Step |
Hyp |
Ref |
Expression |
1 |
|
mbfinf.1 |
⊢ 𝑍 = ( ℤ≥ ‘ 𝑀 ) |
2 |
|
mbfinf.2 |
⊢ 𝐺 = ( 𝑥 ∈ 𝐴 ↦ inf ( ran ( 𝑛 ∈ 𝑍 ↦ 𝐵 ) , ℝ , < ) ) |
3 |
|
mbfinf.3 |
⊢ ( 𝜑 → 𝑀 ∈ ℤ ) |
4 |
|
mbfinf.4 |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ 𝑍 ) → ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ∈ MblFn ) |
5 |
|
mbfinf.5 |
⊢ ( ( 𝜑 ∧ ( 𝑛 ∈ 𝑍 ∧ 𝑥 ∈ 𝐴 ) ) → 𝐵 ∈ ℝ ) |
6 |
|
mbfinf.6 |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → ∃ 𝑦 ∈ ℝ ∀ 𝑛 ∈ 𝑍 𝑦 ≤ 𝐵 ) |
7 |
5
|
anass1rs |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) ∧ 𝑛 ∈ 𝑍 ) → 𝐵 ∈ ℝ ) |
8 |
7
|
fmpttd |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → ( 𝑛 ∈ 𝑍 ↦ 𝐵 ) : 𝑍 ⟶ ℝ ) |
9 |
8
|
frnd |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → ran ( 𝑛 ∈ 𝑍 ↦ 𝐵 ) ⊆ ℝ ) |
10 |
|
uzid |
⊢ ( 𝑀 ∈ ℤ → 𝑀 ∈ ( ℤ≥ ‘ 𝑀 ) ) |
11 |
3 10
|
syl |
⊢ ( 𝜑 → 𝑀 ∈ ( ℤ≥ ‘ 𝑀 ) ) |
12 |
11 1
|
eleqtrrdi |
⊢ ( 𝜑 → 𝑀 ∈ 𝑍 ) |
13 |
12
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → 𝑀 ∈ 𝑍 ) |
14 |
|
eqid |
⊢ ( 𝑛 ∈ 𝑍 ↦ 𝐵 ) = ( 𝑛 ∈ 𝑍 ↦ 𝐵 ) |
15 |
14 7
|
dmmptd |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → dom ( 𝑛 ∈ 𝑍 ↦ 𝐵 ) = 𝑍 ) |
16 |
13 15
|
eleqtrrd |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → 𝑀 ∈ dom ( 𝑛 ∈ 𝑍 ↦ 𝐵 ) ) |
17 |
16
|
ne0d |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → dom ( 𝑛 ∈ 𝑍 ↦ 𝐵 ) ≠ ∅ ) |
18 |
|
dm0rn0 |
⊢ ( dom ( 𝑛 ∈ 𝑍 ↦ 𝐵 ) = ∅ ↔ ran ( 𝑛 ∈ 𝑍 ↦ 𝐵 ) = ∅ ) |
19 |
18
|
necon3bii |
⊢ ( dom ( 𝑛 ∈ 𝑍 ↦ 𝐵 ) ≠ ∅ ↔ ran ( 𝑛 ∈ 𝑍 ↦ 𝐵 ) ≠ ∅ ) |
20 |
17 19
|
sylib |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → ran ( 𝑛 ∈ 𝑍 ↦ 𝐵 ) ≠ ∅ ) |
21 |
8
|
