Step |
Hyp |
Ref |
Expression |
1 |
|
strlem1.1 |
⊢ 𝐴 ∈ Cℋ |
2 |
|
strlem1.2 |
⊢ 𝐵 ∈ Cℋ |
3 |
|
neq0 |
⊢ ( ¬ ( 𝐴 ∖ 𝐵 ) = ∅ ↔ ∃ 𝑥 𝑥 ∈ ( 𝐴 ∖ 𝐵 ) ) |
4 |
|
ssdif0 |
⊢ ( 𝐴 ⊆ 𝐵 ↔ ( 𝐴 ∖ 𝐵 ) = ∅ ) |
5 |
3 4
|
xchnxbir |
⊢ ( ¬ 𝐴 ⊆ 𝐵 ↔ ∃ 𝑥 𝑥 ∈ ( 𝐴 ∖ 𝐵 ) ) |
6 |
|
eldifi |
⊢ ( 𝑥 ∈ ( 𝐴 ∖ 𝐵 ) → 𝑥 ∈ 𝐴 ) |
7 |
1
|
cheli |
⊢ ( 𝑥 ∈ 𝐴 → 𝑥 ∈ ℋ ) |
8 |
|
normcl |
⊢ ( 𝑥 ∈ ℋ → ( normℎ ‘ 𝑥 ) ∈ ℝ ) |
9 |
6 7 8
|
3syl |
⊢ ( 𝑥 ∈ ( 𝐴 ∖ 𝐵 ) → ( normℎ ‘ 𝑥 ) ∈ ℝ ) |
10 |
|
ch0 |
⊢ ( 𝐵 ∈ Cℋ → 0ℎ ∈ 𝐵 ) |
11 |
2 10
|
ax-mp |
⊢ 0ℎ ∈ 𝐵 |
12 |
|
eldifn |
⊢ ( 0ℎ ∈ ( 𝐴 ∖ 𝐵 ) → ¬ 0ℎ ∈ 𝐵 ) |
13 |
11 12
|
mt2 |
⊢ ¬ 0ℎ ∈ ( 𝐴 ∖ 𝐵 ) |
14 |
|
eleq1 |
⊢ ( 𝑥 = 0ℎ → ( 𝑥 ∈ ( 𝐴 ∖ 𝐵 ) ↔ 0ℎ ∈ ( 𝐴 ∖ 𝐵 ) ) ) |
15 |
13 14
|
mtbiri |
⊢ ( 𝑥 = 0ℎ → ¬ 𝑥 ∈ ( 𝐴 ∖ 𝐵 ) ) |
16 |
15
|
con2i |
⊢ ( 𝑥 ∈ ( 𝐴 ∖ 𝐵 ) → ¬ 𝑥 = 0ℎ ) |
17 |
|
norm-i |
⊢ ( 𝑥 ∈ ℋ → ( ( normℎ ‘ 𝑥 ) = 0 ↔ 𝑥 = 0ℎ ) ) |
18 |
6 7 17
|
3syl |
⊢ ( 𝑥 ∈ ( 𝐴 ∖ 𝐵 ) → ( ( normℎ ‘ 𝑥 ) = 0 ↔ 𝑥 = 0ℎ ) ) |
19 |
16 18
|
mtbird |
⊢ ( 𝑥 ∈ ( 𝐴 ∖ 𝐵 ) → ¬ ( normℎ ‘ 𝑥 ) = 0 ) |
20 |
19
|
neqned |
⊢ ( 𝑥 ∈ ( 𝐴 ∖ 𝐵 ) → ( normℎ ‘ 𝑥 ) ≠ 0 ) |
21 |
9 20
|
rereccld |
⊢ ( 𝑥 ∈ ( 𝐴 ∖ 𝐵 ) → ( 1 / ( normℎ ‘ 𝑥 ) ) ∈ ℝ ) |
22 |
21
|
recnd |
⊢ ( 𝑥 ∈ ( 𝐴 ∖ 𝐵 ) → ( 1 / ( normℎ ‘ 𝑥 ) ) ∈ ℂ ) |
23 |
1
|
chshii |
⊢ 𝐴 ∈ Sℋ |
24 |
|
shmulcl |
⊢ ( ( 𝐴 ∈ Sℋ ∧ ( 1 / ( normℎ ‘ 𝑥 ) ) ∈ ℂ ∧ 𝑥 ∈ 𝐴 ) → ( ( 1 / ( normℎ ‘ 𝑥 ) ) ·ℎ 𝑥 ) ∈ 𝐴 ) |
25 |
23 24
|
mp3an1 |
⊢ ( ( ( 1 / ( normℎ ‘ 𝑥 ) ) ∈ ℂ ∧ 𝑥 ∈ 𝐴 ) → ( ( 1 / ( normℎ ‘ 𝑥 ) ) ·ℎ 𝑥 ) ∈ 𝐴 ) |
