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Theorem cdj3i 31976
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 31969 . . 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 31972 . . . 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 233 . . 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 31975 . . . 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 233 . . 3 (∃𝑣 ∈ ℝ (0 < 𝑣 ∧ ∀𝑥𝐴𝑦𝐵 ((norm𝑥) + (norm𝑦)) ≤ (𝑣 · (norm‘(𝑥 + 𝑦)))) → 𝜓)
123, 7, 113jca 1127 . 2 (∃𝑣 ∈ ℝ (0 < 𝑣 ∧ ∀𝑥𝐴𝑦𝐵 ((norm𝑥) + (norm𝑦)) ≤ (𝑣 · (norm‘(𝑥 + 𝑦)))) → ((𝐴𝐵) = 0𝜑𝜓))
13 breq2 5152 . . . . . . . . 9 (𝑣 = 𝑓 → (0 < 𝑣 ↔ 0 < 𝑓))
14 oveq1 7419 . . . . . . . . . . 11 (𝑣 = 𝑓 → (𝑣 · (norm𝑢)) = (𝑓 · (norm𝑢)))
1514breq2d 5160 . . . . . . . . . 10 (𝑣 = 𝑓 → ((norm‘(𝑆𝑢)) ≤ (𝑣 · (norm𝑢)) ↔ (norm‘(𝑆𝑢)) ≤ (𝑓 · (norm𝑢))))
1615ralbidv 3176 . . . . . . . . 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 3234 . . . . . . 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 275 . . . . . 6 (𝜑 ↔ ∃𝑓 ∈ ℝ (0 < 𝑓 ∧ ∀𝑢 ∈ (𝐴 + 𝐵)(norm‘(𝑆𝑢)) ≤ (𝑓 · (norm𝑢))))
20 breq2 5152 . . . . . . . . 9 (𝑣 = 𝑔 → (0 < 𝑣 ↔ 0 < 𝑔))
21 oveq1 7419 . . . . . . . . . . 11 (𝑣 = 𝑔 → (𝑣 · (norm𝑢)) = (𝑔 · (norm𝑢)))
2221breq2d 5160 . . . . . . . . . 10 (𝑣 = 𝑔 → ((norm‘(𝑇𝑢)) ≤ (𝑣 · (norm𝑢)) ↔ (norm‘(𝑇𝑢)) ≤ (𝑔 · (norm𝑢))))
2322ralbidv 3176 . . . . . . . . 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 3234 . . . . . . 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 275 . . . . . 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 3225 . . . . 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 278 . . . 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 653 . . . . . 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 11707 . . . . . . . . 9 (((𝑓 ∈ ℝ ∧ 𝑔 ∈ ℝ) ∧ (0 < 𝑓 ∧ 0 < 𝑔)) → 0 < (𝑓 + 𝑔))
3231ex 412 . . . . . . . 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 30899 . . . . . . . . . . 11 ((𝑡𝐴𝐵) → (𝑡 + ) ∈ (𝐴 + 𝐵))
35 2fveq3 6896 . . . . . . . . . . . . . 14 (𝑢 = (𝑡 + ) → (norm‘(𝑆𝑢)) = (norm‘(𝑆‘(𝑡 + ))))
36 fveq2 6891 . . . . . . . . . . . . . . 15 (𝑢 = (𝑡 + ) → (norm𝑢) = (norm‘(𝑡 + )))
3736oveq2d 7428 . . . . . . . . . . . . . 14 (𝑢 = (𝑡 + ) → (𝑓 · (norm𝑢)) = (𝑓 · (norm‘(𝑡 + ))))
