Step | Hyp | Ref
| Expression |
1 | | cdj3.1 |
. . . 4
⊢ 𝐴 ∈
Sℋ |
2 | | cdj3.2 |
. . . 4
⊢ 𝐵 ∈
Sℋ |
3 | 1, 2 | cdj3lem1 30796 |
. . 3
⊢
(∃𝑣 ∈
ℝ (0 < 𝑣 ∧
∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ((normℎ‘𝑥) +
(normℎ‘𝑦)) ≤ (𝑣 ·
(normℎ‘(𝑥 +ℎ 𝑦)))) → (𝐴 ∩ 𝐵) = 0ℋ) |
4 | | cdj3.3 |
. . . . 5
⊢ 𝑆 = (𝑥 ∈ (𝐴 +ℋ 𝐵) ↦ (℩𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝐵 𝑥 = (𝑧 +ℎ 𝑤))) |
5 | 1, 2, 4 | cdj3lem2b 30799 |
. . . 4
⊢
(∃𝑣 ∈
ℝ (0 < 𝑣 ∧
∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ((normℎ‘𝑥) +
(normℎ‘𝑦)) ≤ (𝑣 ·
(normℎ‘(𝑥 +ℎ 𝑦)))) → ∃𝑣 ∈ ℝ (0 < 𝑣 ∧ ∀𝑢 ∈ (𝐴 +ℋ 𝐵)(normℎ‘(𝑆‘𝑢)) ≤ (𝑣 ·
(normℎ‘𝑢)))) |
6 | | cdj3.5 |
. . . 4
⊢ (𝜑 ↔ ∃𝑣 ∈ ℝ (0 < 𝑣 ∧ ∀𝑢 ∈ (𝐴 +ℋ 𝐵)(normℎ‘(𝑆‘𝑢)) ≤ (𝑣 ·
(normℎ‘𝑢)))) |
7 | 5, 6 | sylibr 233 |
. . 3
⊢
(∃𝑣 ∈
ℝ (0 < 𝑣 ∧
∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ((normℎ‘𝑥) +
(normℎ‘𝑦)) ≤ (𝑣 ·
(normℎ‘(𝑥 +ℎ 𝑦)))) → 𝜑) |
8 | | cdj3.4 |
. . . . 5
⊢ 𝑇 = (𝑥 ∈ (𝐴 +ℋ 𝐵) ↦ (℩𝑤 ∈ 𝐵 ∃𝑧 ∈ 𝐴 𝑥 = (𝑧 +ℎ 𝑤))) |
9 | 1, 2, 8 | cdj3lem3b 30802 |
. . . 4
⊢
(∃𝑣 ∈
ℝ (0 < 𝑣 ∧
∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ((normℎ‘𝑥) +
(normℎ‘𝑦)) ≤ (𝑣 ·
(normℎ‘(𝑥 +ℎ 𝑦)))) → ∃𝑣 ∈ ℝ (0 < 𝑣 ∧ ∀𝑢 ∈ (𝐴 +ℋ 𝐵)(normℎ‘(𝑇‘𝑢)) ≤ (𝑣 ·
(normℎ‘𝑢)))) |
10 | | cdj3.6 |
. . . 4
⊢ (𝜓 ↔ ∃𝑣 ∈ ℝ (0 < 𝑣 ∧ ∀𝑢 ∈ (𝐴 +ℋ 𝐵)(normℎ‘(𝑇‘𝑢)) ≤ (𝑣 ·
(normℎ‘𝑢)))) |
11 | 9, 10 | sylibr 233 |
. . 3
⊢
(∃𝑣 ∈
ℝ (0 < 𝑣 ∧
∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ((normℎ‘𝑥) +
(normℎ‘𝑦)) ≤ (𝑣 ·
(normℎ‘(𝑥 +ℎ 𝑦)))) → 𝜓) |
12 | 3, 7, 11 | 3jca 1127 |
. 2
⊢
(∃𝑣 ∈
ℝ (0 < 𝑣 ∧
∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ((normℎ‘𝑥) +
(normℎ‘𝑦)) ≤ (𝑣 ·
(normℎ‘(𝑥 +ℎ 𝑦)))) → ((𝐴 ∩ 𝐵) = 0ℋ ∧ 𝜑 ∧ 𝜓)) |
13 | | breq2 5078 |
. . . . . . . . 9
⊢ (𝑣 = 𝑓 → (0 < 𝑣 ↔ 0 < 𝑓)) |
14 | | oveq1 7282 |
. . . . . . . . . . 11
⊢ (𝑣 = 𝑓 → (𝑣 ·
(normℎ‘𝑢)) = (𝑓 ·
(normℎ‘𝑢))) |
15 | 14 | breq2d 5086 |
. . . . . . . . . 10
⊢ (𝑣 = 𝑓 →
((normℎ‘(𝑆‘𝑢)) ≤ (𝑣 ·
(normℎ‘𝑢)) ↔
(normℎ‘(𝑆‘𝑢)) ≤ (𝑓 ·
(normℎ‘𝑢)))) |
16 | 15 | ralbidv 3112 |
. . . . . . . . 9
⊢ (𝑣 = 𝑓 → (∀𝑢 ∈ (𝐴 +ℋ 𝐵)(normℎ‘(𝑆‘𝑢)) ≤ (𝑣 ·
(normℎ‘𝑢)) ↔ ∀𝑢 ∈ (𝐴 +ℋ 𝐵)(normℎ‘(𝑆‘𝑢)) ≤ (𝑓 ·
(normℎ‘𝑢)))) |
17 | 13, 16 | anbi12d 631 |
. . . . . . . 8
⊢ (𝑣 = 𝑓 → ((0 < 𝑣 ∧ ∀𝑢 ∈ (𝐴 +ℋ 𝐵)(normℎ‘(𝑆‘𝑢)) ≤ (𝑣 ·
(normℎ‘𝑢))) ↔ (0 < 𝑓 ∧ ∀𝑢 ∈ (𝐴 +ℋ 𝐵)(normℎ‘(𝑆‘𝑢)) ≤ (𝑓 ·
(normℎ‘𝑢))))) |
18 | 17 | cbvrexvw 3384 |
. . . . . . 7
⊢
(∃𝑣 ∈
ℝ (0 < 𝑣 ∧
∀𝑢 ∈ (𝐴 +ℋ 𝐵)(normℎ‘(𝑆‘𝑢)) ≤ (𝑣 ·
