| Step | Hyp | Ref
| Expression |
| 1 | | inss1 4237 |
. . 3
⊢ (𝐷 ∩ 𝐼) ⊆ 𝐷 |
| 2 | 1 | a1i 11 |
. 2
⊢ (𝜑 → (𝐷 ∩ 𝐼) ⊆ 𝐷) |
| 3 | 2 | sselda 3983 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐷 ∩ 𝐼)) → 𝑥 ∈ 𝐷) |
| 4 | | smflimlem4.6 |
. . . . . . . . . 10
⊢ 𝐺 = (𝑥 ∈ 𝐷 ↦ ( ⇝ ‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥)))) |
| 5 | 4 | a1i 11 |
. . . . . . . . 9
⊢ (𝜑 → 𝐺 = (𝑥 ∈ 𝐷 ↦ ( ⇝ ‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))))) |
| 6 | | nfv 1914 |
. . . . . . . . . . 11
⊢
Ⅎ𝑚(𝜑 ∧ 𝑥 ∈ 𝐷) |
| 7 | | nfcv 2905 |
. . . . . . . . . . 11
⊢
Ⅎ𝑚𝐹 |
| 8 | | nfcv 2905 |
. . . . . . . . . . 11
⊢
Ⅎ𝑧𝐹 |
| 9 | | smflimlem4.2 |
. . . . . . . . . . 11
⊢ 𝑍 =
(ℤ≥‘𝑀) |
| 10 | | smflimlem4.3 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → 𝑆 ∈ SAlg) |
| 11 | 10 | adantr 480 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑚 ∈ 𝑍) → 𝑆 ∈ SAlg) |
| 12 | | smflimlem4.4 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → 𝐹:𝑍⟶(SMblFn‘𝑆)) |
| 13 | 12 | ffvelcdmda 7104 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑚 ∈ 𝑍) → (𝐹‘𝑚) ∈ (SMblFn‘𝑆)) |
| 14 | | eqid 2737 |
. . . . . . . . . . . . 13
⊢ dom
(𝐹‘𝑚) = dom (𝐹‘𝑚) |
| 15 | 11, 13, 14 | smff 46747 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑚 ∈ 𝑍) → (𝐹‘𝑚):dom (𝐹‘𝑚)⟶ℝ) |
| 16 | 15 | adantlr 715 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑚 ∈ 𝑍) → (𝐹‘𝑚):dom (𝐹‘𝑚)⟶ℝ) |
| 17 | | smflimlem4.5 |
. . . . . . . . . . . 12
⊢ 𝐷 = {𝑥 ∈ ∪
𝑛 ∈ 𝑍 ∩ 𝑚 ∈
(ℤ≥‘𝑛)dom (𝐹‘𝑚) ∣ (𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥)) ∈ dom ⇝ } |
| 18 | | fveq2 6906 |
. . . . . . . . . . . . . . 15
⊢ (𝑥 = 𝑧 → ((𝐹‘𝑚)‘𝑥) = ((𝐹‘𝑚)‘𝑧)) |
| 19 | 18 | mpteq2dv 5244 |
. . . . . . . . . . . . . 14
⊢ (𝑥 = 𝑧 → (𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥)) = (𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑧))) |
| 20 | 19 | eleq1d 2826 |
. . . . . . . . . . . . 13
⊢ (𝑥 = 𝑧 → ((𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥)) ∈ dom ⇝ ↔ (𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑧)) ∈ dom ⇝ )) |
| 21 | 20 | cbvrabv 3447 |
. . . . . . . . . . . 12
⊢ {𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈
(ℤ≥‘𝑛)dom (𝐹‘𝑚) ∣ (𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥)) ∈ dom ⇝ } = {𝑧 ∈ ∪
𝑛 ∈ 𝑍 ∩ 𝑚 ∈
(ℤ≥‘𝑛)dom (𝐹‘𝑚) ∣ (𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑧)) ∈ dom ⇝ } |
| 22 | 17, 21 | eqtri 2765 |
. . . . . . . . . . 11
⊢ 𝐷 = {𝑧 ∈ ∪
𝑛 ∈ 𝑍 ∩ 𝑚 ∈
(ℤ≥‘𝑛)dom (𝐹‘𝑚) ∣ (𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑧)) ∈ dom ⇝ } |
| 23 | | simpr 484 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → 𝑥 ∈ 𝐷) |
| 24 | 6, 7, 8, 9, 16, 22, 23 | fnlimfvre 45689 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → ( ⇝ ‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))) ∈ ℝ) |
| 25 | 24 | elexd 3504 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → ( ⇝ ‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))) ∈ V) |
| 26 | 5, 25 | fvmpt2d 7029 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → (𝐺‘𝑥) = ( ⇝ ‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥)))) |
| 27 | 26, 24 | eqeltrd 2841 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → (𝐺‘𝑥) ∈ ℝ) |
| 28 | 3, 27 | syldan 591 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐷 ∩ 𝐼)) → (𝐺‘𝑥) ∈ ℝ) |
| 29 | 28 | adantr 480 |
. . . . 5
⊢ (((𝜑 ∧ 𝑥 ∈ (𝐷 ∩ 𝐼)) ∧ 𝑦 ∈ ℝ+) → (𝐺‘𝑥) ∈ ℝ) |
| 30 | | smflimlem4.7 |
. . . . . . . 8
⊢ (𝜑 → 𝐴 ∈ ℝ) |
| 31 | 30 | adantr 480 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑦 ∈ ℝ+) → 𝐴 ∈
ℝ) |
| 32 | | rpre 13043 |
. . . . . . . 8
⊢ (𝑦 ∈ ℝ+
→ 𝑦 ∈
ℝ) |
| 33 | 32 | adantl 481 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑦 ∈ ℝ+) → 𝑦 ∈
ℝ) |
