Step |
Hyp |
Ref |
Expression |
1 |
|
sge0lempt.xph |
⊢ Ⅎ 𝑥 𝜑 |
2 |
|
sge0lempt.a |
⊢ ( 𝜑 → 𝐴 ∈ 𝑉 ) |
3 |
|
sge0lempt.b |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → 𝐵 ∈ ( 0 [,] +∞ ) ) |
4 |
|
sge0lempt.c |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → 𝐶 ∈ ( 0 [,] +∞ ) ) |
5 |
|
sge0lempt.le |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → 𝐵 ≤ 𝐶 ) |
6 |
|
eqid |
⊢ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) = ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) |
7 |
1 3 6
|
fmptdf |
⊢ ( 𝜑 → ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) : 𝐴 ⟶ ( 0 [,] +∞ ) ) |
8 |
|
eqid |
⊢ ( 𝑥 ∈ 𝐴 ↦ 𝐶 ) = ( 𝑥 ∈ 𝐴 ↦ 𝐶 ) |
9 |
1 4 8
|
fmptdf |
⊢ ( 𝜑 → ( 𝑥 ∈ 𝐴 ↦ 𝐶 ) : 𝐴 ⟶ ( 0 [,] +∞ ) ) |
10 |
|
nfv |
⊢ Ⅎ 𝑥 𝑦 ∈ 𝐴 |
11 |
1 10
|
nfan |
⊢ Ⅎ 𝑥 ( 𝜑 ∧ 𝑦 ∈ 𝐴 ) |
12 |
|
nfcv |
⊢ Ⅎ 𝑥 𝑦 |
13 |
12
|
nfcsb1 |
⊢ Ⅎ 𝑥 ⦋ 𝑦 / 𝑥 ⦌ 𝐵 |
14 |
|
nfcv |
⊢ Ⅎ 𝑥 ≤ |
15 |
12
|
nfcsb1 |
⊢ Ⅎ 𝑥 ⦋ 𝑦 / 𝑥 ⦌ 𝐶 |
16 |
13 14 15
|
nfbr |
⊢ Ⅎ 𝑥 ⦋ 𝑦 / 𝑥 ⦌ 𝐵 ≤ ⦋ 𝑦 / 𝑥 ⦌ 𝐶 |
17 |
11 16
|
nfim |
⊢ Ⅎ 𝑥 ( ( 𝜑 ∧ 𝑦 ∈ 𝐴 ) → ⦋ 𝑦 / 𝑥 ⦌ 𝐵 ≤ ⦋ 𝑦 / 𝑥 ⦌ 𝐶 ) |
18 |
|
eleq1w |
⊢ ( 𝑥 = 𝑦 → ( 𝑥 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴 ) ) |
19 |
18
|
anbi2d |
⊢ ( 𝑥 = 𝑦 → ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) ↔ ( 𝜑 ∧ 𝑦 ∈ 𝐴 ) ) ) |
20 |
|
csbeq1a |
⊢ ( 𝑥 = 𝑦 → 𝐵 = ⦋ 𝑦 / 𝑥 ⦌ 𝐵 ) |
21 |
|
csbeq1a |
⊢ ( 𝑥 = 𝑦 → 𝐶 = ⦋ 𝑦 / 𝑥 ⦌ 𝐶 ) |
22 |
20 21
|
breq12d |
⊢ ( 𝑥 = 𝑦 → ( 𝐵 ≤ 𝐶 ↔ ⦋ 𝑦 / 𝑥 ⦌ 𝐵 ≤ ⦋ 𝑦 / 𝑥 ⦌ 𝐶 ) ) |
23 |
19 22
|
imbi12d |
⊢ ( 𝑥 = 𝑦 → ( ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → 𝐵 ≤ 𝐶 ) ↔ ( ( 𝜑 ∧ 𝑦 ∈ 𝐴 ) → ⦋ 𝑦 / 𝑥 ⦌ 𝐵 ≤ ⦋ 𝑦 / 𝑥 ⦌ 𝐶 ) ) ) |
24 |
17 23 5
|
chvarfv |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝐴 ) → ⦋ 𝑦 / 𝑥 ⦌ 𝐵 ≤ ⦋ 𝑦 / 𝑥 ⦌ 𝐶 ) |
25 |
|
simpr |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝐴 ) → 𝑦 ∈ 𝐴 ) |
26 |
13
|
nfel1 |
⊢ Ⅎ 𝑥 ⦋ 𝑦 / 𝑥 ⦌ 𝐵 ∈ ( 0 [,] +∞ ) |
27 |
11 26
|
nfim |
⊢ Ⅎ 𝑥 ( ( 𝜑 ∧ 𝑦 ∈ 𝐴 ) → ⦋ 𝑦 / 𝑥 ⦌ 𝐵 ∈ ( 0 [,] +∞ ) ) |
28 |
20
|
eleq1d |