(normℎ‘𝑢))) ↔ ∃𝑓 ∈ ℝ (0 < 𝑓 ∧ ∀𝑢 ∈ (𝐴 +ℋ 𝐵)(normℎ‘(𝑆‘𝑢)) ≤ (𝑓 ·
(normℎ‘𝑢)))) |
19 | 6, 18 | bitri 274 |
. . . . . 6
⊢ (𝜑 ↔ ∃𝑓 ∈ ℝ (0 < 𝑓 ∧ ∀𝑢 ∈ (𝐴 +ℋ 𝐵)(normℎ‘(𝑆‘𝑢)) ≤ (𝑓 ·
(normℎ‘𝑢)))) |
20 | | breq2 5078 |
. . . . . . . . 9
⊢ (𝑣 = 𝑔 → (0 < 𝑣 ↔ 0 < 𝑔)) |
21 | | oveq1 7282 |
. . . . . . . . . . 11
⊢ (𝑣 = 𝑔 → (𝑣 ·
(normℎ‘𝑢)) = (𝑔 ·
(normℎ‘𝑢))) |
22 | 21 | breq2d 5086 |
. . . . . . . . . 10
⊢ (𝑣 = 𝑔 →
((normℎ‘(𝑇‘𝑢)) ≤ (𝑣 ·
(normℎ‘𝑢)) ↔
(normℎ‘(𝑇‘𝑢)) ≤ (𝑔 ·
(normℎ‘𝑢)))) |
23 | 22 | ralbidv 3112 |
. . . . . . . . 9
⊢ (𝑣 = 𝑔 → (∀𝑢 ∈ (𝐴 +ℋ 𝐵)(normℎ‘(𝑇‘𝑢)) ≤ (𝑣 ·
(normℎ‘𝑢)) ↔ ∀𝑢 ∈ (𝐴 +ℋ 𝐵)(normℎ‘(𝑇‘𝑢)) ≤ (𝑔 ·
(normℎ‘𝑢)))) |
24 | 20, 23 | anbi12d 631 |
. . . . . . . 8
⊢ (𝑣 = 𝑔 → ((0 < 𝑣 ∧ ∀𝑢 ∈ (𝐴 +ℋ 𝐵)(normℎ‘(𝑇‘𝑢)) ≤ (𝑣 ·
(normℎ‘𝑢))) ↔ (0 < 𝑔 ∧ ∀𝑢 ∈ (𝐴 +ℋ 𝐵)(normℎ‘(𝑇‘𝑢)) ≤ (𝑔 ·
(normℎ‘𝑢))))) |
25 | 24 | cbvrexvw 3384 |
. . . . . . 7
⊢
(∃𝑣 ∈
ℝ (0 < 𝑣 ∧
∀𝑢 ∈ (𝐴 +ℋ 𝐵)(normℎ‘(𝑇‘𝑢)) ≤ (𝑣 ·
(normℎ‘𝑢))) ↔ ∃𝑔 ∈ ℝ (0 < 𝑔 ∧ ∀𝑢 ∈ (𝐴 +ℋ 𝐵)(normℎ‘(𝑇‘𝑢)) ≤ (𝑔 ·
(normℎ‘𝑢)))) |
26 | 10, 25 | bitri 274 |
. . . . . 6
⊢ (𝜓 ↔ ∃𝑔 ∈ ℝ (0 < 𝑔 ∧ ∀𝑢 ∈ (𝐴 +ℋ 𝐵)(normℎ‘(𝑇‘𝑢)) ≤ (𝑔 ·
(normℎ‘𝑢)))) |
27 | 19, 26 | anbi12i 627 |
. . . . 5
⊢ ((𝜑 ∧ 𝜓) ↔ (∃𝑓 ∈ ℝ (0 < 𝑓 ∧ ∀𝑢 ∈ (𝐴 +ℋ 𝐵)(normℎ‘(𝑆‘𝑢)) ≤ (𝑓 ·
(normℎ‘𝑢))) ∧ ∃𝑔 ∈ ℝ (0 < 𝑔 ∧ ∀𝑢 ∈ (𝐴 +ℋ 𝐵)(normℎ‘(𝑇‘𝑢)) ≤ (𝑔 ·
(normℎ‘𝑢))))) |
28 | | reeanv 3294 |
. . . . 5
⊢
(∃𝑓 ∈
ℝ ∃𝑔 ∈
ℝ ((0 < 𝑓 ∧
∀𝑢 ∈ (𝐴 +ℋ 𝐵)(normℎ‘(𝑆‘𝑢)) ≤ (𝑓 ·
(normℎ‘𝑢))) ∧ (0 < 𝑔 ∧ ∀𝑢 ∈ (𝐴 +ℋ 𝐵)(normℎ‘(𝑇‘𝑢)) ≤ (𝑔 ·
(normℎ‘𝑢)))) ↔ (∃𝑓 ∈ ℝ (0 < 𝑓 ∧ ∀𝑢 ∈ (𝐴 +ℋ 𝐵)(normℎ‘(𝑆‘𝑢)) ≤ (𝑓 ·
(normℎ‘𝑢))) ∧ ∃𝑔 ∈ ℝ (0 < 𝑔 ∧ ∀𝑢 ∈ (𝐴 +ℋ 𝐵)(normℎ‘(𝑇‘𝑢)) ≤ (𝑔 ·
(normℎ‘𝑢))))) |
29 | 27, 28 | bitr4i 277 |
. . . 4
⊢ ((𝜑 ∧ 𝜓) ↔ ∃𝑓 ∈ ℝ ∃𝑔 ∈ ℝ ((0 < 𝑓 ∧ ∀𝑢 ∈ (𝐴 +ℋ 𝐵)(normℎ‘(𝑆‘𝑢)) ≤ (𝑓 ·
(normℎ‘𝑢))) ∧ (0 < 𝑔 ∧ ∀𝑢 ∈ (𝐴 +ℋ 𝐵)(normℎ‘(𝑇‘𝑢)) ≤ (𝑔 ·
(normℎ‘𝑢))))) |
30 | | an4 653 |
. . . . . 6
⊢ (((0 <
𝑓 ∧ ∀𝑢 ∈ (𝐴 +ℋ 𝐵)(normℎ‘(𝑆‘𝑢)) ≤ (𝑓 ·
(normℎ‘𝑢))) ∧ (0 < 𝑔 ∧ ∀𝑢 ∈ (𝐴 +ℋ 𝐵)(normℎ‘(𝑇‘𝑢)) ≤ (𝑔 ·
(normℎ‘𝑢)))) ↔ ((0 < 𝑓 ∧ 0 < 𝑔) ∧ (∀𝑢 ∈ (𝐴 +ℋ 𝐵)(normℎ‘(𝑆‘𝑢)) ≤ (𝑓 ·
(normℎ‘𝑢)) ∧ ∀𝑢 ∈ (𝐴 +ℋ 𝐵)(normℎ‘(𝑇‘𝑢)) ≤ (𝑔 ·
(normℎ‘𝑢))))) |
31 | | addgt0 11461 |
. . . . . . . . 9
⊢ (((𝑓 ∈ ℝ ∧ 𝑔 ∈ ℝ) ∧ (0 <
𝑓 ∧ 0 < 𝑔)) → 0 < (𝑓 + 𝑔)) |
32 | 31 | ex 413 |
. . . . . . . 8
⊢ ((𝑓 ∈ ℝ ∧ 𝑔 ∈ ℝ) → ((0 <
𝑓 ∧ 0 < 𝑔) → 0 < (𝑓 + 𝑔))) |
33 | 32 | adantl 482 |
. . . . . . 7
⊢ (((𝐴 ∩ 𝐵) = 0ℋ ∧ (𝑓 ∈ ℝ ∧ 𝑔 ∈ ℝ)) → ((0
< 𝑓 ∧ 0 < 𝑔) → 0 < (𝑓 + 𝑔))) |
34 | 1, 2 | shsvai 29726 |
