Step |
Hyp |
Ref |
Expression |
1 |
|
id |
⊢ ( 𝐴 ⊆ ℝ* → 𝐴 ⊆ ℝ* ) |
2 |
|
pnfxr |
⊢ +∞ ∈ ℝ* |
3 |
|
snssi |
⊢ ( +∞ ∈ ℝ* → { +∞ } ⊆ ℝ* ) |
4 |
2 3
|
ax-mp |
⊢ { +∞ } ⊆ ℝ* |
5 |
4
|
a1i |
⊢ ( 𝐴 ⊆ ℝ* → { +∞ } ⊆ ℝ* ) |
6 |
1 5
|
unssd |
⊢ ( 𝐴 ⊆ ℝ* → ( 𝐴 ∪ { +∞ } ) ⊆ ℝ* ) |
7 |
6
|
infxrcld |
⊢ ( 𝐴 ⊆ ℝ* → inf ( ( 𝐴 ∪ { +∞ } ) , ℝ* , < ) ∈ ℝ* ) |
8 |
|
infxrcl |
⊢ ( 𝐴 ⊆ ℝ* → inf ( 𝐴 , ℝ* , < ) ∈ ℝ* ) |
9 |
|
ssun1 |
⊢ 𝐴 ⊆ ( 𝐴 ∪ { +∞ } ) |
10 |
9
|
a1i |
⊢ ( 𝐴 ⊆ ℝ* → 𝐴 ⊆ ( 𝐴 ∪ { +∞ } ) ) |
11 |
|
infxrss |
⊢ ( ( 𝐴 ⊆ ( 𝐴 ∪ { +∞ } ) ∧ ( 𝐴 ∪ { +∞ } ) ⊆ ℝ* ) → inf ( ( 𝐴 ∪ { +∞ } ) , ℝ* , < ) ≤ inf ( 𝐴 , ℝ* , < ) ) |
12 |
10 6 11
|
syl2anc |
⊢ ( 𝐴 ⊆ ℝ* → inf ( ( 𝐴 ∪ { +∞ } ) , ℝ* , < ) ≤ inf ( 𝐴 , ℝ* , < ) ) |
13 |
|
infeq1 |
⊢ ( 𝐴 = ∅ → inf ( 𝐴 , ℝ* , < ) = inf ( ∅ , ℝ* , < ) ) |
14 |
|
xrinf0 |
⊢ inf ( ∅ , ℝ* , < ) = +∞ |
15 |
14 2
|
eqeltri |
⊢ inf ( ∅ , ℝ* , < ) ∈ ℝ* |
16 |
15
|
a1i |
⊢ ( 𝐴 = ∅ → inf ( ∅ , ℝ* , < ) ∈ ℝ* ) |
17 |
13 16
|
eqeltrd |
⊢ ( 𝐴 = ∅ → inf ( 𝐴 , ℝ* , < ) ∈ ℝ* ) |
18 |
|
xrltso |
⊢ < Or ℝ* |
19 |
|
infsn |
⊢ ( ( < Or ℝ* ∧ +∞ ∈ ℝ* ) → inf ( { +∞ } , ℝ* , < ) = +∞ ) |
20 |
18 2 19
|
mp2an |
⊢ inf ( { +∞ } , ℝ* , < ) = +∞ |
21 |
20
|
eqcomi |
⊢ +∞ = inf ( { +∞ } , ℝ* , < ) |
22 |
21
|
a1i |
⊢ ( 𝐴 = ∅ → +∞ = inf ( { +∞ } , ℝ* , < ) ) |
23 |
13 14
|
eqtrdi |
⊢ ( 𝐴 = ∅ → inf ( 𝐴 , ℝ* , < ) = +∞ ) |
24 |
|
uneq1 |
⊢ ( 𝐴 = ∅ → ( 𝐴 ∪ { +∞ } ) = ( ∅ ∪ { +∞ } ) ) |
25 |
|
0un |
⊢ ( ∅ ∪ { +∞ } ) = { +∞ } |
26 |
