| Step | Hyp | Ref | Expression | 
|---|
| 1 |  | cdj3.1 | . . . 4
⊢ 𝐴 ∈
Sℋ | 
| 2 |  | cdj3.2 | . . . 4
⊢ 𝐵 ∈
Sℋ | 
| 3 | 1, 2 | cdj3lem1 32453 | . . 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 32456 | . . . 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 234 | . . 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 32459 | . . . 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 234 | . . 3
⊢
(∃𝑣 ∈
ℝ (0 < 𝑣 ∧
∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ((normℎ‘𝑥) +
(normℎ‘𝑦)) ≤ (𝑣 ·
(normℎ‘(𝑥 +ℎ 𝑦)))) → 𝜓) | 
| 12 | 3, 7, 11 | 3jca 1129 | . 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 5147 | . . . . . . . . 9
⊢ (𝑣 = 𝑓 → (0 < 𝑣 ↔ 0 < 𝑓)) | 
| 14 |  | oveq1 7438 | . . . . . . . . . . 11
⊢ (𝑣 = 𝑓 → (𝑣 ·
(normℎ‘𝑢)) = (𝑓 ·
(normℎ‘𝑢))) | 
| 15 | 14 | breq2d 5155 | . . . . . . . . . 10
⊢ (𝑣 = 𝑓 →
((normℎ‘(𝑆‘𝑢)) ≤ (𝑣 ·
(normℎ‘𝑢)) ↔
(normℎ‘(𝑆‘𝑢)) ≤ (𝑓 ·
(normℎ‘𝑢)))) | 
| 16 | 15 | ralbidv 3178 | . . . . . . . . 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 632 | . . . . . . . 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 3238 | . . . . . . 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 275 | . . . . . 6
⊢ (𝜑 ↔ ∃𝑓 ∈ ℝ (0 < 𝑓 ∧ ∀𝑢 ∈ (𝐴 +ℋ 𝐵)(normℎ‘(𝑆‘𝑢)) ≤ (𝑓 ·
(normℎ‘𝑢)))) | 
| 20 |  | breq2 5147 | . . . . . . . . 9
⊢ (𝑣 = 𝑔 → (0 < 𝑣 ↔ 0 < 𝑔)) | 
| 21 |  | oveq1 7438 | . . . . . . . . . . 11
⊢ (𝑣 = 𝑔 → (𝑣 ·
(normℎ‘𝑢)) = (𝑔 ·
(normℎ‘𝑢))) | 
| 22 | 21 | breq2d 5155 | . . . . . . . . . 10
⊢ (𝑣 = 𝑔 →
((normℎ‘(𝑇‘𝑢)) ≤ (𝑣 ·
(normℎ‘𝑢)) ↔
(normℎ‘(𝑇‘𝑢)) ≤ (𝑔 ·
(normℎ‘𝑢)))) | 
| 23 | 22 | ralbidv 3178 | . . . . . . . . 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 632 | . . . . . . . 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 3238 | . . . . . . 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 275 | . . . . . 6
⊢ (𝜓 ↔ ∃𝑔 ∈ ℝ (0 < 𝑔 ∧ ∀𝑢 ∈ (𝐴 +ℋ 𝐵)(normℎ‘(𝑇‘𝑢)) ≤ (𝑔 ·
(normℎ‘𝑢)))) | 
| 27 | 19, 26 | anbi12i 628 | . . . . 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 3229 | . . . . 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 278 | . . . 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 656 | . . . . . 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 11749 | . . . . . . . . 9
⊢ (((𝑓 ∈ ℝ ∧ 𝑔 ∈ ℝ) ∧ (0 <
𝑓 ∧ 0 < 𝑔)) → 0 < (𝑓 + 𝑔)) | 
| 32 | 31 | ex 412 | . . . . . . . 8
⊢ ((𝑓 ∈ ℝ ∧ 𝑔 ∈ ℝ) → ((0 <
𝑓 ∧ 0 < 𝑔) → 0 < (𝑓 + 𝑔))) | 
