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Theorem cdj3lem3b 29145
Description: Lemma for cdj3i 29146. 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 28068 . . . . 5 (𝐵 + 𝐴) = (𝐴 + 𝐵)
51sheli 27917 . . . . . . . . 9 (𝑤𝐵𝑤 ∈ ℋ)
62sheli 27917 . . . . . . . . 9 (𝑧𝐴𝑧 ∈ ℋ)
7 ax-hvcom 27704 . . . . . . . . 9 ((𝑤 ∈ ℋ ∧ 𝑧 ∈ ℋ) → (𝑤 + 𝑧) = (𝑧 + 𝑤))
85, 6, 7syl2an 494 . . . . . . . 8 ((𝑤𝐵𝑧𝐴) → (𝑤 + 𝑧) = (𝑧 + 𝑤))
98eqeq2d 2631 . . . . . . 7 ((𝑤𝐵𝑧𝐴) → (𝑥 = (𝑤 + 𝑧) ↔ 𝑥 = (𝑧 + 𝑤)))
109rexbidva 3042 . . . . . 6 (𝑤𝐵 → (∃𝑧𝐴 𝑥 = (𝑤 + 𝑧) ↔ ∃𝑧𝐴 𝑥 = (𝑧 + 𝑤)))
1110riotabiia 6582 . . . . 5 (𝑤𝐵𝑧𝐴 𝑥 = (𝑤 + 𝑧)) = (𝑤𝐵𝑧𝐴 𝑥 = (𝑧 + 𝑤))
124, 11mpteq12i 4702 . . . 4 (𝑥 ∈ (𝐵 + 𝐴) ↦ (𝑤𝐵𝑧𝐴 𝑥 = (𝑤 + 𝑧))) = (𝑥 ∈ (𝐴 + 𝐵) ↦ (𝑤𝐵𝑧𝐴 𝑥 = (𝑧 + 𝑤)))
133, 12eqtr4i 2646 . . 3 𝑇 = (𝑥 ∈ (𝐵 + 𝐴) ↦ (𝑤𝐵𝑧𝐴 𝑥 = (𝑤 + 𝑧)))
141, 2, 13cdj3lem2b 29142 . 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 6148 . . . . . . . 8 (𝑥 = 𝑡 → (norm𝑥) = (norm𝑡))
1615oveq1d 6619 . . . . . . 7 (𝑥 = 𝑡 → ((norm𝑥) + (norm𝑦)) = ((norm𝑡) + (norm𝑦)))
17 oveq1 6611 . . . . . . . . 9 (𝑥 = 𝑡 → (𝑥 + 𝑦) = (𝑡 + 𝑦))
1817fveq2d 6152 . . . . . . . 8 (𝑥 = 𝑡 → (norm‘(𝑥 + 𝑦)) = (norm‘(𝑡 + 𝑦)))
1918oveq2d 6620 . . . . . . 7 (𝑥 = 𝑡 → (𝑣 · (norm‘(𝑥 + 𝑦))) = (𝑣 · (norm‘(𝑡 + 𝑦))))
2016, 19breq12d 4626 . . . . . 6 (𝑥 = 𝑡 → (((norm𝑥) + (norm𝑦)) ≤ (𝑣 · (norm‘(𝑥 + 𝑦))) ↔ ((norm𝑡) + (norm𝑦)) ≤ (𝑣 · (norm‘(𝑡 + 𝑦)))))
21 fveq2 6148 . . . . . . . 8 (𝑦 = → (norm𝑦) = (norm))
2221oveq2d 6620 . . . . . . 7 (𝑦 = → ((norm𝑡) + (norm𝑦)) = ((norm𝑡) + (norm)))
23 oveq2 6612 . . . . . . . . 9 (𝑦 = → (𝑡 + 𝑦) = (𝑡 + ))
2423fveq2d 6152 . . . . . . . 8 (𝑦 = → (norm‘(𝑡 + 𝑦)) = (norm‘(𝑡 + )))
2524oveq2d 6620 . . . . . . 7 (𝑦 = → (𝑣 · (norm‘(𝑡 + 𝑦))) = (𝑣 · (norm‘(𝑡 + ))))
2622, 25breq12d 4626 . . . . . 6 (𝑦 = → (((norm𝑡) + (norm𝑦)) ≤ (𝑣 · (norm‘(𝑡 + 𝑦))) ↔ ((norm𝑡) + (norm)) ≤ (𝑣 · (norm‘(𝑡 + )))))
2720, 26cbvral2v 3167 . . . . 5 (∀𝑥𝐴𝑦𝐵 ((norm𝑥) + (norm𝑦)) ≤ (𝑣 · (norm‘(𝑥 + 𝑦))) ↔ ∀𝑡𝐴𝐵 ((norm𝑡) + (norm)) ≤ (𝑣 · (norm‘(𝑡 + ))))