25
|
a1i |
⊢ ( 𝐴 = ∅ → ( ∅ ∪ { +∞ } ) = { +∞ } ) |
27 |
24 26
|
eqtrd |
⊢ ( 𝐴 = ∅ → ( 𝐴 ∪ { +∞ } ) = { +∞ } ) |
28 |
27
|
infeq1d |
⊢ ( 𝐴 = ∅ → inf ( ( 𝐴 ∪ { +∞ } ) , ℝ* , < ) = inf ( { +∞ } , ℝ* , < ) ) |
29 |
22 23 28
|
3eqtr4d |
⊢ ( 𝐴 = ∅ → inf ( 𝐴 , ℝ* , < ) = inf ( ( 𝐴 ∪ { +∞ } ) , ℝ* , < ) ) |
30 |
17 29
|
xreqled |
⊢ ( 𝐴 = ∅ → inf ( 𝐴 , ℝ* , < ) ≤ inf ( ( 𝐴 ∪ { +∞ } ) , ℝ* , < ) ) |
31 |
30
|
adantl |
⊢ ( ( 𝐴 ⊆ ℝ* ∧ 𝐴 = ∅ ) → inf ( 𝐴 , ℝ* , < ) ≤ inf ( ( 𝐴 ∪ { +∞ } ) , ℝ* , < ) ) |
32 |
|
neqne |
⊢ ( ¬ 𝐴 = ∅ → 𝐴 ≠ ∅ ) |
33 |
|
nfv |
⊢ Ⅎ 𝑥 ( 𝐴 ⊆ ℝ* ∧ 𝐴 ≠ ∅ ) |
34 |
|
nfv |
⊢ Ⅎ 𝑦 ( 𝐴 ⊆ ℝ* ∧ 𝐴 ≠ ∅ ) |
35 |
|
simpl |
⊢ ( ( 𝐴 ⊆ ℝ* ∧ 𝐴 ≠ ∅ ) → 𝐴 ⊆ ℝ* ) |
36 |
35 6
|
syl |
⊢ ( ( 𝐴 ⊆ ℝ* ∧ 𝐴 ≠ ∅ ) → ( 𝐴 ∪ { +∞ } ) ⊆ ℝ* ) |
37 |
|
simpr |
⊢ ( ( 𝐴 ⊆ ℝ* ∧ 𝑥 ∈ 𝐴 ) → 𝑥 ∈ 𝐴 ) |
38 |
|
ssel2 |
⊢ ( ( 𝐴 ⊆ ℝ* ∧ 𝑥 ∈ 𝐴 ) → 𝑥 ∈ ℝ* ) |
39 |
38
|
xrleidd |
⊢ ( ( 𝐴 ⊆ ℝ* ∧ 𝑥 ∈ 𝐴 ) → 𝑥 ≤ 𝑥 ) |
40 |
|
breq1 |
⊢ ( 𝑦 = 𝑥 → ( 𝑦 ≤ 𝑥 ↔ 𝑥 ≤ 𝑥 ) ) |
41 |
40
|
rspcev |
⊢ ( ( 𝑥 ∈ 𝐴 ∧ 𝑥 ≤ 𝑥 ) → ∃ 𝑦 ∈ 𝐴 𝑦 ≤ 𝑥 ) |
42 |
37 39 41
|
syl2anc |
⊢ ( ( 𝐴 ⊆ ℝ* ∧ 𝑥 ∈ 𝐴 ) → ∃ 𝑦 ∈ 𝐴 𝑦 ≤ 𝑥 ) |
43 |
42
|
ad4ant14 |
⊢ ( ( ( ( 𝐴 ⊆ ℝ* ∧ 𝐴 ≠ ∅ ) ∧ 𝑥 ∈ ( 𝐴 ∪ { +∞ } ) ) ∧ 𝑥 ∈ 𝐴 ) → ∃ 𝑦 ∈ 𝐴 𝑦 ≤ 𝑥 ) |
44 |
|
simpll |
⊢ ( ( ( ( 𝐴 ⊆ ℝ* ∧ 𝐴 ≠ ∅ ) ∧ 𝑥 ∈ ( 𝐴 ∪ { +∞ } ) ) ∧ ¬ 𝑥 ∈ 𝐴 ) → ( 𝐴 ⊆ ℝ* ∧ 𝐴 ≠ ∅ ) ) |
45 |
|
elunnel1 |
⊢ ( ( 𝑥 ∈ ( 𝐴 ∪ { +∞ } ) ∧ ¬ 𝑥 ∈ 𝐴 ) → 𝑥 ∈ { +∞ } ) |
46 |
|
elsni |
⊢ ( 𝑥 ∈ { +∞ } → 𝑥 = +∞ ) |
47 |
45 46
|
syl |
⊢ ( ( 𝑥 ∈ ( 𝐴 ∪ { +∞ } ) ∧ ¬ 𝑥 ∈ 𝐴 ) → 𝑥 = +∞ ) |
