Step |
Hyp |
Ref |
Expression |
1 |
|
renegcl |
⊢ ( 𝐵 ∈ ℝ → - 𝐵 ∈ ℝ ) |
2 |
1
|
adantr |
⊢ ( ( 𝐵 ∈ ℝ ∧ ( 𝐴 ⊆ { 𝑧 ∈ ℤ ∣ 𝐵 ≤ 𝑧 } ∧ 𝐴 ≠ ∅ ) ) → - 𝐵 ∈ ℝ ) |
3 |
|
arch |
⊢ ( - 𝐵 ∈ ℝ → ∃ 𝑛 ∈ ℕ - 𝐵 < 𝑛 ) |
4 |
2 3
|
syl |
⊢ ( ( 𝐵 ∈ ℝ ∧ ( 𝐴 ⊆ { 𝑧 ∈ ℤ ∣ 𝐵 ≤ 𝑧 } ∧ 𝐴 ≠ ∅ ) ) → ∃ 𝑛 ∈ ℕ - 𝐵 < 𝑛 ) |
5 |
|
simplrl |
⊢ ( ( ( 𝐵 ∈ ℝ ∧ ( 𝐴 ⊆ { 𝑧 ∈ ℤ ∣ 𝐵 ≤ 𝑧 } ∧ 𝐴 ≠ ∅ ) ) ∧ ( 𝑛 ∈ ℕ ∧ - 𝐵 < 𝑛 ) ) → 𝐴 ⊆ { 𝑧 ∈ ℤ ∣ 𝐵 ≤ 𝑧 } ) |
6 |
|
simplrl |
⊢ ( ( ( 𝐵 ∈ ℝ ∧ ( 𝑛 ∈ ℕ ∧ - 𝐵 < 𝑛 ) ) ∧ ( 𝑧 ∈ ℤ ∧ 𝐵 ≤ 𝑧 ) ) → 𝑛 ∈ ℕ ) |
7 |
|
nnnegz |
⊢ ( 𝑛 ∈ ℕ → - 𝑛 ∈ ℤ ) |
8 |
6 7
|
syl |
⊢ ( ( ( 𝐵 ∈ ℝ ∧ ( 𝑛 ∈ ℕ ∧ - 𝐵 < 𝑛 ) ) ∧ ( 𝑧 ∈ ℤ ∧ 𝐵 ≤ 𝑧 ) ) → - 𝑛 ∈ ℤ ) |
9 |
8
|
zred |
⊢ ( ( ( 𝐵 ∈ ℝ ∧ ( 𝑛 ∈ ℕ ∧ - 𝐵 < 𝑛 ) ) ∧ ( 𝑧 ∈ ℤ ∧ 𝐵 ≤ 𝑧 ) ) → - 𝑛 ∈ ℝ ) |
10 |
|
simprl |
⊢ ( ( ( 𝐵 ∈ ℝ ∧ ( 𝑛 ∈ ℕ ∧ - 𝐵 < 𝑛 ) ) ∧ ( 𝑧 ∈ ℤ ∧ 𝐵 ≤ 𝑧 ) ) → 𝑧 ∈ ℤ ) |
11 |
10
|
zred |
⊢ ( ( ( 𝐵 ∈ ℝ ∧ ( 𝑛 ∈ ℕ ∧ - 𝐵 < 𝑛 ) ) ∧ ( 𝑧 ∈ ℤ ∧ 𝐵 ≤ 𝑧 ) ) → 𝑧 ∈ ℝ ) |
12 |
|
simpll |
⊢ ( ( ( 𝐵 ∈ ℝ ∧ ( 𝑛 ∈ ℕ ∧ - 𝐵 < 𝑛 ) ) ∧ ( 𝑧 ∈ ℤ ∧ 𝐵 ≤ 𝑧 ) ) → 𝐵 ∈ ℝ ) |
13 |
6
|
nnred |
⊢ ( ( ( 𝐵 ∈ ℝ ∧ ( 𝑛 ∈ ℕ ∧ - 𝐵 < 𝑛 ) ) ∧ ( 𝑧 ∈ ℤ ∧ 𝐵 ≤ 𝑧 ) ) → 𝑛 ∈ ℝ ) |
