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Theorem cdj3lem3b 30131
Description: Lemma for cdj3i 30132. The second-component function 𝑇 is bounded if the subspaces are completely disjoint. (Contributed by NM, 31-May-2005.) (New usage is discouraged.)
Hypotheses
Ref Expression
cdj3lem2.1 𝐴S
cdj3lem2.2 𝐵S
cdj3lem3.3 𝑇 = (𝑥 ∈ (𝐴 + 𝐵) ↦ (𝑤𝐵𝑧𝐴 𝑥 = (𝑧 + 𝑤)))
Assertion
Ref Expression
cdj3lem3b (∃𝑣 ∈ ℝ (0 < 𝑣 ∧ ∀𝑥𝐴𝑦𝐵 ((norm𝑥) + (norm𝑦)) ≤ (𝑣 · (norm‘(𝑥 + 𝑦)))) → ∃𝑣 ∈ ℝ (0 < 𝑣 ∧ ∀𝑢 ∈ (𝐴 + 𝐵)(norm‘(𝑇𝑢)) ≤ (𝑣 · (norm𝑢))))
Distinct variable groups:   𝑥,𝑦,𝑧,𝑤,𝑣,𝑢,𝐴   𝑥,𝐵,𝑦,𝑧,𝑤,𝑣,𝑢   𝑣,𝑇,𝑢
Allowed substitution hints:   𝑇(𝑥,𝑦,𝑧,𝑤)

Proof of Theorem cdj3lem3b
Dummy variables 𝑡 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 cdj3lem2.2 . . 3 𝐵S
2 cdj3lem2.1 . . 3 𝐴S
3 cdj3lem3.3 . . . 4 𝑇 = (𝑥 ∈ (𝐴 + 𝐵) ↦ (𝑤𝐵𝑧𝐴 𝑥 = (𝑧 + 𝑤)))
41, 2shscomi 29054 . . . . 5 (𝐵 + 𝐴) = (𝐴 + 𝐵)
51sheli 28905 . . . . . . . . 9 (𝑤𝐵𝑤 ∈ ℋ)
62sheli 28905 . . . . . . . . 9 (𝑧𝐴𝑧 ∈ ℋ)
7 ax-hvcom 28692 . . . . . . . . 9 ((𝑤 ∈ ℋ ∧ 𝑧 ∈ ℋ) → (𝑤 + 𝑧) = (𝑧 + 𝑤))
85, 6, 7syl2an 595 . . . . . . . 8 ((𝑤𝐵𝑧𝐴) → (𝑤 + 𝑧) = (𝑧 + 𝑤))
98eqeq2d 2837 . . . . . . 7 ((𝑤𝐵𝑧𝐴) → (𝑥 = (𝑤 + 𝑧) ↔ 𝑥 = (𝑧 + 𝑤)))
109rexbidva 3301 . . . . . 6 (𝑤𝐵 → (∃𝑧𝐴 𝑥 = (𝑤 + 𝑧) ↔ ∃𝑧𝐴 𝑥 = (𝑧 + 𝑤)))
1110riotabiia 7126 . . . . 5 (𝑤𝐵𝑧𝐴 𝑥 = (𝑤 + 𝑧)) = (𝑤𝐵𝑧𝐴 𝑥 = (𝑧 + 𝑤))
124, 11mpteq12i 5156 . . . 4 (𝑥 ∈ (𝐵 + 𝐴) ↦ (𝑤𝐵𝑧𝐴 𝑥 = (𝑤 + 𝑧))) = (𝑥 ∈ (𝐴 + 𝐵) ↦ (𝑤𝐵𝑧𝐴 𝑥 = (𝑧 + 𝑤)))
133, 12eqtr4i 2852 . . 3 𝑇 = (𝑥 ∈ (𝐵 + 𝐴) ↦ (𝑤𝐵𝑧𝐴 𝑥 = (𝑤 + 𝑧)))
141, 2, 13cdj3lem2b 30128 . 2 (∃𝑣 ∈ ℝ (0 < 𝑣 ∧ ∀𝑥𝐵𝑦𝐴 ((norm𝑥) + (norm𝑦)) ≤ (𝑣 · (norm‘(𝑥 + 𝑦)))) → ∃𝑣 ∈ ℝ (0 < 𝑣 ∧ ∀𝑢 ∈ (𝐵 + 𝐴)(norm‘(𝑇𝑢)) ≤ (𝑣 · (norm𝑢))))
15 fveq2 6667 . . . . . . . 8 (𝑥 = 𝑡 → (norm𝑥) = (norm𝑡))
1615oveq1d 7163 . . . . . . 7 (𝑥 = 𝑡 → ((norm𝑥) + (norm𝑦)) = ((norm𝑡) + (norm𝑦)))
