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Theorem cdj3lem3b 29855
Description: Lemma for cdj3i 29856. 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 28778 . . . . 5 (𝐵 + 𝐴) = (𝐴 + 𝐵)
51sheli 28627 . . . . . . . . 9 (𝑤𝐵𝑤 ∈ ℋ)
62sheli 28627 . . . . . . . . 9 (𝑧𝐴𝑧 ∈ ℋ)
7 ax-hvcom 28414 . . . . . . . . 9 ((𝑤 ∈ ℋ ∧ 𝑧 ∈ ℋ) → (𝑤 + 𝑧) = (𝑧 + 𝑤))
85, 6, 7syl2an 591 . . . . . . . 8 ((𝑤𝐵𝑧𝐴) → (𝑤 + 𝑧) = (𝑧 + 𝑤))
98eqeq2d 2836 . . . . . . 7 ((𝑤𝐵𝑧𝐴) → (𝑥 = (𝑤 + 𝑧) ↔ 𝑥 = (𝑧 + 𝑤)))
109rexbidva 3260 . . . . . 6 (𝑤𝐵 → (∃𝑧𝐴 𝑥 = (𝑤 + 𝑧) ↔ ∃𝑧𝐴 𝑥 = (𝑧 + 𝑤)))
1110riotabiia 6884 . . . . 5 (𝑤𝐵𝑧𝐴 𝑥 = (𝑤 + 𝑧)) = (𝑤𝐵𝑧𝐴 𝑥 = (𝑧 + 𝑤))
124, 11mpteq12i 4966 . . . 4 (𝑥 ∈ (𝐵 + 𝐴) ↦ (𝑤𝐵𝑧𝐴 𝑥 = (𝑤 + 𝑧))) = (𝑥 ∈ (𝐴 + 𝐵) ↦ (𝑤𝐵𝑧𝐴 𝑥 = (𝑧 + 𝑤)))
133, 12eqtr4i 2853 . . 3 𝑇 = (𝑥 ∈ (𝐵 + 𝐴) ↦ (𝑤𝐵𝑧𝐴 𝑥 = (𝑤 + 𝑧)))
141, 2, 13cdj3lem2b 29852 . 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 6434 . . . . . . . 8 (𝑥 = 𝑡 → (norm𝑥) = (norm𝑡))
1615oveq1d 6921 . . . . . . 7 (𝑥 = 𝑡 → ((norm𝑥) + (norm𝑦)) = ((norm𝑡) + (norm𝑦)))
17 fvoveq1 6929 . . . . . . . 8 (𝑥 = 𝑡 → (norm‘(𝑥 + 𝑦)) = (norm‘(𝑡 + 𝑦)))
1817oveq2d 6922 . . . . . . 7 (𝑥 = 𝑡 → (𝑣 · (norm‘(𝑥 + 𝑦))) = (𝑣 · (norm‘(𝑡 + 𝑦))))
1916, 18breq12d 4887 . . . . . 6 (𝑥 = 𝑡 → (((norm𝑥) + (norm𝑦)) ≤ (𝑣 · (norm‘(𝑥 + 𝑦))) ↔ ((norm𝑡) + (norm𝑦)) ≤ (𝑣 · (norm‘(𝑡 + 𝑦)))))
20 fveq2 6434 . . . . . . . 8 (𝑦 = → (norm𝑦) = (norm))
2120oveq2d 6922 . . . . . . 7 (𝑦 = → ((norm𝑡) + (norm𝑦)) = ((norm𝑡) + (norm)))
22 oveq2 6914 . . . . . . . . 9 (𝑦 = → (𝑡 + 𝑦) = (𝑡 + ))
2322fveq2d 6438 . . . . . . . 8 (𝑦 = → (norm‘(𝑡 + 𝑦)) = (norm‘(𝑡 + )))
2423oveq2d 6922 . . . . . . 7 (𝑦 = → (𝑣 · (norm‘(𝑡 + 𝑦))) = (𝑣 · (norm‘(𝑡 + ))))
2521, 24breq12d 4887 . . . . . 6 (𝑦 = → (((norm𝑡) + (norm𝑦)) ≤ (𝑣 · (norm‘(𝑡 + 𝑦))) ↔ ((norm𝑡) + (norm)) ≤ (𝑣 · (norm‘(𝑡 + )))))
2619, 25cbvral2v 3392 . . . . 5 (∀𝑥𝐴𝑦𝐵 ((norm𝑥) + (norm𝑦)) ≤ (𝑣 · (norm‘(𝑥 + 𝑦))) ↔ ∀𝑡𝐴𝐵 ((norm𝑡) + (norm)) ≤ (𝑣 · (norm‘(𝑡 + ))))
