HSE Home Hilbert Space Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  HSE Home  >  Th. List  >  cdj3i Structured version   Visualization version   GIF version

Theorem cdj3i 29284
Description: Two ways to express "𝐴 and 𝐵 are completely disjoint subspaces." (1) <=> (3) in Lemma 5 of [Holland] p. 1520. (Contributed by NM, 1-Jun-2005.) (New usage is discouraged.)
Hypotheses
Ref Expression
cdj3.1 𝐴S
cdj3.2 𝐵S
cdj3.3 𝑆 = (𝑥 ∈ (𝐴 + 𝐵) ↦ (𝑧𝐴𝑤𝐵 𝑥 = (𝑧 + 𝑤)))
cdj3.4 𝑇 = (𝑥 ∈ (𝐴 + 𝐵) ↦ (𝑤𝐵𝑧𝐴 𝑥 = (𝑧 + 𝑤)))
cdj3.5 (𝜑 ↔ ∃𝑣 ∈ ℝ (0 < 𝑣 ∧ ∀𝑢 ∈ (𝐴 + 𝐵)(norm‘(𝑆𝑢)) ≤ (𝑣 · (norm𝑢))))
cdj3.6 (𝜓 ↔ ∃𝑣 ∈ ℝ (0 < 𝑣 ∧ ∀𝑢 ∈ (𝐴 + 𝐵)(norm‘(𝑇𝑢)) ≤ (𝑣 · (norm𝑢))))
Assertion
Ref Expression
cdj3i (∃𝑣 ∈ ℝ (0 < 𝑣 ∧ ∀𝑥𝐴𝑦𝐵 ((norm𝑥) + (norm𝑦)) ≤ (𝑣 · (norm‘(𝑥 + 𝑦)))) ↔ ((𝐴𝐵) = 0𝜑𝜓))
Distinct variable groups:   𝑥,𝑦,𝑧,𝑤,𝑣,𝑢,𝐴   𝑥,𝐵,𝑦,𝑧,𝑤,𝑣,𝑢   𝑣,𝑆,𝑢   𝑣,𝑇,𝑢
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧,𝑤,𝑣,𝑢)   𝜓(𝑥,𝑦,𝑧,𝑤,𝑣,𝑢)   𝑆(𝑥,𝑦,𝑧,𝑤)   𝑇(𝑥,𝑦,𝑧,𝑤)

Proof of Theorem cdj3i
Dummy variables 𝑡 𝑓 𝑔 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 cdj3.1 . . . 4 𝐴S
2 cdj3.2 . . . 4 𝐵S
31, 2cdj3lem1 29277 . . 3 (∃𝑣 ∈ ℝ (0 < 𝑣 ∧ ∀𝑥𝐴𝑦𝐵 ((norm𝑥) + (norm𝑦)) ≤ (𝑣 · (norm‘(𝑥 + 𝑦)))) → (𝐴𝐵) = 0)
4 cdj3.3 . . . . 5 𝑆 = (𝑥 ∈ (𝐴 + 𝐵) ↦ (𝑧𝐴𝑤𝐵 𝑥 = (𝑧 + 𝑤)))
51, 2, 4cdj3lem2b 29280 . . . 4 (∃𝑣 ∈ ℝ (0 < 𝑣 ∧ ∀𝑥𝐴𝑦𝐵 ((norm𝑥) + (norm𝑦)) ≤ (𝑣 · (norm‘(𝑥 + 𝑦)))) → ∃𝑣 ∈ ℝ (0 < 𝑣 ∧ ∀𝑢 ∈ (𝐴 + 𝐵)(norm‘(𝑆𝑢)) ≤ (𝑣 · (norm𝑢))))
6 cdj3.5 . . . 4 (𝜑 ↔ ∃𝑣 ∈ ℝ (0 < 𝑣 ∧ ∀𝑢 ∈ (𝐴 + 𝐵)(norm‘(𝑆𝑢)) ≤ (𝑣 · (norm𝑢))))
75, 6sylibr 224 . . 3 (∃𝑣 ∈ ℝ (0 < 𝑣 ∧ ∀𝑥𝐴𝑦𝐵 ((norm𝑥) + (norm𝑦)) ≤ (𝑣 · (norm‘(𝑥 + 𝑦)))) → 𝜑)
8 cdj3.4 . . . . 5 𝑇 = (𝑥 ∈ (𝐴 + 𝐵) ↦ (𝑤𝐵𝑧𝐴 𝑥 = (𝑧 + 𝑤)))
91, 2, 8cdj3lem3b 29283 . . . 4 (∃𝑣 ∈ ℝ (0 < 𝑣 ∧ ∀𝑥𝐴𝑦𝐵 ((norm𝑥) + (norm𝑦)) ≤ (𝑣 · (norm‘(𝑥 + 𝑦)))) → ∃𝑣 ∈ ℝ (0 < 𝑣 ∧ ∀𝑢 ∈ (𝐴 + 𝐵)(norm‘(𝑇𝑢)) ≤ (𝑣 · (norm𝑢))))
10 cdj3.6 . . . 4 (𝜓 ↔ ∃𝑣 ∈ ℝ (0 < 𝑣 ∧ ∀𝑢 ∈ (𝐴 + 𝐵)(norm‘(𝑇𝑢)) ≤ (𝑣 · (norm𝑢))))
119, 10sylibr 224 . . 3 (∃𝑣 ∈ ℝ (0 < 𝑣 ∧ ∀𝑥𝐴𝑦𝐵 ((norm𝑥) + (norm𝑦)) ≤ (𝑣 · (norm‘(𝑥 + 𝑦)))) → 𝜓)
123, 7, 113jca 1241 . 2 (∃𝑣 ∈ ℝ (0 < 𝑣 ∧ ∀𝑥𝐴𝑦𝐵 ((norm𝑥) + (norm𝑦)) ≤ (𝑣 · (norm‘(𝑥 + 𝑦)))) → ((𝐴𝐵) = 0𝜑𝜓))
13 breq2 4655 . . . . . . . . 9 (𝑣 = 𝑓 → (0 < 𝑣 ↔ 0 < 𝑓))
14 oveq1 6654 . . . . . . . . . . 11 (𝑣 = 𝑓 → (𝑣 · (norm𝑢)) = (𝑓 · (norm𝑢)))
1514breq2d 4663 . . . . . . . . . 10 (𝑣 = 𝑓 → ((norm‘(𝑆𝑢)) ≤ (𝑣 · (norm𝑢)) ↔ (norm‘(𝑆𝑢)) ≤ (𝑓 · (norm𝑢))))