17 fvoveq1 7171 . . . . . . . 8 (𝑥 = 𝑡 → (norm‘(𝑥 + 𝑦)) = (norm‘(𝑡 + 𝑦)))
1817oveq2d 7164 . . . . . . 7 (𝑥 = 𝑡 → (𝑣 · (norm‘(𝑥 + 𝑦))) = (𝑣 · (norm‘(𝑡 + 𝑦))))
1916, 18breq12d 5076 . . . . . 6 (𝑥 = 𝑡 → (((norm𝑥) + (norm𝑦)) ≤ (𝑣 · (norm‘(𝑥 + 𝑦))) ↔ ((norm𝑡) + (norm𝑦)) ≤ (𝑣 · (norm‘(𝑡 + 𝑦)))))
20 fveq2 6667 . . . . . . . 8 (𝑦 = → (norm𝑦) = (norm))
2120oveq2d 7164 . . . . . . 7 (𝑦 = → ((norm𝑡) + (norm𝑦)) = ((norm𝑡) + (norm)))
22 oveq2 7156 . . . . . . . . 9 (𝑦 = → (𝑡 + 𝑦) = (𝑡 + ))
2322fveq2d 6671 . . . . . . . 8 (𝑦 = → (norm‘(𝑡 + 𝑦)) = (norm‘(𝑡 + )))
2423oveq2d 7164 . . . . . . 7 (𝑦 = → (𝑣 · (norm‘(𝑡 + 𝑦))) = (𝑣 · (norm‘(𝑡 + ))))
2521, 24breq12d 5076 . . . . . 6 (𝑦 = → (((norm𝑡) + (norm𝑦)) ≤ (𝑣 · (norm‘(𝑡 + 𝑦))) ↔ ((norm𝑡) + (norm)) ≤ (𝑣 · (norm‘(𝑡 + )))))
2619, 25cbvral2v 3470 . . . . 5 (∀𝑥𝐴𝑦𝐵 ((norm𝑥) + (norm𝑦)) ≤ (𝑣 · (norm‘(𝑥 + 𝑦))) ↔ ∀𝑡𝐴𝐵 ((norm𝑡) + (norm)) ≤ (𝑣 · (norm‘(𝑡 + ))))
27 ralcom 3359 . . . . 5 (∀𝑡𝐴𝐵 ((norm𝑡) + (norm)) ≤ (𝑣 · (norm‘(𝑡 + ))) ↔ ∀𝐵𝑡𝐴 ((norm𝑡) + (norm)) ≤ (𝑣 · (norm‘(𝑡 + ))))
281sheli 28905 . . . . . . . . . . . 12 (𝑥𝐵𝑥 ∈ ℋ)
29 normcl 28816 . . . . . . . . . . . 12 (𝑥 ∈ ℋ → (norm𝑥) ∈ ℝ)
3028, 29syl 17 . . . . . . . . . . 11 (𝑥𝐵 → (norm𝑥) ∈ ℝ)
3130recnd 10658 . . . . . . . . . 10 (𝑥𝐵 → (norm𝑥) ∈ ℂ)
322sheli 28905 . . . . . . . . . . . 12 (𝑦𝐴𝑦 ∈ ℋ)
33 normcl 28816 . . . . . . . . . . . 12 (𝑦 ∈ ℋ → (norm𝑦) ∈ ℝ)
3432, 33syl 17 . . . . . . . . . . 11 (𝑦𝐴 → (norm𝑦) ∈ ℝ)
3534recnd 10658 . . . . . . . . . 10 (𝑦𝐴 → (norm𝑦) ∈ ℂ)
36 addcom 10815 . . . . . . . . . 10 (((norm𝑥) ∈ ℂ ∧ (norm𝑦) ∈ ℂ) → ((norm𝑥) + (norm𝑦)) = ((norm𝑦) + (norm𝑥)))
3731, 35, 36syl2an 595 . . . . . . . . 9 ((𝑥𝐵𝑦𝐴) → ((norm𝑥) + (norm𝑦)) = ((norm𝑦) + (norm𝑥)))
38 ax-hvcom 28692 . . . . . . . . . . . 12 ((𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ) → (𝑥 + 𝑦) = (𝑦 + 𝑥))
