Step | Hyp | Ref
| Expression |
1 | | imassrn 5940 |
. . . 4
⊢ (𝑇 “ 𝐴) ⊆ ran 𝑇 |
2 | | rnelsh.1 |
. . . . . 6
⊢ 𝑇 ∈ LinOp |
3 | 2 | lnopfi 30050 |
. . . . 5
⊢ 𝑇: ℋ⟶
ℋ |
4 | | frn 6552 |
. . . . 5
⊢ (𝑇: ℋ⟶ ℋ →
ran 𝑇 ⊆
ℋ) |
5 | 3, 4 | ax-mp 5 |
. . . 4
⊢ ran 𝑇 ⊆
ℋ |
6 | 1, 5 | sstri 3910 |
. . 3
⊢ (𝑇 “ 𝐴) ⊆ ℋ |
7 | 2 | lnop0i 30051 |
. . . 4
⊢ (𝑇‘0ℎ) =
0ℎ |
8 | | imaelsh.2 |
. . . . . 6
⊢ 𝐴 ∈
Sℋ |
9 | | sh0 29297 |
. . . . . 6
⊢ (𝐴 ∈
Sℋ → 0ℎ ∈ 𝐴) |
10 | 8, 9 | ax-mp 5 |
. . . . 5
⊢
0ℎ ∈ 𝐴 |
11 | | ffun 6548 |
. . . . . . 7
⊢ (𝑇: ℋ⟶ ℋ →
Fun 𝑇) |
12 | 3, 11 | ax-mp 5 |
. . . . . 6
⊢ Fun 𝑇 |
13 | 8 | shssii 29294 |
. . . . . . 7
⊢ 𝐴 ⊆
ℋ |
14 | 3 | fdmi 6557 |
. . . . . . 7
⊢ dom 𝑇 = ℋ |
15 | 13, 14 | sseqtrri 3938 |
. . . . . 6
⊢ 𝐴 ⊆ dom 𝑇 |
16 | | funfvima2 7047 |
. . . . . 6
⊢ ((Fun
𝑇 ∧ 𝐴 ⊆ dom 𝑇) → (0ℎ ∈ 𝐴 → (𝑇‘0ℎ) ∈ (𝑇 “ 𝐴))) |
17 | 12, 15, 16 | mp2an 692 |
. . . . 5
⊢
(0ℎ ∈ 𝐴 → (𝑇‘0ℎ) ∈ (𝑇 “ 𝐴)) |
18 | 10, 17 | ax-mp 5 |
. . . 4
⊢ (𝑇‘0ℎ)
∈ (𝑇 “ 𝐴) |
19 | 7, 18 | eqeltrri 2835 |
. . 3
⊢
0ℎ ∈ (𝑇 “ 𝐴) |
20 | 6, 19 | pm3.2i 474 |
. 2
⊢ ((𝑇 “ 𝐴) ⊆ ℋ ∧ 0ℎ
∈ (𝑇 “ 𝐴)) |
21 | | ffn 6545 |
. . . . . 6
⊢ (𝑇: ℋ⟶ ℋ →
𝑇 Fn
ℋ) |
22 | 3, 21 | ax-mp 5 |
. . . . 5
⊢ 𝑇 Fn ℋ |
23 | | oveq1 7220 |
. . . . . . . 8
⊢ (𝑢 = (𝑇‘𝑥) → (𝑢 +ℎ 𝑣) = ((𝑇‘𝑥) +ℎ 𝑣)) |
24 | 23 | eleq1d 2822 |
. . . . . . 7