| 33 | 32 | adantl 481 | . . . . . . 7
⊢ (((𝐴 ∩ 𝐵) = 0ℋ ∧ (𝑓 ∈ ℝ ∧ 𝑔 ∈ ℝ)) → ((0
< 𝑓 ∧ 0 < 𝑔) → 0 < (𝑓 + 𝑔))) | 
| 34 | 1, 2 | shsvai 31383 | . . . . . . . . . . 11
⊢ ((𝑡 ∈ 𝐴 ∧ ℎ ∈ 𝐵) → (𝑡 +ℎ ℎ) ∈ (𝐴 +ℋ 𝐵)) | 
| 35 |  | 2fveq3 6911 | . . . . . . . . . . . . . 14
⊢ (𝑢 = (𝑡 +ℎ ℎ) → (normℎ‘(𝑆‘𝑢)) = (normℎ‘(𝑆‘(𝑡 +ℎ ℎ)))) | 
| 36 |  | fveq2 6906 | . . . . . . . . . . . . . . 15
⊢ (𝑢 = (𝑡 +ℎ ℎ) → (normℎ‘𝑢) =
(normℎ‘(𝑡 +ℎ ℎ))) | 
| 37 | 36 | oveq2d 7447 | . . . . . . . . . . . . . 14
⊢ (𝑢 = (𝑡 +ℎ ℎ) → (𝑓 ·
(normℎ‘𝑢)) = (𝑓 ·
(normℎ‘(𝑡 +ℎ ℎ)))) | 
| 38 | 35, 37 | breq12d 5156 | . . . . . . . . . . . . 13
⊢ (𝑢 = (𝑡 +ℎ ℎ) →
((normℎ‘(𝑆‘𝑢)) ≤ (𝑓 ·
(normℎ‘𝑢)) ↔
(normℎ‘(𝑆‘(𝑡 +ℎ ℎ))) ≤ (𝑓 ·
(normℎ‘(𝑡 +ℎ ℎ))))) | 
| 39 | 38 | rspcv 3618 | . . . . . . . . . . . 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 6911 | . . . . . . . . . . . . . 14
⊢ (𝑢 = (𝑡 +ℎ ℎ) → (normℎ‘(𝑇‘𝑢)) = (normℎ‘(𝑇‘(𝑡 +ℎ ℎ)))) | 
| 41 | 36 | oveq2d 7447 | . . . . . . . . . . . . . 14
⊢ (𝑢 = (𝑡 +ℎ ℎ) → (𝑔 ·
(normℎ‘𝑢)) = (𝑔 ·
(normℎ‘(𝑡 +ℎ ℎ)))) | 
| 42 | 40, 41 | breq12d 5156 | . . . . . . . . . . . . 13
⊢ (𝑢 = (𝑡 +ℎ ℎ) →
((normℎ‘(𝑇‘𝑢)) ≤ (𝑔 ·
(normℎ‘𝑢)) ↔
(normℎ‘(𝑇‘(𝑡 +ℎ ℎ))) ≤ (𝑔 ·
(normℎ‘(𝑡 +ℎ ℎ))))) | 
| 43 | 42 | rspcv 3618 | . . . . . . . . . . . 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 481 | . . . . . . . . 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 31233 | . . . . . . . . . . . . . . 15
⊢ (𝑡 ∈ 𝐴 → 𝑡 ∈ ℋ) | 
| 48 |  | normcl 31144 | . . . . . . . . . . . . . . 15
⊢ (𝑡 ∈ ℋ →
(normℎ‘𝑡) ∈ ℝ) | 
| 49 | 47, 48 | syl 17 | . . . . . . . . . . . . . 14
⊢ (𝑡 ∈ 𝐴 → (normℎ‘𝑡) ∈
ℝ) | 
| 50 | 2 | sheli 31233 | . . . . . . . . . . . . . . 15
⊢ (ℎ ∈ 𝐵 → ℎ ∈ ℋ) | 
| 51 |  | normcl 31144 | . . . . . . . . . . . . . . 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 481 | . . . . . . . . . . . 12
⊢ (((𝑓 ∈ ℝ ∧ 𝑔 ∈ ℝ) ∧ (𝑡 ∈ 𝐴 ∧ ℎ ∈ 𝐵)) →
((normℎ‘𝑡) ∈ ℝ ∧
(normℎ‘ℎ) ∈ ℝ)) | 
| 55 |  | hvaddcl 31031 | . . . . . . . . . . . . . . . 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 31144 | . . . . . . . . . . . . . . 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 11240 | . . . . . . . . . . . . . 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 715 | . . . . . . . . . . . 12
⊢ (((𝑓 ∈ ℝ ∧ 𝑔 ∈ ℝ) ∧ (𝑡 ∈ 𝐴 ∧ ℎ ∈ 𝐵)) → (𝑓 ·
(normℎ‘(𝑡 +ℎ ℎ))) ∈ ℝ) | 
| 62 |  | remulcl 11240 | . . . . . . . . . . . . . 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 714 | . . . . . . . . . . . 12
⊢ (((𝑓 ∈ ℝ ∧ 𝑔 ∈ ℝ) ∧ (𝑡 ∈ 𝐴 ∧ ℎ ∈ 𝐵)) → (𝑔 ·
(normℎ‘(𝑡 +ℎ ℎ))) ∈ ℝ) | 
| 65 |  | le2add 11745 | . . . . . . . . . . . 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 837 | . . . . . . . . . . 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 714 | . . . . . . . . . 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 32454 | . . . . . . . . . . . . . . . 16
⊢ ((𝑡 ∈ 𝐴 ∧ ℎ ∈ 𝐵 ∧ (𝐴 ∩ 𝐵) = 0ℋ) → (𝑆‘(𝑡 +ℎ ℎ)) = 𝑡) | 
| 69 | 68 | fveq2d 6910 | . . . . . . . . . . . . . . 15
⊢ ((𝑡 ∈ 𝐴 ∧ ℎ ∈ 𝐵 ∧ (𝐴 ∩ 𝐵) = 0ℋ) →
(normℎ‘(𝑆‘(𝑡 +ℎ ℎ))) = (normℎ‘𝑡)) | 
| 70 | 69 | breq1d 5153 | . . . . . . . . . . . . . 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 32457 | . . . . . . . . . . . . . . . 16
⊢ ((𝑡 ∈ 𝐴 ∧ ℎ ∈ 𝐵 ∧ (𝐴 ∩ 𝐵) = 0ℋ) → (𝑇‘(𝑡 +ℎ ℎ)) = ℎ) | 
| 72 | 71 | fveq2d 6910 | . . . . . . . . . . . . . . 15
⊢ ((𝑡 ∈ 𝐴 ∧ ℎ ∈ 𝐵 ∧ (𝐴 ∩ 𝐵) = 0ℋ) →
(normℎ‘(𝑇‘(𝑡 +ℎ ℎ))) = (normℎ‘ℎ)) | 
| 73 | 72 | breq1d 5153 | . . . . . . . . . . . . . 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 632 | . . . . . . . . . . . . 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 1119 | . . . . . . . . . . . 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 458 | . . . . . . . . . . 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 715 | . . . . . . . . . 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 11245 | . . . . . . . . . . . . . 14
⊢ (𝑓 ∈ ℝ → 𝑓 ∈
ℂ) | 
| 79 |  | recn 11245 | . . . . . . . . . . . . . 14
⊢ (𝑔 ∈ ℝ → 𝑔 ∈
ℂ) | 
| 80 | 58 | recnd 11289 | . . . . . . . . . . . . . 14
⊢ ((𝑡 ∈ 𝐴 ∧ ℎ ∈ 𝐵) →
(normℎ‘(𝑡 +ℎ ℎ)) ∈ ℂ) | 
| 81 |  | adddir 11252 | . . . . . . . . . . . . . 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 1161 | . . . . . . . . . . . . 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 1119 | . . . . . . . . . . . 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 5155 | . . . . . . . . . . 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 714 | . . . . . . . . . 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 3211 | . . . . . . 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 11238 | . . . . . . . . 9
⊢ ((𝑓 ∈ ℝ ∧ 𝑔 ∈ ℝ) → (𝑓 + 𝑔) ∈ ℝ) | 
| 90 |  | breq2 5147 | . . . . . . . . . . . 12
⊢ (𝑣 = (𝑓 + 𝑔) → (0 < 𝑣 ↔ 0 < (𝑓 + 𝑔))) | 
| 91 |  | fveq2 6906 | . . . . . . . . . . . . . . . 16
⊢ (𝑥 = 𝑡 → (normℎ‘𝑥) =
(normℎ‘𝑡)) | 