ffnd |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → ( 𝑛 ∈ 𝑍 ↦ 𝐵 ) Fn 𝑍 ) |
22 |
|
breq2 |
⊢ ( 𝑧 = ( ( 𝑛 ∈ 𝑍 ↦ 𝐵 ) ‘ 𝑚 ) → ( 𝑦 ≤ 𝑧 ↔ 𝑦 ≤ ( ( 𝑛 ∈ 𝑍 ↦ 𝐵 ) ‘ 𝑚 ) ) ) |
23 |
22
|
ralrn |
⊢ ( ( 𝑛 ∈ 𝑍 ↦ 𝐵 ) Fn 𝑍 → ( ∀ 𝑧 ∈ ran ( 𝑛 ∈ 𝑍 ↦ 𝐵 ) 𝑦 ≤ 𝑧 ↔ ∀ 𝑚 ∈ 𝑍 𝑦 ≤ ( ( 𝑛 ∈ 𝑍 ↦ 𝐵 ) ‘ 𝑚 ) ) ) |
24 |
21 23
|
syl |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → ( ∀ 𝑧 ∈ ran ( 𝑛 ∈ 𝑍 ↦ 𝐵 ) 𝑦 ≤ 𝑧 ↔ ∀ 𝑚 ∈ 𝑍 𝑦 ≤ ( ( 𝑛 ∈ 𝑍 ↦ 𝐵 ) ‘ 𝑚 ) ) ) |
25 |
|
nfcv |
⊢ Ⅎ 𝑛 𝑦 |
26 |
|
nfcv |
⊢ Ⅎ 𝑛 ≤ |
27 |
|
nffvmpt1 |
⊢ Ⅎ 𝑛 ( ( 𝑛 ∈ 𝑍 ↦ 𝐵 ) ‘ 𝑚 ) |
28 |
25 26 27
|
nfbr |
⊢ Ⅎ 𝑛 𝑦 ≤ ( ( 𝑛 ∈ 𝑍 ↦ 𝐵 ) ‘ 𝑚 ) |
29 |
|
nfv |
⊢ Ⅎ 𝑚 𝑦 ≤ ( ( 𝑛 ∈ 𝑍 ↦ 𝐵 ) ‘ 𝑛 ) |
30 |
|
fveq2 |
⊢ ( 𝑚 = 𝑛 → ( ( 𝑛 ∈ 𝑍 ↦ 𝐵 ) ‘ 𝑚 ) = ( ( 𝑛 ∈ 𝑍 ↦ 𝐵 ) ‘ 𝑛 ) ) |
31 |
30
|
breq2d |
⊢ ( 𝑚 = 𝑛 → ( 𝑦 ≤ ( ( 𝑛 ∈ 𝑍 ↦ 𝐵 ) ‘ 𝑚 ) ↔ 𝑦 ≤ ( ( 𝑛 ∈ 𝑍 ↦ 𝐵 ) ‘ 𝑛 ) ) ) |
32 |
28 29 31
|
cbvralw |
⊢ ( ∀ 𝑚 ∈ 𝑍 𝑦 ≤ ( ( 𝑛 ∈ 𝑍 ↦ 𝐵 ) ‘ 𝑚 ) ↔ ∀ 𝑛 ∈ 𝑍 𝑦 ≤ ( ( 𝑛 ∈ 𝑍 ↦ 𝐵 ) ‘ 𝑛 ) ) |
33 |
|
simpr |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) ∧ 𝑛 ∈ 𝑍 ) → 𝑛 ∈ 𝑍 ) |
34 |
14
|
fvmpt2 |
⊢ ( ( 𝑛 ∈ 𝑍 ∧ 𝐵 ∈ ℝ ) → ( ( 𝑛 ∈ 𝑍 ↦ 𝐵 ) ‘ 𝑛 ) = 𝐵 ) |
35 |
33 7 34
|
syl2anc |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) ∧ 𝑛 ∈ 𝑍 ) → ( ( 𝑛 ∈ 𝑍 ↦ 𝐵 ) ‘ 𝑛 ) = 𝐵 ) |
36 |
35
|
breq2d |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) ∧ 𝑛 ∈ 𝑍 ) → ( 𝑦 ≤ ( ( 𝑛 ∈ 𝑍 ↦ 𝐵 ) ‘ 𝑛 ) ↔ 𝑦 ≤ 𝐵 ) ) |
37 |
36
|
ralbidva |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → ( ∀ 𝑛 ∈ 𝑍 𝑦 ≤ ( ( 𝑛 ∈ 𝑍 ↦ 𝐵 ) ‘ 𝑛 ) ↔ ∀ 𝑛 ∈ 𝑍 𝑦 ≤ 𝐵 ) ) |
38 |
32 37
|
syl5bb |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → ( ∀ 𝑚 ∈ 𝑍 𝑦 ≤ ( ( 𝑛 ∈ 𝑍 ↦ 𝐵 ) ‘ 𝑚 ) ↔ ∀ 𝑛 ∈ 𝑍 𝑦 ≤ 𝐵 ) ) |
