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Theorem cdjreui 28675
 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 27559 . . . 4 (𝐶 ∈ (𝐴 + 𝐵) ↔ ∃𝑥𝐴𝑦𝐵 𝐶 = (𝑥 + 𝑦))
43biimpi 205 . . 3 (𝐶 ∈ (𝐴 + 𝐵) → ∃𝑥𝐴𝑦𝐵 𝐶 = (𝑥 + 𝑦))
5 reeanv 3086 . . . . 5 (∃𝑦𝐵𝑤𝐵 (𝐶 = (𝑥 + 𝑦) ∧ 𝐶 = (𝑧 + 𝑤)) ↔ (∃𝑦𝐵 𝐶 = (𝑥 + 𝑦) ∧ ∃𝑤𝐵 𝐶 = (𝑧 + 𝑤)))
6 eqtr2 2630 . . . . . . 7 ((𝐶 = (𝑥 + 𝑦) ∧ 𝐶 = (𝑧 + 𝑤)) → (𝑥 + 𝑦) = (𝑧 + 𝑤))
71sheli 27455 . . . . . . . . . . . 12 (𝑥𝐴𝑥 ∈ ℋ)
82sheli 27455 . . . . . . . . . . . 12 (𝑦𝐵𝑦 ∈ ℋ)
97, 8anim12i 588 . . . . . . . . . . 11 ((𝑥𝐴𝑦𝐵) → (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ))
101sheli 27455 . . . . . . . . . . . 12 (𝑧𝐴𝑧 ∈ ℋ)
112sheli 27455 . . . . . . . . . . . 12 (𝑤𝐵𝑤 ∈ ℋ)
1210, 11anim12i 588 . . . . . . . . . . 11 ((𝑧𝐴𝑤𝐵) → (𝑧 ∈ ℋ ∧ 𝑤 ∈ ℋ))
13 hvaddsub4 27319 . . . . . . . . . . 11 (((𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ) ∧ (𝑧 ∈ ℋ ∧ 𝑤 ∈ ℋ)) → ((𝑥 + 𝑦) = (𝑧 + 𝑤) ↔ (𝑥 𝑧) = (𝑤 𝑦)))
149, 12, 13syl2an 493 . . . . . . . . . 10 (((𝑥𝐴𝑦𝐵) ∧ (𝑧𝐴𝑤𝐵)) → ((𝑥 + 𝑦) = (𝑧 + 𝑤) ↔ (𝑥 𝑧) = (𝑤 𝑦)))
1514an4s 865 . . . . . . . . 9 (((𝑥𝐴𝑧𝐴) ∧ (𝑦𝐵𝑤𝐵)) → ((𝑥 + 𝑦) = (𝑧 + 𝑤) ↔ (𝑥 𝑧) = (𝑤 𝑦)))
1615adantll 746 . . . . . . . 8 ((((𝐴𝐵) = 0 ∧ (𝑥𝐴𝑧𝐴)) ∧ (𝑦𝐵𝑤𝐵)) → ((𝑥 + 𝑦) = (𝑧 + 𝑤) ↔ (𝑥 𝑧) = (𝑤 𝑦)))
17 shsubcl 27461 . . . . . . . . . . . . . . . 16 ((𝐵S𝑤𝐵𝑦𝐵) → (𝑤 𝑦) ∈ 𝐵)
182, 17mp3an1 1403 . . . . . . . . . . . . . . 15 ((𝑤𝐵𝑦𝐵) → (𝑤 𝑦) ∈ 𝐵)
1918ancoms 468 . . . . . . . . . . . . . 14 ((𝑦𝐵𝑤𝐵) → (𝑤 𝑦) ∈ 𝐵)
20 eleq1 2676 . . . . . . . . . . . . . 14 ((𝑥 𝑧) = (𝑤 𝑦) → ((𝑥 𝑧) ∈ 𝐵 ↔ (𝑤 𝑦) ∈ 𝐵))
2119, 20syl5ibrcom 236 . . . . . . . . . . . . 13 ((𝑦𝐵𝑤𝐵) → ((𝑥 𝑧) = (𝑤 𝑦) → (𝑥 𝑧) ∈ 𝐵))
2221adantl 481 . . . . . . . . . . . 12 (((𝑥𝐴𝑧𝐴) ∧ (𝑦𝐵𝑤𝐵)) → ((𝑥 𝑧) = (𝑤 𝑦) → (𝑥 𝑧) ∈ 𝐵))
23 shsubcl 27461 . . . . . . . . . . . . . 14 ((𝐴S𝑥𝐴𝑧𝐴) → (𝑥 𝑧) ∈ 𝐴)
