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Theorem cdj3lem3b 31382
Description: Lemma for cdj3i 31383. 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 30305 . . . . 5 (𝐵 + 𝐴) = (𝐴 + 𝐵)
51sheli 30156 . . . . . . . . 9 (𝑤𝐵𝑤 ∈ ℋ)
62sheli 30156 . . . . . . . . 9 (𝑧𝐴𝑧 ∈ ℋ)
7 ax-hvcom 29943 . . . . . . . . 9 ((𝑤 ∈ ℋ ∧ 𝑧 ∈ ℋ) → (𝑤 + 𝑧) = (𝑧 + 𝑤))
85, 6, 7syl2an 596 . . . . . . . 8 ((𝑤𝐵𝑧𝐴) → (𝑤 + 𝑧) = (𝑧 + 𝑤))
98eqeq2d 2747 . . . . . . 7 ((𝑤𝐵𝑧𝐴) → (𝑥 = (𝑤 + 𝑧) ↔ 𝑥 = (𝑧 + 𝑤)))
109rexbidva 3173 . . . . . 6 (𝑤𝐵 → (∃𝑧𝐴 𝑥 = (𝑤 + 𝑧) ↔ ∃𝑧𝐴 𝑥 = (𝑧 + 𝑤)))
1110riotabiia 7334 . . . . 5 (𝑤𝐵𝑧𝐴 𝑥 = (𝑤 + 𝑧)) = (𝑤𝐵𝑧𝐴 𝑥 = (𝑧 + 𝑤))
124, 11mpteq12i 5211 . . . 4 (𝑥 ∈ (𝐵 + 𝐴) ↦ (𝑤𝐵𝑧𝐴 𝑥 = (𝑤 + 𝑧))) = (𝑥 ∈ (𝐴 + 𝐵) ↦ (𝑤𝐵𝑧𝐴 𝑥 = (𝑧 + 𝑤)))
133, 12eqtr4i 2767 . . 3 𝑇 = (𝑥 ∈ (𝐵 + 𝐴) ↦ (𝑤𝐵𝑧𝐴 𝑥 = (𝑤 + 𝑧)))
141, 2, 13cdj3lem2b 31379 . 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 6842 . . . . . . . 8 (𝑥 = 𝑡 → (norm𝑥) = (norm𝑡))
1615oveq1d 7372 . . . . . . 7 (𝑥 = 𝑡 → ((norm𝑥) + (norm𝑦)) = ((norm𝑡) + (norm𝑦)))
17 fvoveq1 7380 . . . . . . . 8 (𝑥 = 𝑡 → (norm‘(𝑥 + 𝑦)) = (norm‘(𝑡 + 𝑦)))
1817oveq2d 7373 . . . . . . 7 (𝑥 = 𝑡 → (𝑣 · (norm‘(𝑥 + 𝑦))) = (𝑣 · (norm‘(𝑡 + 𝑦))))
1916, 18breq12d 5118 . . . . . 6 (𝑥 = 𝑡 → (((norm𝑥) + (norm𝑦)) ≤ (𝑣 · (norm‘(𝑥 + 𝑦))) ↔ ((norm𝑡) + (norm𝑦)) ≤ (𝑣 · (norm‘(𝑡 + 𝑦)))))
20 fveq2 6842 . . . . . . . 8 (𝑦 = → (norm𝑦) = (norm))
2120oveq2d 7373 . . . . . . 7 (𝑦 = → ((norm𝑡) + (norm𝑦)) = ((norm𝑡) + (norm)))
22 oveq2 7365 . . . . . . . . 9 (𝑦 = → (𝑡 + 𝑦) = (𝑡 + ))
2322fveq2d 6846 . . . . . . . 8 (𝑦 = → (norm‘(𝑡 + 𝑦)) = (norm‘(𝑡 + )))
2423oveq2d 7373 . . . . . . 7 (𝑦 = → (𝑣 · (norm‘(𝑡 + 𝑦))) = (𝑣 · (norm‘(𝑡 + ))))
2521, 24breq12d 5118 . . . . . 6 (𝑦 = → (((norm𝑡) + (norm𝑦)) ≤ (𝑣 · (norm‘(𝑡 + 𝑦))) ↔ ((norm𝑡) + (norm)) ≤ (𝑣 · (norm‘(𝑡 + )))))
2619, 25cbvral2vw 3227 . . . . 5 (∀𝑥𝐴𝑦𝐵 ((norm𝑥) + (norm𝑦)) ≤ (𝑣 · (norm‘(𝑥 + 𝑦))) ↔ ∀𝑡𝐴𝐵 ((norm𝑡) + (norm)) ≤ (𝑣 · (norm‘(𝑡 + ))))
27 ralcom 3272 . . . . 5 (∀𝑡𝐴𝐵 ((norm𝑡) + (norm)) ≤ (𝑣 · (norm‘(𝑡 + ))) ↔ ∀𝐵𝑡𝐴 ((norm𝑡) + (norm)) ≤ (𝑣 · (norm‘(𝑡 + ))))
281sheli 30156 . . . . . . . . . . . 12 (𝑥𝐵𝑥 ∈ ℋ)
29 normcl 30067 . . . . . . . . . . . 12 (𝑥 ∈ ℋ → (norm𝑥) ∈ ℝ)
3028, 29syl 17 . . . . . . . . . . 11 (𝑥𝐵 → (norm𝑥) ∈ ℝ)
3130recnd 11183 . . . . . . . . . 10 (𝑥𝐵 → (norm𝑥) ∈ ℂ)
