Step |
Hyp |
Ref |
Expression |
1 |
|
blocni.8 |
⊢ 𝐶 = ( IndMet ‘ 𝑈 ) |
2 |
|
blocni.d |
⊢ 𝐷 = ( IndMet ‘ 𝑊 ) |
3 |
|
blocni.j |
⊢ 𝐽 = ( MetOpen ‘ 𝐶 ) |
4 |
|
blocni.k |
⊢ 𝐾 = ( MetOpen ‘ 𝐷 ) |
5 |
|
blocni.4 |
⊢ 𝐿 = ( 𝑈 LnOp 𝑊 ) |
6 |
|
blocni.5 |
⊢ 𝐵 = ( 𝑈 BLnOp 𝑊 ) |
7 |
|
blocni.u |
⊢ 𝑈 ∈ NrmCVec |
8 |
|
blocni.w |
⊢ 𝑊 ∈ NrmCVec |
9 |
|
blocni.l |
⊢ 𝑇 ∈ 𝐿 |
10 |
|
blocnilem.1 |
⊢ 𝑋 = ( BaseSet ‘ 𝑈 ) |
11 |
10 1
|
imsxmet |
⊢ ( 𝑈 ∈ NrmCVec → 𝐶 ∈ ( ∞Met ‘ 𝑋 ) ) |
12 |
7 11
|
ax-mp |
⊢ 𝐶 ∈ ( ∞Met ‘ 𝑋 ) |
13 |
|
eqid |
⊢ ( BaseSet ‘ 𝑊 ) = ( BaseSet ‘ 𝑊 ) |
14 |
13 2
|
imsxmet |
⊢ ( 𝑊 ∈ NrmCVec → 𝐷 ∈ ( ∞Met ‘ ( BaseSet ‘ 𝑊 ) ) ) |
15 |
8 14
|
ax-mp |
⊢ 𝐷 ∈ ( ∞Met ‘ ( BaseSet ‘ 𝑊 ) ) |
16 |
|
1rp |
⊢ 1 ∈ ℝ+ |
17 |
3 4
|
metcnpi3 |
⊢ ( ( ( 𝐶 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝐷 ∈ ( ∞Met ‘ ( BaseSet ‘ 𝑊 ) ) ) ∧ ( 𝑇 ∈ ( ( 𝐽 CnP 𝐾 ) ‘ 𝑃 ) ∧ 1 ∈ ℝ+ ) ) → ∃ 𝑦 ∈ ℝ+ ∀ 𝑥 ∈ 𝑋 ( ( 𝑥 𝐶 𝑃 ) ≤ 𝑦 → ( ( 𝑇 ‘ 𝑥 ) 𝐷 ( 𝑇 ‘ 𝑃 ) ) ≤ 1 ) ) |
18 |
16 17
|
mpanr2 |
⊢ ( ( ( 𝐶 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝐷 ∈ ( ∞Met ‘ ( BaseSet ‘ 𝑊 ) ) ) ∧ 𝑇 ∈ ( ( 𝐽 CnP 𝐾 ) ‘ 𝑃 ) ) → ∃ 𝑦 ∈ ℝ+ ∀ 𝑥 ∈ 𝑋 ( ( 𝑥 𝐶 𝑃 ) ≤ 𝑦 → ( ( 𝑇 ‘ 𝑥 ) 𝐷 ( 𝑇 ‘ 𝑃 ) ) ≤ 1 ) ) |
19 |
12 15 18
|
mpanl12 |
⊢ ( 𝑇 ∈ ( ( 𝐽 CnP 𝐾 ) ‘ 𝑃 ) → ∃ 𝑦 ∈ ℝ+ ∀ 𝑥 ∈ 𝑋 ( ( 𝑥 𝐶 𝑃 ) ≤ 𝑦 → ( ( 𝑇 ‘ 𝑥 ) 𝐷 ( 𝑇 ‘ 𝑃 ) ) ≤ 1 ) ) |
20 |
|
rpreccl |
⊢ ( 𝑦 ∈ ℝ+ → ( 1 / 𝑦 ) ∈ ℝ+ ) |
21 |
20
|
rpred |
⊢ ( 𝑦 ∈ ℝ+ → ( 1 / 𝑦 ) ∈ ℝ ) |
22 |
21
|
ad2antlr |
⊢ ( ( ( 𝑃 ∈ 𝑋 ∧ 𝑦 ∈ ℝ+ ) ∧ ∀ 𝑥 ∈ 𝑋 ( ( 𝑥 𝐶 𝑃 ) ≤ 𝑦 → ( ( 𝑇 ‘ 𝑥 ) 𝐷 ( 𝑇 ‘ 𝑃 ) ) ≤ 1 ) ) → ( 1 / 𝑦 ) ∈ ℝ ) |
23 |
|
eqid |
⊢ ( −𝑣 ‘ 𝑈 ) = ( −𝑣 ‘ 𝑈 ) |
24 |
|
eqid |
⊢ ( normCV ‘ 𝑈 ) = ( normCV ‘ 𝑈 ) |
25 |
10 23 24 1
|
imsdval |
⊢ ( ( 𝑈 ∈ NrmCVec ∧ 𝑥 ∈ 𝑋 ∧ 𝑃 ∈ 𝑋 ) → ( 𝑥 𝐶 𝑃 ) = ( ( normCV ‘ 𝑈 ) ‘ ( 𝑥 ( −𝑣 ‘ 𝑈 ) 𝑃 ) ) ) |
26 |
7 25
|
mp3an1 |
⊢ ( ( 𝑥 ∈ 𝑋 ∧ 𝑃 ∈ 𝑋 ) → ( 𝑥 𝐶 𝑃 ) = ( ( normCV ‘ 𝑈 ) ‘ ( 𝑥 ( −𝑣 ‘ 𝑈 ) 𝑃 ) ) ) |
27 |
26
|
breq1d |
⊢ ( ( 𝑥 ∈ 𝑋 ∧ 𝑃 ∈ 𝑋 ) → ( ( 𝑥 𝐶 𝑃 ) ≤ 𝑦 ↔ ( ( normCV ‘ 𝑈 ) ‘ ( 𝑥 ( −𝑣 ‘ 𝑈 ) 𝑃 ) ) ≤ 𝑦 ) ) |
28 |
10 13 5
|
lnof |
⊢ ( ( 𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇 ∈ 𝐿 ) → 𝑇 : 𝑋 ⟶ ( BaseSet ‘ 𝑊 ) ) |
29 |
7 8 9 28
|
mp3an |
⊢ 𝑇 : 𝑋 ⟶ ( BaseSet ‘ 𝑊 ) |
30 |
29
|
ffvelrni |
⊢ ( 𝑥 ∈ 𝑋 → ( 𝑇 ‘ 𝑥 ) ∈ ( BaseSet ‘ 𝑊 ) ) |
31 |
29
|
ffvelrni |
⊢ ( 𝑃 ∈ 𝑋 → ( 𝑇 ‘ 𝑃 ) ∈ ( BaseSet ‘ 𝑊 ) ) |
32 |
|
eqid |
⊢ ( −𝑣 ‘ 𝑊 ) = ( −𝑣 ‘ 𝑊 ) |
33 |
|
eqid |
⊢ ( normCV ‘ 𝑊 ) = ( normCV ‘ 𝑊 ) |
34 |
13 32 33 2
|
imsdval |
⊢ ( ( 𝑊 ∈ NrmCVec ∧ ( 𝑇 ‘ 𝑥 ) ∈ ( BaseSet ‘ 𝑊 ) ∧ ( 𝑇 ‘ 𝑃 ) ∈ ( BaseSet ‘ 𝑊 ) ) → ( ( 𝑇 ‘ 𝑥 ) 𝐷 ( 𝑇 ‘ 𝑃 ) ) = ( ( normCV ‘ 𝑊 ) ‘ ( ( 𝑇 ‘ 𝑥 ) ( −𝑣 ‘ 𝑊 ) ( 𝑇 ‘ 𝑃 ) ) ) ) |
35 |
8 34
|
mp3an1 |
⊢ ( ( ( 𝑇 ‘ 𝑥 ) ∈ ( BaseSet ‘ 𝑊 ) ∧ ( 𝑇 ‘ 𝑃 ) ∈ ( BaseSet ‘ 𝑊 ) ) → ( ( 𝑇 ‘ 𝑥 ) 𝐷 ( 𝑇 ‘ 𝑃 ) ) = ( ( normCV ‘ 𝑊 ) ‘ ( ( 𝑇 ‘ 𝑥 ) ( −𝑣 ‘ 𝑊 ) ( 𝑇 ‘ 𝑃 ) ) ) ) |
36 |
30 31 35
|
syl2an |
⊢ ( ( 𝑥 ∈ 𝑋 ∧ 𝑃 ∈ 𝑋 ) → ( ( 𝑇 ‘ 𝑥 ) 𝐷 ( 𝑇 ‘ 𝑃 ) ) = ( ( normCV ‘ 𝑊 ) ‘ ( ( 𝑇 ‘ 𝑥 ) ( −𝑣 ‘ 𝑊 ) ( 𝑇 ‘ 𝑃 ) ) ) ) |
37 |
7 8 9
|
3pm3.2i |
⊢ ( 𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇 ∈ 𝐿 ) |
38 |
10 23 32 5
|
lnosub |