14 |
|
simplrr |
⊢ ( ( ( 𝐵 ∈ ℝ ∧ ( 𝑛 ∈ ℕ ∧ - 𝐵 < 𝑛 ) ) ∧ ( 𝑧 ∈ ℤ ∧ 𝐵 ≤ 𝑧 ) ) → - 𝐵 < 𝑛 ) |
15 |
12 13 14
|
ltnegcon1d |
⊢ ( ( ( 𝐵 ∈ ℝ ∧ ( 𝑛 ∈ ℕ ∧ - 𝐵 < 𝑛 ) ) ∧ ( 𝑧 ∈ ℤ ∧ 𝐵 ≤ 𝑧 ) ) → - 𝑛 < 𝐵 ) |
16 |
|
simprr |
⊢ ( ( ( 𝐵 ∈ ℝ ∧ ( 𝑛 ∈ ℕ ∧ - 𝐵 < 𝑛 ) ) ∧ ( 𝑧 ∈ ℤ ∧ 𝐵 ≤ 𝑧 ) ) → 𝐵 ≤ 𝑧 ) |
17 |
9 12 11 15 16
|
ltletrd |
⊢ ( ( ( 𝐵 ∈ ℝ ∧ ( 𝑛 ∈ ℕ ∧ - 𝐵 < 𝑛 ) ) ∧ ( 𝑧 ∈ ℤ ∧ 𝐵 ≤ 𝑧 ) ) → - 𝑛 < 𝑧 ) |
18 |
9 11 17
|
ltled |
⊢ ( ( ( 𝐵 ∈ ℝ ∧ ( 𝑛 ∈ ℕ ∧ - 𝐵 < 𝑛 ) ) ∧ ( 𝑧 ∈ ℤ ∧ 𝐵 ≤ 𝑧 ) ) → - 𝑛 ≤ 𝑧 ) |
19 |
|
eluz |
⊢ ( ( - 𝑛 ∈ ℤ ∧ 𝑧 ∈ ℤ ) → ( 𝑧 ∈ ( ℤ≥ ‘ - 𝑛 ) ↔ - 𝑛 ≤ 𝑧 ) ) |
20 |
8 10 19
|
syl2anc |
⊢ ( ( ( 𝐵 ∈ ℝ ∧ ( 𝑛 ∈ ℕ ∧ - 𝐵 < 𝑛 ) ) ∧ ( 𝑧 ∈ ℤ ∧ 𝐵 ≤ 𝑧 ) ) → ( 𝑧 ∈ ( ℤ≥ ‘ - 𝑛 ) ↔ - 𝑛 ≤ 𝑧 ) ) |
21 |
18 20
|
mpbird |
⊢ ( ( ( 𝐵 ∈ ℝ ∧ ( 𝑛 ∈ ℕ ∧ - 𝐵 < 𝑛 ) ) ∧ ( 𝑧 ∈ ℤ ∧ 𝐵 ≤ 𝑧 ) ) → 𝑧 ∈ ( ℤ≥ ‘ - 𝑛 ) ) |
22 |
21
|
expr |
⊢ ( ( ( 𝐵 ∈ ℝ ∧ ( 𝑛 ∈ ℕ ∧ - 𝐵 < 𝑛 ) ) ∧ 𝑧 ∈ ℤ ) → ( 𝐵 ≤ 𝑧 → 𝑧 ∈ ( ℤ≥ ‘ - 𝑛 ) ) ) |
23 |
22
|
ralrimiva |
⊢ ( ( 𝐵 ∈ ℝ ∧ ( 𝑛 ∈ ℕ ∧ - 𝐵 < 𝑛 ) ) → ∀ 𝑧 ∈ ℤ ( 𝐵 ≤ 𝑧 → 𝑧 ∈ ( ℤ≥ ‘ - 𝑛 ) ) ) |
24 |
|
rabss |
⊢ ( { 𝑧 ∈ ℤ ∣ 𝐵 ≤ 𝑧 } ⊆ ( ℤ≥ ‘ - 𝑛 ) ↔ ∀ 𝑧 ∈ ℤ ( 𝐵 ≤ 𝑧 → 𝑧 ∈ ( ℤ≥ ‘ - 𝑛 ) ) ) |