27 ralcom 3309 . . . . 5 (∀𝑡𝐴𝐵 ((norm𝑡) + (norm)) ≤ (𝑣 · (norm‘(𝑡 + ))) ↔ ∀𝐵𝑡𝐴 ((norm𝑡) + (norm)) ≤ (𝑣 · (norm‘(𝑡 + ))))
281sheli 28627 . . . . . . . . . . . 12 (𝑥𝐵𝑥 ∈ ℋ)
29 normcl 28538 . . . . . . . . . . . 12 (𝑥 ∈ ℋ → (norm𝑥) ∈ ℝ)
3028, 29syl 17 . . . . . . . . . . 11 (𝑥𝐵 → (norm𝑥) ∈ ℝ)
3130recnd 10386 . . . . . . . . . 10 (𝑥𝐵 → (norm𝑥) ∈ ℂ)
322sheli 28627 . . . . . . . . . . . 12 (𝑦𝐴𝑦 ∈ ℋ)
33 normcl 28538 . . . . . . . . . . . 12 (𝑦 ∈ ℋ → (norm𝑦) ∈ ℝ)
3432, 33syl 17 . . . . . . . . . . 11 (𝑦𝐴 → (norm𝑦) ∈ ℝ)
3534recnd 10386 . . . . . . . . . 10 (𝑦𝐴 → (norm𝑦) ∈ ℂ)
36 addcom 10542 . . . . . . . . . 10 (((norm𝑥) ∈ ℂ ∧ (norm𝑦) ∈ ℂ) → ((norm𝑥) + (norm𝑦)) = ((norm𝑦) + (norm𝑥)))
3731, 35, 36syl2an 591 . . . . . . . . 9 ((𝑥𝐵𝑦𝐴) → ((norm𝑥) + (norm𝑦)) = ((norm𝑦) + (norm𝑥)))
38 ax-hvcom 28414 . . . . . . . . . . . 12 ((𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ) → (𝑥 + 𝑦) = (𝑦 + 𝑥))
3928, 32, 38syl2an 591 . . . . . . . . . . 11 ((𝑥𝐵𝑦𝐴) → (𝑥 + 𝑦) = (𝑦 + 𝑥))
4039fveq2d 6438 . . . . . . . . . 10 ((𝑥𝐵𝑦𝐴) → (norm‘(𝑥 + 𝑦)) = (norm‘(𝑦 + 𝑥)))
4140oveq2d 6922 . . . . . . . . 9 ((𝑥𝐵𝑦𝐴) → (𝑣 · (norm‘(𝑥 + 𝑦))) = (𝑣 · (norm‘(𝑦 + 𝑥))))
4237, 41breq12d 4887 . . . . . . . 8 ((𝑥𝐵𝑦𝐴) → (((norm𝑥) + (norm𝑦)) ≤ (𝑣 · (norm‘(𝑥 + 𝑦))) ↔ ((norm𝑦) + (norm𝑥)) ≤ (𝑣 · (norm‘(𝑦 + 𝑥)))))
4342ralbidva 3195 . . . . . . 7 (𝑥𝐵 → (∀𝑦𝐴 ((norm𝑥) + (norm𝑦)) ≤ (𝑣 · (norm‘(𝑥 + 𝑦))) ↔ ∀𝑦𝐴 ((norm𝑦) + (norm𝑥)) ≤ (𝑣 · (norm‘(𝑦 + 𝑥)))))
4443ralbiia 3189 . . . . . 6 (∀𝑥𝐵𝑦𝐴 ((norm𝑥) + (norm𝑦)) ≤ (𝑣 · (norm‘(𝑥 + 𝑦))) ↔ ∀𝑥𝐵𝑦𝐴 ((norm𝑦) + (norm𝑥)) ≤ (𝑣 · (norm‘(𝑦 + 𝑥))))
45 fveq2 6434 . . . . . . . . 9 (𝑥 = → (norm𝑥) = (norm))
4645oveq2d 6922 . . . . . . . 8 (𝑥 = → ((norm𝑦) + (norm𝑥)) = ((norm𝑦) + (norm)))
47 oveq2 6914 . . . . . . . . . 10 (𝑥 = → (𝑦 + 𝑥) = (𝑦 + ))
4847fveq2d 6438 . . . . . . . . 9 (𝑥 = → (norm‘(𝑦 + 𝑥)) = (norm‘(𝑦 + )))
4948oveq2d 6922 . . . . . . . 8 (𝑥 = → (𝑣 · (norm‘(𝑦 + 𝑥))) = (𝑣 · (norm‘(𝑦 + ))))
5046, 49breq12d 4887 . . . . . . 7 (𝑥 = → (((norm𝑦) + (norm𝑥)) ≤ (𝑣 · (norm‘(𝑦 + 𝑥))) ↔ ((norm𝑦) + (norm)) ≤ (𝑣 · (norm‘(𝑦 + )))))