| 34 | 31, 33 | readdcld 11290 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑦 ∈ ℝ+) → (𝐴 + 𝑦) ∈ ℝ) |
| 35 | 34 | adantlr 715 |
. . . . 5
⊢ (((𝜑 ∧ 𝑥 ∈ (𝐷 ∩ 𝐼)) ∧ 𝑦 ∈ ℝ+) → (𝐴 + 𝑦) ∈ ℝ) |
| 36 | | nfv 1914 |
. . . . . . . 8
⊢
Ⅎ𝑚((𝜑 ∧ 𝑥 ∈ (𝐷 ∩ 𝐼)) ∧ 𝑦 ∈ ℝ+) |
| 37 | | rphalfcl 13062 |
. . . . . . . . . . 11
⊢ (𝑦 ∈ ℝ+
→ (𝑦 / 2) ∈
ℝ+) |
| 38 | | rpgtrecnn 45391 |
. . . . . . . . . . 11
⊢ ((𝑦 / 2) ∈ ℝ+
→ ∃𝑘 ∈
ℕ (1 / 𝑘) < (𝑦 / 2)) |
| 39 | 37, 38 | syl 17 |
. . . . . . . . . 10
⊢ (𝑦 ∈ ℝ+
→ ∃𝑘 ∈
ℕ (1 / 𝑘) < (𝑦 / 2)) |
| 40 | 39 | adantl 481 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑥 ∈ (𝐷 ∩ 𝐼)) ∧ 𝑦 ∈ ℝ+) →
∃𝑘 ∈ ℕ (1
/ 𝑘) < (𝑦 / 2)) |
| 41 | 10 | ad4antr 732 |
. . . . . . . . . . 11
⊢
(((((𝜑 ∧ 𝑥 ∈ (𝐷 ∩ 𝐼)) ∧ 𝑦 ∈ ℝ+) ∧ 𝑘 ∈ ℕ) ∧ (1 /
𝑘) < (𝑦 / 2)) → 𝑆 ∈ SAlg) |
| 42 | 13 | adantlr 715 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑥 ∈ (𝐷 ∩ 𝐼)) ∧ 𝑚 ∈ 𝑍) → (𝐹‘𝑚) ∈ (SMblFn‘𝑆)) |
| 43 | 42 | ad5ant15 759 |
. . . . . . . . . . 11
⊢
((((((𝜑 ∧ 𝑥 ∈ (𝐷 ∩ 𝐼)) ∧ 𝑦 ∈ ℝ+) ∧ 𝑘 ∈ ℕ) ∧ (1 /
𝑘) < (𝑦 / 2)) ∧ 𝑚 ∈ 𝑍) → (𝐹‘𝑚) ∈ (SMblFn‘𝑆)) |
| 44 | 30 | adantr 480 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐷 ∩ 𝐼)) → 𝐴 ∈ ℝ) |
| 45 | 44 | ad3antrrr 730 |
. . . . . . . . . . 11
⊢
(((((𝜑 ∧ 𝑥 ∈ (𝐷 ∩ 𝐼)) ∧ 𝑦 ∈ ℝ+) ∧ 𝑘 ∈ ℕ) ∧ (1 /
𝑘) < (𝑦 / 2)) → 𝐴 ∈ ℝ) |
| 46 | | smflimlem4.8 |
. . . . . . . . . . . 12
⊢ 𝑃 = (𝑚 ∈ 𝑍, 𝑘 ∈ ℕ ↦ {𝑠 ∈ 𝑆 ∣ {𝑥 ∈ dom (𝐹‘𝑚) ∣ ((𝐹‘𝑚)‘𝑥) < (𝐴 + (1 / 𝑘))} = (𝑠 ∩ dom (𝐹‘𝑚))}) |
| 47 | | nfcv 2905 |
. . . . . . . . . . . . 13
⊢
Ⅎ𝑘𝑍 |
| 48 | | nfcv 2905 |
. . . . . . . . . . . . 13
⊢
Ⅎ𝑗𝑍 |
| 49 | | nfcv 2905 |
. . . . . . . . . . . . 13
⊢
Ⅎ𝑗{𝑠 ∈ 𝑆 ∣ {𝑥 ∈ dom (𝐹‘𝑚) ∣ ((𝐹‘𝑚)‘𝑥) < (𝐴 + (1 / 𝑘))} = (𝑠 ∩ dom (𝐹‘𝑚))} |
| 50 | | nfcv 2905 |
. . . . . . . . . . . . 13
⊢
Ⅎ𝑘{𝑠 ∈ 𝑆 ∣ {𝑧 ∈ dom (𝐹‘𝑚) ∣ ((𝐹‘𝑚)‘𝑧) < (𝐴 + (1 / 𝑗))} = (𝑠 ∩ dom (𝐹‘𝑚))} |
| 51 | 18 | breq1d 5153 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑥 = 𝑧 → (((𝐹‘𝑚)‘𝑥) < (𝐴 + (1 / 𝑘)) ↔ ((𝐹‘𝑚)‘𝑧) < (𝐴 + (1 / 𝑘)))) |
| 52 | 51 | cbvrabv 3447 |
. . . . . . . . . . . . . . . . 17
⊢ {𝑥 ∈ dom (𝐹‘𝑚) ∣ ((𝐹‘𝑚)‘𝑥) < (𝐴 + (1 / 𝑘))} = {𝑧 ∈ dom (𝐹‘𝑚) ∣ ((𝐹‘𝑚)‘𝑧) < (𝐴 + (1 / 𝑘))} |
| 53 | 52 | a1i 11 |
. . . . . . . . . . . . . . . 16
⊢ (𝑘 = 𝑗 → {𝑥 ∈ dom (𝐹‘𝑚) ∣ ((𝐹‘𝑚)‘𝑥) < (𝐴 + (1 / 𝑘))} = {𝑧 ∈ dom (𝐹‘𝑚) ∣ ((𝐹‘𝑚)‘𝑧) < (𝐴 + (1 / 𝑘))}) |
| 54 | | oveq2 7439 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑘 = 𝑗 → (1 / 𝑘) = (1 / 𝑗)) |
| 55 | 54 | oveq2d 7447 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑘 = 𝑗 → (𝐴 + (1 / 𝑘)) = (𝐴 + (1 / 𝑗))) |
| 56 | 55 | breq2d 5155 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑘 = 𝑗 → (((𝐹‘𝑚)‘𝑧) < (𝐴 + (1 / 𝑘)) ↔ ((𝐹‘𝑚)‘𝑧) < (𝐴 + (1 / 𝑗)))) |
| 57 | 56 | rabbidv 3444 |
. . . . . . . . . . . . . . . 16
⊢ (𝑘 = 𝑗 → {𝑧 ∈ dom (𝐹‘𝑚) ∣ ((𝐹‘𝑚)‘𝑧) < (𝐴 + (1 / 𝑘))} = {𝑧 ∈ dom (𝐹‘𝑚) ∣ ((𝐹‘𝑚)‘𝑧) < (𝐴 + (1 / 𝑗))}) |
| 58 | 53, 57 | eqtrd 2777 |
. . . . . . . . . . . . . . 15
⊢ (𝑘 = 𝑗 → {𝑥 ∈ dom (𝐹‘𝑚) ∣ ((𝐹‘𝑚)‘𝑥) < (𝐴 + (1 / 𝑘))} = {𝑧 ∈ dom (𝐹‘𝑚) ∣ ((𝐹‘𝑚)‘𝑧) < (𝐴 + (1 / 𝑗))}) |
| 59 | 58 | eqeq1d 2739 |
. . . . . . . . . . . . . 14
⊢ (𝑘 = 𝑗 → ({𝑥 ∈ dom (𝐹‘𝑚) ∣ ((𝐹‘𝑚)‘𝑥) < (𝐴 + (1 / 𝑘))} = (𝑠 ∩ dom (𝐹‘𝑚)) ↔ {𝑧 ∈ dom (𝐹‘𝑚) ∣ ((𝐹‘𝑚)‘𝑧) < (𝐴 + (1 / 𝑗))} = (𝑠 ∩ dom (𝐹‘𝑚)))) |
| 60 | 59 | rabbidv 3444 |
. . . . . . . . . . . . 13
⊢ (𝑘 = 𝑗 → {𝑠 ∈ 𝑆 ∣ {𝑥 ∈ dom (𝐹‘𝑚) ∣ ((𝐹‘𝑚)‘𝑥) < (𝐴 + (1 / 𝑘))} = (𝑠 ∩ dom (𝐹‘𝑚))} = {𝑠 ∈ 𝑆 ∣ {𝑧 ∈ dom (𝐹‘𝑚) ∣ ((𝐹‘𝑚)‘𝑧) < (𝐴 + (1 / 𝑗))} = (𝑠 ∩ dom (𝐹‘𝑚))}) |
| 61 | 47, 48, 49, 50, 60 | cbvmpo2 45102 |
. . . . . . . . . . . 12
⊢ (𝑚 ∈ 𝑍, 𝑘 ∈ ℕ ↦ {𝑠 ∈ 𝑆 ∣ {𝑥 ∈ dom (𝐹‘𝑚) ∣ ((𝐹‘𝑚)‘𝑥) < (𝐴 + (1 / 𝑘))} = (𝑠 ∩ dom (𝐹‘𝑚))}) = (𝑚 ∈ 𝑍, 𝑗 ∈ ℕ ↦ {𝑠 ∈ 𝑆 ∣ {𝑧 ∈ dom (𝐹‘𝑚) ∣ ((𝐹‘𝑚)‘𝑧) < (𝐴 + (1 / 𝑗))} = (𝑠 ∩ dom (𝐹‘𝑚))}) |
| 62 | 46, 61 | eqtri 2765 |
. . . . . . . . . . 11