39 |
24 38
|
bitrd |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → ( ∀ 𝑧 ∈ ran ( 𝑛 ∈ 𝑍 ↦ 𝐵 ) 𝑦 ≤ 𝑧 ↔ ∀ 𝑛 ∈ 𝑍 𝑦 ≤ 𝐵 ) ) |
40 |
39
|
rexbidv |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → ( ∃ 𝑦 ∈ ℝ ∀ 𝑧 ∈ ran ( 𝑛 ∈ 𝑍 ↦ 𝐵 ) 𝑦 ≤ 𝑧 ↔ ∃ 𝑦 ∈ ℝ ∀ 𝑛 ∈ 𝑍 𝑦 ≤ 𝐵 ) ) |
41 |
6 40
|
mpbird |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → ∃ 𝑦 ∈ ℝ ∀ 𝑧 ∈ ran ( 𝑛 ∈ 𝑍 ↦ 𝐵 ) 𝑦 ≤ 𝑧 ) |
42 |
|
infrenegsup |
⊢ ( ( ran ( 𝑛 ∈ 𝑍 ↦ 𝐵 ) ⊆ ℝ ∧ ran ( 𝑛 ∈ 𝑍 ↦ 𝐵 ) ≠ ∅ ∧ ∃ 𝑦 ∈ ℝ ∀ 𝑧 ∈ ran ( 𝑛 ∈ 𝑍 ↦ 𝐵 ) 𝑦 ≤ 𝑧 ) → inf ( ran ( 𝑛 ∈ 𝑍 ↦ 𝐵 ) , ℝ , < ) = - sup ( { 𝑟 ∈ ℝ ∣ - 𝑟 ∈ ran ( 𝑛 ∈ 𝑍 ↦ 𝐵 ) } , ℝ , < ) ) |
43 |
9 20 41 42
|
syl3anc |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → inf ( ran ( 𝑛 ∈ 𝑍 ↦ 𝐵 ) , ℝ , < ) = - sup ( { 𝑟 ∈ ℝ ∣ - 𝑟 ∈ ran ( 𝑛 ∈ 𝑍 ↦ 𝐵 ) } , ℝ , < ) ) |
44 |
|
rabid |
⊢ ( 𝑟 ∈ { 𝑟 ∈ ℝ ∣ - 𝑟 ∈ ran ( 𝑛 ∈ 𝑍 ↦ 𝐵 ) } ↔ ( 𝑟 ∈ ℝ ∧ - 𝑟 ∈ ran ( 𝑛 ∈ 𝑍 ↦ 𝐵 ) ) ) |
45 |
7
|
recnd |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) ∧ 𝑛 ∈ 𝑍 ) → 𝐵 ∈ ℂ ) |
46 |
45
|
adantlr |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) ∧ 𝑟 ∈ ℝ ) ∧ 𝑛 ∈ 𝑍 ) → 𝐵 ∈ ℂ ) |
47 |
|
simplr |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) ∧ 𝑟 ∈ ℝ ) ∧ 𝑛 ∈ 𝑍 ) → 𝑟 ∈ ℝ ) |
48 |
47
|
recnd |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) ∧ 𝑟 ∈ ℝ ) ∧ 𝑛 ∈ 𝑍 ) → 𝑟 ∈ ℂ ) |
49 |
|
negcon2 |
⊢ ( ( 𝐵 ∈ ℂ ∧ 𝑟 ∈ ℂ ) → ( 𝐵 = - 𝑟 ↔ 𝑟 = - 𝐵 ) ) |
50 |
46 48 49
|
syl2anc |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) ∧ 𝑟 ∈ ℝ ) ∧ 𝑛 ∈ 𝑍 ) → ( 𝐵 = - 𝑟 ↔ 𝑟 = - 𝐵 ) ) |
51 |
|
eqcom |
⊢ ( 𝑟 = - 𝐵 ↔ - 𝐵 = 𝑟 ) |
52 |
50 51
|