28 ralcom 3090 . . . . 5 (∀𝑡𝐴𝐵 ((norm𝑡) + (norm)) ≤ (𝑣 · (norm‘(𝑡 + ))) ↔ ∀𝐵𝑡𝐴 ((norm𝑡) + (norm)) ≤ (𝑣 · (norm‘(𝑡 + ))))
291sheli 27917 . . . . . . . . . . . 12 (𝑥𝐵𝑥 ∈ ℋ)
30 normcl 27828 . . . . . . . . . . . 12 (𝑥 ∈ ℋ → (norm𝑥) ∈ ℝ)
3129, 30syl 17 . . . . . . . . . . 11 (𝑥𝐵 → (norm𝑥) ∈ ℝ)
3231recnd 10012 . . . . . . . . . 10 (𝑥𝐵 → (norm𝑥) ∈ ℂ)
332sheli 27917 . . . . . . . . . . . 12 (𝑦𝐴𝑦 ∈ ℋ)
34 normcl 27828 . . . . . . . . . . . 12 (𝑦 ∈ ℋ → (norm𝑦) ∈ ℝ)
3533, 34syl 17 . . . . . . . . . . 11 (𝑦𝐴 → (norm𝑦) ∈ ℝ)
3635recnd 10012 . . . . . . . . . 10 (𝑦𝐴 → (norm𝑦) ∈ ℂ)
37 addcom 10166 . . . . . . . . . 10 (((norm𝑥) ∈ ℂ ∧ (norm𝑦) ∈ ℂ) → ((norm𝑥) + (norm𝑦)) = ((norm𝑦) + (norm𝑥)))
3832, 36, 37syl2an 494 . . . . . . . . 9 ((𝑥𝐵𝑦𝐴) → ((norm𝑥) + (norm𝑦)) = ((norm𝑦) + (norm𝑥)))
39 ax-hvcom 27704 . . . . . . . . . . . 12 ((𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ) → (𝑥 + 𝑦) = (𝑦 + 𝑥))
4029, 33, 39syl2an 494 . . . . . . . . . . 11 ((𝑥𝐵𝑦𝐴) → (𝑥 + 𝑦) = (𝑦 + 𝑥))
4140fveq2d 6152 . . . . . . . . . 10 ((𝑥𝐵𝑦𝐴) → (norm‘(𝑥 + 𝑦)) = (norm‘(𝑦 + 𝑥)))
4241oveq2d 6620 . . . . . . . . 9 ((𝑥𝐵𝑦𝐴) → (𝑣 · (norm‘(𝑥 + 𝑦))) = (𝑣 · (norm‘(𝑦 + 𝑥))))
4338, 42breq12d 4626 . . . . . . . 8 ((𝑥𝐵𝑦𝐴) → (((norm𝑥) + (norm𝑦)) ≤ (𝑣 · (norm‘(𝑥 + 𝑦))) ↔ ((norm𝑦) + (norm𝑥)) ≤ (𝑣 · (norm‘(𝑦 + 𝑥)))))
4443ralbidva 2979 . . . . . . 7 (𝑥𝐵 → (∀𝑦𝐴 ((norm𝑥) + (norm𝑦)) ≤ (𝑣 · (norm‘(𝑥 + 𝑦))) ↔ ∀𝑦𝐴 ((norm𝑦) + (norm𝑥)) ≤ (𝑣 · (norm‘(𝑦 + 𝑥)))))
4544ralbiia 2973 . . . . . 6 (∀𝑥𝐵𝑦𝐴 ((norm𝑥) + (norm𝑦)) ≤ (𝑣 · (norm‘(𝑥 + 𝑦))) ↔ ∀𝑥𝐵𝑦𝐴 ((norm𝑦) + (norm𝑥)) ≤ (𝑣 · (norm‘(𝑦 + 𝑥))))
46 fveq2 6148 . . . . . . . . 9 (𝑥 = → (norm𝑥) = (norm))
4746oveq2d 6620 . . . . . . . 8 (𝑥 = → ((norm𝑦) + (norm𝑥)) = ((norm𝑦) + (norm)))
48 oveq2 6612 . . . . . . . . . 10 (𝑥 = → (𝑦 + 𝑥) = (𝑦 + ))
4948fveq2d 6152 . . . . . . . . 9 (𝑥 = → (norm‘(𝑦 + 𝑥)) = (norm‘(𝑦 + )))
5049oveq2d 6620 . . . . . . . 8 (𝑥 = → (𝑣 · (norm‘(𝑦 + 𝑥))) = (𝑣 · (norm‘(𝑦 + ))))