⊢ 𝑃 = (𝑚 ∈ 𝑍, 𝑗 ∈ ℕ ↦ {𝑠 ∈ 𝑆 ∣ {𝑧 ∈ dom (𝐹‘𝑚) ∣ ((𝐹‘𝑚)‘𝑧) < (𝐴 + (1 / 𝑗))} = (𝑠 ∩ dom (𝐹‘𝑚))}) |
| 63 | | smflimlem4.9 |
. . . . . . . . . . . 12
⊢ 𝐻 = (𝑚 ∈ 𝑍, 𝑘 ∈ ℕ ↦ (𝐶‘(𝑚𝑃𝑘))) |
| 64 | | nfcv 2905 |
. . . . . . . . . . . . 13
⊢
Ⅎ𝑗(𝐶‘(𝑚𝑃𝑘)) |
| 65 | | nfcv 2905 |
. . . . . . . . . . . . 13
⊢
Ⅎ𝑘(𝐶‘(𝑚𝑃𝑗)) |
| 66 | | oveq2 7439 |
. . . . . . . . . . . . . 14
⊢ (𝑘 = 𝑗 → (𝑚𝑃𝑘) = (𝑚𝑃𝑗)) |
| 67 | 66 | fveq2d 6910 |
. . . . . . . . . . . . 13
⊢ (𝑘 = 𝑗 → (𝐶‘(𝑚𝑃𝑘)) = (𝐶‘(𝑚𝑃𝑗))) |
| 68 | 47, 48, 64, 65, 67 | cbvmpo2 45102 |
. . . . . . . . . . . 12
⊢ (𝑚 ∈ 𝑍, 𝑘 ∈ ℕ ↦ (𝐶‘(𝑚𝑃𝑘))) = (𝑚 ∈ 𝑍, 𝑗 ∈ ℕ ↦ (𝐶‘(𝑚𝑃𝑗))) |
| 69 | 63, 68 | eqtri 2765 |
. . . . . . . . . . 11
⊢ 𝐻 = (𝑚 ∈ 𝑍, 𝑗 ∈ ℕ ↦ (𝐶‘(𝑚𝑃𝑗))) |
| 70 | | smflimlem4.10 |
. . . . . . . . . . . 12
⊢ 𝐼 = ∩ 𝑘 ∈ ℕ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈
(ℤ≥‘𝑛)(𝑚𝐻𝑘) |
| 71 | | simpll 767 |
. . . . . . . . . . . . . . . 16
⊢ (((𝑘 = 𝑗 ∧ 𝑛 ∈ 𝑍) ∧ 𝑚 ∈ (ℤ≥‘𝑛)) → 𝑘 = 𝑗) |
| 72 | 71 | oveq2d 7447 |
. . . . . . . . . . . . . . 15
⊢ (((𝑘 = 𝑗 ∧ 𝑛 ∈ 𝑍) ∧ 𝑚 ∈ (ℤ≥‘𝑛)) → (𝑚𝐻𝑘) = (𝑚𝐻𝑗)) |
| 73 | 72 | iineq2dv 5017 |
. . . . . . . . . . . . . 14
⊢ ((𝑘 = 𝑗 ∧ 𝑛 ∈ 𝑍) → ∩
𝑚 ∈
(ℤ≥‘𝑛)(𝑚𝐻𝑘) = ∩ 𝑚 ∈
(ℤ≥‘𝑛)(𝑚𝐻𝑗)) |
| 74 | 73 | iuneq2dv 5016 |
. . . . . . . . . . . . 13
⊢ (𝑘 = 𝑗 → ∪
𝑛 ∈ 𝑍 ∩ 𝑚 ∈
(ℤ≥‘𝑛)(𝑚𝐻𝑘) = ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈
(ℤ≥‘𝑛)(𝑚𝐻𝑗)) |
| 75 | 74 | cbviinv 5041 |
. . . . . . . . . . . 12
⊢ ∩ 𝑘 ∈ ℕ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈
(ℤ≥‘𝑛)(𝑚𝐻𝑘) = ∩ 𝑗 ∈ ℕ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈
(ℤ≥‘𝑛)(𝑚𝐻𝑗) |
| 76 | 70, 75 | eqtri 2765 |
. . . . . . . . . . 11
⊢ 𝐼 = ∩ 𝑗 ∈ ℕ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈
(ℤ≥‘𝑛)(𝑚𝐻𝑗) |
| 77 | | smflimlem4.11 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑟 ∈ ran 𝑃) → (𝐶‘𝑟) ∈ 𝑟) |
| 78 | 77 | adantlr 715 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑥 ∈ (𝐷 ∩ 𝐼)) ∧ 𝑟 ∈ ran 𝑃) → (𝐶‘𝑟) ∈ 𝑟) |
| 79 | 78 | ad5ant15 759 |
. . . . . . . . . . 11
⊢
((((((𝜑 ∧ 𝑥 ∈ (𝐷 ∩ 𝐼)) ∧ 𝑦 ∈ ℝ+) ∧ 𝑘 ∈ ℕ) ∧ (1 /
𝑘) < (𝑦 / 2)) ∧ 𝑟 ∈ ran 𝑃) → (𝐶‘𝑟) ∈ 𝑟) |
| 80 | | simp-4r 784 |
. . . . . . . . . . 11
⊢
(((((𝜑 ∧ 𝑥 ∈ (𝐷 ∩ 𝐼)) ∧ 𝑦 ∈ ℝ+) ∧ 𝑘 ∈ ℕ) ∧ (1 /
𝑘) < (𝑦 / 2)) → 𝑥 ∈ (𝐷 ∩ 𝐼)) |
| 81 | | simplr 769 |
. . . . . . . . . . 11
⊢
(((((𝜑 ∧ 𝑥 ∈ (𝐷 ∩ 𝐼)) ∧ 𝑦 ∈ ℝ+) ∧ 𝑘 ∈ ℕ) ∧ (1 /
𝑘) < (𝑦 / 2)) → 𝑘 ∈ ℕ) |
| 82 | 37 | ad3antlr 731 |
. . . . . . . . . . 11
⊢
(((((𝜑 ∧ 𝑥 ∈ (𝐷 ∩ 𝐼)) ∧ 𝑦 ∈ ℝ+) ∧ 𝑘 ∈ ℕ) ∧ (1 /
𝑘) < (𝑦 / 2)) → (𝑦 / 2) ∈
ℝ+) |
| 83 | | simpr 484 |
. . . . . . . . . . 11
⊢
(((((𝜑 ∧ 𝑥 ∈ (𝐷 ∩ 𝐼)) ∧ 𝑦 ∈ ℝ+) ∧ 𝑘 ∈ ℕ) ∧ (1 /
𝑘) < (𝑦 / 2)) → (1 / 𝑘) < (𝑦 / 2)) |
| 84 | 9, 41, 43, 22, 45, 62, 69, 76, 79, 80, 81, 82, 83 | smflimlem3 46788 |
. . . . . . . . . 10
⊢
(((((𝜑 ∧ 𝑥 ∈ (𝐷 ∩ 𝐼)) ∧ 𝑦 ∈ ℝ+) ∧ 𝑘 ∈ ℕ) ∧ (1 /
𝑘) < (𝑦 / 2)) → ∃𝑚 ∈ 𝑍 ∀𝑖 ∈ (ℤ≥‘𝑚)(𝑥 ∈ dom (𝐹‘𝑖) ∧ ((𝐹‘𝑖)‘𝑥) < (𝐴 + (𝑦 / 2)))) |
| 85 | 84 | rexlimdva2 3157 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑥 ∈ (𝐷 ∩ 𝐼)) ∧ 𝑦 ∈ ℝ+) →
(∃𝑘 ∈ ℕ (1
/ 𝑘) < (𝑦 / 2) → ∃𝑚 ∈ 𝑍 ∀𝑖 ∈ (ℤ≥‘𝑚)(𝑥 ∈ dom (𝐹‘𝑖) ∧ ((𝐹‘𝑖)‘𝑥) < (𝐴 + (𝑦 / 2))))) |
| 86 | 40, 85 | mpd 15 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑥 ∈ (𝐷 ∩ 𝐼)) ∧ 𝑦 ∈ ℝ+) →
∃𝑚 ∈ 𝑍 ∀𝑖 ∈ (ℤ≥‘𝑚)(𝑥 ∈ dom (𝐹‘𝑖) ∧ ((𝐹‘𝑖)‘𝑥) < (𝐴 + (𝑦 / 2)))) |
| 87 | | nfv 1914 |
. . . . . . . . . 10
⊢
Ⅎ𝑖((𝜑 ∧ 𝑥 ∈ (𝐷 ∩ 𝐼)) ∧ 𝑦 ∈ ℝ+) |
| 88 | | nfcv 2905 |
. . . . . . . . . 10
⊢
Ⅎ𝑖𝐹 |
| 89 | | nfcv 2905 |
. . . . . . . . . 10
⊢
Ⅎ𝑥𝐹 |
| 90 | | smflimlem4.1 |
. . . . . . . . . . 11
⊢ (𝜑 → 𝑀 ∈ ℤ) |
| 91 | 90 | ad2antrr 726 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑥 ∈ (𝐷 ∩ 𝐼)) ∧ 𝑦 ∈ ℝ+) → 𝑀 ∈
ℤ) |
| 92 | | eleq1w 2824 |
. . . . . . . . . . . . . 14
⊢ (𝑚 = 𝑖 → (𝑚 ∈ 𝑍 ↔ 𝑖 ∈ 𝑍)) |
| 93 | 92 | anbi2d 630 |
. . . . . . . . . . . . 13
⊢ (𝑚 = 𝑖 → ((𝜑 ∧ 𝑚 ∈ 𝑍) ↔ (𝜑 ∧ 𝑖 ∈ 𝑍))) |
| 94 | | fveq2 6906 |
. . . . . . . . . . . . . 14