3835, 37breq12d 5161 . . . . . . . . . . . . 13 (𝑢 = (𝑡 + ) → ((norm‘(𝑆𝑢)) ≤ (𝑓 · (norm𝑢)) ↔ (norm‘(𝑆‘(𝑡 + ))) ≤ (𝑓 · (norm‘(𝑡 + )))))
3938rspcv 3608 . . . . . . . . . . . 12 ((𝑡 + ) ∈ (𝐴 + 𝐵) → (∀𝑢 ∈ (𝐴 + 𝐵)(norm‘(𝑆𝑢)) ≤ (𝑓 · (norm𝑢)) → (norm‘(𝑆‘(𝑡 + ))) ≤ (𝑓 · (norm‘(𝑡 + )))))
40 2fveq3 6896 . . . . . . . . . . . . . 14 (𝑢 = (𝑡 + ) → (norm‘(𝑇𝑢)) = (norm‘(𝑇‘(𝑡 + ))))
4136oveq2d 7428 . . . . . . . . . . . . . 14 (𝑢 = (𝑡 + ) → (𝑔 · (norm𝑢)) = (𝑔 · (norm‘(𝑡 + ))))
4240, 41breq12d 5161 . . . . . . . . . . . . 13 (𝑢 = (𝑡 + ) → ((norm‘(𝑇𝑢)) ≤ (𝑔 · (norm𝑢)) ↔ (norm‘(𝑇‘(𝑡 + ))) ≤ (𝑔 · (norm‘(𝑡 + )))))
4342rspcv 3608 . . . . . . . . . . . 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 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 + ))))))
471sheli 30749 . . . . . . . . . . . . . . 15 (𝑡𝐴𝑡 ∈ ℋ)
48 normcl 30660 . . . . . . . . . . . . . . 15 (𝑡 ∈ ℋ → (norm𝑡) ∈ ℝ)
4947, 48syl 17 . . . . . . . . . . . . . 14 (𝑡𝐴 → (norm𝑡) ∈ ℝ)
502sheli 30749 . . . . . . . . . . . . . . 15 (𝐵 ∈ ℋ)
51 normcl 30660 . . . . . . . . . . . . . . 15 ( ∈ ℋ → (norm) ∈ ℝ)
5250, 51syl 17 . . . . . . . . . . . . . 14 (𝐵 → (norm) ∈ ℝ)
5349, 52anim12i 612 . . . . . . . . . . . . 13 ((𝑡𝐴𝐵) → ((norm𝑡) ∈ ℝ ∧ (norm) ∈ ℝ))
5453adantl 481 . . . . . . . . . . . 12 (((𝑓 ∈ ℝ ∧ 𝑔 ∈ ℝ) ∧ (𝑡𝐴𝐵)) → ((norm𝑡) ∈ ℝ ∧ (norm) ∈ ℝ))
55 hvaddcl 30547 . . . . . . . . . . . . . . . 16 ((𝑡 ∈ ℋ ∧ ∈ ℋ) → (𝑡 + ) ∈ ℋ)
5647, 50, 55syl2an 595 . . . . . . . . . . . . . . 15 ((𝑡𝐴𝐵) → (𝑡 + ) ∈ ℋ)
57 normcl 30660 . . . . . . . . . . . . . . 15 ((𝑡 + ) ∈ ℋ → (norm‘(𝑡 + )) ∈ ℝ)
5856, 57syl 17 . . . . . . . . . . . . . 14 ((𝑡𝐴𝐵) → (norm‘(𝑡 + )) ∈ ℝ)
59 remulcl 11201 . . . . . . . . . . . . . 14 ((𝑓 ∈ ℝ ∧ (norm‘(𝑡 + )) ∈ ℝ) → (𝑓 · (norm‘(𝑡 + ))) ∈ ℝ)
6058, 59sylan2 592 . . . . . . . . . . . . 13 ((𝑓 ∈ ℝ ∧ (𝑡𝐴𝐵)) → (𝑓 · (norm‘(𝑡 + ))) ∈ ℝ)
6160adantlr 712 . . . . . . . . . . . 12 (((𝑓 ∈ ℝ ∧ 𝑔 ∈ ℝ) ∧ (𝑡𝐴𝐵)) → (𝑓 · (norm‘(𝑡 + ))) ∈ ℝ)
62 remulcl 11201 . . . . . . . . . . . . . 14 ((𝑔 ∈ ℝ ∧ (norm‘(𝑡 + )) ∈ ℝ) → (𝑔 · (norm‘(𝑡 + ))) ∈ ℝ)
6358, 62sylan2 592 . . . . . . . . . . . . 13 ((𝑔 ∈ ℝ ∧ (𝑡𝐴𝐵)) → (𝑔 · (norm‘(𝑡 + ))) ∈ ℝ)
6463adantll 711 . . . . . . . . . . . 12 (((𝑓 ∈ ℝ ∧ 𝑔 ∈ ℝ) ∧ (𝑡𝐴𝐵)) → (𝑔 · (norm‘(𝑡 + ))) ∈ ℝ)
