Proof of Theorem infrpge
Step | Hyp | Ref
| Expression |
1 | | infrpge.an0 |
. . . . . 6
⊢ (𝜑 → 𝐴 ≠ ∅) |
2 | | n0 4261 |
. . . . . . 7
⊢ (𝐴 ≠ ∅ ↔
∃𝑧 𝑧 ∈ 𝐴) |
3 | 2 | biimpi 219 |
. . . . . 6
⊢ (𝐴 ≠ ∅ →
∃𝑧 𝑧 ∈ 𝐴) |
4 | 1, 3 | syl 17 |
. . . . 5
⊢ (𝜑 → ∃𝑧 𝑧 ∈ 𝐴) |
5 | 4 | adantr 484 |
. . . 4
⊢ ((𝜑 ∧ inf(𝐴, ℝ*, < ) = +∞)
→ ∃𝑧 𝑧 ∈ 𝐴) |
6 | | nfv 1922 |
. . . . 5
⊢
Ⅎ𝑧(𝜑 ∧ inf(𝐴, ℝ*, < ) =
+∞) |
7 | | simpr 488 |
. . . . . . 7
⊢ (((𝜑 ∧ inf(𝐴, ℝ*, < ) = +∞)
∧ 𝑧 ∈ 𝐴) → 𝑧 ∈ 𝐴) |
8 | | infrpge.a |
. . . . . . . . . . . 12
⊢ (𝜑 → 𝐴 ⊆
ℝ*) |
9 | 8 | adantr 484 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑧 ∈ 𝐴) → 𝐴 ⊆
ℝ*) |
10 | | simpr 488 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑧 ∈ 𝐴) → 𝑧 ∈ 𝐴) |
11 | 9, 10 | sseldd 3902 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑧 ∈ 𝐴) → 𝑧 ∈ ℝ*) |
12 | | pnfge 12722 |
. . . . . . . . . 10
⊢ (𝑧 ∈ ℝ*
→ 𝑧 ≤
+∞) |
13 | 11, 12 | syl 17 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑧 ∈ 𝐴) → 𝑧 ≤ +∞) |
14 | 13 | adantlr 715 |
. . . . . . . 8
⊢ (((𝜑 ∧ inf(𝐴, ℝ*, < ) = +∞)
∧ 𝑧 ∈ 𝐴) → 𝑧 ≤ +∞) |
15 | | oveq1 7220 |
. . . . . . . . . . 11
⊢
(inf(𝐴,
ℝ*, < ) = +∞ → (inf(𝐴, ℝ*, < )
+𝑒 𝐵) =
(+∞ +𝑒 𝐵)) |
16 | 15 | adantl 485 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ inf(𝐴, ℝ*, < ) = +∞)
→ (inf(𝐴,
ℝ*, < ) +𝑒 𝐵) = (+∞ +𝑒 𝐵)) |
17 | | infrpge.b |
. . . . . . . . . . . . 13
⊢ (𝜑 → 𝐵 ∈
ℝ+) |
18 | 17 | rpxrd 12629 |
. . . . . . . . . . . 12
⊢ (𝜑 → 𝐵 ∈
ℝ*) |
19 | 17 | rpred 12628 |
. . . . . . . . . . . . 13
⊢ (𝜑 → 𝐵 ∈ ℝ) |
20 | | renemnf 10882 |