. . . . . . . . . . 11
⊢ ((𝑡 ∈ 𝐴 ∧ ℎ ∈ 𝐵) → (𝑡 +ℎ ℎ) ∈ (𝐴 +ℋ 𝐵)) |
35 | | 2fveq3 6779 |
. . . . . . . . . . . . . 14
⊢ (𝑢 = (𝑡 +ℎ ℎ) → (normℎ‘(𝑆‘𝑢)) = (normℎ‘(𝑆‘(𝑡 +ℎ ℎ)))) |
36 | | fveq2 6774 |
. . . . . . . . . . . . . . 15
⊢ (𝑢 = (𝑡 +ℎ ℎ) → (normℎ‘𝑢) =
(normℎ‘(𝑡 +ℎ ℎ))) |
37 | 36 | oveq2d 7291 |
. . . . . . . . . . . . . 14
⊢ (𝑢 = (𝑡 +ℎ ℎ) → (𝑓 ·
(normℎ‘𝑢)) = (𝑓 ·
(normℎ‘(𝑡 +ℎ ℎ)))) |
38 | 35, 37 | breq12d 5087 |
. . . . . . . . . . . . 13
⊢ (𝑢 = (𝑡 +ℎ ℎ) →
((normℎ‘(𝑆‘𝑢)) ≤ (𝑓 ·
(normℎ‘𝑢)) ↔
(normℎ‘(𝑆‘(𝑡 +ℎ ℎ))) ≤ (𝑓 ·
(normℎ‘(𝑡 +ℎ ℎ))))) |
39 | 38 | rspcv 3557 |
. . . . . . . . . . . 12
⊢ ((𝑡 +ℎ ℎ) ∈ (𝐴 +ℋ 𝐵) → (∀𝑢 ∈ (𝐴 +ℋ 𝐵)(normℎ‘(𝑆‘𝑢)) ≤ (𝑓 ·
(normℎ‘𝑢)) →
(normℎ‘(𝑆‘(𝑡 +ℎ ℎ))) ≤ (𝑓 ·
(normℎ‘(𝑡 +ℎ ℎ))))) |
40 | | 2fveq3 6779 |
. . . . . . . . . . . . . 14
⊢ (𝑢 = (𝑡 +ℎ ℎ) → (normℎ‘(𝑇‘𝑢)) = (normℎ‘(𝑇‘(𝑡 +ℎ ℎ)))) |
41 | 36 | oveq2d 7291 |
. . . . . . . . . . . . . 14
⊢ (𝑢 = (𝑡 +ℎ ℎ) → (𝑔 ·
(normℎ‘𝑢)) = (𝑔 ·
(normℎ‘(𝑡 +ℎ ℎ)))) |
42 | 40, 41 | breq12d 5087 |
. . . . . . . . . . . . 13
⊢ (𝑢 = (𝑡 +ℎ ℎ) →
((normℎ‘(𝑇‘𝑢)) ≤ (𝑔 ·
(normℎ‘𝑢)) ↔
(normℎ‘(𝑇‘(𝑡 +ℎ ℎ))) ≤ (𝑔 ·
(normℎ‘(𝑡 +ℎ ℎ))))) |
43 | 42 | rspcv 3557 |
. . . . . . . . . . . 12
⊢ ((𝑡 +ℎ ℎ) ∈ (𝐴 +ℋ 𝐵) → (∀𝑢 ∈ (𝐴 +ℋ 𝐵)(normℎ‘(𝑇‘𝑢)) ≤ (𝑔 ·
(normℎ‘𝑢)) →
(normℎ‘(𝑇‘(𝑡 +ℎ ℎ))) ≤ (𝑔 ·
(normℎ‘(𝑡 +ℎ ℎ))))) |
44 | 39, 43 | anim12d 609 |
. . . . . . . . . . 11
⊢ ((𝑡 +ℎ ℎ) ∈ (𝐴 +ℋ 𝐵) → ((∀𝑢 ∈ (𝐴 +ℋ 𝐵)(normℎ‘(𝑆‘𝑢)) ≤ (𝑓 ·
(normℎ‘𝑢)) ∧ ∀𝑢 ∈ (𝐴 +ℋ 𝐵)(normℎ‘(𝑇‘𝑢)) ≤ (𝑔 ·
(normℎ‘𝑢))) →
((normℎ‘(𝑆‘(𝑡 +ℎ ℎ))) ≤ (𝑓 ·
(normℎ‘(𝑡 +ℎ ℎ))) ∧
(normℎ‘(𝑇‘(𝑡 +ℎ ℎ))) ≤ (𝑔 ·
(normℎ‘(𝑡 +ℎ ℎ)))))) |
45 | 34, 44 | syl 17 |
. . . . . . . . . 10
⊢ ((𝑡 ∈ 𝐴 ∧ ℎ ∈ 𝐵) → ((∀𝑢 ∈ (𝐴 +ℋ 𝐵)(normℎ‘(𝑆‘𝑢)) ≤ (𝑓 ·
(normℎ‘𝑢)) ∧ ∀𝑢 ∈ (𝐴 +ℋ 𝐵)(normℎ‘(𝑇‘𝑢)) ≤ (𝑔 ·
(normℎ‘𝑢))) →
((normℎ‘(𝑆‘(𝑡 +ℎ ℎ))) ≤ (𝑓 ·
(normℎ‘(𝑡 +ℎ ℎ))) ∧
(normℎ‘(𝑇‘(𝑡 +ℎ ℎ))) ≤ (𝑔 ·
(normℎ‘(𝑡 +ℎ ℎ)))))) |
46 | 45 | adantl 482 |
. . . . . . . . 9
⊢ ((((𝐴 ∩ 𝐵) = 0ℋ ∧ (𝑓 ∈ ℝ ∧ 𝑔 ∈ ℝ)) ∧ (𝑡 ∈ 𝐴 ∧ ℎ ∈ 𝐵)) → ((∀𝑢 ∈ (𝐴 +ℋ 𝐵)(normℎ‘(𝑆‘𝑢)) ≤ (𝑓 ·
(normℎ‘𝑢)) ∧ ∀𝑢 ∈ (𝐴 +ℋ 𝐵)(normℎ‘(𝑇‘𝑢)) ≤ (𝑔 ·
(normℎ‘𝑢))) →
((normℎ‘(𝑆‘(𝑡 +ℎ ℎ))) ≤ (𝑓 ·
(normℎ‘(𝑡 +ℎ ℎ))) ∧
(normℎ‘(𝑇‘(𝑡 +ℎ ℎ))) ≤ (𝑔 ·
(normℎ‘(𝑡 +ℎ ℎ)))))) |
47 | 1 | sheli 29576 |
. . . . . . . . . . . . . . 15
⊢ (𝑡 ∈ 𝐴 → 𝑡 ∈ ℋ) |
48 | | normcl 29487 |
. . . . . . . . . . . . . . 15
⊢ (𝑡 ∈ ℋ →
(normℎ‘𝑡) ∈ ℝ) |
49 | 47, 48 | syl 17 |
. . . . . . . . . . . . . 14
⊢ (𝑡 ∈ 𝐴 → (normℎ‘𝑡) ∈
