Step | Hyp | Ref
| Expression |
1 | | cdj3.1 |
. . . 4
⊢ 𝐴 ∈
Sℋ |
2 | | cdj3.2 |
. . . 4
⊢ 𝐵 ∈
Sℋ |
3 | 1, 2 | cdj3lem1 29293 |
. . 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 29296 |
. . . 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 224 |
. . 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 29299 |
. . . 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 224 |
. . 3
⊢
(∃𝑣 ∈
ℝ (0 < 𝑣 ∧
∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ((normℎ‘𝑥) +
(normℎ‘𝑦)) ≤ (𝑣 ·
(normℎ‘(𝑥 +ℎ 𝑦)))) → 𝜓) |
12 | 3, 7, 11 | 3jca 1242 |
. 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 4657 |
. . . . . . . . 9
⊢ (𝑣 = 𝑓 → (0 < 𝑣 ↔ 0 < 𝑓)) |
14 | | oveq1 6657 |
. . . . . . . . . . 11
⊢ (𝑣 = 𝑓 → (𝑣 ·
(normℎ‘𝑢)) = (𝑓 ·
(normℎ‘𝑢))) |
15 | 14 | breq2d 4665 |
. . . . . . . . . 10
⊢ (𝑣 = 𝑓 →
((normℎ‘(𝑆‘𝑢)) ≤ (𝑣 ·
(normℎ‘𝑢)) ↔
(normℎ‘(𝑆‘𝑢)) ≤ (𝑓 ·
(normℎ‘𝑢)))) |
16 | 15 | ralbidv 2986 |
. . . . . . . . 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 747 |
. . . . . . . 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 | cbvrexv 3172 |
. . . . . . 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 264 |
. . . . . 6
⊢ (𝜑 ↔ ∃𝑓 ∈ ℝ (0 < 𝑓 ∧ ∀𝑢 ∈ (𝐴 +ℋ 𝐵)(normℎ‘(𝑆‘𝑢)) ≤ (𝑓 ·
(normℎ‘𝑢)))) |
20 | | breq2 4657 |
. . . . . . . . 9
⊢ (𝑣 = 𝑔 → (0 < 𝑣 ↔ 0 < 𝑔)) |
21 | | oveq1 6657 |
. . . . . . . . . . 11
⊢ (𝑣 = 𝑔 → (𝑣 ·
(normℎ‘𝑢)) = (𝑔 ·
(normℎ‘𝑢))) |
22 | 21 | breq2d 4665 |
. . . . . . . . . 10
⊢ (𝑣 = 𝑔 →
((normℎ‘(𝑇‘𝑢)) ≤ (𝑣 ·
(normℎ‘𝑢)) ↔
(normℎ‘(𝑇‘𝑢)) ≤ (𝑔 ·
(normℎ‘𝑢)))) |
23 | 22 | ralbidv 2986 |
. . . . . . . . 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 747 |
. . . . . . . 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 | cbvrexv 3172 |
. . . . . . 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 264 |
. . . . . 6
⊢ (𝜓 ↔ ∃𝑔 ∈ ℝ (0 < 𝑔 ∧ ∀𝑢 ∈ (𝐴 +ℋ 𝐵)(normℎ‘(𝑇‘𝑢)) ≤ (𝑔 ·
(normℎ‘𝑢)))) |
27 | 19, 26 | anbi12i 733 |
. . . . 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 3107 |
. . . . 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 267 |
. . . 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 865 |
. . . . . 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 10514 |
. . . . . . . . 9
⊢ (((𝑓 ∈ ℝ ∧ 𝑔 ∈ ℝ) ∧ (0 <
𝑓 ∧ 0 < 𝑔)) → 0 < (𝑓 + 𝑔)) |
32 | 31 | ex 450 |
. . . . . . . 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 28223 |
. . . . . . . . . . 11
⊢ ((𝑡 ∈ 𝐴 ∧ ℎ ∈ 𝐵) → (𝑡 +ℎ ℎ) ∈ (𝐴 +ℋ 𝐵)) |
