Proof of Theorem xrge0iifhom
| Step | Hyp | Ref
| Expression |
| 1 | | 0xr 11308 |
. . . . . 6
⊢ 0 ∈
ℝ* |
| 2 | | 1xr 11320 |
. . . . . 6
⊢ 1 ∈
ℝ* |
| 3 | | 0le1 11786 |
. . . . . 6
⊢ 0 ≤
1 |
| 4 | | snunioc 13520 |
. . . . . 6
⊢ ((0
∈ ℝ* ∧ 1 ∈ ℝ* ∧ 0 ≤ 1)
→ ({0} ∪ (0(,]1)) = (0[,]1)) |
| 5 | 1, 2, 3, 4 | mp3an 1463 |
. . . . 5
⊢ ({0}
∪ (0(,]1)) = (0[,]1) |
| 6 | 5 | eleq2i 2833 |
. . . 4
⊢ (𝑌 ∈ ({0} ∪ (0(,]1))
↔ 𝑌 ∈
(0[,]1)) |
| 7 | | elun 4153 |
. . . 4
⊢ (𝑌 ∈ ({0} ∪ (0(,]1))
↔ (𝑌 ∈ {0} ∨
𝑌 ∈
(0(,]1))) |
| 8 | 6, 7 | bitr3i 277 |
. . 3
⊢ (𝑌 ∈ (0[,]1) ↔ (𝑌 ∈ {0} ∨ 𝑌 ∈
(0(,]1))) |
| 9 | | elsni 4643 |
. . . 4
⊢ (𝑌 ∈ {0} → 𝑌 = 0) |
| 10 | 9 | orim1i 910 |
. . 3
⊢ ((𝑌 ∈ {0} ∨ 𝑌 ∈ (0(,]1)) → (𝑌 = 0 ∨ 𝑌 ∈ (0(,]1))) |
| 11 | 8, 10 | sylbi 217 |
. 2
⊢ (𝑌 ∈ (0[,]1) → (𝑌 = 0 ∨ 𝑌 ∈ (0(,]1))) |
| 12 | | 0elunit 13509 |
. . . . . . . 8
⊢ 0 ∈
(0[,]1) |
| 13 | | iftrue 4531 |
. . . . . . . . 9
⊢ (𝑥 = 0 → if(𝑥 = 0, +∞,
-(log‘𝑥)) =
+∞) |
| 14 | | xrge0iifhmeo.1 |
. . . . . . . . 9
⊢ 𝐹 = (𝑥 ∈ (0[,]1) ↦ if(𝑥 = 0, +∞, -(log‘𝑥))) |
| 15 | | pnfex 11314 |
. . . . . . . . 9
⊢ +∞
∈ V |
| 16 | 13, 14, 15 | fvmpt 7016 |
. . . . . . . 8
⊢ (0 ∈
(0[,]1) → (𝐹‘0)
= +∞) |
| 17 | 12, 16 | ax-mp 5 |
. . . . . . 7
⊢ (𝐹‘0) =
+∞ |
| 18 | 17 | oveq2i 7442 |
. . . . . 6
⊢ ((𝐹‘𝑋) +𝑒 (𝐹‘0)) = ((𝐹‘𝑋) +𝑒
+∞) |
| 19 | | eqeq1 2741 |
. . . . . . . . . . 11
⊢ (𝑥 = 𝑋 → (𝑥 = 0 ↔ 𝑋 = 0)) |
| 20 | | fveq2 6906 |
. . . . . . . . . . . 12
⊢ (𝑥 = 𝑋 → (log‘𝑥) = (log‘𝑋)) |
| 21 | 20 | negeqd 11502 |
. . . . . . . . . . 11
⊢ (𝑥 = 𝑋 → -(log‘𝑥) = -(log‘𝑋)) |
| 22 | 19, 21 | ifbieq2d 4552 |
. . . . . . . . . 10
⊢ (𝑥 = 𝑋 → if(𝑥 = 0, +∞, -(log‘𝑥)) = if(𝑋 = 0, +∞, -(log‘𝑋))) |
| 23 | | negex 11506 |