65 le2add 11703 . . . . . . . . . . . 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 834 . . . . . . . . . . 11 (((𝑓 ∈ ℝ ∧ 𝑔 ∈ ℝ) ∧ (𝑡𝐴𝐵)) → (((norm𝑡) ≤ (𝑓 · (norm‘(𝑡 + ))) ∧ (norm) ≤ (𝑔 · (norm‘(𝑡 + )))) → ((norm𝑡) + (norm)) ≤ ((𝑓 · (norm‘(𝑡 + ))) + (𝑔 · (norm‘(𝑡 + ))))))
6766adantll 711 . . . . . . . . . 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 31970 . . . . . . . . . . . . . . . 16 ((𝑡𝐴𝐵 ∧ (𝐴𝐵) = 0) → (𝑆‘(𝑡 + )) = 𝑡)
6968fveq2d 6895 . . . . . . . . . . . . . . 15 ((𝑡𝐴𝐵 ∧ (𝐴𝐵) = 0) → (norm‘(𝑆‘(𝑡 + ))) = (norm𝑡))
7069breq1d 5158 . . . . . . . . . . . . . 14 ((𝑡𝐴𝐵 ∧ (𝐴𝐵) = 0) → ((norm‘(𝑆‘(𝑡 + ))) ≤ (𝑓 · (norm‘(𝑡 + ))) ↔ (norm𝑡) ≤ (𝑓 · (norm‘(𝑡 + )))))
711, 2, 8cdj3lem3 31973 . . . . . . . . . . . . . . . 16 ((𝑡𝐴𝐵 ∧ (𝐴𝐵) = 0) → (𝑇‘(𝑡 + )) = )
7271fveq2d 6895 . . . . . . . . . . . . . . 15 ((𝑡𝐴𝐵 ∧ (𝐴𝐵) = 0) → (norm‘(𝑇‘(𝑡 + ))) = (norm))
7372breq1d 5158 . . . . . . . . . . . . . 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 1117 . . . . . . . . . . . 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 458 . . . . . . . . . . 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 712 . . . . . . . . . 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 11206 . . . . . . . . . . . . . 14 (𝑓 ∈ ℝ → 𝑓 ∈ ℂ)
79 recn 11206 . . . . . . . . . . . . . 14 (𝑔 ∈ ℝ → 𝑔 ∈ ℂ)
8058recnd 11249 . . . . . . . . . . . . . 14 ((𝑡𝐴𝐵) → (norm‘(𝑡 + )) ∈ ℂ)
81 adddir 11212 . . . . . . . . . . . . . 14 ((𝑓 ∈ ℂ ∧ 𝑔 ∈ ℂ ∧ (norm‘(𝑡 + )) ∈ ℂ) → ((𝑓 + 𝑔) · (norm‘(𝑡 + ))) = ((𝑓 · (norm‘(𝑡 + ))) + (𝑔 · (norm‘(𝑡 + )))))
8278, 79, 80, 81syl3an 1159 . . . . . . . . . . . . 13 ((𝑓 ∈ ℝ ∧ 𝑔 ∈ ℝ ∧ (𝑡𝐴𝐵)) → ((𝑓 + 𝑔) · (norm‘(𝑡 + ))) = ((𝑓 · (norm‘(𝑡 + ))) + (𝑔 · (norm‘(𝑡 + )))))
83823expa 1117 . . . . . . . . . . . 12 (((𝑓 ∈ ℝ ∧ 𝑔 ∈ ℝ) ∧ (𝑡𝐴𝐵)) → ((𝑓 + 𝑔) · (norm‘(𝑡 + ))) = ((𝑓 · (norm‘(𝑡 + ))) + (𝑔 · (norm‘(𝑡 + )))))
8483breq2d 5160 . . . . . . . . . . 11 (((𝑓 ∈ ℝ ∧ 𝑔 ∈ ℝ) ∧ (𝑡𝐴𝐵)) → (((norm𝑡) + (norm)) ≤ ((𝑓 + 𝑔) · (norm‘(𝑡 + ))) ↔ ((norm𝑡) + (norm)) ≤ ((𝑓 · (norm‘(𝑡 + ))) + (𝑔 · (norm‘(𝑡 + ))))))
8584adantll 711 . . . . . . . . . 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 294 . . . . . . . . 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 3208 . . . . . . 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 11199 . . . . . . . . 9 ((𝑓 ∈ ℝ ∧ 𝑔 ∈ ℝ) → (𝑓 + 𝑔) ∈ ℝ)