⊢ ( 𝑥 = 𝑦 → ( 𝐵 ∈ ( 0 [,] +∞ ) ↔ ⦋ 𝑦 / 𝑥 ⦌ 𝐵 ∈ ( 0 [,] +∞ ) ) ) |
29 |
19 28
|
imbi12d |
⊢ ( 𝑥 = 𝑦 → ( ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → 𝐵 ∈ ( 0 [,] +∞ ) ) ↔ ( ( 𝜑 ∧ 𝑦 ∈ 𝐴 ) → ⦋ 𝑦 / 𝑥 ⦌ 𝐵 ∈ ( 0 [,] +∞ ) ) ) ) |
30 |
27 29 3
|
chvarfv |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝐴 ) → ⦋ 𝑦 / 𝑥 ⦌ 𝐵 ∈ ( 0 [,] +∞ ) ) |
31 |
12 13 20 6
|
fvmptf |
⊢ ( ( 𝑦 ∈ 𝐴 ∧ ⦋ 𝑦 / 𝑥 ⦌ 𝐵 ∈ ( 0 [,] +∞ ) ) → ( ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ‘ 𝑦 ) = ⦋ 𝑦 / 𝑥 ⦌ 𝐵 ) |
32 |
25 30 31
|
syl2anc |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝐴 ) → ( ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ‘ 𝑦 ) = ⦋ 𝑦 / 𝑥 ⦌ 𝐵 ) |
33 |
|
nfcv |
⊢ Ⅎ 𝑥 ( 0 [,] +∞ ) |
34 |
15 33
|
nfel |
⊢ Ⅎ 𝑥 ⦋ 𝑦 / 𝑥 ⦌ 𝐶 ∈ ( 0 [,] +∞ ) |
35 |
11 34
|
nfim |
⊢ Ⅎ 𝑥 ( ( 𝜑 ∧ 𝑦 ∈ 𝐴 ) → ⦋ 𝑦 / 𝑥 ⦌ 𝐶 ∈ ( 0 [,] +∞ ) ) |
36 |
21
|
eleq1d |
⊢ ( 𝑥 = 𝑦 → ( 𝐶 ∈ ( 0 [,] +∞ ) ↔ ⦋ 𝑦 / 𝑥 ⦌ 𝐶 ∈ ( 0 [,] +∞ ) ) ) |
37 |
19 36
|
imbi12d |
⊢ ( 𝑥 = 𝑦 → ( ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → 𝐶 ∈ ( 0 [,] +∞ ) ) ↔ ( ( 𝜑 ∧ 𝑦 ∈ 𝐴 ) → ⦋ 𝑦 / 𝑥 ⦌ 𝐶 ∈ ( 0 [,] +∞ ) ) ) ) |
38 |
35 37 4
|
chvarfv |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝐴 ) → ⦋ 𝑦 / 𝑥 ⦌ 𝐶 ∈ ( 0 [,] +∞ ) ) |
39 |
12 15 21 8
|
fvmptf |
⊢ ( ( 𝑦 ∈ 𝐴 ∧ ⦋ 𝑦 / 𝑥 ⦌ 𝐶 ∈ ( 0 [,] +∞ ) ) → ( ( 𝑥 ∈ 𝐴 ↦ 𝐶 ) ‘ 𝑦 ) = ⦋ 𝑦 / 𝑥 ⦌ 𝐶 ) |
40 |
25 38 39
|
syl2anc |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝐴 ) → ( ( 𝑥 ∈ 𝐴 ↦ 𝐶 ) ‘ 𝑦 ) = ⦋ 𝑦 / 𝑥 ⦌ 𝐶 ) |
41 |
32 40
|
breq12d |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝐴 ) → ( ( ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ‘ 𝑦 ) ≤ ( ( 𝑥 ∈ 𝐴 ↦ 𝐶 ) ‘ 𝑦 ) ↔ ⦋ 𝑦 / 𝑥 ⦌ 𝐵 ≤ ⦋ 𝑦 / 𝑥 ⦌ 𝐶 ) ) |
42 |
24 41
|
mpbird |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝐴 ) → ( ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ‘ 𝑦 ) ≤ ( ( 𝑥 ∈ 𝐴 ↦ 𝐶 ) ‘ 𝑦 ) ) |
43 |
2 7 9 42
|
sge0le |
⊢ ( 𝜑 → ( Σ^ ‘ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) ≤ ( Σ^ ‘ ( 𝑥 ∈ 𝐴 ↦ 𝐶 ) ) ) |