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Theorem cdj3i 30145
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 30138 . . 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 30141 . . . 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 235 . . 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 30144 . . . 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 235 . . 3 (∃𝑣 ∈ ℝ (0 < 𝑣 ∧ ∀𝑥𝐴𝑦𝐵 ((norm𝑥) + (norm𝑦)) ≤ (𝑣 · (norm‘(𝑥 + 𝑦)))) → 𝜓)
123, 7, 113jca 1120 . 2 (∃𝑣 ∈ ℝ (0 < 𝑣 ∧ ∀𝑥𝐴𝑦𝐵 ((norm𝑥) + (norm𝑦)) ≤ (𝑣 · (norm‘(𝑥 + 𝑦)))) → ((𝐴𝐵) = 0𝜑𝜓))
13 breq2 5061 . . . . . . . . 9 (𝑣 = 𝑓 → (0 < 𝑣 ↔ 0 < 𝑓))
14 oveq1 7152 . . . . . . . . . . 11 (𝑣 = 𝑓 → (𝑣 · (norm𝑢)) = (𝑓 · (norm𝑢)))
1514breq2d 5069 . . . . . . . . . 10 (𝑣 = 𝑓 → ((norm‘(𝑆𝑢)) ≤ (𝑣 · (norm𝑢)) ↔ (norm‘(𝑆𝑢)) ≤ (𝑓 · (norm𝑢))))
1615ralbidv 3194 . . . . . . . . 9 (𝑣 = 𝑓 → (∀𝑢 ∈ (𝐴 + 𝐵)(norm‘(𝑆𝑢)) ≤ (𝑣 · (norm𝑢)) ↔ ∀𝑢 ∈ (𝐴 + 𝐵)(norm‘(𝑆𝑢)) ≤ (𝑓 · (norm𝑢))))
1713, 16anbi12d 630 . . . . . . . 8 (𝑣 = 𝑓 → ((0 < 𝑣 ∧ ∀𝑢 ∈ (𝐴 + 𝐵)(norm‘(𝑆𝑢)) ≤ (𝑣 · (norm𝑢))) ↔ (0 < 𝑓 ∧ ∀𝑢 ∈ (𝐴 + 𝐵)(norm‘(𝑆𝑢)) ≤ (𝑓 · (norm𝑢)))))
1817cbvrexvw 3448 . . . . . . 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 276 . . . . . 6 (𝜑 ↔ ∃𝑓 ∈ ℝ (0 < 𝑓 ∧ ∀𝑢 ∈ (𝐴 + 𝐵)(norm‘(𝑆𝑢)) ≤ (𝑓 · (norm𝑢))))
20 breq2 5061 . . . . . . . . 9 (𝑣 = 𝑔 → (0 < 𝑣 ↔ 0 < 𝑔))
21 oveq1 7152 . . . . . . . . . . 11 (𝑣 = 𝑔 → (𝑣 · (norm𝑢)) = (𝑔 · (norm𝑢)))
2221breq2d 5069 . . . . . . . . . 10 (𝑣 = 𝑔 → ((norm‘(𝑇𝑢)) ≤ (𝑣 · (norm𝑢)) ↔ (norm‘(𝑇𝑢)) ≤ (𝑔 · (norm𝑢))))
2322ralbidv 3194 . . . . . . . . 9 (𝑣 = 𝑔 → (∀𝑢 ∈ (𝐴 + 𝐵)(norm‘(𝑇𝑢)) ≤ (𝑣 · (norm𝑢)) ↔ ∀𝑢 ∈ (𝐴 + 𝐵)(norm‘(𝑇𝑢)) ≤ (𝑔 · (norm𝑢))))
2420, 23anbi12d 630 . . . . . . . 8 (𝑣 = 𝑔 → ((0 < 𝑣 ∧ ∀𝑢 ∈ (𝐴 + 𝐵)(norm‘(𝑇𝑢)) ≤ (𝑣 · (norm𝑢))) ↔ (0 < 𝑔 ∧ ∀𝑢 ∈ (𝐴 + 𝐵)(norm‘(𝑇𝑢)) ≤ (𝑔 · (norm𝑢)))))
2524cbvrexvw 3448 . . . . . . 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 276 . . . . . 6 (𝜓 ↔ ∃𝑔 ∈ ℝ (0 < 𝑔 ∧ ∀𝑢 ∈ (𝐴 + 𝐵)(norm‘(𝑇𝑢)) ≤ (𝑔 · (norm𝑢))))