bitrdi |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) ∧ 𝑟 ∈ ℝ ) ∧ 𝑛 ∈ 𝑍 ) → ( 𝐵 = - 𝑟 ↔ - 𝐵 = 𝑟 ) ) |
53 |
35
|
adantlr |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) ∧ 𝑟 ∈ ℝ ) ∧ 𝑛 ∈ 𝑍 ) → ( ( 𝑛 ∈ 𝑍 ↦ 𝐵 ) ‘ 𝑛 ) = 𝐵 ) |
54 |
53
|
eqeq1d |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) ∧ 𝑟 ∈ ℝ ) ∧ 𝑛 ∈ 𝑍 ) → ( ( ( 𝑛 ∈ 𝑍 ↦ 𝐵 ) ‘ 𝑛 ) = - 𝑟 ↔ 𝐵 = - 𝑟 ) ) |
55 |
|
negex |
⊢ - 𝐵 ∈ V |
56 |
|
eqid |
⊢ ( 𝑛 ∈ 𝑍 ↦ - 𝐵 ) = ( 𝑛 ∈ 𝑍 ↦ - 𝐵 ) |
57 |
56
|
fvmpt2 |
⊢ ( ( 𝑛 ∈ 𝑍 ∧ - 𝐵 ∈ V ) → ( ( 𝑛 ∈ 𝑍 ↦ - 𝐵 ) ‘ 𝑛 ) = - 𝐵 ) |
58 |
55 57
|
mpan2 |
⊢ ( 𝑛 ∈ 𝑍 → ( ( 𝑛 ∈ 𝑍 ↦ - 𝐵 ) ‘ 𝑛 ) = - 𝐵 ) |
59 |
58
|
adantl |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) ∧ 𝑟 ∈ ℝ ) ∧ 𝑛 ∈ 𝑍 ) → ( ( 𝑛 ∈ 𝑍 ↦ - 𝐵 ) ‘ 𝑛 ) = - 𝐵 ) |
60 |
59
|
eqeq1d |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) ∧ 𝑟 ∈ ℝ ) ∧ 𝑛 ∈ 𝑍 ) → ( ( ( 𝑛 ∈ 𝑍 ↦ - 𝐵 ) ‘ 𝑛 ) = 𝑟 ↔ - 𝐵 = 𝑟 ) ) |
61 |
52 54 60
|
3bitr4d |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) ∧ 𝑟 ∈ ℝ ) ∧ 𝑛 ∈ 𝑍 ) → ( ( ( 𝑛 ∈ 𝑍 ↦ 𝐵 ) ‘ 𝑛 ) = - 𝑟 ↔ ( ( 𝑛 ∈ 𝑍 ↦ - 𝐵 ) ‘ 𝑛 ) = 𝑟 ) ) |
62 |
61
|
ralrimiva |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) ∧ 𝑟 ∈ ℝ ) → ∀ 𝑛 ∈ 𝑍 ( ( ( 𝑛 ∈ 𝑍 ↦ 𝐵 ) ‘ 𝑛 ) = - 𝑟 ↔ ( ( 𝑛 ∈ 𝑍 ↦ - 𝐵 ) ‘ 𝑛 ) = 𝑟 ) ) |
63 |
27
|
nfeq1 |
⊢ Ⅎ 𝑛 ( ( 𝑛 ∈ 𝑍 ↦ 𝐵 ) ‘ 𝑚 ) = - 𝑟 |
64 |
|
nffvmpt1 |
⊢ Ⅎ 𝑛 ( ( 𝑛 ∈ 𝑍 ↦ - 𝐵 ) ‘ 𝑚 ) |
65 |
64
|
nfeq1 |
⊢ Ⅎ 𝑛 ( ( 𝑛 ∈ 𝑍 ↦ - 𝐵 ) ‘ 𝑚 ) = 𝑟 |
66 |
63 65
|
nfbi |
⊢ Ⅎ 𝑛 ( ( ( 𝑛 ∈ 𝑍 ↦ 𝐵 ) ‘ 𝑚 ) = - 𝑟 ↔ ( ( 𝑛 ∈ 𝑍 ↦ - 𝐵 ) ‘ 𝑚 ) = 𝑟 ) |
67 |
|
nfv |
⊢ Ⅎ 𝑚 ( ( ( 𝑛 ∈ 𝑍 ↦ 𝐵 ) ‘ 𝑛 ) = - 𝑟 ↔ ( ( 𝑛 ∈ 𝑍 ↦ - 𝐵 ) ‘ 𝑛 ) = 𝑟 ) |
68 |
|
fveqeq2 |