. . . . . . . . . . . . 13
⊢ (𝐵 ∈ ℝ → 𝐵 ≠ -∞) |
21 | 19, 20 | syl 17 |
. . . . . . . . . . . 12
⊢ (𝜑 → 𝐵 ≠ -∞) |
22 | | xaddpnf2 12817 |
. . . . . . . . . . . 12
⊢ ((𝐵 ∈ ℝ*
∧ 𝐵 ≠ -∞)
→ (+∞ +𝑒 𝐵) = +∞) |
23 | 18, 21, 22 | syl2anc 587 |
. . . . . . . . . . 11
⊢ (𝜑 → (+∞
+𝑒 𝐵) =
+∞) |
24 | 23 | adantr 484 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ inf(𝐴, ℝ*, < ) = +∞)
→ (+∞ +𝑒 𝐵) = +∞) |
25 | 16, 24 | eqtr2d 2778 |
. . . . . . . . 9
⊢ ((𝜑 ∧ inf(𝐴, ℝ*, < ) = +∞)
→ +∞ = (inf(𝐴,
ℝ*, < ) +𝑒 𝐵)) |
26 | 25 | adantr 484 |
. . . . . . . 8
⊢ (((𝜑 ∧ inf(𝐴, ℝ*, < ) = +∞)
∧ 𝑧 ∈ 𝐴) → +∞ = (inf(𝐴, ℝ*, < )
+𝑒 𝐵)) |
27 | 14, 26 | breqtrd 5079 |
. . . . . . 7
⊢ (((𝜑 ∧ inf(𝐴, ℝ*, < ) = +∞)
∧ 𝑧 ∈ 𝐴) → 𝑧 ≤ (inf(𝐴, ℝ*, < )
+𝑒 𝐵)) |
28 | 7, 27 | jca 515 |
. . . . . 6
⊢ (((𝜑 ∧ inf(𝐴, ℝ*, < ) = +∞)
∧ 𝑧 ∈ 𝐴) → (𝑧 ∈ 𝐴 ∧ 𝑧 ≤ (inf(𝐴, ℝ*, < )
+𝑒 𝐵))) |
29 | 28 | ex 416 |
. . . . 5
⊢ ((𝜑 ∧ inf(𝐴, ℝ*, < ) = +∞)
→ (𝑧 ∈ 𝐴 → (𝑧 ∈ 𝐴 ∧ 𝑧 ≤ (inf(𝐴, ℝ*, < )
+𝑒 𝐵)))) |
30 | 6, 29 | eximd 2214 |
. . . 4
⊢ ((𝜑 ∧ inf(𝐴, ℝ*, < ) = +∞)
→ (∃𝑧 𝑧 ∈ 𝐴 → ∃𝑧(𝑧 ∈ 𝐴 ∧ 𝑧 ≤ (inf(𝐴, ℝ*, < )
+𝑒 𝐵)))) |
31 | 5, 30 | mpd 15 |
. . 3
⊢ ((𝜑 ∧ inf(𝐴, ℝ*, < ) = +∞)
→ ∃𝑧(𝑧 ∈ 𝐴 ∧ 𝑧 ≤ (inf(𝐴, ℝ*, < )
+𝑒 𝐵))) |
32 | | df-rex 3067 |
. . 3
⊢
(∃𝑧 ∈
𝐴 𝑧 ≤ (inf(𝐴, ℝ*, < )
+𝑒 𝐵)
↔ ∃𝑧(𝑧 ∈ 𝐴 ∧ 𝑧 ≤ (inf(𝐴, ℝ*, < )
+𝑒 𝐵))) |
33 | 31, 32 | sylibr 237 |
. 2
⊢ ((𝜑 ∧ inf(𝐴, ℝ*, < ) = +∞)
→ ∃𝑧 ∈
𝐴 𝑧 ≤ (inf(𝐴, ℝ*, < )
+𝑒 𝐵)) |
34 | | simpl 486 |
. . . 4
⊢ ((𝜑 ∧ ¬ inf(𝐴, ℝ*, < ) = +∞)
→ 𝜑) |
35 | | infrpge.bnd |
. . . . . . . . 9
⊢ (𝜑 → ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑥 ≤ 𝑦) |
