| Step | Hyp | Ref
| Expression |
| 1 | | imassrn 6089 |
. . . 4
⊢ (𝑇 “ 𝐴) ⊆ ran 𝑇 |
| 2 | | rnelsh.1 |
. . . . . 6
⊢ 𝑇 ∈ LinOp |
| 3 | 2 | lnopfi 31988 |
. . . . 5
⊢ 𝑇: ℋ⟶
ℋ |
| 4 | | frn 6743 |
. . . . 5
⊢ (𝑇: ℋ⟶ ℋ →
ran 𝑇 ⊆
ℋ) |
| 5 | 3, 4 | ax-mp 5 |
. . . 4
⊢ ran 𝑇 ⊆
ℋ |
| 6 | 1, 5 | sstri 3993 |
. . 3
⊢ (𝑇 “ 𝐴) ⊆ ℋ |
| 7 | 2 | lnop0i 31989 |
. . . 4
⊢ (𝑇‘0ℎ) =
0ℎ |
| 8 | | imaelsh.2 |
. . . . . 6
⊢ 𝐴 ∈
Sℋ |
| 9 | | sh0 31235 |
. . . . . 6
⊢ (𝐴 ∈
Sℋ → 0ℎ ∈ 𝐴) |
| 10 | 8, 9 | ax-mp 5 |
. . . . 5
⊢
0ℎ ∈ 𝐴 |
| 11 | | ffun 6739 |
. . . . . . 7
⊢ (𝑇: ℋ⟶ ℋ →
Fun 𝑇) |
| 12 | 3, 11 | ax-mp 5 |
. . . . . 6
⊢ Fun 𝑇 |
| 13 | 8 | shssii 31232 |
. . . . . . 7
⊢ 𝐴 ⊆
ℋ |
| 14 | 3 | fdmi 6747 |
. . . . . . 7
⊢ dom 𝑇 = ℋ |
| 15 | 13, 14 | sseqtrri 4033 |
. . . . . 6
⊢ 𝐴 ⊆ dom 𝑇 |
| 16 | | funfvima2 7251 |
. . . . . 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 2838 |
. . 3
⊢
0ℎ ∈ (𝑇 “ 𝐴) |
| 20 | 6, 19 | pm3.2i 470 |
. 2
⊢ ((𝑇 “ 𝐴) ⊆ ℋ ∧ 0ℎ
∈ (𝑇 “ 𝐴)) |
| 21 | | ffn 6736 |
. . . . . 6
⊢ (𝑇: ℋ⟶ ℋ →
𝑇 Fn
ℋ) |
| 22 | 3, 21 | ax-mp 5 |
. . . . 5
⊢ 𝑇 Fn ℋ |
| 23 | | oveq1 7438 |
. . . . . . . 8
⊢ (𝑢 = (𝑇‘𝑥) → (𝑢 +ℎ 𝑣) = ((𝑇‘𝑥) +ℎ 𝑣)) |
| 24 | 23 | eleq1d 2826 |
. . . . . . 7
⊢ (𝑢 = (𝑇‘𝑥) → ((𝑢 +ℎ 𝑣) ∈ (𝑇 “ 𝐴) ↔ ((𝑇‘𝑥) +ℎ 𝑣) ∈ (𝑇 “ 𝐴))) |
| 25 | 24 | ralbidv 3178 |
. . . . . 6
⊢ (𝑢 = (𝑇‘𝑥) → (∀𝑣 ∈ (𝑇 “ 𝐴)(𝑢 +ℎ 𝑣) ∈ (𝑇 “ 𝐴) ↔ ∀𝑣 ∈ (𝑇 “ 𝐴)((𝑇‘𝑥) +ℎ 𝑣) ∈ (𝑇 “ 𝐴))) |
| 26 | 25 | ralima 7257 |
. . . . 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 31233 |
. . . . . . . 8
⊢ (𝑥 ∈ 𝐴 → 𝑥 ∈ ℋ) |
| 29 | 8 | sheli 31233 |
. . . . . . . 8
⊢ (𝑦 ∈ 𝐴 → 𝑦 ∈ ℋ) |
| 30 | 2 | lnopaddi 31990 |
. . . . . . . 8
⊢ ((𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ) → (𝑇‘(𝑥 +ℎ 𝑦)) = ((𝑇‘𝑥) +ℎ (𝑇‘𝑦))) |
| 31 | 28, 29, 30 | syl2an 596 |
. . . . . . 7
⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → (𝑇‘(𝑥 +ℎ 𝑦)) = ((𝑇‘𝑥) +ℎ (𝑇‘𝑦))) |
| 32 | | shaddcl 31236 |
