Step | Hyp | Ref
| Expression |
1 | | cdj3lem2.1 |
. . 3
⊢ 𝐴 ∈
Sℋ |
2 | | cdj3lem2.2 |
. . 3
⊢ 𝐵 ∈
Sℋ |
3 | 1, 2 | cdj3lem1 29293 |
. 2
⊢
(∃𝑣 ∈
ℝ (0 < 𝑣 ∧
∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ((normℎ‘𝑥) +
(normℎ‘𝑦)) ≤ (𝑣 ·
(normℎ‘(𝑥 +ℎ 𝑦)))) → (𝐴 ∩ 𝐵) = 0ℋ) |
4 | 1, 2 | shseli 28175 |
. . . . . . . 8
⊢ (𝑢 ∈ (𝐴 +ℋ 𝐵) ↔ ∃𝑡 ∈ 𝐴 ∃ℎ ∈ 𝐵 𝑢 = (𝑡 +ℎ ℎ)) |
5 | 4 | biimpi 206 |
. . . . . . 7
⊢ (𝑢 ∈ (𝐴 +ℋ 𝐵) → ∃𝑡 ∈ 𝐴 ∃ℎ ∈ 𝐵 𝑢 = (𝑡 +ℎ ℎ)) |
6 | | fveq2 6191 |
. . . . . . . . . . . . . 14
⊢ (𝑥 = 𝑡 → (normℎ‘𝑥) =
(normℎ‘𝑡)) |
7 | 6 | oveq1d 6665 |
. . . . . . . . . . . . 13
⊢ (𝑥 = 𝑡 → ((normℎ‘𝑥) +
(normℎ‘𝑦)) = ((normℎ‘𝑡) +
(normℎ‘𝑦))) |
8 | | oveq1 6657 |
. . . . . . . . . . . . . . 15
⊢ (𝑥 = 𝑡 → (𝑥 +ℎ 𝑦) = (𝑡 +ℎ 𝑦)) |
9 | 8 | fveq2d 6195 |
. . . . . . . . . . . . . 14
⊢ (𝑥 = 𝑡 → (normℎ‘(𝑥 +ℎ 𝑦)) =
(normℎ‘(𝑡 +ℎ 𝑦))) |
10 | 9 | oveq2d 6666 |
. . . . . . . . . . . . 13
⊢ (𝑥 = 𝑡 → (𝑣 ·
(normℎ‘(𝑥 +ℎ 𝑦))) = (𝑣 ·
(normℎ‘(𝑡 +ℎ 𝑦)))) |
11 | 7, 10 | breq12d 4666 |
. . . . . . . . . . . 12
⊢ (𝑥 = 𝑡 →
(((normℎ‘𝑥) + (normℎ‘𝑦)) ≤ (𝑣 ·
(normℎ‘(𝑥 +ℎ 𝑦))) ↔
((normℎ‘𝑡) + (normℎ‘𝑦)) ≤ (𝑣 ·
(normℎ‘(𝑡 +ℎ 𝑦))))) |
12 | | fveq2 6191 |
. . . . . . . . . . . . . 14
⊢ (𝑦 = ℎ → (normℎ‘𝑦) =
(normℎ‘ℎ)) |
13 | 12 | oveq2d 6666 |
. . . . . . . . . . . . 13
⊢ (𝑦 = ℎ → ((normℎ‘𝑡) +
(normℎ‘𝑦)) = ((normℎ‘𝑡) +
(normℎ‘ℎ))) |
14 | | oveq2 6658 |
. . . . . . . . . . . . . . 15
⊢ (𝑦 = ℎ → (𝑡 +ℎ 𝑦) = (𝑡 +ℎ ℎ)) |
15 | 14 | fveq2d 6195 |
. . . . . . . . . . . . . 14
⊢ (𝑦 = ℎ → (normℎ‘(𝑡 +ℎ 𝑦)) =
(normℎ‘(𝑡 +ℎ ℎ))) |
16 | 15 | oveq2d 6666 |
. . . . . . . . . . . . 13
⊢ (𝑦 = ℎ → (𝑣 ·
(normℎ‘(𝑡 +ℎ 𝑦))) = (𝑣 ·
(normℎ‘(𝑡 +ℎ ℎ)))) |
17 | 13, 16 | breq12d 4666 |
. . . . . . . . . . . 12
⊢ (𝑦 = ℎ → (((normℎ‘𝑡) +
