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Theorem cdj3lem2b 28680
Description: Lemma for cdj3i 28684. The first-component function 𝑆 is bounded if the subspaces are completely disjoint. (Contributed by NM, 26-May-2005.) (New usage is discouraged.)
Hypotheses
Ref Expression
cdj3lem2.1 𝐴S
cdj3lem2.2 𝐵S
cdj3lem2.3 𝑆 = (𝑥 ∈ (𝐴 + 𝐵) ↦ (𝑧𝐴𝑤𝐵 𝑥 = (𝑧 + 𝑤)))
Assertion
Ref Expression
cdj3lem2b (∃𝑣 ∈ ℝ (0 < 𝑣 ∧ ∀𝑥𝐴𝑦𝐵 ((norm𝑥) + (norm𝑦)) ≤ (𝑣 · (norm‘(𝑥 + 𝑦)))) → ∃𝑣 ∈ ℝ (0 < 𝑣 ∧ ∀𝑢 ∈ (𝐴 + 𝐵)(norm‘(𝑆𝑢)) ≤ (𝑣 · (norm𝑢))))
Distinct variable groups:   𝑥,𝑦,𝑧,𝑤,𝑣,𝑢,𝐴   𝑥,𝐵,𝑦,𝑧,𝑤,𝑣,𝑢   𝑣,𝑆,𝑢
Allowed substitution hints:   𝑆(𝑥,𝑦,𝑧,𝑤)

Proof of Theorem cdj3lem2b
Dummy variables 𝑡 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 cdj3lem2.1 . . 3 𝐴S
2 cdj3lem2.2 . . 3 𝐵S
31, 2cdj3lem1 28677 . 2 (∃𝑣 ∈ ℝ (0 < 𝑣 ∧ ∀𝑥𝐴𝑦𝐵 ((norm𝑥) + (norm𝑦)) ≤ (𝑣 · (norm‘(𝑥 + 𝑦)))) → (𝐴𝐵) = 0)
41, 2shseli 27559 . . . . . . . 8 (𝑢 ∈ (𝐴 + 𝐵) ↔ ∃𝑡𝐴𝐵 𝑢 = (𝑡 + ))
54biimpi 205 . . . . . . 7 (𝑢 ∈ (𝐴 + 𝐵) → ∃𝑡𝐴𝐵 𝑢 = (𝑡 + ))
6 fveq2 6103 . . . . . . . . . . . . . 14 (𝑥 = 𝑡 → (norm𝑥) = (norm𝑡))
76oveq1d 6564 . . . . . . . . . . . . 13 (𝑥 = 𝑡 → ((norm𝑥) + (norm𝑦)) = ((norm𝑡) + (norm𝑦)))
8 oveq1 6556 . . . . . . . . . . . . . . 15 (𝑥 = 𝑡 → (𝑥 + 𝑦) = (𝑡 + 𝑦))
98fveq2d 6107 . . . . . . . . . . . . . 14 (𝑥 = 𝑡 → (norm‘(𝑥 + 𝑦)) = (norm‘(𝑡 + 𝑦)))
109oveq2d 6565 . . . . . . . . . . . . 13 (𝑥 = 𝑡 → (𝑣 · (norm‘(𝑥 + 𝑦))) = (𝑣 · (norm‘(𝑡 + 𝑦))))
117, 10breq12d 4596 . . . . . . . . . . . 12 (𝑥 = 𝑡 → (((norm𝑥) + (norm𝑦)) ≤ (𝑣 · (norm‘(𝑥 + 𝑦))) ↔ ((norm𝑡) + (norm𝑦)) ≤ (𝑣 · (norm‘(𝑡 + 𝑦)))))
12 fveq2 6103 . . . . . . . . . . . . . 14 (𝑦 = → (norm𝑦) = (norm))
1312oveq2d 6565 . . . . . . . . . . . . 13 (𝑦 = → ((norm𝑡) + (norm𝑦)) = ((norm𝑡) + (norm)))
14 oveq2 6557 . . . . . . . . . . . . . . 15 (𝑦 = → (𝑡 + 𝑦) = (𝑡 + ))
1514fveq2d 6107 . . . . . . . . . . . . . 14 (𝑦 = → (norm‘(𝑡 + 𝑦)) = (norm‘(𝑡 + )))
1615oveq2d 6565 . . . . . . . . . . . . 13 (𝑦 = → (𝑣 · (norm‘(𝑡 + 𝑦))) = (𝑣 · (norm‘(𝑡 + ))))