⊢ ( 𝑚 = 𝑛 → ( ( ( 𝑛 ∈ 𝑍 ↦ 𝐵 ) ‘ 𝑚 ) = - 𝑟 ↔ ( ( 𝑛 ∈ 𝑍 ↦ 𝐵 ) ‘ 𝑛 ) = - 𝑟 ) ) |
69 |
|
fveqeq2 |
⊢ ( 𝑚 = 𝑛 → ( ( ( 𝑛 ∈ 𝑍 ↦ - 𝐵 ) ‘ 𝑚 ) = 𝑟 ↔ ( ( 𝑛 ∈ 𝑍 ↦ - 𝐵 ) ‘ 𝑛 ) = 𝑟 ) ) |
70 |
68 69
|
bibi12d |
⊢ ( 𝑚 = 𝑛 → ( ( ( ( 𝑛 ∈ 𝑍 ↦ 𝐵 ) ‘ 𝑚 ) = - 𝑟 ↔ ( ( 𝑛 ∈ 𝑍 ↦ - 𝐵 ) ‘ 𝑚 ) = 𝑟 ) ↔ ( ( ( 𝑛 ∈ 𝑍 ↦ 𝐵 ) ‘ 𝑛 ) = - 𝑟 ↔ ( ( 𝑛 ∈ 𝑍 ↦ - 𝐵 ) ‘ 𝑛 ) = 𝑟 ) ) ) |
71 |
66 67 70
|
cbvralw |
⊢ ( ∀ 𝑚 ∈ 𝑍 ( ( ( 𝑛 ∈ 𝑍 ↦ 𝐵 ) ‘ 𝑚 ) = - 𝑟 ↔ ( ( 𝑛 ∈ 𝑍 ↦ - 𝐵 ) ‘ 𝑚 ) = 𝑟 ) ↔ ∀ 𝑛 ∈ 𝑍 ( ( ( 𝑛 ∈ 𝑍 ↦ 𝐵 ) ‘ 𝑛 ) = - 𝑟 ↔ ( ( 𝑛 ∈ 𝑍 ↦ - 𝐵 ) ‘ 𝑛 ) = 𝑟 ) ) |
72 |
62 71
|
sylibr |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) ∧ 𝑟 ∈ ℝ ) → ∀ 𝑚 ∈ 𝑍 ( ( ( 𝑛 ∈ 𝑍 ↦ 𝐵 ) ‘ 𝑚 ) = - 𝑟 ↔ ( ( 𝑛 ∈ 𝑍 ↦ - 𝐵 ) ‘ 𝑚 ) = 𝑟 ) ) |
73 |
72
|
r19.21bi |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) ∧ 𝑟 ∈ ℝ ) ∧ 𝑚 ∈ 𝑍 ) → ( ( ( 𝑛 ∈ 𝑍 ↦ 𝐵 ) ‘ 𝑚 ) = - 𝑟 ↔ ( ( 𝑛 ∈ 𝑍 ↦ - 𝐵 ) ‘ 𝑚 ) = 𝑟 ) ) |
74 |
73
|
rexbidva |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) ∧ 𝑟 ∈ ℝ ) → ( ∃ 𝑚 ∈ 𝑍 ( ( 𝑛 ∈ 𝑍 ↦ 𝐵 ) ‘ 𝑚 ) = - 𝑟 ↔ ∃ 𝑚 ∈ 𝑍 ( ( 𝑛 ∈ 𝑍 ↦ - 𝐵 ) ‘ 𝑚 ) = 𝑟 ) ) |
75 |
21
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) ∧ 𝑟 ∈ ℝ ) → ( 𝑛 ∈ 𝑍 ↦ 𝐵 ) Fn 𝑍 ) |
76 |
|
fvelrnb |
⊢ ( ( 𝑛 ∈ 𝑍 ↦ 𝐵 ) Fn 𝑍 → ( - 𝑟 ∈ ran ( 𝑛 ∈ 𝑍 ↦ 𝐵 ) ↔ ∃ 𝑚 ∈ 𝑍 ( ( 𝑛 ∈ 𝑍 ↦ 𝐵 ) ‘ 𝑚 ) = - 𝑟 ) ) |
77 |
75 76
|
syl |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) ∧ 𝑟 ∈ ℝ ) → ( - 𝑟 ∈ ran ( 𝑛 ∈ 𝑍 ↦ 𝐵 ) ↔ ∃ 𝑚 ∈ 𝑍 ( ( 𝑛 ∈ 𝑍 ↦ 𝐵 ) ‘ 𝑚 ) = - 𝑟 ) ) |
78 |
7
|
renegcld |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) ∧ 𝑛 ∈ 𝑍 ) → - 𝐵 ∈ ℝ ) |
79 |
78
|
fmpttd |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → ( 𝑛 ∈ 𝑍 ↦ - 𝐵 ) : 𝑍 ⟶ ℝ ) |
80 |
79
