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Theorem cdj3lem2b 30264
 Description: Lemma for cdj3i 30268. 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 30261 . 2 (∃𝑣 ∈ ℝ (0 < 𝑣 ∧ ∀𝑥𝐴𝑦𝐵 ((norm𝑥) + (norm𝑦)) ≤ (𝑣 · (norm‘(𝑥 + 𝑦)))) → (𝐴𝐵) = 0)
41, 2shseli 29143 . . . . . . . 8 (𝑢 ∈ (𝐴 + 𝐵) ↔ ∃𝑡𝐴𝐵 𝑢 = (𝑡 + ))
54biimpi 219 . . . . . . 7 (𝑢 ∈ (𝐴 + 𝐵) → ∃𝑡𝐴𝐵 𝑢 = (𝑡 + ))
6 fveq2 6655 . . . . . . . . . . . . . 14 (𝑥 = 𝑡 → (norm𝑥) = (norm𝑡))
76oveq1d 7160 . . . . . . . . . . . . 13 (𝑥 = 𝑡 → ((norm𝑥) + (norm𝑦)) = ((norm𝑡) + (norm𝑦)))
8 fvoveq1 7168 . . . . . . . . . . . . . 14 (𝑥 = 𝑡 → (norm‘(𝑥 + 𝑦)) = (norm‘(𝑡 + 𝑦)))
98oveq2d 7161 . . . . . . . . . . . . 13 (𝑥 = 𝑡 → (𝑣 · (norm‘(𝑥 + 𝑦))) = (𝑣 · (norm‘(𝑡 + 𝑦))))
107, 9breq12d 5047 . . . . . . . . . . . 12 (𝑥 = 𝑡 → (((norm𝑥) + (norm𝑦)) ≤ (𝑣 · (norm‘(𝑥 + 𝑦))) ↔ ((norm𝑡) + (norm𝑦)) ≤ (𝑣 · (norm‘(𝑡 + 𝑦)))))
11 fveq2 6655 . . . . . . . . . . . . . 14 (𝑦 = → (norm𝑦) = (norm))
1211oveq2d 7161 . . . . . . . . . . . . 13 (𝑦 = → ((norm𝑡) + (norm𝑦)) = ((norm𝑡) + (norm)))
13 oveq2 7153 . . . . . . . . . . . . . . 15 (𝑦 = → (𝑡 + 𝑦) = (𝑡 + ))
1413fveq2d 6659 . . . . . . . . . . . . . 14 (𝑦 = → (norm‘(𝑡 + 𝑦)) = (norm‘(𝑡 + )))
1514oveq2d 7161 . . . . . . . . . . . . 13 (𝑦 = → (𝑣 · (norm‘(𝑡 + 𝑦))) = (𝑣 · (norm‘(𝑡 + ))))
1612, 15breq12d 5047 . . . . . . . . . . . 12 (𝑦 = → (((norm𝑡) + (norm𝑦)) ≤ (𝑣 · (norm‘(𝑡 + 𝑦))) ↔ ((norm𝑡) + (norm)) ≤ (𝑣 · (norm‘(𝑡 + )))))
1710, 16rspc2v 3582 . . . . . . . . . . 11 ((𝑡𝐴𝐵) → (∀𝑥𝐴𝑦𝐵 ((norm𝑥) + (norm𝑦)) ≤ (𝑣 · (norm‘(𝑥 + 𝑦))) → ((norm𝑡) + (norm)) ≤ (𝑣 · (norm‘(𝑡 + )))))
18 cdj3lem2.3 . . . . . . . . . . . . . . . . . 18 𝑆 = (𝑥 ∈ (𝐴 + 𝐵) ↦ (𝑧𝐴𝑤𝐵 𝑥 = (𝑧 + 𝑤)))
191, 2, 18cdj3lem2 30262 . . . . . . . . . . . . . . . . 17 ((𝑡𝐴𝐵 ∧ (𝐴𝐵) = 0) → (𝑆‘(𝑡 + )) = 𝑡)
20193expa 1115 . . . . . . . . . . . . . . . 16 (((𝑡𝐴𝐵) ∧ (𝐴𝐵) = 0) → (𝑆‘(𝑡 + )) = 𝑡)
2120fveq2d 6659 . . . . . . . . . . . . . . 15 (((𝑡𝐴𝐵) ∧ (𝐴𝐵) = 0) → (norm‘(𝑆‘(𝑡 + ))) = (norm𝑡))