⊢ (𝑚 = 𝑖 → (𝐹‘𝑚) = (𝐹‘𝑖)) |
| 95 | 94 | dmeqd 5916 |
. . . . . . . . . . . . . 14
⊢ (𝑚 = 𝑖 → dom (𝐹‘𝑚) = dom (𝐹‘𝑖)) |
| 96 | 94, 95 | feq12d 6724 |
. . . . . . . . . . . . 13
⊢ (𝑚 = 𝑖 → ((𝐹‘𝑚):dom (𝐹‘𝑚)⟶ℝ ↔ (𝐹‘𝑖):dom (𝐹‘𝑖)⟶ℝ)) |
| 97 | 93, 96 | imbi12d 344 |
. . . . . . . . . . . 12
⊢ (𝑚 = 𝑖 → (((𝜑 ∧ 𝑚 ∈ 𝑍) → (𝐹‘𝑚):dom (𝐹‘𝑚)⟶ℝ) ↔ ((𝜑 ∧ 𝑖 ∈ 𝑍) → (𝐹‘𝑖):dom (𝐹‘𝑖)⟶ℝ))) |
| 98 | 97, 15 | chvarvv 1998 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑖 ∈ 𝑍) → (𝐹‘𝑖):dom (𝐹‘𝑖)⟶ℝ) |
| 99 | 98 | ad4ant14 752 |
. . . . . . . . . 10
⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐷 ∩ 𝐼)) ∧ 𝑦 ∈ ℝ+) ∧ 𝑖 ∈ 𝑍) → (𝐹‘𝑖):dom (𝐹‘𝑖)⟶ℝ) |
| 100 | | fveq2 6906 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑚 = 𝑙 → (𝐹‘𝑚) = (𝐹‘𝑙)) |
| 101 | 100 | dmeqd 5916 |
. . . . . . . . . . . . . . . 16
⊢ (𝑚 = 𝑙 → dom (𝐹‘𝑚) = dom (𝐹‘𝑙)) |
| 102 | 101 | cbviinv 5041 |
. . . . . . . . . . . . . . 15
⊢ ∩ 𝑚 ∈ (ℤ≥‘𝑛)dom (𝐹‘𝑚) = ∩ 𝑙 ∈
(ℤ≥‘𝑛)dom (𝐹‘𝑙) |
| 103 | 102 | a1i 11 |
. . . . . . . . . . . . . 14
⊢ (𝑛 ∈ 𝑍 → ∩
𝑚 ∈
(ℤ≥‘𝑛)dom (𝐹‘𝑚) = ∩ 𝑙 ∈
(ℤ≥‘𝑛)dom (𝐹‘𝑙)) |
| 104 | 103 | iuneq2i 5013 |
. . . . . . . . . . . . 13
⊢ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈
(ℤ≥‘𝑛)dom (𝐹‘𝑚) = ∪ 𝑛 ∈ 𝑍 ∩ 𝑙 ∈
(ℤ≥‘𝑛)dom (𝐹‘𝑙) |
| 105 | | fveq2 6906 |
. . . . . . . . . . . . . . . 16
⊢ (𝑛 = 𝑚 → (ℤ≥‘𝑛) =
(ℤ≥‘𝑚)) |
| 106 | 105 | iineq1d 45095 |
. . . . . . . . . . . . . . 15
⊢ (𝑛 = 𝑚 → ∩
𝑙 ∈
(ℤ≥‘𝑛)dom (𝐹‘𝑙) = ∩ 𝑙 ∈
(ℤ≥‘𝑚)dom (𝐹‘𝑙)) |
| 107 | | fveq2 6906 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑙 = 𝑖 → (𝐹‘𝑙) = (𝐹‘𝑖)) |
| 108 | 107 | dmeqd 5916 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑙 = 𝑖 → dom (𝐹‘𝑙) = dom (𝐹‘𝑖)) |
| 109 | 108 | cbviinv 5041 |
. . . . . . . . . . . . . . . 16
⊢ ∩ 𝑙 ∈ (ℤ≥‘𝑚)dom (𝐹‘𝑙) = ∩ 𝑖 ∈
(ℤ≥‘𝑚)dom (𝐹‘𝑖) |
| 110 | 109 | a1i 11 |
. . . . . . . . . . . . . . 15
⊢ (𝑛 = 𝑚 → ∩
𝑙 ∈
(ℤ≥‘𝑚)dom (𝐹‘𝑙) = ∩ 𝑖 ∈
(ℤ≥‘𝑚)dom (𝐹‘𝑖)) |
| 111 | 106, 110 | eqtrd 2777 |
. . . . . . . . . . . . . 14
⊢ (𝑛 = 𝑚 → ∩
𝑙 ∈
(ℤ≥‘𝑛)dom (𝐹‘𝑙) = ∩ 𝑖 ∈
(ℤ≥‘𝑚)dom (𝐹‘𝑖)) |
| 112 | 111 | cbviunv 5040 |
. . . . . . . . . . . . 13
⊢ ∪ 𝑛 ∈ 𝑍 ∩ 𝑙 ∈
(ℤ≥‘𝑛)dom (𝐹‘𝑙) = ∪ 𝑚 ∈ 𝑍 ∩ 𝑖 ∈
(ℤ≥‘𝑚)dom (𝐹‘𝑖) |
| 113 | 104, 112 | eqtri 2765 |
. . . . . . . . . . . 12
⊢ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈
(ℤ≥‘𝑛)dom (𝐹‘𝑚) = ∪ 𝑚 ∈ 𝑍 ∩ 𝑖 ∈
(ℤ≥‘𝑚)dom (𝐹‘𝑖) |
| 114 | 113 | rabeqi 3450 |
. . . . . . . . . . 11
⊢ {𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈
(ℤ≥‘𝑛)dom (𝐹‘𝑚) ∣ (𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥)) ∈ dom ⇝ } = {𝑥 ∈ ∪
𝑚 ∈ 𝑍 ∩ 𝑖 ∈
(ℤ≥‘𝑚)dom (𝐹‘𝑖) ∣ (𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥)) ∈ dom ⇝ } |
| 115 | | fveq2 6906 |
. . . . . . . . . . . . . . . 16
⊢ (𝑖 = 𝑚 → (𝐹‘𝑖) = (𝐹‘𝑚)) |
| 116 | 115 | fveq1d 6908 |
. . . . . . . . . . . . . . 15
⊢ (𝑖 = 𝑚 → ((𝐹‘𝑖)‘𝑥) = ((𝐹‘𝑚)‘𝑥)) |
| 117 | 116 | cbvmptv 5255 |
. . . . . . . . . . . . . 14
⊢ (𝑖 ∈ 𝑍 ↦ ((𝐹‘𝑖)‘𝑥)) = (𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥)) |
| 118 | 117 | eqcomi 2746 |
. . . . . . . . . . . . 13
⊢ (𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥)) = (𝑖 ∈ 𝑍 ↦ ((𝐹‘𝑖)‘𝑥)) |
| 119 | 118 | eleq1i 2832 |
. . . . . . . . . . . 12
⊢ ((𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥)) ∈ dom ⇝ ↔ (𝑖 ∈ 𝑍 ↦ ((𝐹‘𝑖)‘𝑥)) ∈ dom ⇝ ) |
| 120 | 119 | rabbii 3442 |
. . . . . . . . . . 11
⊢ {𝑥 ∈ ∪ 𝑚 ∈ 𝑍 ∩ 𝑖 ∈
(ℤ≥‘𝑚)dom (𝐹‘𝑖) ∣ (𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥)) ∈ dom ⇝ } = {𝑥 ∈ ∪
𝑚 ∈ 𝑍 ∩ 𝑖 ∈
(ℤ≥‘𝑚)dom (𝐹‘𝑖) ∣ (𝑖 ∈ 𝑍 ↦ ((𝐹‘𝑖)‘𝑥)) ∈ dom ⇝ } |
| 121 | 17, 114, 120 | 3eqtri 2769 |
. . . . . . . . . 10
⊢ 𝐷 = {𝑥 ∈ ∪
𝑚 ∈ 𝑍 ∩ 𝑖 ∈
(ℤ≥‘𝑚)dom (𝐹‘𝑖) ∣ (𝑖 ∈ 𝑍 ↦ ((𝐹‘𝑖)‘𝑥)) ∈ dom ⇝ } |
| 122 | 118 | fveq2i 6909 |
. . . . . . . . . . . 12
⊢ ( ⇝
‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))) = ( ⇝ ‘(𝑖 ∈ 𝑍 ↦ ((𝐹‘𝑖)‘𝑥))) |