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) ∧ 𝑟 ∈ ℝ ) → ( 𝑛 ∈ 𝑍 ↦ - 𝐵 ) : 𝑍 ⟶ ℝ ) |
81 |
80
|
ffnd |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) ∧ 𝑟 ∈ ℝ ) → ( 𝑛 ∈ 𝑍 ↦ - 𝐵 ) Fn 𝑍 ) |
82 |
|
fvelrnb |
⊢ ( ( 𝑛 ∈ 𝑍 ↦ - 𝐵 ) Fn 𝑍 → ( 𝑟 ∈ ran ( 𝑛 ∈ 𝑍 ↦ - 𝐵 ) ↔ ∃ 𝑚 ∈ 𝑍 ( ( 𝑛 ∈ 𝑍 ↦ - 𝐵 ) ‘ 𝑚 ) = 𝑟 ) ) |
83 |
81 82
|
syl |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) ∧ 𝑟 ∈ ℝ ) → ( 𝑟 ∈ ran ( 𝑛 ∈ 𝑍 ↦ - 𝐵 ) ↔ ∃ 𝑚 ∈ 𝑍 ( ( 𝑛 ∈ 𝑍 ↦ - 𝐵 ) ‘ 𝑚 ) = 𝑟 ) ) |
84 |
74 77 83
|
3bitr4d |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) ∧ 𝑟 ∈ ℝ ) → ( - 𝑟 ∈ ran ( 𝑛 ∈ 𝑍 ↦ 𝐵 ) ↔ 𝑟 ∈ ran ( 𝑛 ∈ 𝑍 ↦ - 𝐵 ) ) ) |
85 |
84
|
pm5.32da |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → ( ( 𝑟 ∈ ℝ ∧ - 𝑟 ∈ ran ( 𝑛 ∈ 𝑍 ↦ 𝐵 ) ) ↔ ( 𝑟 ∈ ℝ ∧ 𝑟 ∈ ran ( 𝑛 ∈ 𝑍 ↦ - 𝐵 ) ) ) ) |
86 |
79
|
frnd |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → ran ( 𝑛 ∈ 𝑍 ↦ - 𝐵 ) ⊆ ℝ ) |
87 |
86
|
sseld |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → ( 𝑟 ∈ ran ( 𝑛 ∈ 𝑍 ↦ - 𝐵 ) → 𝑟 ∈ ℝ ) ) |
88 |
87
|
pm4.71rd |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → ( 𝑟 ∈ ran ( 𝑛 ∈ 𝑍 ↦ - 𝐵 ) ↔ ( 𝑟 ∈ ℝ ∧ 𝑟 ∈ ran ( 𝑛 ∈ 𝑍 ↦ - 𝐵 ) ) ) ) |
89 |
85 88
|
bitr4d |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → ( ( 𝑟 ∈ ℝ ∧ - 𝑟 ∈ ran ( 𝑛 ∈ 𝑍 ↦ 𝐵 ) ) ↔ 𝑟 ∈ ran ( 𝑛 ∈ 𝑍 ↦ - 𝐵 ) ) ) |
90 |
44 89
|
syl5bb |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → ( 𝑟 ∈ { 𝑟 ∈ ℝ ∣ - 𝑟 ∈ ran ( 𝑛 ∈ 𝑍 ↦ 𝐵 ) } ↔ 𝑟 ∈ ran ( 𝑛 ∈ 𝑍 ↦ - 𝐵 ) ) ) |
91 |
90
|
alrimiv |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → ∀ 𝑟 ( 𝑟 ∈ { 𝑟 ∈ ℝ ∣ - 𝑟 ∈ ran ( 𝑛 ∈ 𝑍 ↦ 𝐵 ) } ↔ 𝑟 ∈ ran ( 𝑛 ∈ 𝑍 ↦ - 𝐵 ) ) ) |
92 |
|
nfrab1 |
⊢ Ⅎ 𝑟 { 𝑟 ∈ ℝ ∣ - 𝑟 ∈ ran ( 𝑛 ∈ 𝑍 ↦ 𝐵 ) } |
93 |
|
nfcv |
⊢ Ⅎ 𝑟 ran ( 𝑛 ∈ 𝑍 ↦ - 𝐵 ) |
94 |
92 93
|
cleqf |
⊢ ( { 𝑟 ∈ ℝ ∣ - 𝑟 ∈ ran ( 𝑛 ∈ 𝑍 ↦ 𝐵 ) } = ran ( 𝑛 ∈ 𝑍 ↦ - 𝐵 ) ↔ ∀ 𝑟 ( 𝑟 ∈ { 𝑟 ∈ ℝ ∣ - 𝑟 ∈ ran ( 𝑛 ∈ 𝑍 ↦ 𝐵 ) } ↔ 𝑟 ∈ ran ( 𝑛 ∈ 𝑍 ↦ - 𝐵 ) ) ) |
95 |
91 94
|
sylibr |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → { 𝑟 ∈ ℝ ∣ - 𝑟 ∈ ran ( 𝑛 ∈ 𝑍 ↦ 𝐵 ) } = ran ( 𝑛 ∈ 𝑍 ↦ - 𝐵 ) ) |
96 |
95
|
supeq1d |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → sup ( { 𝑟 ∈ ℝ ∣ - 𝑟 ∈ ran ( 𝑛 ∈ 𝑍 ↦ 𝐵 ) } , ℝ , < ) = sup ( ran ( 𝑛 ∈ 𝑍 ↦ - 𝐵 ) , ℝ , < ) ) |
97 |
96
|
negeqd |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → - sup ( { 𝑟 ∈ ℝ ∣ - 𝑟 ∈ ran ( 𝑛 ∈ 𝑍 ↦ 𝐵 ) } , ℝ , < ) = - sup ( ran ( 𝑛 ∈ 𝑍 ↦ - 𝐵 ) , ℝ , < ) ) |
98 |
43 97
|
eqtrd |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → inf ( ran ( 𝑛 ∈ 𝑍 ↦ 𝐵 ) , ℝ , < ) = - sup ( ran ( 𝑛 ∈ 𝑍 ↦ - 𝐵 ) , ℝ , < ) ) |
99 |
98
|
mpteq2dva |
⊢ ( 𝜑 → ( 𝑥 ∈ 𝐴 ↦ inf ( ran ( 𝑛 ∈ 𝑍 ↦ 𝐵 ) , ℝ , < ) ) = ( 𝑥 ∈ 𝐴 ↦ - sup ( ran ( 𝑛 ∈ 𝑍 ↦ - 𝐵 ) , ℝ , < ) ) ) |
100 |
2 99
|
syl5eq |
⊢ ( 𝜑 → 𝐺 = ( 𝑥 ∈ 𝐴 ↦ - sup ( ran ( 𝑛 ∈ 𝑍 ↦ - 𝐵 ) , ℝ , < ) ) ) |
101 |
|
ltso |
⊢ < Or ℝ |
102 |
101
|
supex |
⊢ sup ( ran ( 𝑛 ∈ 𝑍 ↦ - 𝐵 ) , ℝ , < ) ∈ V |
103 |
102
|
a1i |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → sup ( ran ( 𝑛 ∈ 𝑍 ↦ - 𝐵 ) , ℝ , < ) ∈ V ) |
104 |
|
eqid |
⊢ ( 𝑥 ∈ 𝐴 ↦ sup ( ran ( 𝑛 ∈ 𝑍 ↦ - 𝐵 ) , ℝ , < ) ) = ( 𝑥 ∈ 𝐴 ↦ sup ( ran ( 𝑛 ∈ 𝑍 ↦ - 𝐵 ) , ℝ , < ) ) |
105 |
5
|
anassrs |
⊢ ( ( ( 𝜑 ∧ 𝑛 ∈ 𝑍 ) ∧ 𝑥 ∈ 𝐴 ) → 𝐵 ∈ ℝ ) |
106 |
105 4
|
mbfneg |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ 𝑍 ) → ( 𝑥 ∈ 𝐴 ↦ - 𝐵 ) ∈ MblFn ) |