90 breq2 5152 . . . . . . . . . . . 12 (𝑣 = (𝑓 + 𝑔) → (0 < 𝑣 ↔ 0 < (𝑓 + 𝑔)))
91 fveq2 6891 . . . . . . . . . . . . . . . 16 (𝑥 = 𝑡 → (norm𝑥) = (norm𝑡))
9291oveq1d 7427 . . . . . . . . . . . . . . 15 (𝑥 = 𝑡 → ((norm𝑥) + (norm𝑦)) = ((norm𝑡) + (norm𝑦)))
93 fvoveq1 7435 . . . . . . . . . . . . . . . 16 (𝑥 = 𝑡 → (norm‘(𝑥 + 𝑦)) = (norm‘(𝑡 + 𝑦)))
9493oveq2d 7428 . . . . . . . . . . . . . . 15 (𝑥 = 𝑡 → (𝑣 · (norm‘(𝑥 + 𝑦))) = (𝑣 · (norm‘(𝑡 + 𝑦))))
9592, 94breq12d 5161 . . . . . . . . . . . . . 14 (𝑥 = 𝑡 → (((norm𝑥) + (norm𝑦)) ≤ (𝑣 · (norm‘(𝑥 + 𝑦))) ↔ ((norm𝑡) + (norm𝑦)) ≤ (𝑣 · (norm‘(𝑡 + 𝑦)))))
96 fveq2 6891 . . . . . . . . . . . . . . . 16 (𝑦 = → (norm𝑦) = (norm))
9796oveq2d 7428 . . . . . . . . . . . . . . 15 (𝑦 = → ((norm𝑡) + (norm𝑦)) = ((norm𝑡) + (norm)))
98 oveq2 7420 . . . . . . . . . . . . . . . . 17 (𝑦 = → (𝑡 + 𝑦) = (𝑡 + ))
9998fveq2d 6895 . . . . . . . . . . . . . . . 16 (𝑦 = → (norm‘(𝑡 + 𝑦)) = (norm‘(𝑡 + )))
10099oveq2d 7428 . . . . . . . . . . . . . . 15 (𝑦 = → (𝑣 · (norm‘(𝑡 + 𝑦))) = (𝑣 · (norm‘(𝑡 + ))))
10197, 100breq12d 5161 . . . . . . . . . . . . . 14 (𝑦 = → (((norm𝑡) + (norm𝑦)) ≤ (𝑣 · (norm‘(𝑡 + 𝑦))) ↔ ((norm𝑡) + (norm)) ≤ (𝑣 · (norm‘(𝑡 + )))))
10295, 101cbvral2vw 3237 . . . . . . . . . . . . 13 (∀𝑥𝐴𝑦𝐵 ((norm𝑥) + (norm𝑦)) ≤ (𝑣 · (norm‘(𝑥 + 𝑦))) ↔ ∀𝑡𝐴𝐵 ((norm𝑡) + (norm)) ≤ (𝑣 · (norm‘(𝑡 + ))))
103 oveq1 7419 . . . . . . . . . . . . . . 15 (𝑣 = (𝑓 + 𝑔) → (𝑣 · (norm‘(𝑡 + ))) = ((𝑓 + 𝑔) · (norm‘(𝑡 + ))))
104103breq2d 5160 . . . . . . . . . . . . . 14 (𝑣 = (𝑓 + 𝑔) → (((norm𝑡) + (norm)) ≤ (𝑣 · (norm‘(𝑡 + ))) ↔ ((norm𝑡) + (norm)) ≤ ((𝑓 + 𝑔) · (norm‘(𝑡 + )))))
1051042ralbidv 3217 . . . . . . . . . . . . 13 (𝑣 = (𝑓 + 𝑔) → (∀𝑡𝐴𝐵 ((norm𝑡) + (norm)) ≤ (𝑣 · (norm‘(𝑡 + ))) ↔ ∀𝑡𝐴𝐵 ((norm𝑡) + (norm)) ≤ ((𝑓 + 𝑔) · (norm‘(𝑡 + )))))
106102, 105bitrid 283 . . . . . . . . . . . 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 3612 . . . . . . . . . 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 412 . . . . . . . . 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 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))))))