⊢ (𝑢 = (𝑇‘𝑥) → ((𝑢 +ℎ 𝑣) ∈ (𝑇 “ 𝐴) ↔ ((𝑇‘𝑥) +ℎ 𝑣) ∈ (𝑇 “ 𝐴))) |
25 | 24 | ralbidv 3118 |
. . . . . 6
⊢ (𝑢 = (𝑇‘𝑥) → (∀𝑣 ∈ (𝑇 “ 𝐴)(𝑢 +ℎ 𝑣) ∈ (𝑇 “ 𝐴) ↔ ∀𝑣 ∈ (𝑇 “ 𝐴)((𝑇‘𝑥) +ℎ 𝑣) ∈ (𝑇 “ 𝐴))) |
26 | 25 | ralima 7054 |
. . . . 5
⊢ ((𝑇 Fn ℋ ∧ 𝐴 ⊆ ℋ) →
(∀𝑢 ∈ (𝑇 “ 𝐴)∀𝑣 ∈ (𝑇 “ 𝐴)(𝑢 +ℎ 𝑣) ∈ (𝑇 “ 𝐴) ↔ ∀𝑥 ∈ 𝐴 ∀𝑣 ∈ (𝑇 “ 𝐴)((𝑇‘𝑥) +ℎ 𝑣) ∈ (𝑇 “ 𝐴))) |
27 | 22, 13, 26 | mp2an 692 |
. . . 4
⊢
(∀𝑢 ∈
(𝑇 “ 𝐴)∀𝑣 ∈ (𝑇 “ 𝐴)(𝑢 +ℎ 𝑣) ∈ (𝑇 “ 𝐴) ↔ ∀𝑥 ∈ 𝐴 ∀𝑣 ∈ (𝑇 “ 𝐴)((𝑇‘𝑥) +ℎ 𝑣) ∈ (𝑇 “ 𝐴)) |
28 | 8 | sheli 29295 |
. . . . . . . 8
⊢ (𝑥 ∈ 𝐴 → 𝑥 ∈ ℋ) |
29 | 8 | sheli 29295 |
. . . . . . . 8
⊢ (𝑦 ∈ 𝐴 → 𝑦 ∈ ℋ) |
30 | 2 | lnopaddi 30052 |
. . . . . . . 8
⊢ ((𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ) → (𝑇‘(𝑥 +ℎ 𝑦)) = ((𝑇‘𝑥) +ℎ (𝑇‘𝑦))) |
31 | 28, 29, 30 | syl2an 599 |
. . . . . . 7
⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → (𝑇‘(𝑥 +ℎ 𝑦)) = ((𝑇‘𝑥) +ℎ (𝑇‘𝑦))) |
32 | | shaddcl 29298 |
. . . . . . . . 9
⊢ ((𝐴 ∈
Sℋ ∧ 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → (𝑥 +ℎ 𝑦) ∈ 𝐴) |
33 | 8, 32 | mp3an1 1450 |
. . . . . . . 8
⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → (𝑥 +ℎ 𝑦) ∈ 𝐴) |
34 | | funfvima2 7047 |
. . . . . . . . 9
⊢ ((Fun
𝑇 ∧ 𝐴 ⊆ dom 𝑇) → ((𝑥 +ℎ 𝑦) ∈ 𝐴 → (𝑇‘(𝑥 +ℎ 𝑦)) ∈ (𝑇 “ 𝐴))) |
35 | 12, 15, 34 | mp2an 692 |
. . . . . . . 8
⊢ ((𝑥 +ℎ 𝑦) ∈ 𝐴 → (𝑇‘(𝑥 +ℎ 𝑦)) ∈ (𝑇 “ 𝐴)) |
36 | 33, 35 | syl 17 |
