Proof of Theorem infrpge
| Step | Hyp | Ref
| Expression |
| 1 | | infrpge.an0 |
. . . . . 6
⊢ (𝜑 → 𝐴 ≠ ∅) |
| 2 | | n0 4353 |
. . . . . . 7
⊢ (𝐴 ≠ ∅ ↔
∃𝑧 𝑧 ∈ 𝐴) |
| 3 | 2 | biimpi 216 |
. . . . . 6
⊢ (𝐴 ≠ ∅ →
∃𝑧 𝑧 ∈ 𝐴) |
| 4 | 1, 3 | syl 17 |
. . . . 5
⊢ (𝜑 → ∃𝑧 𝑧 ∈ 𝐴) |
| 5 | 4 | adantr 480 |
. . . 4
⊢ ((𝜑 ∧ inf(𝐴, ℝ*, < ) = +∞)
→ ∃𝑧 𝑧 ∈ 𝐴) |
| 6 | | nfv 1914 |
. . . . 5
⊢
Ⅎ𝑧(𝜑 ∧ inf(𝐴, ℝ*, < ) =
+∞) |
| 7 | | simpr 484 |
. . . . . . 7
⊢ (((𝜑 ∧ inf(𝐴, ℝ*, < ) = +∞)
∧ 𝑧 ∈ 𝐴) → 𝑧 ∈ 𝐴) |
| 8 | | infrpge.a |
. . . . . . . . . . . 12
⊢ (𝜑 → 𝐴 ⊆
ℝ*) |
| 9 | 8 | adantr 480 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑧 ∈ 𝐴) → 𝐴 ⊆
ℝ*) |
| 10 | | simpr 484 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑧 ∈ 𝐴) → 𝑧 ∈ 𝐴) |
| 11 | 9, 10 | sseldd 3984 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑧 ∈ 𝐴) → 𝑧 ∈ ℝ*) |
| 12 | | pnfge 13172 |
. . . . . . . . . 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 7438 |
. . . . . . . . . . 11
⊢
(inf(𝐴,
ℝ*, < ) = +∞ → (inf(𝐴, ℝ*, < )
+𝑒 𝐵) =
(+∞ +𝑒 𝐵)) |
| 16 | 15 | adantl 481 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ inf(𝐴, ℝ*, < ) = +∞)
→ (inf(𝐴,
ℝ*, < ) +𝑒 𝐵) = (+∞ +𝑒 𝐵)) |
| 17 | | infrpge.b |
. . . . . . . . . . . . 13
⊢ (𝜑 → 𝐵 ∈
ℝ+) |
| 18 | 17 | rpxrd 13078 |
. . . . . . . . . . . 12
⊢ (𝜑 → 𝐵 ∈
ℝ*) |
| 19 | 17 | rpred 13077 |
. . . . . . . . . . . . 13
⊢ (𝜑 → 𝐵 ∈ ℝ) |
| 20 | | renemnf 11310 |
. . . . . . . . . . . . 13
⊢ (𝐵 ∈ ℝ → 𝐵 ≠ -∞) |
| 21 | 19, 20 | syl 17 |
. . . . . . . . . . . 12
⊢ (𝜑 → 𝐵 ≠ -∞) |
| 22 | | xaddpnf2 13269 |
. . . . . . . . . . . 12
⊢ ((𝐵 ∈ ℝ*
∧ 𝐵 ≠ -∞)
→ (+∞ +𝑒 𝐵) = +∞) |
| 23 | 18, 21, 22 | syl2anc 584 |
. . . . . . . . . . 11
⊢ (𝜑 → (+∞
+𝑒 𝐵) =
+∞) |
| 24 | 23 | adantr 480 |
. . . . . . . . . 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 480 |
