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Theorem cdjreui 29161
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 28045 . . . 4 (𝐶 ∈ (𝐴 + 𝐵) ↔ ∃𝑥𝐴𝑦𝐵 𝐶 = (𝑥 + 𝑦))
43biimpi 206 . . 3 (𝐶 ∈ (𝐴 + 𝐵) → ∃𝑥𝐴𝑦𝐵 𝐶 = (𝑥 + 𝑦))
5 reeanv 3100 . . . . 5 (∃𝑦𝐵𝑤𝐵 (𝐶 = (𝑥 + 𝑦) ∧ 𝐶 = (𝑧 + 𝑤)) ↔ (∃𝑦𝐵 𝐶 = (𝑥 + 𝑦) ∧ ∃𝑤𝐵 𝐶 = (𝑧 + 𝑤)))
6 eqtr2 2641 . . . . . . 7 ((𝐶 = (𝑥 + 𝑦) ∧ 𝐶 = (𝑧 + 𝑤)) → (𝑥 + 𝑦) = (𝑧 + 𝑤))
71sheli 27941 . . . . . . . . . . . 12 (𝑥𝐴𝑥 ∈ ℋ)
82sheli 27941 . . . . . . . . . . . 12 (𝑦𝐵𝑦 ∈ ℋ)
97, 8anim12i 589 . . . . . . . . . . 11 ((𝑥𝐴𝑦𝐵) → (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ))
101sheli 27941 . . . . . . . . . . . 12 (𝑧𝐴𝑧 ∈ ℋ)
112sheli 27941 . . . . . . . . . . . 12 (𝑤𝐵𝑤 ∈ ℋ)
1210, 11anim12i 589 . . . . . . . . . . 11 ((𝑧𝐴𝑤𝐵) → (𝑧 ∈ ℋ ∧ 𝑤 ∈ ℋ))
13 hvaddsub4 27805 . . . . . . . . . . 11 (((𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ) ∧ (𝑧 ∈ ℋ ∧ 𝑤 ∈ ℋ)) → ((𝑥 + 𝑦) = (𝑧 + 𝑤) ↔ (𝑥 𝑧) = (𝑤 𝑦)))
149, 12, 13syl2an 494 . . . . . . . . . 10 (((𝑥𝐴𝑦𝐵) ∧ (𝑧𝐴𝑤𝐵)) → ((𝑥 + 𝑦) = (𝑧 + 𝑤) ↔ (𝑥 𝑧) = (𝑤 𝑦)))
1514an4s 868 . . . . . . . . 9 (((𝑥𝐴𝑧𝐴) ∧ (𝑦𝐵𝑤𝐵)) → ((𝑥 + 𝑦) = (𝑧 + 𝑤) ↔ (𝑥 𝑧) = (𝑤 𝑦)))
1615adantll 749 . . . . . . . 8 ((((𝐴𝐵) = 0 ∧ (𝑥𝐴𝑧𝐴)) ∧ (𝑦𝐵𝑤𝐵)) → ((𝑥 + 𝑦) = (𝑧 + 𝑤) ↔ (𝑥 𝑧) = (𝑤 𝑦)))
17 shsubcl 27947 . . . . . . . . . . . . . . . 16 ((𝐵S𝑤𝐵𝑦𝐵) → (𝑤 𝑦) ∈ 𝐵)
182, 17mp3an1 1408 . . . . . . . . . . . . . . 15 ((𝑤𝐵𝑦𝐵) → (𝑤 𝑦) ∈ 𝐵)
1918ancoms 469 . . . . . . . . . . . . . 14 ((𝑦𝐵𝑤𝐵) → (𝑤 𝑦) ∈ 𝐵)
20 eleq1 2686 . . . . . . . . . . . . . 14 ((𝑥 𝑧) = (𝑤 𝑦) → ((𝑥 𝑧) ∈ 𝐵 ↔ (𝑤 𝑦) ∈ 𝐵))
2119, 20syl5ibrcom 237 . . . . . . . . . . . . 13 ((𝑦𝐵𝑤𝐵) → ((𝑥 𝑧) = (𝑤 𝑦) → (𝑥 𝑧) ∈ 𝐵))
2221adantl 482 . . . . . . . . . . . 12 (((𝑥𝐴𝑧𝐴) ∧ (𝑦𝐵𝑤𝐵)) → ((𝑥 𝑧) = (𝑤 𝑦) → (𝑥 𝑧) ∈ 𝐵))
23 shsubcl 27947 . . . . . . . . . . . . . 14 ((𝐴S𝑥𝐴𝑧𝐴) → (𝑥 𝑧) ∈ 𝐴)
241, 23mp3an1 1408 . . . . . . . . . . . . 13 ((𝑥𝐴𝑧𝐴) → (𝑥 𝑧) ∈ 𝐴)
2524adantr 481 . . . . . . . . . . . 12 (((𝑥𝐴𝑧𝐴) ∧ (𝑦𝐵𝑤𝐵)) → (𝑥 𝑧) ∈ 𝐴)