. . . . . . 7
⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → (𝑇‘(𝑥 +ℎ 𝑦)) ∈ (𝑇 “ 𝐴)) |
37 | 31, 36 | eqeltrrd 2839 |
. . . . . 6
⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → ((𝑇‘𝑥) +ℎ (𝑇‘𝑦)) ∈ (𝑇 “ 𝐴)) |
38 | 37 | ralrimiva 3105 |
. . . . 5
⊢ (𝑥 ∈ 𝐴 → ∀𝑦 ∈ 𝐴 ((𝑇‘𝑥) +ℎ (𝑇‘𝑦)) ∈ (𝑇 “ 𝐴)) |
39 | | oveq2 7221 |
. . . . . . . 8
⊢ (𝑣 = (𝑇‘𝑦) → ((𝑇‘𝑥) +ℎ 𝑣) = ((𝑇‘𝑥) +ℎ (𝑇‘𝑦))) |
40 | 39 | eleq1d 2822 |
. . . . . . 7
⊢ (𝑣 = (𝑇‘𝑦) → (((𝑇‘𝑥) +ℎ 𝑣) ∈ (𝑇 “ 𝐴) ↔ ((𝑇‘𝑥) +ℎ (𝑇‘𝑦)) ∈ (𝑇 “ 𝐴))) |
41 | 40 | ralima 7054 |
. . . . . 6
⊢ ((𝑇 Fn ℋ ∧ 𝐴 ⊆ ℋ) →
(∀𝑣 ∈ (𝑇 “ 𝐴)((𝑇‘𝑥) +ℎ 𝑣) ∈ (𝑇 “ 𝐴) ↔ ∀𝑦 ∈ 𝐴 ((𝑇‘𝑥) +ℎ (𝑇‘𝑦)) ∈ (𝑇 “ 𝐴))) |
42 | 22, 13, 41 | mp2an 692 |
. . . . 5
⊢
(∀𝑣 ∈
(𝑇 “ 𝐴)((𝑇‘𝑥) +ℎ 𝑣) ∈ (𝑇 “ 𝐴) ↔ ∀𝑦 ∈ 𝐴 ((𝑇‘𝑥) +ℎ (𝑇‘𝑦)) ∈ (𝑇 “ 𝐴)) |
43 | 38, 42 | sylibr 237 |
. . . 4
⊢ (𝑥 ∈ 𝐴 → ∀𝑣 ∈ (𝑇 “ 𝐴)((𝑇‘𝑥) +ℎ 𝑣) ∈ (𝑇 “ 𝐴)) |
44 | 27, 43 | mprgbir 3076 |
. . 3
⊢
∀𝑢 ∈
(𝑇 “ 𝐴)∀𝑣 ∈ (𝑇 “ 𝐴)(𝑢 +ℎ 𝑣) ∈ (𝑇 “ 𝐴) |
45 | 2 | lnopmuli 30053 |
. . . . . . . 8
⊢ ((𝑢 ∈ ℂ ∧ 𝑦 ∈ ℋ) → (𝑇‘(𝑢 ·ℎ 𝑦)) = (𝑢 ·ℎ (𝑇‘𝑦))) |
46 | 29, 45 | sylan2 596 |
. . . . . . 7
⊢ ((𝑢 ∈ ℂ ∧ 𝑦 ∈ 𝐴) → (𝑇‘(𝑢 ·ℎ 𝑦)) = (𝑢 ·ℎ (𝑇‘𝑦))) |
47 | | shmulcl 29299 |
. . . . . . . . 9
⊢ ((𝐴 ∈
Sℋ ∧ 𝑢 ∈ ℂ ∧ 𝑦 ∈ 𝐴) → (𝑢 ·ℎ 𝑦) ∈ 𝐴) |
48 | 8, 47 | mp3an1 1450 |
. . . . . . . 8
⊢ ((𝑢 ∈ ℂ ∧ 𝑦 ∈ 𝐴) → (𝑢 ·ℎ 𝑦) ∈ 𝐴) |
49 | | funfvima2 7047 |
. . . . . . . . 9