. . . . . . . . . . 11
⊢
-(log‘𝑋)
∈ V |
| 24 | 15, 23 | ifex 4576 |
. . . . . . . . . 10
⊢ if(𝑋 = 0, +∞,
-(log‘𝑋)) ∈
V |
| 25 | 22, 14, 24 | fvmpt 7016 |
. . . . . . . . 9
⊢ (𝑋 ∈ (0[,]1) → (𝐹‘𝑋) = if(𝑋 = 0, +∞, -(log‘𝑋))) |
| 26 | | pnfxr 11315 |
. . . . . . . . . . 11
⊢ +∞
∈ ℝ* |
| 27 | 26 | a1i 11 |
. . . . . . . . . 10
⊢ ((𝑋 ∈ (0[,]1) ∧ 𝑋 = 0) → +∞ ∈
ℝ*) |
| 28 | | elunitrn 13507 |
. . . . . . . . . . . . . . 15
⊢ (𝑋 ∈ (0[,]1) → 𝑋 ∈
ℝ) |
| 29 | 28 | adantr 480 |
. . . . . . . . . . . . . 14
⊢ ((𝑋 ∈ (0[,]1) ∧ ¬
𝑋 = 0) → 𝑋 ∈
ℝ) |
| 30 | | elunitge0 33898 |
. . . . . . . . . . . . . . . 16
⊢ (𝑋 ∈ (0[,]1) → 0 ≤
𝑋) |
| 31 | 30 | adantr 480 |
. . . . . . . . . . . . . . 15
⊢ ((𝑋 ∈ (0[,]1) ∧ ¬
𝑋 = 0) → 0 ≤ 𝑋) |
| 32 | | simpr 484 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑋 ∈ (0[,]1) ∧ ¬
𝑋 = 0) → ¬ 𝑋 = 0) |
| 33 | 32 | neqned 2947 |
. . . . . . . . . . . . . . 15
⊢ ((𝑋 ∈ (0[,]1) ∧ ¬
𝑋 = 0) → 𝑋 ≠ 0) |
| 34 | 29, 31, 33 | ne0gt0d 11398 |
. . . . . . . . . . . . . 14
⊢ ((𝑋 ∈ (0[,]1) ∧ ¬
𝑋 = 0) → 0 < 𝑋) |
| 35 | 29, 34 | elrpd 13074 |
. . . . . . . . . . . . 13
⊢ ((𝑋 ∈ (0[,]1) ∧ ¬
𝑋 = 0) → 𝑋 ∈
ℝ+) |
| 36 | 35 | relogcld 26665 |
. . . . . . . . . . . 12
⊢ ((𝑋 ∈ (0[,]1) ∧ ¬
𝑋 = 0) →
(log‘𝑋) ∈
ℝ) |
| 37 | 36 | renegcld 11690 |
. . . . . . . . . . 11
⊢ ((𝑋 ∈ (0[,]1) ∧ ¬
𝑋 = 0) →
-(log‘𝑋) ∈
ℝ) |
| 38 | 37 | rexrd 11311 |
. . . . . . . . . 10
⊢ ((𝑋 ∈ (0[,]1) ∧ ¬
𝑋 = 0) →
-(log‘𝑋) ∈
ℝ*) |
| 39 | 27, 38 | ifclda 4561 |
. . . . . . . . 9
⊢ (𝑋 ∈ (0[,]1) → if(𝑋 = 0, +∞,
-(log‘𝑋)) ∈
ℝ*) |
| 40 | 25, 39 | eqeltrd 2841 |
. . . . . . . 8
⊢ (𝑋 ∈ (0[,]1) → (𝐹‘𝑋) ∈
ℝ*) |
| 41 | 40 | adantr 480 |
. . . . . . 7
⊢ ((𝑋 ∈ (0[,]1) ∧ 𝑌 = 0) → (𝐹‘𝑋) ∈
ℝ*) |
| 42 | | neeq1 3003 |