26 |
25
|
ex |
⊢ ( ( 1 / ( normℎ ‘ 𝑥 ) ) ∈ ℂ → ( 𝑥 ∈ 𝐴 → ( ( 1 / ( normℎ ‘ 𝑥 ) ) ·ℎ 𝑥 ) ∈ 𝐴 ) ) |
27 |
22 26
|
syl |
⊢ ( 𝑥 ∈ ( 𝐴 ∖ 𝐵 ) → ( 𝑥 ∈ 𝐴 → ( ( 1 / ( normℎ ‘ 𝑥 ) ) ·ℎ 𝑥 ) ∈ 𝐴 ) ) |
28 |
9
|
recnd |
⊢ ( 𝑥 ∈ ( 𝐴 ∖ 𝐵 ) → ( normℎ ‘ 𝑥 ) ∈ ℂ ) |
29 |
2
|
chshii |
⊢ 𝐵 ∈ Sℋ |
30 |
|
shmulcl |
⊢ ( ( 𝐵 ∈ Sℋ ∧ ( normℎ ‘ 𝑥 ) ∈ ℂ ∧ ( ( 1 / ( normℎ ‘ 𝑥 ) ) ·ℎ 𝑥 ) ∈ 𝐵 ) → ( ( normℎ ‘ 𝑥 ) ·ℎ ( ( 1 / ( normℎ ‘ 𝑥 ) ) ·ℎ 𝑥 ) ) ∈ 𝐵 ) |
31 |
29 30
|
mp3an1 |
⊢ ( ( ( normℎ ‘ 𝑥 ) ∈ ℂ ∧ ( ( 1 / ( normℎ ‘ 𝑥 ) ) ·ℎ 𝑥 ) ∈ 𝐵 ) → ( ( normℎ ‘ 𝑥 ) ·ℎ ( ( 1 / ( normℎ ‘ 𝑥 ) ) ·ℎ 𝑥 ) ) ∈ 𝐵 ) |
32 |
31
|
ex |
⊢ ( ( normℎ ‘ 𝑥 ) ∈ ℂ → ( ( ( 1 / ( normℎ ‘ 𝑥 ) ) ·ℎ 𝑥 ) ∈ 𝐵 → ( ( normℎ ‘ 𝑥 ) ·ℎ ( ( 1 / ( normℎ ‘ 𝑥 ) ) ·ℎ 𝑥 ) ) ∈ 𝐵 ) ) |
33 |
28 32
|
syl |
⊢ ( 𝑥 ∈ ( 𝐴 ∖ 𝐵 ) → ( ( ( 1 / ( normℎ ‘ 𝑥 ) ) ·ℎ 𝑥 ) ∈ 𝐵 → ( ( normℎ ‘ 𝑥 ) ·ℎ ( ( 1 / ( normℎ ‘ 𝑥 ) ) ·ℎ 𝑥 ) ) ∈ 𝐵 ) ) |
34 |
28 20
|
recidd |
⊢ ( 𝑥 ∈ ( 𝐴 ∖ 𝐵 ) → ( ( normℎ ‘ 𝑥 ) · ( 1 / ( normℎ ‘ 𝑥 ) ) ) = 1 ) |
35 |
34
|
oveq1d |
⊢ ( 𝑥 ∈ ( 𝐴 ∖ 𝐵 ) → ( ( ( normℎ ‘ 𝑥 ) · ( 1 / ( normℎ ‘ 𝑥 ) ) ) ·ℎ 𝑥 ) = ( 1 ·ℎ 𝑥 ) ) |
36 |
6 7
|
syl |
⊢ ( 𝑥 ∈ ( 𝐴 ∖ 𝐵 ) → 𝑥 ∈ ℋ ) |
37 |
|
ax-hvmulass |
⊢ ( ( ( 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 |
⊢ ( 𝑥 ∈ ( 𝐴 ∖ 𝐵 ) → ( ( ( normℎ ‘ 𝑥 ) · ( 1 / ( normℎ ‘ 𝑥 ) ) ) ·ℎ 𝑥 ) = ( ( normℎ ‘ 𝑥 ) ·ℎ ( ( 1 / ( normℎ ‘ 𝑥 ) ) ·ℎ 𝑥 ) ) ) |
39 |
|
ax-hvmulid |
⊢ ( 𝑥 ∈ ℋ → ( 1 ·ℎ 𝑥 ) = 𝑥 ) |
40 |
6 7 39
|
3syl |
