Proof of Theorem infleinflem1
| Step | Hyp | Ref
| Expression |
| 1 | | infleinflem1.a |
. . . 4
⊢ (𝜑 → 𝐴 ⊆
ℝ*) |
| 2 | | infxrcl 13375 |
. . . 4
⊢ (𝐴 ⊆ ℝ*
→ inf(𝐴,
ℝ*, < ) ∈ ℝ*) |
| 3 | 1, 2 | syl 17 |
. . 3
⊢ (𝜑 → inf(𝐴, ℝ*, < ) ∈
ℝ*) |
| 4 | | id 22 |
. . 3
⊢
(inf(𝐴,
ℝ*, < ) ∈ ℝ* → inf(𝐴, ℝ*, < )
∈ ℝ*) |
| 5 | 3, 4 | syl 17 |
. 2
⊢ (𝜑 → inf(𝐴, ℝ*, < ) ∈
ℝ*) |
| 6 | | infleinflem1.z |
. . 3
⊢ (𝜑 → 𝑍 ∈ 𝐴) |
| 7 | 1, 6 | sseldd 3984 |
. 2
⊢ (𝜑 → 𝑍 ∈
ℝ*) |
| 8 | | infleinflem1.b |
. . . 4
⊢ (𝜑 → 𝐵 ⊆
ℝ*) |
| 9 | | infxrcl 13375 |
. . . 4
⊢ (𝐵 ⊆ ℝ*
→ inf(𝐵,
ℝ*, < ) ∈ ℝ*) |
| 10 | 8, 9 | syl 17 |
. . 3
⊢ (𝜑 → inf(𝐵, ℝ*, < ) ∈
ℝ*) |
| 11 | | infleinflem1.w |
. . . 4
⊢ (𝜑 → 𝑊 ∈
ℝ+) |
| 12 | | rpxr 13044 |
. . . 4
⊢ (𝑊 ∈ ℝ+
→ 𝑊 ∈
ℝ*) |
| 13 | 11, 12 | syl 17 |
. . 3
⊢ (𝜑 → 𝑊 ∈
ℝ*) |
| 14 | 10, 13 | xaddcld 13343 |
. 2
⊢ (𝜑 → (inf(𝐵, ℝ*, < )
+𝑒 𝑊)
∈ ℝ*) |
| 15 | | infxrlb 13376 |
. . 3
⊢ ((𝐴 ⊆ ℝ*
∧ 𝑍 ∈ 𝐴) → inf(𝐴, ℝ*, < ) ≤ 𝑍) |
| 16 | 1, 6, 15 | syl2anc 584 |
. 2
⊢ (𝜑 → inf(𝐴, ℝ*, < ) ≤ 𝑍) |
| 17 | | infleinflem1.x |
. . . . 5
⊢ (𝜑 → 𝑋 ∈ 𝐵) |
| 18 | 8 | sselda 3983 |
. . . . 5
⊢ ((𝜑 ∧ 𝑋 ∈ 𝐵) → 𝑋 ∈
ℝ*) |
| 19 | 17, 18 | mpdan 687 |
. . . 4
⊢ (𝜑 → 𝑋 ∈
ℝ*) |
| 20 | 11 | rpred 13077 |
. . . . . 6
⊢ (𝜑 → 𝑊 ∈ ℝ) |
| 21 | 20 | rehalfcld 12513 |
. . . . 5
⊢ (𝜑 → (𝑊 / 2) ∈ ℝ) |
| 22 | 21 | rexrd 11311 |
. . . 4
⊢ (𝜑 → (𝑊 / 2) ∈
ℝ*) |
| 23 | 19, 22 | xaddcld 13343 |
. . 3
⊢ (𝜑 → (𝑋 +𝑒 (𝑊 / 2)) ∈
ℝ*) |
| 24 | | infleinflem1.l |
. . 3
⊢ (𝜑 → 𝑍 ≤ (𝑋 +𝑒 (𝑊 / 2))) |
| 25 | | pnfge 13172 |
. . . . . . 7
⊢ ((𝑋 +𝑒 (𝑊 / 2)) ∈
ℝ* → (𝑋 +𝑒 (𝑊 / 2)) ≤ +∞) |
| 26 | 23, 25 | syl 17 |
. . . . . 6
⊢ (𝜑 → (𝑋 +𝑒 (𝑊 / 2)) ≤ +∞) |
| 27 | 26 | adantr 480 |
. . . . 5
⊢ ((𝜑 ∧ inf(𝐵, ℝ*, < ) = +∞)
→ (𝑋
+𝑒 (𝑊 /
2)) ≤ +∞) |
| 28 | | oveq1 7438 |
. . . . . . 7
⊢
(inf(𝐵,
ℝ*, < ) = +∞ → (inf(𝐵, ℝ*, < )
+𝑒 𝑊) =
(+∞ +𝑒 𝑊)) |
| 29 | 28 | adantl 481 |
. . . . . 6
⊢ ((𝜑 ∧ inf(𝐵, ℝ*, < ) = +∞)
→ (inf(𝐵,
ℝ*, < ) +𝑒 𝑊) = (+∞ +𝑒 𝑊)) |
