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Theorem cdjreui 32461
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 31345 . . . 4 (𝐶 ∈ (𝐴 + 𝐵) ↔ ∃𝑥𝐴𝑦𝐵 𝐶 = (𝑥 + 𝑦))
43biimpi 216 . . 3 (𝐶 ∈ (𝐴 + 𝐵) → ∃𝑥𝐴𝑦𝐵 𝐶 = (𝑥 + 𝑦))
5 reeanv 3227 . . . . 5 (∃𝑦𝐵𝑤𝐵 (𝐶 = (𝑥 + 𝑦) ∧ 𝐶 = (𝑧 + 𝑤)) ↔ (∃𝑦𝐵 𝐶 = (𝑥 + 𝑦) ∧ ∃𝑤𝐵 𝐶 = (𝑧 + 𝑤)))
6 eqtr2 2759 . . . . . . 7 ((𝐶 = (𝑥 + 𝑦) ∧ 𝐶 = (𝑧 + 𝑤)) → (𝑥 + 𝑦) = (𝑧 + 𝑤))
71sheli 31243 . . . . . . . . . . . 12 (𝑥𝐴𝑥 ∈ ℋ)
82sheli 31243 . . . . . . . . . . . 12 (𝑦𝐵𝑦 ∈ ℋ)
97, 8anim12i 613 . . . . . . . . . . 11 ((𝑥𝐴𝑦𝐵) → (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ))
101sheli 31243 . . . . . . . . . . . 12 (𝑧𝐴𝑧 ∈ ℋ)
112sheli 31243 . . . . . . . . . . . 12 (𝑤𝐵𝑤 ∈ ℋ)
1210, 11anim12i 613 . . . . . . . . . . 11 ((𝑧𝐴𝑤𝐵) → (𝑧 ∈ ℋ ∧ 𝑤 ∈ ℋ))
13 hvaddsub4 31107 . . . . . . . . . . 11 (((𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ) ∧ (𝑧 ∈ ℋ ∧ 𝑤 ∈ ℋ)) → ((𝑥 + 𝑦) = (𝑧 + 𝑤) ↔ (𝑥 𝑧) = (𝑤 𝑦)))
149, 12, 13syl2an 596 . . . . . . . . . 10 (((𝑥𝐴𝑦𝐵) ∧ (𝑧𝐴𝑤𝐵)) → ((𝑥 + 𝑦) = (𝑧 + 𝑤) ↔ (𝑥 𝑧) = (𝑤 𝑦)))
1514an4s 660 . . . . . . . . 9 (((𝑥𝐴𝑧𝐴) ∧ (𝑦𝐵𝑤𝐵)) → ((𝑥 + 𝑦) = (𝑧 + 𝑤) ↔ (𝑥 𝑧) = (𝑤 𝑦)))
1615adantll 714 . . . . . . . 8 ((((𝐴𝐵) = 0 ∧ (𝑥𝐴𝑧𝐴)) ∧ (𝑦𝐵𝑤𝐵)) → ((𝑥 + 𝑦) = (𝑧 + 𝑤) ↔ (𝑥 𝑧) = (𝑤 𝑦)))
17 shsubcl 31249 . . . . . . . . . . . . . . . 16 ((𝐵S𝑤𝐵𝑦𝐵) → (𝑤 𝑦) ∈ 𝐵)
182, 17mp3an1 1447 . . . . . . . . . . . . . . 15 ((𝑤𝐵𝑦𝐵) → (𝑤 𝑦) ∈ 𝐵)
1918ancoms 458 . . . . . . . . . . . . . 14 ((𝑦𝐵𝑤𝐵) → (𝑤 𝑦) ∈ 𝐵)
20 eleq1 2827 . . . . . . . . . . . . . 14 ((𝑥 𝑧) = (𝑤 𝑦) → ((𝑥 𝑧) ∈ 𝐵 ↔ (𝑤 𝑦) ∈ 𝐵))
2119, 20syl5ibrcom 247 . . . . . . . . . . . . 13 ((𝑦𝐵𝑤𝐵) → ((𝑥 𝑧) = (𝑤 𝑦) → (𝑥 𝑧) ∈ 𝐵))
2221adantl 481 . . . . . . . . . . . 12 (((𝑥𝐴𝑧𝐴) ∧ (𝑦𝐵𝑤𝐵)) → ((𝑥 𝑧) = (𝑤 𝑦) → (𝑥 𝑧) ∈ 𝐵))
23 shsubcl 31249 . . . . . . . . . . . . . 14 ((𝐴S𝑥𝐴𝑧𝐴) → (𝑥 𝑧) ∈ 𝐴)
241, 23mp3an1 1447 . . . . . . . . . . . . 13 ((𝑥𝐴𝑧𝐴) → (𝑥 𝑧) ∈ 𝐴)