2221ad2ant2r 746 . . . . . . . . . . . . . 14 ((((𝑡𝐴𝐵) ∧ ((norm𝑡) + (norm)) ≤ (𝑣 · (norm‘(𝑡 + )))) ∧ ((𝐴𝐵) = 0𝑣 ∈ ℝ)) → (norm‘(𝑆‘(𝑡 + ))) = (norm𝑡))
232sheli 29041 . . . . . . . . . . . . . . . . . . . . . 22 (𝐵 ∈ ℋ)
24 normge0 28953 . . . . . . . . . . . . . . . . . . . . . 22 ( ∈ ℋ → 0 ≤ (norm))
2523, 24syl 17 . . . . . . . . . . . . . . . . . . . . 21 (𝐵 → 0 ≤ (norm))
2625adantl 485 . . . . . . . . . . . . . . . . . . . 20 ((𝑡𝐴𝐵) → 0 ≤ (norm))
271sheli 29041 . . . . . . . . . . . . . . . . . . . . . 22 (𝑡𝐴𝑡 ∈ ℋ)
28 normcl 28952 . . . . . . . . . . . . . . . . . . . . . 22 (𝑡 ∈ ℋ → (norm𝑡) ∈ ℝ)
2927, 28syl 17 . . . . . . . . . . . . . . . . . . . . 21 (𝑡𝐴 → (norm𝑡) ∈ ℝ)
30 normcl 28952 . . . . . . . . . . . . . . . . . . . . . 22 ( ∈ ℋ → (norm) ∈ ℝ)
3123, 30syl 17 . . . . . . . . . . . . . . . . . . . . 21 (𝐵 → (norm) ∈ ℝ)
32 addge01 11157 . . . . . . . . . . . . . . . . . . . . 21 (((norm𝑡) ∈ ℝ ∧ (norm) ∈ ℝ) → (0 ≤ (norm) ↔ (norm𝑡) ≤ ((norm𝑡) + (norm))))
3329, 31, 32syl2an 598 . . . . . . . . . . . . . . . . . . . 20 ((𝑡𝐴𝐵) → (0 ≤ (norm) ↔ (norm𝑡) ≤ ((norm𝑡) + (norm))))
3426, 33mpbid 235 . . . . . . . . . . . . . . . . . . 19 ((𝑡𝐴𝐵) → (norm𝑡) ≤ ((norm𝑡) + (norm)))
3534adantr 484 . . . . . . . . . . . . . . . . . 18 (((𝑡𝐴𝐵) ∧ 𝑣 ∈ ℝ) → (norm𝑡) ≤ ((norm𝑡) + (norm)))
3629ad2antrr 725 . . . . . . . . . . . . . . . . . . 19 (((𝑡𝐴𝐵) ∧ 𝑣 ∈ ℝ) → (norm𝑡) ∈ ℝ)
37 readdcl 10627 . . . . . . . . . . . . . . . . . . . . 21 (((norm𝑡) ∈ ℝ ∧ (norm) ∈ ℝ) → ((norm𝑡) + (norm)) ∈ ℝ)
3829, 31, 37syl2an 598 . . . . . . . . . . . . . . . . . . . 20 ((𝑡𝐴𝐵) → ((norm𝑡) + (norm)) ∈ ℝ)
3938adantr 484 . . . . . . . . . . . . . . . . . . 19 (((𝑡𝐴𝐵) ∧ 𝑣 ∈ ℝ) → ((norm𝑡) + (norm)) ∈ ℝ)
40 hvaddcl 28839 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑡 ∈ ℋ ∧ ∈ ℋ) → (𝑡 + ) ∈ ℋ)
4127, 23, 40syl2an 598 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑡𝐴𝐵) → (𝑡 + ) ∈ ℋ)
42 normcl 28952 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑡 + ) ∈ ℋ → (norm‘(𝑡 + )) ∈ ℝ)
4341, 42syl 17 . . . . . . . . . . . . . . . . . . . . 21 ((𝑡𝐴𝐵) → (norm‘(𝑡 + )) ∈ ℝ)