2719, 26anbi12i 626 . . . . 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 3365 . . . . 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 279 . . . 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 652 . . . . . 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 11114 . . . . . . . . 9 (((𝑓 ∈ ℝ ∧ 𝑔 ∈ ℝ) ∧ (0 < 𝑓 ∧ 0 < 𝑔)) → 0 < (𝑓 + 𝑔))
3231ex 413 . . . . . . . 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 29068 . . . . . . . . . . 11 ((𝑡𝐴𝐵) → (𝑡 + ) ∈ (𝐴 + 𝐵))
35 2fveq3 6668 . . . . . . . . . . . . . 14 (𝑢 = (𝑡 + ) → (norm‘(𝑆𝑢)) = (norm‘(𝑆‘(𝑡 + ))))
36 fveq2 6663 . . . . . . . . . . . . . . 15 (𝑢 = (𝑡 + ) → (norm𝑢) = (norm‘(𝑡 + )))
3736oveq2d 7161 . . . . . . . . . . . . . 14 (𝑢 = (𝑡 + ) → (𝑓 · (norm𝑢)) = (𝑓 · (norm‘(𝑡 + ))))
3835, 37breq12d 5070 . . . . . . . . . . . . 13 (𝑢 = (𝑡 + ) → ((norm‘(𝑆𝑢)) ≤ (𝑓 · (norm𝑢)) ↔ (norm‘(𝑆‘(𝑡 + ))) ≤ (𝑓 · (norm‘(𝑡 + )))))
3938rspcv 3615 . . . . . . . . . . . 12 ((𝑡 + ) ∈ (𝐴 + 𝐵) → (∀𝑢 ∈ (𝐴 + 𝐵)(norm‘(𝑆𝑢)) ≤ (𝑓 · (norm𝑢)) → (norm‘(𝑆‘(𝑡 + ))) ≤ (𝑓 · (norm‘(𝑡 + )))))
40 2fveq3 6668 . . . . . . . . . . . . . 14 (𝑢 = (𝑡 + ) → (norm‘(𝑇𝑢)) = (norm‘(𝑇‘(𝑡 + ))))
4136oveq2d 7161 . . . . . . . . . . . . . 14 (𝑢 = (𝑡 + ) → (𝑔 · (norm𝑢)) = (𝑔 · (norm‘(𝑡 + ))))
4240, 41breq12d 5070 . . . . . . . . . . . . 13 (𝑢 = (𝑡 + ) → ((norm‘(𝑇𝑢)) ≤ (𝑔 · (norm𝑢)) ↔ (norm‘(𝑇‘(𝑡 + ))) ≤ (𝑔 · (norm‘(𝑡 + )))))
4342rspcv 3615 . . . . . . . . . . . 12 ((𝑡 + ) ∈ (𝐴 + 𝐵) → (∀𝑢 ∈ (𝐴 + 𝐵)(norm‘(𝑇𝑢)) ≤ (𝑔 · (norm𝑢)) → (norm‘(𝑇‘(𝑡 + ))) ≤ (𝑔 · (norm‘(𝑡 + )))))
4439, 43anim12d 608 . . . . . . . . . . 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 + ))))))
4534, 44syl 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 + ))))))
4645adantl 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 + ))))))
471sheli 28918 . . . . . . . . . . . . . . 15 (𝑡𝐴𝑡 ∈ ℋ)
48 normcl 28829 . . . . . . . . . . . . . . 15 (𝑡 ∈ ℋ → (norm𝑡) ∈ ℝ)