2524adantr 480 . . . . . . . . . . . 12 (((𝑥𝐴𝑧𝐴) ∧ (𝑦𝐵𝑤𝐵)) → (𝑥 𝑧) ∈ 𝐴)
2622, 25jctild 525 . . . . . . . . . . 11 (((𝑥𝐴𝑧𝐴) ∧ (𝑦𝐵𝑤𝐵)) → ((𝑥 𝑧) = (𝑤 𝑦) → ((𝑥 𝑧) ∈ 𝐴 ∧ (𝑥 𝑧) ∈ 𝐵)))
2726adantll 714 . . . . . . . . . 10 ((((𝐴𝐵) = 0 ∧ (𝑥𝐴𝑧𝐴)) ∧ (𝑦𝐵𝑤𝐵)) → ((𝑥 𝑧) = (𝑤 𝑦) → ((𝑥 𝑧) ∈ 𝐴 ∧ (𝑥 𝑧) ∈ 𝐵)))
28 elin 3979 . . . . . . . . . . . 12 ((𝑥 𝑧) ∈ (𝐴𝐵) ↔ ((𝑥 𝑧) ∈ 𝐴 ∧ (𝑥 𝑧) ∈ 𝐵))
29 eleq2 2828 . . . . . . . . . . . 12 ((𝐴𝐵) = 0 → ((𝑥 𝑧) ∈ (𝐴𝐵) ↔ (𝑥 𝑧) ∈ 0))
3028, 29bitr3id 285 . . . . . . . . . . 11 ((𝐴𝐵) = 0 → (((𝑥 𝑧) ∈ 𝐴 ∧ (𝑥 𝑧) ∈ 𝐵) ↔ (𝑥 𝑧) ∈ 0))
3130ad2antrr 726 . . . . . . . . . 10 ((((𝐴𝐵) = 0 ∧ (𝑥𝐴𝑧𝐴)) ∧ (𝑦𝐵𝑤𝐵)) → (((𝑥 𝑧) ∈ 𝐴 ∧ (𝑥 𝑧) ∈ 𝐵) ↔ (𝑥 𝑧) ∈ 0))
3227, 31sylibd 239 . . . . . . . . 9 ((((𝐴𝐵) = 0 ∧ (𝑥𝐴𝑧𝐴)) ∧ (𝑦𝐵𝑤𝐵)) → ((𝑥 𝑧) = (𝑤 𝑦) → (𝑥 𝑧) ∈ 0))
33 elch0 31283 . . . . . . . . . . . 12 ((𝑥 𝑧) ∈ 0 ↔ (𝑥 𝑧) = 0)
34 hvsubeq0 31097 . . . . . . . . . . . 12 ((𝑥 ∈ ℋ ∧ 𝑧 ∈ ℋ) → ((𝑥 𝑧) = 0𝑥 = 𝑧))
3533, 34bitrid 283 . . . . . . . . . . 11 ((𝑥 ∈ ℋ ∧ 𝑧 ∈ ℋ) → ((𝑥 𝑧) ∈ 0𝑥 = 𝑧))
367, 10, 35syl2an 596 . . . . . . . . . 10 ((𝑥𝐴𝑧𝐴) → ((𝑥 𝑧) ∈ 0𝑥 = 𝑧))
3736ad2antlr 727 . . . . . . . . 9 ((((𝐴𝐵) = 0 ∧ (𝑥𝐴𝑧𝐴)) ∧ (𝑦𝐵𝑤𝐵)) → ((𝑥 𝑧) ∈ 0𝑥 = 𝑧))
3832, 37sylibd 239 . . . . . . . 8 ((((𝐴𝐵) = 0 ∧ (𝑥𝐴𝑧𝐴)) ∧ (𝑦𝐵𝑤𝐵)) → ((𝑥 𝑧) = (𝑤 𝑦) → 𝑥 = 𝑧))
3916, 38sylbid 240 . . . . . . 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 3211 . . . . 5 (((𝐴𝐵) = 0 ∧ (𝑥𝐴𝑧𝐴)) → (∃𝑦𝐵𝑤𝐵 (𝐶 = (𝑥 + 𝑦) ∧ 𝐶 = (𝑧 + 𝑤)) → 𝑥 = 𝑧))
425, 41biimtrrid 243 . . . 4 (((𝐴𝐵) = 0 ∧ (𝑥𝐴𝑧𝐴)) → ((∃𝑦𝐵 𝐶 = (𝑥 + 𝑦) ∧ ∃𝑤𝐵 𝐶 = (𝑧 + 𝑤)) → 𝑥 = 𝑧))
4342ralrimivva 3200 . . 3 ((𝐴𝐵) = 0 → ∀𝑥𝐴𝑧𝐴 ((∃𝑦𝐵 𝐶 = (𝑥 + 𝑦) ∧ ∃𝑤𝐵 𝐶 = (𝑧 + 𝑤)) → 𝑥 = 𝑧))
444, 43anim12i 613 . 2 ((𝐶 ∈ (𝐴 + 𝐵) ∧ (𝐴𝐵) = 0) → (∃𝑥𝐴𝑦𝐵 𝐶 = (𝑥 + 𝑦) ∧ ∀𝑥𝐴𝑧𝐴 ((∃𝑦𝐵 𝐶 = (𝑥 + 𝑦) ∧ ∃𝑤𝐵 𝐶 = (𝑧 + 𝑤)) → 𝑥 = 𝑧)))
45 oveq1 7438 . . . . . 6 (𝑥 = 𝑧 → (𝑥 + 𝑦) = (𝑧 + 𝑦))
4645eqeq2d 2746 . . . . 5 (𝑥 = 𝑧 → (𝐶 = (𝑥 + 𝑦) ↔ 𝐶 = (𝑧 + 𝑦)))