(normℎ‘𝑦)) ≤ (𝑣 ·
(normℎ‘(𝑡 +ℎ 𝑦))) ↔
((normℎ‘𝑡) + (normℎ‘ℎ)) ≤ (𝑣 ·
(normℎ‘(𝑡 +ℎ ℎ))))) |
18 | 11, 17 | rspc2v 3322 |
. . . . . . . . . . 11
⊢ ((𝑡 ∈ 𝐴 ∧ ℎ ∈ 𝐵) → (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ((normℎ‘𝑥) +
(normℎ‘𝑦)) ≤ (𝑣 ·
(normℎ‘(𝑥 +ℎ 𝑦))) →
((normℎ‘𝑡) + (normℎ‘ℎ)) ≤ (𝑣 ·
(normℎ‘(𝑡 +ℎ ℎ))))) |
19 | | cdj3lem2.3 |
. . . . . . . . . . . . . . . . . 18
⊢ 𝑆 = (𝑥 ∈ (𝐴 +ℋ 𝐵) ↦ (℩𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝐵 𝑥 = (𝑧 +ℎ 𝑤))) |
20 | 1, 2, 19 | cdj3lem2 29294 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝑡 ∈ 𝐴 ∧ ℎ ∈ 𝐵 ∧ (𝐴 ∩ 𝐵) = 0ℋ) → (𝑆‘(𝑡 +ℎ ℎ)) = 𝑡) |
21 | 20 | 3expa 1265 |
. . . . . . . . . . . . . . . 16
⊢ (((𝑡 ∈ 𝐴 ∧ ℎ ∈ 𝐵) ∧ (𝐴 ∩ 𝐵) = 0ℋ) → (𝑆‘(𝑡 +ℎ ℎ)) = 𝑡) |
22 | 21 | fveq2d 6195 |
. . . . . . . . . . . . . . 15
⊢ (((𝑡 ∈ 𝐴 ∧ ℎ ∈ 𝐵) ∧ (𝐴 ∩ 𝐵) = 0ℋ) →
(normℎ‘(𝑆‘(𝑡 +ℎ ℎ))) = (normℎ‘𝑡)) |
23 | 22 | ad2ant2r 783 |
. . . . . . . . . . . . . 14
⊢ ((((𝑡 ∈ 𝐴 ∧ ℎ ∈ 𝐵) ∧
((normℎ‘𝑡) + (normℎ‘ℎ)) ≤ (𝑣 ·
(normℎ‘(𝑡 +ℎ ℎ)))) ∧ ((𝐴 ∩ 𝐵) = 0ℋ ∧ 𝑣 ∈ ℝ)) →
(normℎ‘(𝑆‘(𝑡 +ℎ ℎ))) = (normℎ‘𝑡)) |
24 | 2 | sheli 28071 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (ℎ ∈ 𝐵 → ℎ ∈ ℋ) |
25 | | normge0 27983 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (ℎ ∈ ℋ → 0 ≤
(normℎ‘ℎ)) |
26 | 24, 25 | syl 17 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (ℎ ∈ 𝐵 → 0 ≤
(normℎ‘ℎ)) |
27 | 26 | adantl 482 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝑡 ∈ 𝐴 ∧ ℎ ∈ 𝐵) → 0 ≤
(normℎ‘ℎ)) |
28 | 1 | sheli 28071 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑡 ∈ 𝐴 → 𝑡 ∈ ℋ) |
29 | | normcl 27982 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑡 ∈ ℋ →
(normℎ‘𝑡) ∈ ℝ) |