⊢ ( 𝑥 ∈ ( 𝐴 ∖ 𝐵 ) → ( 1 ·ℎ 𝑥 ) = 𝑥 ) |
41 |
35 38 40
|
3eqtr3d |
⊢ ( 𝑥 ∈ ( 𝐴 ∖ 𝐵 ) → ( ( normℎ ‘ 𝑥 ) ·ℎ ( ( 1 / ( normℎ ‘ 𝑥 ) ) ·ℎ 𝑥 ) ) = 𝑥 ) |
42 |
41
|
eleq1d |
⊢ ( 𝑥 ∈ ( 𝐴 ∖ 𝐵 ) → ( ( ( normℎ ‘ 𝑥 ) ·ℎ ( ( 1 / ( normℎ ‘ 𝑥 ) ) ·ℎ 𝑥 ) ) ∈ 𝐵 ↔ 𝑥 ∈ 𝐵 ) ) |
43 |
33 42
|
sylibd |
⊢ ( 𝑥 ∈ ( 𝐴 ∖ 𝐵 ) → ( ( ( 1 / ( normℎ ‘ 𝑥 ) ) ·ℎ 𝑥 ) ∈ 𝐵 → 𝑥 ∈ 𝐵 ) ) |
44 |
43
|
con3d |
⊢ ( 𝑥 ∈ ( 𝐴 ∖ 𝐵 ) → ( ¬ 𝑥 ∈ 𝐵 → ¬ ( ( 1 / ( normℎ ‘ 𝑥 ) ) ·ℎ 𝑥 ) ∈ 𝐵 ) ) |
45 |
27 44
|
anim12d |
⊢ ( 𝑥 ∈ ( 𝐴 ∖ 𝐵 ) → ( ( 𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵 ) → ( ( ( 1 / ( normℎ ‘ 𝑥 ) ) ·ℎ 𝑥 ) ∈ 𝐴 ∧ ¬ ( ( 1 / ( normℎ ‘ 𝑥 ) ) ·ℎ 𝑥 ) ∈ 𝐵 ) ) ) |
46 |
|
eldif |
⊢ ( 𝑥 ∈ ( 𝐴 ∖ 𝐵 ) ↔ ( 𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵 ) ) |
47 |
|
eldif |
⊢ ( ( ( 1 / ( normℎ ‘ 𝑥 ) ) ·ℎ 𝑥 ) ∈ ( 𝐴 ∖ 𝐵 ) ↔ ( ( ( 1 / ( normℎ ‘ 𝑥 ) ) ·ℎ 𝑥 ) ∈ 𝐴 ∧ ¬ ( ( 1 / ( normℎ ‘ 𝑥 ) ) ·ℎ 𝑥 ) ∈ 𝐵 ) ) |
48 |
45 46 47
|
3imtr4g |
⊢ ( 𝑥 ∈ ( 𝐴 ∖ 𝐵 ) → ( 𝑥 ∈ ( 𝐴 ∖ 𝐵 ) → ( ( 1 / ( normℎ ‘ 𝑥 ) ) ·ℎ 𝑥 ) ∈ ( 𝐴 ∖ 𝐵 ) ) ) |
49 |
48
|
pm2.43i |
⊢ ( 𝑥 ∈ ( 𝐴 ∖ 𝐵 ) → ( ( 1 / ( normℎ ‘ 𝑥 ) ) ·ℎ 𝑥 ) ∈ ( 𝐴 ∖ 𝐵 ) ) |
50 |
|
norm-iii |
⊢ ( ( ( 1 / ( normℎ ‘ 𝑥 ) ) ∈ ℂ ∧ 𝑥 ∈ ℋ ) → ( normℎ ‘ ( ( 1 / ( normℎ ‘ 𝑥 ) ) ·ℎ 𝑥 ) ) = ( ( abs ‘ ( 1 / ( normℎ ‘ 𝑥 ) ) ) · ( normℎ ‘ 𝑥 ) ) ) |
51 |
22 36 50
|
syl2anc |
⊢ ( 𝑥 ∈ ( 𝐴 ∖ 𝐵 ) → ( normℎ ‘ ( ( 1 / ( normℎ ‘ 𝑥 ) ) ·ℎ 𝑥 ) ) = ( ( abs ‘ ( 1 / ( normℎ ‘ 𝑥 ) ) ) · ( normℎ ‘ 𝑥 ) ) ) |
52 |
15
|
necon2ai |
⊢ ( 𝑥 ∈ ( 𝐴 ∖ 𝐵 ) → 𝑥 ≠ 0ℎ ) |