1615ralbidv 2985 . . . . . . . . 9 (𝑣 = 𝑓 → (∀𝑢 ∈ (𝐴 + 𝐵)(norm‘(𝑆𝑢)) ≤ (𝑣 · (norm𝑢)) ↔ ∀𝑢 ∈ (𝐴 + 𝐵)(norm‘(𝑆𝑢)) ≤ (𝑓 · (norm𝑢))))
1713, 16anbi12d 747 . . . . . . . 8 (𝑣 = 𝑓 → ((0 < 𝑣 ∧ ∀𝑢 ∈ (𝐴 + 𝐵)(norm‘(𝑆𝑢)) ≤ (𝑣 · (norm𝑢))) ↔ (0 < 𝑓 ∧ ∀𝑢 ∈ (𝐴 + 𝐵)(norm‘(𝑆𝑢)) ≤ (𝑓 · (norm𝑢)))))
1817cbvrexv 3170 . . . . . . 7 (∃𝑣 ∈ ℝ (0 < 𝑣 ∧ ∀𝑢 ∈ (𝐴 + 𝐵)(norm‘(𝑆𝑢)) ≤ (𝑣 · (norm𝑢))) ↔ ∃𝑓 ∈ ℝ (0 < 𝑓 ∧ ∀𝑢 ∈ (𝐴 + 𝐵)(norm‘(𝑆𝑢)) ≤ (𝑓 · (norm𝑢))))
196, 18bitri 264 . . . . . 6 (𝜑 ↔ ∃𝑓 ∈ ℝ (0 < 𝑓 ∧ ∀𝑢 ∈ (𝐴 + 𝐵)(norm‘(𝑆𝑢)) ≤ (𝑓 · (norm𝑢))))
20 breq2 4655 . . . . . . . . 9 (𝑣 = 𝑔 → (0 < 𝑣 ↔ 0 < 𝑔))
21 oveq1 6654 . . . . . . . . . . 11 (𝑣 = 𝑔 → (𝑣 · (norm𝑢)) = (𝑔 · (norm𝑢)))
2221breq2d 4663 . . . . . . . . . 10 (𝑣 = 𝑔 → ((norm‘(𝑇𝑢)) ≤ (𝑣 · (norm𝑢)) ↔ (norm‘(𝑇𝑢)) ≤ (𝑔 · (norm𝑢))))
2322ralbidv 2985 . . . . . . . . 9 (𝑣 = 𝑔 → (∀𝑢 ∈ (𝐴 + 𝐵)(norm‘(𝑇𝑢)) ≤ (𝑣 · (norm𝑢)) ↔ ∀𝑢 ∈ (𝐴 + 𝐵)(norm‘(𝑇𝑢)) ≤ (𝑔 · (norm𝑢))))
2420, 23anbi12d 747 . . . . . . . 8 (𝑣 = 𝑔 → ((0 < 𝑣 ∧ ∀𝑢 ∈ (𝐴 + 𝐵)(norm‘(𝑇𝑢)) ≤ (𝑣 · (norm𝑢))) ↔ (0 < 𝑔 ∧ ∀𝑢 ∈ (𝐴 + 𝐵)(norm‘(𝑇𝑢)) ≤ (𝑔 · (norm𝑢)))))
2524cbvrexv 3170 . . . . . . 7 (∃𝑣 ∈ ℝ (0 < 𝑣 ∧ ∀𝑢 ∈ (𝐴 + 𝐵)(norm‘(𝑇𝑢)) ≤ (𝑣 · (norm𝑢))) ↔ ∃𝑔 ∈ ℝ (0 < 𝑔 ∧ ∀𝑢 ∈ (𝐴 + 𝐵)(norm‘(𝑇𝑢)) ≤ (𝑔 · (norm𝑢))))
2610, 25bitri 264 . . . . . 6 (𝜓 ↔ ∃𝑔 ∈ ℝ (0 < 𝑔 ∧ ∀𝑢 ∈ (𝐴 + 𝐵)(norm‘(𝑇𝑢)) ≤ (𝑔 · (norm𝑢))))
2719, 26anbi12i 733 . . . . 5 ((𝜑𝜓) ↔ (∃𝑓 ∈ ℝ (0 < 𝑓 ∧ ∀𝑢 ∈ (𝐴 + 𝐵)(norm‘(𝑆𝑢)) ≤ (𝑓 · (norm𝑢))) ∧ ∃𝑔 ∈ ℝ (0 < 𝑔 ∧ ∀𝑢 ∈ (𝐴 + 𝐵)(norm‘(𝑇𝑢)) ≤ (𝑔 · (norm𝑢)))))
28 reeanv 3105 . . . . 5 (∃𝑓 ∈ ℝ ∃𝑔 ∈ ℝ ((0 < 𝑓 ∧ ∀𝑢 ∈ (𝐴 + 𝐵)(norm‘(𝑆𝑢)) ≤ (𝑓 · (norm𝑢))) ∧ (0 < 𝑔 ∧ ∀𝑢 ∈ (𝐴 + 𝐵)(norm‘(𝑇𝑢)) ≤ (𝑔 · (norm𝑢)))) ↔ (∃𝑓 ∈ ℝ (0 < 𝑓 ∧ ∀𝑢 ∈ (𝐴 + 𝐵)(norm‘(𝑆𝑢)) ≤ (𝑓 · (norm𝑢))) ∧ ∃𝑔 ∈ ℝ (0 < 𝑔 ∧ ∀𝑢 ∈ (𝐴 + 𝐵)(norm‘(𝑇𝑢)) ≤ (𝑔 · (norm𝑢)))))
2927, 28bitr4i 267 . . . 4 ((𝜑𝜓) ↔ ∃𝑓 ∈ ℝ ∃𝑔 ∈ ℝ ((0 < 𝑓 ∧ ∀𝑢 ∈ (𝐴 + 𝐵)(norm‘(𝑆𝑢)) ≤ (𝑓 · (norm𝑢))) ∧ (0 < 𝑔 ∧ ∀𝑢 ∈ (𝐴 + 𝐵)(norm‘(𝑇𝑢)) ≤ (𝑔 · (norm𝑢)))))
30 an4 865 . . . . . 6 (((0 < 𝑓 ∧ ∀𝑢 ∈ (𝐴 + 𝐵)(norm‘(𝑆𝑢)) ≤ (𝑓 · (norm𝑢))) ∧ (0 < 𝑔 ∧ ∀𝑢 ∈ (𝐴 + 𝐵)(norm‘(𝑇𝑢)) ≤ (𝑔 · (norm𝑢)))) ↔ ((0 < 𝑓 ∧ 0 < 𝑔) ∧ (∀𝑢 ∈ (𝐴 + 𝐵)(norm‘(𝑆𝑢)) ≤ (𝑓 · (norm𝑢)) ∧ ∀𝑢 ∈ (𝐴 + 𝐵)(norm‘(𝑇𝑢)) ≤ (𝑔 · (norm𝑢)))))
31 addgt0 10511 . . . . . . . . 9 (((𝑓 ∈ ℝ ∧ 𝑔 ∈ ℝ) ∧ (0 < 𝑓 ∧ 0 < 𝑔)) → 0 < (𝑓 + 𝑔))
3231ex 450 . . . . . . . 8 ((𝑓 ∈ ℝ ∧ 𝑔 ∈ ℝ) → ((0 < 𝑓 ∧ 0 < 𝑔) → 0 < (𝑓 + 𝑔)))
3332adantl 482 . . . . . . 7 (((𝐴𝐵) = 0 ∧ (𝑓 ∈ ℝ ∧ 𝑔 ∈ ℝ)) → ((0 < 𝑓 ∧ 0 < 𝑔) → 0 < (𝑓 + 𝑔)))