. . . . . . . 8
⊢ (((𝜑 ∧ inf(𝐴, ℝ*, < ) = +∞)
∧ 𝑧 ∈ 𝐴) → +∞ = (inf(𝐴, ℝ*, < )
+𝑒 𝐵)) |
| 27 | 14, 26 | breqtrd 5169 |
. . . . . . 7
⊢ (((𝜑 ∧ inf(𝐴, ℝ*, < ) = +∞)
∧ 𝑧 ∈ 𝐴) → 𝑧 ≤ (inf(𝐴, ℝ*, < )
+𝑒 𝐵)) |
| 28 | 7, 27 | jca 511 |
. . . . . 6
⊢ (((𝜑 ∧ inf(𝐴, ℝ*, < ) = +∞)
∧ 𝑧 ∈ 𝐴) → (𝑧 ∈ 𝐴 ∧ 𝑧 ≤ (inf(𝐴, ℝ*, < )
+𝑒 𝐵))) |
| 29 | 28 | ex 412 |
. . . . 5
⊢ ((𝜑 ∧ inf(𝐴, ℝ*, < ) = +∞)
→ (𝑧 ∈ 𝐴 → (𝑧 ∈ 𝐴 ∧ 𝑧 ≤ (inf(𝐴, ℝ*, < )
+𝑒 𝐵)))) |
| 30 | 6, 29 | eximd 2216 |
. . . 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 3071 |
. . 3
⊢
(∃𝑧 ∈
𝐴 𝑧 ≤ (inf(𝐴, ℝ*, < )
+𝑒 𝐵)
↔ ∃𝑧(𝑧 ∈ 𝐴 ∧ 𝑧 ≤ (inf(𝐴, ℝ*, < )
+𝑒 𝐵))) |
| 33 | 31, 32 | sylibr 234 |
. 2
⊢ ((𝜑 ∧ inf(𝐴, ℝ*, < ) = +∞)
→ ∃𝑧 ∈
𝐴 𝑧 ≤ (inf(𝐴, ℝ*, < )
+𝑒 𝐵)) |
| 34 | | simpl 482 |
. . . 4
⊢ ((𝜑 ∧ ¬ inf(𝐴, ℝ*, < ) = +∞)
→ 𝜑) |
| 35 | | infrpge.bnd |
. . . . . . . . 9
⊢ (𝜑 → ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑥 ≤ 𝑦) |
| 36 | | infrpge.xph |
. . . . . . . . . 10
⊢
Ⅎ𝑥𝜑 |
| 37 | | nfv 1914 |
. . . . . . . . . 10
⊢
Ⅎ𝑥-∞
< inf(𝐴,
ℝ*, < ) |
| 38 | | mnfxr 11318 |
. . . . . . . . . . . . 13
⊢ -∞
∈ ℝ* |
| 39 | 38 | a1i 11 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑥 ∈ ℝ ∧ ∀𝑦 ∈ 𝐴 𝑥 ≤ 𝑦) → -∞ ∈
ℝ*) |
| 40 | | rexr 11307 |
. . . . . . . . . . . . 13
⊢ (𝑥 ∈ ℝ → 𝑥 ∈
ℝ*) |
| 41 | 40 | 3ad2ant2 1135 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑥 ∈ ℝ ∧ ∀𝑦 ∈ 𝐴 𝑥 ≤ 𝑦) → 𝑥 ∈ ℝ*) |
| 42 | | infxrcl 13375 |
. . . . . . . . . . . . . 14
⊢ (𝐴 ⊆ ℝ*
→ inf(𝐴,
ℝ*, < ) ∈ ℝ*) |
| 43 | 8, 42 | syl 17 |
. . . . . . . . . . . . 13
⊢ (𝜑 → inf(𝐴, ℝ*, < ) ∈
ℝ*) |
| 44 | 43 | 3ad2ant1 1134 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑥 ∈ ℝ ∧ ∀𝑦 ∈ 𝐴 𝑥 ≤ 𝑦) → inf(𝐴, ℝ*, < ) ∈
ℝ*) |
| 45 | | mnflt 13165 |
. . . . . . . . . . . . 13
⊢ (𝑥 ∈ ℝ → -∞
< 𝑥) |