241, 23mp3an1 1403 . . . . . . . . . . . . 13 ((𝑥𝐴𝑧𝐴) → (𝑥 𝑧) ∈ 𝐴)
2524adantr 480 . . . . . . . . . . . 12 (((𝑥𝐴𝑧𝐴) ∧ (𝑦𝐵𝑤𝐵)) → (𝑥 𝑧) ∈ 𝐴)
2622, 25jctild 564 . . . . . . . . . . 11 (((𝑥𝐴𝑧𝐴) ∧ (𝑦𝐵𝑤𝐵)) → ((𝑥 𝑧) = (𝑤 𝑦) → ((𝑥 𝑧) ∈ 𝐴 ∧ (𝑥 𝑧) ∈ 𝐵)))
2726adantll 746 . . . . . . . . . 10 ((((𝐴𝐵) = 0 ∧ (𝑥𝐴𝑧𝐴)) ∧ (𝑦𝐵𝑤𝐵)) → ((𝑥 𝑧) = (𝑤 𝑦) → ((𝑥 𝑧) ∈ 𝐴 ∧ (𝑥 𝑧) ∈ 𝐵)))
28 elin 3758 . . . . . . . . . . . 12 ((𝑥 𝑧) ∈ (𝐴𝐵) ↔ ((𝑥 𝑧) ∈ 𝐴 ∧ (𝑥 𝑧) ∈ 𝐵))
29 eleq2 2677 . . . . . . . . . . . 12 ((𝐴𝐵) = 0 → ((𝑥 𝑧) ∈ (𝐴𝐵) ↔ (𝑥 𝑧) ∈ 0))
3028, 29syl5bbr 273 . . . . . . . . . . 11 ((𝐴𝐵) = 0 → (((𝑥 𝑧) ∈ 𝐴 ∧ (𝑥 𝑧) ∈ 𝐵) ↔ (𝑥 𝑧) ∈ 0))
3130ad2antrr 758 . . . . . . . . . 10 ((((𝐴𝐵) = 0 ∧ (𝑥𝐴𝑧𝐴)) ∧ (𝑦𝐵𝑤𝐵)) → (((𝑥 𝑧) ∈ 𝐴 ∧ (𝑥 𝑧) ∈ 𝐵) ↔ (𝑥 𝑧) ∈ 0))
3227, 31sylibd 228 . . . . . . . . 9 ((((𝐴𝐵) = 0 ∧ (𝑥𝐴𝑧𝐴)) ∧ (𝑦𝐵𝑤𝐵)) → ((𝑥 𝑧) = (𝑤 𝑦) → (𝑥 𝑧) ∈ 0))
33 elch0 27495 . . . . . . . . . . . 12 ((𝑥 𝑧) ∈ 0 ↔ (𝑥 𝑧) = 0)
34 hvsubeq0 27309 . . . . . . . . . . . 12 ((𝑥 ∈ ℋ ∧ 𝑧 ∈ ℋ) → ((𝑥 𝑧) = 0𝑥 = 𝑧))
3533, 34syl5bb 271 . . . . . . . . . . 11 ((𝑥 ∈ ℋ ∧ 𝑧 ∈ ℋ) → ((𝑥 𝑧) ∈ 0𝑥 = 𝑧))
367, 10, 35syl2an 493 . . . . . . . . . 10 ((𝑥𝐴𝑧𝐴) → ((𝑥 𝑧) ∈ 0𝑥 = 𝑧))
3736ad2antlr 759 . . . . . . . . 9 ((((𝐴𝐵) = 0 ∧ (𝑥𝐴𝑧𝐴)) ∧ (𝑦𝐵𝑤𝐵)) → ((𝑥 𝑧) ∈ 0𝑥 = 𝑧))
3832, 37sylibd 228 . . . . . . . 8 ((((𝐴𝐵) = 0 ∧ (𝑥𝐴𝑧𝐴)) ∧ (𝑦𝐵𝑤𝐵)) → ((𝑥 𝑧) = (𝑤 𝑦) → 𝑥 = 𝑧))
3916, 38sylbid 229 . . . . . . 7 ((((𝐴𝐵) = 0 ∧ (𝑥𝐴𝑧𝐴)) ∧ (𝑦𝐵𝑤𝐵)) → ((𝑥 + 𝑦) = (𝑧 + 𝑤) → 𝑥 = 𝑧))
406, 39syl5 33 . . . . . 6 ((((𝐴𝐵) = 0 ∧ (𝑥𝐴𝑧𝐴)) ∧ (𝑦𝐵𝑤𝐵)) → ((𝐶 = (𝑥 + 𝑦) ∧ 𝐶 = (𝑧 + 𝑤)) → 𝑥 = 𝑧))
4140rexlimdvva 3020 . . . . 5 (((𝐴𝐵) = 0 ∧ (𝑥𝐴𝑧𝐴)) → (∃𝑦𝐵𝑤𝐵 (𝐶 = (𝑥 + 𝑦) ∧ 𝐶 = (𝑧 + 𝑤)) → 𝑥 = 𝑧))
425, 41syl5bir 232 . . . 4 (((𝐴𝐵) = 0 ∧ (𝑥𝐴𝑧𝐴)) → ((∃𝑦𝐵 𝐶 = (𝑥 + 𝑦) ∧ ∃𝑤𝐵 𝐶 = (𝑧 + 𝑤)) → 𝑥 = 𝑧))
4342ralrimivva 2954 . . 3 ((𝐴𝐵) = 0 → ∀𝑥𝐴𝑧𝐴 ((∃𝑦𝐵 𝐶 = (𝑥 + 𝑦) ∧ ∃𝑤𝐵 𝐶 = (𝑧 + 𝑤)) → 𝑥 = 𝑧))
444, 43anim12i 588 . 2 ((𝐶 ∈ (𝐴 + 𝐵) ∧ (𝐴𝐵) = 0) → (∃𝑥𝐴𝑦𝐵 𝐶 = (𝑥 + 𝑦) ∧ ∀𝑥𝐴𝑧𝐴 ((∃𝑦𝐵 𝐶 = (𝑥 + 𝑦) ∧ ∃𝑤𝐵 𝐶 = (𝑧 + 𝑤)) → 𝑥 = 𝑧)))
45 oveq1 6556 . . . . . 6 (𝑥 = 𝑧 → (𝑥 + 𝑦) = (𝑧 + 𝑦))
4645eqeq2d 2620 . . . . 5 (𝑥 = 𝑧 → (𝐶 = (𝑥 + 𝑦) ↔ 𝐶 = (𝑧 + 𝑦)))
4746rexbidv 3034 . . . 4 (𝑥 = 𝑧 → (∃𝑦𝐵 𝐶 = (𝑥 + 𝑦) ↔ ∃𝑦𝐵 𝐶 = (𝑧 + 𝑦)))
48 oveq2 6557 . . . . . 6 (𝑦 = 𝑤 → (𝑧 + 𝑦) = (𝑧 + 𝑤))
4948eqeq2d 2620 . . . . 5 (𝑦 = 𝑤 → (𝐶 = (𝑧 + 𝑦) ↔ 𝐶 = (𝑧 + 𝑤)))
5049cbvrexv 3148 . . . 4 (∃𝑦𝐵 𝐶 = (𝑧 + 𝑦) ↔ ∃𝑤𝐵 𝐶 = (𝑧 + 𝑤))
5147, 50syl6bb 275 . . 3 (𝑥 = 𝑧 → (∃𝑦𝐵 𝐶 = (𝑥 + 𝑦) ↔ ∃𝑤𝐵 𝐶 = (𝑧 + 𝑤)))
5251reu4 3367 . 2 (∃!𝑥𝐴𝑦𝐵 𝐶 = (𝑥 + 𝑦) ↔ (∃𝑥𝐴𝑦𝐵 𝐶 = (𝑥 + 𝑦) ∧ ∀𝑥𝐴𝑧𝐴 ((∃𝑦𝐵 𝐶 = (𝑥 + 𝑦) ∧ ∃𝑤𝐵 𝐶 = (𝑧 + 𝑤)) → 𝑥 = 𝑧)))
5344, 52sylibr 223 1 ((𝐶 ∈ (𝐴 + 𝐵) ∧ (𝐴𝐵) = 0) → ∃!𝑥𝐴𝑦𝐵 𝐶 = (𝑥 + 𝑦))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∀wral 2896  ∃wrex 2897  ∃!wreu 2898   ∩ cin 3539  (class class class)co 6549   ℋchil 27160   +ℎ cva 27161  0ℎc0v 27165   −ℎ cmv 27166   Sℋ csh 27169   +ℋ cph 27172  0ℋc0h 27176 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892  ax-hilex 27240  ax-hfvadd 27241  ax-hvcom 27242  ax-hvass 27243  ax-hv0cl 27244  ax-hvaddid 27245  ax-hfvmul 27246  ax-hvmulid 27247  ax-hvmulass 27248  ax-hvdistr1 27249  ax-hvdistr2 27250  ax-hvmul0 27251 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-po 4959  df-so 4960  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-er 7629  df-en 7842  df-dom 7843  df-sdom 7844  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-div 10564  df-grpo 26731  df-ablo 26783  df-hvsub 27212  df-sh 27448  df-ch0 27494  df-shs 27551 This theorem is referenced by:  cdj3lem2  28678
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