| 123 | 122 | mpteq2i 5247 |
. . . . . . . . . . 11
⊢ (𝑥 ∈ 𝐷 ↦ ( ⇝ ‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥)))) = (𝑥 ∈ 𝐷 ↦ ( ⇝ ‘(𝑖 ∈ 𝑍 ↦ ((𝐹‘𝑖)‘𝑥)))) |
| 124 | 4, 123 | eqtri 2765 |
. . . . . . . . . 10
⊢ 𝐺 = (𝑥 ∈ 𝐷 ↦ ( ⇝ ‘(𝑖 ∈ 𝑍 ↦ ((𝐹‘𝑖)‘𝑥)))) |
| 125 | 3 | adantr 480 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑥 ∈ (𝐷 ∩ 𝐼)) ∧ 𝑦 ∈ ℝ+) → 𝑥 ∈ 𝐷) |
| 126 | 37 | adantl 481 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑥 ∈ (𝐷 ∩ 𝐼)) ∧ 𝑦 ∈ ℝ+) → (𝑦 / 2) ∈
ℝ+) |
| 127 | 87, 88, 89, 91, 9, 99, 121, 124, 125, 126 | fnlimabslt 45694 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑥 ∈ (𝐷 ∩ 𝐼)) ∧ 𝑦 ∈ ℝ+) →
∃𝑚 ∈ 𝑍 ∀𝑖 ∈ (ℤ≥‘𝑚)(((𝐹‘𝑖)‘𝑥) ∈ ℝ ∧ (abs‘(((𝐹‘𝑖)‘𝑥) − (𝐺‘𝑥))) < (𝑦 / 2))) |
| 128 | 29 | adantr 480 |
. . . . . . . . . . . . . . . . . 18
⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐷 ∩ 𝐼)) ∧ 𝑦 ∈ ℝ+) ∧ ((𝐹‘𝑖)‘𝑥) ∈ ℝ) → (𝐺‘𝑥) ∈ ℝ) |
| 129 | | simpr 484 |
. . . . . . . . . . . . . . . . . 18
⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐷 ∩ 𝐼)) ∧ 𝑦 ∈ ℝ+) ∧ ((𝐹‘𝑖)‘𝑥) ∈ ℝ) → ((𝐹‘𝑖)‘𝑥) ∈ ℝ) |
| 130 | 128, 129 | resubcld 11691 |
. . . . . . . . . . . . . . . . 17
⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐷 ∩ 𝐼)) ∧ 𝑦 ∈ ℝ+) ∧ ((𝐹‘𝑖)‘𝑥) ∈ ℝ) → ((𝐺‘𝑥) − ((𝐹‘𝑖)‘𝑥)) ∈ ℝ) |
| 131 | 130 | adantrr 717 |
. . . . . . . . . . . . . . . 16
⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐷 ∩ 𝐼)) ∧ 𝑦 ∈ ℝ+) ∧ (((𝐹‘𝑖)‘𝑥) ∈ ℝ ∧ (abs‘(((𝐹‘𝑖)‘𝑥) − (𝐺‘𝑥))) < (𝑦 / 2))) → ((𝐺‘𝑥) − ((𝐹‘𝑖)‘𝑥)) ∈ ℝ) |
| 132 | 130 | recnd 11289 |
. . . . . . . . . . . . . . . . . 18
⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐷 ∩ 𝐼)) ∧ 𝑦 ∈ ℝ+) ∧ ((𝐹‘𝑖)‘𝑥) ∈ ℝ) → ((𝐺‘𝑥) − ((𝐹‘𝑖)‘𝑥)) ∈ ℂ) |
| 133 | 132 | abscld 15475 |
. . . . . . . . . . . . . . . . 17
⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐷 ∩ 𝐼)) ∧ 𝑦 ∈ ℝ+) ∧ ((𝐹‘𝑖)‘𝑥) ∈ ℝ) → (abs‘((𝐺‘𝑥) − ((𝐹‘𝑖)‘𝑥))) ∈ ℝ) |
| 134 | 133 | adantrr 717 |
. . . . . . . . . . . . . . . 16
⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐷 ∩ 𝐼)) ∧ 𝑦 ∈ ℝ+) ∧ (((𝐹‘𝑖)‘𝑥) ∈ ℝ ∧ (abs‘(((𝐹‘𝑖)‘𝑥) − (𝐺‘𝑥))) < (𝑦 / 2))) → (abs‘((𝐺‘𝑥) − ((𝐹‘𝑖)‘𝑥))) ∈ ℝ) |
| 135 | 32 | rehalfcld 12513 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑦 ∈ ℝ+
→ (𝑦 / 2) ∈
ℝ) |
| 136 | 135 | ad2antlr 727 |
. . . . . . . . . . . . . . . 16
⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐷 ∩ 𝐼)) ∧ 𝑦 ∈ ℝ+) ∧ (((𝐹‘𝑖)‘𝑥) ∈ ℝ ∧ (abs‘(((𝐹‘𝑖)‘𝑥) − (𝐺‘𝑥))) < (𝑦 / 2))) → (𝑦 / 2) ∈ ℝ) |
| 137 | 131 | leabsd 15453 |
. . . . . . . . . . . . . . . 16
⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐷 ∩ 𝐼)) ∧ 𝑦 ∈ ℝ+) ∧ (((𝐹‘𝑖)‘𝑥) ∈ ℝ ∧ (abs‘(((𝐹‘𝑖)‘𝑥) − (𝐺‘𝑥))) < (𝑦 / 2))) → ((𝐺‘𝑥) − ((𝐹‘𝑖)‘𝑥)) ≤ (abs‘((𝐺‘𝑥) − ((𝐹‘𝑖)‘𝑥)))) |
| 138 | 28 | recnd 11289 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐷 ∩ 𝐼)) → (𝐺‘𝑥) ∈ ℂ) |
| 139 | 138 | adantr 480 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (((𝜑 ∧ 𝑥 ∈ (𝐷 ∩ 𝐼)) ∧ ((𝐹‘𝑖)‘𝑥) ∈ ℝ) → (𝐺‘𝑥) ∈ ℂ) |
| 140 | | recn 11245 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (((𝐹‘𝑖)‘𝑥) ∈ ℝ → ((𝐹‘𝑖)‘𝑥) ∈ ℂ) |
| 141 | 140 | adantl 481 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (((𝜑 ∧ 𝑥 ∈ (𝐷 ∩ 𝐼)) ∧ ((𝐹‘𝑖)‘𝑥) ∈ ℝ) → ((𝐹‘𝑖)‘𝑥) ∈ ℂ) |
| 142 | 139, 141 | abssubd 15492 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝜑 ∧ 𝑥 ∈ (𝐷 ∩ 𝐼)) ∧ ((𝐹‘𝑖)‘𝑥) ∈ ℝ) → (abs‘((𝐺‘𝑥) − ((𝐹‘𝑖)‘𝑥))) = (abs‘(((𝐹‘𝑖)‘𝑥) − (𝐺‘𝑥)))) |
| 143 | 142 | adantrr 717 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝜑 ∧ 𝑥 ∈ (𝐷 ∩ 𝐼)) ∧ (((𝐹‘𝑖)‘𝑥) ∈ ℝ ∧ (abs‘(((𝐹‘𝑖)‘𝑥) − (𝐺‘𝑥))) < (𝑦 / 2))) → (abs‘((𝐺‘𝑥) − ((𝐹‘𝑖)‘𝑥))) = (abs‘(((𝐹‘𝑖)‘𝑥) − (𝐺‘𝑥)))) |
| 144 | | simprr 773 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝜑 ∧ 𝑥 ∈ (𝐷 ∩ 𝐼)) ∧ (((𝐹‘𝑖)‘𝑥) ∈ ℝ ∧ (abs‘(((𝐹‘𝑖)‘𝑥) − (𝐺‘𝑥))) < (𝑦 / 2))) → (abs‘(((𝐹‘𝑖)‘𝑥) − (𝐺‘𝑥))) < (𝑦 / 2)) |