30 | 28, 29 | syl 17 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑡 ∈ 𝐴 → (normℎ‘𝑡) ∈
ℝ) |
31 | | normcl 27982 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (ℎ ∈ ℋ →
(normℎ‘ℎ) ∈ ℝ) |
32 | 24, 31 | syl 17 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (ℎ ∈ 𝐵 → (normℎ‘ℎ) ∈
ℝ) |
33 | | addge01 10538 |
. . . . . . . . . . . . . . . . . . . . 21
⊢
(((normℎ‘𝑡) ∈ ℝ ∧
(normℎ‘ℎ) ∈ ℝ) → (0 ≤
(normℎ‘ℎ) ↔ (normℎ‘𝑡) ≤
((normℎ‘𝑡) + (normℎ‘ℎ)))) |
34 | 30, 32, 33 | syl2an 494 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝑡 ∈ 𝐴 ∧ ℎ ∈ 𝐵) → (0 ≤
(normℎ‘ℎ) ↔ (normℎ‘𝑡) ≤
((normℎ‘𝑡) + (normℎ‘ℎ)))) |
35 | 27, 34 | mpbid 222 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝑡 ∈ 𝐴 ∧ ℎ ∈ 𝐵) →
(normℎ‘𝑡) ≤ ((normℎ‘𝑡) +
(normℎ‘ℎ))) |
36 | 35 | adantr 481 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝑡 ∈ 𝐴 ∧ ℎ ∈ 𝐵) ∧ 𝑣 ∈ ℝ) →
(normℎ‘𝑡) ≤ ((normℎ‘𝑡) +
(normℎ‘ℎ))) |
37 | 30 | ad2antrr 762 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝑡 ∈ 𝐴 ∧ ℎ ∈ 𝐵) ∧ 𝑣 ∈ ℝ) →
(normℎ‘𝑡) ∈ ℝ) |
38 | | readdcl 10019 |
. . . . . . . . . . . . . . . . . . . . 21
⊢
(((normℎ‘𝑡) ∈ ℝ ∧
(normℎ‘ℎ) ∈ ℝ) →
((normℎ‘𝑡) + (normℎ‘ℎ)) ∈
ℝ) |
39 | 30, 32, 38 | syl2an 494 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝑡 ∈ 𝐴 ∧ ℎ ∈ 𝐵) →
((normℎ‘𝑡) + (normℎ‘ℎ)) ∈
ℝ) |
40 | 39 | adantr 481 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝑡 ∈ 𝐴 ∧ ℎ ∈ 𝐵) ∧ 𝑣 ∈ ℝ) →
((normℎ‘𝑡) + (normℎ‘ℎ)) ∈
ℝ) |
41 | | hvaddcl 27869 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝑡 ∈ ℋ ∧ ℎ ∈ ℋ) → (𝑡 +ℎ ℎ) ∈
ℋ) |
42 | 28, 24, 41 | syl2an 494 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝑡 ∈ 𝐴 ∧ ℎ ∈ 𝐵) → (𝑡 +ℎ ℎ) ∈ ℋ) |
43 | | normcl 27982 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝑡 +ℎ ℎ) ∈ ℋ →