1713, 16breq12d 4596 . . . . . . . . . . . 12 (𝑦 = → (((norm𝑡) + (norm𝑦)) ≤ (𝑣 · (norm‘(𝑡 + 𝑦))) ↔ ((norm𝑡) + (norm)) ≤ (𝑣 · (norm‘(𝑡 + )))))
1811, 17rspc2v 3293 . . . . . . . . . . 11 ((𝑡𝐴𝐵) → (∀𝑥𝐴𝑦𝐵 ((norm𝑥) + (norm𝑦)) ≤ (𝑣 · (norm‘(𝑥 + 𝑦))) → ((norm𝑡) + (norm)) ≤ (𝑣 · (norm‘(𝑡 + )))))
19 cdj3lem2.3 . . . . . . . . . . . . . . . . . 18 𝑆 = (𝑥 ∈ (𝐴 + 𝐵) ↦ (𝑧𝐴𝑤𝐵 𝑥 = (𝑧 + 𝑤)))
201, 2, 19cdj3lem2 28678 . . . . . . . . . . . . . . . . 17 ((𝑡𝐴𝐵 ∧ (𝐴𝐵) = 0) → (𝑆‘(𝑡 + )) = 𝑡)
21203expa 1257 . . . . . . . . . . . . . . . 16 (((𝑡𝐴𝐵) ∧ (𝐴𝐵) = 0) → (𝑆‘(𝑡 + )) = 𝑡)
2221fveq2d 6107 . . . . . . . . . . . . . . 15 (((𝑡𝐴𝐵) ∧ (𝐴𝐵) = 0) → (norm‘(𝑆‘(𝑡 + ))) = (norm𝑡))
2322ad2ant2r 779 . . . . . . . . . . . . . 14 ((((𝑡𝐴𝐵) ∧ ((norm𝑡) + (norm)) ≤ (𝑣 · (norm‘(𝑡 + )))) ∧ ((𝐴𝐵) = 0𝑣 ∈ ℝ)) → (norm‘(𝑆‘(𝑡 + ))) = (norm𝑡))
242sheli 27455 . . . . . . . . . . . . . . . . . . . . . 22 (𝐵 ∈ ℋ)
25 normge0 27367 . . . . . . . . . . . . . . . . . . . . . 22 ( ∈ ℋ → 0 ≤ (norm))
2624, 25syl 17 . . . . . . . . . . . . . . . . . . . . 21 (𝐵 → 0 ≤ (norm))
2726adantl 481 . . . . . . . . . . . . . . . . . . . 20 ((𝑡𝐴𝐵) → 0 ≤ (norm))
281sheli 27455 . . . . . . . . . . . . . . . . . . . . . 22 (𝑡𝐴𝑡 ∈ ℋ)
29 normcl 27366 . . . . . . . . . . . . . . . . . . . . . 22 (𝑡 ∈ ℋ → (norm𝑡) ∈ ℝ)
3028, 29syl 17 . . . . . . . . . . . . . . . . . . . . 21 (𝑡𝐴 → (norm𝑡) ∈ ℝ)
31 normcl 27366 . . . . . . . . . . . . . . . . . . . . . 22 ( ∈ ℋ → (norm) ∈ ℝ)
3224, 31syl 17 . . . . . . . . . . . . . . . . . . . . 21 (𝐵 → (norm) ∈ ℝ)
33 addge01 10417 . . . . . . . . . . . . . . . . . . . . 21 (((norm𝑡) ∈ ℝ ∧ (norm) ∈ ℝ) → (0 ≤ (norm) ↔ (norm𝑡) ≤ ((norm𝑡) + (norm))))
3430, 32, 33syl2an 493 . . . . . . . . . . . . . . . . . . . 20 ((𝑡𝐴𝐵) → (0 ≤ (norm) ↔ (norm𝑡) ≤ ((norm𝑡) + (norm))))
3527, 34mpbid 221 . . . . . . . . . . . . . . . . . . 19 ((𝑡𝐴𝐵) → (norm𝑡) ≤ ((norm𝑡) + (norm)))
3635adantr 480 . . . . . . . . . . . . . . . . . 18 (((𝑡𝐴𝐵) ∧ 𝑣 ∈ ℝ) → (norm𝑡) ≤ ((norm𝑡) + (norm)))