| 145 | 143, 144 | eqbrtrd 5165 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑥 ∈ (𝐷 ∩ 𝐼)) ∧ (((𝐹‘𝑖)‘𝑥) ∈ ℝ ∧ (abs‘(((𝐹‘𝑖)‘𝑥) − (𝐺‘𝑥))) < (𝑦 / 2))) → (abs‘((𝐺‘𝑥) − ((𝐹‘𝑖)‘𝑥))) < (𝑦 / 2)) |
| 146 | 145 | adantlr 715 |
. . . . . . . . . . . . . . . 16
⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐷 ∩ 𝐼)) ∧ 𝑦 ∈ ℝ+) ∧ (((𝐹‘𝑖)‘𝑥) ∈ ℝ ∧ (abs‘(((𝐹‘𝑖)‘𝑥) − (𝐺‘𝑥))) < (𝑦 / 2))) → (abs‘((𝐺‘𝑥) − ((𝐹‘𝑖)‘𝑥))) < (𝑦 / 2)) |
| 147 | 131, 134,
136, 137, 146 | lelttrd 11419 |
. . . . . . . . . . . . . . 15
⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐷 ∩ 𝐼)) ∧ 𝑦 ∈ ℝ+) ∧ (((𝐹‘𝑖)‘𝑥) ∈ ℝ ∧ (abs‘(((𝐹‘𝑖)‘𝑥) − (𝐺‘𝑥))) < (𝑦 / 2))) → ((𝐺‘𝑥) − ((𝐹‘𝑖)‘𝑥)) < (𝑦 / 2)) |
| 148 | 29 | adantr 480 |
. . . . . . . . . . . . . . . 16
⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐷 ∩ 𝐼)) ∧ 𝑦 ∈ ℝ+) ∧ (((𝐹‘𝑖)‘𝑥) ∈ ℝ ∧ (abs‘(((𝐹‘𝑖)‘𝑥) − (𝐺‘𝑥))) < (𝑦 / 2))) → (𝐺‘𝑥) ∈ ℝ) |
| 149 | | simprl 771 |
. . . . . . . . . . . . . . . 16
⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐷 ∩ 𝐼)) ∧ 𝑦 ∈ ℝ+) ∧ (((𝐹‘𝑖)‘𝑥) ∈ ℝ ∧ (abs‘(((𝐹‘𝑖)‘𝑥) − (𝐺‘𝑥))) < (𝑦 / 2))) → ((𝐹‘𝑖)‘𝑥) ∈ ℝ) |
| 150 | 148, 149,
136 | ltsubadd2d 11861 |
. . . . . . . . . . . . . . 15
⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐷 ∩ 𝐼)) ∧ 𝑦 ∈ ℝ+) ∧ (((𝐹‘𝑖)‘𝑥) ∈ ℝ ∧ (abs‘(((𝐹‘𝑖)‘𝑥) − (𝐺‘𝑥))) < (𝑦 / 2))) → (((𝐺‘𝑥) − ((𝐹‘𝑖)‘𝑥)) < (𝑦 / 2) ↔ (𝐺‘𝑥) < (((𝐹‘𝑖)‘𝑥) + (𝑦 / 2)))) |
| 151 | 147, 150 | mpbid 232 |
. . . . . . . . . . . . . 14
⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐷 ∩ 𝐼)) ∧ 𝑦 ∈ ℝ+) ∧ (((𝐹‘𝑖)‘𝑥) ∈ ℝ ∧ (abs‘(((𝐹‘𝑖)‘𝑥) − (𝐺‘𝑥))) < (𝑦 / 2))) → (𝐺‘𝑥) < (((𝐹‘𝑖)‘𝑥) + (𝑦 / 2))) |
| 152 | 151 | ex 412 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑥 ∈ (𝐷 ∩ 𝐼)) ∧ 𝑦 ∈ ℝ+) → ((((𝐹‘𝑖)‘𝑥) ∈ ℝ ∧ (abs‘(((𝐹‘𝑖)‘𝑥) − (𝐺‘𝑥))) < (𝑦 / 2)) → (𝐺‘𝑥) < (((𝐹‘𝑖)‘𝑥) + (𝑦 / 2)))) |
| 153 | 152 | ad2antrr 726 |
. . . . . . . . . . . 12
⊢
(((((𝜑 ∧ 𝑥 ∈ (𝐷 ∩ 𝐼)) ∧ 𝑦 ∈ ℝ+) ∧ 𝑚 ∈ 𝑍) ∧ 𝑖 ∈ (ℤ≥‘𝑚)) → ((((𝐹‘𝑖)‘𝑥) ∈ ℝ ∧ (abs‘(((𝐹‘𝑖)‘𝑥) − (𝐺‘𝑥))) < (𝑦 / 2)) → (𝐺‘𝑥) < (((𝐹‘𝑖)‘𝑥) + (𝑦 / 2)))) |
| 154 | 153 | ralimdva 3167 |
. . . . . . . . . . 11
⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐷 ∩ 𝐼)) ∧ 𝑦 ∈ ℝ+) ∧ 𝑚 ∈ 𝑍) → (∀𝑖 ∈ (ℤ≥‘𝑚)(((𝐹‘𝑖)‘𝑥) ∈ ℝ ∧ (abs‘(((𝐹‘𝑖)‘𝑥) − (𝐺‘𝑥))) < (𝑦 / 2)) → ∀𝑖 ∈ (ℤ≥‘𝑚)(𝐺‘𝑥) < (((𝐹‘𝑖)‘𝑥) + (𝑦 / 2)))) |
| 155 | 154 | ex 412 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑥 ∈ (𝐷 ∩ 𝐼)) ∧ 𝑦 ∈ ℝ+) → (𝑚 ∈ 𝑍 → (∀𝑖 ∈ (ℤ≥‘𝑚)(((𝐹‘𝑖)‘𝑥) ∈ ℝ ∧ (abs‘(((𝐹‘𝑖)‘𝑥) − (𝐺‘𝑥))) < (𝑦 / 2)) → ∀𝑖 ∈ (ℤ≥‘𝑚)(𝐺‘𝑥) < (((𝐹‘𝑖)‘𝑥) + (𝑦 / 2))))) |
| 156 | 36, 155 | reximdai 3261 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑥 ∈ (𝐷 ∩ 𝐼)) ∧ 𝑦 ∈ ℝ+) →
(∃𝑚 ∈ 𝑍 ∀𝑖 ∈ (ℤ≥‘𝑚)(((𝐹‘𝑖)‘𝑥) ∈ ℝ ∧ (abs‘(((𝐹‘𝑖)‘𝑥) − (𝐺‘𝑥))) < (𝑦 / 2)) → ∃𝑚 ∈ 𝑍 ∀𝑖 ∈ (ℤ≥‘𝑚)(𝐺‘𝑥) < (((𝐹‘𝑖)‘𝑥) + (𝑦 / 2)))) |
| 157 | 127, 156 | mpd 15 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑥 ∈ (𝐷 ∩ 𝐼)) ∧ 𝑦 ∈ ℝ+) →
∃𝑚 ∈ 𝑍 ∀𝑖 ∈ (ℤ≥‘𝑚)(𝐺‘𝑥) < (((𝐹‘𝑖)‘𝑥) + (𝑦 / 2))) |
| 158 | 115 | dmeqd 5916 |
. . . . . . . . . 10
⊢ (𝑖 = 𝑚 → dom (𝐹‘𝑖) = dom (𝐹‘𝑚)) |
| 159 | 158 | eleq2d 2827 |
. . . . . . . . 9
⊢ (𝑖 = 𝑚 → (𝑥 ∈ dom (𝐹‘𝑖) ↔ 𝑥 ∈ dom (𝐹‘𝑚))) |
| 160 | 116 | breq1d 5153 |
. . . . . . . . 9
⊢ (𝑖 = 𝑚 → (((𝐹‘𝑖)‘𝑥) < (𝐴 + (𝑦 / 2)) ↔ ((𝐹‘𝑚)‘𝑥) < (𝐴 + (𝑦 / 2)))) |
| 161 | 159, 160 | anbi12d 632 |
. . . . . . . 8
⊢ (𝑖 = 𝑚 → ((𝑥 ∈ dom (𝐹‘𝑖) ∧ ((𝐹‘𝑖)‘𝑥) < (𝐴 + (𝑦 / 2))) ↔ (𝑥 ∈ dom (𝐹‘𝑚) ∧ ((𝐹‘𝑚)‘𝑥) < (𝐴 + (𝑦 / 2))))) |
| 162 | 116 | oveq1d 7446 |
. . . . . . . . 9
⊢ (𝑖 = 𝑚 → (((𝐹‘𝑖)‘𝑥) + (𝑦 / 2)) = (((𝐹‘𝑚)‘𝑥) + (𝑦 / 2))) |
| 163 | 162 | breq2d 5155 |
. . . . . . . 8
⊢ (𝑖 = 𝑚 → ((𝐺‘𝑥) < (((𝐹‘𝑖)‘𝑥) + (𝑦 / 2)) ↔ (𝐺‘𝑥) < (((𝐹‘𝑚)‘𝑥) + (𝑦 / 2)))) |
| 164 | 36, 9, 86, 157, 161, 163 | rexanuz3 45101 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑥 ∈ (𝐷 ∩ 𝐼)) ∧ 𝑦 ∈ ℝ+) →