(normℎ‘(𝑡 +ℎ ℎ)) ∈ ℝ) |
44 | 42, 43 | syl 17 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝑡 ∈ 𝐴 ∧ ℎ ∈ 𝐵) →
(normℎ‘(𝑡 +ℎ ℎ)) ∈ ℝ) |
45 | | remulcl 10021 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝑣 ∈ ℝ ∧
(normℎ‘(𝑡 +ℎ ℎ)) ∈ ℝ) → (𝑣 ·
(normℎ‘(𝑡 +ℎ ℎ))) ∈ ℝ) |
46 | 44, 45 | sylan2 491 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝑣 ∈ ℝ ∧ (𝑡 ∈ 𝐴 ∧ ℎ ∈ 𝐵)) → (𝑣 ·
(normℎ‘(𝑡 +ℎ ℎ))) ∈ ℝ) |
47 | 46 | ancoms 469 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝑡 ∈ 𝐴 ∧ ℎ ∈ 𝐵) ∧ 𝑣 ∈ ℝ) → (𝑣 ·
(normℎ‘(𝑡 +ℎ ℎ))) ∈ ℝ) |
48 | | letr 10131 |
. . . . . . . . . . . . . . . . . . 19
⊢
(((normℎ‘𝑡) ∈ ℝ ∧
((normℎ‘𝑡) + (normℎ‘ℎ)) ∈ ℝ ∧ (𝑣 ·
(normℎ‘(𝑡 +ℎ ℎ))) ∈ ℝ) →
(((normℎ‘𝑡) ≤ ((normℎ‘𝑡) +
(normℎ‘ℎ)) ∧ ((normℎ‘𝑡) +
(normℎ‘ℎ)) ≤ (𝑣 ·
(normℎ‘(𝑡 +ℎ ℎ)))) →
(normℎ‘𝑡) ≤ (𝑣 ·
(normℎ‘(𝑡 +ℎ ℎ))))) |
49 | 37, 40, 47, 48 | syl3anc 1326 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝑡 ∈ 𝐴 ∧ ℎ ∈ 𝐵) ∧ 𝑣 ∈ ℝ) →
(((normℎ‘𝑡) ≤ ((normℎ‘𝑡) +
(normℎ‘ℎ)) ∧ ((normℎ‘𝑡) +
(normℎ‘ℎ)) ≤ (𝑣 ·
(normℎ‘(𝑡 +ℎ ℎ)))) →
(normℎ‘𝑡) ≤ (𝑣 ·
(normℎ‘(𝑡 +ℎ ℎ))))) |
50 | 36, 49 | mpand 711 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝑡 ∈ 𝐴 ∧ ℎ ∈ 𝐵) ∧ 𝑣 ∈ ℝ) →
(((normℎ‘𝑡) + (normℎ‘ℎ)) ≤ (𝑣 ·
(normℎ‘(𝑡 +ℎ ℎ))) →
(normℎ‘𝑡) ≤ (𝑣 ·
(normℎ‘(𝑡 +ℎ ℎ))))) |
51 | 50 | imp 445 |
. . . . . . . . . . . . . . . 16
⊢ ((((𝑡 ∈ 𝐴 ∧ ℎ ∈ 𝐵) ∧ 𝑣 ∈ ℝ) ∧
((normℎ‘𝑡) + (normℎ‘ℎ)) ≤ (𝑣 ·
(normℎ‘(𝑡 +ℎ ℎ)))) →
(normℎ‘𝑡) ≤ (𝑣 ·
(normℎ‘(𝑡 +ℎ ℎ)))) |
52 | 51 | an32s 846 |
. . . . . . . . . . . . . . 15
⊢ ((((𝑡 ∈ 𝐴 ∧ ℎ ∈ 𝐵) ∧
((normℎ‘𝑡) + (normℎ‘ℎ)) ≤ (𝑣 ·
(normℎ‘(𝑡 +ℎ ℎ)))) ∧ 𝑣 ∈ ℝ) →
(normℎ‘𝑡) ≤ (𝑣 ·
(normℎ‘(𝑡 +ℎ ℎ)))) |
53 | 52 | adantrl 752 |
. . . . . . . . . . . . . 14
⊢ ((((𝑡 ∈ 𝐴 ∧ ℎ ∈ 𝐵) ∧
