Step | Hyp | Ref
| Expression |
1 | | neq0 4305 |
. . 3
⊢ (¬
(𝐴 ∖ 𝐵) = ∅ ↔ ∃𝑥 𝑥 ∈ (𝐴 ∖ 𝐵)) |
2 | | ssdif0 4323 |
. . 3
⊢ (𝐴 ⊆ 𝐵 ↔ (𝐴 ∖ 𝐵) = ∅) |
3 | 1, 2 | xchnxbir 332 |
. 2
⊢ (¬
𝐴 ⊆ 𝐵 ↔ ∃𝑥 𝑥 ∈ (𝐴 ∖ 𝐵)) |
4 | | eldifi 4086 |
. . . . . . . . . . 11
⊢ (𝑥 ∈ (𝐴 ∖ 𝐵) → 𝑥 ∈ 𝐴) |
5 | | strlem1.1 |
. . . . . . . . . . . 12
⊢ 𝐴 ∈
Cℋ |
6 | 5 | cheli 30174 |
. . . . . . . . . . 11
⊢ (𝑥 ∈ 𝐴 → 𝑥 ∈ ℋ) |
7 | | normcl 30067 |
. . . . . . . . . . 11
⊢ (𝑥 ∈ ℋ →
(normℎ‘𝑥) ∈ ℝ) |
8 | 4, 6, 7 | 3syl 18 |
. . . . . . . . . 10
⊢ (𝑥 ∈ (𝐴 ∖ 𝐵) →
(normℎ‘𝑥) ∈ ℝ) |
9 | | strlem1.2 |
. . . . . . . . . . . . . . . 16
⊢ 𝐵 ∈
Cℋ |
10 | | ch0 30170 |
. . . . . . . . . . . . . . . 16
⊢ (𝐵 ∈
Cℋ → 0ℎ ∈ 𝐵) |
11 | 9, 10 | ax-mp 5 |
. . . . . . . . . . . . . . 15
⊢
0ℎ ∈ 𝐵 |
12 | | eldifn 4087 |
. . . . . . . . . . . . . . 15
⊢
(0ℎ ∈ (𝐴 ∖ 𝐵) → ¬ 0ℎ ∈
𝐵) |
13 | 11, 12 | mt2 199 |
. . . . . . . . . . . . . 14
⊢ ¬
0ℎ ∈ (𝐴 ∖ 𝐵) |
14 | | eleq1 2825 |
. . . . . . . . . . . . . 14
⊢ (𝑥 = 0ℎ →
(𝑥 ∈ (𝐴 ∖ 𝐵) ↔ 0ℎ ∈ (𝐴 ∖ 𝐵))) |
15 | 13, 14 | mtbiri 326 |
. . . . . . . . . . . . 13
⊢ (𝑥 = 0ℎ →
¬ 𝑥 ∈ (𝐴 ∖ 𝐵)) |
16 | 15 | con2i 139 |
. . . . . . . . . . . 12
⊢ (𝑥 ∈ (𝐴 ∖ 𝐵) → ¬ 𝑥 = 0ℎ) |
17 | | norm-i 30071 |
. . . . . . . . . . . . 13
⊢ (𝑥 ∈ ℋ →
((normℎ‘𝑥) = 0 ↔ 𝑥 = 0ℎ)) |
18 | 4, 6, 17 | 3syl 18 |
. . . . . . . . . . . 12
⊢ (𝑥 ∈ (𝐴 ∖ 𝐵) →
((normℎ‘𝑥) = 0 ↔ 𝑥 = 0ℎ)) |
19 | 16, 18 | mtbird 324 |
. . . . . . . . . . 11
⊢ (𝑥 ∈ (𝐴 ∖ 𝐵) → ¬
(normℎ‘𝑥) = 0) |
20 | 19 | neqned 2950 |
. . . . . . . . . 10
⊢ (𝑥 ∈ (𝐴 ∖ 𝐵) →
(normℎ‘𝑥) ≠ 0) |
21 | 8, 20 | rereccld 11982 |
. . . . . . . . 9
⊢ (𝑥 ∈ (𝐴 ∖ 𝐵) → (1 /
(normℎ‘𝑥)) ∈ ℝ) |
22 | 21 | recnd 11183 |
. . . . . . . 8
⊢ (𝑥 ∈ (𝐴 ∖ 𝐵) → (1 /