4947, 48syl 17 . . . . . . . . . . . . . 14 (𝑡𝐴 → (norm𝑡) ∈ ℝ)
502sheli 28918 . . . . . . . . . . . . . . 15 (𝐵 ∈ ℋ)
51 normcl 28829 . . . . . . . . . . . . . . 15 ( ∈ ℋ → (norm) ∈ ℝ)
5250, 51syl 17 . . . . . . . . . . . . . 14 (𝐵 → (norm) ∈ ℝ)
5349, 52anim12i 612 . . . . . . . . . . . . 13 ((𝑡𝐴𝐵) → ((norm𝑡) ∈ ℝ ∧ (norm) ∈ ℝ))
5453adantl 482 . . . . . . . . . . . 12 (((𝑓 ∈ ℝ ∧ 𝑔 ∈ ℝ) ∧ (𝑡𝐴𝐵)) → ((norm𝑡) ∈ ℝ ∧ (norm) ∈ ℝ))
55 hvaddcl 28716 . . . . . . . . . . . . . . . 16 ((𝑡 ∈ ℋ ∧ ∈ ℋ) → (𝑡 + ) ∈ ℋ)
5647, 50, 55syl2an 595 . . . . . . . . . . . . . . 15 ((𝑡𝐴𝐵) → (𝑡 + ) ∈ ℋ)
57 normcl 28829 . . . . . . . . . . . . . . 15 ((𝑡 + ) ∈ ℋ → (norm‘(𝑡 + )) ∈ ℝ)
5856, 57syl 17 . . . . . . . . . . . . . 14 ((𝑡𝐴𝐵) → (norm‘(𝑡 + )) ∈ ℝ)
59 remulcl 10610 . . . . . . . . . . . . . 14 ((𝑓 ∈ ℝ ∧ (norm‘(𝑡 + )) ∈ ℝ) → (𝑓 · (norm‘(𝑡 + ))) ∈ ℝ)
6058, 59sylan2 592 . . . . . . . . . . . . 13 ((𝑓 ∈ ℝ ∧ (𝑡𝐴𝐵)) → (𝑓 · (norm‘(𝑡 + ))) ∈ ℝ)
6160adantlr 711 . . . . . . . . . . . 12 (((𝑓 ∈ ℝ ∧ 𝑔 ∈ ℝ) ∧ (𝑡𝐴𝐵)) → (𝑓 · (norm‘(𝑡 + ))) ∈ ℝ)
62 remulcl 10610 . . . . . . . . . . . . . 14 ((𝑔 ∈ ℝ ∧ (norm‘(𝑡 + )) ∈ ℝ) → (𝑔 · (norm‘(𝑡 + ))) ∈ ℝ)
6358, 62sylan2 592 . . . . . . . . . . . . 13 ((𝑔 ∈ ℝ ∧ (𝑡𝐴𝐵)) → (𝑔 · (norm‘(𝑡 + ))) ∈ ℝ)
6463adantll 710 . . . . . . . . . . . 12 (((𝑓 ∈ ℝ ∧ 𝑔 ∈ ℝ) ∧ (𝑡𝐴𝐵)) → (𝑔 · (norm‘(𝑡 + ))) ∈ ℝ)
65 le2add 11110 . . . . . . . . . . . 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 + ))))))
6654, 61, 64, 65syl12anc 832 . . . . . . . . . . 11 (((𝑓 ∈ ℝ ∧ 𝑔 ∈ ℝ) ∧ (𝑡𝐴𝐵)) → (((norm𝑡) ≤ (𝑓 · (norm‘(𝑡 + ))) ∧ (norm) ≤ (𝑔 · (norm‘(𝑡 + )))) → ((norm𝑡) + (norm)) ≤ ((𝑓 · (norm‘(𝑡 + ))) + (𝑔 · (norm‘(𝑡 + ))))))
6766adantll 710 . . . . . . . . . 10 ((((𝐴𝐵) = 0 ∧ (𝑓 ∈ ℝ ∧ 𝑔 ∈ ℝ)) ∧ (𝑡𝐴𝐵)) → (((norm𝑡) ≤ (𝑓 · (norm‘(𝑡 + ))) ∧ (norm) ≤ (𝑔 · (norm‘(𝑡 + )))) → ((norm𝑡) + (norm)) ≤ ((𝑓 · (norm‘(𝑡 + ))) + (𝑔 · (norm‘(𝑡 + ))))))
681, 2, 4cdj3lem2 30139 . . . . . . . . . . . . . . . 16 ((𝑡𝐴𝐵 ∧ (𝐴𝐵) = 0) → (𝑆‘(𝑡 + )) = 𝑡)