36 | | infrpge.xph |
. . . . . . . . . 10
⊢
Ⅎ𝑥𝜑 |
37 | | nfv 1922 |
. . . . . . . . . 10
⊢
Ⅎ𝑥-∞
< inf(𝐴,
ℝ*, < ) |
38 | | mnfxr 10890 |
. . . . . . . . . . . . 13
⊢ -∞
∈ ℝ* |
39 | 38 | a1i 11 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑥 ∈ ℝ ∧ ∀𝑦 ∈ 𝐴 𝑥 ≤ 𝑦) → -∞ ∈
ℝ*) |
40 | | rexr 10879 |
. . . . . . . . . . . . 13
⊢ (𝑥 ∈ ℝ → 𝑥 ∈
ℝ*) |
41 | 40 | 3ad2ant2 1136 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑥 ∈ ℝ ∧ ∀𝑦 ∈ 𝐴 𝑥 ≤ 𝑦) → 𝑥 ∈ ℝ*) |
42 | | infxrcl 12923 |
. . . . . . . . . . . . . 14
⊢ (𝐴 ⊆ ℝ*
→ inf(𝐴,
ℝ*, < ) ∈ ℝ*) |
43 | 8, 42 | syl 17 |
. . . . . . . . . . . . 13
⊢ (𝜑 → inf(𝐴, ℝ*, < ) ∈
ℝ*) |
44 | 43 | 3ad2ant1 1135 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑥 ∈ ℝ ∧ ∀𝑦 ∈ 𝐴 𝑥 ≤ 𝑦) → inf(𝐴, ℝ*, < ) ∈
ℝ*) |
45 | | mnflt 12715 |
. . . . . . . . . . . . 13
⊢ (𝑥 ∈ ℝ → -∞
< 𝑥) |
46 | 45 | 3ad2ant2 1136 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑥 ∈ ℝ ∧ ∀𝑦 ∈ 𝐴 𝑥 ≤ 𝑦) → -∞ < 𝑥) |
47 | | simp3 1140 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑥 ∈ ℝ ∧ ∀𝑦 ∈ 𝐴 𝑥 ≤ 𝑦) → ∀𝑦 ∈ 𝐴 𝑥 ≤ 𝑦) |
48 | 8 | adantr 484 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → 𝐴 ⊆
ℝ*) |
49 | 40 | adantl 485 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → 𝑥 ∈ ℝ*) |
50 | | infxrgelb 12925 |
. . . . . . . . . . . . . . 15
⊢ ((𝐴 ⊆ ℝ*
∧ 𝑥 ∈
ℝ*) → (𝑥 ≤ inf(𝐴, ℝ*, < ) ↔
∀𝑦 ∈ 𝐴 𝑥 ≤ 𝑦)) |
51 | 48, 49, 50 | syl2anc 587 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (𝑥 ≤ inf(𝐴, ℝ*, < ) ↔
∀𝑦 ∈ 𝐴 𝑥 ≤ 𝑦)) |
52 | 51 | 3adant3 1134 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑥 ∈ ℝ ∧ ∀𝑦 ∈ 𝐴 𝑥 ≤ 𝑦) → (𝑥 ≤ inf(𝐴, ℝ*, < ) ↔