3730ad2antrr 758 . . . . . . . . . . . . . . . . . . 19 (((𝑡𝐴𝐵) ∧ 𝑣 ∈ ℝ) → (norm𝑡) ∈ ℝ)
38 readdcl 9898 . . . . . . . . . . . . . . . . . . . . 21 (((norm𝑡) ∈ ℝ ∧ (norm) ∈ ℝ) → ((norm𝑡) + (norm)) ∈ ℝ)
3930, 32, 38syl2an 493 . . . . . . . . . . . . . . . . . . . 20 ((𝑡𝐴𝐵) → ((norm𝑡) + (norm)) ∈ ℝ)
4039adantr 480 . . . . . . . . . . . . . . . . . . 19 (((𝑡𝐴𝐵) ∧ 𝑣 ∈ ℝ) → ((norm𝑡) + (norm)) ∈ ℝ)
41 hvaddcl 27253 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑡 ∈ ℋ ∧ ∈ ℋ) → (𝑡 + ) ∈ ℋ)
4228, 24, 41syl2an 493 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑡𝐴𝐵) → (𝑡 + ) ∈ ℋ)
43 normcl 27366 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑡 + ) ∈ ℋ → (norm‘(𝑡 + )) ∈ ℝ)
4442, 43syl 17 . . . . . . . . . . . . . . . . . . . . 21 ((𝑡𝐴𝐵) → (norm‘(𝑡 + )) ∈ ℝ)
45 remulcl 9900 . . . . . . . . . . . . . . . . . . . . 21 ((𝑣 ∈ ℝ ∧ (norm‘(𝑡 + )) ∈ ℝ) → (𝑣 · (norm‘(𝑡 + ))) ∈ ℝ)
4644, 45sylan2 490 . . . . . . . . . . . . . . . . . . . 20 ((𝑣 ∈ ℝ ∧ (𝑡𝐴𝐵)) → (𝑣 · (norm‘(𝑡 + ))) ∈ ℝ)
4746ancoms 468 . . . . . . . . . . . . . . . . . . 19 (((𝑡𝐴𝐵) ∧ 𝑣 ∈ ℝ) → (𝑣 · (norm‘(𝑡 + ))) ∈ ℝ)
48 letr 10010 . . . . . . . . . . . . . . . . . . 19 (((norm𝑡) ∈ ℝ ∧ ((norm𝑡) + (norm)) ∈ ℝ ∧ (𝑣 · (norm‘(𝑡 + ))) ∈ ℝ) → (((norm𝑡) ≤ ((norm𝑡) + (norm)) ∧ ((norm𝑡) + (norm)) ≤ (𝑣 · (norm‘(𝑡 + )))) → (norm𝑡) ≤ (𝑣 · (norm‘(𝑡 + )))))
4937, 40, 47, 48syl3anc 1318 . . . . . . . . . . . . . . . . . 18 (((𝑡𝐴𝐵) ∧ 𝑣 ∈ ℝ) → (((norm𝑡) ≤ ((norm𝑡) + (norm)) ∧ ((norm𝑡) + (norm)) ≤ (𝑣 · (norm‘(𝑡 + )))) → (norm𝑡) ≤ (𝑣 · (norm‘(𝑡 + )))))
5036, 49mpand 707 . . . . . . . . . . . . . . . . 17 (((𝑡𝐴𝐵) ∧ 𝑣 ∈ ℝ) → (((norm𝑡) + (norm)) ≤ (𝑣 · (norm‘(𝑡 + ))) → (norm𝑡) ≤ (𝑣 · (norm‘(𝑡 + )))))
5150imp 444 . . . . . . . . . . . . . . . 16 ((((𝑡𝐴𝐵) ∧ 𝑣 ∈ ℝ) ∧ ((norm𝑡) + (norm)) ≤ (𝑣 · (norm‘(𝑡 + )))) → (norm𝑡) ≤ (𝑣 · (norm‘(𝑡 + ))))
5251an32s 842 . . . . . . . . . . . . . . 15 ((((𝑡𝐴𝐵) ∧ ((norm𝑡) + (norm)) ≤ (𝑣 · (norm‘(𝑡 + )))) ∧ 𝑣 ∈ ℝ) → (norm𝑡) ≤ (𝑣 · (norm‘(𝑡 + ))))