44 remulcl 10629 . . . . . . . . . . . . . . . . . . . . 21 ((𝑣 ∈ ℝ ∧ (norm‘(𝑡 + )) ∈ ℝ) → (𝑣 · (norm‘(𝑡 + ))) ∈ ℝ)
4543, 44sylan2 595 . . . . . . . . . . . . . . . . . . . 20 ((𝑣 ∈ ℝ ∧ (𝑡𝐴𝐵)) → (𝑣 · (norm‘(𝑡 + ))) ∈ ℝ)
4645ancoms 462 . . . . . . . . . . . . . . . . . . 19 (((𝑡𝐴𝐵) ∧ 𝑣 ∈ ℝ) → (𝑣 · (norm‘(𝑡 + ))) ∈ ℝ)
47 letr 10741 . . . . . . . . . . . . . . . . . . 19 (((norm𝑡) ∈ ℝ ∧ ((norm𝑡) + (norm)) ∈ ℝ ∧ (𝑣 · (norm‘(𝑡 + ))) ∈ ℝ) → (((norm𝑡) ≤ ((norm𝑡) + (norm)) ∧ ((norm𝑡) + (norm)) ≤ (𝑣 · (norm‘(𝑡 + )))) → (norm𝑡) ≤ (𝑣 · (norm‘(𝑡 + )))))
4836, 39, 46, 47syl3anc 1368 . . . . . . . . . . . . . . . . . 18 (((𝑡𝐴𝐵) ∧ 𝑣 ∈ ℝ) → (((norm𝑡) ≤ ((norm𝑡) + (norm)) ∧ ((norm𝑡) + (norm)) ≤ (𝑣 · (norm‘(𝑡 + )))) → (norm𝑡) ≤ (𝑣 · (norm‘(𝑡 + )))))
4935, 48mpand 694 . . . . . . . . . . . . . . . . 17 (((𝑡𝐴𝐵) ∧ 𝑣 ∈ ℝ) → (((norm𝑡) + (norm)) ≤ (𝑣 · (norm‘(𝑡 + ))) → (norm𝑡) ≤ (𝑣 · (norm‘(𝑡 + )))))
5049imp 410 . . . . . . . . . . . . . . . 16 ((((𝑡𝐴𝐵) ∧ 𝑣 ∈ ℝ) ∧ ((norm𝑡) + (norm)) ≤ (𝑣 · (norm‘(𝑡 + )))) → (norm𝑡) ≤ (𝑣 · (norm‘(𝑡 + ))))
5150an32s 651 . . . . . . . . . . . . . . 15 ((((𝑡𝐴𝐵) ∧ ((norm𝑡) + (norm)) ≤ (𝑣 · (norm‘(𝑡 + )))) ∧ 𝑣 ∈ ℝ) → (norm𝑡) ≤ (𝑣 · (norm‘(𝑡 + ))))
5251adantrl 715 . . . . . . . . . . . . . 14 ((((𝑡𝐴𝐵) ∧ ((norm𝑡) + (norm)) ≤ (𝑣 · (norm‘(𝑡 + )))) ∧ ((𝐴𝐵) = 0𝑣 ∈ ℝ)) → (norm𝑡) ≤ (𝑣 · (norm‘(𝑡 + ))))
5322, 52eqbrtrd 5056 . . . . . . . . . . . . 13 ((((𝑡𝐴𝐵) ∧ ((norm𝑡) + (norm)) ≤ (𝑣 · (norm‘(𝑡 + )))) ∧ ((𝐴𝐵) = 0𝑣 ∈ ℝ)) → (norm‘(𝑆‘(𝑡 + ))) ≤ (𝑣 · (norm‘(𝑡 + ))))
54 2fveq3 6660 . . . . . . . . . . . . . 14 (𝑢 = (𝑡 + ) → (norm‘(𝑆𝑢)) = (norm‘(𝑆‘(𝑡 + ))))
55 fveq2 6655 . . . . . . . . . . . . . . 15 (𝑢 = (𝑡 + ) → (norm𝑢) = (norm‘(𝑡 + )))
5655oveq2d 7161 . . . . . . . . . . . . . 14 (𝑢 = (𝑡 + ) → (𝑣 · (norm𝑢)) = (𝑣 · (norm‘(𝑡 + ))))
5754, 56breq12d 5047 . . . . . . . . . . . . 13 (𝑢 = (𝑡 + ) → ((norm‘(𝑆𝑢)) ≤ (𝑣 · (norm𝑢)) ↔ (norm‘(𝑆‘(𝑡 + ))) ≤ (𝑣 · (norm‘(𝑡 + )))))
5853, 57syl5ibrcom 250 . . . . . . . . . . . 12 ((((𝑡𝐴𝐵) ∧ ((norm𝑡) + (norm)) ≤ (𝑣 · (norm‘(𝑡 + )))) ∧ ((𝐴𝐵) = 0𝑣 ∈ ℝ)) → (𝑢 = (𝑡 + ) → (norm‘(𝑆𝑢)) ≤ (𝑣 · (norm𝑢))))