2622, 25jctild 565 . . . . . . . . . . 11 (((𝑥𝐴𝑧𝐴) ∧ (𝑦𝐵𝑤𝐵)) → ((𝑥 𝑧) = (𝑤 𝑦) → ((𝑥 𝑧) ∈ 𝐴 ∧ (𝑥 𝑧) ∈ 𝐵)))
2726adantll 749 . . . . . . . . . 10 ((((𝐴𝐵) = 0 ∧ (𝑥𝐴𝑧𝐴)) ∧ (𝑦𝐵𝑤𝐵)) → ((𝑥 𝑧) = (𝑤 𝑦) → ((𝑥 𝑧) ∈ 𝐴 ∧ (𝑥 𝑧) ∈ 𝐵)))
28 elin 3779 . . . . . . . . . . . 12 ((𝑥 𝑧) ∈ (𝐴𝐵) ↔ ((𝑥 𝑧) ∈ 𝐴 ∧ (𝑥 𝑧) ∈ 𝐵))
29 eleq2 2687 . . . . . . . . . . . 12 ((𝐴𝐵) = 0 → ((𝑥 𝑧) ∈ (𝐴𝐵) ↔ (𝑥 𝑧) ∈ 0))
3028, 29syl5bbr 274 . . . . . . . . . . 11 ((𝐴𝐵) = 0 → (((𝑥 𝑧) ∈ 𝐴 ∧ (𝑥 𝑧) ∈ 𝐵) ↔ (𝑥 𝑧) ∈ 0))
3130ad2antrr 761 . . . . . . . . . 10 ((((𝐴𝐵) = 0 ∧ (𝑥𝐴𝑧𝐴)) ∧ (𝑦𝐵𝑤𝐵)) → (((𝑥 𝑧) ∈ 𝐴 ∧ (𝑥 𝑧) ∈ 𝐵) ↔ (𝑥 𝑧) ∈ 0))
3227, 31sylibd 229 . . . . . . . . 9 ((((𝐴𝐵) = 0 ∧ (𝑥𝐴𝑧𝐴)) ∧ (𝑦𝐵𝑤𝐵)) → ((𝑥 𝑧) = (𝑤 𝑦) → (𝑥 𝑧) ∈ 0))
33 elch0 27981 . . . . . . . . . . . 12 ((𝑥 𝑧) ∈ 0 ↔ (𝑥 𝑧) = 0)
34 hvsubeq0 27795 . . . . . . . . . . . 12 ((𝑥 ∈ ℋ ∧ 𝑧 ∈ ℋ) → ((𝑥 𝑧) = 0𝑥 = 𝑧))
3533, 34syl5bb 272 . . . . . . . . . . 11 ((𝑥 ∈ ℋ ∧ 𝑧 ∈ ℋ) → ((𝑥 𝑧) ∈ 0𝑥 = 𝑧))
367, 10, 35syl2an 494 . . . . . . . . . 10 ((𝑥𝐴𝑧𝐴) → ((𝑥 𝑧) ∈ 0𝑥 = 𝑧))
3736ad2antlr 762 . . . . . . . . 9 ((((𝐴𝐵) = 0 ∧ (𝑥𝐴𝑧𝐴)) ∧ (𝑦𝐵𝑤𝐵)) → ((𝑥 𝑧) ∈ 0𝑥 = 𝑧))
3832, 37sylibd 229 . . . . . . . 8 ((((𝐴𝐵) = 0 ∧ (𝑥𝐴𝑧𝐴)) ∧ (𝑦𝐵𝑤𝐵)) → ((𝑥 𝑧) = (𝑤 𝑦) → 𝑥 = 𝑧))
3916, 38sylbid 230 . . . . . . 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 3032 . . . . 5 (((𝐴𝐵) = 0 ∧ (𝑥𝐴𝑧𝐴)) → (∃𝑦𝐵𝑤𝐵 (𝐶 = (𝑥 + 𝑦) ∧ 𝐶 = (𝑧 + 𝑤)) → 𝑥 = 𝑧))
425, 41syl5bir 233 . . . 4 (((𝐴𝐵) = 0 ∧ (𝑥𝐴𝑧𝐴)) → ((∃𝑦𝐵 𝐶 = (𝑥 + 𝑦) ∧ ∃𝑤𝐵 𝐶 = (𝑧 + 𝑤)) → 𝑥 = 𝑧))
4342ralrimivva 2966 . . 3 ((𝐴𝐵) = 0 → ∀𝑥𝐴𝑧𝐴 ((∃𝑦𝐵 𝐶 = (𝑥 + 𝑦) ∧ ∃𝑤𝐵 𝐶 = (𝑧 + 𝑤)) → 𝑥 = 𝑧))
444, 43anim12i 589 . 2 ((𝐶 ∈ (𝐴 + 𝐵) ∧ (𝐴𝐵) = 0) → (∃𝑥𝐴𝑦𝐵 𝐶 = (𝑥 + 𝑦) ∧ ∀𝑥𝐴𝑧𝐴 ((∃𝑦𝐵 𝐶 = (𝑥 + 𝑦) ∧ ∃𝑤𝐵 𝐶 = (𝑧 + 𝑤)) → 𝑥 = 𝑧)))
45 oveq1 6617 . . . . . 6 (𝑥 = 𝑧 → (𝑥 + 𝑦) = (𝑧 + 𝑦))
4645eqeq2d 2631 . . . . 5 (𝑥 = 𝑧 → (𝐶 = (𝑥 + 𝑦) ↔ 𝐶 = (𝑧 + 𝑦)))
4746rexbidv 3046 . . . 4 (𝑥 = 𝑧 → (∃𝑦𝐵 𝐶 = (𝑥 + 𝑦) ↔ ∃𝑦𝐵 𝐶 = (𝑧 + 𝑦)))
48 oveq2 6618 . . . . . 6 (𝑦 = 𝑤 → (𝑧 + 𝑦) = (𝑧 + 𝑤))
4948eqeq2d 2631 . . . . 5 (𝑦 = 𝑤 → (𝐶 = (𝑧 + 𝑦) ↔ 𝐶 = (𝑧 + 𝑤)))
