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Theorem cdj3lem3b 29639
Description: Lemma for cdj3i 29640. 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 28562 . . . . 5 (𝐵 + 𝐴) = (𝐴 + 𝐵)
51sheli 28411 . . . . . . . . 9 (𝑤𝐵𝑤 ∈ ℋ)
62sheli 28411 . . . . . . . . 9 (𝑧𝐴𝑧 ∈ ℋ)
7 ax-hvcom 28198 . . . . . . . . 9 ((𝑤 ∈ ℋ ∧ 𝑧 ∈ ℋ) → (𝑤 + 𝑧) = (𝑧 + 𝑤))
85, 6, 7syl2an 583 . . . . . . . 8 ((𝑤𝐵𝑧𝐴) → (𝑤 + 𝑧) = (𝑧 + 𝑤))
98eqeq2d 2781 . . . . . . 7 ((𝑤𝐵𝑧𝐴) → (𝑥 = (𝑤 + 𝑧) ↔ 𝑥 = (𝑧 + 𝑤)))
109rexbidva 3197 . . . . . 6 (𝑤𝐵 → (∃𝑧𝐴 𝑥 = (𝑤 + 𝑧) ↔ ∃𝑧𝐴 𝑥 = (𝑧 + 𝑤)))
1110riotabiia 6774 . . . . 5 (𝑤𝐵𝑧𝐴 𝑥 = (𝑤 + 𝑧)) = (𝑤𝐵𝑧𝐴 𝑥 = (𝑧 + 𝑤))
124, 11mpteq12i 4877 . . . 4 (𝑥 ∈ (𝐵 + 𝐴) ↦ (𝑤𝐵𝑧𝐴 𝑥 = (𝑤 + 𝑧))) = (𝑥 ∈ (𝐴 + 𝐵) ↦ (𝑤𝐵𝑧𝐴 𝑥 = (𝑧 + 𝑤)))
133, 12eqtr4i 2796 . . 3 𝑇 = (𝑥 ∈ (𝐵 + 𝐴) ↦ (𝑤𝐵𝑧𝐴 𝑥 = (𝑤 + 𝑧)))
141, 2, 13cdj3lem2b 29636 . 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 6333 . . . . . . . 8 (𝑥 = 𝑡 → (norm𝑥) = (norm𝑡))
1615oveq1d 6811 . . . . . . 7 (𝑥 = 𝑡 → ((norm𝑥) + (norm𝑦)) = ((norm𝑡) + (norm𝑦)))
17 fvoveq1 6819 . . . . . . . 8 (𝑥 = 𝑡 → (norm‘(𝑥 + 𝑦)) = (norm‘(𝑡 + 𝑦)))
1817oveq2d 6812 . . . . . . 7 (𝑥 = 𝑡 → (𝑣 · (norm‘(𝑥 + 𝑦))) = (𝑣 · (norm‘(𝑡 + 𝑦))))
1916, 18breq12d 4800 . . . . . 6 (𝑥 = 𝑡 → (((norm𝑥) + (norm𝑦)) ≤ (𝑣 · (norm‘(𝑥 + 𝑦))) ↔ ((norm𝑡) + (norm𝑦)) ≤ (𝑣 · (norm‘(𝑡 + 𝑦)))))
20 fveq2 6333 . . . . . . . 8 (𝑦 = → (norm𝑦) = (norm))
2120oveq2d 6812 . . . . . . 7 (𝑦 = → ((norm𝑡) + (norm𝑦)) = ((norm𝑡) + (norm)))
22 oveq2 6804 . . . . . . . . 9 (𝑦 = → (𝑡 + 𝑦) = (𝑡 + ))
2322fveq2d 6337 . . . . . . . 8 (𝑦 = → (norm‘(𝑡 + 𝑦)) = (norm‘(𝑡 + )))
2423oveq2d 6812 . . . . . . 7 (𝑦 = → (𝑣 · (norm‘(𝑡 + 𝑦))) = (𝑣 · (norm‘(𝑡 + ))))
2521, 24breq12d 4800 . . . . . 6 (𝑦 = → (((norm𝑡) + (norm𝑦)) ≤ (𝑣 · (norm‘(𝑡 + 𝑦))) ↔ ((norm𝑡) + (norm)) ≤ (𝑣 · (norm‘(𝑡 + )))))
2619, 25cbvral2v 3328 . . . . 5 (∀𝑥𝐴𝑦𝐵 ((norm𝑥) + (norm𝑦)) ≤ (𝑣 · (norm‘(𝑥 + 𝑦))) ↔ ∀𝑡𝐴𝐵 ((norm𝑡) + (norm)) ≤ (𝑣 · (norm‘(𝑡 + ))))
27 ralcom 3246 . . . . 5 (∀𝑡𝐴𝐵 ((norm𝑡) + (norm)) ≤ (𝑣 · (norm‘(𝑡 + ))) ↔ ∀𝐵𝑡𝐴 ((norm𝑡) + (norm)) ≤ (𝑣 · (norm‘(𝑡 + ))))