. . . . . . . . . 10
⊢ (+∞
= if(𝑋 = 0, +∞,
-(log‘𝑋)) →
(+∞ ≠ -∞ ↔ if(𝑋 = 0, +∞, -(log‘𝑋)) ≠
-∞)) |
| 43 | | neeq1 3003 |
. . . . . . . . . 10
⊢
(-(log‘𝑋) =
if(𝑋 = 0, +∞,
-(log‘𝑋)) →
(-(log‘𝑋) ≠
-∞ ↔ if(𝑋 = 0,
+∞, -(log‘𝑋))
≠ -∞)) |
| 44 | | pnfnemnf 11316 |
. . . . . . . . . . 11
⊢ +∞
≠ -∞ |
| 45 | 44 | a1i 11 |
. . . . . . . . . 10
⊢ ((𝑋 ∈ (0[,]1) ∧ 𝑋 = 0) → +∞ ≠
-∞) |
| 46 | 37 | renemnfd 11313 |
. . . . . . . . . 10
⊢ ((𝑋 ∈ (0[,]1) ∧ ¬
𝑋 = 0) →
-(log‘𝑋) ≠
-∞) |
| 47 | 42, 43, 45, 46 | ifbothda 4564 |
. . . . . . . . 9
⊢ (𝑋 ∈ (0[,]1) → if(𝑋 = 0, +∞,
-(log‘𝑋)) ≠
-∞) |
| 48 | 25, 47 | eqnetrd 3008 |
. . . . . . . 8
⊢ (𝑋 ∈ (0[,]1) → (𝐹‘𝑋) ≠ -∞) |
| 49 | 48 | adantr 480 |
. . . . . . 7
⊢ ((𝑋 ∈ (0[,]1) ∧ 𝑌 = 0) → (𝐹‘𝑋) ≠ -∞) |
| 50 | | xaddpnf1 13268 |
. . . . . . 7
⊢ (((𝐹‘𝑋) ∈ ℝ* ∧ (𝐹‘𝑋) ≠ -∞) → ((𝐹‘𝑋) +𝑒 +∞) =
+∞) |
| 51 | 41, 49, 50 | syl2anc 584 |
. . . . . 6
⊢ ((𝑋 ∈ (0[,]1) ∧ 𝑌 = 0) → ((𝐹‘𝑋) +𝑒 +∞) =
+∞) |
| 52 | 18, 51 | eqtrid 2789 |
. . . . 5
⊢ ((𝑋 ∈ (0[,]1) ∧ 𝑌 = 0) → ((𝐹‘𝑋) +𝑒 (𝐹‘0)) = +∞) |
| 53 | | unitsscn 13540 |
. . . . . . . . 9
⊢ (0[,]1)
⊆ ℂ |
| 54 | | simpl 482 |
. . . . . . . . 9
⊢ ((𝑋 ∈ (0[,]1) ∧ 𝑌 = 0) → 𝑋 ∈ (0[,]1)) |
| 55 | 53, 54 | sselid 3981 |
. . . . . . . 8
⊢ ((𝑋 ∈ (0[,]1) ∧ 𝑌 = 0) → 𝑋 ∈ ℂ) |
| 56 | 55 | mul01d 11460 |
. . . . . . 7
⊢ ((𝑋 ∈ (0[,]1) ∧ 𝑌 = 0) → (𝑋 · 0) = 0) |
| 57 | 56 | fveq2d 6910 |
. . . . . 6
⊢ ((𝑋 ∈ (0[,]1) ∧ 𝑌 = 0) → (𝐹‘(𝑋 · 0)) = (𝐹‘0)) |
| 58 | 57, 17 | eqtrdi 2793 |
. . . . 5
⊢ ((𝑋 ∈ (0[,]1) ∧ 𝑌 = 0) → (𝐹‘(𝑋 · 0)) = +∞) |
| 59 | 52, 58 | eqtr4d 2780 |
. . . 4
⊢ ((𝑋 ∈ (0[,]1) ∧ 𝑌 = 0) → ((𝐹‘𝑋) +𝑒 (𝐹‘0)) = (𝐹‘(𝑋 · 0))) |
| 60 | | simpr 484 |
. . . . . 6
⊢ ((𝑋 ∈ (0[,]1) ∧ 𝑌 = 0) → 𝑌 = 0) |
| 61 | 60 | fveq2d 6910 |