6968fveq2d 6667 . . . . . . . . . . . . . . 15 ((𝑡𝐴𝐵 ∧ (𝐴𝐵) = 0) → (norm‘(𝑆‘(𝑡 + ))) = (norm𝑡))
7069breq1d 5067 . . . . . . . . . . . . . 14 ((𝑡𝐴𝐵 ∧ (𝐴𝐵) = 0) → ((norm‘(𝑆‘(𝑡 + ))) ≤ (𝑓 · (norm‘(𝑡 + ))) ↔ (norm𝑡) ≤ (𝑓 · (norm‘(𝑡 + )))))
711, 2, 8cdj3lem3 30142 . . . . . . . . . . . . . . . 16 ((𝑡𝐴𝐵 ∧ (𝐴𝐵) = 0) → (𝑇‘(𝑡 + )) = )
7271fveq2d 6667 . . . . . . . . . . . . . . 15 ((𝑡𝐴𝐵 ∧ (𝐴𝐵) = 0) → (norm‘(𝑇‘(𝑡 + ))) = (norm))
7372breq1d 5067 . . . . . . . . . . . . . 14 ((𝑡𝐴𝐵 ∧ (𝐴𝐵) = 0) → ((norm‘(𝑇‘(𝑡 + ))) ≤ (𝑔 · (norm‘(𝑡 + ))) ↔ (norm) ≤ (𝑔 · (norm‘(𝑡 + )))))
7470, 73anbi12d 630 . . . . . . . . . . . . 13 ((𝑡𝐴𝐵 ∧ (𝐴𝐵) = 0) → (((norm‘(𝑆‘(𝑡 + ))) ≤ (𝑓 · (norm‘(𝑡 + ))) ∧ (norm‘(𝑇‘(𝑡 + ))) ≤ (𝑔 · (norm‘(𝑡 + )))) ↔ ((norm𝑡) ≤ (𝑓 · (norm‘(𝑡 + ))) ∧ (norm) ≤ (𝑔 · (norm‘(𝑡 + ))))))
75743expa 1110 . . . . . . . . . . . 12 (((𝑡𝐴𝐵) ∧ (𝐴𝐵) = 0) → (((norm‘(𝑆‘(𝑡 + ))) ≤ (𝑓 · (norm‘(𝑡 + ))) ∧ (norm‘(𝑇‘(𝑡 + ))) ≤ (𝑔 · (norm‘(𝑡 + )))) ↔ ((norm𝑡) ≤ (𝑓 · (norm‘(𝑡 + ))) ∧ (norm) ≤ (𝑔 · (norm‘(𝑡 + ))))))
7675ancoms 459 . . . . . . . . . . 11 (((𝐴𝐵) = 0 ∧ (𝑡𝐴𝐵)) → (((norm‘(𝑆‘(𝑡 + ))) ≤ (𝑓 · (norm‘(𝑡 + ))) ∧ (norm‘(𝑇‘(𝑡 + ))) ≤ (𝑔 · (norm‘(𝑡 + )))) ↔ ((norm𝑡) ≤ (𝑓 · (norm‘(𝑡 + ))) ∧ (norm) ≤ (𝑔 · (norm‘(𝑡 + ))))))
7776adantlr 711 . . . . . . . . . 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 + ))))))
78 recn 10615 . . . . . . . . . . . . . 14 (𝑓 ∈ ℝ → 𝑓 ∈ ℂ)
79 recn 10615 . . . . . . . . . . . . . 14 (𝑔 ∈ ℝ → 𝑔 ∈ ℂ)
8058recnd 10657 . . . . . . . . . . . . . 14 ((𝑡𝐴𝐵) → (norm‘(𝑡 + )) ∈ ℂ)
81 adddir 10620 . . . . . . . . . . . . . 14 ((𝑓 ∈ ℂ ∧ 𝑔 ∈ ℂ ∧ (norm‘(𝑡 + )) ∈ ℂ) → ((𝑓 + 𝑔) · (norm‘(𝑡 + ))) = ((𝑓 · (norm‘(𝑡 + ))) + (𝑔 · (norm‘(𝑡 + )))))
8278, 79, 80, 81syl3an 1152 . . . . . . . . . . . . 13 ((𝑓 ∈ ℝ ∧ 𝑔 ∈ ℝ ∧ (𝑡𝐴𝐵)) → ((𝑓 + 𝑔) · (norm‘(𝑡 + ))) = ((𝑓 · (norm‘(𝑡 + ))) + (𝑔 · (norm‘(𝑡 + )))))
83823expa 1110 . . . . . . . . . . . 12 (((𝑓 ∈ ℝ ∧ 𝑔 ∈ ℝ) ∧ (𝑡𝐴𝐵)) → ((𝑓 + 𝑔) · (norm‘(𝑡 + ))) = ((𝑓 · (norm‘(𝑡 + ))) + (𝑔 · (norm‘(𝑡 + )))))