| 46 | 45 | 3ad2ant2 1135 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑥 ∈ ℝ ∧ ∀𝑦 ∈ 𝐴 𝑥 ≤ 𝑦) → -∞ < 𝑥) |
| 47 | | simp3 1139 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑥 ∈ ℝ ∧ ∀𝑦 ∈ 𝐴 𝑥 ≤ 𝑦) → ∀𝑦 ∈ 𝐴 𝑥 ≤ 𝑦) |
| 48 | 8 | adantr 480 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → 𝐴 ⊆
ℝ*) |
| 49 | 40 | adantl 481 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → 𝑥 ∈ ℝ*) |
| 50 | | infxrgelb 13377 |
. . . . . . . . . . . . . . 15
⊢ ((𝐴 ⊆ ℝ*
∧ 𝑥 ∈
ℝ*) → (𝑥 ≤ inf(𝐴, ℝ*, < ) ↔
∀𝑦 ∈ 𝐴 𝑥 ≤ 𝑦)) |
| 51 | 48, 49, 50 | syl2anc 584 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (𝑥 ≤ inf(𝐴, ℝ*, < ) ↔
∀𝑦 ∈ 𝐴 𝑥 ≤ 𝑦)) |
| 52 | 51 | 3adant3 1133 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑥 ∈ ℝ ∧ ∀𝑦 ∈ 𝐴 𝑥 ≤ 𝑦) → (𝑥 ≤ inf(𝐴, ℝ*, < ) ↔
∀𝑦 ∈ 𝐴 𝑥 ≤ 𝑦)) |
| 53 | 47, 52 | mpbird 257 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑥 ∈ ℝ ∧ ∀𝑦 ∈ 𝐴 𝑥 ≤ 𝑦) → 𝑥 ≤ inf(𝐴, ℝ*, <
)) |
| 54 | 39, 41, 44, 46, 53 | xrltletrd 13203 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑥 ∈ ℝ ∧ ∀𝑦 ∈ 𝐴 𝑥 ≤ 𝑦) → -∞ < inf(𝐴, ℝ*, <
)) |
| 55 | 54 | 3exp 1120 |
. . . . . . . . . 10
⊢ (𝜑 → (𝑥 ∈ ℝ → (∀𝑦 ∈ 𝐴 𝑥 ≤ 𝑦 → -∞ < inf(𝐴, ℝ*, <
)))) |
| 56 | 36, 37, 55 | rexlimd 3266 |
. . . . . . . . 9
⊢ (𝜑 → (∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑥 ≤ 𝑦 → -∞ < inf(𝐴, ℝ*, <
))) |
| 57 | 35, 56 | mpd 15 |
. . . . . . . 8
⊢ (𝜑 → -∞ < inf(𝐴, ℝ*, <
)) |
| 58 | 57 | adantr 480 |
. . . . . . 7
⊢ ((𝜑 ∧ ¬ inf(𝐴, ℝ*, < ) = +∞)
→ -∞ < inf(𝐴,
ℝ*, < )) |
| 59 | | neqne 2948 |
. . . . . . . . 9
⊢ (¬
inf(𝐴, ℝ*,
< ) = +∞ → inf(𝐴, ℝ*, < ) ≠
+∞) |
| 60 | 59 | adantl 481 |
. . . . . . . 8
⊢ ((𝜑 ∧ ¬ inf(𝐴, ℝ*, < ) = +∞)
→ inf(𝐴,
ℝ*, < ) ≠ +∞) |
| 61 | 43 | adantr 480 |
. . . . . . . 8
⊢ ((𝜑 ∧ ¬ inf(𝐴, ℝ*, < ) = +∞)
→ inf(𝐴,
ℝ*, < ) ∈ ℝ*) |