(normℎ‘𝑥)) ∈ ℂ) |
23 | 5 | chshii 30169 |
. . . . . . . . . 10
⊢ 𝐴 ∈
Sℋ |
24 | | shmulcl 30160 |
. . . . . . . . . 10
⊢ ((𝐴 ∈
Sℋ ∧ (1 /
(normℎ‘𝑥)) ∈ ℂ ∧ 𝑥 ∈ 𝐴) → ((1 /
(normℎ‘𝑥)) ·ℎ 𝑥) ∈ 𝐴) |
25 | 23, 24 | mp3an1 1448 |
. . . . . . . . 9
⊢ (((1 /
(normℎ‘𝑥)) ∈ ℂ ∧ 𝑥 ∈ 𝐴) → ((1 /
(normℎ‘𝑥)) ·ℎ 𝑥) ∈ 𝐴) |
26 | 25 | ex 413 |
. . . . . . . 8
⊢ ((1 /
(normℎ‘𝑥)) ∈ ℂ → (𝑥 ∈ 𝐴 → ((1 /
(normℎ‘𝑥)) ·ℎ 𝑥) ∈ 𝐴)) |
27 | 22, 26 | syl 17 |
. . . . . . 7
⊢ (𝑥 ∈ (𝐴 ∖ 𝐵) → (𝑥 ∈ 𝐴 → ((1 /
(normℎ‘𝑥)) ·ℎ 𝑥) ∈ 𝐴)) |
28 | 8 | recnd 11183 |
. . . . . . . . . 10
⊢ (𝑥 ∈ (𝐴 ∖ 𝐵) →
(normℎ‘𝑥) ∈ ℂ) |
29 | 9 | chshii 30169 |
. . . . . . . . . . . 12
⊢ 𝐵 ∈
Sℋ |
30 | | shmulcl 30160 |
. . . . . . . . . . . 12
⊢ ((𝐵 ∈
Sℋ ∧ (normℎ‘𝑥) ∈ ℂ ∧ ((1 /
(normℎ‘𝑥)) ·ℎ 𝑥) ∈ 𝐵) →
((normℎ‘𝑥) ·ℎ ((1 /
(normℎ‘𝑥)) ·ℎ 𝑥)) ∈ 𝐵) |
31 | 29, 30 | mp3an1 1448 |
. . . . . . . . . . 11
⊢
(((normℎ‘𝑥) ∈ ℂ ∧ ((1 /
(normℎ‘𝑥)) ·ℎ 𝑥) ∈ 𝐵) →
((normℎ‘𝑥) ·ℎ ((1 /
(normℎ‘𝑥)) ·ℎ 𝑥)) ∈ 𝐵) |
32 | 31 | ex 413 |
. . . . . . . . . 10
⊢
((normℎ‘𝑥) ∈ ℂ → (((1 /
(normℎ‘𝑥)) ·ℎ 𝑥) ∈ 𝐵 →
((normℎ‘𝑥) ·ℎ ((1 /
(normℎ‘𝑥)) ·ℎ 𝑥)) ∈ 𝐵)) |
33 | 28, 32 | syl 17 |
. . . . . . . . 9
⊢ (𝑥 ∈ (𝐴 ∖ 𝐵) → (((1 /
(normℎ‘𝑥)) ·ℎ 𝑥) ∈ 𝐵 →
((normℎ‘𝑥) ·ℎ ((1 /
(normℎ‘𝑥)) ·ℎ 𝑥)) ∈ 𝐵)) |
34 | 28, 20 | recidd 11926 |
. . . . . . . . . . . 12
⊢ (𝑥 ∈ (𝐴 ∖ 𝐵) →
((normℎ‘𝑥) · (1 /
(normℎ‘𝑥))) = 1) |
35 | 34 | oveq1d 7372 |
. . . . . . . . . . 11
⊢ (𝑥 ∈ (𝐴 ∖ 𝐵) →
(((normℎ‘𝑥) · (1 /
(normℎ‘𝑥))) ·ℎ 𝑥) = (1
·ℎ 𝑥)) |
36 | 4, 6 | syl 17 |
. . . . . . . . . . . 12
⊢ (𝑥 ∈ (𝐴 ∖ 𝐵) → 𝑥 ∈ ℋ) |
37 | | ax-hvmulass 29949 |
. . . . . . . . . . . 12
⊢
(((normℎ‘𝑥) ∈ ℂ ∧ (1 /
(normℎ‘𝑥)) ∈ ℂ ∧ 𝑥 ∈ ℋ) →
(((normℎ‘𝑥) · (1 /
(normℎ‘𝑥))) ·ℎ 𝑥) =