. . . . 5
⊢ ((𝑋 ∈ (0[,]1) ∧ 𝑌 = 0) → (𝐹‘𝑌) = (𝐹‘0)) |
| 62 | 61 | oveq2d 7447 |
. . . 4
⊢ ((𝑋 ∈ (0[,]1) ∧ 𝑌 = 0) → ((𝐹‘𝑋) +𝑒 (𝐹‘𝑌)) = ((𝐹‘𝑋) +𝑒 (𝐹‘0))) |
| 63 | 60 | oveq2d 7447 |
. . . . 5
⊢ ((𝑋 ∈ (0[,]1) ∧ 𝑌 = 0) → (𝑋 · 𝑌) = (𝑋 · 0)) |
| 64 | 63 | fveq2d 6910 |
. . . 4
⊢ ((𝑋 ∈ (0[,]1) ∧ 𝑌 = 0) → (𝐹‘(𝑋 · 𝑌)) = (𝐹‘(𝑋 · 0))) |
| 65 | 59, 62, 64 | 3eqtr4rd 2788 |
. . 3
⊢ ((𝑋 ∈ (0[,]1) ∧ 𝑌 = 0) → (𝐹‘(𝑋 · 𝑌)) = ((𝐹‘𝑋) +𝑒 (𝐹‘𝑌))) |
| 66 | 5 | eleq2i 2833 |
. . . . . 6
⊢ (𝑋 ∈ ({0} ∪ (0(,]1))
↔ 𝑋 ∈
(0[,]1)) |
| 67 | | elun 4153 |
. . . . . 6
⊢ (𝑋 ∈ ({0} ∪ (0(,]1))
↔ (𝑋 ∈ {0} ∨
𝑋 ∈
(0(,]1))) |
| 68 | 66, 67 | bitr3i 277 |
. . . . 5
⊢ (𝑋 ∈ (0[,]1) ↔ (𝑋 ∈ {0} ∨ 𝑋 ∈
(0(,]1))) |
| 69 | | elsni 4643 |
. . . . . 6
⊢ (𝑋 ∈ {0} → 𝑋 = 0) |
| 70 | 69 | orim1i 910 |
. . . . 5
⊢ ((𝑋 ∈ {0} ∨ 𝑋 ∈ (0(,]1)) → (𝑋 = 0 ∨ 𝑋 ∈ (0(,]1))) |
| 71 | 68, 70 | sylbi 217 |
. . . 4
⊢ (𝑋 ∈ (0[,]1) → (𝑋 = 0 ∨ 𝑋 ∈ (0(,]1))) |
| 72 | 17 | oveq1i 7441 |
. . . . . . . 8
⊢ ((𝐹‘0) +𝑒
(𝐹‘𝑌)) = (+∞ +𝑒 (𝐹‘𝑌)) |
| 73 | | simpr 484 |
. . . . . . . . . 10
⊢ ((𝑋 = 0 ∧ 𝑌 ∈ (0(,]1)) → 𝑌 ∈ (0(,]1)) |
| 74 | 14 | xrge0iifcv 33933 |
. . . . . . . . . . . 12
⊢ (𝑌 ∈ (0(,]1) → (𝐹‘𝑌) = -(log‘𝑌)) |
| 75 | | 0le0 12367 |
. . . . . . . . . . . . . . . . 17
⊢ 0 ≤
0 |
| 76 | | 1re 11261 |
. . . . . . . . . . . . . . . . . 18
⊢ 1 ∈
ℝ |
| 77 | | ltpnf 13162 |
. . . . . . . . . . . . . . . . . 18
⊢ (1 ∈
ℝ → 1 < +∞) |
| 78 | 76, 77 | ax-mp 5 |
. . . . . . . . . . . . . . . . 17
⊢ 1 <
+∞ |
| 79 | | iocssioo 13479 |
. . . . . . . . . . . . . . . . 17
⊢ (((0
∈ ℝ* ∧ +∞ ∈ ℝ*) ∧ (0
≤ 0 ∧ 1 < +∞)) → (0(,]1) ⊆
(0(,)+∞)) |
| 80 | 1, 26, 75, 78, 79 | mp4an 693 |
. . . . . . . . . . . . . . . 16
⊢ (0(,]1)
⊆ (0(,)+∞) |