⊢ ((Fun
𝑇 ∧ 𝐴 ⊆ dom 𝑇) → ((𝑢 ·ℎ 𝑦) ∈ 𝐴 → (𝑇‘(𝑢 ·ℎ 𝑦)) ∈ (𝑇 “ 𝐴))) |
50 | 12, 15, 49 | mp2an 692 |
. . . . . . . 8
⊢ ((𝑢
·ℎ 𝑦) ∈ 𝐴 → (𝑇‘(𝑢 ·ℎ 𝑦)) ∈ (𝑇 “ 𝐴)) |
51 | 48, 50 | syl 17 |
. . . . . . 7
⊢ ((𝑢 ∈ ℂ ∧ 𝑦 ∈ 𝐴) → (𝑇‘(𝑢 ·ℎ 𝑦)) ∈ (𝑇 “ 𝐴)) |
52 | 46, 51 | eqeltrrd 2839 |
. . . . . 6
⊢ ((𝑢 ∈ ℂ ∧ 𝑦 ∈ 𝐴) → (𝑢 ·ℎ (𝑇‘𝑦)) ∈ (𝑇 “ 𝐴)) |
53 | 52 | ralrimiva 3105 |
. . . . 5
⊢ (𝑢 ∈ ℂ →
∀𝑦 ∈ 𝐴 (𝑢 ·ℎ (𝑇‘𝑦)) ∈ (𝑇 “ 𝐴)) |
54 | | oveq2 7221 |
. . . . . . . 8
⊢ (𝑣 = (𝑇‘𝑦) → (𝑢 ·ℎ 𝑣) = (𝑢 ·ℎ (𝑇‘𝑦))) |
55 | 54 | eleq1d 2822 |
. . . . . . 7
⊢ (𝑣 = (𝑇‘𝑦) → ((𝑢 ·ℎ 𝑣) ∈ (𝑇 “ 𝐴) ↔ (𝑢 ·ℎ (𝑇‘𝑦)) ∈ (𝑇 “ 𝐴))) |
56 | 55 | ralima 7054 |
. . . . . 6
⊢ ((𝑇 Fn ℋ ∧ 𝐴 ⊆ ℋ) →
(∀𝑣 ∈ (𝑇 “ 𝐴)(𝑢 ·ℎ 𝑣) ∈ (𝑇 “ 𝐴) ↔ ∀𝑦 ∈ 𝐴 (𝑢 ·ℎ (𝑇‘𝑦)) ∈ (𝑇 “ 𝐴))) |
57 | 22, 13, 56 | mp2an 692 |
. . . . 5
⊢
(∀𝑣 ∈
(𝑇 “ 𝐴)(𝑢 ·ℎ 𝑣) ∈ (𝑇 “ 𝐴) ↔ ∀𝑦 ∈ 𝐴 (𝑢 ·ℎ (𝑇‘𝑦)) ∈ (𝑇 “ 𝐴)) |
58 | 53, 57 | sylibr 237 |
. . . 4
⊢ (𝑢 ∈ ℂ →
∀𝑣 ∈ (𝑇 “ 𝐴)(𝑢 ·ℎ 𝑣) ∈ (𝑇 “ 𝐴)) |
59 | 58 | rgen 3071 |
. . 3
⊢
∀𝑢 ∈
ℂ ∀𝑣 ∈
(𝑇 “ 𝐴)(𝑢 ·ℎ 𝑣) ∈ (𝑇 “ 𝐴) |
60 | 44, 59 | pm3.2i 474 |
. 2
⊢
(∀𝑢 ∈
(𝑇 “ 𝐴)∀𝑣 ∈ (𝑇 “ 𝐴)(𝑢 +ℎ 𝑣) ∈ (𝑇 “ 𝐴) ∧ ∀𝑢 ∈ ℂ ∀𝑣 ∈ (𝑇 “ 𝐴)(𝑢 ·ℎ 𝑣) ∈ (𝑇 “ 𝐴)) |
61 | | issh2 29290 |
. 2
⊢ ((𝑇 “ 𝐴) ∈ Sℋ
↔ (((𝑇 “ 𝐴) ⊆ ℋ ∧
0ℎ ∈ (𝑇 “ 𝐴)) ∧ (∀𝑢 ∈ (𝑇 “ 𝐴)∀𝑣 ∈ (𝑇 “ 𝐴)(𝑢 +ℎ 𝑣) ∈ (𝑇 “ 𝐴) ∧ ∀𝑢 ∈ ℂ ∀𝑣 ∈ (𝑇 “ 𝐴)(𝑢 ·ℎ 𝑣) ∈ (𝑇 “ 𝐴)))) |
62 | 20, 60, 61 | mpbir2an 711 |
1
⊢ (𝑇 “ 𝐴) ∈
Sℋ |