107 |
5
|
renegcld |
⊢ ( ( 𝜑 ∧ ( 𝑛 ∈ 𝑍 ∧ 𝑥 ∈ 𝐴 ) ) → - 𝐵 ∈ ℝ ) |
108 |
|
renegcl |
⊢ ( 𝑦 ∈ ℝ → - 𝑦 ∈ ℝ ) |
109 |
108
|
ad2antrl |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) ∧ ( 𝑦 ∈ ℝ ∧ ∀ 𝑛 ∈ 𝑍 𝑦 ≤ 𝐵 ) ) → - 𝑦 ∈ ℝ ) |
110 |
|
simplr |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) ∧ 𝑦 ∈ ℝ ) ∧ 𝑛 ∈ 𝑍 ) → 𝑦 ∈ ℝ ) |
111 |
7
|
adantlr |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) ∧ 𝑦 ∈ ℝ ) ∧ 𝑛 ∈ 𝑍 ) → 𝐵 ∈ ℝ ) |
112 |
110 111
|
lenegd |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) ∧ 𝑦 ∈ ℝ ) ∧ 𝑛 ∈ 𝑍 ) → ( 𝑦 ≤ 𝐵 ↔ - 𝐵 ≤ - 𝑦 ) ) |
113 |
112
|
ralbidva |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) ∧ 𝑦 ∈ ℝ ) → ( ∀ 𝑛 ∈ 𝑍 𝑦 ≤ 𝐵 ↔ ∀ 𝑛 ∈ 𝑍 - 𝐵 ≤ - 𝑦 ) ) |
114 |
113
|
biimpd |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) ∧ 𝑦 ∈ ℝ ) → ( ∀ 𝑛 ∈ 𝑍 𝑦 ≤ 𝐵 → ∀ 𝑛 ∈ 𝑍 - 𝐵 ≤ - 𝑦 ) ) |
115 |
114
|
impr |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) ∧ ( 𝑦 ∈ ℝ ∧ ∀ 𝑛 ∈ 𝑍 𝑦 ≤ 𝐵 ) ) → ∀ 𝑛 ∈ 𝑍 - 𝐵 ≤ - 𝑦 ) |
116 |
|
brralrspcev |
⊢ ( ( - 𝑦 ∈ ℝ ∧ ∀ 𝑛 ∈ 𝑍 - 𝐵 ≤ - 𝑦 ) → ∃ 𝑧 ∈ ℝ ∀ 𝑛 ∈ 𝑍 - 𝐵 ≤ 𝑧 ) |
117 |
109 115 116
|
syl2anc |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) ∧ ( 𝑦 ∈ ℝ ∧ ∀ 𝑛 ∈ 𝑍 𝑦 ≤ 𝐵 ) ) → ∃ 𝑧 ∈ ℝ ∀ 𝑛 ∈ 𝑍 - 𝐵 ≤ 𝑧 ) |
118 |
6 117
|
rexlimddv |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → ∃ 𝑧 ∈ ℝ ∀ 𝑛 ∈ 𝑍 - 𝐵 ≤ 𝑧 ) |
119 |
1 104 3 106 107 118
|
mbfsup |
⊢ ( 𝜑 → ( 𝑥 ∈ 𝐴 ↦ sup ( ran ( 𝑛 ∈ 𝑍 ↦ - 𝐵 ) , ℝ , < ) ) ∈ MblFn ) |
120 |
103 119
|
mbfneg |
⊢ ( 𝜑 → ( 𝑥 ∈ 𝐴 ↦ - sup ( ran ( 𝑛 ∈ 𝑍 ↦ - 𝐵 ) , ℝ , < ) ) ∈ MblFn ) |
121 |
100 120
|
eqeltrd |
⊢ ( 𝜑 → 𝐺 ∈ MblFn ) |