341, 2shsvai 28207 . . . . . . . . . . 11 ((𝑡𝐴𝐵) → (𝑡 + ) ∈ (𝐴 + 𝐵))
35 fveq2 6189 . . . . . . . . . . . . . . 15 (𝑢 = (𝑡 + ) → (𝑆𝑢) = (𝑆‘(𝑡 + )))
3635fveq2d 6193 . . . . . . . . . . . . . 14 (𝑢 = (𝑡 + ) → (norm‘(𝑆𝑢)) = (norm‘(𝑆‘(𝑡 + ))))
37 fveq2 6189 . . . . . . . . . . . . . . 15 (𝑢 = (𝑡 + ) → (norm𝑢) = (norm‘(𝑡 + )))
3837oveq2d 6663 . . . . . . . . . . . . . 14 (𝑢 = (𝑡 + ) → (𝑓 · (norm𝑢)) = (𝑓 · (norm‘(𝑡 + ))))
3936, 38breq12d 4664 . . . . . . . . . . . . 13 (𝑢 = (𝑡 + ) → ((norm‘(𝑆𝑢)) ≤ (𝑓 · (norm𝑢)) ↔ (norm‘(𝑆‘(𝑡 + ))) ≤ (𝑓 · (norm‘(𝑡 + )))))
4039rspcv 3303 . . . . . . . . . . . 12 ((𝑡 + ) ∈ (𝐴 + 𝐵) → (∀𝑢 ∈ (𝐴 + 𝐵)(norm‘(𝑆𝑢)) ≤ (𝑓 · (norm𝑢)) → (norm‘(𝑆‘(𝑡 + ))) ≤ (𝑓 · (norm‘(𝑡 + )))))
41 fveq2 6189 . . . . . . . . . . . . . . 15 (𝑢 = (𝑡 + ) → (𝑇𝑢) = (𝑇‘(𝑡 + )))
4241fveq2d 6193 . . . . . . . . . . . . . 14 (𝑢 = (𝑡 + ) → (norm‘(𝑇𝑢)) = (norm‘(𝑇‘(𝑡 + ))))
4337oveq2d 6663 . . . . . . . . . . . . . 14 (𝑢 = (𝑡 + ) → (𝑔 · (norm𝑢)) = (𝑔 · (norm‘(𝑡 + ))))
4442, 43breq12d 4664 . . . . . . . . . . . . 13 (𝑢 = (𝑡 + ) → ((norm‘(𝑇𝑢)) ≤ (𝑔 · (norm𝑢)) ↔ (norm‘(𝑇‘(𝑡 + ))) ≤ (𝑔 · (norm‘(𝑡 + )))))
4544rspcv 3303 . . . . . . . . . . . 12 ((𝑡 + ) ∈ (𝐴 + 𝐵) → (∀𝑢 ∈ (𝐴 + 𝐵)(norm‘(𝑇𝑢)) ≤ (𝑔 · (norm𝑢)) → (norm‘(𝑇‘(𝑡 + ))) ≤ (𝑔 · (norm‘(𝑡 + )))))
4640, 45anim12d 586 . . . . . . . . . . 11 ((𝑡 + ) ∈ (𝐴 + 𝐵) → ((∀𝑢 ∈ (𝐴 + 𝐵)(norm‘(𝑆𝑢)) ≤ (𝑓 · (norm𝑢)) ∧ ∀𝑢 ∈ (𝐴 + 𝐵)(norm‘(𝑇𝑢)) ≤ (𝑔 · (norm𝑢))) → ((norm‘(𝑆‘(𝑡 + ))) ≤ (𝑓 · (norm‘(𝑡 + ))) ∧ (norm‘(𝑇‘(𝑡 + ))) ≤ (𝑔 · (norm‘(𝑡 + ))))))
4734, 46syl 17 . . . . . . . . . 10 ((𝑡𝐴𝐵) → ((∀𝑢 ∈ (𝐴 + 𝐵)(norm‘(𝑆𝑢)) ≤ (𝑓 · (norm𝑢)) ∧ ∀𝑢 ∈ (𝐴 + 𝐵)(norm‘(𝑇𝑢)) ≤ (𝑔 · (norm𝑢))) → ((norm‘(𝑆‘(𝑡 + ))) ≤ (𝑓 · (norm‘(𝑡 + ))) ∧ (norm‘(𝑇‘(𝑡 + ))) ≤ (𝑔 · (norm‘(𝑡 + ))))))
4847adantl 482 . . . . . . . . 9 ((((𝐴𝐵) = 0 ∧ (𝑓 ∈ ℝ ∧ 𝑔 ∈ ℝ)) ∧ (𝑡𝐴𝐵)) → ((∀𝑢 ∈ (𝐴 + 𝐵)(norm‘(𝑆𝑢)) ≤ (𝑓 · (norm𝑢)) ∧ ∀𝑢 ∈ (𝐴 + 𝐵)(norm‘(𝑇𝑢)) ≤ (𝑔 · (norm𝑢))) → ((norm‘(𝑆‘(𝑡 + ))) ≤ (𝑓 · (norm‘(𝑡 + ))) ∧ (norm‘(𝑇‘(𝑡 + ))) ≤ (𝑔 · (norm‘(𝑡 + ))))))
491sheli 28055 . . . . . . . . . . . . . . 15 (𝑡𝐴𝑡 ∈ ℋ)
50 normcl 27966 . . . . . . . . . . . . . . 15 (𝑡 ∈ ℋ → (norm𝑡) ∈ ℝ)
5149, 50syl 17 . . . . . . . . . . . . . 14 (𝑡𝐴 → (norm𝑡) ∈ ℝ)
522sheli 28055 . . . . . . . . . . . . . . 15 (𝐵 ∈ ℋ)
53 normcl 27966 . . . . . . . . . . . . . . 15 ( ∈ ℋ → (norm) ∈ ℝ)
5452, 53syl 17 . . . . . . . . . . . . . 14 (𝐵 → (norm) ∈ ℝ)