| 81 | | ioorp 13465 |
. . . . . . . . . . . . . . . 16
⊢
(0(,)+∞) = ℝ+ |
| 82 | 80, 81 | sseqtri 4032 |
. . . . . . . . . . . . . . 15
⊢ (0(,]1)
⊆ ℝ+ |
| 83 | 82 | sseli 3979 |
. . . . . . . . . . . . . 14
⊢ (𝑌 ∈ (0(,]1) → 𝑌 ∈
ℝ+) |
| 84 | 83 | relogcld 26665 |
. . . . . . . . . . . . 13
⊢ (𝑌 ∈ (0(,]1) →
(log‘𝑌) ∈
ℝ) |
| 85 | 84 | renegcld 11690 |
. . . . . . . . . . . 12
⊢ (𝑌 ∈ (0(,]1) →
-(log‘𝑌) ∈
ℝ) |
| 86 | 74, 85 | eqeltrd 2841 |
. . . . . . . . . . 11
⊢ (𝑌 ∈ (0(,]1) → (𝐹‘𝑌) ∈ ℝ) |
| 87 | 86 | rexrd 11311 |
. . . . . . . . . 10
⊢ (𝑌 ∈ (0(,]1) → (𝐹‘𝑌) ∈
ℝ*) |
| 88 | 73, 87 | syl 17 |
. . . . . . . . 9
⊢ ((𝑋 = 0 ∧ 𝑌 ∈ (0(,]1)) → (𝐹‘𝑌) ∈
ℝ*) |
| 89 | 86 | renemnfd 11313 |
. . . . . . . . . 10
⊢ (𝑌 ∈ (0(,]1) → (𝐹‘𝑌) ≠ -∞) |
| 90 | 73, 89 | syl 17 |
. . . . . . . . 9
⊢ ((𝑋 = 0 ∧ 𝑌 ∈ (0(,]1)) → (𝐹‘𝑌) ≠ -∞) |
| 91 | | xaddpnf2 13269 |
. . . . . . . . 9
⊢ (((𝐹‘𝑌) ∈ ℝ* ∧ (𝐹‘𝑌) ≠ -∞) → (+∞
+𝑒 (𝐹‘𝑌)) = +∞) |
| 92 | 88, 90, 91 | syl2anc 584 |
. . . . . . . 8
⊢ ((𝑋 = 0 ∧ 𝑌 ∈ (0(,]1)) → (+∞
+𝑒 (𝐹‘𝑌)) = +∞) |
| 93 | 72, 92 | eqtrid 2789 |
. . . . . . 7
⊢ ((𝑋 = 0 ∧ 𝑌 ∈ (0(,]1)) → ((𝐹‘0) +𝑒 (𝐹‘𝑌)) = +∞) |
| 94 | | rpssre 13042 |
. . . . . . . . . . . . 13
⊢
ℝ+ ⊆ ℝ |
| 95 | 82, 94 | sstri 3993 |
. . . . . . . . . . . 12
⊢ (0(,]1)
⊆ ℝ |
| 96 | | ax-resscn 11212 |
. . . . . . . . . . . 12
⊢ ℝ
⊆ ℂ |
| 97 | 95, 96 | sstri 3993 |
. . . . . . . . . . 11
⊢ (0(,]1)
⊆ ℂ |
| 98 | 97, 73 | sselid 3981 |
. . . . . . . . . 10
⊢ ((𝑋 = 0 ∧ 𝑌 ∈ (0(,]1)) → 𝑌 ∈ ℂ) |
| 99 | 98 | mul02d 11459 |
. . . . . . . . 9
⊢ ((𝑋 = 0 ∧ 𝑌 ∈ (0(,]1)) → (0 · 𝑌) = 0) |
| 100 | 99 | fveq2d 6910 |
. . . . . . . 8
⊢ ((𝑋 = 0 ∧ 𝑌 ∈ (0(,]1)) → (𝐹‘(0 · 𝑌)) = (𝐹‘0)) |
| 101 | 100, 17 | eqtrdi 2793 |