5049cbvrexv 3163 . . . 4 (∃𝑦𝐵 𝐶 = (𝑧 + 𝑦) ↔ ∃𝑤𝐵 𝐶 = (𝑧 + 𝑤))
5147, 50syl6bb 276 . . 3 (𝑥 = 𝑧 → (∃𝑦𝐵 𝐶 = (𝑥 + 𝑦) ↔ ∃𝑤𝐵 𝐶 = (𝑧 + 𝑤)))
5251reu4 3386 . 2 (∃!𝑥𝐴𝑦𝐵 𝐶 = (𝑥 + 𝑦) ↔ (∃𝑥𝐴𝑦𝐵 𝐶 = (𝑥 + 𝑦) ∧ ∀𝑥𝐴𝑧𝐴 ((∃𝑦𝐵 𝐶 = (𝑥 + 𝑦) ∧ ∃𝑤𝐵 𝐶 = (𝑧 + 𝑤)) → 𝑥 = 𝑧)))
5344, 52sylibr 224 1 ((𝐶 ∈ (𝐴 + 𝐵) ∧ (𝐴𝐵) = 0) → ∃!𝑥𝐴𝑦𝐵 𝐶 = (𝑥 + 𝑦))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384   = wceq 1480  wcel 1987  wral 2907  wrex 2908  ∃!wreu 2909  cin 3558  (class class class)co 6610  chil 27646   + cva 27647  0c0v 27651   cmv 27652   S csh 27655   + cph 27658  0c0h 27662
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6909  ax-resscn 9945  ax-1cn 9946  ax-icn 9947  ax-addcl 9948  ax-addrcl 9949  ax-mulcl 9950  ax-mulrcl 9951  ax-mulcom 9952  ax-addass 9953  ax-mulass 9954  ax-distr 9955  ax-i2m1 9956  ax-1ne0 9957  ax-1rid 9958  ax-rnegex 9959  ax-rrecex 9960  ax-cnre 9961  ax-pre-lttri 9962  ax-pre-lttrn 9963  ax-pre-ltadd 9964  ax-pre-mulgt0 9965  ax-hilex 27726  ax-hfvadd 27727  ax-hvcom 27728  ax-hvass 27729  ax-hv0cl 27730  ax-hvaddid 27731  ax-hfvmul 27732  ax-hvmulid 27733  ax-hvmulass 27734  ax-hvdistr1 27735  ax-hvdistr2 27736  ax-hvmul0 27737
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3191  df-sbc 3422  df-csb 3519  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-id 4994  df-po 5000  df-so 5001  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-iota 5815  df-fun 5854  df-fn 5855  df-f 5856  df-f1 5857  df-fo 5858  df-f1o 5859  df-fv 5860  df-riota 6571  df-ov 6613  df-oprab 6614  df-mpt2 6615  df-er 7694  df-en 7908  df-dom 7909  df-sdom 7910  df-pnf 10028  df-mnf 10029  df-xr 10030  df-ltxr 10031  df-le 10032  df-sub 10220  df-neg 10221  df-div 10637  df-grpo 27217  df-ablo 27269  df-hvsub 27698  df-sh 27934  df-ch0 27980  df-shs 28037
This theorem is referenced by:  cdj3lem2  29164
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