8483breq2d 5069 . . . . . . . . . . 11 (((𝑓 ∈ ℝ ∧ 𝑔 ∈ ℝ) ∧ (𝑡𝐴𝐵)) → (((norm𝑡) + (norm)) ≤ ((𝑓 + 𝑔) · (norm‘(𝑡 + ))) ↔ ((norm𝑡) + (norm)) ≤ ((𝑓 · (norm‘(𝑡 + ))) + (𝑔 · (norm‘(𝑡 + ))))))
8584adantll 710 . . . . . . . . . 10 ((((𝐴𝐵) = 0 ∧ (𝑓 ∈ ℝ ∧ 𝑔 ∈ ℝ)) ∧ (𝑡𝐴𝐵)) → (((norm𝑡) + (norm)) ≤ ((𝑓 + 𝑔) · (norm‘(𝑡 + ))) ↔ ((norm𝑡) + (norm)) ≤ ((𝑓 · (norm‘(𝑡 + ))) + (𝑔 · (norm‘(𝑡 + ))))))
8667, 77, 853imtr4d 295 . . . . . . . . 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 + )))))
8746, 86syld 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 + )))))
8887ralrimdvva 3191 . . . . . . 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 + )))))
89 readdcl 10608 . . . . . . . . 9 ((𝑓 ∈ ℝ ∧ 𝑔 ∈ ℝ) → (𝑓 + 𝑔) ∈ ℝ)
90 breq2 5061 . . . . . . . . . . . 12 (𝑣 = (𝑓 + 𝑔) → (0 < 𝑣 ↔ 0 < (𝑓 + 𝑔)))
91 fveq2 6663 . . . . . . . . . . . . . . . 16 (𝑥 = 𝑡 → (norm𝑥) = (norm𝑡))
9291oveq1d 7160 . . . . . . . . . . . . . . 15 (𝑥 = 𝑡 → ((norm𝑥) + (norm𝑦)) = ((norm𝑡) + (norm𝑦)))
93 fvoveq1 7168 . . . . . . . . . . . . . . . 16 (𝑥 = 𝑡 → (norm‘(𝑥 + 𝑦)) = (norm‘(𝑡 + 𝑦)))
9493oveq2d 7161 . . . . . . . . . . . . . . 15 (𝑥 = 𝑡 → (𝑣 · (norm‘(𝑥 + 𝑦))) = (𝑣 · (norm‘(𝑡 + 𝑦))))
9592, 94breq12d 5070 . . . . . . . . . . . . . 14 (𝑥 = 𝑡 → (((norm𝑥) + (norm𝑦)) ≤ (𝑣 · (norm‘(𝑥 + 𝑦))) ↔ ((norm𝑡) + (norm𝑦)) ≤ (𝑣 · (norm‘(𝑡 + 𝑦)))))
96 fveq2 6663 . . . . . . . . . . . . . . . 16 (𝑦 = → (norm𝑦) = (norm))
9796oveq2d 7161 . . . . . . . . . . . . . . 15 (𝑦 = → ((norm𝑡) + (norm𝑦)) = ((norm𝑡) + (norm)))
98 oveq2 7153 . . . . . . . . . . . . . . . . 17 (𝑦 = → (𝑡 + 𝑦) = (𝑡 + ))
9998fveq2d 6667 . . . . . . . . . . . . . . . 16 (𝑦 = → (norm‘(𝑡 + 𝑦)) = (norm‘(𝑡 + )))
10099oveq2d 7161 . . . . . . . . . . . . . . 15 (𝑦 = → (𝑣 · (norm‘(𝑡 + 𝑦))) = (𝑣 · (norm‘(𝑡 + ))))
10197, 100breq12d 5070 . . . . . . . . . . . . . 14 (𝑦 = → (((norm𝑡) + (norm𝑦)) ≤ (𝑣 · (norm‘(𝑡 + 𝑦))) ↔ ((norm𝑡) + (norm)) ≤ (𝑣 · (norm‘(𝑡 + )))))