25 |
23 24
|
sylibr |
⊢ ( ( 𝐵 ∈ ℝ ∧ ( 𝑛 ∈ ℕ ∧ - 𝐵 < 𝑛 ) ) → { 𝑧 ∈ ℤ ∣ 𝐵 ≤ 𝑧 } ⊆ ( ℤ≥ ‘ - 𝑛 ) ) |
26 |
25
|
adantlr |
⊢ ( ( ( 𝐵 ∈ ℝ ∧ ( 𝐴 ⊆ { 𝑧 ∈ ℤ ∣ 𝐵 ≤ 𝑧 } ∧ 𝐴 ≠ ∅ ) ) ∧ ( 𝑛 ∈ ℕ ∧ - 𝐵 < 𝑛 ) ) → { 𝑧 ∈ ℤ ∣ 𝐵 ≤ 𝑧 } ⊆ ( ℤ≥ ‘ - 𝑛 ) ) |
27 |
5 26
|
sstrd |
⊢ ( ( ( 𝐵 ∈ ℝ ∧ ( 𝐴 ⊆ { 𝑧 ∈ ℤ ∣ 𝐵 ≤ 𝑧 } ∧ 𝐴 ≠ ∅ ) ) ∧ ( 𝑛 ∈ ℕ ∧ - 𝐵 < 𝑛 ) ) → 𝐴 ⊆ ( ℤ≥ ‘ - 𝑛 ) ) |
28 |
|
simplrr |
⊢ ( ( ( 𝐵 ∈ ℝ ∧ ( 𝐴 ⊆ { 𝑧 ∈ ℤ ∣ 𝐵 ≤ 𝑧 } ∧ 𝐴 ≠ ∅ ) ) ∧ ( 𝑛 ∈ ℕ ∧ - 𝐵 < 𝑛 ) ) → 𝐴 ≠ ∅ ) |
29 |
|
infssuzcl |
⊢ ( ( 𝐴 ⊆ ( ℤ≥ ‘ - 𝑛 ) ∧ 𝐴 ≠ ∅ ) → inf ( 𝐴 , ℝ , < ) ∈ 𝐴 ) |
30 |
27 28 29
|
syl2anc |
⊢ ( ( ( 𝐵 ∈ ℝ ∧ ( 𝐴 ⊆ { 𝑧 ∈ ℤ ∣ 𝐵 ≤ 𝑧 } ∧ 𝐴 ≠ ∅ ) ) ∧ ( 𝑛 ∈ ℕ ∧ - 𝐵 < 𝑛 ) ) → inf ( 𝐴 , ℝ , < ) ∈ 𝐴 ) |
31 |
|
infssuzle |
⊢ ( ( 𝐴 ⊆ ( ℤ≥ ‘ - 𝑛 ) ∧ 𝑦 ∈ 𝐴 ) → inf ( 𝐴 , ℝ , < ) ≤ 𝑦 ) |
32 |
27 31
|
sylan |
⊢ ( ( ( ( 𝐵 ∈ ℝ ∧ ( 𝐴 ⊆ { 𝑧 ∈ ℤ ∣ 𝐵 ≤ 𝑧 } ∧ 𝐴 ≠ ∅ ) ) ∧ ( 𝑛 ∈ ℕ ∧ - 𝐵 < 𝑛 ) ) ∧ 𝑦 ∈ 𝐴 ) → inf ( 𝐴 , ℝ , < ) ≤ 𝑦 ) |
33 |
32
|
ralrimiva |
⊢ ( ( ( 𝐵 ∈ ℝ ∧ ( 𝐴 ⊆ { 𝑧 ∈ ℤ ∣ 𝐵 ≤ 𝑧 } ∧ 𝐴 ≠ ∅ ) ) ∧ ( 𝑛 ∈ ℕ ∧ - 𝐵 < 𝑛 ) ) → ∀ 𝑦 ∈ 𝐴 inf ( 𝐴 , ℝ , < ) ≤ 𝑦 ) |
34 |
|
breq2 |