. . . . . . 7
⊢ ((𝑋 = 0 ∧ 𝑌 ∈ (0(,]1)) → (𝐹‘(0 · 𝑌)) = +∞) |
| 102 | 93, 101 | eqtr4d 2780 |
. . . . . 6
⊢ ((𝑋 = 0 ∧ 𝑌 ∈ (0(,]1)) → ((𝐹‘0) +𝑒 (𝐹‘𝑌)) = (𝐹‘(0 · 𝑌))) |
| 103 | | simpl 482 |
. . . . . . . 8
⊢ ((𝑋 = 0 ∧ 𝑌 ∈ (0(,]1)) → 𝑋 = 0) |
| 104 | 103 | fveq2d 6910 |
. . . . . . 7
⊢ ((𝑋 = 0 ∧ 𝑌 ∈ (0(,]1)) → (𝐹‘𝑋) = (𝐹‘0)) |
| 105 | 104 | oveq1d 7446 |
. . . . . 6
⊢ ((𝑋 = 0 ∧ 𝑌 ∈ (0(,]1)) → ((𝐹‘𝑋) +𝑒 (𝐹‘𝑌)) = ((𝐹‘0) +𝑒 (𝐹‘𝑌))) |
| 106 | 103 | fvoveq1d 7453 |
. . . . . 6
⊢ ((𝑋 = 0 ∧ 𝑌 ∈ (0(,]1)) → (𝐹‘(𝑋 · 𝑌)) = (𝐹‘(0 · 𝑌))) |
| 107 | 102, 105,
106 | 3eqtr4rd 2788 |
. . . . 5
⊢ ((𝑋 = 0 ∧ 𝑌 ∈ (0(,]1)) → (𝐹‘(𝑋 · 𝑌)) = ((𝐹‘𝑋) +𝑒 (𝐹‘𝑌))) |
| 108 | | simpl 482 |
. . . . . . . . . 10
⊢ ((𝑋 ∈ (0(,]1) ∧ 𝑌 ∈ (0(,]1)) → 𝑋 ∈
(0(,]1)) |
| 109 | 82, 108 | sselid 3981 |
. . . . . . . . 9
⊢ ((𝑋 ∈ (0(,]1) ∧ 𝑌 ∈ (0(,]1)) → 𝑋 ∈
ℝ+) |
| 110 | 109 | relogcld 26665 |
. . . . . . . 8
⊢ ((𝑋 ∈ (0(,]1) ∧ 𝑌 ∈ (0(,]1)) →
(log‘𝑋) ∈
ℝ) |
| 111 | 110 | renegcld 11690 |
. . . . . . 7
⊢ ((𝑋 ∈ (0(,]1) ∧ 𝑌 ∈ (0(,]1)) →
-(log‘𝑋) ∈
ℝ) |
| 112 | | simpr 484 |
. . . . . . . . . 10
⊢ ((𝑋 ∈ (0(,]1) ∧ 𝑌 ∈ (0(,]1)) → 𝑌 ∈
(0(,]1)) |
| 113 | 82, 112 | sselid 3981 |
. . . . . . . . 9
⊢ ((𝑋 ∈ (0(,]1) ∧ 𝑌 ∈ (0(,]1)) → 𝑌 ∈
ℝ+) |
| 114 | 113 | relogcld 26665 |
. . . . . . . 8
⊢ ((𝑋 ∈ (0(,]1) ∧ 𝑌 ∈ (0(,]1)) →
(log‘𝑌) ∈
ℝ) |
| 115 | 114 | renegcld 11690 |
. . . . . . 7
⊢ ((𝑋 ∈ (0(,]1) ∧ 𝑌 ∈ (0(,]1)) →
-(log‘𝑌) ∈
ℝ) |
| 116 | | rexadd 13274 |
. . . . . . 7
⊢
((-(log‘𝑋)
∈ ℝ ∧ -(log‘𝑌) ∈ ℝ) → (-(log‘𝑋) +𝑒
-(log‘𝑌)) =
(-(log‘𝑋) +
-(log‘𝑌))) |
| 117 | 111, 115,
116 | syl2anc 584 |
. . . . . 6
⊢ ((𝑋 ∈ (0(,]1) ∧ 𝑌 ∈ (0(,]1)) →
(-(log‘𝑋)
+𝑒 -(log‘𝑌)) = (-(log‘𝑋) + -(log‘𝑌))) |
| 118 | 14 | xrge0iifcv 33933 |