ℝ) |
50 | 2 | sheli 29576 |
. . . . . . . . . . . . . . 15
⊢ (ℎ ∈ 𝐵 → ℎ ∈ ℋ) |
51 | | normcl 29487 |
. . . . . . . . . . . . . . 15
⊢ (ℎ ∈ ℋ →
(normℎ‘ℎ) ∈ ℝ) |
52 | 50, 51 | syl 17 |
. . . . . . . . . . . . . 14
⊢ (ℎ ∈ 𝐵 → (normℎ‘ℎ) ∈
ℝ) |
53 | 49, 52 | anim12i 613 |
. . . . . . . . . . . . 13
⊢ ((𝑡 ∈ 𝐴 ∧ ℎ ∈ 𝐵) →
((normℎ‘𝑡) ∈ ℝ ∧
(normℎ‘ℎ) ∈ ℝ)) |
54 | 53 | adantl 482 |
. . . . . . . . . . . 12
⊢ (((𝑓 ∈ ℝ ∧ 𝑔 ∈ ℝ) ∧ (𝑡 ∈ 𝐴 ∧ ℎ ∈ 𝐵)) →
((normℎ‘𝑡) ∈ ℝ ∧
(normℎ‘ℎ) ∈ ℝ)) |
55 | | hvaddcl 29374 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑡 ∈ ℋ ∧ ℎ ∈ ℋ) → (𝑡 +ℎ ℎ) ∈
ℋ) |
56 | 47, 50, 55 | syl2an 596 |
. . . . . . . . . . . . . . 15
⊢ ((𝑡 ∈ 𝐴 ∧ ℎ ∈ 𝐵) → (𝑡 +ℎ ℎ) ∈ ℋ) |
57 | | normcl 29487 |
. . . . . . . . . . . . . . 15
⊢ ((𝑡 +ℎ ℎ) ∈ ℋ →
(normℎ‘(𝑡 +ℎ ℎ)) ∈ ℝ) |
58 | 56, 57 | syl 17 |
. . . . . . . . . . . . . 14
⊢ ((𝑡 ∈ 𝐴 ∧ ℎ ∈ 𝐵) →
(normℎ‘(𝑡 +ℎ ℎ)) ∈ ℝ) |
59 | | remulcl 10956 |
. . . . . . . . . . . . . 14
⊢ ((𝑓 ∈ ℝ ∧
(normℎ‘(𝑡 +ℎ ℎ)) ∈ ℝ) → (𝑓 ·
(normℎ‘(𝑡 +ℎ ℎ))) ∈ ℝ) |
60 | 58, 59 | sylan2 593 |
. . . . . . . . . . . . 13
⊢ ((𝑓 ∈ ℝ ∧ (𝑡 ∈ 𝐴 ∧ ℎ ∈ 𝐵)) → (𝑓 ·
(normℎ‘(𝑡 +ℎ ℎ))) ∈ ℝ) |
61 | 60 | adantlr 712 |
. . . . . . . . . . . 12
⊢ (((𝑓 ∈ ℝ ∧ 𝑔 ∈ ℝ) ∧ (𝑡 ∈ 𝐴 ∧ ℎ ∈ 𝐵)) → (𝑓 ·
(normℎ‘(𝑡 +ℎ ℎ))) ∈ ℝ) |
62 | | remulcl 10956 |
. . . . . . . . . . . . . 14
⊢ ((𝑔 ∈ ℝ ∧
(normℎ‘(𝑡 +ℎ ℎ)) ∈ ℝ) → (𝑔 ·
(normℎ‘(𝑡 +ℎ ℎ))) ∈ ℝ) |
63 | 58, 62 | sylan2 593 |
. . . . . . . . . . . . 13
⊢ ((𝑔 ∈ ℝ ∧ (𝑡 ∈ 𝐴 ∧ ℎ ∈ 𝐵)) → (𝑔 ·
(normℎ‘(𝑡 +ℎ ℎ))) ∈ ℝ) |
64 | 63 | adantll 711 |
. . . . . . . . . . . 12
⊢ (((𝑓 ∈ ℝ ∧ 𝑔 ∈ ℝ) ∧ (𝑡 ∈ 𝐴 ∧ ℎ ∈ 𝐵)) → (𝑔 ·
(normℎ‘(𝑡 +ℎ ℎ))) ∈ ℝ) |
65 | | le2add 11457 |
. . . . . . . . . . . 12
⊢
((((normℎ‘𝑡) ∈ ℝ ∧
(normℎ‘ℎ) ∈ ℝ) ∧ ((𝑓 ·
(normℎ‘(𝑡 +ℎ ℎ))) ∈ ℝ ∧ (𝑔 ·
(normℎ‘(𝑡 +ℎ ℎ))) ∈ ℝ)) →
(((normℎ‘𝑡) ≤ (𝑓 ·
(normℎ‘(𝑡 +ℎ ℎ))) ∧ (normℎ‘ℎ) ≤ (𝑔 ·
(normℎ‘(𝑡 +ℎ ℎ)))) →
((normℎ‘𝑡) + (normℎ‘ℎ)) ≤ ((𝑓 ·
(normℎ‘(𝑡 +ℎ ℎ))) + (𝑔 ·
(normℎ‘(𝑡 +ℎ ℎ)))))) |
66 | 54, 61, 64, 65 | syl12anc 834 |
. . . . . . . . . . 11
⊢ (((𝑓 ∈ ℝ ∧ 𝑔 ∈ ℝ) ∧ (𝑡 ∈ 𝐴 ∧ ℎ ∈ 𝐵)) →
(((normℎ‘𝑡) ≤ (𝑓 ·
(normℎ‘(𝑡 +ℎ ℎ))) ∧ (normℎ‘ℎ) ≤ (𝑔 ·
(normℎ‘(𝑡 +ℎ ℎ)))) →
((normℎ‘𝑡) + (normℎ‘ℎ)) ≤ ((𝑓 ·
(normℎ‘(𝑡 +ℎ ℎ))) + (𝑔 ·
(normℎ‘(𝑡 +ℎ ℎ)))))) |
67 | 66 | adantll 711 |
. . . . . . . . . 10
⊢ ((((𝐴 ∩ 𝐵) = 0ℋ ∧ (𝑓 ∈ ℝ ∧ 𝑔 ∈ ℝ)) ∧ (𝑡 ∈ 𝐴 ∧ ℎ ∈ 𝐵)) →
(((normℎ‘𝑡) ≤ (𝑓 ·
(normℎ‘(𝑡 +ℎ ℎ))) ∧ (normℎ‘ℎ) ≤ (𝑔 ·
(normℎ‘(𝑡 +ℎ ℎ)))) →