⊢ ( 𝑦 = inf ( 𝐴 , ℝ , < ) → ( 𝑥 ≤ 𝑦 ↔ 𝑥 ≤ inf ( 𝐴 , ℝ , < ) ) ) |
35 |
|
simprr |
⊢ ( ( ( ( 𝐵 ∈ ℝ ∧ ( 𝐴 ⊆ { 𝑧 ∈ ℤ ∣ 𝐵 ≤ 𝑧 } ∧ 𝐴 ≠ ∅ ) ) ∧ ( 𝑛 ∈ ℕ ∧ - 𝐵 < 𝑛 ) ) ∧ ( 𝑥 ∈ 𝐴 ∧ ∀ 𝑦 ∈ 𝐴 𝑥 ≤ 𝑦 ) ) → ∀ 𝑦 ∈ 𝐴 𝑥 ≤ 𝑦 ) |
36 |
30
|
adantr |
⊢ ( ( ( ( 𝐵 ∈ ℝ ∧ ( 𝐴 ⊆ { 𝑧 ∈ ℤ ∣ 𝐵 ≤ 𝑧 } ∧ 𝐴 ≠ ∅ ) ) ∧ ( 𝑛 ∈ ℕ ∧ - 𝐵 < 𝑛 ) ) ∧ ( 𝑥 ∈ 𝐴 ∧ ∀ 𝑦 ∈ 𝐴 𝑥 ≤ 𝑦 ) ) → inf ( 𝐴 , ℝ , < ) ∈ 𝐴 ) |
37 |
34 35 36
|
rspcdva |
⊢ ( ( ( ( 𝐵 ∈ ℝ ∧ ( 𝐴 ⊆ { 𝑧 ∈ ℤ ∣ 𝐵 ≤ 𝑧 } ∧ 𝐴 ≠ ∅ ) ) ∧ ( 𝑛 ∈ ℕ ∧ - 𝐵 < 𝑛 ) ) ∧ ( 𝑥 ∈ 𝐴 ∧ ∀ 𝑦 ∈ 𝐴 𝑥 ≤ 𝑦 ) ) → 𝑥 ≤ inf ( 𝐴 , ℝ , < ) ) |
38 |
27
|
adantr |
⊢ ( ( ( ( 𝐵 ∈ ℝ ∧ ( 𝐴 ⊆ { 𝑧 ∈ ℤ ∣ 𝐵 ≤ 𝑧 } ∧ 𝐴 ≠ ∅ ) ) ∧ ( 𝑛 ∈ ℕ ∧ - 𝐵 < 𝑛 ) ) ∧ ( 𝑥 ∈ 𝐴 ∧ ∀ 𝑦 ∈ 𝐴 𝑥 ≤ 𝑦 ) ) → 𝐴 ⊆ ( ℤ≥ ‘ - 𝑛 ) ) |
39 |
|
simprl |
⊢ ( ( ( ( 𝐵 ∈ ℝ ∧ ( 𝐴 ⊆ { 𝑧 ∈ ℤ ∣ 𝐵 ≤ 𝑧 } ∧ 𝐴 ≠ ∅ ) ) ∧ ( 𝑛 ∈ ℕ ∧ - 𝐵 < 𝑛 ) ) ∧ ( 𝑥 ∈ 𝐴 ∧ ∀ 𝑦 ∈ 𝐴 𝑥 ≤ 𝑦 ) ) → 𝑥 ∈ 𝐴 ) |
40 |
|
infssuzle |
⊢ ( ( 𝐴 ⊆ ( ℤ≥ ‘ - 𝑛 ) ∧ 𝑥 ∈ 𝐴 ) → inf ( 𝐴 , ℝ , < ) ≤ 𝑥 ) |
41 |
38 39 40
|
syl2anc |
⊢ ( ( ( ( 𝐵 ∈ ℝ ∧ ( 𝐴 ⊆ { 𝑧 ∈ ℤ ∣ 𝐵 ≤ 𝑧 } ∧ 𝐴 ≠ ∅ ) ) ∧ ( 𝑛 ∈ ℕ ∧ - 𝐵 < 𝑛 ) ) ∧ ( 𝑥 ∈ 𝐴 ∧ ∀ 𝑦 ∈ 𝐴 𝑥 ≤ 𝑦 ) ) → inf ( 𝐴 , ℝ , < ) ≤ 𝑥 ) |
42 |
|
uzssz |
⊢ ( ℤ≥ ‘ - 𝑛 ) ⊆ ℤ |