4746rexbidv 3177 . . . 4 (𝑥 = 𝑧 → (∃𝑦𝐵 𝐶 = (𝑥 + 𝑦) ↔ ∃𝑦𝐵 𝐶 = (𝑧 + 𝑦)))
48 oveq2 7439 . . . . . 6 (𝑦 = 𝑤 → (𝑧 + 𝑦) = (𝑧 + 𝑤))
4948eqeq2d 2746 . . . . 5 (𝑦 = 𝑤 → (𝐶 = (𝑧 + 𝑦) ↔ 𝐶 = (𝑧 + 𝑤)))
5049cbvrexvw 3236 . . . 4 (∃𝑦𝐵 𝐶 = (𝑧 + 𝑦) ↔ ∃𝑤𝐵 𝐶 = (𝑧 + 𝑤))
5147, 50bitrdi 287 . . 3 (𝑥 = 𝑧 → (∃𝑦𝐵 𝐶 = (𝑥 + 𝑦) ↔ ∃𝑤𝐵 𝐶 = (𝑧 + 𝑤)))
5251reu4 3740 . 2 (∃!𝑥𝐴𝑦𝐵 𝐶 = (𝑥 + 𝑦) ↔ (∃𝑥𝐴𝑦𝐵 𝐶 = (𝑥 + 𝑦) ∧ ∀𝑥𝐴𝑧𝐴 ((∃𝑦𝐵 𝐶 = (𝑥 + 𝑦) ∧ ∃𝑤𝐵 𝐶 = (𝑧 + 𝑤)) → 𝑥 = 𝑧)))
5344, 52sylibr 234 1 ((𝐶 ∈ (𝐴 + 𝐵) ∧ (𝐴𝐵) = 0) → ∃!𝑥𝐴𝑦𝐵 𝐶 = (𝑥 + 𝑦))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1537  wcel 2106  wral 3059  wrex 3068  ∃!wreu 3376  cin 3962  (class class class)co 7431  chba 30948   + cva 30949  0c0v 30953   cmv 30954   S csh 30957   + cph 30960  0c0h 30964
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-resscn 11210  ax-1cn 11211  ax-icn 11212  ax-addcl 11213  ax-addrcl 11214  ax-mulcl 11215  ax-mulrcl 11216  ax-mulcom 11217  ax-addass 11218  ax-mulass 11219  ax-distr 11220  ax-i2m1 11221  ax-1ne0 11222  ax-1rid 11223  ax-rnegex 11224  ax-rrecex 11225  ax-cnre 11226  ax-pre-lttri 11227  ax-pre-lttrn 11228  ax-pre-ltadd 11229  ax-pre-mulgt0 11230  ax-hilex 31028  ax-hfvadd 31029  ax-hvcom 31030  ax-hvass 31031  ax-hv0cl 31032  ax-hvaddid 31033  ax-hfvmul 31034  ax-hvmulid 31035  ax-hvmulass 31036  ax-hvdistr1 31037  ax-hvdistr2 31038  ax-hvmul0 31039
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-po 5597  df-so 5598  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-er 8744  df-en 8985  df-dom 8986  df-sdom 8987  df-pnf 11295  df-mnf 11296  df-xr 11297  df-ltxr 11298  df-le 11299  df-sub 11492  df-neg 11493  df-div 11919  df-grpo 30522  df-ablo 30574  df-hvsub 31000  df-sh 31236  df-ch0 31282  df-shs 31337
This theorem is referenced by:  cdj3lem2  32464
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