((normℎ‘𝑡) + (normℎ‘ℎ)) ≤ (𝑣 ·
(normℎ‘(𝑡 +ℎ ℎ)))) ∧ ((𝐴 ∩ 𝐵) = 0ℋ ∧ 𝑣 ∈ ℝ)) →
(normℎ‘𝑡) ≤ (𝑣 ·
(normℎ‘(𝑡 +ℎ ℎ)))) |
54 | 23, 53 | eqbrtrd 4675 |
. . . . . . . . . . . . 13
⊢ ((((𝑡 ∈ 𝐴 ∧ ℎ ∈ 𝐵) ∧
((normℎ‘𝑡) + (normℎ‘ℎ)) ≤ (𝑣 ·
(normℎ‘(𝑡 +ℎ ℎ)))) ∧ ((𝐴 ∩ 𝐵) = 0ℋ ∧ 𝑣 ∈ ℝ)) →
(normℎ‘(𝑆‘(𝑡 +ℎ ℎ))) ≤ (𝑣 ·
(normℎ‘(𝑡 +ℎ ℎ)))) |
55 | | fveq2 6191 |
. . . . . . . . . . . . . . 15
⊢ (𝑢 = (𝑡 +ℎ ℎ) → (𝑆‘𝑢) = (𝑆‘(𝑡 +ℎ ℎ))) |
56 | 55 | fveq2d 6195 |
. . . . . . . . . . . . . 14
⊢ (𝑢 = (𝑡 +ℎ ℎ) → (normℎ‘(𝑆‘𝑢)) = (normℎ‘(𝑆‘(𝑡 +ℎ ℎ)))) |
57 | | fveq2 6191 |
. . . . . . . . . . . . . . 15
⊢ (𝑢 = (𝑡 +ℎ ℎ) → (normℎ‘𝑢) =
(normℎ‘(𝑡 +ℎ ℎ))) |
58 | 57 | oveq2d 6666 |
. . . . . . . . . . . . . 14
⊢ (𝑢 = (𝑡 +ℎ ℎ) → (𝑣 ·
(normℎ‘𝑢)) = (𝑣 ·
(normℎ‘(𝑡 +ℎ ℎ)))) |
59 | 56, 58 | breq12d 4666 |
. . . . . . . . . . . . 13
⊢ (𝑢 = (𝑡 +ℎ ℎ) →
((normℎ‘(𝑆‘𝑢)) ≤ (𝑣 ·
(normℎ‘𝑢)) ↔
(normℎ‘(𝑆‘(𝑡 +ℎ ℎ))) ≤ (𝑣 ·
(normℎ‘(𝑡 +ℎ ℎ))))) |
60 | 54, 59 | syl5ibrcom 237 |
. . . . . . . . . . . 12
⊢ ((((𝑡 ∈ 𝐴 ∧ ℎ ∈ 𝐵) ∧
((normℎ‘𝑡) + (normℎ‘ℎ)) ≤ (𝑣 ·
(normℎ‘(𝑡 +ℎ ℎ)))) ∧ ((𝐴 ∩ 𝐵) = 0ℋ ∧ 𝑣 ∈ ℝ)) → (𝑢 = (𝑡 +ℎ ℎ) → (normℎ‘(𝑆‘𝑢)) ≤ (𝑣 ·
(normℎ‘𝑢)))) |
61 | 60 | exp31 630 |
. . . . . . . . . . 11
⊢ ((𝑡 ∈ 𝐴 ∧ ℎ ∈ 𝐵) →
(((normℎ‘𝑡) + (normℎ‘ℎ)) ≤ (𝑣 ·
(normℎ‘(𝑡 +ℎ ℎ))) → (((𝐴 ∩ 𝐵) = 0ℋ ∧ 𝑣 ∈ ℝ) → (𝑢 = (𝑡 +ℎ ℎ) → (normℎ‘(𝑆‘𝑢)) ≤ (𝑣 ·
(normℎ‘𝑢)))))) |
62 | 18, 61 | syld 47 |
. . . . . . . . . 10
⊢ ((𝑡 ∈ 𝐴 ∧ ℎ ∈ 𝐵) → (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ((normℎ‘𝑥) +
(normℎ‘𝑦)) ≤ (𝑣 ·
(normℎ‘(𝑥 +ℎ 𝑦))) → (((𝐴 ∩ 𝐵) = 0ℋ ∧ 𝑣 ∈ ℝ) → (𝑢 = (𝑡 +ℎ ℎ) → (normℎ‘(𝑆‘𝑢)) ≤ (𝑣 ·
(normℎ‘𝑢)))))) |
63 | 62 | com14 96 |
. . . . . . . . 9
⊢ (𝑢 = (𝑡 +ℎ ℎ) → (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ((normℎ‘𝑥) +
(normℎ‘𝑦)) ≤ (𝑣 ·
(normℎ‘(𝑥 +ℎ 𝑦))) → (((𝐴 ∩ 𝐵) = 0ℋ ∧ 𝑣 ∈ ℝ) → ((𝑡 ∈ 𝐴 ∧ ℎ ∈ 𝐵) →
(normℎ‘(𝑆‘𝑢)) ≤ (𝑣 ·
(normℎ‘𝑢)))))) |
64 | 63 | com4t 93 |
. . . . . . . 8
⊢ (((𝐴 ∩ 𝐵) = 0ℋ ∧ 𝑣 ∈ ℝ) → ((𝑡 ∈ 𝐴 ∧ ℎ ∈ 𝐵) → (𝑢 = (𝑡 +ℎ ℎ) → (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ((normℎ‘𝑥) +
(normℎ‘𝑦)) ≤ (𝑣 ·
(normℎ‘(𝑥 +ℎ 𝑦))) →
(normℎ‘(𝑆‘𝑢)) ≤ (𝑣 ·
(normℎ‘𝑢)))))) |
65 | 64 | rexlimdvv 3037 |
. . . . . . 7
⊢ (((𝐴 ∩ 𝐵) = 0ℋ ∧ 𝑣 ∈ ℝ) →
(∃𝑡 ∈ 𝐴 ∃ℎ ∈ 𝐵 𝑢 = (𝑡 +ℎ ℎ) → (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ((normℎ‘𝑥) +
(normℎ‘𝑦)) ≤ (𝑣 ·
(normℎ‘(𝑥 +ℎ 𝑦))) →
(normℎ‘(𝑆‘𝑢)) ≤ (𝑣 ·
(normℎ‘𝑢))))) |
66 | 5, 65 | syl5com 31 |
. . . . . 6
⊢ (𝑢 ∈ (𝐴 +ℋ 𝐵) → (((𝐴 ∩ 𝐵) = 0ℋ ∧ 𝑣 ∈ ℝ) →
(∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ((normℎ‘𝑥) +
(normℎ‘𝑦)) ≤ (𝑣 ·
(normℎ‘(𝑥 +ℎ 𝑦))) →
(normℎ‘(𝑆‘𝑢)) ≤ (𝑣 ·
(normℎ‘𝑢))))) |
67 | 66 | com3l 89 |
. . . . 5
⊢ (((𝐴 ∩ 𝐵) = 0ℋ ∧ 𝑣 ∈ ℝ) →
(∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ((normℎ‘𝑥) +
(normℎ‘𝑦)) ≤ (𝑣 ·
(normℎ‘(𝑥 +ℎ 𝑦))) → (𝑢 ∈ (𝐴 +ℋ 𝐵) →
(normℎ‘(𝑆‘𝑢)) ≤ (𝑣 ·
(normℎ‘𝑢))))) |
68 | 67 | ralrimdv 2968 |
. . . 4
⊢ (((𝐴 ∩ 𝐵) = 0ℋ ∧ 𝑣 ∈ ℝ) →
(∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ((normℎ‘𝑥) +
(normℎ‘𝑦)) ≤ (𝑣 ·
(normℎ‘(𝑥 +ℎ 𝑦))) → ∀𝑢 ∈ (𝐴 +ℋ 𝐵)(normℎ‘(𝑆‘𝑢)) ≤ (𝑣 ·
(normℎ‘𝑢)))) |
69 | 68 | anim2d 589 |
. . 3
⊢ (((𝐴 ∩ 𝐵) = 0ℋ ∧ 𝑣 ∈ ℝ) → ((0 <
𝑣 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ((normℎ‘𝑥) +
(normℎ‘𝑦)) ≤ (𝑣 ·
(normℎ‘(𝑥 +ℎ 𝑦)))) → (0 < 𝑣 ∧ ∀𝑢 ∈ (𝐴 +ℋ 𝐵)(normℎ‘(𝑆‘𝑢)) ≤ (𝑣 ·
(normℎ‘𝑢))))) |
70 | 69 | reximdva 3017 |
. 2
⊢ ((𝐴 ∩ 𝐵) = 0ℋ → (∃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))))) |
71 | 3, 70 | mpcom 38 |
1
⊢
(∃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)))) |