322sheli 30156 . . . . . . . . . . . 12 (𝑦𝐴𝑦 ∈ ℋ)
33 normcl 30067 . . . . . . . . . . . 12 (𝑦 ∈ ℋ → (norm𝑦) ∈ ℝ)
3432, 33syl 17 . . . . . . . . . . 11 (𝑦𝐴 → (norm𝑦) ∈ ℝ)
3534recnd 11183 . . . . . . . . . 10 (𝑦𝐴 → (norm𝑦) ∈ ℂ)
36 addcom 11341 . . . . . . . . . 10 (((norm𝑥) ∈ ℂ ∧ (norm𝑦) ∈ ℂ) → ((norm𝑥) + (norm𝑦)) = ((norm𝑦) + (norm𝑥)))
3731, 35, 36syl2an 596 . . . . . . . . 9 ((𝑥𝐵𝑦𝐴) → ((norm𝑥) + (norm𝑦)) = ((norm𝑦) + (norm𝑥)))
38 ax-hvcom 29943 . . . . . . . . . . . 12 ((𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ) → (𝑥 + 𝑦) = (𝑦 + 𝑥))
3928, 32, 38syl2an 596 . . . . . . . . . . 11 ((𝑥𝐵𝑦𝐴) → (𝑥 + 𝑦) = (𝑦 + 𝑥))
4039fveq2d 6846 . . . . . . . . . 10 ((𝑥𝐵𝑦𝐴) → (norm‘(𝑥 + 𝑦)) = (norm‘(𝑦 + 𝑥)))
4140oveq2d 7373 . . . . . . . . 9 ((𝑥𝐵𝑦𝐴) → (𝑣 · (norm‘(𝑥 + 𝑦))) = (𝑣 · (norm‘(𝑦 + 𝑥))))
4237, 41breq12d 5118 . . . . . . . 8 ((𝑥𝐵𝑦𝐴) → (((norm𝑥) + (norm𝑦)) ≤ (𝑣 · (norm‘(𝑥 + 𝑦))) ↔ ((norm𝑦) + (norm𝑥)) ≤ (𝑣 · (norm‘(𝑦 + 𝑥)))))
4342ralbidva 3172 . . . . . . 7 (𝑥𝐵 → (∀𝑦𝐴 ((norm𝑥) + (norm𝑦)) ≤ (𝑣 · (norm‘(𝑥 + 𝑦))) ↔ ∀𝑦𝐴 ((norm𝑦) + (norm𝑥)) ≤ (𝑣 · (norm‘(𝑦 + 𝑥)))))
4443ralbiia 3094 . . . . . 6 (∀𝑥𝐵𝑦𝐴 ((norm𝑥) + (norm𝑦)) ≤ (𝑣 · (norm‘(𝑥 + 𝑦))) ↔ ∀𝑥𝐵𝑦𝐴 ((norm𝑦) + (norm𝑥)) ≤ (𝑣 · (norm‘(𝑦 + 𝑥))))
45 fveq2 6842 . . . . . . . . 9 (𝑥 = → (norm𝑥) = (norm))
4645oveq2d 7373 . . . . . . . 8 (𝑥 = → ((norm𝑦) + (norm𝑥)) = ((norm𝑦) + (norm)))
47 oveq2 7365 . . . . . . . . . 10 (𝑥 = → (𝑦 + 𝑥) = (𝑦 + ))
4847fveq2d 6846 . . . . . . . . 9 (𝑥 = → (norm‘(𝑦 + 𝑥)) = (norm‘(𝑦 + )))
4948oveq2d 7373 . . . . . . . 8 (𝑥 = → (𝑣 · (norm‘(𝑦 + 𝑥))) = (𝑣 · (norm‘(𝑦 + ))))
5046, 49breq12d 5118 . . . . . . 7 (𝑥 = → (((norm𝑦) + (norm𝑥)) ≤ (𝑣 · (norm‘(𝑦 + 𝑥))) ↔ ((norm𝑦) + (norm)) ≤ (𝑣 · (norm‘(𝑦 + )))))
51 fveq2 6842 . . . . . . . . 9 (𝑦 = 𝑡 → (norm𝑦) = (norm𝑡))
5251oveq1d 7372 . . . . . . . 8 (𝑦 = 𝑡 → ((norm𝑦) + (norm)) = ((norm𝑡) + (norm)))
53 fvoveq1 7380 . . . . . . . . 9 (𝑦 = 𝑡 → (norm‘(𝑦 + )) = (norm‘(𝑡 + )))
5453oveq2d 7373 . . . . . . . 8 (𝑦 = 𝑡 → (𝑣 · (norm‘(𝑦 + ))) = (𝑣 · (norm‘(𝑡 + ))))
5552, 54breq12d 5118 . . . . . . 7 (𝑦 = 𝑡 → (((norm𝑦) + (norm)) ≤ (𝑣 · (norm‘(𝑦 + ))) ↔ ((norm𝑡) + (norm)) ≤ (𝑣 · (norm‘(𝑡 + )))))
5650, 55cbvral2vw 3227 . . . . . 6 (∀𝑥𝐵𝑦𝐴 ((norm𝑦) + (norm𝑥)) ≤ (𝑣 · (norm‘(𝑦 + 𝑥))) ↔ ∀𝐵𝑡𝐴 ((norm𝑡) + (norm)) ≤ (𝑣 · (norm‘(𝑡 + ))))