5551, 54anim12i 590 . . . . . . . . . . . . 13 ((𝑡𝐴𝐵) → ((norm𝑡) ∈ ℝ ∧ (norm) ∈ ℝ))
5655adantl 482 . . . . . . . . . . . 12 (((𝑓 ∈ ℝ ∧ 𝑔 ∈ ℝ) ∧ (𝑡𝐴𝐵)) → ((norm𝑡) ∈ ℝ ∧ (norm) ∈ ℝ))
57 hvaddcl 27853 . . . . . . . . . . . . . . . 16 ((𝑡 ∈ ℋ ∧ ∈ ℋ) → (𝑡 + ) ∈ ℋ)
5849, 52, 57syl2an 494 . . . . . . . . . . . . . . 15 ((𝑡𝐴𝐵) → (𝑡 + ) ∈ ℋ)
59 normcl 27966 . . . . . . . . . . . . . . 15 ((𝑡 + ) ∈ ℋ → (norm‘(𝑡 + )) ∈ ℝ)
6058, 59syl 17 . . . . . . . . . . . . . 14 ((𝑡𝐴𝐵) → (norm‘(𝑡 + )) ∈ ℝ)
61 remulcl 10018 . . . . . . . . . . . . . 14 ((𝑓 ∈ ℝ ∧ (norm‘(𝑡 + )) ∈ ℝ) → (𝑓 · (norm‘(𝑡 + ))) ∈ ℝ)
6260, 61sylan2 491 . . . . . . . . . . . . 13 ((𝑓 ∈ ℝ ∧ (𝑡𝐴𝐵)) → (𝑓 · (norm‘(𝑡 + ))) ∈ ℝ)
6362adantlr 751 . . . . . . . . . . . 12 (((𝑓 ∈ ℝ ∧ 𝑔 ∈ ℝ) ∧ (𝑡𝐴𝐵)) → (𝑓 · (norm‘(𝑡 + ))) ∈ ℝ)
64 remulcl 10018 . . . . . . . . . . . . . 14 ((𝑔 ∈ ℝ ∧ (norm‘(𝑡 + )) ∈ ℝ) → (𝑔 · (norm‘(𝑡 + ))) ∈ ℝ)
6560, 64sylan2 491 . . . . . . . . . . . . 13 ((𝑔 ∈ ℝ ∧ (𝑡𝐴𝐵)) → (𝑔 · (norm‘(𝑡 + ))) ∈ ℝ)
6665adantll 750 . . . . . . . . . . . 12 (((𝑓 ∈ ℝ ∧ 𝑔 ∈ ℝ) ∧ (𝑡𝐴𝐵)) → (𝑔 · (norm‘(𝑡 + ))) ∈ ℝ)
67 le2add 10507 . . . . . . . . . . . 12 ((((norm𝑡) ∈ ℝ ∧ (norm) ∈ ℝ) ∧ ((𝑓 · (norm‘(𝑡 + ))) ∈ ℝ ∧ (𝑔 · (norm‘(𝑡 + ))) ∈ ℝ)) → (((norm𝑡) ≤ (𝑓 · (norm‘(𝑡 + ))) ∧ (norm) ≤ (𝑔 · (norm‘(𝑡 + )))) → ((norm𝑡) + (norm)) ≤ ((𝑓 · (norm‘(𝑡 + ))) + (𝑔 · (norm‘(𝑡 + ))))))
6856, 63, 66, 67syl12anc 1323 . . . . . . . . . . 11 (((𝑓 ∈ ℝ ∧ 𝑔 ∈ ℝ) ∧ (𝑡𝐴𝐵)) → (((norm𝑡) ≤ (𝑓 · (norm‘(𝑡 + ))) ∧ (norm) ≤ (𝑔 · (norm‘(𝑡 + )))) → ((norm𝑡) + (norm)) ≤ ((𝑓 · (norm‘(𝑡 + ))) + (𝑔 · (norm‘(𝑡 + ))))))
6968adantll 750 . . . . . . . . . 10 ((((𝐴𝐵) = 0 ∧ (𝑓 ∈ ℝ ∧ 𝑔 ∈ ℝ)) ∧ (𝑡𝐴𝐵)) → (((norm𝑡) ≤ (𝑓 · (norm‘(𝑡 + ))) ∧ (norm) ≤ (𝑔 · (norm‘(𝑡 + )))) → ((norm𝑡) + (norm)) ≤ ((𝑓 · (norm‘(𝑡 + ))) + (𝑔 · (norm‘(𝑡 + ))))))
701, 2, 4cdj3lem2 29278 . . . . . . . . . . . . . . . 16 ((𝑡𝐴𝐵 ∧ (𝐴𝐵) = 0) → (𝑆‘(𝑡 + )) = 𝑡)
7170fveq2d 6193 . . . . . . . . . . . . . . 15 ((𝑡𝐴𝐵 ∧ (𝐴𝐵) = 0) → (norm‘(𝑆‘(𝑡 + ))) = (norm𝑡))
7271breq1d 4661 . . . . . . . . . . . . . 14 ((𝑡𝐴𝐵 ∧ (𝐴𝐵) = 0) → ((norm‘(𝑆‘(𝑡 + ))) ≤ (𝑓 · (norm‘(𝑡 + ))) ↔ (norm𝑡) ≤ (𝑓 · (norm‘(𝑡 + )))))
731, 2, 8cdj3lem3 29281 . . . . . . . . . . . . . . . 16 ((𝑡𝐴𝐵 ∧ (𝐴𝐵) = 0) → (𝑇‘(𝑡 + )) = )
7473fveq2d 6193 . . . . . . . . . . . . . . 15 ((𝑡𝐴𝐵 ∧ (𝐴𝐵) = 0) → (norm‘(𝑇‘(𝑡 + ))) = (norm))
7574breq1d 4661 . . . . . . . . . . . . . 14 ((𝑡𝐴𝐵 ∧ (𝐴𝐵) = 0) → ((norm‘(𝑇‘(𝑡 + ))) ≤ (𝑔 · (norm‘(𝑡 + ))) ↔ (norm) ≤ (𝑔 · (norm‘(𝑡 + )))))