281sheli 28411 . . . . . . . . . . . 12 (𝑥𝐵𝑥 ∈ ℋ)
29 normcl 28322 . . . . . . . . . . . 12 (𝑥 ∈ ℋ → (norm𝑥) ∈ ℝ)
3028, 29syl 17 . . . . . . . . . . 11 (𝑥𝐵 → (norm𝑥) ∈ ℝ)
3130recnd 10274 . . . . . . . . . 10 (𝑥𝐵 → (norm𝑥) ∈ ℂ)
322sheli 28411 . . . . . . . . . . . 12 (𝑦𝐴𝑦 ∈ ℋ)
33 normcl 28322 . . . . . . . . . . . 12 (𝑦 ∈ ℋ → (norm𝑦) ∈ ℝ)
3432, 33syl 17 . . . . . . . . . . 11 (𝑦𝐴 → (norm𝑦) ∈ ℝ)
3534recnd 10274 . . . . . . . . . 10 (𝑦𝐴 → (norm𝑦) ∈ ℂ)
36 addcom 10428 . . . . . . . . . 10 (((norm𝑥) ∈ ℂ ∧ (norm𝑦) ∈ ℂ) → ((norm𝑥) + (norm𝑦)) = ((norm𝑦) + (norm𝑥)))
3731, 35, 36syl2an 583 . . . . . . . . 9 ((𝑥𝐵𝑦𝐴) → ((norm𝑥) + (norm𝑦)) = ((norm𝑦) + (norm𝑥)))
38 ax-hvcom 28198 . . . . . . . . . . . 12 ((𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ) → (𝑥 + 𝑦) = (𝑦 + 𝑥))
3928, 32, 38syl2an 583 . . . . . . . . . . 11 ((𝑥𝐵𝑦𝐴) → (𝑥 + 𝑦) = (𝑦 + 𝑥))
4039fveq2d 6337 . . . . . . . . . 10 ((𝑥𝐵𝑦𝐴) → (norm‘(𝑥 + 𝑦)) = (norm‘(𝑦 + 𝑥)))
4140oveq2d 6812 . . . . . . . . 9 ((𝑥𝐵𝑦𝐴) → (𝑣 · (norm‘(𝑥 + 𝑦))) = (𝑣 · (norm‘(𝑦 + 𝑥))))
4237, 41breq12d 4800 . . . . . . . 8 ((𝑥𝐵𝑦𝐴) → (((norm𝑥) + (norm𝑦)) ≤ (𝑣 · (norm‘(𝑥 + 𝑦))) ↔ ((norm𝑦) + (norm𝑥)) ≤ (𝑣 · (norm‘(𝑦 + 𝑥)))))
4342ralbidva 3134 . . . . . . 7 (𝑥𝐵 → (∀𝑦𝐴 ((norm𝑥) + (norm𝑦)) ≤ (𝑣 · (norm‘(𝑥 + 𝑦))) ↔ ∀𝑦𝐴 ((norm𝑦) + (norm𝑥)) ≤ (𝑣 · (norm‘(𝑦 + 𝑥)))))
4443ralbiia 3128 . . . . . 6 (∀𝑥𝐵𝑦𝐴 ((norm𝑥) + (norm𝑦)) ≤ (𝑣 · (norm‘(𝑥 + 𝑦))) ↔ ∀𝑥𝐵𝑦𝐴 ((norm𝑦) + (norm𝑥)) ≤ (𝑣 · (norm‘(𝑦 + 𝑥))))
45 fveq2 6333 . . . . . . . . 9 (𝑥 = → (norm𝑥) = (norm))
4645oveq2d 6812 . . . . . . . 8 (𝑥 = → ((norm𝑦) + (norm𝑥)) = ((norm𝑦) + (norm)))
47 oveq2 6804 . . . . . . . . . 10 (𝑥 = → (𝑦 + 𝑥) = (𝑦 + ))
4847fveq2d 6337 . . . . . . . . 9 (𝑥 = → (norm‘(𝑦 + 𝑥)) = (norm‘(𝑦 + )))
4948oveq2d 6812 . . . . . . . 8 (𝑥 = → (𝑣 · (norm‘(𝑦 + 𝑥))) = (𝑣 · (norm‘(𝑦 + ))))
5046, 49breq12d 4800 . . . . . . 7 (𝑥 = → (((norm𝑦) + (norm𝑥)) ≤ (𝑣 · (norm‘(𝑦 + 𝑥))) ↔ ((norm𝑦) + (norm)) ≤ (𝑣 · (norm‘(𝑦 + )))))