51 fveq2 6434 . . . . . . . . 9 (𝑦 = 𝑡 → (norm𝑦) = (norm𝑡))
5251oveq1d 6921 . . . . . . . 8 (𝑦 = 𝑡 → ((norm𝑦) + (norm)) = ((norm𝑡) + (norm)))
53 fvoveq1 6929 . . . . . . . . 9 (𝑦 = 𝑡 → (norm‘(𝑦 + )) = (norm‘(𝑡 + )))
5453oveq2d 6922 . . . . . . . 8 (𝑦 = 𝑡 → (𝑣 · (norm‘(𝑦 + ))) = (𝑣 · (norm‘(𝑡 + ))))
5552, 54breq12d 4887 . . . . . . 7 (𝑦 = 𝑡 → (((norm𝑦) + (norm)) ≤ (𝑣 · (norm‘(𝑦 + ))) ↔ ((norm𝑡) + (norm)) ≤ (𝑣 · (norm‘(𝑡 + )))))
5650, 55cbvral2v 3392 . . . . . 6 (∀𝑥𝐵𝑦𝐴 ((norm𝑦) + (norm𝑥)) ≤ (𝑣 · (norm‘(𝑦 + 𝑥))) ↔ ∀𝐵𝑡𝐴 ((norm𝑡) + (norm)) ≤ (𝑣 · (norm‘(𝑡 + ))))
5744, 56bitr2i 268 . . . . 5 (∀𝐵𝑡𝐴 ((norm𝑡) + (norm)) ≤ (𝑣 · (norm‘(𝑡 + ))) ↔ ∀𝑥𝐵𝑦𝐴 ((norm𝑥) + (norm𝑦)) ≤ (𝑣 · (norm‘(𝑥 + 𝑦))))
5826, 27, 573bitri 289 . . . 4 (∀𝑥𝐴𝑦𝐵 ((norm𝑥) + (norm𝑦)) ≤ (𝑣 · (norm‘(𝑥 + 𝑦))) ↔ ∀𝑥𝐵𝑦𝐴 ((norm𝑥) + (norm𝑦)) ≤ (𝑣 · (norm‘(𝑥 + 𝑦))))
5958anbi2i 618 . . 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 28778 . . . . 5 (𝐴 + 𝐵) = (𝐵 + 𝐴)
6261raleqi 3355 . . . 4 (∀𝑢 ∈ (𝐴 + 𝐵)(norm‘(𝑇𝑢)) ≤ (𝑣 · (norm𝑢)) ↔ ∀𝑢 ∈ (𝐵 + 𝐴)(norm‘(𝑇𝑢)) ≤ (𝑣 · (norm𝑢)))
6362anbi2i 618 . . 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 284 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 386   = wceq 1658  wcel 2166  wral 3118  wrex 3119   class class class wbr 4874  cmpt 4953  cfv 6124  crio 6866  (class class class)co 6906  cc 10251  cr 10252  0cc0 10253   + caddc 10256   · cmul 10258   < clt 10392  cle 10393  chba 28332   + cva 28333  normcno 28336   S csh 28341   + cph 28344
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-8 2168  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2391  ax-ext 2804  ax-rep 4995  ax-sep 5006  ax-nul 5014  ax-pow 5066  ax-pr 5128  ax-un 7210  ax-cnex 10309  ax-resscn 10310  ax-1cn 10311  ax-icn 10312  ax-addcl 10313  ax-addrcl 10314  ax-mulcl 10315  ax-mulrcl 10316  ax-mulcom 10317  ax-addass 10318  ax-mulass 10319  ax-distr 10320  ax-i2m1 10321  ax-1ne0 10322  ax-1rid 10323  ax-rnegex 10324  ax-rrecex 10325  