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Theorem pjhthmo 29006
Description: Projection Theorem, uniqueness part. Any two disjoint subspaces yield a unique decomposition of vectors into each subspace. (Contributed by Mario Carneiro, 15-May-2014.) (New usage is discouraged.)
Assertion
Ref Expression
pjhthmo ((𝐴S𝐵S ∧ (𝐴𝐵) = 0) → ∃*𝑥(𝑥𝐴 ∧ ∃𝑦𝐵 𝐶 = (𝑥 + 𝑦)))
Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐵,𝑦   𝑥,𝐶,𝑦

Proof of Theorem pjhthmo
Dummy variables 𝑤 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 an4 652 . . . 4 (((𝑥𝐴𝑧𝐴) ∧ (∃𝑦𝐵 𝐶 = (𝑥 + 𝑦) ∧ ∃𝑤𝐵 𝐶 = (𝑧 + 𝑤))) ↔ ((𝑥𝐴 ∧ ∃𝑦𝐵 𝐶 = (𝑥 + 𝑦)) ∧ (𝑧𝐴 ∧ ∃𝑤𝐵 𝐶 = (𝑧 + 𝑤))))
2 reeanv 3365 . . . . . 6 (∃𝑦𝐵𝑤𝐵 (𝐶 = (𝑥 + 𝑦) ∧ 𝐶 = (𝑧 + 𝑤)) ↔ (∃𝑦𝐵 𝐶 = (𝑥 + 𝑦) ∧ ∃𝑤𝐵 𝐶 = (𝑧 + 𝑤)))
3 simpll1 1204 . . . . . . . . . 10 ((((𝐴S𝐵S ∧ (𝐴𝐵) = 0) ∧ (𝑥𝐴𝑧𝐴)) ∧ ((𝑦𝐵𝑤𝐵) ∧ (𝐶 = (𝑥 + 𝑦) ∧ 𝐶 = (𝑧 + 𝑤)))) → 𝐴S )
4 simpll2 1205 . . . . . . . . . 10 ((((𝐴S𝐵S ∧ (𝐴𝐵) = 0) ∧ (𝑥𝐴𝑧𝐴)) ∧ ((𝑦𝐵𝑤𝐵) ∧ (𝐶 = (𝑥 + 𝑦) ∧ 𝐶 = (𝑧 + 𝑤)))) → 𝐵S )
5 simpll3 1206 . . . . . . . . . 10 ((((𝐴S𝐵S ∧ (𝐴𝐵) = 0) ∧ (𝑥𝐴𝑧𝐴)) ∧ ((𝑦𝐵𝑤𝐵) ∧ (𝐶 = (𝑥 + 𝑦) ∧ 𝐶 = (𝑧 + 𝑤)))) → (𝐴𝐵) = 0)
6 simplrl 773 . . . . . . . . . 10 ((((𝐴S𝐵S ∧ (𝐴𝐵) = 0) ∧ (𝑥𝐴𝑧𝐴)) ∧ ((𝑦𝐵𝑤𝐵) ∧ (𝐶 = (𝑥 + 𝑦) ∧ 𝐶 = (𝑧 + 𝑤)))) → 𝑥𝐴)
7 simprll 775 . . . . . . . . . 10 ((((𝐴S𝐵S ∧ (𝐴𝐵) = 0) ∧ (𝑥𝐴𝑧𝐴)) ∧ ((𝑦𝐵𝑤𝐵) ∧ (𝐶 = (𝑥 + 𝑦) ∧ 𝐶 = (𝑧 + 𝑤)))) → 𝑦𝐵)
