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Theorem cdj3i 29428
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 29421 . . 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 29424 . . . 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 29427 . . . 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 1261 . 2 (∃𝑣 ∈ ℝ (0 < 𝑣 ∧ ∀𝑥𝐴𝑦𝐵 ((norm𝑥) + (norm𝑦)) ≤ (𝑣 · (norm‘(𝑥 + 𝑦)))) → ((𝐴𝐵) = 0𝜑𝜓))
13 breq2 4689 . . . . . . . . 9 (𝑣 = 𝑓 → (0 < 𝑣 ↔ 0 < 𝑓))
14 oveq1 6697 . . . . . . . . . . 11 (𝑣 = 𝑓 → (𝑣 · (norm𝑢)) = (𝑓 · (norm𝑢)))
1514breq2d 4697 . . . . . . . . . 10 (𝑣 = 𝑓 → ((norm‘(𝑆𝑢)) ≤ (𝑣 · (norm𝑢)) ↔ (norm‘(𝑆𝑢)) ≤ (𝑓 · (norm𝑢))))
1615ralbidv 3015 . . . . . . . . 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 3202 . . . . . . 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 4689 . . . . . . . . 9 (𝑣 = 𝑔 → (0 < 𝑣 ↔ 0 < 𝑔))
21 oveq1 6697 . . . . . . . . . . 11 (𝑣 = 𝑔 → (𝑣 · (norm𝑢)) = (𝑔 · (norm𝑢)))
2221breq2d 4697 . . . . . . . . . 10 (𝑣 = 𝑔 → ((norm‘(𝑇𝑢)) ≤ (𝑣 · (norm𝑢)) ↔ (norm‘(𝑇𝑢)) ≤ (𝑔 · (norm𝑢))))
2322ralbidv 3015 . . . . . . . . 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 3202 . . . . . . 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 3136 . . . . 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 882 . . . . . 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 10552 . . . . . . . . 9 (((𝑓 ∈ ℝ ∧ 𝑔 ∈ ℝ) ∧ (0 < 𝑓 ∧ 0 < 𝑔)) → 0 < (𝑓 + 𝑔))
3231ex 449 . . . . . . . 8 ((𝑓 ∈ ℝ ∧ 𝑔 ∈ ℝ) → ((0 < 𝑓 ∧ 0 < 𝑔) → 0 < (𝑓 + 𝑔)))
3332adantl 481 . . . . . . 7 (((𝐴𝐵) = 0 ∧ (𝑓 ∈ ℝ ∧ 𝑔 ∈ ℝ)) → ((0 < 𝑓 ∧ 0 < 𝑔) → 0 < (𝑓 + 𝑔)))
341, 2shsvai 28351 . . . . . . . . . . 11 ((𝑡𝐴𝐵) → (𝑡 + ) ∈ (𝐴 + 𝐵))
35 fveq2 6229 . . . . . . . . . . . . . . 15 (𝑢 = (𝑡 + ) → (𝑆𝑢) = (𝑆‘(𝑡 + )))
3635fveq2d 6233 . . . . . . . . . . . . . 14 (𝑢 = (𝑡 + ) → (norm‘(𝑆𝑢)) = (norm‘(𝑆‘(𝑡 + ))))
37 fveq2 6229 . . . . . . . . . . . . . . 15 (𝑢 = (𝑡 + ) → (norm𝑢) = (norm‘(𝑡 + )))
3837oveq2d 6706 . . . . . . . . . . . . . 14 (𝑢 = (𝑡 + ) → (𝑓 · (norm𝑢)) = (𝑓 · (norm‘(𝑡 + ))))
3936, 38breq12d 4698 . . . . . . . . . . . . 13 (𝑢 = (𝑡 + ) → ((norm‘(𝑆𝑢)) ≤ (𝑓 · (norm𝑢)) ↔ (norm‘(𝑆‘(𝑡 + ))) ≤ (𝑓 · (norm‘(𝑡 + )))))
4039rspcv 3336 . . . . . . . . . . . 12 ((𝑡 + ) ∈ (𝐴 + 𝐵) → (∀𝑢 ∈ (𝐴 + 𝐵)(norm‘(𝑆𝑢)) ≤ (𝑓 · (norm𝑢)) → (norm‘(𝑆‘(𝑡 + ))) ≤ (𝑓 · (norm‘(𝑡 + )))))
41 fveq2 6229 . . . . . . . . . . . . . . 15 (𝑢 = (𝑡 + ) → (𝑇𝑢) = (𝑇‘(𝑡 + )))
4241fveq2d 6233 . . . . . . . . . . . . . 14 (𝑢 = (𝑡 + ) → (norm‘(𝑇𝑢)) = (norm‘(𝑇‘(𝑡 + ))))
4337oveq2d 6706 . . . . . . . . . . . . . 14 (𝑢 = (𝑡 + ) → (𝑔 · (norm𝑢)) = (𝑔 · (norm‘(𝑡 + ))))