51 fveq2 6333 . . . . . . . . 9 (𝑦 = 𝑡 → (norm𝑦) = (norm𝑡))
5251oveq1d 6811 . . . . . . . 8 (𝑦 = 𝑡 → ((norm𝑦) + (norm)) = ((norm𝑡) + (norm)))
53 fvoveq1 6819 . . . . . . . . 9 (𝑦 = 𝑡 → (norm‘(𝑦 + )) = (norm‘(𝑡 + )))
5453oveq2d 6812 . . . . . . . 8 (𝑦 = 𝑡 → (𝑣 · (norm‘(𝑦 + ))) = (𝑣 · (norm‘(𝑡 + ))))
5552, 54breq12d 4800 . . . . . . 7 (𝑦 = 𝑡 → (((norm𝑦) + (norm)) ≤ (𝑣 · (norm‘(𝑦 + ))) ↔ ((norm𝑡) + (norm)) ≤ (𝑣 · (norm‘(𝑡 + )))))
5650, 55cbvral2v 3328 . . . . . 6 (∀𝑥𝐵𝑦𝐴 ((norm𝑦) + (norm𝑥)) ≤ (𝑣 · (norm‘(𝑦 + 𝑥))) ↔ ∀𝐵𝑡𝐴 ((norm𝑡) + (norm)) ≤ (𝑣 · (norm‘(𝑡 + ))))
5744, 56bitr2i 265 . . . . 5 (∀𝐵𝑡𝐴 ((norm𝑡) + (norm)) ≤ (𝑣 · (norm‘(𝑡 + ))) ↔ ∀𝑥𝐵𝑦𝐴 ((norm𝑥) + (norm𝑦)) ≤ (𝑣 · (norm‘(𝑥 + 𝑦))))
5826, 27, 573bitri 286 . . . 4 (∀𝑥𝐴𝑦𝐵 ((norm𝑥) + (norm𝑦)) ≤ (𝑣 · (norm‘(𝑥 + 𝑦))) ↔ ∀𝑥𝐵𝑦𝐴 ((norm𝑥) + (norm𝑦)) ≤ (𝑣 · (norm‘(𝑥 + 𝑦))))
5958anbi2i 609 . . 3 ((0 < 𝑣 ∧ ∀𝑥𝐴𝑦𝐵 ((norm𝑥) + (norm𝑦)) ≤ (𝑣 · (norm‘(𝑥 + 𝑦)))) ↔ (0 < 𝑣 ∧ ∀𝑥𝐵𝑦𝐴 ((norm𝑥) + (norm𝑦)) ≤ (𝑣 · (norm‘(𝑥 + 𝑦)))))
6059rexbii 3189 . 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 28562 . . . . 5 (𝐴 + 𝐵) = (𝐵 + 𝐴)
6261raleqi 3291 . . . 4 (∀𝑢 ∈ (𝐴 + 𝐵)(norm‘(𝑇𝑢)) ≤ (𝑣 · (norm𝑢)) ↔ ∀𝑢 ∈ (𝐵 + 𝐴)(norm‘(𝑇𝑢)) ≤ (𝑣 · (norm𝑢)))
6362anbi2i 609 . . 3 ((0 < 𝑣 ∧ ∀𝑢 ∈ (𝐴 + 𝐵)(norm‘(𝑇𝑢)) ≤ (𝑣 · (norm𝑢))) ↔ (0 < 𝑣 ∧ ∀𝑢 ∈ (𝐵 + 𝐴)(norm‘(𝑇𝑢)) ≤ (𝑣 · (norm𝑢))))
6463rexbii 3189 . 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 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 382   = wceq 1631  wcel 2145  wral 3061  wrex 3062   class class class wbr 4787  cmpt 4864  cfv 6030  crio 6756  (class class class)co 6796  cc 10140  cr 10141  0cc0 10142   + caddc 10145   · cmul 10147   < clt 10280  cle 10281  chil 28116   + cva 28117  normcno 28120   S csh 28125   + cph 28128
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4905  ax-sep 4916  ax-nul 4924  ax-pow 4975  ax-pr 5035  ax-un 7100  ax-cnex 10198  ax-resscn 10199  ax-1cn 10200  ax-icn 10201  ax-addcl 10202  ax-addrcl 10203  ax-mulcl 10204  ax-mulrcl 10205  ax-mulcom 10206  ax-addass 10207  ax-mulass 10208  ax-distr 10209  ax-i2m1 10210  ax-1ne0 10211  ax-1rid 10212  ax-rnegex 10213  ax-rrecex 10214  ax-cnre 10215  ax-pre-lttri 10216  ax-pre-lttrn 10217  ax-pre-ltadd 10218  ax-pre-mulgt0 10219  ax-pre-sup 10220  ax-hilex 28196  ax-hfvadd 28197  ax-hvcom 28198  ax-hvass 28199  ax-hv0cl 28200  ax-hvaddid 28201  ax-hfvmul 28202  ax-hvmulid 28203  ax-hvmulass 28204  ax-hvdistr1 28205  ax-hvdistr2 28206  ax-hvmul0 28207  ax-hfi 28276  ax-his1 28279  ax-his3 28281  ax-his4 28282
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3or 1072  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-nel 3047  df-ral 3066  df-rex 3067  df-reu 3068  df-rmo 3069  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-pss 3739  df-nul 4064  df-if 4227  df-pw 4300  df-sn 4318  df-pr 4320  df-tp 4322  df-op 4324  df-uni 4576  df-int 4613  df-iun 4657  df-br 4788  df-opab 4848  df-mpt 4865  df-tr 4888  df-id 5158  df-eprel 5163  df-po 5171  df-so 5172  df-fr 5209  df-we 5211  df-xp 5256  df-rel 5257  df-cnv 5258  df-co 5259  df-dm 5260  df-rn 5261  df-res 5262  df-ima 5263  df-pred 5822  df-ord 5868  df-on 5869  df-lim 5870  df-suc 5871  df-iota 5993  df-fun 6032  df-fn 6033  df-f 6034  df-f1 6035  df-fo 6036  df-f1o 6037  df-fv 6038  df-riota 6757  df-ov 6799  df-oprab 6800  df-mpt2 6801  df-om 7217  df-2nd 7320  df-wrecs 7563  df-recs 7625  df-rdg 7663  df-er 7900  df-en 8114  df-dom 8115  df-sdom 8116  df-sup 8508  df-pnf 10282  df-mnf 10283  df-xr 10284  df-ltxr 10285  df-le 10286  df-sub 10474  df-neg 10475  df-div 10891  df-nn 11227  df-2 11285  df-3 11286  df-n0 11500  df-z 11585  df-uz 11894  df-rp 12036  df-seq 13009  df-exp 13068  df-cj 14047  df-re 14048  df-im 14049  df-sqrt 14183  df-abs 14184  df-grpo 27687  df-ablo 27739  df-hnorm 28165  df-hvsub 28168  df-sh 28404  df-ch0 28450  df-shs 28507
This theorem is referenced by:  cdj3i  29640
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