10295, 101cbvral2vw 3459 . . . . . . . . . . . . 13 (∀𝑥𝐴𝑦𝐵 ((norm𝑥) + (norm𝑦)) ≤ (𝑣 · (norm‘(𝑥 + 𝑦))) ↔ ∀𝑡𝐴𝐵 ((norm𝑡) + (norm)) ≤ (𝑣 · (norm‘(𝑡 + ))))
103 oveq1 7152 . . . . . . . . . . . . . . 15 (𝑣 = (𝑓 + 𝑔) → (𝑣 · (norm‘(𝑡 + ))) = ((𝑓 + 𝑔) · (norm‘(𝑡 + ))))
104103breq2d 5069 . . . . . . . . . . . . . 14 (𝑣 = (𝑓 + 𝑔) → (((norm𝑡) + (norm)) ≤ (𝑣 · (norm‘(𝑡 + ))) ↔ ((norm𝑡) + (norm)) ≤ ((𝑓 + 𝑔) · (norm‘(𝑡 + )))))
1051042ralbidv 3196 . . . . . . . . . . . . 13 (𝑣 = (𝑓 + 𝑔) → (∀𝑡𝐴𝐵 ((norm𝑡) + (norm)) ≤ (𝑣 · (norm‘(𝑡 + ))) ↔ ∀𝑡𝐴𝐵 ((norm𝑡) + (norm)) ≤ ((𝑓 + 𝑔) · (norm‘(𝑡 + )))))
106102, 105syl5bb 284 . . . . . . . . . . . 12 (𝑣 = (𝑓 + 𝑔) → (∀𝑥𝐴𝑦𝐵 ((norm𝑥) + (norm𝑦)) ≤ (𝑣 · (norm‘(𝑥 + 𝑦))) ↔ ∀𝑡𝐴𝐵 ((norm𝑡) + (norm)) ≤ ((𝑓 + 𝑔) · (norm‘(𝑡 + )))))
10790, 106anbi12d 630 . . . . . . . . . . 11 (𝑣 = (𝑓 + 𝑔) → ((0 < 𝑣 ∧ ∀𝑥𝐴𝑦𝐵 ((norm𝑥) + (norm𝑦)) ≤ (𝑣 · (norm‘(𝑥 + 𝑦)))) ↔ (0 < (𝑓 + 𝑔) ∧ ∀𝑡𝐴𝐵 ((norm𝑡) + (norm)) ≤ ((𝑓 + 𝑔) · (norm‘(𝑡 + ))))))
108107rspcev 3620 . . . . . . . . . 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)))))
109108ex 413 . . . . . . . . 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))))))
11089, 109syl 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))))))
111110adantl 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))))))
11233, 88, 111syl2and 607 . . . . . 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))))))
11330, 112syl5bi 243 . . . . 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))))))
114113rexlimdvva 3291 . . . 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))))))
11529, 114syl5bi 243 . . 3 ((𝐴𝐵) = 0 → ((𝜑𝜓) → ∃𝑣 ∈ ℝ (0 < 𝑣 ∧ ∀𝑥𝐴𝑦𝐵 ((norm𝑥) + (norm𝑦)) ≤ (𝑣 · (norm‘(𝑥 + 𝑦))))))
1161153impib 1108 . 2 (((𝐴𝐵) = 0𝜑𝜓) → ∃𝑣 ∈ ℝ (0 < 𝑣 ∧ ∀𝑥𝐴𝑦𝐵 ((norm𝑥) + (norm𝑦)) ≤ (𝑣 · (norm‘(𝑥 + 𝑦)))))