ax-cnre 10326  ax-pre-lttri 10327  ax-pre-lttrn 10328  ax-pre-ltadd 10329  ax-pre-mulgt0 10330  ax-pre-sup 10331  ax-hilex 28412  ax-hfvadd 28413  ax-hvcom 28414  ax-hvass 28415  ax-hv0cl 28416  ax-hvaddid 28417  ax-hfvmul 28418  ax-hvmulid 28419  ax-hvmulass 28420  ax-hvdistr1 28421  ax-hvdistr2 28422  ax-hvmul0 28423  ax-hfi 28492  ax-his1 28495  ax-his3 28497  ax-his4 28498
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3or 1114  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-mo 2606  df-eu 2641  df-clab 2813  df-cleq 2819  df-clel 2822  df-nfc 2959  df-ne 3001  df-nel 3104  df-ral 3123  df-rex 3124  df-reu 3125  df-rmo 3126  df-rab 3127  df-v 3417  df-sbc 3664  df-csb 3759  df-dif 3802  df-un 3804  df-in 3806  df-ss 3813  df-pss 3815  df-nul 4146  df-if 4308  df-pw 4381  df-sn 4399  df-pr 4401  df-tp 4403  df-op 4405  df-uni 4660  df-int 4699  df-iun 4743  df-br 4875  df-opab 4937  df-mpt 4954  df-tr 4977  df-id 5251  df-eprel 5256  df-po 5264  df-so 5265  df-fr 5302  df-we 5304  df-xp 5349  df-rel 5350  df-cnv 5351  df-co 5352  df-dm 5353  df-rn 5354  df-res 5355  df-ima 5356  df-pred 5921  df-ord 5967  df-on 5968  df-lim 5969  df-suc 5970  df-iota 6087  df-fun 6126  df-fn 6127  df-f 6128  df-f1 6129  df-fo 6130  df-f1o 6131  df-fv 6132  df-riota 6867  df-ov 6909  df-oprab 6910  df-mpt2 6911  df-om 7328  df-2nd 7430  df-wrecs 7673  df-recs 7735  df-rdg 7773  df-er 8010  df-en 8224  df-dom 8225  df-sdom 8226  df-sup 8618  df-pnf 10394  df-mnf 10395  df-xr 10396  df-ltxr 10397  df-le 10398  df-sub 10588  df-neg 10589  df-div 11011  df-nn 11352  df-2 11415  df-3 11416  df-n0 11620  df-z 11706  df-uz 11970  df-rp 12114  df-seq 13097  df-exp 13156  df-cj 14217  df-re 14218  df-im 14219  df-sqrt 14353  df-abs 14354  df-grpo 27904  df-ablo 27956  df-hnorm 28381  df-hvsub 28384  df-sh 28620  df-ch0 28666  df-shs 28723
This theorem is referenced by:  cdj3i  29856
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