35 | | fveq2 6191 |
. . . . . . . . . . . . . . 15
⊢ (𝑢 = (𝑡 +ℎ ℎ) → (𝑆‘𝑢) = (𝑆‘(𝑡 +ℎ ℎ))) |
36 | 35 | fveq2d 6195 |
. . . . . . . . . . . . . 14
⊢ (𝑢 = (𝑡 +ℎ ℎ) → (normℎ‘(𝑆‘𝑢)) = (normℎ‘(𝑆‘(𝑡 +ℎ ℎ)))) |
37 | | fveq2 6191 |
. . . . . . . . . . . . . . 15
⊢ (𝑢 = (𝑡 +ℎ ℎ) → (normℎ‘𝑢) =
(normℎ‘(𝑡 +ℎ ℎ))) |
38 | 37 | oveq2d 6666 |
. . . . . . . . . . . . . 14
⊢ (𝑢 = (𝑡 +ℎ ℎ) → (𝑓 ·
(normℎ‘𝑢)) = (𝑓 ·
(normℎ‘(𝑡 +ℎ ℎ)))) |
39 | 36, 38 | breq12d 4666 |
. . . . . . . . . . . . 13
⊢ (𝑢 = (𝑡 +ℎ ℎ) →
((normℎ‘(𝑆‘𝑢)) ≤ (𝑓 ·
(normℎ‘𝑢)) ↔
(normℎ‘(𝑆‘(𝑡 +ℎ ℎ))) ≤ (𝑓 ·
(normℎ‘(𝑡 +ℎ ℎ))))) |
40 | 39 | rspcv 3305 |
. . . . . . . . . . . 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))))) |
41 | | fveq2 6191 |
. . . . . . . . . . . . . . 15
⊢ (𝑢 = (𝑡 +ℎ ℎ) → (𝑇‘𝑢) = (𝑇‘(𝑡 +ℎ ℎ))) |
42 | 41 | fveq2d 6195 |
. . . . . . . . . . . . . 14
⊢ (𝑢 = (𝑡 +ℎ ℎ) → (normℎ‘(𝑇‘𝑢)) = (normℎ‘(𝑇‘(𝑡 +ℎ ℎ)))) |
43 | 37 | oveq2d 6666 |
. . . . . . . . . . . . . 14
⊢ (𝑢 = (𝑡 +ℎ ℎ) → (𝑔 ·
(normℎ‘𝑢)) = (𝑔 ·
(normℎ‘(𝑡 +ℎ ℎ)))) |
44 | 42, 43 | breq12d 4666 |
. . . . . . . . . . . . 13
⊢ (𝑢 = (𝑡 +ℎ ℎ) →
((normℎ‘(𝑇‘𝑢)) ≤ (𝑔 ·
(normℎ‘𝑢)) ↔
(normℎ‘(𝑇‘(𝑡 +ℎ ℎ))) ≤ (𝑔 ·
(normℎ‘(𝑡 +ℎ ℎ))))) |
45 | 44 | rspcv 3305 |
. . . . . . . . . . . 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))))) |
46 | 40, 45 | anim12d 586 |
. . . . . . . . . . 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)))))) |
47 | 34, 46 | 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)))))) |
48 | 47 | 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)))))) |
49 | 1 | sheli 28071 |
. . . . . . . . . . . . . . 15
⊢ (𝑡 ∈ 𝐴 → 𝑡 ∈ ℋ) |
50 | | normcl 27982 |
. . . . . . . . . . . . . . 15
⊢ (𝑡 ∈ ℋ →
(normℎ‘𝑡) ∈ ℝ) |
51 | 49, 50 | syl 17 |
. . . . . . . . . . . . . 14
⊢ (𝑡 ∈ 𝐴 → (normℎ‘𝑡) ∈
ℝ) |
52 | 2 | sheli 28071 |
. . . . . . . . . . . . . . 15
⊢ (ℎ ∈ 𝐵 → ℎ ∈ ℋ) |
53 | | normcl 27982 |
. . . . . . . . . . . . . . 15
⊢ (ℎ ∈ ℋ →
(normℎ‘ℎ) ∈ ℝ) |
54 | 52, 53 | syl 17 |
. . . . . . . . . . . . . 14
⊢ (ℎ ∈ 𝐵 → (normℎ‘ℎ) ∈
ℝ) |
55 | 51, 54 | anim12i 590 |
. . . . . . . . . . . . 13
⊢ ((𝑡 ∈ 𝐴 ∧ ℎ ∈ 𝐵) →
((normℎ‘𝑡) ∈ ℝ ∧
(normℎ‘ℎ) ∈ ℝ)) |
56 | 55 | adantl 482 |
. . . . . . . . . . . 12
⊢ (((𝑓 ∈ ℝ ∧ 𝑔 ∈ ℝ) ∧ (𝑡 ∈ 𝐴 ∧ ℎ ∈ 𝐵)) →
((normℎ‘𝑡) ∈ ℝ ∧
(normℎ‘ℎ) ∈ ℝ)) |
57 | | hvaddcl 27869 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑡 ∈ ℋ ∧ ℎ ∈ ℋ) → (𝑡 +ℎ ℎ) ∈
ℋ) |