⊢ ( ( ( 𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇 ∈ 𝐿 ) ∧ ( 𝑥 ∈ 𝑋 ∧ 𝑃 ∈ 𝑋 ) ) → ( 𝑇 ‘ ( 𝑥 ( −𝑣 ‘ 𝑈 ) 𝑃 ) ) = ( ( 𝑇 ‘ 𝑥 ) ( −𝑣 ‘ 𝑊 ) ( 𝑇 ‘ 𝑃 ) ) ) |
39 |
37 38
|
mpan |
⊢ ( ( 𝑥 ∈ 𝑋 ∧ 𝑃 ∈ 𝑋 ) → ( 𝑇 ‘ ( 𝑥 ( −𝑣 ‘ 𝑈 ) 𝑃 ) ) = ( ( 𝑇 ‘ 𝑥 ) ( −𝑣 ‘ 𝑊 ) ( 𝑇 ‘ 𝑃 ) ) ) |
40 |
39
|
fveq2d |
⊢ ( ( 𝑥 ∈ 𝑋 ∧ 𝑃 ∈ 𝑋 ) → ( ( normCV ‘ 𝑊 ) ‘ ( 𝑇 ‘ ( 𝑥 ( −𝑣 ‘ 𝑈 ) 𝑃 ) ) ) = ( ( normCV ‘ 𝑊 ) ‘ ( ( 𝑇 ‘ 𝑥 ) ( −𝑣 ‘ 𝑊 ) ( 𝑇 ‘ 𝑃 ) ) ) ) |
41 |
36 40
|
eqtr4d |
⊢ ( ( 𝑥 ∈ 𝑋 ∧ 𝑃 ∈ 𝑋 ) → ( ( 𝑇 ‘ 𝑥 ) 𝐷 ( 𝑇 ‘ 𝑃 ) ) = ( ( normCV ‘ 𝑊 ) ‘ ( 𝑇 ‘ ( 𝑥 ( −𝑣 ‘ 𝑈 ) 𝑃 ) ) ) ) |
42 |
41
|
breq1d |
⊢ ( ( 𝑥 ∈ 𝑋 ∧ 𝑃 ∈ 𝑋 ) → ( ( ( 𝑇 ‘ 𝑥 ) 𝐷 ( 𝑇 ‘ 𝑃 ) ) ≤ 1 ↔ ( ( normCV ‘ 𝑊 ) ‘ ( 𝑇 ‘ ( 𝑥 ( −𝑣 ‘ 𝑈 ) 𝑃 ) ) ) ≤ 1 ) ) |
43 |
27 42
|
imbi12d |
⊢ ( ( 𝑥 ∈ 𝑋 ∧ 𝑃 ∈ 𝑋 ) → ( ( ( 𝑥 𝐶 𝑃 ) ≤ 𝑦 → ( ( 𝑇 ‘ 𝑥 ) 𝐷 ( 𝑇 ‘ 𝑃 ) ) ≤ 1 ) ↔ ( ( ( normCV ‘ 𝑈 ) ‘ ( 𝑥 ( −𝑣 ‘ 𝑈 ) 𝑃 ) ) ≤ 𝑦 → ( ( normCV ‘ 𝑊 ) ‘ ( 𝑇 ‘ ( 𝑥 ( −𝑣 ‘ 𝑈 ) 𝑃 ) ) ) ≤ 1 ) ) ) |
44 |
43
|
ancoms |
⊢ ( ( 𝑃 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋 ) → ( ( ( 𝑥 𝐶 𝑃 ) ≤ 𝑦 → ( ( 𝑇 ‘ 𝑥 ) 𝐷 ( 𝑇 ‘ 𝑃 ) ) ≤ 1 ) ↔ ( ( ( normCV ‘ 𝑈 ) ‘ ( 𝑥 ( −𝑣 ‘ 𝑈 ) 𝑃 ) ) ≤ 𝑦 → ( ( normCV ‘ 𝑊 ) ‘ ( 𝑇 ‘ ( 𝑥 ( −𝑣 ‘ 𝑈 ) 𝑃 ) ) ) ≤ 1 ) ) ) |
45 |
44
|
adantlr |
⊢ ( ( ( 𝑃 ∈ 𝑋 ∧ 𝑦 ∈ ℝ+ ) ∧ 𝑥 ∈ 𝑋 ) → ( ( ( 𝑥 𝐶 𝑃 ) ≤ 𝑦 → ( ( 𝑇 ‘ 𝑥 ) 𝐷 ( 𝑇 ‘ 𝑃 ) ) ≤ 1 ) ↔ ( ( ( normCV ‘ 𝑈 ) ‘ ( 𝑥 ( −𝑣 ‘ 𝑈 ) 𝑃 ) ) ≤ 𝑦 → ( ( normCV ‘ 𝑊 ) ‘ ( 𝑇 ‘ ( 𝑥 ( −𝑣 ‘ 𝑈 ) 𝑃 ) ) ) ≤ 1 ) ) ) |
46 |
45
|
ralbidva |
⊢ ( ( 𝑃 ∈ 𝑋 ∧ 𝑦 ∈ ℝ+ ) → ( ∀ 𝑥 ∈ 𝑋 ( ( 𝑥 𝐶 𝑃 ) ≤ 𝑦 → ( ( 𝑇 ‘ 𝑥 ) 𝐷 ( 𝑇 ‘ 𝑃 ) ) ≤ 1 ) ↔ ∀ 𝑥 ∈ 𝑋 ( ( ( normCV ‘ 𝑈 ) ‘ ( 𝑥 ( −𝑣 ‘ 𝑈 ) 𝑃 ) ) ≤ 𝑦 → ( ( normCV ‘ 𝑊 ) ‘ ( 𝑇 ‘ ( 𝑥 ( −𝑣 ‘ 𝑈 ) 𝑃 ) ) ) ≤ 1 ) ) ) |
47 |
|
2fveq3 |
⊢ ( 𝑧 = ( 0vec ‘ 𝑈 ) → ( ( normCV ‘ 𝑊 ) ‘ ( 𝑇 ‘ 𝑧 ) ) = ( ( normCV ‘ 𝑊 ) ‘ ( 𝑇 ‘ ( 0vec ‘ 𝑈 ) ) ) ) |
48 |
|
fveq2 |
⊢ ( 𝑧 = ( 0vec ‘ 𝑈 ) → ( ( normCV ‘ 𝑈 ) ‘ 𝑧 ) = ( ( normCV ‘ 𝑈 ) ‘ ( 0vec ‘ 𝑈 ) ) ) |
49 |
48
|
oveq2d |
⊢ ( 𝑧 = ( 0vec ‘ 𝑈 ) → ( ( 1 / 𝑦 ) · ( ( normCV ‘ 𝑈 ) ‘ 𝑧 ) ) = ( ( 1 / 𝑦 ) · ( ( normCV ‘ 𝑈 ) ‘ ( 0vec ‘ 𝑈 ) ) ) ) |
50 |
47 49
|
breq12d |
⊢ ( 𝑧 = ( 0vec ‘ 𝑈 ) → ( ( ( normCV ‘ 𝑊 ) ‘ ( 𝑇 ‘ 𝑧 ) ) ≤ ( ( 1 / 𝑦 ) · ( ( normCV ‘ 𝑈 ) ‘ 𝑧 ) ) ↔ ( ( normCV ‘ 𝑊 ) ‘ ( 𝑇 ‘ ( 0vec ‘ 𝑈 ) ) ) ≤ ( ( 1 / 𝑦 ) · ( ( normCV ‘ 𝑈 ) ‘ ( 0vec ‘ 𝑈 ) ) ) ) ) |
51 |
7
|
a1i |
⊢ ( ( ( 𝑃 ∈ 𝑋 ∧ 𝑦 ∈ ℝ+ ) ∧ ( 𝑧 ∈ 𝑋 ∧ 𝑧 ≠ ( 0vec ‘ 𝑈 ) ) ) → 𝑈 ∈ NrmCVec ) |
52 |
|
simpll |
⊢ ( ( ( 𝑃 ∈ 𝑋 ∧ 𝑦 ∈ ℝ+ ) ∧ ( 𝑧 ∈ 𝑋 ∧ 𝑧 ≠ ( 0vec ‘ 𝑈 ) ) ) → 𝑃 ∈ 𝑋 ) |
53 |
|
simpr |
⊢ ( ( 𝑃 ∈ 𝑋 ∧ 𝑦 ∈ ℝ+ ) → 𝑦 ∈ ℝ+ ) |
54 |
10 24
|
nvcl |
⊢ ( ( 𝑈 ∈ NrmCVec ∧ 𝑧 ∈ 𝑋 ) → ( ( normCV ‘ 𝑈 ) ‘ 𝑧 ) ∈ ℝ ) |
55 |
7 54
|
mpan |
⊢ ( 𝑧 ∈ 𝑋 → ( ( normCV ‘ 𝑈 ) ‘ 𝑧 ) ∈ ℝ ) |
56 |
55
|
adantr |
⊢ ( ( 𝑧 ∈ 𝑋 ∧ 𝑧 ≠ ( 0vec ‘ 𝑈 ) ) → ( ( normCV ‘ 𝑈 ) ‘ 𝑧 ) ∈ ℝ ) |
57 |
|
eqid |
⊢ ( 0vec ‘ 𝑈 ) = ( 0vec ‘ 𝑈 ) |
58 |
10 57 24
|
nvgt0 |
⊢ ( ( 𝑈 ∈ NrmCVec ∧ 𝑧 ∈ 𝑋 ) → ( 𝑧 ≠ ( 0vec ‘ 𝑈 ) ↔ 0 < ( ( normCV ‘ 𝑈 ) ‘ 𝑧 ) ) ) |
59 |
7 58
|
mpan |
⊢ ( 𝑧 ∈ 𝑋 → ( 𝑧 ≠ ( 0vec ‘ 𝑈 ) ↔ 0 < ( ( normCV ‘ 𝑈 ) ‘ 𝑧 ) ) ) |
60 |
59
|
biimpa |
⊢ ( ( 𝑧 ∈ 𝑋 ∧ 𝑧 ≠ ( 0vec ‘ 𝑈 ) ) → 0 < ( ( normCV ‘ 𝑈 ) ‘ 𝑧 ) ) |
61 |
56 60
|
elrpd |