∃𝑚 ∈ 𝑍 ((𝑥 ∈ dom (𝐹‘𝑚) ∧ ((𝐹‘𝑚)‘𝑥) < (𝐴 + (𝑦 / 2))) ∧ (𝐺‘𝑥) < (((𝐹‘𝑚)‘𝑥) + (𝑦 / 2)))) |
| 165 | | df-3an 1089 |
. . . . . . . . 9
⊢ ((𝑥 ∈ dom (𝐹‘𝑚) ∧ ((𝐹‘𝑚)‘𝑥) < (𝐴 + (𝑦 / 2)) ∧ (𝐺‘𝑥) < (((𝐹‘𝑚)‘𝑥) + (𝑦 / 2))) ↔ ((𝑥 ∈ dom (𝐹‘𝑚) ∧ ((𝐹‘𝑚)‘𝑥) < (𝐴 + (𝑦 / 2))) ∧ (𝐺‘𝑥) < (((𝐹‘𝑚)‘𝑥) + (𝑦 / 2)))) |
| 166 | | 3ancomb 1099 |
. . . . . . . . 9
⊢ ((𝑥 ∈ dom (𝐹‘𝑚) ∧ ((𝐹‘𝑚)‘𝑥) < (𝐴 + (𝑦 / 2)) ∧ (𝐺‘𝑥) < (((𝐹‘𝑚)‘𝑥) + (𝑦 / 2))) ↔ (𝑥 ∈ dom (𝐹‘𝑚) ∧ (𝐺‘𝑥) < (((𝐹‘𝑚)‘𝑥) + (𝑦 / 2)) ∧ ((𝐹‘𝑚)‘𝑥) < (𝐴 + (𝑦 / 2)))) |
| 167 | 165, 166 | bitr3i 277 |
. . . . . . . 8
⊢ (((𝑥 ∈ dom (𝐹‘𝑚) ∧ ((𝐹‘𝑚)‘𝑥) < (𝐴 + (𝑦 / 2))) ∧ (𝐺‘𝑥) < (((𝐹‘𝑚)‘𝑥) + (𝑦 / 2))) ↔ (𝑥 ∈ dom (𝐹‘𝑚) ∧ (𝐺‘𝑥) < (((𝐹‘𝑚)‘𝑥) + (𝑦 / 2)) ∧ ((𝐹‘𝑚)‘𝑥) < (𝐴 + (𝑦 / 2)))) |
| 168 | 167 | rexbii 3094 |
. . . . . . 7
⊢
(∃𝑚 ∈
𝑍 ((𝑥 ∈ dom (𝐹‘𝑚) ∧ ((𝐹‘𝑚)‘𝑥) < (𝐴 + (𝑦 / 2))) ∧ (𝐺‘𝑥) < (((𝐹‘𝑚)‘𝑥) + (𝑦 / 2))) ↔ ∃𝑚 ∈ 𝑍 (𝑥 ∈ dom (𝐹‘𝑚) ∧ (𝐺‘𝑥) < (((𝐹‘𝑚)‘𝑥) + (𝑦 / 2)) ∧ ((𝐹‘𝑚)‘𝑥) < (𝐴 + (𝑦 / 2)))) |
| 169 | 164, 168 | sylib 218 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑥 ∈ (𝐷 ∩ 𝐼)) ∧ 𝑦 ∈ ℝ+) →
∃𝑚 ∈ 𝑍 (𝑥 ∈ dom (𝐹‘𝑚) ∧ (𝐺‘𝑥) < (((𝐹‘𝑚)‘𝑥) + (𝑦 / 2)) ∧ ((𝐹‘𝑚)‘𝑥) < (𝐴 + (𝑦 / 2)))) |
| 170 | 29 | ad2antrr 726 |
. . . . . . . . 9
⊢
(((((𝜑 ∧ 𝑥 ∈ (𝐷 ∩ 𝐼)) ∧ 𝑦 ∈ ℝ+) ∧ 𝑚 ∈ 𝑍) ∧ (𝑥 ∈ dom (𝐹‘𝑚) ∧ (𝐺‘𝑥) < (((𝐹‘𝑚)‘𝑥) + (𝑦 / 2)) ∧ ((𝐹‘𝑚)‘𝑥) < (𝐴 + (𝑦 / 2)))) → (𝐺‘𝑥) ∈ ℝ) |
| 171 | 15 | 3adant3 1133 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑚 ∈ 𝑍 ∧ 𝑥 ∈ dom (𝐹‘𝑚)) → (𝐹‘𝑚):dom (𝐹‘𝑚)⟶ℝ) |
| 172 | | simp3 1139 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑚 ∈ 𝑍 ∧ 𝑥 ∈ dom (𝐹‘𝑚)) → 𝑥 ∈ dom (𝐹‘𝑚)) |
| 173 | 171, 172 | ffvelcdmd 7105 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑚 ∈ 𝑍 ∧ 𝑥 ∈ dom (𝐹‘𝑚)) → ((𝐹‘𝑚)‘𝑥) ∈ ℝ) |
| 174 | 173 | ad4ant134 1175 |
. . . . . . . . . . . 12
⊢ ((((𝜑 ∧ 𝑦 ∈ ℝ+) ∧ 𝑚 ∈ 𝑍) ∧ 𝑥 ∈ dom (𝐹‘𝑚)) → ((𝐹‘𝑚)‘𝑥) ∈ ℝ) |
| 175 | | simpllr 776 |
. . . . . . . . . . . . 13
⊢ ((((𝜑 ∧ 𝑦 ∈ ℝ+) ∧ 𝑚 ∈ 𝑍) ∧ 𝑥 ∈ dom (𝐹‘𝑚)) → 𝑦 ∈ ℝ+) |
| 176 | 175, 135 | syl 17 |
. . . . . . . . . . . 12
⊢ ((((𝜑 ∧ 𝑦 ∈ ℝ+) ∧ 𝑚 ∈ 𝑍) ∧ 𝑥 ∈ dom (𝐹‘𝑚)) → (𝑦 / 2) ∈ ℝ) |
| 177 | 174, 176 | readdcld 11290 |
. . . . . . . . . . 11
⊢ ((((𝜑 ∧ 𝑦 ∈ ℝ+) ∧ 𝑚 ∈ 𝑍) ∧ 𝑥 ∈ dom (𝐹‘𝑚)) → (((𝐹‘𝑚)‘𝑥) + (𝑦 / 2)) ∈ ℝ) |
| 178 | 177 | adantl3r 750 |
. . . . . . . . . 10
⊢
(((((𝜑 ∧ 𝑥 ∈ (𝐷 ∩ 𝐼)) ∧ 𝑦 ∈ ℝ+) ∧ 𝑚 ∈ 𝑍) ∧ 𝑥 ∈ dom (𝐹‘𝑚)) → (((𝐹‘𝑚)‘𝑥) + (𝑦 / 2)) ∈ ℝ) |
| 179 | 178 | 3ad2antr1 1189 |
. . . . . . . . 9
⊢
(((((𝜑 ∧ 𝑥 ∈ (𝐷 ∩ 𝐼)) ∧ 𝑦 ∈ ℝ+) ∧ 𝑚 ∈ 𝑍) ∧ (𝑥 ∈ dom (𝐹‘𝑚) ∧ (𝐺‘𝑥) < (((𝐹‘𝑚)‘𝑥) + (𝑦 / 2)) ∧ ((𝐹‘𝑚)‘𝑥) < (𝐴 + (𝑦 / 2)))) → (((𝐹‘𝑚)‘𝑥) + (𝑦 / 2)) ∈ ℝ) |
| 180 | | rehalfcl 12492 |
. . . . . . . . . . . . . 14
⊢ (𝑦 ∈ ℝ → (𝑦 / 2) ∈
ℝ) |
| 181 | 33, 180 | syl 17 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑦 ∈ ℝ+) → (𝑦 / 2) ∈
ℝ) |
| 182 | 31, 181 | jca 511 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑦 ∈ ℝ+) → (𝐴 ∈ ℝ ∧ (𝑦 / 2) ∈
ℝ)) |
| 183 | | readdcl 11238 |
. . . . . . . . . . . 12
⊢ ((𝐴 ∈ ℝ ∧ (𝑦 / 2) ∈ ℝ) →
(𝐴 + (𝑦 / 2)) ∈ ℝ) |
| 184 | 182, 183 | syl 17 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑦 ∈ ℝ+) → (𝐴 + (𝑦 / 2)) ∈ ℝ) |
| 185 | 184, 181 | readdcld 11290 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑦 ∈ ℝ+) → ((𝐴 + (𝑦 / 2)) + (𝑦 / 2)) ∈ ℝ) |
| 186 | 185 | ad5ant13 757 |
. . . . . . . . 9
⊢
(((((𝜑 ∧ 𝑥 ∈ (𝐷 ∩ 𝐼)) ∧ 𝑦 ∈ ℝ+) ∧ 𝑚 ∈ 𝑍) ∧ (𝑥 ∈ dom (𝐹‘𝑚) ∧ (𝐺‘𝑥) < (((𝐹‘𝑚)‘𝑥) + (𝑦 / 2)) ∧ ((𝐹‘𝑚)‘𝑥) < (𝐴 + (𝑦 / 2)))) → ((𝐴 + (𝑦 / 2)) + (𝑦 / 2)) ∈ ℝ) |
| 187 | | simpr2 1196 |
. . . . . . . . 9
⊢
(((((𝜑 ∧ 𝑥 ∈ (𝐷 ∩ 𝐼)) ∧ 𝑦 ∈ ℝ+) ∧ 𝑚 ∈ 𝑍) ∧ (𝑥 ∈ dom (𝐹‘𝑚) ∧ (𝐺‘𝑥) < (((𝐹‘𝑚)‘𝑥) + (𝑦 / 2)) ∧ ((𝐹‘𝑚)‘𝑥) < (𝐴 + (𝑦 / 2)))) → (𝐺‘𝑥) < (((𝐹‘𝑚)‘𝑥) + (𝑦 / 2))) |
| 188 | 174 | adantrr 717 |
. . . . . . . . . . . 12
⊢ ((((𝜑 ∧ 𝑦 ∈ ℝ+) ∧ 𝑚 ∈ 𝑍) ∧ (𝑥 ∈ dom (𝐹‘𝑚) ∧ ((𝐹‘𝑚)‘𝑥) < (𝐴 + (𝑦 / 2)))) → ((𝐹‘𝑚)‘𝑥) ∈ ℝ) |
