HSE Home Hilbert Space Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  HSE Home  >  Th. List  >  cdjreui Structured version   Visualization version   GIF version

Theorem cdjreui 30136
Description: A member of the sum of disjoint subspaces has a unique decomposition. Part of Lemma 5 of [Holland] p. 1520. (Contributed by NM, 20-May-2005.) (New usage is discouraged.)
Hypotheses
Ref Expression
cdjreu.1 𝐴S
cdjreu.2 𝐵S
Assertion
Ref Expression
cdjreui ((𝐶 ∈ (𝐴 + 𝐵) ∧ (𝐴𝐵) = 0) → ∃!𝑥𝐴𝑦𝐵 𝐶 = (𝑥 + 𝑦))
Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐵,𝑦   𝑥,𝐶,𝑦

Proof of Theorem cdjreui
Dummy variables 𝑧 𝑤 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 cdjreu.1 . . . . 5 𝐴S
2 cdjreu.2 . . . . 5 𝐵S
31, 2shseli 29020 . . . 4 (𝐶 ∈ (𝐴 + 𝐵) ↔ ∃𝑥𝐴𝑦𝐵 𝐶 = (𝑥 + 𝑦))
43biimpi 217 . . 3 (𝐶 ∈ (𝐴 + 𝐵) → ∃𝑥𝐴𝑦𝐵 𝐶 = (𝑥 + 𝑦))
5 reeanv 3365 . . . . 5 (∃𝑦𝐵𝑤𝐵 (𝐶 = (𝑥 + 𝑦) ∧ 𝐶 = (𝑧 + 𝑤)) ↔ (∃𝑦𝐵 𝐶 = (𝑥 + 𝑦) ∧ ∃𝑤𝐵 𝐶 = (𝑧 + 𝑤)))
6 eqtr2 2839 . . . . . . 7 ((𝐶 = (𝑥 + 𝑦) ∧ 𝐶 = (𝑧 + 𝑤)) → (𝑥 + 𝑦) = (𝑧 + 𝑤))
71sheli 28918 . . . . . . . . . . . 12 (𝑥𝐴𝑥 ∈ ℋ)
82sheli 28918 . . . . . . . . . . . 12 (𝑦𝐵𝑦 ∈ ℋ)
97, 8anim12i 612 . . . . . . . . . . 11 ((𝑥𝐴𝑦𝐵) → (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ))
101sheli 28918 . . . . . . . . . . . 12 (𝑧𝐴𝑧 ∈ ℋ)
112sheli 28918 . . . . . . . . . . . 12 (𝑤𝐵𝑤 ∈ ℋ)
1210, 11anim12i 612 . . . . . . . . . . 11 ((𝑧𝐴𝑤𝐵) → (𝑧 ∈ ℋ ∧ 𝑤 ∈ ℋ))
13 hvaddsub4 28782 . . . . . . . . . . 11 (((𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ) ∧ (𝑧 ∈ ℋ ∧ 𝑤 ∈ ℋ)) → ((𝑥 + 𝑦) = (𝑧 + 𝑤) ↔ (𝑥 𝑧) = (𝑤 𝑦)))
149, 12, 13syl2an 595 . . . . . . . . . 10 (((𝑥𝐴𝑦𝐵) ∧ (𝑧𝐴𝑤𝐵)) → ((𝑥 + 𝑦) = (𝑧 + 𝑤) ↔ (𝑥 𝑧) = (𝑤 𝑦)))