⊢ ( ( 𝑧 ∈ 𝑋 ∧ 𝑧 ≠ ( 0vec ‘ 𝑈 ) ) → ( ( normCV ‘ 𝑈 ) ‘ 𝑧 ) ∈ ℝ+ ) |
62 |
|
rpdivcl |
⊢ ( ( 𝑦 ∈ ℝ+ ∧ ( ( normCV ‘ 𝑈 ) ‘ 𝑧 ) ∈ ℝ+ ) → ( 𝑦 / ( ( normCV ‘ 𝑈 ) ‘ 𝑧 ) ) ∈ ℝ+ ) |
63 |
53 61 62
|
syl2an |
⊢ ( ( ( 𝑃 ∈ 𝑋 ∧ 𝑦 ∈ ℝ+ ) ∧ ( 𝑧 ∈ 𝑋 ∧ 𝑧 ≠ ( 0vec ‘ 𝑈 ) ) ) → ( 𝑦 / ( ( normCV ‘ 𝑈 ) ‘ 𝑧 ) ) ∈ ℝ+ ) |
64 |
63
|
rpcnd |
⊢ ( ( ( 𝑃 ∈ 𝑋 ∧ 𝑦 ∈ ℝ+ ) ∧ ( 𝑧 ∈ 𝑋 ∧ 𝑧 ≠ ( 0vec ‘ 𝑈 ) ) ) → ( 𝑦 / ( ( normCV ‘ 𝑈 ) ‘ 𝑧 ) ) ∈ ℂ ) |
65 |
|
simprl |
⊢ ( ( ( 𝑃 ∈ 𝑋 ∧ 𝑦 ∈ ℝ+ ) ∧ ( 𝑧 ∈ 𝑋 ∧ 𝑧 ≠ ( 0vec ‘ 𝑈 ) ) ) → 𝑧 ∈ 𝑋 ) |
66 |
|
eqid |
⊢ ( ·𝑠OLD ‘ 𝑈 ) = ( ·𝑠OLD ‘ 𝑈 ) |
67 |
10 66
|
nvscl |
⊢ ( ( 𝑈 ∈ NrmCVec ∧ ( 𝑦 / ( ( normCV ‘ 𝑈 ) ‘ 𝑧 ) ) ∈ ℂ ∧ 𝑧 ∈ 𝑋 ) → ( ( 𝑦 / ( ( normCV ‘ 𝑈 ) ‘ 𝑧 ) ) ( ·𝑠OLD ‘ 𝑈 ) 𝑧 ) ∈ 𝑋 ) |
68 |
51 64 65 67
|
syl3anc |
⊢ ( ( ( 𝑃 ∈ 𝑋 ∧ 𝑦 ∈ ℝ+ ) ∧ ( 𝑧 ∈ 𝑋 ∧ 𝑧 ≠ ( 0vec ‘ 𝑈 ) ) ) → ( ( 𝑦 / ( ( normCV ‘ 𝑈 ) ‘ 𝑧 ) ) ( ·𝑠OLD ‘ 𝑈 ) 𝑧 ) ∈ 𝑋 ) |
69 |
|
eqid |
⊢ ( +𝑣 ‘ 𝑈 ) = ( +𝑣 ‘ 𝑈 ) |
70 |
10 69 23
|
nvpncan2 |
⊢ ( ( 𝑈 ∈ NrmCVec ∧ 𝑃 ∈ 𝑋 ∧ ( ( 𝑦 / ( ( normCV ‘ 𝑈 ) ‘ 𝑧 ) ) ( ·𝑠OLD ‘ 𝑈 ) 𝑧 ) ∈ 𝑋 ) → ( ( 𝑃 ( +𝑣 ‘ 𝑈 ) ( ( 𝑦 / ( ( normCV ‘ 𝑈 ) ‘ 𝑧 ) ) ( ·𝑠OLD ‘ 𝑈 ) 𝑧 ) ) ( −𝑣 ‘ 𝑈 ) 𝑃 ) = ( ( 𝑦 / ( ( normCV ‘ 𝑈 ) ‘ 𝑧 ) ) ( ·𝑠OLD ‘ 𝑈 ) 𝑧 ) ) |
71 |
51 52 68 70
|
syl3anc |
⊢ ( ( ( 𝑃 ∈ 𝑋 ∧ 𝑦 ∈ ℝ+ ) ∧ ( 𝑧 ∈ 𝑋 ∧ 𝑧 ≠ ( 0vec ‘ 𝑈 ) ) ) → ( ( 𝑃 ( +𝑣 ‘ 𝑈 ) ( ( 𝑦 / ( ( normCV ‘ 𝑈 ) ‘ 𝑧 ) ) ( ·𝑠OLD ‘ 𝑈 ) 𝑧 ) ) ( −𝑣 ‘ 𝑈 ) 𝑃 ) = ( ( 𝑦 / ( ( normCV ‘ 𝑈 ) ‘ 𝑧 ) ) ( ·𝑠OLD ‘ 𝑈 ) 𝑧 ) ) |
72 |
71
|
fveq2d |
⊢ ( ( ( 𝑃 ∈ 𝑋 ∧ 𝑦 ∈ ℝ+ ) ∧ ( 𝑧 ∈ 𝑋 ∧ 𝑧 ≠ ( 0vec ‘ 𝑈 ) ) ) → ( ( normCV ‘ 𝑈 ) ‘ ( ( 𝑃 ( +𝑣 ‘ 𝑈 ) ( ( 𝑦 / ( ( normCV ‘ 𝑈 ) ‘ 𝑧 ) ) ( ·𝑠OLD ‘ 𝑈 ) 𝑧 ) ) ( −𝑣 ‘ 𝑈 ) 𝑃 ) ) = ( ( normCV ‘ 𝑈 ) ‘ ( ( 𝑦 / ( ( normCV ‘ 𝑈 ) ‘ 𝑧 ) ) ( ·𝑠OLD ‘ 𝑈 ) 𝑧 ) ) ) |
73 |
63
|
rprege0d |
⊢ ( ( ( 𝑃 ∈ 𝑋 ∧ 𝑦 ∈ ℝ+ ) ∧ ( 𝑧 ∈ 𝑋 ∧ 𝑧 ≠ ( 0vec ‘ 𝑈 ) ) ) → ( ( 𝑦 / ( ( normCV ‘ 𝑈 ) ‘ 𝑧 ) ) ∈ ℝ ∧ 0 ≤ ( 𝑦 / ( ( normCV ‘ 𝑈 ) ‘ 𝑧 ) ) ) ) |
74 |
10 66 24
|
nvsge0 |
⊢ ( ( 𝑈 ∈ NrmCVec ∧ ( ( 𝑦 / ( ( normCV ‘ 𝑈 ) ‘ 𝑧 ) ) ∈ ℝ ∧ 0 ≤ ( 𝑦 / ( ( normCV ‘ 𝑈 ) ‘ 𝑧 ) ) ) ∧ 𝑧 ∈ 𝑋 ) → ( ( normCV ‘ 𝑈 ) ‘ ( ( 𝑦 / ( ( normCV ‘ 𝑈 ) ‘ 𝑧 ) ) ( ·𝑠OLD ‘ 𝑈 ) 𝑧 ) ) = ( ( 𝑦 / ( ( normCV ‘ 𝑈 ) ‘ 𝑧 ) ) · ( ( normCV ‘ 𝑈 ) ‘ 𝑧 ) ) ) |
75 |
51 73 65 74
|
syl3anc |
⊢ ( ( ( 𝑃 ∈ 𝑋 ∧ 𝑦 ∈ ℝ+ ) ∧ ( 𝑧 ∈ 𝑋 ∧ 𝑧 ≠ ( 0vec ‘ 𝑈 ) ) ) → ( ( normCV ‘ 𝑈 ) ‘ ( ( 𝑦 / ( ( normCV ‘ 𝑈 ) ‘ 𝑧 ) ) ( ·𝑠OLD ‘ 𝑈 ) 𝑧 ) ) = ( ( 𝑦 / ( ( normCV ‘ 𝑈 ) ‘ 𝑧 ) ) · ( ( normCV ‘ 𝑈 ) ‘ 𝑧 ) ) ) |
76 |
|
rpcn |
⊢ ( 𝑦 ∈ ℝ+ → 𝑦 ∈ ℂ ) |
77 |
76
|
ad2antlr |
⊢ ( ( ( 𝑃 ∈ 𝑋 ∧ 𝑦 ∈ ℝ+ ) ∧ ( 𝑧 ∈ 𝑋 ∧ 𝑧 ≠ ( 0vec ‘ 𝑈 ) ) ) → 𝑦 ∈ ℂ ) |
78 |
55
|
ad2antrl |
⊢ ( ( ( 𝑃 ∈ 𝑋 ∧ 𝑦 ∈ ℝ+ ) ∧ ( 𝑧 ∈ 𝑋 ∧ 𝑧 ≠ ( 0vec ‘ 𝑈 ) ) ) → ( ( normCV ‘ 𝑈 ) ‘ 𝑧 ) ∈ ℝ ) |
79 |
78
|
recnd |
⊢ ( ( ( 𝑃 ∈ 𝑋 ∧ 𝑦 ∈ ℝ+ ) ∧ ( 𝑧 ∈ 𝑋 ∧ 𝑧 ≠ ( 0vec ‘ 𝑈 ) ) ) → ( ( normCV ‘ 𝑈 ) ‘ 𝑧 ) ∈ ℂ ) |
80 |
10 57 24
|
nvz |
⊢ ( ( 𝑈 ∈ NrmCVec ∧ 𝑧 ∈ 𝑋 ) → ( ( ( normCV ‘ 𝑈 ) ‘ 𝑧 ) = 0 ↔ 𝑧 = ( 0vec ‘ 𝑈 ) ) ) |
81 |
7 80
|
mpan |
⊢ ( 𝑧 ∈ 𝑋 → ( ( ( normCV ‘ 𝑈 ) ‘ 𝑧 ) = 0 ↔ 𝑧 = ( 0vec ‘ 𝑈 ) ) ) |
82 |
81
|
necon3bid |
⊢ ( 𝑧 ∈ 𝑋 → ( ( ( normCV ‘ 𝑈 ) ‘ 𝑧 ) ≠ 0 ↔ 𝑧 ≠ ( 0vec ‘ 𝑈 ) ) ) |
83 |
82
|
biimpar |
⊢ ( ( 𝑧 ∈ 𝑋 ∧ 𝑧 ≠ ( 0vec ‘ 𝑈 ) ) → ( ( normCV ‘ 𝑈 ) ‘ 𝑧 ) ≠ 0 ) |