3928, 32, 38syl2an 595 . . . . . . . . . . 11 ((𝑥𝐵𝑦𝐴) → (𝑥 + 𝑦) = (𝑦 + 𝑥))
4039fveq2d 6671 . . . . . . . . . 10 ((𝑥𝐵𝑦𝐴) → (norm‘(𝑥 + 𝑦)) = (norm‘(𝑦 + 𝑥)))
4140oveq2d 7164 . . . . . . . . 9 ((𝑥𝐵𝑦𝐴) → (𝑣 · (norm‘(𝑥 + 𝑦))) = (𝑣 · (norm‘(𝑦 + 𝑥))))
4237, 41breq12d 5076 . . . . . . . 8 ((𝑥𝐵𝑦𝐴) → (((norm𝑥) + (norm𝑦)) ≤ (𝑣 · (norm‘(𝑥 + 𝑦))) ↔ ((norm𝑦) + (norm𝑥)) ≤ (𝑣 · (norm‘(𝑦 + 𝑥)))))
4342ralbidva 3201 . . . . . . 7 (𝑥𝐵 → (∀𝑦𝐴 ((norm𝑥) + (norm𝑦)) ≤ (𝑣 · (norm‘(𝑥 + 𝑦))) ↔ ∀𝑦𝐴 ((norm𝑦) + (norm𝑥)) ≤ (𝑣 · (norm‘(𝑦 + 𝑥)))))
4443ralbiia 3169 . . . . . 6 (∀𝑥𝐵𝑦𝐴 ((norm𝑥) + (norm𝑦)) ≤ (𝑣 · (norm‘(𝑥 + 𝑦))) ↔ ∀𝑥𝐵𝑦𝐴 ((norm𝑦) + (norm𝑥)) ≤ (𝑣 · (norm‘(𝑦 + 𝑥))))
45 fveq2 6667 . . . . . . . . 9 (𝑥 = → (norm𝑥) = (norm))
4645oveq2d 7164 . . . . . . . 8 (𝑥 = → ((norm𝑦) + (norm𝑥)) = ((norm𝑦) + (norm)))
47 oveq2 7156 . . . . . . . . . 10 (𝑥 = → (𝑦 + 𝑥) = (𝑦 + ))
4847fveq2d 6671 . . . . . . . . 9 (𝑥 = → (norm‘(𝑦 + 𝑥)) = (norm‘(𝑦 + )))
4948oveq2d 7164 . . . . . . . 8 (𝑥 = → (𝑣 · (norm‘(𝑦 + 𝑥))) = (𝑣 · (norm‘(𝑦 + ))))
5046, 49breq12d 5076 . . . . . . 7 (𝑥 = → (((norm𝑦) + (norm𝑥)) ≤ (𝑣 · (norm‘(𝑦 + 𝑥))) ↔ ((norm𝑦) + (norm)) ≤ (𝑣 · (norm‘(𝑦 + )))))
51 fveq2 6667 . . . . . . . . 9 (𝑦 = 𝑡 → (norm𝑦) = (norm𝑡))
5251oveq1d 7163 . . . . . . . 8 (𝑦 = 𝑡 → ((norm𝑦) + (norm)) = ((norm𝑡) + (norm)))
53 fvoveq1 7171 . . . . . . . . 9 (𝑦 = 𝑡 → (norm‘(𝑦 + )) = (norm‘(𝑡 + )))
5453oveq2d 7164 . . . . . . . 8 (𝑦 = 𝑡 → (𝑣 · (norm‘(𝑦 + ))) = (𝑣 · (norm‘(𝑡 + ))))
5552, 54breq12d 5076 . . . . . . 7 (𝑦 = 𝑡 → (((norm𝑦) + (norm)) ≤ (𝑣 · (norm‘(𝑦 + ))) ↔ ((norm𝑡) + (norm)) ≤ (𝑣 · (norm‘(𝑡 + )))))
5650, 55cbvral2v 3470 . . . . . 6 (∀𝑥𝐵𝑦𝐴 ((norm𝑦) + (norm𝑥)) ≤ (𝑣 · (norm‘(𝑦 + 𝑥))) ↔ ∀𝐵𝑡𝐴 ((norm𝑡) + (norm)) ≤ (𝑣 · (norm‘(𝑡 + ))))
5744, 56bitr2i 277 . . . . 5 (∀𝐵𝑡𝐴 ((norm𝑡) + (norm)) ≤ (𝑣 · (norm‘(𝑡 + ))) ↔ ∀𝑥𝐵𝑦𝐴 ((norm𝑥) + (norm𝑦)) ≤ (𝑣 · (norm‘(𝑥 + 𝑦))))