((normℎ‘𝑡) + (normℎ‘ℎ)) ≤ ((𝑓 ·
(normℎ‘(𝑡 +ℎ ℎ))) + (𝑔 ·
(normℎ‘(𝑡 +ℎ ℎ)))))) |
68 | 1, 2, 4 | cdj3lem2 30797 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑡 ∈ 𝐴 ∧ ℎ ∈ 𝐵 ∧ (𝐴 ∩ 𝐵) = 0ℋ) → (𝑆‘(𝑡 +ℎ ℎ)) = 𝑡) |
69 | 68 | fveq2d 6778 |
. . . . . . . . . . . . . . 15
⊢ ((𝑡 ∈ 𝐴 ∧ ℎ ∈ 𝐵 ∧ (𝐴 ∩ 𝐵) = 0ℋ) →
(normℎ‘(𝑆‘(𝑡 +ℎ ℎ))) = (normℎ‘𝑡)) |
70 | 69 | breq1d 5084 |
. . . . . . . . . . . . . 14
⊢ ((𝑡 ∈ 𝐴 ∧ ℎ ∈ 𝐵 ∧ (𝐴 ∩ 𝐵) = 0ℋ) →
((normℎ‘(𝑆‘(𝑡 +ℎ ℎ))) ≤ (𝑓 ·
(normℎ‘(𝑡 +ℎ ℎ))) ↔
(normℎ‘𝑡) ≤ (𝑓 ·
(normℎ‘(𝑡 +ℎ ℎ))))) |
71 | 1, 2, 8 | cdj3lem3 30800 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑡 ∈ 𝐴 ∧ ℎ ∈ 𝐵 ∧ (𝐴 ∩ 𝐵) = 0ℋ) → (𝑇‘(𝑡 +ℎ ℎ)) = ℎ) |
72 | 71 | fveq2d 6778 |
. . . . . . . . . . . . . . 15
⊢ ((𝑡 ∈ 𝐴 ∧ ℎ ∈ 𝐵 ∧ (𝐴 ∩ 𝐵) = 0ℋ) →
(normℎ‘(𝑇‘(𝑡 +ℎ ℎ))) = (normℎ‘ℎ)) |
73 | 72 | breq1d 5084 |
. . . . . . . . . . . . . 14
⊢ ((𝑡 ∈ 𝐴 ∧ ℎ ∈ 𝐵 ∧ (𝐴 ∩ 𝐵) = 0ℋ) →
((normℎ‘(𝑇‘(𝑡 +ℎ ℎ))) ≤ (𝑔 ·
(normℎ‘(𝑡 +ℎ ℎ))) ↔
(normℎ‘ℎ) ≤ (𝑔 ·
(normℎ‘(𝑡 +ℎ ℎ))))) |
74 | 70, 73 | anbi12d 631 |
. . . . . . . . . . . . 13
⊢ ((𝑡 ∈ 𝐴 ∧ ℎ ∈ 𝐵 ∧ (𝐴 ∩ 𝐵) = 0ℋ) →
(((normℎ‘(𝑆‘(𝑡 +ℎ ℎ))) ≤ (𝑓 ·
(normℎ‘(𝑡 +ℎ ℎ))) ∧
(normℎ‘(𝑇‘(𝑡 +ℎ ℎ))) ≤ (𝑔 ·
(normℎ‘(𝑡 +ℎ ℎ)))) ↔
((normℎ‘𝑡) ≤ (𝑓 ·
(normℎ‘(𝑡 +ℎ ℎ))) ∧ (normℎ‘ℎ) ≤ (𝑔 ·
(normℎ‘(𝑡 +ℎ ℎ)))))) |
75 | 74 | 3expa 1117 |
. . . . . . . . . . . 12
⊢ (((𝑡 ∈ 𝐴 ∧ ℎ ∈ 𝐵) ∧ (𝐴 ∩ 𝐵) = 0ℋ) →
(((normℎ‘(𝑆‘(𝑡 +ℎ ℎ))) ≤ (𝑓 ·
(normℎ‘(𝑡 +ℎ ℎ))) ∧
(normℎ‘(𝑇‘(𝑡 +ℎ ℎ))) ≤ (𝑔 ·
(normℎ‘(𝑡 +ℎ ℎ)))) ↔
((normℎ‘𝑡) ≤ (𝑓 ·
(normℎ‘(𝑡 +ℎ ℎ))) ∧ (normℎ‘ℎ) ≤ (𝑔 ·
(normℎ‘(𝑡 +ℎ ℎ)))))) |
76 | 75 | ancoms 459 |
. . . . . . . . . . 11
⊢ (((𝐴 ∩ 𝐵) = 0ℋ ∧ (𝑡 ∈ 𝐴 ∧ ℎ ∈ 𝐵)) →
(((normℎ‘(𝑆‘(𝑡 +ℎ ℎ))) ≤ (𝑓 ·
(normℎ‘(𝑡 +ℎ ℎ))) ∧
(normℎ‘(𝑇‘(𝑡 +ℎ ℎ))) ≤ (𝑔 ·
(normℎ‘(𝑡 +ℎ ℎ)))) ↔
((normℎ‘𝑡) ≤ (𝑓 ·
(normℎ‘(𝑡 +ℎ ℎ))) ∧ (normℎ‘ℎ) ≤ (𝑔 ·
(normℎ‘(𝑡 +ℎ ℎ)))))) |
77 | 76 | adantlr 712 |
. . . . . . . . . 10
⊢ ((((𝐴 ∩ 𝐵) = 0ℋ ∧ (𝑓 ∈ ℝ ∧ 𝑔 ∈ ℝ)) ∧ (𝑡 ∈ 𝐴 ∧ ℎ ∈ 𝐵)) →
(((normℎ‘(𝑆‘(𝑡 +ℎ ℎ))) ≤ (𝑓 ·
(normℎ‘(𝑡 +ℎ ℎ))) ∧
(normℎ‘(𝑇‘(𝑡 +ℎ ℎ))) ≤ (𝑔 ·
(normℎ‘(𝑡 +ℎ ℎ)))) ↔
((normℎ‘𝑡) ≤ (𝑓 ·
(normℎ‘(𝑡 +ℎ ℎ))) ∧ (normℎ‘ℎ) ≤ (𝑔 ·
(normℎ‘(𝑡 +ℎ ℎ)))))) |
78 | | recn 10961 |
. . . . . . . . . . . . . 14
⊢ (𝑓 ∈ ℝ → 𝑓 ∈
ℂ) |
79 | | recn 10961 |
. . . . . . . . . . . . . 14
⊢ (𝑔 ∈ ℝ → 𝑔 ∈
ℂ) |
80 | 58 | recnd 11003 |
. . . . . . . . . . . . . 14
⊢ ((𝑡 ∈ 𝐴 ∧ ℎ ∈ 𝐵) →
(normℎ‘(𝑡 +ℎ ℎ)) ∈ ℂ) |
81 | | adddir 10966 |
. . . . . . . . . . . . . 14
⊢ ((𝑓 ∈ ℂ ∧ 𝑔 ∈ ℂ ∧