| 92 | 91 | oveq1d 7446 | . . . . . . . . . . . . . . 15
⊢ (𝑥 = 𝑡 → ((normℎ‘𝑥) +
(normℎ‘𝑦)) = ((normℎ‘𝑡) +
(normℎ‘𝑦))) | 
| 93 |  | fvoveq1 7454 | . . . . . . . . . . . . . . . 16
⊢ (𝑥 = 𝑡 → (normℎ‘(𝑥 +ℎ 𝑦)) =
(normℎ‘(𝑡 +ℎ 𝑦))) | 
| 94 | 93 | oveq2d 7447 | . . . . . . . . . . . . . . 15
⊢ (𝑥 = 𝑡 → (𝑣 ·
(normℎ‘(𝑥 +ℎ 𝑦))) = (𝑣 ·
(normℎ‘(𝑡 +ℎ 𝑦)))) | 
| 95 | 92, 94 | breq12d 5156 | . . . . . . . . . . . . . 14
⊢ (𝑥 = 𝑡 →
(((normℎ‘𝑥) + (normℎ‘𝑦)) ≤ (𝑣 ·
(normℎ‘(𝑥 +ℎ 𝑦))) ↔
((normℎ‘𝑡) + (normℎ‘𝑦)) ≤ (𝑣 ·
(normℎ‘(𝑡 +ℎ 𝑦))))) | 
| 96 |  | fveq2 6906 | . . . . . . . . . . . . . . . 16
⊢ (𝑦 = ℎ → (normℎ‘𝑦) =
(normℎ‘ℎ)) | 
| 97 | 96 | oveq2d 7447 | . . . . . . . . . . . . . . 15
⊢ (𝑦 = ℎ → ((normℎ‘𝑡) +
(normℎ‘𝑦)) = ((normℎ‘𝑡) +
(normℎ‘ℎ))) | 
| 98 |  | oveq2 7439 | . . . . . . . . . . . . . . . . 17
⊢ (𝑦 = ℎ → (𝑡 +ℎ 𝑦) = (𝑡 +ℎ ℎ)) | 
| 99 | 98 | fveq2d 6910 | . . . . . . . . . . . . . . . 16
⊢ (𝑦 = ℎ → (normℎ‘(𝑡 +ℎ 𝑦)) =
(normℎ‘(𝑡 +ℎ ℎ))) | 
| 100 | 99 | oveq2d 7447 | . . . . . . . . . . . . . . 15
⊢ (𝑦 = ℎ → (𝑣 ·
(normℎ‘(𝑡 +ℎ 𝑦))) = (𝑣 ·
(normℎ‘(𝑡 +ℎ ℎ)))) | 
| 101 | 97, 100 | breq12d 5156 | . . . . . . . . . . . . . 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 3241 | . . . . . . . . . . . . 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 7438 | . . . . . . . . . . . . . . 15
⊢ (𝑣 = (𝑓 + 𝑔) → (𝑣 ·
(normℎ‘(𝑡 +ℎ ℎ))) = ((𝑓 + 𝑔) ·
(normℎ‘(𝑡 +ℎ ℎ)))) | 
| 104 | 103 | breq2d 5155 | . . . . . . . . . . . . . 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 3221 | . . . . . . . . . . . . 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 | bitrid 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 632 | . . . . . . . . . . 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 3622 | . . . . . . . . . 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 412 | . . . . . . . . 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 481 | . . . . . . 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 | biimtrid 242 | . . . . 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 3213 | . . . 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 | biimtrid 242 | . . 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 1117 | . 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 209 | 1
⊢
(∃𝑣 ∈
ℝ (0 < 𝑣 ∧
∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ((normℎ‘𝑥) +
(normℎ‘𝑦)) ≤ (𝑣 ·
(normℎ‘(𝑥 +ℎ 𝑦)))) ↔ ((𝐴 ∩ 𝐵) = 0ℋ ∧ 𝜑 ∧ 𝜓)) |