1514an4s 656 . . . . . . . . 9 (((𝑥𝐴𝑧𝐴) ∧ (𝑦𝐵𝑤𝐵)) → ((𝑥 + 𝑦) = (𝑧 + 𝑤) ↔ (𝑥 𝑧) = (𝑤 𝑦)))
1615adantll 710 . . . . . . . 8 ((((𝐴𝐵) = 0 ∧ (𝑥𝐴𝑧𝐴)) ∧ (𝑦𝐵𝑤𝐵)) → ((𝑥 + 𝑦) = (𝑧 + 𝑤) ↔ (𝑥 𝑧) = (𝑤 𝑦)))
17 shsubcl 28924 . . . . . . . . . . . . . . . 16 ((𝐵S𝑤𝐵𝑦𝐵) → (𝑤 𝑦) ∈ 𝐵)
182, 17mp3an1 1439 . . . . . . . . . . . . . . 15 ((𝑤𝐵𝑦𝐵) → (𝑤 𝑦) ∈ 𝐵)
1918ancoms 459 . . . . . . . . . . . . . 14 ((𝑦𝐵𝑤𝐵) → (𝑤 𝑦) ∈ 𝐵)
20 eleq1 2897 . . . . . . . . . . . . . 14 ((𝑥 𝑧) = (𝑤 𝑦) → ((𝑥 𝑧) ∈ 𝐵 ↔ (𝑤 𝑦) ∈ 𝐵))
2119, 20syl5ibrcom 248 . . . . . . . . . . . . 13 ((𝑦𝐵𝑤𝐵) → ((𝑥 𝑧) = (𝑤 𝑦) → (𝑥 𝑧) ∈ 𝐵))
2221adantl 482 . . . . . . . . . . . 12 (((𝑥𝐴𝑧𝐴) ∧ (𝑦𝐵𝑤𝐵)) → ((𝑥 𝑧) = (𝑤 𝑦) → (𝑥 𝑧) ∈ 𝐵))
23 shsubcl 28924 . . . . . . . . . . . . . 14 ((𝐴S𝑥𝐴𝑧𝐴) → (𝑥 𝑧) ∈ 𝐴)
241, 23mp3an1 1439 . . . . . . . . . . . . 13 ((𝑥𝐴𝑧𝐴) → (𝑥 𝑧) ∈ 𝐴)
2524adantr 481 . . . . . . . . . . . 12 (((𝑥𝐴𝑧𝐴) ∧ (𝑦𝐵𝑤𝐵)) → (𝑥 𝑧) ∈ 𝐴)
2622, 25jctild 526 . . . . . . . . . . 11 (((𝑥𝐴𝑧𝐴) ∧ (𝑦𝐵𝑤𝐵)) → ((𝑥 𝑧) = (𝑤 𝑦) → ((𝑥 𝑧) ∈ 𝐴 ∧ (𝑥 𝑧) ∈ 𝐵)))
2726adantll 710 . . . . . . . . . 10 ((((𝐴𝐵) = 0 ∧ (𝑥𝐴𝑧𝐴)) ∧ (𝑦𝐵𝑤𝐵)) → ((𝑥 𝑧) = (𝑤 𝑦) → ((𝑥 𝑧) ∈ 𝐴 ∧ (𝑥 𝑧) ∈ 𝐵)))
28 elin 4166 . . . . . . . . . . . 12 ((𝑥 𝑧) ∈ (𝐴𝐵) ↔ ((𝑥 𝑧) ∈ 𝐴 ∧ (𝑥 𝑧) ∈ 𝐵))
29 eleq2 2898 . . . . . . . . . . . 12 ((𝐴𝐵) = 0 → ((𝑥 𝑧) ∈ (𝐴𝐵) ↔ (𝑥 𝑧) ∈ 0))
3028, 29syl5bbr 286 . . . . . . . . . . 11 ((𝐴𝐵) = 0 → (((𝑥 𝑧) ∈ 𝐴 ∧ (𝑥 𝑧) ∈ 𝐵) ↔ (𝑥 𝑧) ∈ 0))
3130ad2antrr 722 . . . . . . . . . 10 ((((𝐴𝐵) = 0 ∧ (𝑥𝐴𝑧𝐴)) ∧ (𝑦𝐵𝑤𝐵)) → (((𝑥 𝑧) ∈ 𝐴 ∧ (𝑥 𝑧) ∈ 𝐵) ↔ (𝑥 𝑧) ∈ 0))
3227, 31sylibd 240 . . . . . . . . 9 ((((𝐴𝐵) = 0 ∧ (𝑥𝐴𝑧𝐴)) ∧ (𝑦𝐵𝑤𝐵)) → ((𝑥 𝑧) = (𝑤 𝑦) → (𝑥 𝑧) ∈ 0))