5958exp31 423 . . . . . . . . . . 11 ((𝑡𝐴𝐵) → (((norm𝑡) + (norm)) ≤ (𝑣 · (norm‘(𝑡 + ))) → (((𝐴𝐵) = 0𝑣 ∈ ℝ) → (𝑢 = (𝑡 + ) → (norm‘(𝑆𝑢)) ≤ (𝑣 · (norm𝑢))))))
6017, 59syld 47 . . . . . . . . . 10 ((𝑡𝐴𝐵) → (∀𝑥𝐴𝑦𝐵 ((norm𝑥) + (norm𝑦)) ≤ (𝑣 · (norm‘(𝑥 + 𝑦))) → (((𝐴𝐵) = 0𝑣 ∈ ℝ) → (𝑢 = (𝑡 + ) → (norm‘(𝑆𝑢)) ≤ (𝑣 · (norm𝑢))))))
6160com14 96 . . . . . . . . 9 (𝑢 = (𝑡 + ) → (∀𝑥𝐴𝑦𝐵 ((norm𝑥) + (norm𝑦)) ≤ (𝑣 · (norm‘(𝑥 + 𝑦))) → (((𝐴𝐵) = 0𝑣 ∈ ℝ) → ((𝑡𝐴𝐵) → (norm‘(𝑆𝑢)) ≤ (𝑣 · (norm𝑢))))))
6261com4t 93 . . . . . . . 8 (((𝐴𝐵) = 0𝑣 ∈ ℝ) → ((𝑡𝐴𝐵) → (𝑢 = (𝑡 + ) → (∀𝑥𝐴𝑦𝐵 ((norm𝑥) + (norm𝑦)) ≤ (𝑣 · (norm‘(𝑥 + 𝑦))) → (norm‘(𝑆𝑢)) ≤ (𝑣 · (norm𝑢))))))
6362rexlimdvv 3253 . . . . . . 7 (((𝐴𝐵) = 0𝑣 ∈ ℝ) → (∃𝑡𝐴𝐵 𝑢 = (𝑡 + ) → (∀𝑥𝐴𝑦𝐵 ((norm𝑥) + (norm𝑦)) ≤ (𝑣 · (norm‘(𝑥 + 𝑦))) → (norm‘(𝑆𝑢)) ≤ (𝑣 · (norm𝑢)))))
645, 63syl5com 31 . . . . . 6 (𝑢 ∈ (𝐴 + 𝐵) → (((𝐴𝐵) = 0𝑣 ∈ ℝ) → (∀𝑥𝐴𝑦𝐵 ((norm𝑥) + (norm𝑦)) ≤ (𝑣 · (norm‘(𝑥 + 𝑦))) → (norm‘(𝑆𝑢)) ≤ (𝑣 · (norm𝑢)))))
6564com3l 89 . . . . 5 (((𝐴𝐵) = 0𝑣 ∈ ℝ) → (∀𝑥𝐴𝑦𝐵 ((norm𝑥) + (norm𝑦)) ≤ (𝑣 · (norm‘(𝑥 + 𝑦))) → (𝑢 ∈ (𝐴 + 𝐵) → (norm‘(𝑆𝑢)) ≤ (𝑣 · (norm𝑢)))))
6665ralrimdv 3153 . . . 4 (((𝐴𝐵) = 0𝑣 ∈ ℝ) → (∀𝑥𝐴𝑦𝐵 ((norm𝑥) + (norm𝑦)) ≤ (𝑣 · (norm‘(𝑥 + 𝑦))) → ∀𝑢 ∈ (𝐴 + 𝐵)(norm‘(𝑆𝑢)) ≤ (𝑣 · (norm𝑢))))
6766anim2d 614 . . 3 (((𝐴𝐵) = 0𝑣 ∈ ℝ) → ((0 < 𝑣 ∧ ∀𝑥𝐴𝑦𝐵 ((norm𝑥) + (norm𝑦)) ≤ (𝑣 · (norm‘(𝑥 + 𝑦)))) → (0 < 𝑣 ∧ ∀𝑢 ∈ (𝐴 + 𝐵)(norm‘(𝑆𝑢)) ≤ (𝑣 · (norm𝑢)))))
6867reximdva 3234 . 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)))))
693, 68mpcom 38 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 209   ∧ wa 399   = wceq 1538   ∈ wcel 2111  ∀wral 3106  ∃wrex 3107   ∩ cin 3882   class class class wbr 5034   ↦ cmpt 5114  ‘cfv 6332  ℩crio 7102  (class class class)co 7145  ℝcr 10543  0cc0 10544   + caddc 10547   · cmul 10549   < clt 10682   ≤ cle 10683   ℋchba 28746   +ℎ cva 28747  normℎcno 28750   Sℋ csh 28755   +ℋ cph 28758  0ℋc0h 28762 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5158  ax-sep 5171  ax-nul 5178  ax-pow 5235  ax-pr 5299  ax-un 7454  ax-cnex 10600  ax-resscn 10601  ax-1cn 10602  ax-icn 10603  ax-addcl 10604  ax-addrcl 10605  ax-mulcl 10606  ax-mulrcl 10607  ax-mulcom 10608  ax-addass 10609  ax-mulass 10610  ax-distr 10611  ax-i2m1 10612  ax-1ne0 10613  ax-1rid 10614  ax-rnegex 10615  ax-rrecex 10616  ax-cnre 10617  ax-pre-lttri 10618  ax-pre-lttrn 10619  ax-pre-ltadd 10620  ax-pre-mulgt0 10621  ax-pre-sup 10622  ax-hilex 28826  ax-hfvadd 28827  ax-hvcom 28828  ax-hvass 28829  ax-hv0cl 28830  ax-hvaddid 28831  ax-hfvmul 28832  ax-hvmulid 28833  ax-hvmulass 28834  ax-hvdistr1 28835  ax-hvdistr2 28836  ax-hvmul0 28837  ax-hfi 28906  ax-his1 28909  ax-his3 28911  ax-his4 28912 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-nel 3092  df-ral 3111  df-rex 3112  df-reu 3113  df-rmo 3114  df-rab 3115  df-v 3444  df-sbc 3723  df-csb 3831  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-tp 4533  df-op 4535  df-uni 4805  df-int 4843  df-iun 4887  df-br 5035  df-opab 5097  df-mpt 5115  df-tr 5141  df-id 5429  df-eprel 5434  df-po 5442  df-so 5443  df-fr 5482  df-we 5484  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-pred 6123  df-ord 6169  df-on 6170  df-lim 6171  df-suc 6172  df-iota 6291  df-fun 6334  df-fn 6335  df-f 6336  df-f1 6337  df-fo 6338  df-f1o 6339  df-fv 6340  df-riota 7103  df-ov 7148  df-oprab 7149  df-mpo 7150  df-om 7574  df-2nd 7685  df-wrecs 7948  df-recs 8009  df-rdg 8047  df-er 8290  df-en 8511  df-dom 8512  df-sdom 8513  df-sup 8908  df-pnf 10684  df-mnf 10685  df-xr 10686  df-ltxr 10687  df-le 10688  df-sub 10879  df-neg 10880  df-div 11305  df-nn 11644  df-2 11706  df-3 11707  df-n0 11904  df-z 11990  df-uz 12252  df-rp 12398  df-seq 13385  df-exp 13446  df-cj 14470  df-re 14471  df-im 14472  df-sqrt 14606  df-abs 14607  df-grpo 28320  df-ablo 28372  df-hnorm 28795  df-hvsub 28798  df-sh 29034  df-ch0 29080  df-shs 29135 This theorem is referenced by:  cdj3lem3b  30267  cdj3i  30268
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