5352adantrl 748 . . . . . . . . . . . . . 14 ((((𝑡𝐴𝐵) ∧ ((norm𝑡) + (norm)) ≤ (𝑣 · (norm‘(𝑡 + )))) ∧ ((𝐴𝐵) = 0𝑣 ∈ ℝ)) → (norm𝑡) ≤ (𝑣 · (norm‘(𝑡 + ))))
5423, 53eqbrtrd 4605 . . . . . . . . . . . . 13 ((((𝑡𝐴𝐵) ∧ ((norm𝑡) + (norm)) ≤ (𝑣 · (norm‘(𝑡 + )))) ∧ ((𝐴𝐵) = 0𝑣 ∈ ℝ)) → (norm‘(𝑆‘(𝑡 + ))) ≤ (𝑣 · (norm‘(𝑡 + ))))
55 fveq2 6103 . . . . . . . . . . . . . . 15 (𝑢 = (𝑡 + ) → (𝑆𝑢) = (𝑆‘(𝑡 + )))
5655fveq2d 6107 . . . . . . . . . . . . . 14 (𝑢 = (𝑡 + ) → (norm‘(𝑆𝑢)) = (norm‘(𝑆‘(𝑡 + ))))
57 fveq2 6103 . . . . . . . . . . . . . . 15 (𝑢 = (𝑡 + ) → (norm𝑢) = (norm‘(𝑡 + )))
5857oveq2d 6565 . . . . . . . . . . . . . 14 (𝑢 = (𝑡 + ) → (𝑣 · (norm𝑢)) = (𝑣 · (norm‘(𝑡 + ))))
5956, 58breq12d 4596 . . . . . . . . . . . . 13 (𝑢 = (𝑡 + ) → ((norm‘(𝑆𝑢)) ≤ (𝑣 · (norm𝑢)) ↔ (norm‘(𝑆‘(𝑡 + ))) ≤ (𝑣 · (norm‘(𝑡 + )))))
6054, 59syl5ibrcom 236 . . . . . . . . . . . 12 ((((𝑡𝐴𝐵) ∧ ((norm𝑡) + (norm)) ≤ (𝑣 · (norm‘(𝑡 + )))) ∧ ((𝐴𝐵) = 0𝑣 ∈ ℝ)) → (𝑢 = (𝑡 + ) → (norm‘(𝑆𝑢)) ≤ (𝑣 · (norm𝑢))))
6160exp31 628 . . . . . . . . . . 11 ((𝑡𝐴𝐵) → (((norm𝑡) + (norm)) ≤ (𝑣 · (norm‘(𝑡 + ))) → (((𝐴𝐵) = 0𝑣 ∈ ℝ) → (𝑢 = (𝑡 + ) → (norm‘(𝑆𝑢)) ≤ (𝑣 · (norm𝑢))))))
6218, 61syld 46 . . . . . . . . . 10 ((𝑡𝐴𝐵) → (∀𝑥𝐴𝑦𝐵 ((norm𝑥) + (norm𝑦)) ≤ (𝑣 · (norm‘(𝑥 + 𝑦))) → (((𝐴𝐵) = 0𝑣 ∈ ℝ) → (𝑢 = (𝑡 + ) → (norm‘(𝑆𝑢)) ≤ (𝑣 · (norm𝑢))))))
6362com14 94 . . . . . . . . 9 (𝑢 = (𝑡 + ) → (∀𝑥𝐴𝑦𝐵 ((norm𝑥) + (norm𝑦)) ≤ (𝑣 · (norm‘(𝑥 + 𝑦))) → (((𝐴𝐵) = 0𝑣 ∈ ℝ) → ((𝑡𝐴𝐵) → (norm‘(𝑆𝑢)) ≤ (𝑣 · (norm𝑢))))))
6463com4t 91 . . . . . . . 8 (((𝐴𝐵) = 0𝑣 ∈ ℝ) → ((𝑡𝐴𝐵) → (𝑢 = (𝑡 + ) → (∀𝑥𝐴𝑦𝐵 ((norm𝑥) + (norm𝑦)) ≤ (𝑣 · (norm‘(𝑥 + 𝑦))) → (norm‘(𝑆𝑢)) ≤ (𝑣 · (norm𝑢))))))
6564rexlimdvv 3019 . . . . . . 7 (((𝐴𝐵) = 0𝑣 ∈ ℝ) → (∃𝑡𝐴𝐵 𝑢 = (𝑡 + ) → (∀𝑥𝐴𝑦𝐵 ((norm𝑥) + (norm𝑦)) ≤ (𝑣 · (norm‘(𝑥 + 𝑦))) → (norm‘(𝑆𝑢)) ≤ (𝑣 · (norm𝑢)))))
665, 65syl5com 31 . . . . . 6 (𝑢 ∈ (𝐴 + 𝐵) → (((𝐴𝐵) = 0𝑣 ∈ ℝ) → (∀𝑥𝐴𝑦𝐵 ((norm𝑥) + (norm𝑦)) ≤ (𝑣 · (norm‘(𝑥 + 𝑦))) → (norm‘(𝑆𝑢)) ≤ (𝑣 · (norm𝑢)))))