33 elch0 28958 . . . . . . . . . . . 12 ((𝑥 𝑧) ∈ 0 ↔ (𝑥 𝑧) = 0)
34 hvsubeq0 28772 . . . . . . . . . . . 12 ((𝑥 ∈ ℋ ∧ 𝑧 ∈ ℋ) → ((𝑥 𝑧) = 0𝑥 = 𝑧))
3533, 34syl5bb 284 . . . . . . . . . . 11 ((𝑥 ∈ ℋ ∧ 𝑧 ∈ ℋ) → ((𝑥 𝑧) ∈ 0𝑥 = 𝑧))
367, 10, 35syl2an 595 . . . . . . . . . 10 ((𝑥𝐴𝑧𝐴) → ((𝑥 𝑧) ∈ 0𝑥 = 𝑧))
3736ad2antlr 723 . . . . . . . . 9 ((((𝐴𝐵) = 0 ∧ (𝑥𝐴𝑧𝐴)) ∧ (𝑦𝐵𝑤𝐵)) → ((𝑥 𝑧) ∈ 0𝑥 = 𝑧))
3832, 37sylibd 240 . . . . . . . 8 ((((𝐴𝐵) = 0 ∧ (𝑥𝐴𝑧𝐴)) ∧ (𝑦𝐵𝑤𝐵)) → ((𝑥 𝑧) = (𝑤 𝑦) → 𝑥 = 𝑧))
3916, 38sylbid 241 . . . . . . 7 ((((𝐴𝐵) = 0 ∧ (𝑥𝐴𝑧𝐴)) ∧ (𝑦𝐵𝑤𝐵)) → ((𝑥 + 𝑦) = (𝑧 + 𝑤) → 𝑥 = 𝑧))
406, 39syl5 34 . . . . . 6 ((((𝐴𝐵) = 0 ∧ (𝑥𝐴𝑧𝐴)) ∧ (𝑦𝐵𝑤𝐵)) → ((𝐶 = (𝑥 + 𝑦) ∧ 𝐶 = (𝑧 + 𝑤)) → 𝑥 = 𝑧))
4140rexlimdvva 3291 . . . . 5 (((𝐴𝐵) = 0 ∧ (𝑥𝐴𝑧𝐴)) → (∃𝑦𝐵𝑤𝐵 (𝐶 = (𝑥 + 𝑦) ∧ 𝐶 = (𝑧 + 𝑤)) → 𝑥 = 𝑧))
425, 41syl5bir 244 . . . 4 (((𝐴𝐵) = 0 ∧ (𝑥𝐴𝑧𝐴)) → ((∃𝑦𝐵 𝐶 = (𝑥 + 𝑦) ∧ ∃𝑤𝐵 𝐶 = (𝑧 + 𝑤)) → 𝑥 = 𝑧))
4342ralrimivva 3188 . . 3 ((𝐴𝐵) = 0 → ∀𝑥𝐴𝑧𝐴 ((∃𝑦𝐵 𝐶 = (𝑥 + 𝑦) ∧ ∃𝑤𝐵 𝐶 = (𝑧 + 𝑤)) → 𝑥 = 𝑧))
444, 43anim12i 612 . 2 ((𝐶 ∈ (𝐴 + 𝐵) ∧ (𝐴𝐵) = 0) → (∃𝑥𝐴𝑦𝐵 𝐶 = (𝑥 + 𝑦) ∧ ∀𝑥𝐴𝑧𝐴 ((∃𝑦𝐵 𝐶 = (𝑥 + 𝑦) ∧ ∃𝑤𝐵 𝐶 = (𝑧 + 𝑤)) → 𝑥 = 𝑧)))
45 oveq1 7152 . . . . . 6 (𝑥 = 𝑧 → (𝑥 + 𝑦) = (𝑧 + 𝑦))
4645eqeq2d 2829 . . . . 5 (𝑥 = 𝑧 → (𝐶 = (𝑥 + 𝑦) ↔ 𝐶 = (𝑧 + 𝑦)))
4746rexbidv 3294 . . . 4 (𝑥 = 𝑧 → (∃𝑦𝐵 𝐶 = (𝑥 + 𝑦) ↔ ∃𝑦𝐵 𝐶 = (𝑧 + 𝑦)))
48 oveq2 7153 . . . . . 6 (𝑦 = 𝑤 → (𝑧 + 𝑦) = (𝑧 + 𝑤))
4948eqeq2d 2829 . . . . 5 (𝑦 = 𝑤 → (𝐶 = (𝑧 + 𝑦) ↔ 𝐶 = (𝑧 + 𝑤)))
5049cbvrexvw 3448 . . . 4 (∃𝑦𝐵 𝐶 = (𝑧 + 𝑦) ↔ ∃𝑤𝐵 𝐶 = (𝑧 + 𝑤))
5147, 50syl6bb 288 . . 3 (𝑥 = 𝑧 → (∃𝑦𝐵 𝐶 = (𝑥 + 𝑦) ↔ ∃𝑤𝐵 𝐶 = (𝑧 + 𝑤)))
5251reu4 3719 . 2 (∃!𝑥𝐴𝑦𝐵 𝐶 = (𝑥 + 𝑦) ↔ (∃𝑥𝐴𝑦𝐵 𝐶 = (𝑥 + 𝑦) ∧ ∀𝑥𝐴𝑧𝐴 ((∃𝑦𝐵 𝐶 = (𝑥 + 𝑦) ∧ ∃𝑤𝐵 𝐶 = (𝑧 + 𝑤)) → 𝑥 = 𝑧)))