. . . . . . 7
⊢ (𝑋 ∈ (0(,]1) → (𝐹‘𝑋) = -(log‘𝑋)) |
| 119 | 118, 74 | oveqan12d 7450 |
. . . . . 6
⊢ ((𝑋 ∈ (0(,]1) ∧ 𝑌 ∈ (0(,]1)) → ((𝐹‘𝑋) +𝑒 (𝐹‘𝑌)) = (-(log‘𝑋) +𝑒 -(log‘𝑌))) |
| 120 | 109 | rpred 13077 |
. . . . . . . . . 10
⊢ ((𝑋 ∈ (0(,]1) ∧ 𝑌 ∈ (0(,]1)) → 𝑋 ∈
ℝ) |
| 121 | 113 | rpred 13077 |
. . . . . . . . . 10
⊢ ((𝑋 ∈ (0(,]1) ∧ 𝑌 ∈ (0(,]1)) → 𝑌 ∈
ℝ) |
| 122 | 120, 121 | remulcld 11291 |
. . . . . . . . 9
⊢ ((𝑋 ∈ (0(,]1) ∧ 𝑌 ∈ (0(,]1)) → (𝑋 · 𝑌) ∈ ℝ) |
| 123 | 109 | rpgt0d 13080 |
. . . . . . . . . 10
⊢ ((𝑋 ∈ (0(,]1) ∧ 𝑌 ∈ (0(,]1)) → 0 <
𝑋) |
| 124 | 113 | rpgt0d 13080 |
. . . . . . . . . 10
⊢ ((𝑋 ∈ (0(,]1) ∧ 𝑌 ∈ (0(,]1)) → 0 <
𝑌) |
| 125 | 120, 121,
123, 124 | mulgt0d 11416 |
. . . . . . . . 9
⊢ ((𝑋 ∈ (0(,]1) ∧ 𝑌 ∈ (0(,]1)) → 0 <
(𝑋 · 𝑌)) |
| 126 | | iocssicc 13477 |
. . . . . . . . . . . 12
⊢ (0(,]1)
⊆ (0[,]1) |
| 127 | 126, 108 | sselid 3981 |
. . . . . . . . . . 11
⊢ ((𝑋 ∈ (0(,]1) ∧ 𝑌 ∈ (0(,]1)) → 𝑋 ∈
(0[,]1)) |
| 128 | 126, 112 | sselid 3981 |
. . . . . . . . . . 11
⊢ ((𝑋 ∈ (0(,]1) ∧ 𝑌 ∈ (0(,]1)) → 𝑌 ∈
(0[,]1)) |
| 129 | | iimulcl 24966 |
. . . . . . . . . . 11
⊢ ((𝑋 ∈ (0[,]1) ∧ 𝑌 ∈ (0[,]1)) → (𝑋 · 𝑌) ∈ (0[,]1)) |
| 130 | 127, 128,
129 | syl2anc 584 |
. . . . . . . . . 10
⊢ ((𝑋 ∈ (0(,]1) ∧ 𝑌 ∈ (0(,]1)) → (𝑋 · 𝑌) ∈ (0[,]1)) |
| 131 | | elicc01 13506 |
. . . . . . . . . . 11
⊢ ((𝑋 · 𝑌) ∈ (0[,]1) ↔ ((𝑋 · 𝑌) ∈ ℝ ∧ 0 ≤ (𝑋 · 𝑌) ∧ (𝑋 · 𝑌) ≤ 1)) |
| 132 | 131 | simp3bi 1148 |
. . . . . . . . . 10
⊢ ((𝑋 · 𝑌) ∈ (0[,]1) → (𝑋 · 𝑌) ≤ 1) |
| 133 | 130, 132 | syl 17 |
. . . . . . . . 9
⊢ ((𝑋 ∈ (0(,]1) ∧ 𝑌 ∈ (0(,]1)) → (𝑋 · 𝑌) ≤ 1) |
| 134 | | elioc2 13450 |
. . . . . . . . . 10
⊢ ((0
∈ ℝ* ∧ 1 ∈ ℝ) → ((𝑋 · 𝑌) ∈ (0(,]1) ↔ ((𝑋 · 𝑌) ∈ ℝ ∧ 0 < (𝑋 · 𝑌) ∧ (𝑋 · 𝑌) ≤ 1))) |