7672, 75anbi12d 747 . . . . . . . . . . . . 13 ((𝑡𝐴𝐵 ∧ (𝐴𝐵) = 0) → (((norm‘(𝑆‘(𝑡 + ))) ≤ (𝑓 · (norm‘(𝑡 + ))) ∧ (norm‘(𝑇‘(𝑡 + ))) ≤ (𝑔 · (norm‘(𝑡 + )))) ↔ ((norm𝑡) ≤ (𝑓 · (norm‘(𝑡 + ))) ∧ (norm) ≤ (𝑔 · (norm‘(𝑡 + ))))))
77763expa 1264 . . . . . . . . . . . 12 (((𝑡𝐴𝐵) ∧ (𝐴𝐵) = 0) → (((norm‘(𝑆‘(𝑡 + ))) ≤ (𝑓 · (norm‘(𝑡 + ))) ∧ (norm‘(𝑇‘(𝑡 + ))) ≤ (𝑔 · (norm‘(𝑡 + )))) ↔ ((norm𝑡) ≤ (𝑓 · (norm‘(𝑡 + ))) ∧ (norm) ≤ (𝑔 · (norm‘(𝑡 + ))))))
7877ancoms 469 . . . . . . . . . . 11 (((𝐴𝐵) = 0 ∧ (𝑡𝐴𝐵)) → (((norm‘(𝑆‘(𝑡 + ))) ≤ (𝑓 · (norm‘(𝑡 + ))) ∧ (norm‘(𝑇‘(𝑡 + ))) ≤ (𝑔 · (norm‘(𝑡 + )))) ↔ ((norm𝑡) ≤ (𝑓 · (norm‘(𝑡 + ))) ∧ (norm) ≤ (𝑔 · (norm‘(𝑡 + ))))))
7978adantlr 751 . . . . . . . . . 10 ((((𝐴𝐵) = 0 ∧ (𝑓 ∈ ℝ ∧ 𝑔 ∈ ℝ)) ∧ (𝑡𝐴𝐵)) → (((norm‘(𝑆‘(𝑡 + ))) ≤ (𝑓 · (norm‘(𝑡 + ))) ∧ (norm‘(𝑇‘(𝑡 + ))) ≤ (𝑔 · (norm‘(𝑡 + )))) ↔ ((norm𝑡) ≤ (𝑓 · (norm‘(𝑡 + ))) ∧ (norm) ≤ (𝑔 · (norm‘(𝑡 + ))))))
80 recn 10023 . . . . . . . . . . . . . 14 (𝑓 ∈ ℝ → 𝑓 ∈ ℂ)
81 recn 10023 . . . . . . . . . . . . . 14 (𝑔 ∈ ℝ → 𝑔 ∈ ℂ)
8260recnd 10065 . . . . . . . . . . . . . 14 ((𝑡𝐴𝐵) → (norm‘(𝑡 + )) ∈ ℂ)
83 adddir 10028 . . . . . . . . . . . . . 14 ((𝑓 ∈ ℂ ∧ 𝑔 ∈ ℂ ∧ (norm‘(𝑡 + )) ∈ ℂ) → ((𝑓 + 𝑔) · (norm‘(𝑡 + ))) = ((𝑓 · (norm‘(𝑡 + ))) + (𝑔 · (norm‘(𝑡 + )))))
8480, 81, 82, 83syl3an 1367 . . . . . . . . . . . . 13 ((𝑓 ∈ ℝ ∧ 𝑔 ∈ ℝ ∧ (𝑡𝐴𝐵)) → ((𝑓 + 𝑔) · (norm‘(𝑡 + ))) = ((𝑓 · (norm‘(𝑡 + ))) + (𝑔 · (norm‘(𝑡 + )))))
85843expa 1264 . . . . . . . . . . . 12 (((𝑓 ∈ ℝ ∧ 𝑔 ∈ ℝ) ∧ (𝑡𝐴𝐵)) → ((𝑓 + 𝑔) · (norm‘(𝑡 + ))) = ((𝑓 · (norm‘(𝑡 + ))) + (𝑔 · (norm‘(𝑡 + )))))
8685breq2d 4663 . . . . . . . . . . 11 (((𝑓 ∈ ℝ ∧ 𝑔 ∈ ℝ) ∧ (𝑡𝐴𝐵)) → (((norm𝑡) + (norm)) ≤ ((𝑓 + 𝑔) · (norm‘(𝑡 + ))) ↔ ((norm𝑡) + (norm)) ≤ ((𝑓 · (norm‘(𝑡 + ))) + (𝑔 · (norm‘(𝑡 + ))))))
8786adantll 750 . . . . . . . . . 10 ((((𝐴𝐵) = 0 ∧ (𝑓 ∈ ℝ ∧ 𝑔 ∈ ℝ)) ∧ (𝑡𝐴𝐵)) → (((norm𝑡) + (norm)) ≤ ((𝑓 + 𝑔) · (norm‘(𝑡 + ))) ↔ ((norm𝑡) + (norm)) ≤ ((𝑓 · (norm‘(𝑡 + ))) + (𝑔 · (norm‘(𝑡 + ))))))
8869, 79, 873imtr4d 283 . . . . . . . . 9 ((((𝐴𝐵) = 0 ∧ (𝑓 ∈ ℝ ∧ 𝑔 ∈ ℝ)) ∧ (𝑡𝐴𝐵)) → (((norm‘(𝑆‘(𝑡 + ))) ≤ (𝑓 · (norm‘(𝑡 + ))) ∧ (norm‘(𝑇‘(𝑡 + ))) ≤ (𝑔 · (norm‘(𝑡 + )))) → ((norm𝑡) + (norm)) ≤ ((𝑓 + 𝑔) · (norm‘(𝑡 + )))))
8948, 88syld 47 . . . . . . . 8 ((((𝐴𝐵) = 0 ∧ (𝑓 ∈ ℝ ∧ 𝑔 ∈ ℝ)) ∧ (𝑡𝐴𝐵)) → ((∀𝑢 ∈ (𝐴 + 𝐵)(norm‘(𝑆𝑢)) ≤ (𝑓 · (norm𝑢)) ∧ ∀𝑢 ∈ (𝐴 + 𝐵)(norm‘(𝑇𝑢)) ≤ (𝑔 · (norm𝑢))) → ((norm𝑡) + (norm)) ≤ ((𝑓 + 𝑔) · (norm‘(𝑡 + )))))
9089ralrimdvva 2973 . . . . . . 7 (((𝐴𝐵) = 0 ∧ (𝑓 ∈ ℝ ∧ 𝑔 ∈ ℝ)) → ((∀𝑢 ∈ (𝐴 + 𝐵)(norm‘(𝑆𝑢)) ≤ (𝑓 · (norm𝑢)) ∧ ∀𝑢 ∈ (𝐴 + 𝐵)(norm‘(𝑇𝑢)) ≤ (𝑔 · (norm𝑢))) → ∀𝑡𝐴𝐵 ((norm𝑡) + (norm)) ≤ ((𝑓 + 𝑔) · (norm‘(𝑡 + )))))
91 readdcl 10016 . . . . . . . . 9 ((𝑓 ∈ ℝ ∧ 𝑔 ∈ ℝ) → (𝑓 + 𝑔) ∈ ℝ)
92 breq2 4655 . . . . . . . . . . . 12 (𝑣 = (𝑓 + 𝑔) → (0 < 𝑣 ↔ 0 < (𝑓 + 𝑔)))
93 fveq2 6189 . . . . . . . . . . . . . . . 16 (𝑥 = 𝑡 → (norm𝑥) = (norm𝑡))
9493oveq1d 6662 . . . . . . . . . . . . . . 15 (𝑥 = 𝑡 → ((norm𝑥) + (norm𝑦)) = ((norm𝑡) + (norm𝑦)))
95 oveq1 6654 . . . . . . . . . . . . . . . . 17 (𝑥 = 𝑡 → (𝑥 + 𝑦) = (𝑡 + 𝑦))
9695fveq2d 6193 . . . . . . . . . . . . . . . 16 (𝑥 = 𝑡 → (norm‘(𝑥 + 𝑦)) = (norm‘(𝑡 + 𝑦)))
9796oveq2d 6663 . . . . . . . . . . . . . . 15 (𝑥 = 𝑡 → (𝑣 · (norm‘(𝑥 + 𝑦))) = (𝑣 · (norm‘(𝑡 + 𝑦))))
9894, 97breq12d 4664 . . . . . . . . . . . . . 14 (𝑥 = 𝑡 → (((norm𝑥) + (norm𝑦)) ≤ (𝑣 · (norm‘(𝑥 + 𝑦))) ↔ ((norm𝑡) + (norm𝑦)) ≤ (𝑣 · (norm‘(𝑡 + 𝑦)))))
99 fveq2 6189 . . . . . . . . . . . . . . . 16 (𝑦 = → (norm𝑦) = (norm))
10099oveq2d 6663 . . . . . . . . . . . . . . 15 (𝑦 = → ((norm𝑡) + (norm𝑦)) = ((norm𝑡) + (norm)))
101 oveq2 6655 . . . . . . . . . . . . . . . . 17 (𝑦 = → (𝑡 + 𝑦) = (𝑡 + ))
102101fveq2d 6193 . . . . . . . . . . . . . . . 16 (𝑦 = → (norm‘(𝑡 + 𝑦)) = (norm‘(𝑡 + )))
103102oveq2d 6663 . . . . . . . . . . . . . . 15 (𝑦 = → (𝑣 · (norm‘(𝑡 + 𝑦))) = (𝑣 · (norm‘(𝑡 + ))))
104100, 103breq12d 4664 . . . . . . . . . . . . . 14 (𝑦 = → (((norm𝑡) + (norm𝑦)) ≤ (𝑣 · (norm‘(𝑡 + 𝑦))) ↔ ((norm𝑡) + (norm)) ≤ (𝑣 · (norm‘(𝑡 + )))))
10598, 104cbvral2v 3177 . . . . . . . . . . . . 13 (∀𝑥𝐴𝑦𝐵 ((norm𝑥) + (norm𝑦)) ≤ (𝑣 · (norm‘(𝑥 + 𝑦))) ↔ ∀𝑡𝐴𝐵 ((norm𝑡) + (norm)) ≤ (𝑣 · (norm‘(𝑡 + ))))
106 oveq1 6654 . . . . . . . . . . . . . . 15 (𝑣 = (𝑓 + 𝑔) → (𝑣 · (norm‘(𝑡 + ))) = ((𝑓 + 𝑔) · (norm‘(𝑡 + ))))
107106breq2d 4663 . . . . . . . . . . . . . 14 (𝑣 = (𝑓 + 𝑔) → (((norm𝑡) + (norm)) ≤ (𝑣 · (norm‘(𝑡 + ))) ↔ ((norm𝑡) + (norm)) ≤ ((𝑓 + 𝑔) · (norm‘(𝑡 + )))))
1081072ralbidv 2988 . . . . . . . . . . . . 13 (𝑣 = (𝑓 + 𝑔) → (∀𝑡𝐴𝐵 ((norm𝑡) + (norm)) ≤ (𝑣 · (norm‘(𝑡 + ))) ↔ ∀𝑡𝐴𝐵 ((norm𝑡) + (norm)) ≤ ((𝑓 + 𝑔) · (norm‘(𝑡 + )))))
109105, 108syl5bb 272 . . . . . . . . . . . 12 (𝑣 = (𝑓 + 𝑔) → (∀𝑥𝐴𝑦𝐵 ((norm𝑥) + (norm𝑦)) ≤ (𝑣 · (norm‘(𝑥 + 𝑦))) ↔ ∀𝑡𝐴𝐵 ((norm𝑡) + (norm)) ≤ ((𝑓 + 𝑔) · (norm‘(𝑡 + )))))
11092, 109anbi12d 747 . . . . . . . . . . 11 (𝑣 = (𝑓 + 𝑔) → ((0 < 𝑣 ∧ ∀𝑥𝐴𝑦𝐵 ((norm𝑥) + (norm𝑦)) ≤ (𝑣 · (norm‘(𝑥 + 𝑦)))) ↔ (0 < (𝑓 + 𝑔) ∧ ∀𝑡𝐴𝐵 ((norm𝑡) + (norm)) ≤ ((𝑓 + 𝑔) · (norm‘(𝑡 + ))))))
111110rspcev 3307 . . . . . . . . . 10 (((𝑓 + 𝑔) ∈ ℝ ∧ (0 < (𝑓 + 𝑔) ∧ ∀𝑡𝐴𝐵 ((norm𝑡) + (norm)) ≤ ((𝑓 + 𝑔) · (norm‘(𝑡 + ))))) → ∃𝑣 ∈ ℝ (0 < 𝑣 ∧ ∀𝑥𝐴𝑦𝐵 ((norm𝑥) + (norm𝑦)) ≤ (𝑣 · (norm‘(𝑥 + 𝑦)))))
112111ex 450 . . . . . . . . 9 ((𝑓 + 𝑔) ∈ ℝ → ((0 < (𝑓 + 𝑔) ∧ ∀𝑡𝐴𝐵 ((norm𝑡) + (norm)) ≤ ((𝑓 + 𝑔) · (norm‘(𝑡 + )))) → ∃𝑣 ∈ ℝ (0 < 𝑣 ∧ ∀𝑥𝐴𝑦𝐵 ((norm𝑥) + (norm𝑦)) ≤ (𝑣 · (norm‘(𝑥 + 𝑦))))))
11391, 112syl 17 . . . . . . . 8 ((𝑓 ∈ ℝ ∧ 𝑔 ∈ ℝ) → ((0 < (𝑓 + 𝑔) ∧ ∀𝑡𝐴𝐵 ((norm𝑡) + (norm)) ≤ ((𝑓 + 𝑔) · (norm‘(𝑡 + )))) → ∃𝑣 ∈ ℝ (0 < 𝑣 ∧ ∀𝑥𝐴𝑦𝐵 ((norm𝑥) + (norm𝑦)) ≤ (𝑣 · (norm‘(𝑥 + 𝑦))))))
114113adantl 482 . . . . . . 7 (((𝐴𝐵) = 0 ∧ (𝑓 ∈ ℝ ∧ 𝑔 ∈ ℝ)) → ((0 < (𝑓 + 𝑔) ∧ ∀𝑡𝐴𝐵 ((norm𝑡) + (norm)) ≤ ((𝑓 + 𝑔) · (norm‘(𝑡 + )))) → ∃𝑣 ∈ ℝ (0 < 𝑣 ∧ ∀𝑥𝐴𝑦𝐵 ((norm𝑥) + (norm𝑦)) ≤ (𝑣 · (norm‘(𝑥 + 𝑦))))))
11533, 90, 114syl2and 500 . . . . . 6 (((𝐴𝐵) = 0 ∧ (𝑓 ∈ ℝ ∧ 𝑔 ∈ ℝ)) → (((0 < 𝑓 ∧ 0 < 𝑔) ∧ (∀𝑢 ∈ (𝐴 + 𝐵)(norm‘(𝑆𝑢)) ≤ (𝑓 · (norm𝑢)) ∧ ∀𝑢 ∈ (𝐴 + 𝐵)(norm‘(𝑇𝑢)) ≤ (𝑔 · (norm𝑢)))) → ∃𝑣 ∈ ℝ (0 < 𝑣 ∧ ∀𝑥𝐴𝑦𝐵 ((norm𝑥) + (norm𝑦)) ≤ (𝑣 · (norm‘(𝑥 + 𝑦))))))
11630, 115syl5bi 232 . . . . 5 (((𝐴𝐵) = 0 ∧ (𝑓 ∈ ℝ ∧ 𝑔 ∈ ℝ)) → (((0 < 𝑓 ∧ ∀𝑢 ∈ (𝐴 + 𝐵)(norm‘(𝑆𝑢)) ≤ (𝑓 · (norm𝑢))) ∧ (0 < 𝑔 ∧ ∀𝑢 ∈ (𝐴 + 𝐵)(norm‘(𝑇𝑢)) ≤ (𝑔 · (norm𝑢)))) → ∃𝑣 ∈ ℝ (0 < 𝑣 ∧ ∀𝑥𝐴𝑦𝐵 ((norm𝑥) + (norm𝑦)) ≤ (𝑣 · (norm‘(𝑥 + 𝑦))))))
117116rexlimdvva 3036 . . . 4 ((𝐴𝐵) = 0 → (∃𝑓 ∈ ℝ ∃𝑔 ∈ ℝ ((0 < 𝑓 ∧ ∀𝑢 ∈ (𝐴 + 𝐵)(norm‘(𝑆𝑢)) ≤ (𝑓 · (norm𝑢))) ∧ (0 < 𝑔 ∧ ∀𝑢 ∈ (𝐴 + 𝐵)(norm‘(𝑇𝑢)) ≤ (𝑔 · (norm𝑢)))) → ∃𝑣 ∈ ℝ (0 < 𝑣 ∧ ∀𝑥𝐴𝑦𝐵 ((norm𝑥) + (norm𝑦)) ≤ (𝑣 · (norm‘(𝑥 + 𝑦))))))
11829, 117syl5bi 232 . . 3 ((𝐴𝐵) = 0 → ((𝜑𝜓) → ∃𝑣 ∈ ℝ (0 < 𝑣 ∧ ∀𝑥𝐴𝑦𝐵 ((norm𝑥) + (norm𝑦)) ≤ (𝑣 · (norm‘(𝑥 + 𝑦))))))
1191183impib 1261 . 2 (((𝐴𝐵) = 0𝜑𝜓) → ∃𝑣 ∈ ℝ (0 < 𝑣 ∧ ∀𝑥𝐴𝑦𝐵 ((norm𝑥) + (norm𝑦)) ≤ (𝑣 · (norm‘(𝑥 + 𝑦)))))
12012, 119impbii 199 1 (∃𝑣 ∈ ℝ (0 < 𝑣 ∧ ∀𝑥𝐴𝑦𝐵 ((norm𝑥) + (norm𝑦)) ≤ (𝑣 · (norm‘(𝑥 + 𝑦)))) ↔ ((𝐴𝐵) = 0𝜑𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384  w3a 1037   = wceq 1482  wcel 1989  wral 2911  wrex 2912  cin 3571   class class class wbr 4651  cmpt 4727  cfv 5886  crio 6607  (class class class)co 6647  cc 9931  cr 9932  0cc0 9933   + caddc 9936   · cmul 9938   < clt 10071  cle 10072  chil 27760   + cva 27761  normcno 27764   S csh 27769   + cph 27772  0c0h 27776