4442, 43breq12d 4698 . . . . . . . . . . . . 13 (𝑢 = (𝑡 + ) → ((norm‘(𝑇𝑢)) ≤ (𝑔 · (norm𝑢)) ↔ (norm‘(𝑇‘(𝑡 + ))) ≤ (𝑔 · (norm‘(𝑡 + )))))
4544rspcv 3336 . . . . . . . . . . . 12 ((𝑡 + ) ∈ (𝐴 + 𝐵) → (∀𝑢 ∈ (𝐴 + 𝐵)(norm‘(𝑇𝑢)) ≤ (𝑔 · (norm𝑢)) → (norm‘(𝑇‘(𝑡 + ))) ≤ (𝑔 · (norm‘(𝑡 + )))))
4640, 45anim12d 585 . . . . . . . . . . 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 481 . . . . . . . . 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 28199 . . . . . . . . . . . . . . 15 (𝑡𝐴𝑡 ∈ ℋ)
50 normcl 28110 . . . . . . . . . . . . . . 15 (𝑡 ∈ ℋ → (norm𝑡) ∈ ℝ)
5149, 50syl 17 . . . . . . . . . . . . . 14 (𝑡𝐴 → (norm𝑡) ∈ ℝ)
522sheli 28199 . . . . . . . . . . . . . . 15 (𝐵 ∈ ℋ)
53 normcl 28110 . . . . . . . . . . . . . . 15 ( ∈ ℋ → (norm) ∈ ℝ)
5452, 53syl 17 . . . . . . . . . . . . . 14 (𝐵 → (norm) ∈ ℝ)
5551, 54anim12i 589 . . . . . . . . . . . . 13 ((𝑡𝐴𝐵) → ((norm𝑡) ∈ ℝ ∧ (norm) ∈ ℝ))
5655adantl 481 . . . . . . . . . . . 12 (((𝑓 ∈ ℝ ∧ 𝑔 ∈ ℝ) ∧ (𝑡𝐴𝐵)) → ((norm𝑡) ∈ ℝ ∧ (norm) ∈ ℝ))
57 hvaddcl 27997 . . . . . . . . . . . . . . . 16 ((𝑡 ∈ ℋ ∧ ∈ ℋ) → (𝑡 + ) ∈ ℋ)
5849, 52, 57syl2an 493 . . . . . . . . . . . . . . 15 ((𝑡𝐴𝐵) → (𝑡 + ) ∈ ℋ)
59 normcl 28110 . . . . . . . . . . . . . . 15 ((𝑡 + ) ∈ ℋ → (norm‘(𝑡 + )) ∈ ℝ)
6058, 59syl 17 . . . . . . . . . . . . . 14 ((𝑡𝐴𝐵) → (norm‘(𝑡 + )) ∈ ℝ)
61 remulcl 10059 . . . . . . . . . . . . . 14 ((𝑓 ∈ ℝ ∧ (norm‘(𝑡 + )) ∈ ℝ) → (𝑓 · (norm‘(𝑡 + ))) ∈ ℝ)
6260, 61sylan2 490 . . . . . . . . . . . . 13 ((𝑓 ∈ ℝ ∧ (𝑡𝐴𝐵)) → (𝑓 · (norm‘(𝑡 + ))) ∈ ℝ)
6362adantlr 751 . . . . . . . . . . . 12 (((𝑓 ∈ ℝ ∧ 𝑔 ∈ ℝ) ∧ (𝑡𝐴𝐵)) → (𝑓 · (norm‘(𝑡 + ))) ∈ ℝ)
64 remulcl 10059 . . . . . . . . . . . . . 14 ((𝑔 ∈ ℝ ∧ (norm‘(𝑡 + )) ∈ ℝ) → (𝑔 · (norm‘(𝑡 + ))) ∈ ℝ)
6560, 64sylan2 490 . . . . . . . . . . . . 13 ((𝑔 ∈ ℝ ∧ (𝑡𝐴𝐵)) → (𝑔 · (norm‘(𝑡 + ))) ∈ ℝ)
6665adantll 750 . . . . . . . . . . . 12 (((𝑓 ∈ ℝ ∧ 𝑔 ∈ ℝ) ∧ (𝑡𝐴𝐵)) → (𝑔 · (norm‘(𝑡 + ))) ∈ ℝ)
67 le2add 10548 . . . . . . . . . . . 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 1364 . . . . . . . . . . 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 29422 . . . . . . . . . . . . . . . 16 ((𝑡𝐴𝐵 ∧ (𝐴𝐵) = 0) → (𝑆‘(𝑡 + )) = 𝑡)
7170fveq2d 6233 . . . . . . . . . . . . . . 15 ((𝑡𝐴𝐵 ∧ (𝐴𝐵) = 0) → (norm‘(𝑆‘(𝑡 + ))) = (norm𝑡))
7271breq1d 4695 . . . . . . . . . . . . . 14 ((𝑡𝐴𝐵 ∧ (𝐴𝐵) = 0) → ((norm‘(𝑆‘(𝑡 + ))) ≤ (𝑓 · (norm‘(𝑡 + ))) ↔ (norm𝑡) ≤ (𝑓 · (norm‘(𝑡 + )))))
731, 2, 8cdj3lem3 29425 . . . . . . . . . . . . . . . 16 ((𝑡𝐴𝐵 ∧ (𝐴𝐵) = 0) → (𝑇‘(𝑡 + )) = )