8 simplrr 774 . . . . . . . . . 10 ((((𝐴S𝐵S ∧ (𝐴𝐵) = 0) ∧ (𝑥𝐴𝑧𝐴)) ∧ ((𝑦𝐵𝑤𝐵) ∧ (𝐶 = (𝑥 + 𝑦) ∧ 𝐶 = (𝑧 + 𝑤)))) → 𝑧𝐴)
9 simprlr 776 . . . . . . . . . 10 ((((𝐴S𝐵S ∧ (𝐴𝐵) = 0) ∧ (𝑥𝐴𝑧𝐴)) ∧ ((𝑦𝐵𝑤𝐵) ∧ (𝐶 = (𝑥 + 𝑦) ∧ 𝐶 = (𝑧 + 𝑤)))) → 𝑤𝐵)
10 simprrl 777 . . . . . . . . . . 11 ((((𝐴S𝐵S ∧ (𝐴𝐵) = 0) ∧ (𝑥𝐴𝑧𝐴)) ∧ ((𝑦𝐵𝑤𝐵) ∧ (𝐶 = (𝑥 + 𝑦) ∧ 𝐶 = (𝑧 + 𝑤)))) → 𝐶 = (𝑥 + 𝑦))
11 simprrr 778 . . . . . . . . . . 11 ((((𝐴S𝐵S ∧ (𝐴𝐵) = 0) ∧ (𝑥𝐴𝑧𝐴)) ∧ ((𝑦𝐵𝑤𝐵) ∧ (𝐶 = (𝑥 + 𝑦) ∧ 𝐶 = (𝑧 + 𝑤)))) → 𝐶 = (𝑧 + 𝑤))
1210, 11eqtr3d 2855 . . . . . . . . . 10 ((((𝐴S𝐵S ∧ (𝐴𝐵) = 0) ∧ (𝑥𝐴𝑧𝐴)) ∧ ((𝑦𝐵𝑤𝐵) ∧ (𝐶 = (𝑥 + 𝑦) ∧ 𝐶 = (𝑧 + 𝑤)))) → (𝑥 + 𝑦) = (𝑧 + 𝑤))
133, 4, 5, 6, 7, 8, 9, 12shuni 29004 . . . . . . . . 9 ((((𝐴S𝐵S ∧ (𝐴𝐵) = 0) ∧ (𝑥𝐴𝑧𝐴)) ∧ ((𝑦𝐵𝑤𝐵) ∧ (𝐶 = (𝑥 + 𝑦) ∧ 𝐶 = (𝑧 + 𝑤)))) → (𝑥 = 𝑧𝑦 = 𝑤))
1413simpld 495 . . . . . . . 8 ((((𝐴S𝐵S ∧ (𝐴𝐵) = 0) ∧ (𝑥𝐴𝑧𝐴)) ∧ ((𝑦𝐵𝑤𝐵) ∧ (𝐶 = (𝑥 + 𝑦) ∧ 𝐶 = (𝑧 + 𝑤)))) → 𝑥 = 𝑧)
1514exp32 421 . . . . . . 7 (((𝐴S𝐵S ∧ (𝐴𝐵) = 0) ∧ (𝑥𝐴𝑧𝐴)) → ((𝑦𝐵𝑤𝐵) → ((𝐶 = (𝑥 + 𝑦) ∧ 𝐶 = (𝑧 + 𝑤)) → 𝑥 = 𝑧)))
1615rexlimdvv 3290 . . . . . 6 (((𝐴S𝐵S ∧ (𝐴𝐵) = 0) ∧ (𝑥𝐴𝑧𝐴)) → (∃𝑦𝐵𝑤𝐵 (𝐶 = (𝑥 + 𝑦) ∧ 𝐶 = (𝑧 + 𝑤)) → 𝑥 = 𝑧))
172, 16syl5bir 244 . . . . 5 (((𝐴S𝐵S ∧ (𝐴𝐵) = 0) ∧ (𝑥𝐴𝑧𝐴)) → ((∃𝑦𝐵 𝐶 = (𝑥 + 𝑦) ∧ ∃𝑤𝐵 𝐶 = (𝑧 + 𝑤)) → 𝑥 = 𝑧))
1817expimpd 454 . . . 4 ((𝐴S𝐵S ∧ (𝐴𝐵) = 0) → (((𝑥𝐴𝑧𝐴) ∧ (∃𝑦𝐵 𝐶 = (𝑥 + 𝑦) ∧ ∃𝑤𝐵 𝐶 = (𝑧 + 𝑤))) → 𝑥 = 𝑧))
191, 18syl5bir 244 . . 3 ((𝐴S𝐵S ∧ (𝐴𝐵) = 0) → (((𝑥𝐴 ∧ ∃𝑦𝐵 𝐶 = (𝑥 + 𝑦)) ∧ (𝑧𝐴 ∧ ∃𝑤𝐵 𝐶 = (𝑧 + 𝑤))) → 𝑥 = 𝑧))