84 |
83
|
adantl |
⊢ ( ( ( 𝑃 ∈ 𝑋 ∧ 𝑦 ∈ ℝ+ ) ∧ ( 𝑧 ∈ 𝑋 ∧ 𝑧 ≠ ( 0vec ‘ 𝑈 ) ) ) → ( ( normCV ‘ 𝑈 ) ‘ 𝑧 ) ≠ 0 ) |
85 |
77 79 84
|
divcan1d |
⊢ ( ( ( 𝑃 ∈ 𝑋 ∧ 𝑦 ∈ ℝ+ ) ∧ ( 𝑧 ∈ 𝑋 ∧ 𝑧 ≠ ( 0vec ‘ 𝑈 ) ) ) → ( ( 𝑦 / ( ( normCV ‘ 𝑈 ) ‘ 𝑧 ) ) · ( ( normCV ‘ 𝑈 ) ‘ 𝑧 ) ) = 𝑦 ) |
86 |
72 75 85
|
3eqtrd |
⊢ ( ( ( 𝑃 ∈ 𝑋 ∧ 𝑦 ∈ ℝ+ ) ∧ ( 𝑧 ∈ 𝑋 ∧ 𝑧 ≠ ( 0vec ‘ 𝑈 ) ) ) → ( ( normCV ‘ 𝑈 ) ‘ ( ( 𝑃 ( +𝑣 ‘ 𝑈 ) ( ( 𝑦 / ( ( normCV ‘ 𝑈 ) ‘ 𝑧 ) ) ( ·𝑠OLD ‘ 𝑈 ) 𝑧 ) ) ( −𝑣 ‘ 𝑈 ) 𝑃 ) ) = 𝑦 ) |
87 |
|
rpre |
⊢ ( 𝑦 ∈ ℝ+ → 𝑦 ∈ ℝ ) |
88 |
87
|
leidd |
⊢ ( 𝑦 ∈ ℝ+ → 𝑦 ≤ 𝑦 ) |
89 |
88
|
ad2antlr |
⊢ ( ( ( 𝑃 ∈ 𝑋 ∧ 𝑦 ∈ ℝ+ ) ∧ ( 𝑧 ∈ 𝑋 ∧ 𝑧 ≠ ( 0vec ‘ 𝑈 ) ) ) → 𝑦 ≤ 𝑦 ) |
90 |
86 89
|
eqbrtrd |
⊢ ( ( ( 𝑃 ∈ 𝑋 ∧ 𝑦 ∈ ℝ+ ) ∧ ( 𝑧 ∈ 𝑋 ∧ 𝑧 ≠ ( 0vec ‘ 𝑈 ) ) ) → ( ( normCV ‘ 𝑈 ) ‘ ( ( 𝑃 ( +𝑣 ‘ 𝑈 ) ( ( 𝑦 / ( ( normCV ‘ 𝑈 ) ‘ 𝑧 ) ) ( ·𝑠OLD ‘ 𝑈 ) 𝑧 ) ) ( −𝑣 ‘ 𝑈 ) 𝑃 ) ) ≤ 𝑦 ) |
91 |
10 69
|
nvgcl |
⊢ ( ( 𝑈 ∈ NrmCVec ∧ 𝑃 ∈ 𝑋 ∧ ( ( 𝑦 / ( ( normCV ‘ 𝑈 ) ‘ 𝑧 ) ) ( ·𝑠OLD ‘ 𝑈 ) 𝑧 ) ∈ 𝑋 ) → ( 𝑃 ( +𝑣 ‘ 𝑈 ) ( ( 𝑦 / ( ( normCV ‘ 𝑈 ) ‘ 𝑧 ) ) ( ·𝑠OLD ‘ 𝑈 ) 𝑧 ) ) ∈ 𝑋 ) |
92 |
51 52 68 91
|
syl3anc |
⊢ ( ( ( 𝑃 ∈ 𝑋 ∧ 𝑦 ∈ ℝ+ ) ∧ ( 𝑧 ∈ 𝑋 ∧ 𝑧 ≠ ( 0vec ‘ 𝑈 ) ) ) → ( 𝑃 ( +𝑣 ‘ 𝑈 ) ( ( 𝑦 / ( ( normCV ‘ 𝑈 ) ‘ 𝑧 ) ) ( ·𝑠OLD ‘ 𝑈 ) 𝑧 ) ) ∈ 𝑋 ) |
93 |
|
fvoveq1 |
⊢ ( 𝑥 = ( 𝑃 ( +𝑣 ‘ 𝑈 ) ( ( 𝑦 / ( ( normCV ‘ 𝑈 ) ‘ 𝑧 ) ) ( ·𝑠OLD ‘ 𝑈 ) 𝑧 ) ) → ( ( normCV ‘ 𝑈 ) ‘ ( 𝑥 ( −𝑣 ‘ 𝑈 ) 𝑃 ) ) = ( ( normCV ‘ 𝑈 ) ‘ ( ( 𝑃 ( +𝑣 ‘ 𝑈 ) ( ( 𝑦 / ( ( normCV ‘ 𝑈 ) ‘ 𝑧 ) ) ( ·𝑠OLD ‘ 𝑈 ) 𝑧 ) ) ( −𝑣 ‘ 𝑈 ) 𝑃 ) ) ) |
94 |
93
|
breq1d |
⊢ ( 𝑥 = ( 𝑃 ( +𝑣 ‘ 𝑈 ) ( ( 𝑦 / ( ( normCV ‘ 𝑈 ) ‘ 𝑧 ) ) ( ·𝑠OLD ‘ 𝑈 ) 𝑧 ) ) → ( ( ( normCV ‘ 𝑈 ) ‘ ( 𝑥 ( −𝑣 ‘ 𝑈 ) 𝑃 ) ) ≤ 𝑦 ↔ ( ( normCV ‘ 𝑈 ) ‘ ( ( 𝑃 ( +𝑣 ‘ 𝑈 ) ( ( 𝑦 / ( ( normCV ‘ 𝑈 ) ‘ 𝑧 ) ) ( ·𝑠OLD ‘ 𝑈 ) 𝑧 ) ) ( −𝑣 ‘ 𝑈 ) 𝑃 ) ) ≤ 𝑦 ) ) |
95 |
|
fvoveq1 |
⊢ ( 𝑥 = ( 𝑃 ( +𝑣 ‘ 𝑈 ) ( ( 𝑦 / ( ( normCV ‘ 𝑈 ) ‘ 𝑧 ) ) ( ·𝑠OLD ‘ 𝑈 ) 𝑧 ) ) → ( 𝑇 ‘ ( 𝑥 ( −𝑣 ‘ 𝑈 ) 𝑃 ) ) = ( 𝑇 ‘ ( ( 𝑃 ( +𝑣 ‘ 𝑈 ) ( ( 𝑦 / ( ( normCV ‘ 𝑈 ) ‘ 𝑧 ) ) ( ·𝑠OLD ‘ 𝑈 ) 𝑧 ) ) ( −𝑣 ‘ 𝑈 ) 𝑃 ) ) ) |
96 |
95
|
fveq2d |
⊢ ( 𝑥 = ( 𝑃 ( +𝑣 ‘ 𝑈 ) ( ( 𝑦 / ( ( normCV ‘ 𝑈 ) ‘ 𝑧 ) ) ( ·𝑠OLD ‘ 𝑈 ) 𝑧 ) ) → ( ( normCV ‘ 𝑊 ) ‘ ( 𝑇 ‘ ( 𝑥 ( −𝑣 ‘ 𝑈 ) 𝑃 ) ) ) = ( ( normCV ‘ 𝑊 ) ‘ ( 𝑇 ‘ ( ( 𝑃 ( +𝑣 ‘ 𝑈 ) ( ( 𝑦 / ( ( normCV ‘ 𝑈 ) ‘ 𝑧 ) ) ( ·𝑠OLD ‘ 𝑈 ) 𝑧 ) ) ( −𝑣 ‘ 𝑈 ) 𝑃 ) ) ) ) |
97 |
96
|
breq1d |
⊢ ( 𝑥 = ( 𝑃 ( +𝑣 ‘ 𝑈 ) ( ( 𝑦 / ( ( normCV ‘ 𝑈 ) ‘ 𝑧 ) ) ( ·𝑠OLD ‘ 𝑈 ) 𝑧 ) ) → ( ( ( normCV ‘ 𝑊 ) ‘ ( 𝑇 ‘ ( 𝑥 ( −𝑣 ‘ 𝑈 ) 𝑃 ) ) ) ≤ 1 ↔ ( ( normCV ‘ 𝑊 ) ‘ ( 𝑇 ‘ ( ( 𝑃 ( +𝑣 ‘ 𝑈 ) ( ( 𝑦 / ( ( normCV ‘ 𝑈 ) ‘ 𝑧 ) ) ( ·𝑠OLD ‘ 𝑈 ) 𝑧 ) ) ( −𝑣 ‘ 𝑈 ) 𝑃 ) ) ) ≤ 1 ) ) |
98 |
94 97
|
imbi12d |
⊢ ( 𝑥 = ( 𝑃 ( +𝑣 ‘ 𝑈 ) ( ( 𝑦 / ( ( normCV ‘ 𝑈 ) ‘ 𝑧 ) ) ( ·𝑠OLD ‘ 𝑈 ) 𝑧 ) ) → ( ( ( ( normCV ‘ 𝑈 ) ‘ ( 𝑥 ( −𝑣 ‘ 𝑈 ) 𝑃 ) ) ≤ 𝑦 → ( ( normCV ‘ 𝑊 ) ‘ ( 𝑇 ‘ ( 𝑥 ( −𝑣 ‘ 𝑈 ) 𝑃 ) ) ) ≤ 1 ) ↔ ( ( ( normCV ‘ 𝑈 ) ‘ ( ( 𝑃 ( +𝑣 ‘ 𝑈 ) ( ( 𝑦 / ( ( normCV ‘ 𝑈 ) ‘ 𝑧 ) ) ( ·𝑠OLD ‘ 𝑈 ) 𝑧 ) ) ( −𝑣 ‘ 𝑈 ) 𝑃 ) ) ≤ 𝑦 → ( ( normCV ‘ 𝑊 ) ‘ ( 𝑇 ‘ ( ( 𝑃 ( +𝑣 ‘ 𝑈 ) ( ( 𝑦 / ( ( normCV ‘ 𝑈 ) ‘ 𝑧 ) ) ( ·𝑠OLD ‘ 𝑈 ) 𝑧 ) ) ( −𝑣 ‘ 𝑈 ) 𝑃 ) ) ) ≤ 1 ) ) ) |
99 |
98
|
rspcv |