7473fveq2d 6233 . . . . . . . . . . . . . . 15 ((𝑡𝐴𝐵 ∧ (𝐴𝐵) = 0) → (norm‘(𝑇‘(𝑡 + ))) = (norm))
7574breq1d 4695 . . . . . . . . . . . . . 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 1284 . . . . . . . . . . . 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 468 . . . . . . . . . . 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 10064 . . . . . . . . . . . . . 14 (𝑓 ∈ ℝ → 𝑓 ∈ ℂ)
81 recn 10064 . . . . . . . . . . . . . 14 (𝑔 ∈ ℝ → 𝑔 ∈ ℂ)
8260recnd 10106 . . . . . . . . . . . . . 14 ((𝑡𝐴𝐵) → (norm‘(𝑡 + )) ∈ ℂ)
83 adddir 10069 . . . . . . . . . . . . . 14 ((𝑓 ∈ ℂ ∧ 𝑔 ∈ ℂ ∧ (norm‘(𝑡 + )) ∈ ℂ) → ((𝑓 + 𝑔) · (norm‘(𝑡 + ))) = ((𝑓 · (norm‘(𝑡 + ))) + (𝑔 · (norm‘(𝑡 + )))))
8480, 81, 82, 83syl3an 1408 . . . . . . . . . . . . 13 ((𝑓 ∈ ℝ ∧ 𝑔 ∈ ℝ ∧ (𝑡𝐴𝐵)) → ((𝑓 + 𝑔) · (norm‘(𝑡 + ))) = ((𝑓 · (norm‘(𝑡 + ))) + (𝑔 · (norm‘(𝑡 + )))))
85843expa 1284 . . . . . . . . . . . 12 (((𝑓 ∈ ℝ ∧ 𝑔 ∈ ℝ) ∧ (𝑡𝐴𝐵)) → ((𝑓 + 𝑔) · (norm‘(𝑡 + ))) = ((𝑓 · (norm‘(𝑡 + ))) + (𝑔 · (norm‘(𝑡 + )))))
8685breq2d 4697 . . . . . . . . . . 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 3003 . . . . . . 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 10057 . . . . . . . . 9 ((𝑓 ∈ ℝ ∧ 𝑔 ∈ ℝ) → (𝑓 + 𝑔) ∈ ℝ)
92 breq2 4689 . . . . . . . . . . . 12 (𝑣 = (𝑓 + 𝑔) → (0 < 𝑣 ↔ 0 < (𝑓 + 𝑔)))
93 fveq2 6229 . . . . . . . . . . . . . . . 16 (𝑥 = 𝑡 → (norm𝑥) = (norm𝑡))
9493oveq1d 6705 . . . . . . . . . . . . . . 15 (𝑥 = 𝑡 → ((norm𝑥) + (norm𝑦)) = ((norm𝑡) + (norm𝑦)))
95 oveq1 6697 . . . . . . . . . . . . . . . . 17 (𝑥 = 𝑡 → (𝑥 + 𝑦) = (𝑡 + 𝑦))
9695fveq2d 6233 . . . . . . . . . . . . . . . 16 (𝑥 = 𝑡 → (norm‘(𝑥 + 𝑦)) = (norm‘(𝑡 + 𝑦)))
9796oveq2d 6706 . . . . . . . . . . . . . . 15 (𝑥 = 𝑡 → (𝑣 · (norm‘(𝑥 + 𝑦))) = (𝑣 · (norm‘(𝑡 + 𝑦))))
9894, 97breq12d 4698 . . . . . . . . . . . . . 14 (𝑥 = 𝑡 → (((norm𝑥) + (norm𝑦)) ≤ (𝑣 · (norm‘(𝑥 + 𝑦))) ↔ ((norm𝑡) + (norm𝑦)) ≤ (𝑣 · (norm‘(𝑡 + 𝑦)))))
99 fveq2 6229 . . . . . . . . . . . . . . . 16 (𝑦 = → (norm𝑦) = (norm))
10099oveq2d 6706 . . . . . . . . . . . . . . 15 (𝑦 = → ((norm𝑡) + (norm𝑦)) = ((norm𝑡) + (norm)))
101 oveq2 6698 . . . . . . . . . . . . . . . . 17 (𝑦 = → (𝑡 + 𝑦) = (𝑡 + ))
102101fveq2d 6233 . . . . . . . . . . . . . . . 16 (𝑦 = → (norm‘(𝑡 + 𝑦)) = (norm‘(𝑡 + )))
103102oveq2d 6706 . . . . . . . . . . . . . . 15 (𝑦 = → (𝑣 · (norm‘(𝑡 + 𝑦))) = (𝑣 · (norm‘(𝑡 + ))))
104100, 103breq12d 4698 . . . . . . . . . . . . . 14 (𝑦 = → (((norm𝑡) + (norm𝑦)) ≤ (𝑣 · (norm‘(𝑡 + 𝑦))) ↔ ((norm𝑡) + (norm)) ≤ (𝑣 · (norm‘(𝑡 + )))))
10598, 104cbvral2v 3209 . . . . . . . . . . . . 13 (∀𝑥𝐴𝑦𝐵 ((norm𝑥) + (norm𝑦)) ≤ (𝑣 · (norm‘(𝑥 + 𝑦))) ↔ ∀𝑡𝐴𝐵 ((norm𝑡) + (norm)) ≤ (𝑣 · (norm‘(𝑡 + ))))