53 |
|
normgt0 |
⊢ ( 𝑥 ∈ ℋ → ( 𝑥 ≠ 0ℎ ↔ 0 < ( normℎ ‘ 𝑥 ) ) ) |
54 |
6 7 53
|
3syl |
⊢ ( 𝑥 ∈ ( 𝐴 ∖ 𝐵 ) → ( 𝑥 ≠ 0ℎ ↔ 0 < ( normℎ ‘ 𝑥 ) ) ) |
55 |
52 54
|
mpbid |
⊢ ( 𝑥 ∈ ( 𝐴 ∖ 𝐵 ) → 0 < ( normℎ ‘ 𝑥 ) ) |
56 |
|
1re |
⊢ 1 ∈ ℝ |
57 |
|
0le1 |
⊢ 0 ≤ 1 |
58 |
|
divge0 |
⊢ ( ( ( 1 ∈ ℝ ∧ 0 ≤ 1 ) ∧ ( ( normℎ ‘ 𝑥 ) ∈ ℝ ∧ 0 < ( normℎ ‘ 𝑥 ) ) ) → 0 ≤ ( 1 / ( normℎ ‘ 𝑥 ) ) ) |
59 |
56 57 58
|
mpanl12 |
⊢ ( ( ( normℎ ‘ 𝑥 ) ∈ ℝ ∧ 0 < ( normℎ ‘ 𝑥 ) ) → 0 ≤ ( 1 / ( normℎ ‘ 𝑥 ) ) ) |
60 |
9 55 59
|
syl2anc |
⊢ ( 𝑥 ∈ ( 𝐴 ∖ 𝐵 ) → 0 ≤ ( 1 / ( normℎ ‘ 𝑥 ) ) ) |
61 |
21 60
|
absidd |
⊢ ( 𝑥 ∈ ( 𝐴 ∖ 𝐵 ) → ( abs ‘ ( 1 / ( normℎ ‘ 𝑥 ) ) ) = ( 1 / ( normℎ ‘ 𝑥 ) ) ) |
62 |
61
|
oveq1d |
⊢ ( 𝑥 ∈ ( 𝐴 ∖ 𝐵 ) → ( ( abs ‘ ( 1 / ( normℎ ‘ 𝑥 ) ) ) · ( normℎ ‘ 𝑥 ) ) = ( ( 1 / ( normℎ ‘ 𝑥 ) ) · ( normℎ ‘ 𝑥 ) ) ) |
63 |
28 20
|
recid2d |
⊢ ( 𝑥 ∈ ( 𝐴 ∖ 𝐵 ) → ( ( 1 / ( normℎ ‘ 𝑥 ) ) · ( normℎ ‘ 𝑥 ) ) = 1 ) |
64 |
51 62 63
|
3eqtrd |
⊢ ( 𝑥 ∈ ( 𝐴 ∖ 𝐵 ) → ( normℎ ‘ ( ( 1 / ( normℎ ‘ 𝑥 ) ) ·ℎ 𝑥 ) ) = 1 ) |
65 |
|
fveqeq2 |
⊢ ( 𝑢 = ( ( 1 / ( normℎ ‘ 𝑥 ) ) ·ℎ 𝑥 ) → ( ( normℎ ‘ 𝑢 ) = 1 ↔ ( normℎ ‘ ( ( 1 / ( normℎ ‘ 𝑥 ) ) ·ℎ 𝑥 ) ) = 1 ) ) |
66 |
65
|
rspcev |
⊢ ( ( ( ( 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 |
⊢ ( 𝑥 ∈ ( 𝐴 ∖ 𝐵 ) → ∃ 𝑢 ∈ ( 𝐴 ∖ 𝐵 ) ( normℎ ‘ 𝑢 ) = 1 ) |
68 |
67
|
exlimiv |
⊢ ( ∃ 𝑥 𝑥 ∈ ( 𝐴 ∖ 𝐵 ) → ∃ 𝑢 ∈ ( 𝐴 ∖ 𝐵 ) ( normℎ ‘ 𝑢 ) = 1 ) |
69 |
5 68
|
sylbi |
⊢ ( ¬ 𝐴 ⊆ 𝐵 → ∃ 𝑢 ∈ ( 𝐴 ∖ 𝐵 ) ( normℎ ‘ 𝑢 ) = 1 ) |