∀𝑦 ∈ 𝐴 𝑥 ≤ 𝑦)) |
53 | 47, 52 | mpbird 260 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑥 ∈ ℝ ∧ ∀𝑦 ∈ 𝐴 𝑥 ≤ 𝑦) → 𝑥 ≤ inf(𝐴, ℝ*, <
)) |
54 | 39, 41, 44, 46, 53 | xrltletrd 12751 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑥 ∈ ℝ ∧ ∀𝑦 ∈ 𝐴 𝑥 ≤ 𝑦) → -∞ < inf(𝐴, ℝ*, <
)) |
55 | 54 | 3exp 1121 |
. . . . . . . . . 10
⊢ (𝜑 → (𝑥 ∈ ℝ → (∀𝑦 ∈ 𝐴 𝑥 ≤ 𝑦 → -∞ < inf(𝐴, ℝ*, <
)))) |
56 | 36, 37, 55 | rexlimd 3236 |
. . . . . . . . 9
⊢ (𝜑 → (∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑥 ≤ 𝑦 → -∞ < inf(𝐴, ℝ*, <
))) |
57 | 35, 56 | mpd 15 |
. . . . . . . 8
⊢ (𝜑 → -∞ < inf(𝐴, ℝ*, <
)) |
58 | 57 | adantr 484 |
. . . . . . 7
⊢ ((𝜑 ∧ ¬ inf(𝐴, ℝ*, < ) = +∞)
→ -∞ < inf(𝐴,
ℝ*, < )) |
59 | | neqne 2948 |
. . . . . . . . 9
⊢ (¬
inf(𝐴, ℝ*,
< ) = +∞ → inf(𝐴, ℝ*, < ) ≠
+∞) |
60 | 59 | adantl 485 |
. . . . . . . 8
⊢ ((𝜑 ∧ ¬ inf(𝐴, ℝ*, < ) = +∞)
→ inf(𝐴,
ℝ*, < ) ≠ +∞) |
61 | 43 | adantr 484 |
. . . . . . . 8
⊢ ((𝜑 ∧ ¬ inf(𝐴, ℝ*, < ) = +∞)
→ inf(𝐴,
ℝ*, < ) ∈ ℝ*) |
62 | 60, 61 | nepnfltpnf 42554 |
. . . . . . 7
⊢ ((𝜑 ∧ ¬ inf(𝐴, ℝ*, < ) = +∞)
→ inf(𝐴,
ℝ*, < ) < +∞) |
63 | 58, 62 | jca 515 |
. . . . . 6
⊢ ((𝜑 ∧ ¬ inf(𝐴, ℝ*, < ) = +∞)
→ (-∞ < inf(𝐴, ℝ*, < ) ∧
inf(𝐴, ℝ*,
< ) < +∞)) |
64 | | xrrebnd 12758 |
. . . . . . . 8
⊢
(inf(𝐴,
ℝ*, < ) ∈ ℝ* → (inf(𝐴, ℝ*, < )
∈ ℝ ↔ (-∞ < inf(𝐴, ℝ*, < ) ∧
inf(𝐴, ℝ*,
< ) < +∞))) |
65 | 43, 64 | syl 17 |
. . . . . . 7
⊢ (𝜑 → (inf(𝐴, ℝ*, < ) ∈ ℝ
↔ (-∞ < inf(𝐴, ℝ*, < ) ∧
inf(𝐴, ℝ*,
< ) < +∞))) |
66 | 65 | adantr 484 |
. . . . . 6
⊢ ((𝜑 ∧ ¬ inf(𝐴, ℝ*, < ) = +∞)