(normℎ‘(𝑡 +ℎ ℎ)) ∈ ℂ) → ((𝑓 + 𝑔) ·
(normℎ‘(𝑡 +ℎ ℎ))) = ((𝑓 ·
(normℎ‘(𝑡 +ℎ ℎ))) + (𝑔 ·
(normℎ‘(𝑡 +ℎ ℎ))))) |
82 | 78, 79, 80, 81 | syl3an 1159 |
. . . . . . . . . . . . 13
⊢ ((𝑓 ∈ ℝ ∧ 𝑔 ∈ ℝ ∧ (𝑡 ∈ 𝐴 ∧ ℎ ∈ 𝐵)) → ((𝑓 + 𝑔) ·
(normℎ‘(𝑡 +ℎ ℎ))) = ((𝑓 ·
(normℎ‘(𝑡 +ℎ ℎ))) + (𝑔 ·
(normℎ‘(𝑡 +ℎ ℎ))))) |
83 | 82 | 3expa 1117 |
. . . . . . . . . . . 12
⊢ (((𝑓 ∈ ℝ ∧ 𝑔 ∈ ℝ) ∧ (𝑡 ∈ 𝐴 ∧ ℎ ∈ 𝐵)) → ((𝑓 + 𝑔) ·
(normℎ‘(𝑡 +ℎ ℎ))) = ((𝑓 ·
(normℎ‘(𝑡 +ℎ ℎ))) + (𝑔 ·
(normℎ‘(𝑡 +ℎ ℎ))))) |
84 | 83 | breq2d 5086 |
. . . . . . . . . . 11
⊢ (((𝑓 ∈ ℝ ∧ 𝑔 ∈ ℝ) ∧ (𝑡 ∈ 𝐴 ∧ ℎ ∈ 𝐵)) →
(((normℎ‘𝑡) + (normℎ‘ℎ)) ≤ ((𝑓 + 𝑔) ·
(normℎ‘(𝑡 +ℎ ℎ))) ↔
((normℎ‘𝑡) + (normℎ‘ℎ)) ≤ ((𝑓 ·
(normℎ‘(𝑡 +ℎ ℎ))) + (𝑔 ·
(normℎ‘(𝑡 +ℎ ℎ)))))) |
85 | 84 | adantll 711 |
. . . . . . . . . 10
⊢ ((((𝐴 ∩ 𝐵) = 0ℋ ∧ (𝑓 ∈ ℝ ∧ 𝑔 ∈ ℝ)) ∧ (𝑡 ∈ 𝐴 ∧ ℎ ∈ 𝐵)) →
(((normℎ‘𝑡) + (normℎ‘ℎ)) ≤ ((𝑓 + 𝑔) ·
(normℎ‘(𝑡 +ℎ ℎ))) ↔
((normℎ‘𝑡) + (normℎ‘ℎ)) ≤ ((𝑓 ·
(normℎ‘(𝑡 +ℎ ℎ))) + (𝑔 ·
(normℎ‘(𝑡 +ℎ ℎ)))))) |
86 | 67, 77, 85 | 3imtr4d 294 |
. . . . . . . . 9
⊢ ((((𝐴 ∩ 𝐵) = 0ℋ ∧ (𝑓 ∈ ℝ ∧ 𝑔 ∈ ℝ)) ∧ (𝑡 ∈ 𝐴 ∧ ℎ ∈ 𝐵)) →
(((normℎ‘(𝑆‘(𝑡 +ℎ ℎ))) ≤ (𝑓 ·
(normℎ‘(𝑡 +ℎ ℎ))) ∧
(normℎ‘(𝑇‘(𝑡 +ℎ ℎ))) ≤ (𝑔 ·
(normℎ‘(𝑡 +ℎ ℎ)))) →
((normℎ‘𝑡) + (normℎ‘ℎ)) ≤ ((𝑓 + 𝑔) ·
(normℎ‘(𝑡 +ℎ ℎ))))) |
87 | 46, 86 | syld 47 |
. . . . . . . 8
⊢ ((((𝐴 ∩ 𝐵) = 0ℋ ∧ (𝑓 ∈ ℝ ∧ 𝑔 ∈ ℝ)) ∧ (𝑡 ∈ 𝐴 ∧ ℎ ∈ 𝐵)) → ((∀𝑢 ∈ (𝐴 +ℋ 𝐵)(normℎ‘(𝑆‘𝑢)) ≤ (𝑓 ·
(normℎ‘𝑢)) ∧ ∀𝑢 ∈ (𝐴 +ℋ 𝐵)(normℎ‘(𝑇‘𝑢)) ≤ (𝑔 ·
(normℎ‘𝑢))) →
((normℎ‘𝑡) + (normℎ‘ℎ)) ≤ ((𝑓 + 𝑔) ·
(normℎ‘(𝑡 +ℎ ℎ))))) |
88 | 87 | ralrimdvva 3125 |
. . . . . . 7
⊢ (((𝐴 ∩ 𝐵) = 0ℋ ∧ (𝑓 ∈ ℝ ∧ 𝑔 ∈ ℝ)) →
((∀𝑢 ∈ (𝐴 +ℋ 𝐵)(normℎ‘(𝑆‘𝑢)) ≤ (𝑓 ·
(normℎ‘𝑢)) ∧ ∀𝑢 ∈ (𝐴 +ℋ 𝐵)(normℎ‘(𝑇‘𝑢)) ≤ (𝑔 ·
(normℎ‘𝑢))) → ∀𝑡 ∈ 𝐴 ∀ℎ ∈ 𝐵 ((normℎ‘𝑡) +
(normℎ‘ℎ)) ≤ ((𝑓 + 𝑔) ·
(normℎ‘(𝑡 +ℎ ℎ))))) |
89 | | readdcl 10954 |
. . . . . . . . 9
⊢ ((𝑓 ∈ ℝ ∧ 𝑔 ∈ ℝ) → (𝑓 + 𝑔) ∈ ℝ) |
90 | | breq2 5078 |
. . . . . . . . . . . 12
⊢ (𝑣 = (𝑓 + 𝑔) → (0 < 𝑣 ↔ 0 < (𝑓 + 𝑔))) |
91 | | fveq2 6774 |
. . . . . . . . . . . . . . . 16
⊢ (𝑥 = 𝑡 → (normℎ‘𝑥) =
(normℎ‘𝑡)) |
92 | 91 | oveq1d 7290 |
. . . . . . . . . . . . . . 15
⊢ (𝑥 = 𝑡 → ((normℎ‘𝑥) +
(normℎ‘𝑦)) = ((normℎ‘𝑡) +
(normℎ‘𝑦))) |
93 | | fvoveq1 7298 |
. . . . . . . . . . . . . . . 16
⊢ (𝑥 = 𝑡 → (normℎ‘(𝑥 +ℎ 𝑦)) =
(normℎ‘(𝑡 +ℎ 𝑦))) |
94 | 93 | oveq2d 7291 |
. . . . . . . . . . . . . . 15
⊢ (𝑥 = 𝑡 → (𝑣 ·
(normℎ‘(𝑥 +ℎ 𝑦))) = (𝑣 ·
(normℎ‘(𝑡 +ℎ 𝑦)))) |
95 | 92, 94 | breq12d 5087 |