5744, 56bitr2i 275 . . . . 5 (∀𝐵𝑡𝐴 ((norm𝑡) + (norm)) ≤ (𝑣 · (norm‘(𝑡 + ))) ↔ ∀𝑥𝐵𝑦𝐴 ((norm𝑥) + (norm𝑦)) ≤ (𝑣 · (norm‘(𝑥 + 𝑦))))
5826, 27, 573bitri 296 . . . 4 (∀𝑥𝐴𝑦𝐵 ((norm𝑥) + (norm𝑦)) ≤ (𝑣 · (norm‘(𝑥 + 𝑦))) ↔ ∀𝑥𝐵𝑦𝐴 ((norm𝑥) + (norm𝑦)) ≤ (𝑣 · (norm‘(𝑥 + 𝑦))))
5958anbi2i 623 . . 3 ((0 < 𝑣 ∧ ∀𝑥𝐴𝑦𝐵 ((norm𝑥) + (norm𝑦)) ≤ (𝑣 · (norm‘(𝑥 + 𝑦)))) ↔ (0 < 𝑣 ∧ ∀𝑥𝐵𝑦𝐴 ((norm𝑥) + (norm𝑦)) ≤ (𝑣 · (norm‘(𝑥 + 𝑦)))))
6059rexbii 3097 . 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 30305 . . . . 5 (𝐴 + 𝐵) = (𝐵 + 𝐴)
6261raleqi 3311 . . . 4 (∀𝑢 ∈ (𝐴 + 𝐵)(norm‘(𝑇𝑢)) ≤ (𝑣 · (norm𝑢)) ↔ ∀𝑢 ∈ (𝐵 + 𝐴)(norm‘(𝑇𝑢)) ≤ (𝑣 · (norm𝑢)))
6362anbi2i 623 . . 3 ((0 < 𝑣 ∧ ∀𝑢 ∈ (𝐴 + 𝐵)(norm‘(𝑇𝑢)) ≤ (𝑣 · (norm𝑢))) ↔ (0 < 𝑣 ∧ ∀𝑢 ∈ (𝐵 + 𝐴)(norm‘(𝑇𝑢)) ≤ (𝑣 · (norm𝑢))))
6463rexbii 3097 . 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 291 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 1541  wcel 2106  wral 3064  wrex 3073   class class class wbr 5105  cmpt 5188  cfv 6496  crio 7312  (class class class)co 7357  cc 11049  cr 11050  0cc0 11051   + caddc 11054   · cmul 11056   < clt 11189  cle 11190  chba 29861   + cva 29862  normcno 29865   S csh 29870   + cph 29873
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128  ax-pre-sup 11129  ax-hilex 29941  ax-hfvadd 29942  ax-hvcom 29943  ax-hvass 29944  ax-hv0cl 29945  ax-hvaddid 29946  ax-hfvmul 29947  ax-hvmulid 29948  ax-hvmulass 29949  ax-hvdistr1 29950  ax-hvdistr2 29951  ax-hvmul0 29952  ax-hfi 30021  ax-his1 30024  ax-his3 30026  ax-his4 30027
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-int 4908  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7803  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-er 8648  df-en 8884  df-dom 8885  df-sdom 8886  df-sup 9378  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-div 11813  df-nn 12154  df-2 12216  df-3 12217  df-n0 12414  df-z 12500  df-uz 12764  df-rp 12916  df-seq 13907  df-exp 13968  df-cj 14984  df-re 14985  df-im 14986  df-sqrt 15120  df-abs 15121  df-grpo 29435  df-ablo 29487  df-hnorm 29910  df-hvsub 29913  df-sh 30149  df-ch0 30195  df-shs 30250
This theorem is referenced by:  cdj3i  31383
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