→ (inf(𝐴,
ℝ*, < ) ∈ ℝ ↔ (-∞ < inf(𝐴, ℝ*, < )
∧ inf(𝐴,
ℝ*, < ) < +∞))) |
67 | 63, 66 | mpbird 260 |
. . . . 5
⊢ ((𝜑 ∧ ¬ inf(𝐴, ℝ*, < ) = +∞)
→ inf(𝐴,
ℝ*, < ) ∈ ℝ) |
68 | | simpr 488 |
. . . . . . . 8
⊢ ((𝜑 ∧ inf(𝐴, ℝ*, < ) ∈
ℝ) → inf(𝐴,
ℝ*, < ) ∈ ℝ) |
69 | 17 | adantr 484 |
. . . . . . . 8
⊢ ((𝜑 ∧ inf(𝐴, ℝ*, < ) ∈
ℝ) → 𝐵 ∈
ℝ+) |
70 | 68, 69 | ltaddrpd 12661 |
. . . . . . 7
⊢ ((𝜑 ∧ inf(𝐴, ℝ*, < ) ∈
ℝ) → inf(𝐴,
ℝ*, < ) < (inf(𝐴, ℝ*, < ) + 𝐵)) |
71 | 19 | adantr 484 |
. . . . . . . . 9
⊢ ((𝜑 ∧ inf(𝐴, ℝ*, < ) ∈
ℝ) → 𝐵 ∈
ℝ) |
72 | | rexadd 12822 |
. . . . . . . . 9
⊢
((inf(𝐴,
ℝ*, < ) ∈ ℝ ∧ 𝐵 ∈ ℝ) → (inf(𝐴, ℝ*, < )
+𝑒 𝐵) =
(inf(𝐴,
ℝ*, < ) + 𝐵)) |
73 | 68, 71, 72 | syl2anc 587 |
. . . . . . . 8
⊢ ((𝜑 ∧ inf(𝐴, ℝ*, < ) ∈
ℝ) → (inf(𝐴,
ℝ*, < ) +𝑒 𝐵) = (inf(𝐴, ℝ*, < ) + 𝐵)) |
74 | 73 | eqcomd 2743 |
. . . . . . 7
⊢ ((𝜑 ∧ inf(𝐴, ℝ*, < ) ∈
ℝ) → (inf(𝐴,
ℝ*, < ) + 𝐵) = (inf(𝐴, ℝ*, < )
+𝑒 𝐵)) |
75 | 70, 74 | breqtrd 5079 |
. . . . . 6
⊢ ((𝜑 ∧ inf(𝐴, ℝ*, < ) ∈
ℝ) → inf(𝐴,
ℝ*, < ) < (inf(𝐴, ℝ*, < )
+𝑒 𝐵)) |
76 | 43 | adantr 484 |
. . . . . . 7
⊢ ((𝜑 ∧ inf(𝐴, ℝ*, < ) ∈
ℝ) → inf(𝐴,
ℝ*, < ) ∈ ℝ*) |
77 | 43, 18 | xaddcld 12891 |
. . . . . . . 8
⊢ (𝜑 → (inf(𝐴, ℝ*, < )
+𝑒 𝐵)
∈ ℝ*) |
78 | 77 | adantr 484 |
. . . . . . 7
⊢ ((𝜑 ∧ inf(𝐴, ℝ*, < ) ∈
ℝ) → (inf(𝐴,
ℝ*, < ) +𝑒 𝐵) ∈
ℝ*) |
79 | | xrltnle 10900 |
. . . . . . 7
⊢
((inf(𝐴,
ℝ*, < ) ∈ ℝ* ∧ (inf(𝐴, ℝ*, < )
+𝑒 𝐵)
∈ ℝ*) → (inf(𝐴, ℝ*, < ) <
(inf(𝐴,
ℝ*, < ) +𝑒 𝐵) ↔ ¬ (inf(𝐴, ℝ*, < )
+𝑒 𝐵)
≤ inf(𝐴,
ℝ*, < ))) |
80 | 76, 78, 79 | syl2anc 587 |
. . . . . 6
⊢ ((𝜑 ∧ inf(𝐴, ℝ*, < ) ∈