58 | 49, 52, 57 | syl2an 494 |
. . . . . . . . . . . . . . 15
⊢ ((𝑡 ∈ 𝐴 ∧ ℎ ∈ 𝐵) → (𝑡 +ℎ ℎ) ∈ ℋ) |
59 | | normcl 27982 |
. . . . . . . . . . . . . . 15
⊢ ((𝑡 +ℎ ℎ) ∈ ℋ →
(normℎ‘(𝑡 +ℎ ℎ)) ∈ ℝ) |
60 | 58, 59 | syl 17 |
. . . . . . . . . . . . . 14
⊢ ((𝑡 ∈ 𝐴 ∧ ℎ ∈ 𝐵) →
(normℎ‘(𝑡 +ℎ ℎ)) ∈ ℝ) |
61 | | remulcl 10021 |
. . . . . . . . . . . . . 14
⊢ ((𝑓 ∈ ℝ ∧
(normℎ‘(𝑡 +ℎ ℎ)) ∈ ℝ) → (𝑓 ·
(normℎ‘(𝑡 +ℎ ℎ))) ∈ ℝ) |
62 | 60, 61 | sylan2 491 |
. . . . . . . . . . . . 13
⊢ ((𝑓 ∈ ℝ ∧ (𝑡 ∈ 𝐴 ∧ ℎ ∈ 𝐵)) → (𝑓 ·
(normℎ‘(𝑡 +ℎ ℎ))) ∈ ℝ) |
63 | 62 | adantlr 751 |
. . . . . . . . . . . 12
⊢ (((𝑓 ∈ ℝ ∧ 𝑔 ∈ ℝ) ∧ (𝑡 ∈ 𝐴 ∧ ℎ ∈ 𝐵)) → (𝑓 ·
(normℎ‘(𝑡 +ℎ ℎ))) ∈ ℝ) |
64 | | remulcl 10021 |
. . . . . . . . . . . . . 14
⊢ ((𝑔 ∈ ℝ ∧
(normℎ‘(𝑡 +ℎ ℎ)) ∈ ℝ) → (𝑔 ·
(normℎ‘(𝑡 +ℎ ℎ))) ∈ ℝ) |
65 | 60, 64 | sylan2 491 |
. . . . . . . . . . . . 13
⊢ ((𝑔 ∈ ℝ ∧ (𝑡 ∈ 𝐴 ∧ ℎ ∈ 𝐵)) → (𝑔 ·
(normℎ‘(𝑡 +ℎ ℎ))) ∈ ℝ) |
66 | 65 | adantll 750 |
. . . . . . . . . . . 12
⊢ (((𝑓 ∈ ℝ ∧ 𝑔 ∈ ℝ) ∧ (𝑡 ∈ 𝐴 ∧ ℎ ∈ 𝐵)) → (𝑔 ·
(normℎ‘(𝑡 +ℎ ℎ))) ∈ ℝ) |
67 | | le2add 10510 |
. . . . . . . . . . . 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)))))) |
68 | 56, 63, 66, 67 | syl12anc 1324 |
. . . . . . . . . . 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)))))) |
69 | 68 | adantll 750 |
. . . . . . . . . 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)))))) |
70 | 1, 2, 4 | cdj3lem2 29294 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑡 ∈ 𝐴 ∧ ℎ ∈ 𝐵 ∧ (𝐴 ∩ 𝐵) = 0ℋ) → (𝑆‘(𝑡 +ℎ ℎ)) = 𝑡) |
71 | 70 | fveq2d 6195 |
. . . . . . . . . . . . . . 15
⊢ ((𝑡 ∈ 𝐴 ∧ ℎ ∈ 𝐵 ∧ (𝐴 ∩ 𝐵) = 0ℋ) →
(normℎ‘(𝑆‘(𝑡 +ℎ ℎ))) = (normℎ‘𝑡)) |
72 | 71 | breq1d 4663 |
. . . . . . . . . . . . . 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))))) |
73 | 1, 2, 8 | cdj3lem3 29297 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑡 ∈ 𝐴 ∧ ℎ ∈ 𝐵 ∧ (𝐴 ∩ 𝐵) = 0ℋ) → (𝑇‘(𝑡 +ℎ ℎ)) = ℎ) |
74 | 73 | fveq2d 6195 |
. . . . . . . . . . . . . . 15
⊢ ((𝑡 ∈ 𝐴 ∧ ℎ ∈ 𝐵 ∧ (𝐴 ∩ 𝐵) = 0ℋ) →
(normℎ‘(𝑇‘(𝑡 +ℎ ℎ))) = (normℎ‘ℎ)) |
75 | 74 | breq1d 4663 |
. . . . . . . . . . . . . 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))))) |
76 | 72, 75 | anbi12d 747 |
. . . . . . . . . . . . 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)))))) |
77 | 76 | 3expa 1265 |
. . . . . . . . . . . 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)))))) |
78 | 77 | ancoms 469 |
. . . . . . . . . . 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)))))) |