⊢ ( ( 𝑃 ( +𝑣 ‘ 𝑈 ) ( ( 𝑦 / ( ( normCV ‘ 𝑈 ) ‘ 𝑧 ) ) ( ·𝑠OLD ‘ 𝑈 ) 𝑧 ) ) ∈ 𝑋 → ( ∀ 𝑥 ∈ 𝑋 ( ( ( normCV ‘ 𝑈 ) ‘ ( 𝑥 ( −𝑣 ‘ 𝑈 ) 𝑃 ) ) ≤ 𝑦 → ( ( normCV ‘ 𝑊 ) ‘ ( 𝑇 ‘ ( 𝑥 ( −𝑣 ‘ 𝑈 ) 𝑃 ) ) ) ≤ 1 ) → ( ( ( normCV ‘ 𝑈 ) ‘ ( ( 𝑃 ( +𝑣 ‘ 𝑈 ) ( ( 𝑦 / ( ( normCV ‘ 𝑈 ) ‘ 𝑧 ) ) ( ·𝑠OLD ‘ 𝑈 ) 𝑧 ) ) ( −𝑣 ‘ 𝑈 ) 𝑃 ) ) ≤ 𝑦 → ( ( normCV ‘ 𝑊 ) ‘ ( 𝑇 ‘ ( ( 𝑃 ( +𝑣 ‘ 𝑈 ) ( ( 𝑦 / ( ( normCV ‘ 𝑈 ) ‘ 𝑧 ) ) ( ·𝑠OLD ‘ 𝑈 ) 𝑧 ) ) ( −𝑣 ‘ 𝑈 ) 𝑃 ) ) ) ≤ 1 ) ) ) |
100 |
92 99
|
syl |
⊢ ( ( ( 𝑃 ∈ 𝑋 ∧ 𝑦 ∈ ℝ+ ) ∧ ( 𝑧 ∈ 𝑋 ∧ 𝑧 ≠ ( 0vec ‘ 𝑈 ) ) ) → ( ∀ 𝑥 ∈ 𝑋 ( ( ( normCV ‘ 𝑈 ) ‘ ( 𝑥 ( −𝑣 ‘ 𝑈 ) 𝑃 ) ) ≤ 𝑦 → ( ( normCV ‘ 𝑊 ) ‘ ( 𝑇 ‘ ( 𝑥 ( −𝑣 ‘ 𝑈 ) 𝑃 ) ) ) ≤ 1 ) → ( ( ( normCV ‘ 𝑈 ) ‘ ( ( 𝑃 ( +𝑣 ‘ 𝑈 ) ( ( 𝑦 / ( ( normCV ‘ 𝑈 ) ‘ 𝑧 ) ) ( ·𝑠OLD ‘ 𝑈 ) 𝑧 ) ) ( −𝑣 ‘ 𝑈 ) 𝑃 ) ) ≤ 𝑦 → ( ( normCV ‘ 𝑊 ) ‘ ( 𝑇 ‘ ( ( 𝑃 ( +𝑣 ‘ 𝑈 ) ( ( 𝑦 / ( ( normCV ‘ 𝑈 ) ‘ 𝑧 ) ) ( ·𝑠OLD ‘ 𝑈 ) 𝑧 ) ) ( −𝑣 ‘ 𝑈 ) 𝑃 ) ) ) ≤ 1 ) ) ) |
101 |
90 100
|
mpid |
⊢ ( ( ( 𝑃 ∈ 𝑋 ∧ 𝑦 ∈ ℝ+ ) ∧ ( 𝑧 ∈ 𝑋 ∧ 𝑧 ≠ ( 0vec ‘ 𝑈 ) ) ) → ( ∀ 𝑥 ∈ 𝑋 ( ( ( normCV ‘ 𝑈 ) ‘ ( 𝑥 ( −𝑣 ‘ 𝑈 ) 𝑃 ) ) ≤ 𝑦 → ( ( normCV ‘ 𝑊 ) ‘ ( 𝑇 ‘ ( 𝑥 ( −𝑣 ‘ 𝑈 ) 𝑃 ) ) ) ≤ 1 ) → ( ( normCV ‘ 𝑊 ) ‘ ( 𝑇 ‘ ( ( 𝑃 ( +𝑣 ‘ 𝑈 ) ( ( 𝑦 / ( ( normCV ‘ 𝑈 ) ‘ 𝑧 ) ) ( ·𝑠OLD ‘ 𝑈 ) 𝑧 ) ) ( −𝑣 ‘ 𝑈 ) 𝑃 ) ) ) ≤ 1 ) ) |
102 |
29
|
ffvelrni |
⊢ ( 𝑧 ∈ 𝑋 → ( 𝑇 ‘ 𝑧 ) ∈ ( BaseSet ‘ 𝑊 ) ) |
103 |
13 33
|
nvcl |
⊢ ( ( 𝑊 ∈ NrmCVec ∧ ( 𝑇 ‘ 𝑧 ) ∈ ( BaseSet ‘ 𝑊 ) ) → ( ( normCV ‘ 𝑊 ) ‘ ( 𝑇 ‘ 𝑧 ) ) ∈ ℝ ) |
104 |
8 102 103
|
sylancr |
⊢ ( 𝑧 ∈ 𝑋 → ( ( normCV ‘ 𝑊 ) ‘ ( 𝑇 ‘ 𝑧 ) ) ∈ ℝ ) |
105 |
104
|
ad2antrl |
⊢ ( ( ( 𝑃 ∈ 𝑋 ∧ 𝑦 ∈ ℝ+ ) ∧ ( 𝑧 ∈ 𝑋 ∧ 𝑧 ≠ ( 0vec ‘ 𝑈 ) ) ) → ( ( normCV ‘ 𝑊 ) ‘ ( 𝑇 ‘ 𝑧 ) ) ∈ ℝ ) |
106 |
|
1red |
⊢ ( ( ( 𝑃 ∈ 𝑋 ∧ 𝑦 ∈ ℝ+ ) ∧ ( 𝑧 ∈ 𝑋 ∧ 𝑧 ≠ ( 0vec ‘ 𝑈 ) ) ) → 1 ∈ ℝ ) |
107 |
105 106 63
|
lemuldiv2d |
⊢ ( ( ( 𝑃 ∈ 𝑋 ∧ 𝑦 ∈ ℝ+ ) ∧ ( 𝑧 ∈ 𝑋 ∧ 𝑧 ≠ ( 0vec ‘ 𝑈 ) ) ) → ( ( ( 𝑦 / ( ( normCV ‘ 𝑈 ) ‘ 𝑧 ) ) · ( ( normCV ‘ 𝑊 ) ‘ ( 𝑇 ‘ 𝑧 ) ) ) ≤ 1 ↔ ( ( normCV ‘ 𝑊 ) ‘ ( 𝑇 ‘ 𝑧 ) ) ≤ ( 1 / ( 𝑦 / ( ( normCV ‘ 𝑈 ) ‘ 𝑧 ) ) ) ) ) |
108 |
71
|
fveq2d |
⊢ ( ( ( 𝑃 ∈ 𝑋 ∧ 𝑦 ∈ ℝ+ ) ∧ ( 𝑧 ∈ 𝑋 ∧ 𝑧 ≠ ( 0vec ‘ 𝑈 ) ) ) → ( 𝑇 ‘ ( ( 𝑃 ( +𝑣 ‘ 𝑈 ) ( ( 𝑦 / ( ( normCV ‘ 𝑈 ) ‘ 𝑧 ) ) ( ·𝑠OLD ‘ 𝑈 ) 𝑧 ) ) ( −𝑣 ‘ 𝑈 ) 𝑃 ) ) = ( 𝑇 ‘ ( ( 𝑦 / ( ( normCV ‘ 𝑈 ) ‘ 𝑧 ) ) ( ·𝑠OLD ‘ 𝑈 ) 𝑧 ) ) ) |
109 |
|
eqid |
⊢ ( ·𝑠OLD ‘ 𝑊 ) = ( ·𝑠OLD ‘ 𝑊 ) |
110 |
10 66 109 5
|
lnomul |
⊢ ( ( ( 𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇 ∈ 𝐿 ) ∧ ( ( 𝑦 / ( ( normCV ‘ 𝑈 ) ‘ 𝑧 ) ) ∈ ℂ ∧ 𝑧 ∈ 𝑋 ) ) → ( 𝑇 ‘ ( ( 𝑦 / ( ( normCV ‘ 𝑈 ) ‘ 𝑧 ) ) ( ·𝑠OLD ‘ 𝑈 ) 𝑧 ) ) = ( ( 𝑦 / ( ( normCV ‘ 𝑈 ) ‘ 𝑧 ) ) ( ·𝑠OLD ‘ 𝑊 ) ( 𝑇 ‘ 𝑧 ) ) ) |
111 |
37 110
|
mpan |
⊢ ( ( ( 𝑦 / ( ( normCV ‘ 𝑈 ) ‘ 𝑧 ) ) ∈ ℂ ∧ 𝑧 ∈ 𝑋 ) → ( 𝑇 ‘ ( ( 𝑦 / ( ( normCV ‘ 𝑈 ) ‘ 𝑧 ) ) ( ·𝑠OLD ‘ 𝑈 ) 𝑧 ) ) = ( ( 𝑦 / ( ( normCV ‘ 𝑈 ) ‘ 𝑧 ) ) ( ·𝑠OLD ‘ 𝑊 ) ( 𝑇 ‘ 𝑧 ) ) ) |
112 |
64 65 111
|
syl2anc |
⊢ ( ( ( 𝑃 ∈ 𝑋 ∧ 𝑦 ∈ ℝ+ ) ∧ ( 𝑧 ∈ 𝑋 ∧ 𝑧 ≠ ( 0vec ‘ 𝑈 ) ) ) → ( 𝑇 ‘ ( ( 𝑦 / ( ( normCV ‘ 𝑈 ) ‘ 𝑧 ) ) ( ·𝑠OLD ‘ 𝑈 ) 𝑧 ) ) = ( ( 𝑦 / ( ( normCV ‘ 𝑈 ) ‘ 𝑧 ) ) ( ·𝑠OLD ‘ 𝑊 ) ( 𝑇 ‘ 𝑧 ) ) ) |
113 |
108 112
|
eqtrd |
⊢ ( ( ( 𝑃 ∈ 𝑋 ∧ 𝑦 ∈ ℝ+ ) ∧ ( 𝑧 ∈ 𝑋 ∧ 𝑧 ≠ ( 0vec ‘ 𝑈 ) ) ) → ( 𝑇 ‘ ( ( 𝑃 ( +𝑣 ‘ 𝑈 ) ( ( 𝑦 / ( ( normCV ‘ 𝑈 ) ‘ 𝑧 ) ) ( ·𝑠OLD ‘ 𝑈 ) 𝑧 ) ) ( −𝑣 ‘ 𝑈 ) 𝑃 ) ) = ( ( 𝑦 / ( ( normCV ‘ 𝑈 ) ‘ 𝑧 ) ) ( ·𝑠OLD ‘ 𝑊 ) ( 𝑇 ‘ 𝑧 ) ) ) |