| 30 | | rpre 13043 |
. . . . . . . . . 10
⊢ (𝑊 ∈ ℝ+
→ 𝑊 ∈
ℝ) |
| 31 | | renemnf 11310 |
. . . . . . . . . 10
⊢ (𝑊 ∈ ℝ → 𝑊 ≠ -∞) |
| 32 | 30, 31 | syl 17 |
. . . . . . . . 9
⊢ (𝑊 ∈ ℝ+
→ 𝑊 ≠
-∞) |
| 33 | | xaddpnf2 13269 |
. . . . . . . . 9
⊢ ((𝑊 ∈ ℝ*
∧ 𝑊 ≠ -∞)
→ (+∞ +𝑒 𝑊) = +∞) |
| 34 | 12, 32, 33 | syl2anc 584 |
. . . . . . . 8
⊢ (𝑊 ∈ ℝ+
→ (+∞ +𝑒 𝑊) = +∞) |
| 35 | 11, 34 | syl 17 |
. . . . . . 7
⊢ (𝜑 → (+∞
+𝑒 𝑊) =
+∞) |
| 36 | 35 | adantr 480 |
. . . . . 6
⊢ ((𝜑 ∧ inf(𝐵, ℝ*, < ) = +∞)
→ (+∞ +𝑒 𝑊) = +∞) |
| 37 | 29, 36 | eqtr2d 2778 |
. . . . 5
⊢ ((𝜑 ∧ inf(𝐵, ℝ*, < ) = +∞)
→ +∞ = (inf(𝐵,
ℝ*, < ) +𝑒 𝑊)) |
| 38 | 27, 37 | breqtrd 5169 |
. . . 4
⊢ ((𝜑 ∧ inf(𝐵, ℝ*, < ) = +∞)
→ (𝑋
+𝑒 (𝑊 /
2)) ≤ (inf(𝐵,
ℝ*, < ) +𝑒 𝑊)) |
| 39 | 8, 17 | sseldd 3984 |
. . . . . . 7
⊢ (𝜑 → 𝑋 ∈
ℝ*) |
| 40 | 10, 22 | xaddcld 13343 |
. . . . . . 7
⊢ (𝜑 → (inf(𝐵, ℝ*, < )
+𝑒 (𝑊 /
2)) ∈ ℝ*) |
| 41 | | rphalfcl 13062 |
. . . . . . . . 9
⊢ (𝑊 ∈ ℝ+
→ (𝑊 / 2) ∈
ℝ+) |
| 42 | 11, 41 | syl 17 |
. . . . . . . 8
⊢ (𝜑 → (𝑊 / 2) ∈
ℝ+) |
| 43 | 42 | rpxrd 13078 |
. . . . . . 7
⊢ (𝜑 → (𝑊 / 2) ∈
ℝ*) |
| 44 | | infleinflem1.i |
. . . . . . 7
⊢ (𝜑 → 𝑋 ≤ (inf(𝐵, ℝ*, < )
+𝑒 (𝑊 /
2))) |
| 45 | 39, 40, 43, 44 | xleadd1d 45340 |
. . . . . 6
⊢ (𝜑 → (𝑋 +𝑒 (𝑊 / 2)) ≤ ((inf(𝐵, ℝ*, < )
+𝑒 (𝑊 /
2)) +𝑒 (𝑊 / 2))) |
| 46 | 45 | adantr 480 |
. . . . 5
⊢ ((𝜑 ∧ ¬ inf(𝐵, ℝ*, < ) = +∞)
→ (𝑋
+𝑒 (𝑊 /
2)) ≤ ((inf(𝐵,
ℝ*, < ) +𝑒 (𝑊 / 2)) +𝑒 (𝑊 / 2))) |
| 47 | 10 | adantr 480 |
. . . . . . 7
⊢ ((𝜑 ∧ ¬ inf(𝐵, ℝ*, < ) = +∞)
→ inf(𝐵,
ℝ*, < ) ∈ ℝ*) |
| 48 | | neqne 2948 |
. . . . . . . 8
⊢ (¬
inf(𝐵, ℝ*,
< ) = +∞ → inf(𝐵, ℝ*, < ) ≠
+∞) |
| 49 | 48 | adantl 481 |
. . . . . . 7
⊢ ((𝜑 ∧ ¬ inf(𝐵, ℝ*, < ) = +∞)
→ inf(𝐵,
ℝ*, < ) ≠ +∞) |
| 50 | 43 | adantr 480 |
. . . . . . 7
⊢ ((𝜑 ∧ ¬ inf(𝐵, ℝ*, < ) = +∞)
→ (𝑊 / 2) ∈
ℝ*) |
| 51 | 11 | adantr 480 |
. . . . . . . 8
⊢ ((𝜑 ∧ ¬ inf(𝐵, ℝ*, < ) = +∞)
→ 𝑊 ∈
ℝ+) |
| 52 | | rpre 13043 |
. . . . . . . 8
⊢ ((𝑊 / 2) ∈ ℝ+
→ (𝑊 / 2) ∈
ℝ) |
| 53 | | renepnf 11309 |
. . . . . . . 8
⊢ ((𝑊 / 2) ∈ ℝ →
(𝑊 / 2) ≠
+∞) |
| 54 | 51, 41, 52, 53 | 4syl 19 |
. . . . . . 7
⊢ ((𝜑 ∧ ¬ inf(𝐵, ℝ*, < ) = +∞)
→ (𝑊 / 2) ≠
+∞) |
| 55 | | xaddass2 13292 |
. . . . . . 7
⊢
(((inf(𝐵,
ℝ*, < ) ∈ ℝ* ∧ inf(𝐵, ℝ*, < )
≠ +∞) ∧ ((𝑊 /
2) ∈ ℝ* ∧ (𝑊 / 2) ≠ +∞) ∧ ((𝑊 / 2) ∈ ℝ*
∧ (𝑊 / 2) ≠
+∞)) → ((inf(𝐵,
ℝ*, < ) +𝑒 (𝑊 / 2)) +𝑒 (𝑊 / 2)) = (inf(𝐵, ℝ*, < )
+𝑒 ((𝑊 /
2) +𝑒 (𝑊
/ 2)))) |
| 56 | 47, 49, 50, 54, 50, 54, 55 | syl222anc 1388 |
. . . . . 6
⊢ ((𝜑 ∧ ¬ inf(𝐵, ℝ*, < ) = +∞)
→ ((inf(𝐵,
ℝ*, < ) +𝑒 (𝑊 / 2)) +𝑒 (𝑊 / 2)) = (inf(𝐵, ℝ*, < )
+𝑒 ((𝑊 /
2) +𝑒 (𝑊
/ 2)))) |
| 57 | | rehalfcl 12492 |
. . . . . . . . . 10
⊢ (𝑊 ∈ ℝ → (𝑊 / 2) ∈
ℝ) |
| 58 | 57, 57 | rexaddd 13276 |
. . . . . . . . 9
⊢ (𝑊 ∈ ℝ → ((𝑊 / 2) +𝑒
(𝑊 / 2)) = ((𝑊 / 2) + (𝑊 / 2))) |
| 59 | | recn 11245 |
. . . . . . . . . 10
⊢ (𝑊 ∈ ℝ → 𝑊 ∈
ℂ) |
| 60 | | 2halves 12494 |
. . . . . . . . . 10
⊢ (𝑊 ∈ ℂ → ((𝑊 / 2) + (𝑊 / 2)) = 𝑊) |
| 61 | 59, 60 | syl 17 |
. . . . . . . . 9
⊢ (𝑊 ∈ ℝ → ((𝑊 / 2) + (𝑊 / 2)) = 𝑊) |
| 62 | 58, 61 | eqtrd 2777 |
. . . . . . . 8
⊢ (𝑊 ∈ ℝ → ((𝑊 / 2) +𝑒
(𝑊 / 2)) = 𝑊) |
| 63 | 62 | oveq2d 7447 |
. . . . . . 7
⊢ (𝑊 ∈ ℝ →
(inf(𝐵,
ℝ*, < ) +𝑒 ((𝑊 / 2) +𝑒 (𝑊 / 2))) = (inf(𝐵, ℝ*, < )
+𝑒 𝑊)) |
| 64 | 51, 30, 63 | 3syl 18 |
. . . . . 6
⊢ ((𝜑 ∧ ¬ inf(𝐵, ℝ*, < ) = +∞)
→ (inf(𝐵,
ℝ*, < ) +𝑒 ((𝑊 / 2) +𝑒 (𝑊 / 2))) = (inf(𝐵, ℝ*, < )
+𝑒 𝑊)) |
| 65 | 56, 64 | eqtrd 2777 |
. . . . 5
⊢ ((𝜑 ∧ ¬ inf(𝐵, ℝ*, < ) = +∞)
→ ((inf(𝐵,
ℝ*, < ) +𝑒 (𝑊 / 2)) +𝑒 (𝑊 / 2)) = (inf(𝐵, ℝ*, < )
+𝑒 𝑊)) |
| 66 | 46, 65 | breqtrd 5169 |
. . . 4
⊢ ((𝜑 ∧ ¬ inf(𝐵, ℝ*, < ) = +∞)
→ (𝑋
+𝑒 (𝑊 /
2)) ≤ (inf(𝐵,
ℝ*, < ) +𝑒 𝑊)) |
| 67 | 38, 66 | pm2.61dan 813 |
. . 3
⊢ (𝜑 → (𝑋 +𝑒 (𝑊 / 2)) ≤ (inf(𝐵, ℝ*, < )
+𝑒 𝑊)) |
| 68 | 7, 23, 14, 24, 67 | xrletrd 13204 |
. 2
⊢ (𝜑 → 𝑍 ≤ (inf(𝐵, ℝ*, < )
+𝑒 𝑊)) |
| 69 | 5, 7, 14, 16, 68 | xrletrd 13204 |
1
⊢ (𝜑 → inf(𝐴, ℝ*, < ) ≤
(inf(𝐵,
ℝ*, < ) +𝑒 𝑊)) |