106 oveq1 6697 . . . . . . . . . . . . . . 15 (𝑣 = (𝑓 + 𝑔) → (𝑣 · (norm‘(𝑡 + ))) = ((𝑓 + 𝑔) · (norm‘(𝑡 + ))))
107106breq2d 4697 . . . . . . . . . . . . . 14 (𝑣 = (𝑓 + 𝑔) → (((norm𝑡) + (norm)) ≤ (𝑣 · (norm‘(𝑡 + ))) ↔ ((norm𝑡) + (norm)) ≤ ((𝑓 + 𝑔) · (norm‘(𝑡 + )))))
1081072ralbidv 3018 . . . . . . . . . . . . 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 3340 . . . . . . . . . 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 449 . . . . . . . . 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 481 . . . . . . 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 499 . . . . . 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 3067 . . . 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 1281 . 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 383  w3a 1054   = wceq 1523  wcel 2030  wral 2941  wrex 2942  cin 3606   class class class wbr 4685  cmpt 4762  cfv 5926  crio 6650  (class class class)co 6690  cc 9972  cr 9973  0cc0 9974   + caddc 9977   · cmul 9979   < clt 10112  cle 10113  chil 27904   + cva 27905  normcno 27908   S csh 27913   + cph 27916  0c0h 27920
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-cnex 10030  ax-resscn 10031  ax-1cn 10032  ax-icn 10033  ax-addcl 10034  ax-addrcl 10035  ax-mulcl 10036  ax-mulrcl 10037  ax-mulcom 10038  ax-addass 10039  ax-mulass 10040  ax-distr 10041  ax-i2m1 10042  ax-1ne0 10043  ax-1rid 10044  ax-rnegex 10045  ax-rrecex 10046  ax-cnre 10047  ax-pre-lttri 10048  ax-pre-lttrn 10049  ax-pre-ltadd 10050  ax-pre-mulgt0 10051  ax-pre-sup 10052  ax-hilex 27984  ax-hfvadd 27985  ax-hvcom 27986  ax-hvass 27987  ax-hv0cl 27988  ax-hvaddid 27989  ax-hfvmul 27990  ax-hvmulid 27991  ax-hvmulass 27992  ax-hvdistr1 27993  ax-hvdistr2 27994  ax-hvmul0 27995  ax-hfi 28064  ax-his1 28067  ax-his3 28069  ax-his4 28070
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-2nd 7211  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-er 7787  df-en 7998  df-dom 7999  df-sdom 8000  df-sup 8389  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117  df-le 10118  df-sub 10306  df-neg 10307  df-div 10723  df-nn 11059  df-2 11117  df-3 11118  df-n0 11331  df-z 11416  df-uz 11726  df-rp 11871  df-seq 12842  df-exp 12901  df-cj 13883  df-re 13884  df-im 13885  df-sqrt 14019  df-abs 14020  df-grpo 27475  df-ablo 27527  df-hnorm 27953  df-hvsub 27956  df-sh 28192  df-ch0 28238  df-shs 28295
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator