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Theorem pjhthmo 29083
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 655 . . . 4 (((𝑥𝐴𝑧𝐴) ∧ (∃𝑦𝐵 𝐶 = (𝑥 + 𝑦) ∧ ∃𝑤𝐵 𝐶 = (𝑧 + 𝑤))) ↔ ((𝑥𝐴 ∧ ∃𝑦𝐵 𝐶 = (𝑥 + 𝑦)) ∧ (𝑧𝐴 ∧ ∃𝑤𝐵 𝐶 = (𝑧 + 𝑤))))
2 reeanv 3348 . . . . . 6 (∃𝑦𝐵𝑤𝐵 (𝐶 = (𝑥 + 𝑦) ∧ 𝐶 = (𝑧 + 𝑤)) ↔ (∃𝑦𝐵 𝐶 = (𝑥 + 𝑦) ∧ ∃𝑤𝐵 𝐶 = (𝑧 + 𝑤)))
3 simpll1 1209 . . . . . . . . . 10 ((((𝐴S𝐵S ∧ (𝐴𝐵) = 0) ∧ (𝑥𝐴𝑧𝐴)) ∧ ((𝑦𝐵𝑤𝐵) ∧ (𝐶 = (𝑥 + 𝑦) ∧ 𝐶 = (𝑧 + 𝑤)))) → 𝐴S )
4 simpll2 1210 . . . . . . . . . 10 ((((𝐴S𝐵S ∧ (𝐴𝐵) = 0) ∧ (𝑥𝐴𝑧𝐴)) ∧ ((𝑦𝐵𝑤𝐵) ∧ (𝐶 = (𝑥 + 𝑦) ∧ 𝐶 = (𝑧 + 𝑤)))) → 𝐵S )
5 simpll3 1211 . . . . . . . . . 10 ((((𝐴S𝐵S ∧ (𝐴𝐵) = 0) ∧ (𝑥𝐴𝑧𝐴)) ∧ ((𝑦𝐵𝑤𝐵) ∧ (𝐶 = (𝑥 + 𝑦) ∧ 𝐶 = (𝑧 + 𝑤)))) → (𝐴𝐵) = 0)
6 simplrl 776 . . . . . . . . . 10 ((((𝐴S𝐵S ∧ (𝐴𝐵) = 0) ∧ (𝑥𝐴𝑧𝐴)) ∧ ((𝑦𝐵𝑤𝐵) ∧ (𝐶 = (𝑥 + 𝑦) ∧ 𝐶 = (𝑧 + 𝑤)))) → 𝑥𝐴)
7 simprll 778 . . . . . . . . . 10 ((((𝐴S𝐵S ∧ (𝐴𝐵) = 0) ∧ (𝑥𝐴𝑧𝐴)) ∧ ((𝑦𝐵𝑤𝐵) ∧ (𝐶 = (𝑥 + 𝑦) ∧ 𝐶 = (𝑧 + 𝑤)))) → 𝑦𝐵)
8 simplrr 777 . . . . . . . . . 10 ((((𝐴S𝐵S ∧ (𝐴𝐵) = 0) ∧ (𝑥𝐴𝑧𝐴)) ∧ ((𝑦𝐵𝑤𝐵) ∧ (𝐶 = (𝑥 + 𝑦) ∧ 𝐶 = (𝑧 + 𝑤)))) → 𝑧𝐴)
9 simprlr 779 . . . . . . . . . 10 ((((𝐴S𝐵S ∧ (𝐴𝐵) = 0) ∧ (𝑥𝐴𝑧𝐴)) ∧ ((𝑦𝐵𝑤𝐵) ∧ (𝐶 = (𝑥 + 𝑦) ∧ 𝐶 = (𝑧 + 𝑤)))) → 𝑤𝐵)
10 simprrl 780 . . . . . . . . . . 11 ((((𝐴S𝐵S ∧ (𝐴𝐵) = 0) ∧ (𝑥𝐴𝑧𝐴)) ∧ ((𝑦𝐵𝑤𝐵) ∧ (𝐶 = (𝑥 + 𝑦) ∧ 𝐶 = (𝑧 + 𝑤)))) → 𝐶 = (𝑥 + 𝑦))
11 simprrr 781 . . . . . . . . . . 11 ((((𝐴S𝐵S ∧ (𝐴𝐵) = 0) ∧ (𝑥𝐴𝑧𝐴)) ∧ ((𝑦𝐵𝑤𝐵) ∧ (𝐶 = (𝑥 + 𝑦) ∧ 𝐶 = (𝑧 + 𝑤)))) → 𝐶 = (𝑧 + 𝑤))
1210, 11eqtr3d 2859 . . . . . . . . . 10 ((((𝐴S𝐵S ∧ (𝐴𝐵) = 0) ∧ (𝑥𝐴𝑧𝐴)) ∧ ((𝑦𝐵𝑤𝐵) ∧ (𝐶 = (𝑥 + 𝑦) ∧ 𝐶 = (𝑧 + 𝑤)))) → (𝑥 + 𝑦) = (𝑧 + 𝑤))
133, 4, 5, 6, 7, 8, 9, 12shuni 29081 . . . . . . . . 9 ((((𝐴S𝐵S ∧ (𝐴𝐵) = 0) ∧ (𝑥𝐴𝑧𝐴)) ∧ ((𝑦𝐵𝑤𝐵) ∧ (𝐶 = (𝑥 + 𝑦) ∧ 𝐶 = (𝑧 + 𝑤)))) → (𝑥 = 𝑧𝑦 = 𝑤))
1413simpld 498 . . . . . . . 8 ((((𝐴S𝐵S ∧ (𝐴𝐵) = 0) ∧ (𝑥𝐴𝑧𝐴)) ∧ ((𝑦𝐵𝑤𝐵) ∧ (𝐶 = (𝑥 + 𝑦) ∧ 𝐶 = (𝑧 + 𝑤)))) → 𝑥 = 𝑧)
1514exp32 424 . . . . . . 7 (((𝐴S𝐵S ∧ (𝐴𝐵) = 0) ∧ (𝑥𝐴𝑧𝐴)) → ((𝑦𝐵𝑤𝐵) → ((𝐶 = (𝑥 + 𝑦) ∧ 𝐶 = (𝑧 + 𝑤)) → 𝑥 = 𝑧)))
1615rexlimdvv 3279 . . . . . 6 (((𝐴S𝐵S ∧ (𝐴𝐵) = 0) ∧ (𝑥𝐴𝑧𝐴)) → (∃𝑦𝐵𝑤𝐵 (𝐶 = (𝑥 + 𝑦) ∧ 𝐶 = (𝑧 + 𝑤)) → 𝑥 = 𝑧))
172, 16syl5bir 246 . . . . 5 (((𝐴S𝐵S ∧ (𝐴𝐵) = 0) ∧ (𝑥𝐴𝑧𝐴)) → ((∃𝑦𝐵 𝐶 = (𝑥 + 𝑦) ∧ ∃𝑤𝐵 𝐶 = (𝑧 + 𝑤)) → 𝑥 = 𝑧))
1817expimpd 457 . . . 4 ((𝐴S𝐵S ∧ (𝐴𝐵) = 0) → (((𝑥𝐴𝑧𝐴) ∧ (∃𝑦𝐵 𝐶 = (𝑥 + 𝑦) ∧ ∃𝑤𝐵 𝐶 = (𝑧 + 𝑤))) → 𝑥 = 𝑧))
191, 18syl5bir 246 . . 3 ((𝐴S𝐵S ∧ (𝐴𝐵) = 0) → (((𝑥𝐴 ∧ ∃𝑦𝐵 𝐶 = (𝑥 + 𝑦)) ∧ (𝑧𝐴 ∧ ∃𝑤𝐵 𝐶 = (𝑧 + 𝑤))) → 𝑥 = 𝑧))
2019alrimivv 1929 . 2 ((𝐴S𝐵S ∧ (𝐴𝐵) = 0) → ∀𝑥𝑧(((𝑥𝐴 ∧ ∃𝑦𝐵 𝐶 = (𝑥 + 𝑦)) ∧ (𝑧𝐴 ∧ ∃𝑤𝐵 𝐶 = (𝑧 + 𝑤))) → 𝑥 = 𝑧))
21 eleq1w 2896 . . . 4 (𝑥 = 𝑧 → (𝑥𝐴𝑧𝐴))
22 oveq1 7147 . . . . . . 7 (𝑥 = 𝑧 → (𝑥 + 𝑦) = (𝑧 + 𝑦))
2322eqeq2d 2833 . . . . . 6 (𝑥 = 𝑧 → (𝐶 = (𝑥 + 𝑦) ↔ 𝐶 = (𝑧 + 𝑦)))
2423rexbidv 3283 . . . . 5 (𝑥 = 𝑧 → (∃𝑦𝐵 𝐶 = (𝑥 + 𝑦) ↔ ∃𝑦𝐵 𝐶 = (𝑧 + 𝑦)))
25 oveq2 7148 . . . . . . 7 (𝑦 = 𝑤 → (𝑧 + 𝑦) = (𝑧 + 𝑤))
2625eqeq2d 2833 . . . . . 6 (𝑦 = 𝑤 → (𝐶 = (𝑧 + 𝑦) ↔ 𝐶 = (𝑧 + 𝑤)))
2726cbvrexvw 3425 . . . . 5 (∃𝑦𝐵 𝐶 = (𝑧 + 𝑦) ↔ ∃𝑤𝐵 𝐶 = (𝑧 + 𝑤))
2824, 27syl6bb 290 . . . 4 (𝑥 = 𝑧 → (∃𝑦𝐵 𝐶 = (𝑥 + 𝑦) ↔ ∃𝑤𝐵 𝐶 = (𝑧 + 𝑤)))
2921, 28anbi12d 633 . . 3 (𝑥 = 𝑧 → ((𝑥𝐴 ∧ ∃𝑦𝐵 𝐶 = (𝑥 + 𝑦)) ↔ (𝑧𝐴 ∧ ∃𝑤𝐵 𝐶 = (𝑧 + 𝑤))))
3029mo4 2649 . 2 (∃*𝑥(𝑥𝐴 ∧ ∃𝑦𝐵 𝐶 = (𝑥 + 𝑦)) ↔ ∀𝑥𝑧(((𝑥𝐴 ∧ ∃𝑦𝐵 𝐶 = (𝑥 + 𝑦)) ∧ (𝑧𝐴 ∧ ∃𝑤𝐵 𝐶 = (𝑧 + 𝑤))) → 𝑥 = 𝑧))
3120, 30sylibr 237 1 ((𝐴S𝐵S ∧ (𝐴𝐵) = 0) → ∃*𝑥(𝑥𝐴 ∧ ∃𝑦𝐵 𝐶 = (𝑥 + 𝑦)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399  w3a 1084  wal 1536   = wceq 1538  wcel 2114  ∃*wmo 2620  wrex 3131  cin 3907  (class class class)co 7140   + cva 28701   S csh 28709  0c0h 28716
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-sep 5179  ax-nul 5186  ax-pow 5243  ax-pr 5307  ax-un 7446  ax-resscn 10583  ax-1cn 10584  ax-icn 10585  ax-addcl 10586  ax-addrcl 10587  ax-mulcl 10588  ax-mulrcl 10589  ax-mulcom 10590  ax-addass 10591  ax-mulass 10592  ax-distr 10593  ax-i2m1 10594  ax-1ne0 10595  ax-1rid 10596  ax-rnegex 10597  ax-rrecex 10598  ax-cnre 10599  ax-pre-lttri 10600  ax-pre-lttrn 10601  ax-pre-ltadd 10602  ax-pre-mulgt0 10603  ax-hilex 28780  ax-hfvadd 28781  ax-hvcom 28782  ax-hvass 28783  ax-hv0cl 28784  ax-hvaddid 28785  ax-hfvmul 28786  ax-hvmulid 28787  ax-hvmulass 28788  ax-hvdistr1 28789  ax-hvdistr2 28790  ax-hvmul0 28791
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ne 3012  df-nel 3116  df-ral 3135  df-rex 3136  df-reu 3137  df-rmo 3138  df-rab 3139  df-v 3471  df-sbc 3748  df-csb 3856  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4266  df-if 4440  df-pw 4513  df-sn 4540  df-pr 4542  df-op 4546  df-uni 4814  df-iun 4896  df-br 5043  df-opab 5105  df-mpt 5123  df-id 5437  df-po 5451  df-so 5452  df-xp 5538  df-rel 5539  df-cnv 5540  df-co 5541  df-dm 5542  df-rn 5543  df-res 5544  df-ima 5545  df-iota 6293  df-fun 6336  df-fn 6337  df-f 6338  df-f1 6339  df-fo 6340  df-f1o 6341  df-fv 6342  df-riota 7098  df-ov 7143  df-oprab 7144  df-mpo 7145  df-er 8276  df-en 8497  df-dom 8498  df-sdom 8499  df-pnf 10666  df-mnf 10667  df-xr 10668  df-ltxr 10669  df-le 10670  df-sub 10861  df-neg 10862  df-div 11287  df-hvsub 28752  df-sh 28988  df-ch0 29034
This theorem is referenced by:  pjhtheu  29175  pjpreeq  29179
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