5147, 50breq12d 4626 . . . . . . 7 (𝑥 = → (((norm𝑦) + (norm𝑥)) ≤ (𝑣 · (norm‘(𝑦 + 𝑥))) ↔ ((norm𝑦) + (norm)) ≤ (𝑣 · (norm‘(𝑦 + )))))
52 fveq2 6148 . . . . . . . . 9 (𝑦 = 𝑡 → (norm𝑦) = (norm𝑡))
5352oveq1d 6619 . . . . . . . 8 (𝑦 = 𝑡 → ((norm𝑦) + (norm)) = ((norm𝑡) + (norm)))
54 oveq1 6611 . . . . . . . . . 10 (𝑦 = 𝑡 → (𝑦 + ) = (𝑡 + ))
5554fveq2d 6152 . . . . . . . . 9 (𝑦 = 𝑡 → (norm‘(𝑦 + )) = (norm‘(𝑡 + )))
5655oveq2d 6620 . . . . . . . 8 (𝑦 = 𝑡 → (𝑣 · (norm‘(𝑦 + ))) = (𝑣 · (norm‘(𝑡 + ))))
5753, 56breq12d 4626 . . . . . . 7 (𝑦 = 𝑡 → (((norm𝑦) + (norm)) ≤ (𝑣 · (norm‘(𝑦 + ))) ↔ ((norm𝑡) + (norm)) ≤ (𝑣 · (norm‘(𝑡 + )))))
5851, 57cbvral2v 3167 . . . . . 6 (∀𝑥𝐵𝑦𝐴 ((norm𝑦) + (norm𝑥)) ≤ (𝑣 · (norm‘(𝑦 + 𝑥))) ↔ ∀𝐵𝑡𝐴 ((norm𝑡) + (norm)) ≤ (𝑣 · (norm‘(𝑡 + ))))
5945, 58bitr2i 265 . . . . 5 (∀𝐵𝑡𝐴 ((norm𝑡) + (norm)) ≤ (𝑣 · (norm‘(𝑡 + ))) ↔ ∀𝑥𝐵𝑦𝐴 ((norm𝑥) + (norm𝑦)) ≤ (𝑣 · (norm‘(𝑥 + 𝑦))))
6027, 28, 593bitri 286 . . . 4 (∀𝑥𝐴𝑦𝐵 ((norm𝑥) + (norm𝑦)) ≤ (𝑣 · (norm‘(𝑥 + 𝑦))) ↔ ∀𝑥𝐵𝑦𝐴 ((norm𝑥) + (norm𝑦)) ≤ (𝑣 · (norm‘(𝑥 + 𝑦))))
6160anbi2i 729 . . 3 ((0 < 𝑣 ∧ ∀𝑥𝐴𝑦𝐵 ((norm𝑥) + (norm𝑦)) ≤ (𝑣 · (norm‘(𝑥 + 𝑦)))) ↔ (0 < 𝑣 ∧ ∀𝑥𝐵𝑦𝐴 ((norm𝑥) + (norm𝑦)) ≤ (𝑣 · (norm‘(𝑥 + 𝑦)))))
6261rexbii 3034 . 2 (∃𝑣 ∈ ℝ (0 < 𝑣 ∧ ∀𝑥𝐴𝑦𝐵 ((norm𝑥) + (norm𝑦)) ≤ (𝑣 · (norm‘(𝑥 + 𝑦)))) ↔ ∃𝑣 ∈ ℝ (0 < 𝑣 ∧ ∀𝑥𝐵𝑦𝐴 ((norm𝑥) + (norm𝑦)) ≤ (𝑣 · (norm‘(𝑥 + 𝑦)))))
632, 1shscomi 28068 . . . . 5 (𝐴 + 𝐵) = (𝐵 + 𝐴)
6463raleqi 3131 . . . 4 (∀𝑢 ∈ (𝐴 + 𝐵)(norm‘(𝑇𝑢)) ≤ (𝑣 · (norm𝑢)) ↔ ∀𝑢 ∈ (𝐵 + 𝐴)(norm‘(𝑇𝑢)) ≤ (𝑣 · (norm𝑢)))
6564anbi2i 729 . . 3 ((0 < 𝑣 ∧ ∀𝑢 ∈ (𝐴 + 𝐵)(norm‘(𝑇𝑢)) ≤ (𝑣 · (norm𝑢))) ↔ (0 < 𝑣 ∧ ∀𝑢 ∈ (𝐵 + 𝐴)(norm‘(𝑇𝑢)) ≤ (𝑣 · (norm𝑢))))
6665rexbii 3034 . 2 (∃𝑣 ∈ ℝ (0 < 𝑣 ∧ ∀𝑢 ∈ (𝐴 + 𝐵)(norm‘(𝑇𝑢)) ≤ (𝑣 · (norm𝑢))) ↔ ∃𝑣 ∈ ℝ (0 < 𝑣 ∧ ∀𝑢 ∈ (𝐵 + 𝐴)(norm‘(𝑇𝑢)) ≤ (𝑣 · (norm𝑢))))
6714, 62, 663imtr4i 281 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 384   = wceq 1480  wcel 1987  wral 2907  wrex 2908   class class class wbr 4613  cmpt 4673  cfv 5847  crio 6564  (class class class)co 6604  cc 9878  cr 9879  0cc0 9880   + caddc 9883   · cmul 9885   < clt 10018  cle 10019  chil 27622   + cva 27623  normcno 27626   S csh 27631   + cph 27634