43 |
|
zssre |
⊢ ℤ ⊆ ℝ |
44 |
42 43
|
sstri |
⊢ ( ℤ≥ ‘ - 𝑛 ) ⊆ ℝ |
45 |
27 44
|
sstrdi |
⊢ ( ( ( 𝐵 ∈ ℝ ∧ ( 𝐴 ⊆ { 𝑧 ∈ ℤ ∣ 𝐵 ≤ 𝑧 } ∧ 𝐴 ≠ ∅ ) ) ∧ ( 𝑛 ∈ ℕ ∧ - 𝐵 < 𝑛 ) ) → 𝐴 ⊆ ℝ ) |
46 |
45
|
adantr |
⊢ ( ( ( ( 𝐵 ∈ ℝ ∧ ( 𝐴 ⊆ { 𝑧 ∈ ℤ ∣ 𝐵 ≤ 𝑧 } ∧ 𝐴 ≠ ∅ ) ) ∧ ( 𝑛 ∈ ℕ ∧ - 𝐵 < 𝑛 ) ) ∧ ( 𝑥 ∈ 𝐴 ∧ ∀ 𝑦 ∈ 𝐴 𝑥 ≤ 𝑦 ) ) → 𝐴 ⊆ ℝ ) |
47 |
46 39
|
sseldd |
⊢ ( ( ( ( 𝐵 ∈ ℝ ∧ ( 𝐴 ⊆ { 𝑧 ∈ ℤ ∣ 𝐵 ≤ 𝑧 } ∧ 𝐴 ≠ ∅ ) ) ∧ ( 𝑛 ∈ ℕ ∧ - 𝐵 < 𝑛 ) ) ∧ ( 𝑥 ∈ 𝐴 ∧ ∀ 𝑦 ∈ 𝐴 𝑥 ≤ 𝑦 ) ) → 𝑥 ∈ ℝ ) |
48 |
45 30
|
sseldd |
⊢ ( ( ( 𝐵 ∈ ℝ ∧ ( 𝐴 ⊆ { 𝑧 ∈ ℤ ∣ 𝐵 ≤ 𝑧 } ∧ 𝐴 ≠ ∅ ) ) ∧ ( 𝑛 ∈ ℕ ∧ - 𝐵 < 𝑛 ) ) → inf ( 𝐴 , ℝ , < ) ∈ ℝ ) |
49 |
48
|
adantr |
⊢ ( ( ( ( 𝐵 ∈ ℝ ∧ ( 𝐴 ⊆ { 𝑧 ∈ ℤ ∣ 𝐵 ≤ 𝑧 } ∧ 𝐴 ≠ ∅ ) ) ∧ ( 𝑛 ∈ ℕ ∧ - 𝐵 < 𝑛 ) ) ∧ ( 𝑥 ∈ 𝐴 ∧ ∀ 𝑦 ∈ 𝐴 𝑥 ≤ 𝑦 ) ) → inf ( 𝐴 , ℝ , < ) ∈ ℝ ) |
50 |
47 49
|
letri3d |
⊢ ( ( ( ( 𝐵 ∈ ℝ ∧ ( 𝐴 ⊆ { 𝑧 ∈ ℤ ∣ 𝐵 ≤ 𝑧 } ∧ 𝐴 ≠ ∅ ) ) ∧ ( 𝑛 ∈ ℕ ∧ - 𝐵 < 𝑛 ) ) ∧ ( 𝑥 ∈ 𝐴 ∧ ∀ 𝑦 ∈ 𝐴 𝑥 ≤ 𝑦 ) ) → ( 𝑥 = inf ( 𝐴 , ℝ , < ) ↔ ( 𝑥 ≤ inf ( 𝐴 , ℝ , < ) ∧ inf ( 𝐴 , ℝ , < ) ≤ 𝑥 ) ) ) |
51 |
37 41 50
|
mpbir2and |
⊢ ( ( ( ( 𝐵 ∈ ℝ ∧ ( 𝐴 ⊆ { 𝑧 ∈ ℤ ∣ 𝐵 ≤ 𝑧 } ∧ 𝐴 ≠ ∅ ) ) ∧ ( 𝑛 ∈ ℕ ∧ - 𝐵 < 𝑛 ) ) ∧ ( 𝑥 ∈ 𝐴 ∧ ∀ 𝑦 ∈ 𝐴 𝑥 ≤ 𝑦 ) ) → 𝑥 = inf ( 𝐴 , ℝ , < ) ) |