. . . . . . . . 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 7251 |
. . . . . . . . 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 2842 |
. . . . . 6
⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → ((𝑇‘𝑥) +ℎ (𝑇‘𝑦)) ∈ (𝑇 “ 𝐴)) |
| 38 | 37 | ralrimiva 3146 |
. . . . 5
⊢ (𝑥 ∈ 𝐴 → ∀𝑦 ∈ 𝐴 ((𝑇‘𝑥) +ℎ (𝑇‘𝑦)) ∈ (𝑇 “ 𝐴)) |
| 39 | | oveq2 7439 |
. . . . . . . 8
⊢ (𝑣 = (𝑇‘𝑦) → ((𝑇‘𝑥) +ℎ 𝑣) = ((𝑇‘𝑥) +ℎ (𝑇‘𝑦))) |
| 40 | 39 | eleq1d 2826 |
. . . . . . 7
⊢ (𝑣 = (𝑇‘𝑦) → (((𝑇‘𝑥) +ℎ 𝑣) ∈ (𝑇 “ 𝐴) ↔ ((𝑇‘𝑥) +ℎ (𝑇‘𝑦)) ∈ (𝑇 “ 𝐴))) |
| 41 | 40 | ralima 7257 |
. . . . . 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 234 |
. . . 4
⊢ (𝑥 ∈ 𝐴 → ∀𝑣 ∈ (𝑇 “ 𝐴)((𝑇‘𝑥) +ℎ 𝑣) ∈ (𝑇 “ 𝐴)) |
| 44 | 27, 43 | mprgbir 3068 |
. . 3
⊢
∀𝑢 ∈
(𝑇 “ 𝐴)∀𝑣 ∈ (𝑇 “ 𝐴)(𝑢 +ℎ 𝑣) ∈ (𝑇 “ 𝐴) |
| 45 | 2 | lnopmuli 31991 |
. . . . . . . 8
⊢ ((𝑢 ∈ ℂ ∧ 𝑦 ∈ ℋ) → (𝑇‘(𝑢 ·ℎ 𝑦)) = (𝑢 ·ℎ (𝑇‘𝑦))) |
| 46 | 29, 45 | sylan2 593 |
. . . . . . 7
⊢ ((𝑢 ∈ ℂ ∧ 𝑦 ∈ 𝐴) → (𝑇‘(𝑢 ·ℎ 𝑦)) = (𝑢 ·ℎ (𝑇‘𝑦))) |
| 47 | | shmulcl 31237 |
. . . . . . . . 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 7251 |
. . . . . . . . 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 2842 |
. . . . . 6
⊢ ((𝑢 ∈ ℂ ∧ 𝑦 ∈ 𝐴) → (𝑢 ·ℎ (𝑇‘𝑦)) ∈ (𝑇 “ 𝐴)) |
| 53 | 52 | ralrimiva 3146 |
. . . . 5
⊢ (𝑢 ∈ ℂ →
∀𝑦 ∈ 𝐴 (𝑢 ·ℎ (𝑇‘𝑦)) ∈ (𝑇 “ 𝐴)) |
| 54 | | oveq2 7439 |
. . . . . . . 8
⊢ (𝑣 = (𝑇‘𝑦) → (𝑢 ·ℎ 𝑣) = (𝑢 ·ℎ (𝑇‘𝑦))) |
| 55 | 54 | eleq1d 2826 |
. . . . . . 7
⊢ (𝑣 = (𝑇‘𝑦) → ((𝑢 ·ℎ 𝑣) ∈ (𝑇 “ 𝐴) ↔ (𝑢 ·ℎ (𝑇‘𝑦)) ∈ (𝑇 “ 𝐴))) |
| 56 | 55 | ralima 7257 |
. . . . . 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 234 |
. . . 4
⊢ (𝑢 ∈ ℂ →
∀𝑣 ∈ (𝑇 “ 𝐴)(𝑢 ·ℎ 𝑣) ∈ (𝑇 “ 𝐴)) |
| 59 | 58 | rgen 3063 |
. . 3
⊢
∀𝑢 ∈
ℂ ∀𝑣 ∈
(𝑇 “ 𝐴)(𝑢 ·ℎ 𝑣) ∈ (𝑇 “ 𝐴) |
| 60 | 44, 59 | pm3.2i 470 |
. 2
⊢
(∀𝑢 ∈
(𝑇 “ 𝐴)∀𝑣 ∈ (𝑇 “ 𝐴)(𝑢 +ℎ 𝑣) ∈ (𝑇 “ 𝐴) ∧ ∀𝑢 ∈ ℂ ∀𝑣 ∈ (𝑇 “ 𝐴)(𝑢 ·ℎ 𝑣) ∈ (𝑇 “ 𝐴)) |
| 61 | | issh2 31228 |
. 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 |