ℝ) → (inf(𝐴,
ℝ*, < ) < (inf(𝐴, ℝ*, < )
+𝑒 𝐵)
↔ ¬ (inf(𝐴,
ℝ*, < ) +𝑒 𝐵) ≤ inf(𝐴, ℝ*, <
))) |
81 | 75, 80 | mpbid 235 |
. . . . 5
⊢ ((𝜑 ∧ inf(𝐴, ℝ*, < ) ∈
ℝ) → ¬ (inf(𝐴, ℝ*, < )
+𝑒 𝐵)
≤ inf(𝐴,
ℝ*, < )) |
82 | 34, 67, 81 | syl2anc 587 |
. . . 4
⊢ ((𝜑 ∧ ¬ inf(𝐴, ℝ*, < ) = +∞)
→ ¬ (inf(𝐴,
ℝ*, < ) +𝑒 𝐵) ≤ inf(𝐴, ℝ*, <
)) |
83 | | simpr 488 |
. . . . . 6
⊢ ((𝜑 ∧ ¬ (inf(𝐴, ℝ*, < )
+𝑒 𝐵)
≤ inf(𝐴,
ℝ*, < )) → ¬ (inf(𝐴, ℝ*, < )
+𝑒 𝐵)
≤ inf(𝐴,
ℝ*, < )) |
84 | | simpl 486 |
. . . . . . 7
⊢ ((𝜑 ∧ ¬ (inf(𝐴, ℝ*, < )
+𝑒 𝐵)
≤ inf(𝐴,
ℝ*, < )) → 𝜑) |
85 | | infxrgelb 12925 |
. . . . . . . 8
⊢ ((𝐴 ⊆ ℝ*
∧ (inf(𝐴,
ℝ*, < ) +𝑒 𝐵) ∈ ℝ*) →
((inf(𝐴,
ℝ*, < ) +𝑒 𝐵) ≤ inf(𝐴, ℝ*, < ) ↔
∀𝑧 ∈ 𝐴 (inf(𝐴, ℝ*, < )
+𝑒 𝐵)
≤ 𝑧)) |
86 | 8, 77, 85 | syl2anc 587 |
. . . . . . 7
⊢ (𝜑 → ((inf(𝐴, ℝ*, < )
+𝑒 𝐵)
≤ inf(𝐴,
ℝ*, < ) ↔ ∀𝑧 ∈ 𝐴 (inf(𝐴, ℝ*, < )
+𝑒 𝐵)
≤ 𝑧)) |
87 | 84, 86 | syl 17 |
. . . . . 6
⊢ ((𝜑 ∧ ¬ (inf(𝐴, ℝ*, < )
+𝑒 𝐵)
≤ inf(𝐴,
ℝ*, < )) → ((inf(𝐴, ℝ*, < )
+𝑒 𝐵)
≤ inf(𝐴,
ℝ*, < ) ↔ ∀𝑧 ∈ 𝐴 (inf(𝐴, ℝ*, < )
+𝑒 𝐵)
≤ 𝑧)) |
88 | 83, 87 | mtbid 327 |
. . . . 5
⊢ ((𝜑 ∧ ¬ (inf(𝐴, ℝ*, < )
+𝑒 𝐵)
≤ inf(𝐴,
ℝ*, < )) → ¬ ∀𝑧 ∈ 𝐴 (inf(𝐴, ℝ*, < )
+𝑒 𝐵)
≤ 𝑧) |
89 | | rexnal 3160 |
. . . . 5
⊢
(∃𝑧 ∈
𝐴 ¬ (inf(𝐴, ℝ*, < )
+𝑒 𝐵)
≤ 𝑧 ↔ ¬
∀𝑧 ∈ 𝐴 (inf(𝐴, ℝ*, < )
+𝑒 𝐵)
≤ 𝑧) |
90 | 88, 89 | sylibr 237 |
. . . 4
⊢ ((𝜑 ∧ ¬ (inf(𝐴, ℝ*, < )
+𝑒 𝐵)
≤ inf(𝐴,
ℝ*, < )) → ∃𝑧 ∈ 𝐴 ¬ (inf(𝐴, ℝ*, < )
+𝑒 𝐵)
≤ 𝑧) |
91 | 34, 82, 90 | syl2anc 587 |
. . 3
⊢ ((𝜑 ∧ ¬ inf(𝐴, ℝ*, < ) = +∞)
→ ∃𝑧 ∈
𝐴 ¬ (inf(𝐴, ℝ*, < )
+𝑒 𝐵)
≤ 𝑧) |
92 | 11 | adantr 484 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑧 ∈ 𝐴) ∧ ¬ (inf(𝐴, ℝ*, < )