5344, 52sylibr 235 1 ((𝐶 ∈ (𝐴 + 𝐵) ∧ (𝐴𝐵) = 0) → ∃!𝑥𝐴𝑦𝐵 𝐶 = (𝑥 + 𝑦))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396   = wceq 1528  wcel 2105  wral 3135  wrex 3136  ∃!wreu 3137  cin 3932  (class class class)co 7145  chba 28623   + cva 28624  0c0v 28628   cmv 28629   S csh 28632   + cph 28635  0c0h 28639
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450  ax-resscn 10582  ax-1cn 10583  ax-icn 10584  ax-addcl 10585  ax-addrcl 10586  ax-mulcl 10587  ax-mulrcl 10588  ax-mulcom 10589  ax-addass 10590  ax-mulass 10591  ax-distr 10592  ax-i2m1 10593  ax-1ne0 10594  ax-1rid 10595  ax-rnegex 10596  ax-rrecex 10597  ax-cnre 10598  ax-pre-lttri 10599  ax-pre-lttrn 10600  ax-pre-ltadd 10601  ax-pre-mulgt0 10602  ax-hilex 28703  ax-hfvadd 28704  ax-hvcom 28705  ax-hvass 28706  ax-hv0cl 28707  ax-hvaddid 28708  ax-hfvmul 28709  ax-hvmulid 28710  ax-hvmulass 28711  ax-hvdistr1 28712  ax-hvdistr2 28713  ax-hvmul0 28714
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-nel 3121  df-ral 3140  df-rex 3141  df-reu 3142  df-rmo 3143  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-po 5467  df-so 5468  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-riota 7103  df-ov 7148  df-oprab 7149  df-mpo 7150  df-er 8278  df-en 8498  df-dom 8499  df-sdom 8500  df-pnf 10665  df-mnf 10666  df-xr 10667  df-ltxr 10668  df-le 10669  df-sub 10860  df-neg 10861  df-div 11286  df-grpo 28197  df-ablo 28249  df-hvsub 28675  df-sh 28911  df-ch0 28957  df-shs 29012
This theorem is referenced by:  cdj3lem2  30139
  Copyright terms: Public domain W3C validator