((normℎ‘𝑥) ·ℎ ((1 /
(normℎ‘𝑥)) ·ℎ 𝑥))) |
38 | 28, 22, 36, 37 | syl3anc 1371 |
. . . . . . . . . . 11
⊢ (𝑥 ∈ (𝐴 ∖ 𝐵) →
(((normℎ‘𝑥) · (1 /
(normℎ‘𝑥))) ·ℎ 𝑥) =
((normℎ‘𝑥) ·ℎ ((1 /
(normℎ‘𝑥)) ·ℎ 𝑥))) |
39 | | ax-hvmulid 29948 |
. . . . . . . . . . . 12
⊢ (𝑥 ∈ ℋ → (1
·ℎ 𝑥) = 𝑥) |
40 | 4, 6, 39 | 3syl 18 |
. . . . . . . . . . 11
⊢ (𝑥 ∈ (𝐴 ∖ 𝐵) → (1
·ℎ 𝑥) = 𝑥) |
41 | 35, 38, 40 | 3eqtr3d 2784 |
. . . . . . . . . 10
⊢ (𝑥 ∈ (𝐴 ∖ 𝐵) →
((normℎ‘𝑥) ·ℎ ((1 /
(normℎ‘𝑥)) ·ℎ 𝑥)) = 𝑥) |
42 | 41 | eleq1d 2822 |
. . . . . . . . 9
⊢ (𝑥 ∈ (𝐴 ∖ 𝐵) →
(((normℎ‘𝑥) ·ℎ ((1 /
(normℎ‘𝑥)) ·ℎ 𝑥)) ∈ 𝐵 ↔ 𝑥 ∈ 𝐵)) |
43 | 33, 42 | sylibd 238 |
. . . . . . . 8
⊢ (𝑥 ∈ (𝐴 ∖ 𝐵) → (((1 /
(normℎ‘𝑥)) ·ℎ 𝑥) ∈ 𝐵 → 𝑥 ∈ 𝐵)) |
44 | 43 | con3d 152 |
. . . . . . 7
⊢ (𝑥 ∈ (𝐴 ∖ 𝐵) → (¬ 𝑥 ∈ 𝐵 → ¬ ((1 /
(normℎ‘𝑥)) ·ℎ 𝑥) ∈ 𝐵)) |
45 | 27, 44 | anim12d 609 |
. . . . . 6
⊢ (𝑥 ∈ (𝐴 ∖ 𝐵) → ((𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵) → (((1 /
(normℎ‘𝑥)) ·ℎ 𝑥) ∈ 𝐴 ∧ ¬ ((1 /
(normℎ‘𝑥)) ·ℎ 𝑥) ∈ 𝐵))) |
46 | | eldif 3920 |
. . . . . 6
⊢ (𝑥 ∈ (𝐴 ∖ 𝐵) ↔ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵)) |
47 | | eldif 3920 |
. . . . . 6
⊢ (((1 /
(normℎ‘𝑥)) ·ℎ 𝑥) ∈ (𝐴 ∖ 𝐵) ↔ (((1 /
(normℎ‘𝑥)) ·ℎ 𝑥) ∈ 𝐴 ∧ ¬ ((1 /
(normℎ‘𝑥)) ·ℎ 𝑥) ∈ 𝐵)) |
48 | 45, 46, 47 | 3imtr4g 295 |
. . . . 5
⊢ (𝑥 ∈ (𝐴 ∖ 𝐵) → (𝑥 ∈ (𝐴 ∖ 𝐵) → ((1 /
(normℎ‘𝑥)) ·ℎ 𝑥) ∈ (𝐴 ∖ 𝐵))) |
49 | 48 | pm2.43i 52 |
. . . 4
⊢ (𝑥 ∈ (𝐴 ∖ 𝐵) → ((1 /
(normℎ‘𝑥)) ·ℎ 𝑥) ∈ (𝐴 ∖ 𝐵)) |
50 | | norm-iii 30082 |
. . . . . 6
⊢ (((1 /
(normℎ‘𝑥)) ∈ ℂ ∧ 𝑥 ∈ ℋ) →
(normℎ‘((1 / (normℎ‘𝑥))
·ℎ 𝑥)) = ((abs‘(1 /
(normℎ‘𝑥))) ·
(normℎ‘𝑥))) |
51 | 22, 36, 50 | syl2anc 584 |
. . . . 5
⊢ (𝑥 ∈ (𝐴 ∖ 𝐵) → (normℎ‘((1
/ (normℎ‘𝑥)) ·ℎ 𝑥)) = ((abs‘(1 /