. . . . . . . . . . . . . 14
⊢ (𝑥 = 𝑡 →
(((normℎ‘𝑥) + (normℎ‘𝑦)) ≤ (𝑣 ·
(normℎ‘(𝑥 +ℎ 𝑦))) ↔
((normℎ‘𝑡) + (normℎ‘𝑦)) ≤ (𝑣 ·
(normℎ‘(𝑡 +ℎ 𝑦))))) |
96 | | fveq2 6774 |
. . . . . . . . . . . . . . . 16
⊢ (𝑦 = ℎ → (normℎ‘𝑦) =
(normℎ‘ℎ)) |
97 | 96 | oveq2d 7291 |
. . . . . . . . . . . . . . 15
⊢ (𝑦 = ℎ → ((normℎ‘𝑡) +
(normℎ‘𝑦)) = ((normℎ‘𝑡) +
(normℎ‘ℎ))) |
98 | | oveq2 7283 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑦 = ℎ → (𝑡 +ℎ 𝑦) = (𝑡 +ℎ ℎ)) |
99 | 98 | fveq2d 6778 |
. . . . . . . . . . . . . . . 16
⊢ (𝑦 = ℎ → (normℎ‘(𝑡 +ℎ 𝑦)) =
(normℎ‘(𝑡 +ℎ ℎ))) |
100 | 99 | oveq2d 7291 |
. . . . . . . . . . . . . . 15
⊢ (𝑦 = ℎ → (𝑣 ·
(normℎ‘(𝑡 +ℎ 𝑦))) = (𝑣 ·
(normℎ‘(𝑡 +ℎ ℎ)))) |
101 | 97, 100 | breq12d 5087 |
. . . . . . . . . . . . . 14
⊢ (𝑦 = ℎ → (((normℎ‘𝑡) +
(normℎ‘𝑦)) ≤ (𝑣 ·
(normℎ‘(𝑡 +ℎ 𝑦))) ↔
((normℎ‘𝑡) + (normℎ‘ℎ)) ≤ (𝑣 ·
(normℎ‘(𝑡 +ℎ ℎ))))) |
102 | 95, 101 | cbvral2vw 3396 |
. . . . . . . . . . . . 13
⊢
(∀𝑥 ∈
𝐴 ∀𝑦 ∈ 𝐵 ((normℎ‘𝑥) +
(normℎ‘𝑦)) ≤ (𝑣 ·
(normℎ‘(𝑥 +ℎ 𝑦))) ↔ ∀𝑡 ∈ 𝐴 ∀ℎ ∈ 𝐵 ((normℎ‘𝑡) +
(normℎ‘ℎ)) ≤ (𝑣 ·
(normℎ‘(𝑡 +ℎ ℎ)))) |
103 | | oveq1 7282 |
. . . . . . . . . . . . . . 15
⊢ (𝑣 = (𝑓 + 𝑔) → (𝑣 ·
(normℎ‘(𝑡 +ℎ ℎ))) = ((𝑓 + 𝑔) ·
(normℎ‘(𝑡 +ℎ ℎ)))) |
104 | 103 | breq2d 5086 |
. . . . . . . . . . . . . 14
⊢ (𝑣 = (𝑓 + 𝑔) →
(((normℎ‘𝑡) + (normℎ‘ℎ)) ≤ (𝑣 ·
(normℎ‘(𝑡 +ℎ ℎ))) ↔
((normℎ‘𝑡) + (normℎ‘ℎ)) ≤ ((𝑓 + 𝑔) ·
(normℎ‘(𝑡 +ℎ ℎ))))) |
105 | 104 | 2ralbidv 3129 |
. . . . . . . . . . . . 13
⊢ (𝑣 = (𝑓 + 𝑔) → (∀𝑡 ∈ 𝐴 ∀ℎ ∈ 𝐵 ((normℎ‘𝑡) +
(normℎ‘ℎ)) ≤ (𝑣 ·
(normℎ‘(𝑡 +ℎ ℎ))) ↔ ∀𝑡 ∈ 𝐴 ∀ℎ ∈ 𝐵 ((normℎ‘𝑡) +
(normℎ‘ℎ)) ≤ ((𝑓 + 𝑔) ·
(normℎ‘(𝑡 +ℎ ℎ))))) |
106 | 102, 105 | syl5bb 283 |
. . . . . . . . . . . 12
⊢ (𝑣 = (𝑓 + 𝑔) → (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ((normℎ‘𝑥) +
(normℎ‘𝑦)) ≤ (𝑣 ·
(normℎ‘(𝑥 +ℎ 𝑦))) ↔ ∀𝑡 ∈ 𝐴 ∀ℎ ∈ 𝐵 ((normℎ‘𝑡) +
(normℎ‘ℎ)) ≤ ((𝑓 + 𝑔) ·
(normℎ‘(𝑡 +ℎ ℎ))))) |
107 | 90, 106 | anbi12d 631 |
. . . . . . . . . . 11
⊢ (𝑣 = (𝑓 + 𝑔) → ((0 < 𝑣 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ((normℎ‘𝑥) +
(normℎ‘𝑦)) ≤ (𝑣 ·
(normℎ‘(𝑥 +ℎ 𝑦)))) ↔ (0 < (𝑓 + 𝑔) ∧ ∀𝑡 ∈ 𝐴 ∀ℎ ∈ 𝐵 ((normℎ‘𝑡) +
(normℎ‘ℎ)) ≤ ((𝑓 + 𝑔) ·
(normℎ‘(𝑡 +ℎ ℎ)))))) |
108 | 107 | rspcev 3561 |
. . . . . . . . . 10
⊢ (((𝑓 + 𝑔) ∈ ℝ ∧ (0 < (𝑓 + 𝑔) ∧ ∀𝑡 ∈ 𝐴 ∀ℎ ∈ 𝐵 ((normℎ‘𝑡) +
(normℎ‘ℎ)) ≤ ((𝑓 + 𝑔) ·
(normℎ‘(𝑡 +ℎ ℎ))))) → ∃𝑣 ∈ ℝ (0 < 𝑣 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ((normℎ‘𝑥) +
(normℎ‘𝑦)) ≤ (𝑣 ·