79 | 78 | adantlr 751 |
. . . . . . . . . 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)))))) |
80 | | recn 10026 |
. . . . . . . . . . . . . 14
⊢ (𝑓 ∈ ℝ → 𝑓 ∈
ℂ) |
81 | | recn 10026 |
. . . . . . . . . . . . . 14
⊢ (𝑔 ∈ ℝ → 𝑔 ∈
ℂ) |
82 | 60 | recnd 10068 |
. . . . . . . . . . . . . 14
⊢ ((𝑡 ∈ 𝐴 ∧ ℎ ∈ 𝐵) →
(normℎ‘(𝑡 +ℎ ℎ)) ∈ ℂ) |
83 | | adddir 10031 |
. . . . . . . . . . . . . 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))))) |
84 | 80, 81, 82, 83 | syl3an 1368 |
. . . . . . . . . . . . 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))))) |
85 | 84 | 3expa 1265 |
. . . . . . . . . . . 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))))) |
86 | 85 | breq2d 4665 |
. . . . . . . . . . 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)))))) |
87 | 86 | adantll 750 |
. . . . . . . . . 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)))))) |
88 | 69, 79, 87 | 3imtr4d 283 |
. . . . . . . . 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))))) |
89 | 48, 88 | 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))))) |
90 | 89 | ralrimdvva 2974 |
. . . . . . 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))))) |
91 | | readdcl 10019 |
. . . . . . . . 9
⊢ ((𝑓 ∈ ℝ ∧ 𝑔 ∈ ℝ) → (𝑓 + 𝑔) ∈ ℝ) |
92 | | breq2 4657 |
. . . . . . . . . . . 12
⊢ (𝑣 = (𝑓 + 𝑔) → (0 < 𝑣 ↔ 0 < (𝑓 + 𝑔))) |
93 | | fveq2 6191 |
. . . . . . . . . . . . . . . 16
⊢ (𝑥 = 𝑡 → (normℎ‘𝑥) =
(normℎ‘𝑡)) |
94 | 93 | oveq1d 6665 |
. . . . . . . . . . . . . . 15
⊢ (𝑥 = 𝑡 → ((normℎ‘𝑥) +
(normℎ‘𝑦)) = ((normℎ‘𝑡) +
(normℎ‘𝑦))) |
95 | | oveq1 6657 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑥 = 𝑡 → (𝑥 +ℎ 𝑦) = (𝑡 +ℎ 𝑦)) |
96 | 95 | fveq2d 6195 |
. . . . . . . . . . . . . . . 16
⊢ (𝑥 = 𝑡 → (normℎ‘(𝑥 +ℎ 𝑦)) =
(normℎ‘(𝑡 +ℎ 𝑦))) |
97 | 96 | oveq2d 6666 |
. . . . . . . . . . . . . . 15
⊢ (𝑥 = 𝑡 → (𝑣 ·
(normℎ‘(𝑥 +ℎ 𝑦))) = (𝑣 ·
(normℎ‘(𝑡 +ℎ 𝑦)))) |
98 | 94, 97 | breq12d 4666 |
. . . . . . . . . . . . . 14
⊢ (𝑥 = 𝑡 →
(((normℎ‘𝑥) + (normℎ‘𝑦)) ≤ (𝑣 ·
(normℎ‘(𝑥 +ℎ 𝑦))) ↔
((normℎ‘𝑡) + (normℎ‘𝑦)) ≤ (𝑣 ·
(normℎ‘(𝑡 +ℎ 𝑦))))) |
99 | | fveq2 6191 |
. . . . . . . . . . . . . . . 16
⊢ (𝑦 = ℎ → (normℎ‘𝑦) =
(normℎ‘ℎ)) |
100 | 99 | oveq2d 6666 |
. . . . . . . . . . . . . . 15
⊢ (𝑦 = ℎ → ((normℎ‘𝑡) +
(normℎ‘𝑦)) = ((normℎ‘𝑡) +
(normℎ‘ℎ))) |
101 | | oveq2 6658 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑦 = ℎ → (𝑡 +ℎ 𝑦) = (𝑡 +ℎ ℎ)) |