(normℎ‘𝑥))) ·
(normℎ‘𝑥))) |
52 | 15 | necon2ai 2973 |
. . . . . . . . 9
⊢ (𝑥 ∈ (𝐴 ∖ 𝐵) → 𝑥 ≠ 0ℎ) |
53 | | normgt0 30069 |
. . . . . . . . . 10
⊢ (𝑥 ∈ ℋ → (𝑥 ≠ 0ℎ
↔ 0 < (normℎ‘𝑥))) |
54 | 4, 6, 53 | 3syl 18 |
. . . . . . . . 9
⊢ (𝑥 ∈ (𝐴 ∖ 𝐵) → (𝑥 ≠ 0ℎ ↔ 0 <
(normℎ‘𝑥))) |
55 | 52, 54 | mpbid 231 |
. . . . . . . 8
⊢ (𝑥 ∈ (𝐴 ∖ 𝐵) → 0 <
(normℎ‘𝑥)) |
56 | | 1re 11155 |
. . . . . . . . 9
⊢ 1 ∈
ℝ |
57 | | 0le1 11678 |
. . . . . . . . 9
⊢ 0 ≤
1 |
58 | | divge0 12024 |
. . . . . . . . 9
⊢ (((1
∈ ℝ ∧ 0 ≤ 1) ∧ ((normℎ‘𝑥) ∈ ℝ ∧ 0 <
(normℎ‘𝑥))) → 0 ≤ (1 /
(normℎ‘𝑥))) |
59 | 56, 57, 58 | mpanl12 700 |
. . . . . . . 8
⊢
(((normℎ‘𝑥) ∈ ℝ ∧ 0 <
(normℎ‘𝑥)) → 0 ≤ (1 /
(normℎ‘𝑥))) |
60 | 8, 55, 59 | syl2anc 584 |
. . . . . . 7
⊢ (𝑥 ∈ (𝐴 ∖ 𝐵) → 0 ≤ (1 /
(normℎ‘𝑥))) |
61 | 21, 60 | absidd 15307 |
. . . . . 6
⊢ (𝑥 ∈ (𝐴 ∖ 𝐵) → (abs‘(1 /
(normℎ‘𝑥))) = (1 /
(normℎ‘𝑥))) |
62 | 61 | oveq1d 7372 |
. . . . 5
⊢ (𝑥 ∈ (𝐴 ∖ 𝐵) → ((abs‘(1 /
(normℎ‘𝑥))) ·
(normℎ‘𝑥)) = ((1 /
(normℎ‘𝑥)) ·
(normℎ‘𝑥))) |
63 | 28, 20 | recid2d 11927 |
. . . . 5
⊢ (𝑥 ∈ (𝐴 ∖ 𝐵) → ((1 /
(normℎ‘𝑥)) ·
(normℎ‘𝑥)) = 1) |
64 | 51, 62, 63 | 3eqtrd 2780 |
. . . 4
⊢ (𝑥 ∈ (𝐴 ∖ 𝐵) → (normℎ‘((1
/ (normℎ‘𝑥)) ·ℎ 𝑥)) = 1) |
65 | | fveqeq2 6851 |
. . . . 5
⊢ (𝑢 = ((1 /
(normℎ‘𝑥)) ·ℎ 𝑥) →
((normℎ‘𝑢) = 1 ↔
(normℎ‘((1 / (normℎ‘𝑥))
·ℎ 𝑥)) = 1)) |
66 | 65 | rspcev 3581 |
. . . 4
⊢ ((((1 /
(normℎ‘𝑥)) ·ℎ 𝑥) ∈ (𝐴 ∖ 𝐵) ∧ (normℎ‘((1 /
(normℎ‘𝑥)) ·ℎ 𝑥)) = 1) → ∃𝑢 ∈ (𝐴 ∖ 𝐵)(normℎ‘𝑢) = 1) |
67 | 49, 64, 66 | syl2anc 584 |
. . 3
⊢ (𝑥 ∈ (𝐴 ∖ 𝐵) → ∃𝑢 ∈ (𝐴 ∖ 𝐵)(normℎ‘𝑢) = 1) |
68 | 67 | exlimiv 1933 |
. 2
⊢
(∃𝑥 𝑥 ∈ (𝐴 ∖ 𝐵) → ∃𝑢 ∈ (𝐴 ∖ 𝐵)(normℎ‘𝑢) = 1) |
69 | 3, 68 | sylbi 216 |
1
⊢ (¬
𝐴 ⊆ 𝐵 → ∃𝑢 ∈ (𝐴 ∖ 𝐵)(normℎ‘𝑢) = 1) |