| 62 | 60, 61 | nepnfltpnf 45353 |
. . . . . . 7
⊢ ((𝜑 ∧ ¬ inf(𝐴, ℝ*, < ) = +∞)
→ inf(𝐴,
ℝ*, < ) < +∞) |
| 63 | 58, 62 | jca 511 |
. . . . . 6
⊢ ((𝜑 ∧ ¬ inf(𝐴, ℝ*, < ) = +∞)
→ (-∞ < inf(𝐴, ℝ*, < ) ∧
inf(𝐴, ℝ*,
< ) < +∞)) |
| 64 | | xrrebnd 13210 |
. . . . . . . 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 480 |
. . . . . 6
⊢ ((𝜑 ∧ ¬ inf(𝐴, ℝ*, < ) = +∞)
→ (inf(𝐴,
ℝ*, < ) ∈ ℝ ↔ (-∞ < inf(𝐴, ℝ*, < )
∧ inf(𝐴,
ℝ*, < ) < +∞))) |
| 67 | 63, 66 | mpbird 257 |
. . . . 5
⊢ ((𝜑 ∧ ¬ inf(𝐴, ℝ*, < ) = +∞)
→ inf(𝐴,
ℝ*, < ) ∈ ℝ) |
| 68 | | simpr 484 |
. . . . . . . 8
⊢ ((𝜑 ∧ inf(𝐴, ℝ*, < ) ∈
ℝ) → inf(𝐴,
ℝ*, < ) ∈ ℝ) |
| 69 | 17 | adantr 480 |
. . . . . . . 8
⊢ ((𝜑 ∧ inf(𝐴, ℝ*, < ) ∈
ℝ) → 𝐵 ∈
ℝ+) |
| 70 | 68, 69 | ltaddrpd 13110 |
. . . . . . 7
⊢ ((𝜑 ∧ inf(𝐴, ℝ*, < ) ∈
ℝ) → inf(𝐴,
ℝ*, < ) < (inf(𝐴, ℝ*, < ) + 𝐵)) |
| 71 | 19 | adantr 480 |
. . . . . . . . 9
⊢ ((𝜑 ∧ inf(𝐴, ℝ*, < ) ∈
ℝ) → 𝐵 ∈
ℝ) |
| 72 | | rexadd 13274 |
. . . . . . . . 9
⊢
((inf(𝐴,
ℝ*, < ) ∈ ℝ ∧ 𝐵 ∈ ℝ) → (inf(𝐴, ℝ*, < )
+𝑒 𝐵) =
(inf(𝐴,
ℝ*, < ) + 𝐵)) |
| 73 | 68, 71, 72 | syl2anc 584 |
. . . . . . . 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 5169 |
. . . . . 6
⊢ ((𝜑 ∧ inf(𝐴, ℝ*, < ) ∈
ℝ) → inf(𝐴,
ℝ*, < ) < (inf(𝐴, ℝ*, < )
+𝑒 𝐵)) |
| 76 | 43 | adantr 480 |
. . . . . . 7
⊢ ((𝜑 ∧ inf(𝐴, ℝ*, < ) ∈
ℝ) → inf(𝐴,
ℝ*, < ) ∈ ℝ*) |
| 77 | 43, 18 | xaddcld 13343 |
. . . . . . . 8
⊢ (𝜑 → (inf(𝐴, ℝ*, < )
+𝑒 𝐵)
∈ ℝ*) |
| 78 | 77 | adantr 480 |
. . . . . . 7
⊢ ((𝜑 ∧ inf(𝐴, ℝ*, < ) ∈
ℝ) → (inf(𝐴,
ℝ*, < ) +𝑒 𝐵) ∈
ℝ*) |
| 79 | | xrltnle 11328 |
. . . . . . 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 584 |
. . . . . 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 232 |
. . . . 5
⊢ ((𝜑 ∧ inf(𝐴, ℝ*, < ) ∈
ℝ) → ¬ (inf(𝐴, ℝ*, < )
+𝑒 𝐵)
≤ inf(𝐴,
ℝ*, < )) |
| 82 | 34, 67, 81 | syl2anc 584 |