2019alrimivv 1920 . 2 ((𝐴S𝐵S ∧ (𝐴𝐵) = 0) → ∀𝑥𝑧(((𝑥𝐴 ∧ ∃𝑦𝐵 𝐶 = (𝑥 + 𝑦)) ∧ (𝑧𝐴 ∧ ∃𝑤𝐵 𝐶 = (𝑧 + 𝑤))) → 𝑥 = 𝑧))
21 eleq1w 2892 . . . 4 (𝑥 = 𝑧 → (𝑥𝐴𝑧𝐴))
22 oveq1 7152 . . . . . . 7 (𝑥 = 𝑧 → (𝑥 + 𝑦) = (𝑧 + 𝑦))
2322eqeq2d 2829 . . . . . 6 (𝑥 = 𝑧 → (𝐶 = (𝑥 + 𝑦) ↔ 𝐶 = (𝑧 + 𝑦)))
2423rexbidv 3294 . . . . 5 (𝑥 = 𝑧 → (∃𝑦𝐵 𝐶 = (𝑥 + 𝑦) ↔ ∃𝑦𝐵 𝐶 = (𝑧 + 𝑦)))
25 oveq2 7153 . . . . . . 7 (𝑦 = 𝑤 → (𝑧 + 𝑦) = (𝑧 + 𝑤))
2625eqeq2d 2829 . . . . . 6 (𝑦 = 𝑤 → (𝐶 = (𝑧 + 𝑦) ↔ 𝐶 = (𝑧 + 𝑤)))
2726cbvrexvw 3448 . . . . 5 (∃𝑦𝐵 𝐶 = (𝑧 + 𝑦) ↔ ∃𝑤𝐵 𝐶 = (𝑧 + 𝑤))
2824, 27syl6bb 288 . . . 4 (𝑥 = 𝑧 → (∃𝑦𝐵 𝐶 = (𝑥 + 𝑦) ↔ ∃𝑤𝐵 𝐶 = (𝑧 + 𝑤)))
2921, 28anbi12d 630 . . 3 (𝑥 = 𝑧 → ((𝑥𝐴 ∧ ∃𝑦𝐵 𝐶 = (𝑥 + 𝑦)) ↔ (𝑧𝐴 ∧ ∃𝑤𝐵 𝐶 = (𝑧 + 𝑤))))
3029mo4 2643 . 2 (∃*𝑥(𝑥𝐴 ∧ ∃𝑦𝐵 𝐶 = (𝑥 + 𝑦)) ↔ ∀𝑥𝑧(((𝑥𝐴 ∧ ∃𝑦𝐵 𝐶 = (𝑥 + 𝑦)) ∧ (𝑧𝐴 ∧ ∃𝑤𝐵 𝐶 = (𝑧 + 𝑤))) → 𝑥 = 𝑧))
3120, 30sylibr 235 1 ((𝐴S𝐵S ∧ (𝐴𝐵) = 0) → ∃*𝑥(𝑥𝐴 ∧ ∃𝑦𝐵 𝐶 = (𝑥 + 𝑦)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  w3a 1079  wal 1526   = wceq 1528  wcel 2105  ∃*wmo 2613  wrex 3136  cin 3932  (class class class)co 7145   + cva 28624   S csh 28632  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-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-hvsub 28675  df-sh 28911  df-ch0 28957
This theorem is referenced by:  pjhtheu  29098  pjpreeq  29102
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