(normℎ‘(𝑥 +ℎ 𝑦))))) |
109 | 108 | ex 413 |
. . . . . . . . 9
⊢ ((𝑓 + 𝑔) ∈ ℝ → ((0 < (𝑓 + 𝑔) ∧ ∀𝑡 ∈ 𝐴 ∀ℎ ∈ 𝐵 ((normℎ‘𝑡) +
(normℎ‘ℎ)) ≤ ((𝑓 + 𝑔) ·
(normℎ‘(𝑡 +ℎ ℎ)))) → ∃𝑣 ∈ ℝ (0 < 𝑣 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ((normℎ‘𝑥) +
(normℎ‘𝑦)) ≤ (𝑣 ·
(normℎ‘(𝑥 +ℎ 𝑦)))))) |
110 | 89, 109 | syl 17 |
. . . . . . . 8
⊢ ((𝑓 ∈ ℝ ∧ 𝑔 ∈ ℝ) → ((0 <
(𝑓 + 𝑔) ∧ ∀𝑡 ∈ 𝐴 ∀ℎ ∈ 𝐵 ((normℎ‘𝑡) +
(normℎ‘ℎ)) ≤ ((𝑓 + 𝑔) ·
(normℎ‘(𝑡 +ℎ ℎ)))) → ∃𝑣 ∈ ℝ (0 < 𝑣 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ((normℎ‘𝑥) +
(normℎ‘𝑦)) ≤ (𝑣 ·
(normℎ‘(𝑥 +ℎ 𝑦)))))) |
111 | 110 | adantl 482 |
. . . . . . 7
⊢ (((𝐴 ∩ 𝐵) = 0ℋ ∧ (𝑓 ∈ ℝ ∧ 𝑔 ∈ ℝ)) → ((0
< (𝑓 + 𝑔) ∧ ∀𝑡 ∈ 𝐴 ∀ℎ ∈ 𝐵 ((normℎ‘𝑡) +
(normℎ‘ℎ)) ≤ ((𝑓 + 𝑔) ·
(normℎ‘(𝑡 +ℎ ℎ)))) → ∃𝑣 ∈ ℝ (0 < 𝑣 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ((normℎ‘𝑥) +
(normℎ‘𝑦)) ≤ (𝑣 ·
(normℎ‘(𝑥 +ℎ 𝑦)))))) |
112 | 33, 88, 111 | syl2and 608 |
. . . . . 6
⊢ (((𝐴 ∩ 𝐵) = 0ℋ ∧ (𝑓 ∈ ℝ ∧ 𝑔 ∈ ℝ)) → (((0
< 𝑓 ∧ 0 < 𝑔) ∧ (∀𝑢 ∈ (𝐴 +ℋ 𝐵)(normℎ‘(𝑆‘𝑢)) ≤ (𝑓 ·
(normℎ‘𝑢)) ∧ ∀𝑢 ∈ (𝐴 +ℋ 𝐵)(normℎ‘(𝑇‘𝑢)) ≤ (𝑔 ·
(normℎ‘𝑢)))) → ∃𝑣 ∈ ℝ (0 < 𝑣 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ((normℎ‘𝑥) +
(normℎ‘𝑦)) ≤ (𝑣 ·
(normℎ‘(𝑥 +ℎ 𝑦)))))) |
113 | 30, 112 | syl5bi 241 |
. . . . 5
⊢ (((𝐴 ∩ 𝐵) = 0ℋ ∧ (𝑓 ∈ ℝ ∧ 𝑔 ∈ ℝ)) → (((0
< 𝑓 ∧ ∀𝑢 ∈ (𝐴 +ℋ 𝐵)(normℎ‘(𝑆‘𝑢)) ≤ (𝑓 ·
(normℎ‘𝑢))) ∧ (0 < 𝑔 ∧ ∀𝑢 ∈ (𝐴 +ℋ 𝐵)(normℎ‘(𝑇‘𝑢)) ≤ (𝑔 ·
(normℎ‘𝑢)))) → ∃𝑣 ∈ ℝ (0 < 𝑣 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ((normℎ‘𝑥) +
(normℎ‘𝑦)) ≤ (𝑣 ·
(normℎ‘(𝑥 +ℎ 𝑦)))))) |
114 | 113 | rexlimdvva 3223 |
. . . 4
⊢ ((𝐴 ∩ 𝐵) = 0ℋ → (∃𝑓 ∈ ℝ ∃𝑔 ∈ ℝ ((0 < 𝑓 ∧ ∀𝑢 ∈ (𝐴 +ℋ 𝐵)(normℎ‘(𝑆‘𝑢)) ≤ (𝑓 ·
(normℎ‘𝑢))) ∧ (0 < 𝑔 ∧ ∀𝑢 ∈ (𝐴 +ℋ 𝐵)(normℎ‘(𝑇‘𝑢)) ≤ (𝑔 ·
(normℎ‘𝑢)))) → ∃𝑣 ∈ ℝ (0 < 𝑣 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ((normℎ‘𝑥) +
(normℎ‘𝑦)) ≤ (𝑣 ·
(normℎ‘(𝑥 +ℎ 𝑦)))))) |
115 | 29, 114 | syl5bi 241 |
. . 3
⊢ ((𝐴 ∩ 𝐵) = 0ℋ → ((𝜑 ∧ 𝜓) → ∃𝑣 ∈ ℝ (0 < 𝑣 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ((normℎ‘𝑥) +
(normℎ‘𝑦)) ≤ (𝑣 ·
(normℎ‘(𝑥 +ℎ 𝑦)))))) |
116 | 115 | 3impib 1115 |
. 2
⊢ (((𝐴 ∩ 𝐵) = 0ℋ ∧ 𝜑 ∧ 𝜓) → ∃𝑣 ∈ ℝ (0 < 𝑣 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ((normℎ‘𝑥) +
(normℎ‘𝑦)) ≤ (𝑣 ·
(normℎ‘(𝑥 +ℎ 𝑦))))) |
117 | 12, 116 | impbii 208 |
1
⊢
(∃𝑣 ∈
ℝ (0 < 𝑣 ∧
∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ((normℎ‘𝑥) +
(normℎ‘𝑦)) ≤ (𝑣 ·
(normℎ‘(𝑥 +ℎ 𝑦)))) ↔ ((𝐴 ∩ 𝐵) = 0ℋ ∧ 𝜑 ∧ 𝜓)) |