. . . 4
⊢ ((𝜑 ∧ ¬ inf(𝐴, ℝ*, < ) = +∞)
→ ¬ (inf(𝐴,
ℝ*, < ) +𝑒 𝐵) ≤ inf(𝐴, ℝ*, <
)) |
| 83 | | simpr 484 |
. . . . . 6
⊢ ((𝜑 ∧ ¬ (inf(𝐴, ℝ*, < )
+𝑒 𝐵)
≤ inf(𝐴,
ℝ*, < )) → ¬ (inf(𝐴, ℝ*, < )
+𝑒 𝐵)
≤ inf(𝐴,
ℝ*, < )) |
| 84 | | simpl 482 |
. . . . . . 7
⊢ ((𝜑 ∧ ¬ (inf(𝐴, ℝ*, < )
+𝑒 𝐵)
≤ inf(𝐴,
ℝ*, < )) → 𝜑) |
| 85 | | infxrgelb 13377 |
. . . . . . . 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 584 |
. . . . . . 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 324 |
. . . . 5
⊢ ((𝜑 ∧ ¬ (inf(𝐴, ℝ*, < )
+𝑒 𝐵)
≤ inf(𝐴,
ℝ*, < )) → ¬ ∀𝑧 ∈ 𝐴 (inf(𝐴, ℝ*, < )
+𝑒 𝐵)
≤ 𝑧) |
| 89 | | rexnal 3100 |
. . . . 5
⊢
(∃𝑧 ∈
𝐴 ¬ (inf(𝐴, ℝ*, < )
+𝑒 𝐵)
≤ 𝑧 ↔ ¬
∀𝑧 ∈ 𝐴 (inf(𝐴, ℝ*, < )
+𝑒 𝐵)
≤ 𝑧) |
| 90 | 88, 89 | sylibr 234 |
. . . 4
⊢ ((𝜑 ∧ ¬ (inf(𝐴, ℝ*, < )
+𝑒 𝐵)
≤ inf(𝐴,
ℝ*, < )) → ∃𝑧 ∈ 𝐴 ¬ (inf(𝐴, ℝ*, < )
+𝑒 𝐵)
≤ 𝑧) |
| 91 | 34, 82, 90 | syl2anc 584 |
. . 3
⊢ ((𝜑 ∧ ¬ inf(𝐴, ℝ*, < ) = +∞)
→ ∃𝑧 ∈
𝐴 ¬ (inf(𝐴, ℝ*, < )
+𝑒 𝐵)
≤ 𝑧) |
| 92 | 11 | adantr 480 |
. . . . . . 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 484 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑧 ∈ 𝐴) ∧ ¬ (inf(𝐴, ℝ*, < )
+𝑒 𝐵)
≤ 𝑧) → ¬
(inf(𝐴,
ℝ*, < ) +𝑒 𝐵) ≤ 𝑧) |
| 95 | | xrltnle 11328 |
. . . . . . . . 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 584 |
. . . . . . . 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 257 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑧 ∈ 𝐴) ∧ ¬ (inf(𝐴, ℝ*, < )
+𝑒 𝐵)
≤ 𝑧) → 𝑧 < (inf(𝐴, ℝ*, < )
+𝑒 𝐵)) |
| 98 | 92, 93, 97 | xrltled 13192 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑧 ∈ 𝐴) ∧ ¬ (inf(𝐴, ℝ*, < )
+𝑒 𝐵)
≤ 𝑧) → 𝑧 ≤ (inf(𝐴, ℝ*, < )
+𝑒 𝐵)) |
| 99 | 98 | ex 412 |
. . . . 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 3168 |
. . 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)) |