114 |
113
|
fveq2d |
⊢ ( ( ( 𝑃 ∈ 𝑋 ∧ 𝑦 ∈ ℝ+ ) ∧ ( 𝑧 ∈ 𝑋 ∧ 𝑧 ≠ ( 0vec ‘ 𝑈 ) ) ) → ( ( normCV ‘ 𝑊 ) ‘ ( 𝑇 ‘ ( ( 𝑃 ( +𝑣 ‘ 𝑈 ) ( ( 𝑦 / ( ( normCV ‘ 𝑈 ) ‘ 𝑧 ) ) ( ·𝑠OLD ‘ 𝑈 ) 𝑧 ) ) ( −𝑣 ‘ 𝑈 ) 𝑃 ) ) ) = ( ( normCV ‘ 𝑊 ) ‘ ( ( 𝑦 / ( ( normCV ‘ 𝑈 ) ‘ 𝑧 ) ) ( ·𝑠OLD ‘ 𝑊 ) ( 𝑇 ‘ 𝑧 ) ) ) ) |
115 |
8
|
a1i |
⊢ ( ( ( 𝑃 ∈ 𝑋 ∧ 𝑦 ∈ ℝ+ ) ∧ ( 𝑧 ∈ 𝑋 ∧ 𝑧 ≠ ( 0vec ‘ 𝑈 ) ) ) → 𝑊 ∈ NrmCVec ) |
116 |
102
|
ad2antrl |
⊢ ( ( ( 𝑃 ∈ 𝑋 ∧ 𝑦 ∈ ℝ+ ) ∧ ( 𝑧 ∈ 𝑋 ∧ 𝑧 ≠ ( 0vec ‘ 𝑈 ) ) ) → ( 𝑇 ‘ 𝑧 ) ∈ ( BaseSet ‘ 𝑊 ) ) |
117 |
13 109 33
|
nvsge0 |
⊢ ( ( 𝑊 ∈ NrmCVec ∧ ( ( 𝑦 / ( ( normCV ‘ 𝑈 ) ‘ 𝑧 ) ) ∈ ℝ ∧ 0 ≤ ( 𝑦 / ( ( normCV ‘ 𝑈 ) ‘ 𝑧 ) ) ) ∧ ( 𝑇 ‘ 𝑧 ) ∈ ( BaseSet ‘ 𝑊 ) ) → ( ( normCV ‘ 𝑊 ) ‘ ( ( 𝑦 / ( ( normCV ‘ 𝑈 ) ‘ 𝑧 ) ) ( ·𝑠OLD ‘ 𝑊 ) ( 𝑇 ‘ 𝑧 ) ) ) = ( ( 𝑦 / ( ( normCV ‘ 𝑈 ) ‘ 𝑧 ) ) · ( ( normCV ‘ 𝑊 ) ‘ ( 𝑇 ‘ 𝑧 ) ) ) ) |
118 |
115 73 116 117
|
syl3anc |
⊢ ( ( ( 𝑃 ∈ 𝑋 ∧ 𝑦 ∈ ℝ+ ) ∧ ( 𝑧 ∈ 𝑋 ∧ 𝑧 ≠ ( 0vec ‘ 𝑈 ) ) ) → ( ( normCV ‘ 𝑊 ) ‘ ( ( 𝑦 / ( ( normCV ‘ 𝑈 ) ‘ 𝑧 ) ) ( ·𝑠OLD ‘ 𝑊 ) ( 𝑇 ‘ 𝑧 ) ) ) = ( ( 𝑦 / ( ( normCV ‘ 𝑈 ) ‘ 𝑧 ) ) · ( ( normCV ‘ 𝑊 ) ‘ ( 𝑇 ‘ 𝑧 ) ) ) ) |
119 |
114 118
|
eqtrd |
⊢ ( ( ( 𝑃 ∈ 𝑋 ∧ 𝑦 ∈ ℝ+ ) ∧ ( 𝑧 ∈ 𝑋 ∧ 𝑧 ≠ ( 0vec ‘ 𝑈 ) ) ) → ( ( normCV ‘ 𝑊 ) ‘ ( 𝑇 ‘ ( ( 𝑃 ( +𝑣 ‘ 𝑈 ) ( ( 𝑦 / ( ( normCV ‘ 𝑈 ) ‘ 𝑧 ) ) ( ·𝑠OLD ‘ 𝑈 ) 𝑧 ) ) ( −𝑣 ‘ 𝑈 ) 𝑃 ) ) ) = ( ( 𝑦 / ( ( normCV ‘ 𝑈 ) ‘ 𝑧 ) ) · ( ( normCV ‘ 𝑊 ) ‘ ( 𝑇 ‘ 𝑧 ) ) ) ) |
120 |
119
|
breq1d |
⊢ ( ( ( 𝑃 ∈ 𝑋 ∧ 𝑦 ∈ ℝ+ ) ∧ ( 𝑧 ∈ 𝑋 ∧ 𝑧 ≠ ( 0vec ‘ 𝑈 ) ) ) → ( ( ( normCV ‘ 𝑊 ) ‘ ( 𝑇 ‘ ( ( 𝑃 ( +𝑣 ‘ 𝑈 ) ( ( 𝑦 / ( ( normCV ‘ 𝑈 ) ‘ 𝑧 ) ) ( ·𝑠OLD ‘ 𝑈 ) 𝑧 ) ) ( −𝑣 ‘ 𝑈 ) 𝑃 ) ) ) ≤ 1 ↔ ( ( 𝑦 / ( ( normCV ‘ 𝑈 ) ‘ 𝑧 ) ) · ( ( normCV ‘ 𝑊 ) ‘ ( 𝑇 ‘ 𝑧 ) ) ) ≤ 1 ) ) |
121 |
|
rpcnne0 |
⊢ ( 𝑦 ∈ ℝ+ → ( 𝑦 ∈ ℂ ∧ 𝑦 ≠ 0 ) ) |
122 |
|
rpcnne0 |
⊢ ( ( ( normCV ‘ 𝑈 ) ‘ 𝑧 ) ∈ ℝ+ → ( ( ( normCV ‘ 𝑈 ) ‘ 𝑧 ) ∈ ℂ ∧ ( ( normCV ‘ 𝑈 ) ‘ 𝑧 ) ≠ 0 ) ) |
123 |
|
recdiv |
⊢ ( ( ( 𝑦 ∈ ℂ ∧ 𝑦 ≠ 0 ) ∧ ( ( ( normCV ‘ 𝑈 ) ‘ 𝑧 ) ∈ ℂ ∧ ( ( normCV ‘ 𝑈 ) ‘ 𝑧 ) ≠ 0 ) ) → ( 1 / ( 𝑦 / ( ( normCV ‘ 𝑈 ) ‘ 𝑧 ) ) ) = ( ( ( normCV ‘ 𝑈 ) ‘ 𝑧 ) / 𝑦 ) ) |
124 |
121 122 123
|
syl2an |
⊢ ( ( 𝑦 ∈ ℝ+ ∧ ( ( normCV ‘ 𝑈 ) ‘ 𝑧 ) ∈ ℝ+ ) → ( 1 / ( 𝑦 / ( ( normCV ‘ 𝑈 ) ‘ 𝑧 ) ) ) = ( ( ( normCV ‘ 𝑈 ) ‘ 𝑧 ) / 𝑦 ) ) |
125 |
53 61 124
|
syl2an |
⊢ ( ( ( 𝑃 ∈ 𝑋 ∧ 𝑦 ∈ ℝ+ ) ∧ ( 𝑧 ∈ 𝑋 ∧ 𝑧 ≠ ( 0vec ‘ 𝑈 ) ) ) → ( 1 / ( 𝑦 / ( ( normCV ‘ 𝑈 ) ‘ 𝑧 ) ) ) = ( ( ( normCV ‘ 𝑈 ) ‘ 𝑧 ) / 𝑦 ) ) |
126 |
|
rpne0 |
⊢ ( 𝑦 ∈ ℝ+ → 𝑦 ≠ 0 ) |
127 |
126
|
ad2antlr |
⊢ ( ( ( 𝑃 ∈ 𝑋 ∧ 𝑦 ∈ ℝ+ ) ∧ ( 𝑧 ∈ 𝑋 ∧ 𝑧 ≠ ( 0vec ‘ 𝑈 ) ) ) → 𝑦 ≠ 0 ) |
128 |
79 77 127
|
divrec2d |
⊢ ( ( ( 𝑃 ∈ 𝑋 ∧ 𝑦 ∈ ℝ+ ) ∧ ( 𝑧 ∈ 𝑋 ∧ 𝑧 ≠ ( 0vec ‘ 𝑈 ) ) ) → ( ( ( normCV ‘ 𝑈 ) ‘ 𝑧 ) / 𝑦 ) = ( ( 1 / 𝑦 ) · ( ( normCV ‘ 𝑈 ) ‘ 𝑧 ) ) ) |
129 |
125 128
|
eqtr2d |
⊢ ( ( ( 𝑃 ∈ 𝑋 ∧ 𝑦 ∈ ℝ+ ) ∧ ( 𝑧 ∈ 𝑋 ∧ 𝑧 ≠ ( 0vec ‘ 𝑈 ) ) ) → ( ( 1 / 𝑦 ) · ( ( normCV ‘ 𝑈 ) ‘ 𝑧 ) ) = ( 1 / ( 𝑦 / ( ( normCV ‘ 𝑈 ) ‘ 𝑧 ) ) ) ) |
130 |
129
|
breq2d |
⊢ ( ( ( 𝑃 ∈ 𝑋 ∧ 𝑦 ∈ ℝ+ ) ∧ ( 𝑧 ∈ 𝑋 ∧ 𝑧 ≠ ( 0vec ‘ 𝑈 ) ) ) → ( ( ( normCV ‘ 𝑊 ) ‘ ( 𝑇 ‘ 𝑧 ) ) ≤ ( ( 1 / 𝑦 ) · ( ( normCV ‘ 𝑈 ) ‘ 𝑧 ) ) ↔ ( ( normCV ‘ 𝑊 ) ‘ ( 𝑇 ‘ 𝑧 ) ) ≤ ( 1 / ( 𝑦 / ( ( normCV ‘ 𝑈 ) ‘ 𝑧 ) ) ) ) ) |
131 |
107 120 130
|
3bitr4d |
⊢ ( ( ( 𝑃 ∈ 𝑋 ∧ 𝑦 ∈ ℝ+ ) ∧ ( 𝑧 ∈ 𝑋 ∧ 𝑧 ≠ ( 0vec ‘ 𝑈 ) ) ) → ( ( ( normCV ‘ 𝑊 ) ‘ ( 𝑇 ‘ ( ( 𝑃 ( +𝑣 ‘ 𝑈 ) ( ( 𝑦 / ( ( normCV ‘ 𝑈 ) ‘ 𝑧 ) ) ( ·𝑠OLD ‘ 𝑈 ) 𝑧 ) ) ( −𝑣 ‘ 𝑈 ) 𝑃 ) ) ) ≤ 1 ↔ ( ( normCV ‘ 𝑊 ) ‘ ( 𝑇 ‘ 𝑧 ) ) ≤ ( ( 1 / 𝑦 ) · ( ( normCV ‘ 𝑈 ) ‘ 𝑧 ) ) ) ) |
132 |
101 131
|
sylibd |
⊢ ( ( ( 𝑃 ∈ 𝑋 ∧ 𝑦 ∈ ℝ+ ) ∧ ( 𝑧 ∈ 𝑋 ∧ 𝑧 ≠ ( 0vec ‘ 𝑈 ) ) ) → ( ∀ 𝑥 ∈ 𝑋 ( ( ( normCV ‘ 𝑈 ) ‘ ( 𝑥 ( −𝑣 ‘ 𝑈 ) 𝑃 ) ) ≤ 𝑦 → ( ( normCV ‘ 𝑊 ) ‘ ( 𝑇 ‘ ( 𝑥 ( −𝑣 ‘ 𝑈 ) 𝑃 ) ) ) ≤ 1 ) → ( ( normCV ‘ 𝑊 ) ‘ ( 𝑇 ‘ 𝑧 ) ) ≤ ( ( 1 / 𝑦 ) · ( ( normCV ‘ 𝑈 ) ‘ 𝑧 ) ) ) ) |
133 |
132
|
anassrs |
⊢ ( ( ( ( 𝑃 ∈ 𝑋 ∧ 𝑦 ∈ ℝ+ ) ∧ 𝑧 ∈ 𝑋 ) ∧ 𝑧 ≠ ( 0vec ‘ 𝑈 ) ) → ( ∀ 𝑥 ∈ 𝑋 ( ( ( normCV ‘ 𝑈 ) ‘ ( 𝑥 ( −𝑣 ‘ 𝑈 ) 𝑃 ) ) ≤ 𝑦 → ( ( normCV ‘ 𝑊 ) ‘ ( 𝑇 ‘ ( 𝑥 ( −𝑣 ‘ 𝑈 ) 𝑃 ) ) ) ≤ 1 ) → ( ( normCV ‘ 𝑊 ) ‘ ( 𝑇 ‘ 𝑧 ) ) ≤ ( ( 1 / 𝑦 ) · ( ( normCV ‘ 𝑈 ) ‘ 𝑧 ) ) ) ) |
134 |
133
|
imp |
⊢ ( ( ( ( ( 𝑃 ∈ 𝑋 ∧ 𝑦 ∈ ℝ+ ) ∧ 𝑧 ∈ 𝑋 ) ∧ 𝑧 ≠ ( 0vec ‘ 𝑈 ) ) ∧ ∀ 𝑥 ∈ 𝑋 ( ( ( normCV ‘ 𝑈 ) ‘ ( 𝑥 ( −𝑣 ‘ 𝑈 ) 𝑃 ) ) ≤ 𝑦 → ( ( normCV ‘ 𝑊 ) ‘ ( 𝑇 ‘ ( 𝑥 ( −𝑣 ‘ 𝑈 ) 𝑃 ) ) ) ≤ 1 ) ) → ( ( normCV ‘ 𝑊 ) ‘ ( 𝑇 ‘ 𝑧 ) ) ≤ ( ( 1 / 𝑦 ) · ( ( normCV ‘ 𝑈 ) ‘ 𝑧 ) ) ) |
135 |
134
|
an32s |
⊢ ( ( ( ( ( 𝑃 ∈ 𝑋 ∧ 𝑦 ∈ ℝ+ ) ∧ 𝑧 ∈ 𝑋 ) ∧ ∀ 𝑥 ∈ 𝑋 ( ( ( normCV ‘ 𝑈 ) ‘ ( 𝑥 ( −𝑣 ‘ 𝑈 ) 𝑃 ) ) ≤ 𝑦 → ( ( normCV ‘ 𝑊 ) ‘ ( 𝑇 ‘ ( 𝑥 ( −𝑣 ‘ 𝑈 ) 𝑃 ) ) ) ≤ 1 ) ) ∧ 𝑧 ≠ ( 0vec ‘ 𝑈 ) ) → ( ( normCV ‘ 𝑊 ) ‘ ( 𝑇 ‘ 𝑧 ) ) ≤ ( ( 1 / 𝑦 ) · ( ( normCV ‘ 𝑈 ) ‘ 𝑧 ) ) ) |
136 |
|
eqid |
⊢ ( 0vec ‘ 𝑊 ) = ( 0vec ‘ 𝑊 ) |
137 |
10 13 57 136 5
|
lno0 |
⊢ ( ( 𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇 ∈ 𝐿 ) → ( 𝑇 ‘ ( 0vec ‘ 𝑈 ) ) = ( 0vec ‘ 𝑊 ) ) |
138 |
7 8 9 137
|
mp3an |
⊢ ( 𝑇 ‘ ( 0vec ‘ 𝑈 ) ) = ( 0vec ‘ 𝑊 ) |
139 |
138
|
fveq2i |
⊢ ( ( normCV ‘ 𝑊 ) ‘ ( 𝑇 ‘ ( 0vec ‘ 𝑈 ) ) ) = ( ( normCV ‘ 𝑊 ) ‘ ( 0vec ‘ 𝑊 ) ) |
140 |
136 33
|
nvz0 |
⊢ ( 𝑊 ∈ NrmCVec → ( ( normCV ‘ 𝑊 ) ‘ ( 0vec ‘ 𝑊 ) ) = 0 ) |
141 |
8 140
|
ax-mp |
⊢ ( ( normCV ‘ 𝑊 ) ‘ ( 0vec ‘ 𝑊 ) ) = 0 |
142 |
139 141
|
eqtri |
⊢ ( ( normCV ‘ 𝑊 ) ‘ ( 𝑇 ‘ ( 0vec ‘ 𝑈 ) ) ) = 0 |
143 |
|
0le0 |
⊢ 0 ≤ 0 |
144 |
142 143
|
eqbrtri |
⊢ ( ( normCV ‘ 𝑊 ) ‘ ( 𝑇 ‘ ( 0vec ‘ 𝑈 ) ) ) ≤ 0 |
145 |
20
|
rpcnd |
⊢ ( 𝑦 ∈ ℝ+ → ( 1 / 𝑦 ) ∈ ℂ ) |
146 |
57 24
|
nvz0 |
⊢ ( 𝑈 ∈ NrmCVec → ( ( normCV ‘ 𝑈 ) ‘ ( 0vec ‘ 𝑈 ) ) = 0 ) |
147 |
7 146
|
ax-mp |
⊢ ( ( normCV ‘ 𝑈 ) ‘ ( 0vec ‘ 𝑈 ) ) = 0 |
148 |
147
|
oveq2i |
⊢ ( ( 1 / 𝑦 ) · ( ( normCV ‘ 𝑈 ) ‘ ( 0vec ‘ 𝑈 ) ) ) = ( ( 1 / 𝑦 ) · 0 ) |
149 |
|
mul01 |
⊢ ( ( 1 / 𝑦 ) ∈ ℂ → ( ( 1 / 𝑦 ) · 0 ) = 0 ) |
150 |
148 149
|
eqtrid |
⊢ ( ( 1 / 𝑦 ) ∈ ℂ → ( ( 1 / 𝑦 ) · ( ( normCV ‘ 𝑈 ) ‘ ( 0vec ‘ 𝑈 ) ) ) = 0 ) |
151 |
145 150
|
syl |
⊢ ( 𝑦 ∈ ℝ+ → ( ( 1 / 𝑦 ) · ( ( normCV ‘ 𝑈 ) ‘ ( 0vec ‘ 𝑈 ) ) ) = 0 ) |
152 |
144 151
|
breqtrrid |
⊢ ( 𝑦 ∈ ℝ+ → ( ( normCV ‘ 𝑊 ) ‘ ( 𝑇 ‘ ( 0vec ‘ 𝑈 ) ) ) ≤ ( ( 1 / 𝑦 ) · ( ( normCV ‘ 𝑈 ) ‘ ( 0vec ‘ 𝑈 ) ) ) ) |
153 |
152
|
ad3antlr |
⊢ ( ( ( ( 𝑃 ∈ 𝑋 ∧ 𝑦 ∈ ℝ+ ) ∧ 𝑧 ∈ 𝑋 ) ∧ ∀ 𝑥 ∈ 𝑋 ( ( ( normCV ‘ 𝑈 ) ‘ ( 𝑥 ( −𝑣 ‘ 𝑈 ) 𝑃 ) ) ≤ 𝑦 → ( ( normCV ‘ 𝑊 ) ‘ ( 𝑇 ‘ ( 𝑥 ( −𝑣 ‘ 𝑈 ) 𝑃 ) ) ) ≤ 1 ) ) → ( ( normCV ‘ 𝑊 ) ‘ ( 𝑇 ‘ ( 0vec ‘ 𝑈 ) ) ) ≤ ( ( 1 / 𝑦 ) · ( ( normCV ‘ 𝑈 ) ‘ ( 0vec ‘ 𝑈 ) ) ) ) |
154 |
50 135 153
|
pm2.61ne |
⊢ ( ( ( ( 𝑃 ∈ 𝑋 ∧ 𝑦 ∈ ℝ+ ) ∧ 𝑧 ∈ 𝑋 ) ∧ ∀ 𝑥 ∈ 𝑋 ( ( ( normCV ‘ 𝑈 ) ‘ ( 𝑥 ( −𝑣 ‘ 𝑈 ) 𝑃 ) ) ≤ 𝑦 → ( ( normCV ‘ 𝑊 ) ‘ ( 𝑇 ‘ ( 𝑥 ( −𝑣 ‘ 𝑈 ) 𝑃 ) ) ) ≤ 1 ) ) → ( ( normCV ‘ 𝑊 ) ‘ ( 𝑇 ‘ 𝑧 ) ) ≤ ( ( 1 / 𝑦 ) · ( ( normCV ‘ 𝑈 ) ‘ 𝑧 ) ) ) |
155 |
154
|
ex |
⊢ ( ( ( 𝑃 ∈ 𝑋 ∧ 𝑦 ∈ ℝ+ ) ∧ 𝑧 ∈ 𝑋 ) → ( ∀ 𝑥 ∈ 𝑋 ( ( ( normCV ‘ 𝑈 ) ‘ ( 𝑥 ( −𝑣 ‘ 𝑈 ) 𝑃 ) ) ≤ 𝑦 → ( ( normCV ‘ 𝑊 ) ‘ ( 𝑇 ‘ ( 𝑥 ( −𝑣 ‘ 𝑈 ) 𝑃 ) ) ) ≤ 1 ) → ( ( normCV ‘ 𝑊 ) ‘ ( 𝑇 ‘ 𝑧 ) ) ≤ ( ( 1 / 𝑦 ) · ( ( normCV ‘ 𝑈 ) ‘ 𝑧 ) ) ) ) |
156 |
155
|
ralrimdva |
⊢ ( ( 𝑃 ∈ 𝑋 ∧ 𝑦 ∈ ℝ+ ) → ( ∀ 𝑥 ∈ 𝑋 ( ( ( normCV ‘ 𝑈 ) ‘ ( 𝑥 ( −𝑣 ‘ 𝑈 ) 𝑃 ) ) ≤ 𝑦 → ( ( normCV ‘ 𝑊 ) ‘ ( 𝑇 ‘ ( 𝑥 ( −𝑣 ‘ 𝑈 ) 𝑃 ) ) ) ≤ 1 ) → ∀ 𝑧 ∈ 𝑋 ( ( normCV ‘ 𝑊 ) ‘ ( 𝑇 ‘ 𝑧 ) ) ≤ ( ( 1 / 𝑦 ) · ( ( normCV ‘ 𝑈 ) ‘ 𝑧 ) ) ) ) |
157 |
46 156
|
sylbid |
⊢ ( ( 𝑃 ∈ 𝑋 ∧ 𝑦 ∈ ℝ+ ) → ( ∀ 𝑥 ∈ 𝑋 ( ( 𝑥 𝐶 𝑃 ) ≤ 𝑦 → ( ( 𝑇 ‘ 𝑥 ) 𝐷 ( 𝑇 ‘ 𝑃 ) ) ≤ 1 ) → ∀ 𝑧 ∈ 𝑋 ( ( normCV ‘ 𝑊 ) ‘ ( 𝑇 ‘ 𝑧 ) ) ≤ ( ( 1 / 𝑦 ) · ( ( normCV ‘ 𝑈 ) ‘ 𝑧 ) ) ) ) |
158 |
157
|
imp |
⊢ ( ( ( 𝑃 ∈ 𝑋 ∧ 𝑦 ∈ ℝ+ ) ∧ ∀ 𝑥 ∈ 𝑋 ( ( 𝑥 𝐶 𝑃 ) ≤ 𝑦 → ( ( 𝑇 ‘ 𝑥 ) 𝐷 ( 𝑇 ‘ 𝑃 ) ) ≤ 1 ) ) → ∀ 𝑧 ∈ 𝑋 ( ( normCV ‘ 𝑊 ) ‘ ( 𝑇 ‘ 𝑧 ) ) ≤ ( ( 1 / 𝑦 ) · ( ( normCV ‘ 𝑈 ) ‘ 𝑧 ) ) ) |
159 |
|
oveq1 |
⊢ ( 𝑥 = ( 1 / 𝑦 ) → ( 𝑥 · ( ( normCV ‘ 𝑈 ) ‘ 𝑧 ) ) = ( ( 1 / 𝑦 ) · ( ( normCV ‘ 𝑈 ) ‘ 𝑧 ) ) ) |
160 |
159
|
breq2d |
⊢ ( 𝑥 = ( 1 / 𝑦 ) → ( ( ( normCV ‘ 𝑊 ) ‘ ( 𝑇 ‘ 𝑧 ) ) ≤ ( 𝑥 · ( ( normCV ‘ 𝑈 ) ‘ 𝑧 ) ) ↔ ( ( normCV ‘ 𝑊 ) ‘ ( 𝑇 ‘ 𝑧 ) ) ≤ ( ( 1 / 𝑦 ) · ( ( normCV ‘ 𝑈 ) ‘ 𝑧 ) ) ) ) |
161 |
160
|
ralbidv |
⊢ ( 𝑥 = ( 1 / 𝑦 ) → ( ∀ 𝑧 ∈ 𝑋 ( ( normCV ‘ 𝑊 ) ‘ ( 𝑇 ‘ 𝑧 ) ) ≤ ( 𝑥 · ( ( normCV ‘ 𝑈 ) ‘ 𝑧 ) ) ↔ ∀ 𝑧 ∈ 𝑋 ( ( normCV ‘ 𝑊 ) ‘ ( 𝑇 ‘ 𝑧 ) ) ≤ ( ( 1 / 𝑦 ) · ( ( normCV ‘ 𝑈 ) ‘ 𝑧 ) ) ) ) |
162 |
161
|
rspcev |
⊢ ( ( ( 1 / 𝑦 ) ∈ ℝ ∧ ∀ 𝑧 ∈ 𝑋 ( ( normCV ‘ 𝑊 ) ‘ ( 𝑇 ‘ 𝑧 ) ) ≤ ( ( 1 / 𝑦 ) · ( ( normCV ‘ 𝑈 ) ‘ 𝑧 ) ) ) → ∃ 𝑥 ∈ ℝ ∀ 𝑧 ∈ 𝑋 ( ( normCV ‘ 𝑊 ) ‘ ( 𝑇 ‘ 𝑧 ) ) ≤ ( 𝑥 · ( ( normCV ‘ 𝑈 ) ‘ 𝑧 ) ) ) |
163 |
22 158 162
|
syl2anc |
⊢ ( ( ( 𝑃 ∈ 𝑋 ∧ 𝑦 ∈ ℝ+ ) ∧ ∀ 𝑥 ∈ 𝑋 ( ( 𝑥 𝐶 𝑃 ) ≤ 𝑦 → ( ( 𝑇 ‘ 𝑥 ) 𝐷 ( 𝑇 ‘ 𝑃 ) ) ≤ 1 ) ) → ∃ 𝑥 ∈ ℝ ∀ 𝑧 ∈ 𝑋 ( ( normCV ‘ 𝑊 ) ‘ ( 𝑇 ‘ 𝑧 ) ) ≤ ( 𝑥 · ( ( normCV ‘ 𝑈 ) ‘ 𝑧 ) ) ) |
164 |
163
|
rexlimdva2 |
⊢ ( 𝑃 ∈ 𝑋 → ( ∃ 𝑦 ∈ ℝ+ ∀ 𝑥 ∈ 𝑋 ( ( 𝑥 𝐶 𝑃 ) ≤ 𝑦 → ( ( 𝑇 ‘ 𝑥 ) 𝐷 ( 𝑇 ‘ 𝑃 ) ) ≤ 1 ) → ∃ 𝑥 ∈ ℝ ∀ 𝑧 ∈ 𝑋 ( ( normCV ‘ 𝑊 ) ‘ ( 𝑇 ‘ 𝑧 ) ) ≤ ( 𝑥 · ( ( normCV ‘ 𝑈 ) ‘ 𝑧 ) ) ) ) |
165 |
19 164
|
syl5 |
⊢ ( 𝑃 ∈ 𝑋 → ( 𝑇 ∈ ( ( 𝐽 CnP 𝐾 ) ‘ 𝑃 ) → ∃ 𝑥 ∈ ℝ ∀ 𝑧 ∈ 𝑋 ( ( normCV ‘ 𝑊 ) ‘ ( 𝑇 ‘ 𝑧 ) ) ≤ ( 𝑥 · ( ( normCV ‘ 𝑈 ) ‘ 𝑧 ) ) ) ) |
166 |
165
|
imp |
⊢ ( ( 𝑃 ∈ 𝑋 ∧ 𝑇 ∈ ( ( 𝐽 CnP 𝐾 ) ‘ 𝑃 ) ) → ∃ 𝑥 ∈ ℝ ∀ 𝑧 ∈ 𝑋 ( ( normCV ‘ 𝑊 ) ‘ ( 𝑇 ‘ 𝑧 ) ) ≤ ( 𝑥 · ( ( normCV ‘ 𝑈 ) ‘ 𝑧 ) ) ) |
167 |
10 24 33 5 6 7 8
|
isblo3i |
⊢ ( 𝑇 ∈ 𝐵 ↔ ( 𝑇 ∈ 𝐿 ∧ ∃ 𝑥 ∈ ℝ ∀ 𝑧 ∈ 𝑋 ( ( normCV ‘ 𝑊 ) ‘ ( 𝑇 ‘ 𝑧 ) ) ≤ ( 𝑥 · ( ( normCV ‘ 𝑈 ) ‘ 𝑧 ) ) ) ) |
168 |
9 167
|
mpbiran |
⊢ ( 𝑇 ∈ 𝐵 ↔ ∃ 𝑥 ∈ ℝ ∀ 𝑧 ∈ 𝑋 ( ( normCV ‘ 𝑊 ) ‘ ( 𝑇 ‘ 𝑧 ) ) ≤ ( 𝑥 · ( ( normCV ‘ 𝑈 ) ‘ 𝑧 ) ) ) |
169 |
166 168
|
sylibr |
⊢ ( ( 𝑃 ∈ 𝑋 ∧ 𝑇 ∈ ( ( 𝐽 CnP 𝐾 ) ‘ 𝑃 ) ) → 𝑇 ∈ 𝐵 ) |