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-8 1991  ax-9 1998  ax-10 2018  ax-11 2033  ax-12 2046  ax-13 2245  ax-ext 2601  ax-rep 4769  ax-sep 4779  ax-nul 4787  ax-pow 4841  ax-pr 4904  ax-un 6946  ax-cnex 9989  ax-resscn 9990  ax-1cn 9991  ax-icn 9992  ax-addcl 9993  ax-addrcl 9994  ax-mulcl 9995  ax-mulrcl 9996  ax-mulcom 9997  ax-addass 9998  ax-mulass 9999  ax-distr 10000  ax-i2m1 10001  ax-1ne0 10002  ax-1rid 10003  ax-rnegex 10004  ax-rrecex 10005  ax-cnre 10006  ax-pre-lttri 10007  ax-pre-lttrn 10008  ax-pre-ltadd 10009  ax-pre-mulgt0 10010  ax-pre-sup 10011  ax-hilex 27840  ax-hfvadd 27841  ax-hvcom 27842  ax-hvass 27843  ax-hv0cl 27844  ax-hvaddid 27845  ax-hfvmul 27846  ax-hvmulid 27847  ax-hvmulass 27848  ax-hvdistr1 27849  ax-hvdistr2 27850  ax-hvmul0 27851  ax-hfi 27920  ax-his1 27923  ax-his3 27925  ax-his4 27926
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1485  df-ex 1704  df-nf 1709  df-sb 1880  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2752  df-ne 2794  df-nel 2897  df-ral 2916  df-rex 2917  df-reu 2918  df-rmo 2919  df-rab 2920  df-v 3200  df-sbc 3434  df-csb 3532  df-dif 3575  df-un 3577  df-in 3579  df-ss 3586  df-pss 3588  df-nul 3914  df-if 4085  df-pw 4158  df-sn 4176  df-pr 4178  df-tp 4180  df-op 4182  df-uni 4435  df-int 4474  df-iun 4520  df-br 4652  df-opab 4711  df-mpt 4728  df-tr 4751  df-id 5022  df-eprel 5027  df-po 5033  df-so 5034  df-fr 5071  df-we 5073  df-xp 5118  df-rel 5119  df-cnv 5120  df-co 5121  df-dm 5122  df-rn 5123  df-res 5124  df-ima 5125  df-pred 5678  df-ord 5724  df-on 5725  df-lim 5726  df-suc 5727  df-iota 5849  df-fun 5888  df-fn 5889  df-f 5890  df-f1 5891  df-fo 5892  df-f1o 5893  df-fv 5894  df-riota 6608  df-ov 6650  df-oprab 6651  df-mpt2 6652  df-om 7063  df-2nd 7166  df-wrecs 7404  df-recs 7465  df-rdg 7503  df-er 7739  df-en 7953  df-dom 7954  df-sdom 7955  df-sup 8345  df-pnf 10073  df-mnf 10074  df-xr 10075  df-ltxr 10076  df-le 10077  df-sub 10265  df-neg 10266  df-div 10682  df-nn 11018  df-2 11076  df-3 11077  df-n0 11290  df-z 11375  df-uz 11685  df-rp 11830  df-seq 12797  df-exp 12856  df-cj 13833  df-re 13834  df-im 13835  df-sqrt 13969  df-abs 13970  df-grpo 27331  df-ablo 27383  df-hnorm 27809  df-hvsub 27812  df-sh 28048  df-ch0 28094  df-shs 28151
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator