ℎ ∈ 𝑉) |
13 | 4, 5, 6 | frlmbasmap 20467 |
. . . . . . . 8
⊢ ((𝐼 ∈ 𝑊 ∧ ℎ ∈ 𝑉) → ℎ ∈ (𝐵 ↑𝑚 𝐼)) |
14 | 2, 12, 13 | syl2anc 581 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑔 ∈ 𝑉 ∧ ℎ ∈ 𝑉) → ℎ ∈ (𝐵 ↑𝑚 𝐼)) |
15 | | elmapi 8145 |
. . . . . . 7
⊢ (ℎ ∈ (𝐵 ↑𝑚 𝐼) → ℎ:𝐼⟶𝐵) |
16 | 14, 15 | syl 17 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑔 ∈ 𝑉 ∧ ℎ ∈ 𝑉) → ℎ:𝐼⟶𝐵) |
17 | 16 | ffnd 6280 |
. . . . 5
⊢ ((𝜑 ∧ 𝑔 ∈ 𝑉 ∧ ℎ ∈ 𝑉) → ℎ Fn 𝐼) |
18 | | inidm 4048 |
. . . . 5
⊢ (𝐼 ∩ 𝐼) = 𝐼 |
19 | | eqidd 2827 |
. . . . 5
⊢ (((𝜑 ∧ 𝑔 ∈ 𝑉 ∧ ℎ ∈ 𝑉) ∧ 𝑥 ∈ 𝐼) → (𝑔‘𝑥) = (𝑔‘𝑥)) |
20 | | eqidd 2827 |
. . . . 5
⊢ (((𝜑 ∧ 𝑔 ∈ 𝑉 ∧ ℎ ∈ 𝑉) ∧ 𝑥 ∈ 𝐼) → (ℎ‘𝑥) = (ℎ‘𝑥)) |
21 | 11, 17, 2, 2, 18, 19, 20 | offval 7165 |
. . . 4
⊢ ((𝜑 ∧ 𝑔 ∈ 𝑉 ∧ ℎ ∈ 𝑉) → (𝑔 ∘𝑓 · ℎ) = (𝑥 ∈ 𝐼 ↦ ((𝑔‘𝑥) · (ℎ‘𝑥)))) |
22 | 21 | oveq1d 6921 |
. . 3
⊢ ((𝜑 ∧ 𝑔 ∈ 𝑉 ∧ ℎ ∈ 𝑉) → ((𝑔 ∘𝑓 · ℎ) supp 0 ) = ((𝑥 ∈ 𝐼 ↦ ((𝑔‘𝑥) · (ℎ‘𝑥))) supp 0 )) |
23 | | ovexd 6940 |
. . . 4
⊢ ((𝜑 ∧ 𝑔 ∈ 𝑉 ∧ ℎ ∈ 𝑉) → (𝑔 ∘𝑓 · ℎ) ∈ V) |
24 | | funmpt 6162 |
. . . . . . 7
⊢ Fun
(𝑥 ∈ 𝐼 ↦ ((𝑔‘𝑥) · (ℎ‘𝑥))) |
25 | 24 | a1i 11 |
. . . . . 6
⊢ ((𝑔 ∘𝑓
·
ℎ) = (𝑥 ∈ 𝐼 ↦ ((𝑔‘𝑥) · (ℎ‘𝑥))) → Fun (𝑥 ∈ 𝐼 ↦ ((𝑔‘𝑥) · (ℎ‘𝑥)))) |
26 | | funeq 6144 |
. . . . . 6
⊢ ((𝑔 ∘𝑓
·
ℎ) = (𝑥 ∈ 𝐼 ↦ ((𝑔‘𝑥) · (ℎ‘𝑥))) → (Fun (𝑔 ∘𝑓 · ℎ) ↔ Fun (𝑥 ∈ 𝐼 ↦ ((𝑔‘𝑥) · (ℎ‘𝑥))))) |
27 | 25, 26 | mpbird 249 |
. . . . 5
⊢ ((𝑔 ∘𝑓
·
ℎ) = (𝑥 ∈ 𝐼 ↦ ((𝑔‘𝑥) · (ℎ‘𝑥))) → Fun (𝑔 ∘𝑓 · ℎ)) |
28 | 21, 27 | syl 17 |
. . . 4
⊢ ((𝜑 ∧ 𝑔 ∈ 𝑉 ∧ ℎ ∈ 𝑉) → Fun (𝑔 ∘𝑓 · ℎ)) |
29 | | frlmphl.0 |
. . . . . 6
⊢ 0 =
(0g‘𝑅) |
30 | 4, 29, 6 | frlmbasfsupp 20466 |
. . . . 5
⊢ ((𝐼 ∈ 𝑊 ∧ 𝑔 ∈ 𝑉) → 𝑔 finSupp 0 ) |
31 | 2, 3, 30 | syl2anc 581 |
. . . 4
⊢ ((𝜑 ∧ 𝑔 ∈ 𝑉 ∧ ℎ ∈ 𝑉) → 𝑔 finSupp 0 ) |
32 | | frlmphl.f |
. . . . . . . . . 10
⊢ (𝜑 → 𝑅 ∈ Field) |
33 | | isfld 19113 |
. . . . . . . . . 10
⊢ (𝑅 ∈ Field ↔ (𝑅 ∈ DivRing ∧ 𝑅 ∈ CRing)) |
34 | 32, 33 | sylib 210 |
. . . . . . . . 9
⊢ (𝜑 → (𝑅 ∈ DivRing ∧ 𝑅 ∈ CRing)) |
35 | 34 | simpld 490 |
. . . . . . . 8
⊢ (𝜑 → 𝑅 ∈ DivRing) |
36 | | drngring 19111 |
. . . . . . . 8
⊢ (𝑅 ∈ DivRing → 𝑅 ∈ Ring) |
37 | 35, 36 | syl 17 |
. . . . . . 7
⊢ (𝜑 → 𝑅 ∈ Ring) |
38 | 37 | 3ad2ant1 1169 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑔 ∈ 𝑉 ∧ ℎ ∈ 𝑉) → 𝑅 ∈ Ring) |
39 | 5, 29 | ring0cl 18924 |
. . . . . 6
⊢ (𝑅 ∈ Ring → 0 ∈ 𝐵) |
40 | 38, 39 | syl 17 |
. . . . 5
⊢ ((𝜑 ∧ 𝑔 ∈ 𝑉 ∧ ℎ ∈ 𝑉) → 0 ∈ 𝐵) |
41 | | frlmphl.t |
. . . . . . 7
⊢ · =
(.r‘𝑅) |
42 | 5, 41, 29 | ringlz 18942 |
. . . . . 6
⊢ ((𝑅 ∈ Ring ∧ 𝑥 ∈ 𝐵) → ( 0 · 𝑥) = 0 ) |
43 | 38, 42 | sylan 577 |
. . . . 5
⊢ (((𝜑 ∧ 𝑔 ∈ 𝑉 ∧ ℎ ∈ 𝑉) ∧ 𝑥 ∈ 𝐵) → ( 0 · 𝑥) = 0 ) |
44 | 2, 40, 10, 16, 43 | suppofss1d 7598 |
. . . 4
⊢ ((𝜑 ∧ 𝑔 ∈ 𝑉 ∧ ℎ ∈ 𝑉) → ((𝑔 ∘𝑓 · ℎ) supp 0 ) ⊆ (𝑔 supp 0 )) |
45 | | fsuppsssupp 8561 |
. . . . 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
) |
46 | 45 | fsuppimpd 8552 |
. . . 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) |
47 | 23, 28, 31, 44, 46 | syl22anc 874 |
. . 3
⊢ ((𝜑 ∧ 𝑔 ∈ 𝑉 ∧ ℎ ∈ 𝑉) → ((𝑔 ∘𝑓 · ℎ) supp 0 ) ∈
Fin) |
48 | 22, 47 | eqeltrrd 2908 |
. 2
⊢ ((𝜑 ∧ 𝑔 ∈ 𝑉 ∧ ℎ ∈ 𝑉) → ((𝑥 ∈ 𝐼 ↦ ((𝑔‘𝑥) · (ℎ‘𝑥))) supp 0 ) ∈
Fin) |
49 | 24 | a1i 11 |
. . 3
⊢ ((𝜑 ∧ 𝑔 ∈ 𝑉 ∧ ℎ ∈ 𝑉) → Fun (𝑥 ∈ 𝐼 ↦ ((𝑔‘𝑥) · (ℎ‘𝑥)))) |
50 | | mptexg 6741 |
. . . 4
⊢ (𝐼 ∈ 𝑊 → (𝑥 ∈ 𝐼 ↦ ((𝑔‘𝑥) · (ℎ‘𝑥))) ∈ V) |
51 | 2, 50 | syl 17 |
. . 3
⊢ ((𝜑 ∧ 𝑔 ∈ 𝑉 ∧ ℎ ∈ 𝑉) → (𝑥 ∈ 𝐼 ↦ ((𝑔‘𝑥) · (ℎ‘𝑥))) ∈ V) |
52 | 29 | fvexi 6448 |
. . . 4
⊢ 0 ∈
V |
53 | 52 | a1i 11 |
. . 3
⊢ ((𝜑 ∧ 𝑔 ∈ 𝑉 ∧ ℎ ∈ 𝑉) → 0 ∈ V) |
54 | | funisfsupp 8550 |
. . 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)) |
55 | 49, 51, 53, 54 | syl3anc 1496 |
. 2
⊢ ((𝜑 ∧ 𝑔 ∈ 𝑉 ∧ ℎ ∈ 𝑉) → ((𝑥 ∈ 𝐼 ↦ ((𝑔‘𝑥) · (ℎ‘𝑥))) finSupp 0 ↔ ((𝑥 ∈ 𝐼 ↦ ((𝑔‘𝑥) · (ℎ‘𝑥))) supp 0 ) ∈
Fin)) |
56 | 48, 55 | mpbird 249 |
1
⊢ ((𝜑 ∧ 𝑔 ∈ 𝑉 ∧ ℎ ∈ 𝑉) → (𝑥 ∈ 𝐼 ↦ ((𝑔‘𝑥) · (ℎ‘𝑥))) finSupp 0 ) |