6766com3l 87 . . . . 5 (((𝐴𝐵) = 0𝑣 ∈ ℝ) → (∀𝑥𝐴𝑦𝐵 ((norm𝑥) + (norm𝑦)) ≤ (𝑣 · (norm‘(𝑥 + 𝑦))) → (𝑢 ∈ (𝐴 + 𝐵) → (norm‘(𝑆𝑢)) ≤ (𝑣 · (norm𝑢)))))
6867ralrimdv 2951 . . . 4 (((𝐴𝐵) = 0𝑣 ∈ ℝ) → (∀𝑥𝐴𝑦𝐵 ((norm𝑥) + (norm𝑦)) ≤ (𝑣 · (norm‘(𝑥 + 𝑦))) → ∀𝑢 ∈ (𝐴 + 𝐵)(norm‘(𝑆𝑢)) ≤ (𝑣 · (norm𝑢))))
6968anim2d 587 . . 3 (((𝐴𝐵) = 0𝑣 ∈ ℝ) → ((0 < 𝑣 ∧ ∀𝑥𝐴𝑦𝐵 ((norm𝑥) + (norm𝑦)) ≤ (𝑣 · (norm‘(𝑥 + 𝑦)))) → (0 < 𝑣 ∧ ∀𝑢 ∈ (𝐴 + 𝐵)(norm‘(𝑆𝑢)) ≤ (𝑣 · (norm𝑢)))))
7069reximdva 3000 . 2 ((𝐴𝐵) = 0 → (∃𝑣 ∈ ℝ (0 < 𝑣 ∧ ∀𝑥𝐴𝑦𝐵 ((norm𝑥) + (norm𝑦)) ≤ (𝑣 · (norm‘(𝑥 + 𝑦)))) → ∃𝑣 ∈ ℝ (0 < 𝑣 ∧ ∀𝑢 ∈ (𝐴 + 𝐵)(norm‘(𝑆𝑢)) ≤ (𝑣 · (norm𝑢)))))
713, 70mpcom 37 1 (∃𝑣 ∈ ℝ (0 < 𝑣 ∧ ∀𝑥𝐴𝑦𝐵 ((norm𝑥) + (norm𝑦)) ≤ (𝑣 · (norm‘(𝑥 + 𝑦)))) → ∃𝑣 ∈ ℝ (0 < 𝑣 ∧ ∀𝑢 ∈ (𝐴 + 𝐵)(norm‘(𝑆𝑢)) ≤ (𝑣 · (norm𝑢))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 195  wa 383   = wceq 1475  wcel 1977  wral 2896  wrex 2897  cin 3539   class class class wbr 4583  cmpt 4643  cfv 5804  crio 6510  (class class class)co 6549  cr 9814  0cc0 9815   + caddc 9818   · cmul 9820   < clt 9953  cle 9954  chil 27160   + cva 27161  normcno 27164   S csh 27169   + cph 27172  0c0h 27176
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892  ax-pre-sup 9893  ax-hilex 27240  ax-hfvadd 27241  ax-hvcom 27242  ax-hvass 27243  ax-hv0cl 27244  ax-hvaddid 27245  ax-hfvmul 27246  ax-hvmulid 27247  ax-hvmulass 27248  ax-hvdistr1 27249  ax-hvdistr2 27250  ax-hvmul0 27251  ax-hfi 27320  ax-his1 27323  ax-his3 27325  ax-his4 27326
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-er 7629  df-en 7842  df-dom 7843  df-sdom 7844  df-sup 8231  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-div 10564  df-nn 10898  df-2 10956  df-3 10957  df-n0 11170  df-z 11255  df-uz 11564  df-rp 11709  df-seq 12664  df-exp 12723  df-cj 13687  df-re 13688  df-im 13689  df-sqrt 13823  df-abs 13824  df-grpo 26731  df-ablo 26783  df-hnorm 27209  df-hvsub 27212  df-sh 27448  df-ch0 27494  df-shs 27551
This theorem is referenced by:  cdj3lem3b  28683  cdj3i  28684
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