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902  ax-cnex 9936  ax-resscn 9937  ax-1cn 9938  ax-icn 9939  ax-addcl 9940  ax-addrcl 9941  ax-mulcl 9942  ax-mulrcl 9943  ax-mulcom 9944  ax-addass 9945  ax-mulass 9946  ax-distr 9947  ax-i2m1 9948  ax-1ne0 9949  ax-1rid 9950  ax-rnegex 9951  ax-rrecex 9952  ax-cnre 9953  ax-pre-lttri 9954  ax-pre-lttrn 9955  ax-pre-ltadd 9956  ax-pre-mulgt0 9957  ax-pre-sup 9958  ax-hilex 27702  ax-hfvadd 27703  ax-hvcom 27704  ax-hvass 27705  ax-hv0cl 27706  ax-hvaddid 27707  ax-hfvmul 27708  ax-hvmulid 27709  ax-hvmulass 27710  ax-hvdistr1 27711  ax-hvdistr2 27712  ax-hvmul0 27713  ax-hfi 27782  ax-his1 27785  ax-his3 27787  ax-his4 27788
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-tp 4153  df-op 4155  df-uni 4403  df-int 4441  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-tr 4713  df-eprel 4985  df-id 4989  df-po 4995  df-so 4996  df-fr 5033  df-we 5035  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-pred 5639  df-ord 5685  df-on 5686  df-lim 5687  df-suc 5688  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-riota 6565  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-om 7013  df-2nd 7114  df-wrecs 7352  df-recs 7413  df-rdg 7451  df-er 7687  df-en 7900  df-dom 7901  df-sdom 7902  df-sup 8292  df-pnf 10020  df-mnf 10021  df-xr 10022  df-ltxr 10023  df-le 10024  df-sub 10212  df-neg 10213  df-div 10629  df-nn 10965  df-2 11023  df-3 11024  df-n0 11237  df-z 11322  df-uz 11632  df-rp 11777  df-seq 12742  df-exp 12801  df-cj 13773  df-re 13774  df-im 13775  df-sqrt 13909  df-abs 13910  df-grpo 27193  df-ablo 27245  df-hnorm 27671  df-hvsub 27674  df-sh 27910  df-ch0 27956  df-shs 28013
This theorem is referenced by:  cdj3i  29146
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