5826, 27, 573bitri 298 . . . 4 (∀𝑥𝐴𝑦𝐵 ((norm𝑥) + (norm𝑦)) ≤ (𝑣 · (norm‘(𝑥 + 𝑦))) ↔ ∀𝑥𝐵𝑦𝐴 ((norm𝑥) + (norm𝑦)) ≤ (𝑣 · (norm‘(𝑥 + 𝑦))))
5958anbi2i 622 . . 3 ((0 < 𝑣 ∧ ∀𝑥𝐴𝑦𝐵 ((norm𝑥) + (norm𝑦)) ≤ (𝑣 · (norm‘(𝑥 + 𝑦)))) ↔ (0 < 𝑣 ∧ ∀𝑥𝐵𝑦𝐴 ((norm𝑥) + (norm𝑦)) ≤ (𝑣 · (norm‘(𝑥 + 𝑦)))))
6059rexbii 3252 . 2 (∃𝑣 ∈ ℝ (0 < 𝑣 ∧ ∀𝑥𝐴𝑦𝐵 ((norm𝑥) + (norm𝑦)) ≤ (𝑣 · (norm‘(𝑥 + 𝑦)))) ↔ ∃𝑣 ∈ ℝ (0 < 𝑣 ∧ ∀𝑥𝐵𝑦𝐴 ((norm𝑥) + (norm𝑦)) ≤ (𝑣 · (norm‘(𝑥 + 𝑦)))))
612, 1shscomi 29054 . . . . 5 (𝐴 + 𝐵) = (𝐵 + 𝐴)
6261raleqi 3419 . . . 4 (∀𝑢 ∈ (𝐴 + 𝐵)(norm‘(𝑇𝑢)) ≤ (𝑣 · (norm𝑢)) ↔ ∀𝑢 ∈ (𝐵 + 𝐴)(norm‘(𝑇𝑢)) ≤ (𝑣 · (norm𝑢)))
6362anbi2i 622 . . 3 ((0 < 𝑣 ∧ ∀𝑢 ∈ (𝐴 + 𝐵)(norm‘(𝑇𝑢)) ≤ (𝑣 · (norm𝑢))) ↔ (0 < 𝑣 ∧ ∀𝑢 ∈ (𝐵 + 𝐴)(norm‘(𝑇𝑢)) ≤ (𝑣 · (norm𝑢))))
6463rexbii 3252 . 2 (∃𝑣 ∈ ℝ (0 < 𝑣 ∧ ∀𝑢 ∈ (𝐴 + 𝐵)(norm‘(𝑇𝑢)) ≤ (𝑣 · (norm𝑢))) ↔ ∃𝑣 ∈ ℝ (0 < 𝑣 ∧ ∀𝑢 ∈ (𝐵 + 𝐴)(norm‘(𝑇𝑢)) ≤ (𝑣 · (norm𝑢))))
6514, 60, 643imtr4i 293 1 (∃𝑣 ∈ ℝ (0 < 𝑣 ∧ ∀𝑥𝐴𝑦𝐵 ((norm𝑥) + (norm𝑦)) ≤ (𝑣 · (norm‘(𝑥 + 𝑦)))) → ∃𝑣 ∈ ℝ (0 < 𝑣 ∧ ∀𝑢 ∈ (𝐴 + 𝐵)(norm‘(𝑇𝑢)) ≤ (𝑣 · (norm𝑢))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1530  wcel 2107  wral 3143  wrex 3144   class class class wbr 5063  cmpt 5143  cfv 6352  crio 7105  (class class class)co 7148  cc 10524  cr 10525  0cc0 10526   + caddc 10529   · cmul 10531   < clt 10664  cle 10665  chba 28610   + cva 28611  normcno 28614   S csh 28619   + cph 28622
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-13 2385  ax-ext 2798  ax-rep 5187  ax-sep 5200  ax-nul 5207  ax-pow 5263  ax-pr 5326  ax-un 7451  ax-cnex 10582  ax-resscn 10583  ax-1cn 10584  ax-icn 10585  ax-addcl 10586  ax-addrcl 10587  ax-mulcl 10588  ax-mulrcl 10589  ax-mulcom 10590  ax-addass 10591  ax-mulass 10592  ax-distr 10593  ax-i2m1 10594  ax-1ne0 10595  ax-1rid 10596  ax-rnegex 10597  ax-rrecex 10598  ax-cnre 10599  ax-pre-lttri 10600  ax-pre-lttrn 10601  ax-pre-ltadd 10602  ax-pre-mulgt0 10603  ax-pre-sup 10604  ax-hilex 28690  ax-hfvadd 28691  ax-hvcom 28692  ax-hvass 28693  ax-hv0cl 28694  ax-hvaddid 28695  ax-hfvmul 28696  ax-hvmulid 28697  ax-hvmulass 28698  ax-hvdistr1 28699  ax-hvdistr2 28700  ax-hvmul0 28701  ax-hfi 28770  ax-his1 28773  ax-his3 28775  ax-his4 28776
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3or 1082  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ne 3022  df-nel 3129  df-ral 3148  df-rex 3149  df-reu 3150  df-rmo 3151  df-rab 3152  df-v 3502  df-sbc 3777  df-csb 3888  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-pss 3958  df-nul 4296  df-if 4471  df-pw 4544  df-sn 4565  df-pr 4567  df-tp 4569  df-op 4571  df-uni 4838  df-int 4875  df-iun 4919  df-br 5064  df-opab 5126  df-mpt 5144  df-tr 5170  df-id 5459  df-eprel 5464  df-po 5473  df-so 5474  df-fr 5513  df-we 5515  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-pred 6146  df-ord 6192  df-on 6193  df-lim 6194  df-suc 6195  df-iota 6312  df-fun 6354  df-fn 6355  df-f 6356  df-f1 6357  df-fo 6358  df-f1o 6359  df-fv 6360  df-riota 7106  df-ov 7151  df-oprab 7152  df-mpo 7153  df-om 7569  df-2nd 7681  df-wrecs 7938  df-recs 7999  df-rdg 8037  df-er 8279  df-en 8499  df-dom 8500  df-sdom 8501  df-sup 8895  df-pnf 10666  df-mnf 10667  df-xr 10668  df-ltxr 10669  df-le 10670  df-sub 10861  df-neg 10862  df-div 11287  df-nn 11628  df-2 11689  df-3 11690  df-n0 11887  df-z 11971  df-uz 12233  df-rp 12380  df-seq 13360  df-exp 13420  df-cj 14448  df-re 14449  df-im 14450  df-sqrt 14584  df-abs 14585  df-grpo 28184  df-ablo 28236  df-hnorm 28659  df-hvsub 28662  df-sh 28898  df-ch0 28944  df-shs 28999
This theorem is referenced by:  cdj3i  30132
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