| 135 | 1, 76, 134 | mp2an 692 |
. . . . . . . . 9
⊢ ((𝑋 · 𝑌) ∈ (0(,]1) ↔ ((𝑋 · 𝑌) ∈ ℝ ∧ 0 < (𝑋 · 𝑌) ∧ (𝑋 · 𝑌) ≤ 1)) |
| 136 | 122, 125,
133, 135 | syl3anbrc 1344 |
. . . . . . . 8
⊢ ((𝑋 ∈ (0(,]1) ∧ 𝑌 ∈ (0(,]1)) → (𝑋 · 𝑌) ∈ (0(,]1)) |
| 137 | 14 | xrge0iifcv 33933 |
. . . . . . . 8
⊢ ((𝑋 · 𝑌) ∈ (0(,]1) → (𝐹‘(𝑋 · 𝑌)) = -(log‘(𝑋 · 𝑌))) |
| 138 | 136, 137 | syl 17 |
. . . . . . 7
⊢ ((𝑋 ∈ (0(,]1) ∧ 𝑌 ∈ (0(,]1)) → (𝐹‘(𝑋 · 𝑌)) = -(log‘(𝑋 · 𝑌))) |
| 139 | 109, 113 | relogmuld 26667 |
. . . . . . . 8
⊢ ((𝑋 ∈ (0(,]1) ∧ 𝑌 ∈ (0(,]1)) →
(log‘(𝑋 ·
𝑌)) = ((log‘𝑋) + (log‘𝑌))) |
| 140 | 139 | negeqd 11502 |
. . . . . . 7
⊢ ((𝑋 ∈ (0(,]1) ∧ 𝑌 ∈ (0(,]1)) →
-(log‘(𝑋 ·
𝑌)) = -((log‘𝑋) + (log‘𝑌))) |
| 141 | 110 | recnd 11289 |
. . . . . . . 8
⊢ ((𝑋 ∈ (0(,]1) ∧ 𝑌 ∈ (0(,]1)) →
(log‘𝑋) ∈
ℂ) |
| 142 | 114 | recnd 11289 |
. . . . . . . 8
⊢ ((𝑋 ∈ (0(,]1) ∧ 𝑌 ∈ (0(,]1)) →
(log‘𝑌) ∈
ℂ) |
| 143 | 141, 142 | negdid 11633 |
. . . . . . 7
⊢ ((𝑋 ∈ (0(,]1) ∧ 𝑌 ∈ (0(,]1)) →
-((log‘𝑋) +
(log‘𝑌)) =
(-(log‘𝑋) +
-(log‘𝑌))) |
| 144 | 138, 140,
143 | 3eqtrd 2781 |
. . . . . 6
⊢ ((𝑋 ∈ (0(,]1) ∧ 𝑌 ∈ (0(,]1)) → (𝐹‘(𝑋 · 𝑌)) = (-(log‘𝑋) + -(log‘𝑌))) |
| 145 | 117, 119,
144 | 3eqtr4rd 2788 |
. . . . 5
⊢ ((𝑋 ∈ (0(,]1) ∧ 𝑌 ∈ (0(,]1)) → (𝐹‘(𝑋 · 𝑌)) = ((𝐹‘𝑋) +𝑒 (𝐹‘𝑌))) |
| 146 | 107, 145 | jaoian 959 |
. . . 4
⊢ (((𝑋 = 0 ∨ 𝑋 ∈ (0(,]1)) ∧ 𝑌 ∈ (0(,]1)) → (𝐹‘(𝑋 · 𝑌)) = ((𝐹‘𝑋) +𝑒 (𝐹‘𝑌))) |
| 147 | 71, 146 | sylan 580 |
. . 3
⊢ ((𝑋 ∈ (0[,]1) ∧ 𝑌 ∈ (0(,]1)) → (𝐹‘(𝑋 · 𝑌)) = ((𝐹‘𝑋) +𝑒 (𝐹‘𝑌))) |
| 148 | 65, 147 | jaodan 960 |
. 2
⊢ ((𝑋 ∈ (0[,]1) ∧ (𝑌 = 0 ∨ 𝑌 ∈ (0(,]1))) → (𝐹‘(𝑋 · 𝑌)) = ((𝐹‘𝑋) +𝑒 (𝐹‘𝑌))) |
| 149 | 11, 148 | sylan2 593 |
1
⊢ ((𝑋 ∈ (0[,]1) ∧ 𝑌 ∈ (0[,]1)) → (𝐹‘(𝑋 · 𝑌)) = ((𝐹‘𝑋) +𝑒 (𝐹‘𝑌))) |