48 |
47
|
adantll |
⊢ ( ( ( ( 𝐴 ⊆ ℝ* ∧ 𝐴 ≠ ∅ ) ∧ 𝑥 ∈ ( 𝐴 ∪ { +∞ } ) ) ∧ ¬ 𝑥 ∈ 𝐴 ) → 𝑥 = +∞ ) |
49 |
|
simplr |
⊢ ( ( ( 𝐴 ⊆ ℝ* ∧ 𝐴 ≠ ∅ ) ∧ 𝑥 = +∞ ) → 𝐴 ≠ ∅ ) |
50 |
|
ssel2 |
⊢ ( ( 𝐴 ⊆ ℝ* ∧ 𝑦 ∈ 𝐴 ) → 𝑦 ∈ ℝ* ) |
51 |
|
pnfge |
⊢ ( 𝑦 ∈ ℝ* → 𝑦 ≤ +∞ ) |
52 |
50 51
|
syl |
⊢ ( ( 𝐴 ⊆ ℝ* ∧ 𝑦 ∈ 𝐴 ) → 𝑦 ≤ +∞ ) |
53 |
52
|
adantlr |
⊢ ( ( ( 𝐴 ⊆ ℝ* ∧ 𝑥 = +∞ ) ∧ 𝑦 ∈ 𝐴 ) → 𝑦 ≤ +∞ ) |
54 |
|
simplr |
⊢ ( ( ( 𝐴 ⊆ ℝ* ∧ 𝑥 = +∞ ) ∧ 𝑦 ∈ 𝐴 ) → 𝑥 = +∞ ) |
55 |
53 54
|
breqtrrd |
⊢ ( ( ( 𝐴 ⊆ ℝ* ∧ 𝑥 = +∞ ) ∧ 𝑦 ∈ 𝐴 ) → 𝑦 ≤ 𝑥 ) |
56 |
55
|
ralrimiva |
⊢ ( ( 𝐴 ⊆ ℝ* ∧ 𝑥 = +∞ ) → ∀ 𝑦 ∈ 𝐴 𝑦 ≤ 𝑥 ) |
57 |
56
|
adantlr |
⊢ ( ( ( 𝐴 ⊆ ℝ* ∧ 𝐴 ≠ ∅ ) ∧ 𝑥 = +∞ ) → ∀ 𝑦 ∈ 𝐴 𝑦 ≤ 𝑥 ) |
58 |
|
r19.2z |
⊢ ( ( 𝐴 ≠ ∅ ∧ ∀ 𝑦 ∈ 𝐴 𝑦 ≤ 𝑥 ) → ∃ 𝑦 ∈ 𝐴 𝑦 ≤ 𝑥 ) |
59 |
49 57 58
|
syl2anc |
⊢ ( ( ( 𝐴 ⊆ ℝ* ∧ 𝐴 ≠ ∅ ) ∧ 𝑥 = +∞ ) → ∃ 𝑦 ∈ 𝐴 𝑦 ≤ 𝑥 ) |
60 |
44 48 59
|
syl2anc |
⊢ ( ( ( ( 𝐴 ⊆ ℝ* ∧ 𝐴 ≠ ∅ ) ∧ 𝑥 ∈ ( 𝐴 ∪ { +∞ } ) ) ∧ ¬ 𝑥 ∈ 𝐴 ) → ∃ 𝑦 ∈ 𝐴 𝑦 ≤ 𝑥 ) |
61 |
43 60
|
pm2.61dan |
⊢ ( ( ( 𝐴 ⊆ ℝ* ∧ 𝐴 ≠ ∅ ) ∧ 𝑥 ∈ ( 𝐴 ∪ { +∞ } ) ) → ∃ 𝑦 ∈ 𝐴 𝑦 ≤ 𝑥 ) |
62 |
33 34 35 36 61
|
infleinf2 |
⊢ ( ( 𝐴 ⊆ ℝ* ∧ 𝐴 ≠ ∅ ) → inf ( 𝐴 , ℝ* , < ) ≤ inf ( ( 𝐴 ∪ { +∞ } ) , ℝ* , < ) ) |
63 |
32 62
|
sylan2 |
⊢ ( ( 𝐴 ⊆ ℝ* ∧ ¬ 𝐴 = ∅ ) → inf ( 𝐴 , ℝ* , < ) ≤ inf ( ( 𝐴 ∪ { +∞ } ) , ℝ* , < ) ) |
64 |
31 63
|
pm2.61dan |
⊢ ( 𝐴 ⊆ ℝ* → inf ( 𝐴 , ℝ* , < ) ≤ inf ( ( 𝐴 ∪ { +∞ } ) , ℝ* , < ) ) |
65 |
7 8 12 64
|
xrletrid |
⊢ ( 𝐴 ⊆ ℝ* → inf ( ( 𝐴 ∪ { +∞ } ) , ℝ* , < ) = inf ( 𝐴 , ℝ* , < ) ) |