52 |
51
|
expr |
⊢ ( ( ( ( 𝐵 ∈ ℝ ∧ ( 𝐴 ⊆ { 𝑧 ∈ ℤ ∣ 𝐵 ≤ 𝑧 } ∧ 𝐴 ≠ ∅ ) ) ∧ ( 𝑛 ∈ ℕ ∧ - 𝐵 < 𝑛 ) ) ∧ 𝑥 ∈ 𝐴 ) → ( ∀ 𝑦 ∈ 𝐴 𝑥 ≤ 𝑦 → 𝑥 = inf ( 𝐴 , ℝ , < ) ) ) |
53 |
52
|
ralrimiva |
⊢ ( ( ( 𝐵 ∈ ℝ ∧ ( 𝐴 ⊆ { 𝑧 ∈ ℤ ∣ 𝐵 ≤ 𝑧 } ∧ 𝐴 ≠ ∅ ) ) ∧ ( 𝑛 ∈ ℕ ∧ - 𝐵 < 𝑛 ) ) → ∀ 𝑥 ∈ 𝐴 ( ∀ 𝑦 ∈ 𝐴 𝑥 ≤ 𝑦 → 𝑥 = inf ( 𝐴 , ℝ , < ) ) ) |
54 |
|
breq1 |
⊢ ( 𝑥 = inf ( 𝐴 , ℝ , < ) → ( 𝑥 ≤ 𝑦 ↔ inf ( 𝐴 , ℝ , < ) ≤ 𝑦 ) ) |
55 |
54
|
ralbidv |
⊢ ( 𝑥 = inf ( 𝐴 , ℝ , < ) → ( ∀ 𝑦 ∈ 𝐴 𝑥 ≤ 𝑦 ↔ ∀ 𝑦 ∈ 𝐴 inf ( 𝐴 , ℝ , < ) ≤ 𝑦 ) ) |
56 |
55
|
eqreu |
⊢ ( ( inf ( 𝐴 , ℝ , < ) ∈ 𝐴 ∧ ∀ 𝑦 ∈ 𝐴 inf ( 𝐴 , ℝ , < ) ≤ 𝑦 ∧ ∀ 𝑥 ∈ 𝐴 ( ∀ 𝑦 ∈ 𝐴 𝑥 ≤ 𝑦 → 𝑥 = inf ( 𝐴 , ℝ , < ) ) ) → ∃! 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 𝑥 ≤ 𝑦 ) |
57 |
30 33 53 56
|
syl3anc |
⊢ ( ( ( 𝐵 ∈ ℝ ∧ ( 𝐴 ⊆ { 𝑧 ∈ ℤ ∣ 𝐵 ≤ 𝑧 } ∧ 𝐴 ≠ ∅ ) ) ∧ ( 𝑛 ∈ ℕ ∧ - 𝐵 < 𝑛 ) ) → ∃! 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 𝑥 ≤ 𝑦 ) |
58 |
4 57
|
rexlimddv |
⊢ ( ( 𝐵 ∈ ℝ ∧ ( 𝐴 ⊆ { 𝑧 ∈ ℤ ∣ 𝐵 ≤ 𝑧 } ∧ 𝐴 ≠ ∅ ) ) → ∃! 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 𝑥 ≤ 𝑦 ) |