102 | 101 | fveq2d 6195 |
. . . . . . . . . . . . . . . 16
⊢ (𝑦 = ℎ → (normℎ‘(𝑡 +ℎ 𝑦)) =
(normℎ‘(𝑡 +ℎ ℎ))) |
103 | 102 | oveq2d 6666 |
. . . . . . . . . . . . . . 15
⊢ (𝑦 = ℎ → (𝑣 ·
(normℎ‘(𝑡 +ℎ 𝑦))) = (𝑣 ·
(normℎ‘(𝑡 +ℎ ℎ)))) |
104 | 100, 103 | breq12d 4666 |
. . . . . . . . . . . . . 14
⊢ (𝑦 = ℎ → (((normℎ‘𝑡) +
(normℎ‘𝑦)) ≤ (𝑣 ·
(normℎ‘(𝑡 +ℎ 𝑦))) ↔
((normℎ‘𝑡) + (normℎ‘ℎ)) ≤ (𝑣 ·
(normℎ‘(𝑡 +ℎ ℎ))))) |
105 | 98, 104 | cbvral2v 3179 |
. . . . . . . . . . . . 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)))) |
106 | | oveq1 6657 |
. . . . . . . . . . . . . . 15
⊢ (𝑣 = (𝑓 + 𝑔) → (𝑣 ·
(normℎ‘(𝑡 +ℎ ℎ))) = ((𝑓 + 𝑔) ·
(normℎ‘(𝑡 +ℎ ℎ)))) |
107 | 106 | breq2d 4665 |
. . . . . . . . . . . . . 14
⊢ (𝑣 = (𝑓 + 𝑔) →
(((normℎ‘𝑡) + (normℎ‘ℎ)) ≤ (𝑣 ·
(normℎ‘(𝑡 +ℎ ℎ))) ↔
((normℎ‘𝑡) + (normℎ‘ℎ)) ≤ ((𝑓 + 𝑔) ·
(normℎ‘(𝑡 +ℎ ℎ))))) |
108 | 107 | 2ralbidv 2989 |
. . . . . . . . . . . . 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))))) |
109 | 105, 108 | syl5bb 272 |
. . . . . . . . . . . 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))))) |
110 | 92, 109 | anbi12d 747 |
. . . . . . . . . . 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)))))) |
111 | 110 | rspcev 3309 |
. . . . . . . . . 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))))) |
112 | 111 | ex 450 |
. . . . . . . . 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)))))) |
113 | 91, 112 | 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)))))) |
114 | 113 | 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)))))) |
115 | 33, 90, 114 | syl2and 500 |
. . . . . 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)))))) |
116 | 30, 115 | syl5bi 232 |
. . . . 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)))))) |
117 | 116 | rexlimdvva 3038 |
. . . 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)))))) |
118 | 29, 117 | syl5bi 232 |
. . 3
⊢ ((𝐴 ∩ 𝐵) = 0ℋ → ((𝜑 ∧ 𝜓) → ∃𝑣 ∈ ℝ (0 < 𝑣 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ((normℎ‘𝑥) +
(normℎ‘𝑦)) ≤ (𝑣 ·
(normℎ‘(𝑥 +ℎ 𝑦)))))) |
119 | 118 | 3impib 1262 |
. 2
⊢ (((𝐴 ∩ 𝐵) = 0ℋ ∧ 𝜑 ∧ 𝜓) → ∃𝑣 ∈ ℝ (0 < 𝑣 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ((normℎ‘𝑥) +
(normℎ‘𝑦)) ≤ (𝑣 ·
(normℎ‘(𝑥 +ℎ 𝑦))))) |
120 | 12, 119 | impbii 199 |
1
⊢
(∃𝑣 ∈
ℝ (0 < 𝑣 ∧
∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ((normℎ‘𝑥) +
(normℎ‘𝑦)) ≤ (𝑣 ·
(normℎ‘(𝑥 +ℎ 𝑦)))) ↔ ((𝐴 ∩ 𝐵) = 0ℋ ∧ 𝜑 ∧ 𝜓)) |