11712, 116impbii 210 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 207  wa 396  w3a 1079   = wceq 1528  wcel 2105  wral 3135  wrex 3136  cin 3932   class class class wbr 5057  cmpt 5137  cfv 6348  crio 7102  (class class class)co 7145  cc 10523  cr 10524  0cc0 10525   + caddc 10528   · cmul 10530   < clt 10663  cle 10664  chba 28623   + cva 28624  normcno 28627   S csh 28632   + cph 28635  0c0h 28639
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450  ax-cnex 10581  ax-resscn 10582  ax-1cn 10583  ax-icn 10584  ax-addcl 10585  ax-addrcl 10586  ax-mulcl 10587  ax-mulrcl 10588  ax-mulcom 10589  ax-addass 10590  ax-mulass 10591  ax-distr 10592  ax-i2m1 10593  ax-1ne0 10594  ax-1rid 10595  ax-rnegex 10596  ax-rrecex 10597  ax-cnre 10598  ax-pre-lttri 10599  ax-pre-lttrn 10600  ax-pre-ltadd 10601  ax-pre-mulgt0 10602  ax-pre-sup 10603  ax-hilex 28703  ax-hfvadd 28704  ax-hvcom 28705  ax-hvass 28706  ax-hv0cl 28707  ax-hvaddid 28708  ax-hfvmul 28709  ax-hvmulid 28710  ax-hvmulass 28711  ax-hvdistr1 28712  ax-hvdistr2 28713  ax-hvmul0 28714  ax-hfi 28783  ax-his1 28786  ax-his3 28788  ax-his4 28789
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-nel 3121  df-ral 3140  df-rex 3141  df-reu 3142  df-rmo 3143  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-pss 3951  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-tp 4562  df-op 4564  df-uni 4831  df-int 4868  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-riota 7103  df-ov 7148  df-oprab 7149  df-mpo 7150  df-om 7570  df-2nd 7679  df-wrecs 7936  df-recs 7997  df-rdg 8035  df-er 8278  df-en 8498  df-dom 8499  df-sdom 8500  df-sup 8894  df-pnf 10665  df-mnf 10666  df-xr 10667  df-ltxr 10668  df-le 10669  df-sub 10860  df-neg 10861  df-div 11286  df-nn 11627  df-2 11688  df-3 11689  df-n0 11886  df-z 11970  df-uz 12232  df-rp 12378  df-seq 13358  df-exp 13418  df-cj 14446  df-re 14447  df-im 14448  df-sqrt 14582  df-abs 14583  df-grpo 28197  df-ablo 28249  df-hnorm 28672  df-hvsub 28675  df-sh 28911  df-ch0 28957  df-shs 29012
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator