Step |
Hyp |
Ref |
Expression |
1 |
|
imasf1oxmet.u |
⊢ ( 𝜑 → 𝑈 = ( 𝐹 “s 𝑅 ) ) |
2 |
|
imasf1oxmet.v |
⊢ ( 𝜑 → 𝑉 = ( Base ‘ 𝑅 ) ) |
3 |
|
imasf1oxmet.f |
⊢ ( 𝜑 → 𝐹 : 𝑉 –1-1-onto→ 𝐵 ) |
4 |
|
imasf1oxmet.r |
⊢ ( 𝜑 → 𝑅 ∈ 𝑍 ) |
5 |
|
imasf1oxmet.e |
⊢ 𝐸 = ( ( dist ‘ 𝑅 ) ↾ ( 𝑉 × 𝑉 ) ) |
6 |
|
imasf1oxmet.d |
⊢ 𝐷 = ( dist ‘ 𝑈 ) |
7 |
|
imasf1omet.m |
⊢ ( 𝜑 → 𝐸 ∈ ( Met ‘ 𝑉 ) ) |
8 |
|
metxmet |
⊢ ( 𝐸 ∈ ( Met ‘ 𝑉 ) → 𝐸 ∈ ( ∞Met ‘ 𝑉 ) ) |
9 |
7 8
|
syl |
⊢ ( 𝜑 → 𝐸 ∈ ( ∞Met ‘ 𝑉 ) ) |
10 |
1 2 3 4 5 6 9
|
imasf1oxmet |
⊢ ( 𝜑 → 𝐷 ∈ ( ∞Met ‘ 𝐵 ) ) |
11 |
|
f1ofo |
⊢ ( 𝐹 : 𝑉 –1-1-onto→ 𝐵 → 𝐹 : 𝑉 –onto→ 𝐵 ) |
12 |
3 11
|
syl |
⊢ ( 𝜑 → 𝐹 : 𝑉 –onto→ 𝐵 ) |
13 |
|
eqid |
⊢ ( dist ‘ 𝑅 ) = ( dist ‘ 𝑅 ) |
14 |
1 2 12 4 13 6
|
imasdsfn |
⊢ ( 𝜑 → 𝐷 Fn ( 𝐵 × 𝐵 ) ) |
15 |
1
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉 ) ) → 𝑈 = ( 𝐹 “s 𝑅 ) ) |
16 |
2
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉 ) ) → 𝑉 = ( Base ‘ 𝑅 ) ) |
17 |
3
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉 ) ) → 𝐹 : 𝑉 –1-1-onto→ 𝐵 ) |
18 |
4
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉 ) ) → 𝑅 ∈ 𝑍 ) |
19 |
9
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉 ) ) → 𝐸 ∈ ( ∞Met ‘ 𝑉 ) ) |
20 |
|
simprl |
⊢ ( ( 𝜑 ∧ ( 𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉 ) ) → 𝑎 ∈ 𝑉 ) |
21 |
|
simprr |
⊢ ( ( 𝜑 ∧ ( 𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉 ) ) → 𝑏 ∈ 𝑉 ) |
22 |
15 16 17 18 5 6 19 20 21
|
imasdsf1o |
⊢ ( ( 𝜑 ∧ ( 𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉 ) ) → ( ( 𝐹 ‘ 𝑎 ) 𝐷 ( 𝐹 ‘ 𝑏 ) ) = ( 𝑎 𝐸 𝑏 ) ) |
23 |
|
metcl |
⊢ ( ( 𝐸 ∈ ( Met ‘ 𝑉 ) ∧ 𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉 ) → ( 𝑎 𝐸 𝑏 ) ∈ ℝ ) |
24 |
23
|
3expb |
⊢ ( ( 𝐸 ∈ ( Met ‘ 𝑉 ) ∧ ( 𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉 ) ) → ( 𝑎 𝐸 𝑏 ) ∈ ℝ ) |
25 |
7 24
|
sylan |
⊢ ( ( 𝜑 ∧ ( 𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉 ) ) → ( 𝑎 𝐸 𝑏 ) ∈ ℝ ) |
26 |
22 25
|
eqeltrd |
⊢ ( ( 𝜑 ∧ ( 𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉 ) ) → ( ( 𝐹 ‘ 𝑎 ) 𝐷 ( 𝐹 ‘ 𝑏 ) ) ∈ ℝ ) |
27 |
26
|
ralrimivva |
⊢ ( 𝜑 → ∀ 𝑎 ∈ 𝑉 ∀ 𝑏 ∈ 𝑉 ( ( 𝐹 ‘ 𝑎 ) 𝐷 ( 𝐹 ‘ 𝑏 ) ) ∈ ℝ ) |
28 |
|
f1ofn |
⊢ ( 𝐹 : 𝑉 –1-1-onto→ 𝐵 → 𝐹 Fn 𝑉 ) |
29 |
3 28
|
syl |
⊢ ( 𝜑 → 𝐹 Fn 𝑉 ) |
30 |
|
oveq2 |
⊢ ( 𝑦 = ( 𝐹 ‘ 𝑏 ) → ( ( 𝐹 ‘ 𝑎 ) 𝐷 𝑦 ) = ( ( 𝐹 ‘ 𝑎 ) 𝐷 ( 𝐹 ‘ 𝑏 ) ) ) |
31 |
30
|
eleq1d |
⊢ ( 𝑦 = ( 𝐹 ‘ 𝑏 ) → ( ( ( 𝐹 ‘ 𝑎 ) 𝐷 𝑦 ) ∈ ℝ ↔ ( ( 𝐹 ‘ 𝑎 ) 𝐷 ( 𝐹 ‘ 𝑏 ) ) ∈ ℝ ) ) |
32 |
31
|
ralrn |
⊢ ( 𝐹 Fn 𝑉 → ( ∀ 𝑦 ∈ ran 𝐹 ( ( 𝐹 ‘ 𝑎 ) 𝐷 𝑦 ) ∈ ℝ ↔ ∀ 𝑏 ∈ 𝑉 ( ( 𝐹 ‘ 𝑎 ) 𝐷 ( 𝐹 ‘ 𝑏 ) ) ∈ ℝ ) ) |
33 |
29 32
|
syl |
⊢ ( 𝜑 → ( ∀ 𝑦 ∈ ran 𝐹 ( ( 𝐹 ‘ 𝑎 ) 𝐷 𝑦 ) ∈ ℝ ↔ ∀ 𝑏 ∈ 𝑉 ( ( 𝐹 ‘ 𝑎 ) 𝐷 ( 𝐹 ‘ 𝑏 ) ) ∈ ℝ ) ) |
34 |
|
forn |
⊢ ( 𝐹 : 𝑉 –onto→ 𝐵 → ran 𝐹 = 𝐵 ) |
35 |
12 34
|
syl |
⊢ ( 𝜑 → ran 𝐹 = 𝐵 ) |
36 |
35
|
raleqdv |
⊢ ( 𝜑 → ( ∀ 𝑦 ∈ ran 𝐹 ( ( 𝐹 ‘ 𝑎 ) 𝐷 𝑦 ) ∈ ℝ ↔ ∀ 𝑦 ∈ 𝐵 ( ( 𝐹 ‘ 𝑎 ) 𝐷 𝑦 ) ∈ ℝ ) ) |
37 |
33 36
|
bitr3d |
⊢ ( 𝜑 → ( ∀ 𝑏 ∈ 𝑉 ( ( 𝐹 ‘ 𝑎 ) 𝐷 ( 𝐹 ‘ 𝑏 ) ) ∈ ℝ ↔ ∀ 𝑦 ∈ 𝐵 ( ( 𝐹 ‘ 𝑎 ) 𝐷 𝑦 ) ∈ ℝ ) ) |
38 |
37
|
ralbidv |
⊢ ( 𝜑 → ( ∀ 𝑎 ∈ 𝑉 ∀ 𝑏 ∈ 𝑉 ( ( 𝐹 ‘ 𝑎 ) 𝐷 ( 𝐹 ‘ 𝑏 ) ) ∈ ℝ ↔ ∀ 𝑎 ∈ 𝑉 ∀ 𝑦 ∈ 𝐵 ( ( 𝐹 ‘ 𝑎 ) 𝐷 𝑦 ) ∈ ℝ ) ) |
39 |
27 38
|
mpbid |
⊢ ( 𝜑 → ∀ 𝑎 ∈ 𝑉 ∀ 𝑦 ∈ 𝐵 ( ( 𝐹 ‘ 𝑎 ) 𝐷 𝑦 ) ∈ ℝ ) |
40 |
|
oveq1 |
⊢ ( 𝑥 = ( 𝐹 ‘ 𝑎 ) → ( 𝑥 𝐷 𝑦 ) = ( ( 𝐹 ‘ 𝑎 ) 𝐷 𝑦 ) ) |
41 |
40
|
eleq1d |
⊢ ( 𝑥 = ( 𝐹 ‘ 𝑎 ) → ( ( 𝑥 𝐷 𝑦 ) ∈ ℝ ↔ ( ( 𝐹 ‘ 𝑎 ) 𝐷 𝑦 ) ∈ ℝ ) ) |
42 |
41
|
ralbidv |
⊢ ( 𝑥 = ( 𝐹 ‘ 𝑎 ) → ( ∀ 𝑦 ∈ 𝐵 ( 𝑥 𝐷 𝑦 ) ∈ ℝ ↔ ∀ 𝑦 ∈ 𝐵 ( ( 𝐹 ‘ 𝑎 ) 𝐷 𝑦 ) ∈ ℝ ) ) |
43 |
42
|
ralrn |
⊢ ( 𝐹 Fn 𝑉 → ( ∀ 𝑥 ∈ ran 𝐹 ∀ 𝑦 ∈ 𝐵 ( 𝑥 𝐷 𝑦 ) ∈ ℝ ↔ ∀ 𝑎 ∈ 𝑉 ∀ 𝑦 ∈ 𝐵 ( ( 𝐹 ‘ 𝑎 ) 𝐷 𝑦 ) ∈ ℝ ) ) |
44 |
29 43
|
syl |
⊢ ( 𝜑 → ( ∀ 𝑥 ∈ ran 𝐹 ∀ 𝑦 ∈ 𝐵 ( 𝑥 𝐷 𝑦 ) ∈ ℝ ↔ ∀ 𝑎 ∈ 𝑉 ∀ 𝑦 ∈ 𝐵 ( ( 𝐹 ‘ 𝑎 ) 𝐷 𝑦 ) ∈ ℝ ) ) |
45 |
35
|
raleqdv |
⊢ ( 𝜑 → ( ∀ 𝑥 ∈ ran 𝐹 ∀ 𝑦 ∈ 𝐵 ( 𝑥 𝐷 𝑦 ) ∈ ℝ ↔ ∀ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ( 𝑥 𝐷 𝑦 ) ∈ ℝ ) ) |
46 |
44 45
|
bitr3d |
⊢ ( 𝜑 → ( ∀ 𝑎 ∈ 𝑉 ∀ 𝑦 ∈ 𝐵 ( ( 𝐹 ‘ 𝑎 ) 𝐷 𝑦 ) ∈ ℝ ↔ ∀ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ( 𝑥 𝐷 𝑦 ) ∈ ℝ ) ) |
47 |
39 46
|
mpbid |
⊢ ( 𝜑 → ∀ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ( 𝑥 𝐷 𝑦 ) ∈ ℝ ) |
48 |
|
ffnov |
⊢ ( 𝐷 : ( 𝐵 × 𝐵 ) ⟶ ℝ ↔ ( 𝐷 Fn ( 𝐵 × 𝐵 ) ∧ ∀ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ( 𝑥 𝐷 𝑦 ) ∈ ℝ ) ) |
49 |
14 47 48
|
sylanbrc |
⊢ ( 𝜑 → 𝐷 : ( 𝐵 × 𝐵 ) ⟶ ℝ ) |
50 |
|
ismet2 |
⊢ ( 𝐷 ∈ ( Met ‘ 𝐵 ) ↔ ( 𝐷 ∈ ( ∞Met ‘ 𝐵 ) ∧ 𝐷 : ( 𝐵 × 𝐵 ) ⟶ ℝ ) ) |
51 |
10 49 50
|
sylanbrc |
⊢ ( 𝜑 → 𝐷 ∈ ( Met ‘ 𝐵 ) ) |