+𝑒 𝐵)
≤ 𝑧) → 𝑧 ∈
ℝ*) |
93 | 77 | ad2antrr 726 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑧 ∈ 𝐴) ∧ ¬ (inf(𝐴, ℝ*, < )
+𝑒 𝐵)
≤ 𝑧) → (inf(𝐴, ℝ*, < )
+𝑒 𝐵)
∈ ℝ*) |
94 | | simpr 488 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑧 ∈ 𝐴) ∧ ¬ (inf(𝐴, ℝ*, < )
+𝑒 𝐵)
≤ 𝑧) → ¬
(inf(𝐴,
ℝ*, < ) +𝑒 𝐵) ≤ 𝑧) |
95 | | xrltnle 10900 |
. . . . . . . . 9
⊢ ((𝑧 ∈ ℝ*
∧ (inf(𝐴,
ℝ*, < ) +𝑒 𝐵) ∈ ℝ*) → (𝑧 < (inf(𝐴, ℝ*, < )
+𝑒 𝐵)
↔ ¬ (inf(𝐴,
ℝ*, < ) +𝑒 𝐵) ≤ 𝑧)) |
96 | 92, 93, 95 | syl2anc 587 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑧 ∈ 𝐴) ∧ ¬ (inf(𝐴, ℝ*, < )
+𝑒 𝐵)
≤ 𝑧) → (𝑧 < (inf(𝐴, ℝ*, < )
+𝑒 𝐵)
↔ ¬ (inf(𝐴,
ℝ*, < ) +𝑒 𝐵) ≤ 𝑧)) |
97 | 94, 96 | mpbird 260 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑧 ∈ 𝐴) ∧ ¬ (inf(𝐴, ℝ*, < )
+𝑒 𝐵)
≤ 𝑧) → 𝑧 < (inf(𝐴, ℝ*, < )
+𝑒 𝐵)) |
98 | 92, 93, 97 | xrltled 12740 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑧 ∈ 𝐴) ∧ ¬ (inf(𝐴, ℝ*, < )
+𝑒 𝐵)
≤ 𝑧) → 𝑧 ≤ (inf(𝐴, ℝ*, < )
+𝑒 𝐵)) |
99 | 98 | ex 416 |
. . . . 5
⊢ ((𝜑 ∧ 𝑧 ∈ 𝐴) → (¬ (inf(𝐴, ℝ*, < )
+𝑒 𝐵)
≤ 𝑧 → 𝑧 ≤ (inf(𝐴, ℝ*, < )
+𝑒 𝐵))) |
100 | 99 | adantlr 715 |
. . . 4
⊢ (((𝜑 ∧ ¬ inf(𝐴, ℝ*, < ) = +∞)
∧ 𝑧 ∈ 𝐴) → (¬ (inf(𝐴, ℝ*, < )
+𝑒 𝐵)
≤ 𝑧 → 𝑧 ≤ (inf(𝐴, ℝ*, < )
+𝑒 𝐵))) |
101 | 100 | reximdva 3193 |
. . 3
⊢ ((𝜑 ∧ ¬ inf(𝐴, ℝ*, < ) = +∞)
→ (∃𝑧 ∈
𝐴 ¬ (inf(𝐴, ℝ*, < )
+𝑒 𝐵)
≤ 𝑧 → ∃𝑧 ∈ 𝐴 𝑧 ≤ (inf(𝐴, ℝ*, < )
+𝑒 𝐵))) |
102 | 91, 101 | mpd 15 |
. 2
⊢ ((𝜑 ∧ ¬ inf(𝐴, ℝ*, < ) = +∞)
→ ∃𝑧 ∈
𝐴 𝑧 ≤ (inf(𝐴, ℝ*, < )
+𝑒 𝐵)) |
103 | 33, 102 | pm2.61dan 813 |
1
⊢ (𝜑 → ∃𝑧 ∈ 𝐴 𝑧 ≤ (inf(𝐴, ℝ*, < )
+𝑒 𝐵)) |