| 189 | 184 | ad2antrr 726 |
. . . . . . . . . . . 12
⊢ ((((𝜑 ∧ 𝑦 ∈ ℝ+) ∧ 𝑚 ∈ 𝑍) ∧ (𝑥 ∈ dom (𝐹‘𝑚) ∧ ((𝐹‘𝑚)‘𝑥) < (𝐴 + (𝑦 / 2)))) → (𝐴 + (𝑦 / 2)) ∈ ℝ) |
| 190 | 176 | adantrr 717 |
. . . . . . . . . . . 12
⊢ ((((𝜑 ∧ 𝑦 ∈ ℝ+) ∧ 𝑚 ∈ 𝑍) ∧ (𝑥 ∈ dom (𝐹‘𝑚) ∧ ((𝐹‘𝑚)‘𝑥) < (𝐴 + (𝑦 / 2)))) → (𝑦 / 2) ∈ ℝ) |
| 191 | | simprr 773 |
. . . . . . . . . . . 12
⊢ ((((𝜑 ∧ 𝑦 ∈ ℝ+) ∧ 𝑚 ∈ 𝑍) ∧ (𝑥 ∈ dom (𝐹‘𝑚) ∧ ((𝐹‘𝑚)‘𝑥) < (𝐴 + (𝑦 / 2)))) → ((𝐹‘𝑚)‘𝑥) < (𝐴 + (𝑦 / 2))) |
| 192 | 188, 189,
190, 191 | ltadd1dd 11874 |
. . . . . . . . . . 11
⊢ ((((𝜑 ∧ 𝑦 ∈ ℝ+) ∧ 𝑚 ∈ 𝑍) ∧ (𝑥 ∈ dom (𝐹‘𝑚) ∧ ((𝐹‘𝑚)‘𝑥) < (𝐴 + (𝑦 / 2)))) → (((𝐹‘𝑚)‘𝑥) + (𝑦 / 2)) < ((𝐴 + (𝑦 / 2)) + (𝑦 / 2))) |
| 193 | 192 | adantl3r 750 |
. . . . . . . . . 10
⊢
(((((𝜑 ∧ 𝑥 ∈ (𝐷 ∩ 𝐼)) ∧ 𝑦 ∈ ℝ+) ∧ 𝑚 ∈ 𝑍) ∧ (𝑥 ∈ dom (𝐹‘𝑚) ∧ ((𝐹‘𝑚)‘𝑥) < (𝐴 + (𝑦 / 2)))) → (((𝐹‘𝑚)‘𝑥) + (𝑦 / 2)) < ((𝐴 + (𝑦 / 2)) + (𝑦 / 2))) |
| 194 | 193 | 3adantr2 1171 |
. . . . . . . . 9
⊢
(((((𝜑 ∧ 𝑥 ∈ (𝐷 ∩ 𝐼)) ∧ 𝑦 ∈ ℝ+) ∧ 𝑚 ∈ 𝑍) ∧ (𝑥 ∈ dom (𝐹‘𝑚) ∧ (𝐺‘𝑥) < (((𝐹‘𝑚)‘𝑥) + (𝑦 / 2)) ∧ ((𝐹‘𝑚)‘𝑥) < (𝐴 + (𝑦 / 2)))) → (((𝐹‘𝑚)‘𝑥) + (𝑦 / 2)) < ((𝐴 + (𝑦 / 2)) + (𝑦 / 2))) |
| 195 | 170, 179,
186, 187, 194 | lttrd 11422 |
. . . . . . . 8
⊢
(((((𝜑 ∧ 𝑥 ∈ (𝐷 ∩ 𝐼)) ∧ 𝑦 ∈ ℝ+) ∧ 𝑚 ∈ 𝑍) ∧ (𝑥 ∈ dom (𝐹‘𝑚) ∧ (𝐺‘𝑥) < (((𝐹‘𝑚)‘𝑥) + (𝑦 / 2)) ∧ ((𝐹‘𝑚)‘𝑥) < (𝐴 + (𝑦 / 2)))) → (𝐺‘𝑥) < ((𝐴 + (𝑦 / 2)) + (𝑦 / 2))) |
| 196 | 31 | recnd 11289 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑦 ∈ ℝ+) → 𝐴 ∈
ℂ) |
| 197 | 181 | recnd 11289 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑦 ∈ ℝ+) → (𝑦 / 2) ∈
ℂ) |
| 198 | 196, 197,
197 | addassd 11283 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑦 ∈ ℝ+) → ((𝐴 + (𝑦 / 2)) + (𝑦 / 2)) = (𝐴 + ((𝑦 / 2) + (𝑦 / 2)))) |
| 199 | 32 | recnd 11289 |
. . . . . . . . . . . . 13
⊢ (𝑦 ∈ ℝ+
→ 𝑦 ∈
ℂ) |
| 200 | | 2halves 12494 |
. . . . . . . . . . . . 13
⊢ (𝑦 ∈ ℂ → ((𝑦 / 2) + (𝑦 / 2)) = 𝑦) |
| 201 | 199, 200 | syl 17 |
. . . . . . . . . . . 12
⊢ (𝑦 ∈ ℝ+
→ ((𝑦 / 2) + (𝑦 / 2)) = 𝑦) |
| 202 | 201 | oveq2d 7447 |
. . . . . . . . . . 11
⊢ (𝑦 ∈ ℝ+
→ (𝐴 + ((𝑦 / 2) + (𝑦 / 2))) = (𝐴 + 𝑦)) |
| 203 | 202 | adantl 481 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑦 ∈ ℝ+) → (𝐴 + ((𝑦 / 2) + (𝑦 / 2))) = (𝐴 + 𝑦)) |
| 204 | 198, 203 | eqtrd 2777 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑦 ∈ ℝ+) → ((𝐴 + (𝑦 / 2)) + (𝑦 / 2)) = (𝐴 + 𝑦)) |
| 205 | 204 | ad5ant13 757 |
. . . . . . . 8
⊢
(((((𝜑 ∧ 𝑥 ∈ (𝐷 ∩ 𝐼)) ∧ 𝑦 ∈ ℝ+) ∧ 𝑚 ∈ 𝑍) ∧ (𝑥 ∈ dom (𝐹‘𝑚) ∧ (𝐺‘𝑥) < (((𝐹‘𝑚)‘𝑥) + (𝑦 / 2)) ∧ ((𝐹‘𝑚)‘𝑥) < (𝐴 + (𝑦 / 2)))) → ((𝐴 + (𝑦 / 2)) + (𝑦 / 2)) = (𝐴 + 𝑦)) |
| 206 | 195, 205 | breqtrd 5169 |
. . . . . . 7
⊢
(((((𝜑 ∧ 𝑥 ∈ (𝐷 ∩ 𝐼)) ∧ 𝑦 ∈ ℝ+) ∧ 𝑚 ∈ 𝑍) ∧ (𝑥 ∈ dom (𝐹‘𝑚) ∧ (𝐺‘𝑥) < (((𝐹‘𝑚)‘𝑥) + (𝑦 / 2)) ∧ ((𝐹‘𝑚)‘𝑥) < (𝐴 + (𝑦 / 2)))) → (𝐺‘𝑥) < (𝐴 + 𝑦)) |
| 207 | 206 | rexlimdva2 3157 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑥 ∈ (𝐷 ∩ 𝐼)) ∧ 𝑦 ∈ ℝ+) →
(∃𝑚 ∈ 𝑍 (𝑥 ∈ dom (𝐹‘𝑚) ∧ (𝐺‘𝑥) < (((𝐹‘𝑚)‘𝑥) + (𝑦 / 2)) ∧ ((𝐹‘𝑚)‘𝑥) < (𝐴 + (𝑦 / 2))) → (𝐺‘𝑥) < (𝐴 + 𝑦))) |
| 208 | 169, 207 | mpd 15 |
. . . . 5
⊢ (((𝜑 ∧ 𝑥 ∈ (𝐷 ∩ 𝐼)) ∧ 𝑦 ∈ ℝ+) → (𝐺‘𝑥) < (𝐴 + 𝑦)) |
| 209 | 29, 35, 208 | ltled 11409 |
. . . 4
⊢ (((𝜑 ∧ 𝑥 ∈ (𝐷 ∩ 𝐼)) ∧ 𝑦 ∈ ℝ+) → (𝐺‘𝑥) ≤ (𝐴 + 𝑦)) |
| 210 | 209 | ralrimiva 3146 |
. . 3
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐷 ∩ 𝐼)) → ∀𝑦 ∈ ℝ+ (𝐺‘𝑥) ≤ (𝐴 + 𝑦)) |
| 211 | | alrple 13248 |
. . . 4
⊢ (((𝐺‘𝑥) ∈ ℝ ∧ 𝐴 ∈ ℝ) → ((𝐺‘𝑥) ≤ 𝐴 ↔ ∀𝑦 ∈ ℝ+ (𝐺‘𝑥) ≤ (𝐴 + 𝑦))) |
| 212 | 28, 44, 211 | syl2anc 584 |
. . 3
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐷 ∩ 𝐼)) → ((𝐺‘𝑥) ≤ 𝐴 ↔ ∀𝑦 ∈ ℝ+ (𝐺‘𝑥) ≤ (𝐴 + 𝑦))) |
| 213 | 210, 212 | mpbird 257 |
. 2
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐷 ∩ 𝐼)) → (𝐺‘𝑥) ≤ 𝐴) |
| 214 | 2, 213 | ssrabdv 4074 |
1
⊢ (𝜑 → (𝐷 ∩ 𝐼) ⊆ {𝑥 ∈ 𝐷 ∣ (𝐺‘𝑥) ≤ 𝐴}) |