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, 112biimtrid 241 . . . . 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 3210 . . . 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, 114biimtrid 241 . . 3 ((𝐴𝐵) = 0 → ((𝜑𝜓) → ∃𝑣 ∈ ℝ (0 < 𝑣 ∧ ∀𝑥𝐴𝑦𝐵 ((norm𝑥) + (norm𝑦)) ≤ (𝑣 · (norm‘(𝑥 + 𝑦))))))
1161153impib 1115 . 2 (((𝐴𝐵) = 0𝜑𝜓) → ∃𝑣 ∈ ℝ (0 < 𝑣 ∧ ∀𝑥𝐴𝑦𝐵 ((norm𝑥) + (norm𝑦)) ≤ (𝑣 · (norm‘(𝑥 + 𝑦)))))
11712, 116impbii 208 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 205  wa 395  w3a 1086   = wceq 1540  wcel 2105  wral 3060  wrex 3069  cin 3947   class class class wbr 5148  cmpt 5231  cfv 6543  crio 7367  (class class class)co 7412  cc 11114  cr 11115  0cc0 11116   + caddc 11119   · cmul 11121   < clt 11255  cle 11256  chba 30454   + cva 30455  normcno 30458   S csh 30463   + cph 30466  0c0h 30470
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7729  ax-cnex 11172  ax-resscn 11173  ax-1cn 11174  ax-icn 11175  ax-addcl 11176  ax-addrcl 11177  ax-mulcl 11178  ax-mulrcl 11179  ax-mulcom 11180  ax-addass 11181  ax-mulass 11182  ax-distr 11183  ax-i2m1 11184  ax-1ne0 11185  ax-1rid 11186  ax-rnegex 11187  ax-rrecex 11188  ax-cnre 11189  ax-pre-lttri 11190  ax-pre-lttrn 11191  ax-pre-ltadd 11192  ax-pre-mulgt0 11193  ax-pre-sup 11194  ax-hilex 30534  ax-hfvadd 30535  ax-hvcom 30536  ax-hvass 30537  ax-hv0cl 30538  ax-hvaddid 30539  ax-hfvmul 30540  ax-hvmulid 30541  ax-hvmulass 30542  ax-hvdistr1 30543  ax-hvdistr2 30544  ax-hvmul0 30545  ax-hfi 30614  ax-his1 30617  ax-his3 30619  ax-his4 30620
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-rmo 3375  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-int 4951  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7368  df-ov 7415  df-oprab 7416  df-mpo 7417  df-om 7860  df-2nd 7980  df-frecs 8272  df-wrecs 8303  df-recs 8377  df-rdg 8416  df-er 8709  df-en 8946  df-dom 8947  df-sdom 8948  df-sup 9443  df-pnf 11257  df-mnf 11258  df-xr 11259  df-ltxr 11260  df-le 11261  df-sub 11453  df-neg 11454  df-div 11879  df-nn 12220  df-2 12282  df-3 12283  df-n0 12480  df-z 12566  df-uz 12830  df-rp 12982  df-seq 13974  df-exp 14035  df-cj 15053  df-re 15054  df-im 15055  df-sqrt 15189  df-abs 15190  df-grpo 30028  df-ablo 30080  df-hnorm 30503  df-hvsub 30506  df-sh 30742  df-ch0 30788  df-shs 30843
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator