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Theorem pjhthmo 28747
 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 646 . . . 4 (((𝑥𝐴𝑧𝐴) ∧ (∃𝑦𝐵 𝐶 = (𝑥 + 𝑦) ∧ ∃𝑤𝐵 𝐶 = (𝑧 + 𝑤))) ↔ ((𝑥𝐴 ∧ ∃𝑦𝐵 𝐶 = (𝑥 + 𝑦)) ∧ (𝑧𝐴 ∧ ∃𝑤𝐵 𝐶 = (𝑧 + 𝑤))))
2 reeanv 3292 . . . . . 6 (∃𝑦𝐵𝑤𝐵 (𝐶 = (𝑥 + 𝑦) ∧ 𝐶 = (𝑧 + 𝑤)) ↔ (∃𝑦𝐵 𝐶 = (𝑥 + 𝑦) ∧ ∃𝑤𝐵 𝐶 = (𝑧 + 𝑤)))
3 simpll1 1226 . . . . . . . . . 10 ((((𝐴S𝐵S ∧ (𝐴𝐵) = 0) ∧ (𝑥𝐴𝑧𝐴)) ∧ ((𝑦𝐵𝑤𝐵) ∧ (𝐶 = (𝑥 + 𝑦) ∧ 𝐶 = (𝑧 + 𝑤)))) → 𝐴S )
4 simpll2 1228 . . . . . . . . . 10 ((((𝐴S𝐵S ∧ (𝐴𝐵) = 0) ∧ (𝑥𝐴𝑧𝐴)) ∧ ((𝑦𝐵𝑤𝐵) ∧ (𝐶 = (𝑥 + 𝑦) ∧ 𝐶 = (𝑧 + 𝑤)))) → 𝐵S )
5 simpll3 1230 . . . . . . . . . 10 ((((𝐴S𝐵S ∧ (𝐴𝐵) = 0) ∧ (𝑥𝐴𝑧𝐴)) ∧ ((𝑦𝐵𝑤𝐵) ∧ (𝐶 = (𝑥 + 𝑦) ∧ 𝐶 = (𝑧 + 𝑤)))) → (𝐴𝐵) = 0)
6 simplrl 767 . . . . . . . . . 10 ((((𝐴S𝐵S ∧ (𝐴𝐵) = 0) ∧ (𝑥𝐴𝑧𝐴)) ∧ ((𝑦𝐵𝑤𝐵) ∧ (𝐶 = (𝑥 + 𝑦) ∧ 𝐶 = (𝑧 + 𝑤)))) → 𝑥𝐴)
7 simprll 769 . . . . . . . . . 10 ((((𝐴S𝐵S ∧ (𝐴𝐵) = 0) ∧ (𝑥𝐴𝑧𝐴)) ∧ ((𝑦𝐵𝑤𝐵) ∧ (𝐶 = (𝑥 + 𝑦) ∧ 𝐶 = (𝑧 + 𝑤)))) → 𝑦𝐵)
8 simplrr 768 . . . . . . . . . 10 ((((𝐴S𝐵S ∧ (𝐴𝐵) = 0) ∧ (𝑥𝐴𝑧𝐴)) ∧ ((𝑦𝐵𝑤𝐵) ∧ (𝐶 = (𝑥 + 𝑦) ∧ 𝐶 = (𝑧 + 𝑤)))) → 𝑧𝐴)
9 simprlr 770 . . . . . . . . . 10 ((((𝐴S𝐵S ∧ (𝐴𝐵) = 0) ∧ (𝑥𝐴𝑧𝐴)) ∧ ((𝑦𝐵𝑤𝐵) ∧ (𝐶 = (𝑥 + 𝑦) ∧ 𝐶 = (𝑧 + 𝑤)))) → 𝑤𝐵)
10 simprrl 771 . . . . . . . . . . 11 ((((𝐴S𝐵S ∧ (𝐴𝐵) = 0) ∧ (𝑥𝐴𝑧𝐴)) ∧ ((𝑦𝐵𝑤𝐵) ∧ (𝐶 = (𝑥 + 𝑦) ∧ 𝐶 = (𝑧 + 𝑤)))) → 𝐶 = (𝑥 + 𝑦))
11 simprrr 772 . . . . . . . . . . 11 ((((𝐴S𝐵S ∧ (𝐴𝐵) = 0) ∧ (𝑥𝐴𝑧𝐴)) ∧ ((𝑦𝐵𝑤𝐵) ∧ (𝐶 = (𝑥 + 𝑦) ∧ 𝐶 = (𝑧 + 𝑤)))) → 𝐶 = (𝑧 + 𝑤))
1210, 11eqtr3d 2815 . . . . . . . . . 10 ((((𝐴S𝐵S ∧ (𝐴𝐵) = 0) ∧ (𝑥𝐴𝑧𝐴)) ∧ ((𝑦𝐵𝑤𝐵) ∧ (𝐶 = (𝑥 + 𝑦) ∧ 𝐶 = (𝑧 + 𝑤)))) → (𝑥 + 𝑦) = (𝑧 + 𝑤))
133, 4, 5, 6, 7, 8, 9, 12shuni 28745 . . . . . . . . 9 ((((𝐴S𝐵S ∧ (𝐴𝐵) = 0) ∧ (𝑥𝐴𝑧𝐴)) ∧ ((𝑦𝐵𝑤𝐵) ∧ (𝐶 = (𝑥 + 𝑦) ∧ 𝐶 = (𝑧 + 𝑤)))) → (𝑥 = 𝑧𝑦 = 𝑤))
1413simpld 490 . . . . . . . 8 ((((𝐴S𝐵S ∧ (𝐴𝐵) = 0) ∧ (𝑥𝐴𝑧𝐴)) ∧ ((𝑦𝐵𝑤𝐵) ∧ (𝐶 = (𝑥 + 𝑦) ∧ 𝐶 = (𝑧 + 𝑤)))) → 𝑥 = 𝑧)
1514exp32 413 . . . . . . 7 (((𝐴S𝐵S ∧ (𝐴𝐵) = 0) ∧ (𝑥𝐴𝑧𝐴)) → ((𝑦𝐵𝑤𝐵) → ((𝐶 = (𝑥 + 𝑦) ∧ 𝐶 = (𝑧 + 𝑤)) → 𝑥 = 𝑧)))
1615rexlimdvv 3219 . . . . . 6 (((𝐴S𝐵S ∧ (𝐴𝐵) = 0) ∧ (𝑥𝐴𝑧𝐴)) → (∃𝑦𝐵𝑤𝐵 (𝐶 = (𝑥 + 𝑦) ∧ 𝐶 = (𝑧 + 𝑤)) → 𝑥 = 𝑧))
172, 16syl5bir 235 . . . . 5 (((𝐴S𝐵S ∧ (𝐴𝐵) = 0) ∧ (𝑥𝐴𝑧𝐴)) → ((∃𝑦𝐵 𝐶 = (𝑥 + 𝑦) ∧ ∃𝑤𝐵 𝐶 = (𝑧 + 𝑤)) → 𝑥 = 𝑧))
1817expimpd 447 . . . 4 ((𝐴S𝐵S ∧ (𝐴𝐵) = 0) → (((𝑥𝐴𝑧𝐴) ∧ (∃𝑦𝐵 𝐶 = (𝑥 + 𝑦) ∧ ∃𝑤𝐵 𝐶 = (𝑧 + 𝑤))) → 𝑥 = 𝑧))
191, 18syl5bir 235 . . 3 ((𝐴S𝐵S ∧ (𝐴𝐵) = 0) → (((𝑥𝐴 ∧ ∃𝑦𝐵 𝐶 = (𝑥 + 𝑦)) ∧ (𝑧𝐴 ∧ ∃𝑤𝐵 𝐶 = (𝑧 + 𝑤))) → 𝑥 = 𝑧))
2019alrimivv 1971 . 2 ((𝐴S𝐵S ∧ (𝐴𝐵) = 0) → ∀𝑥𝑧(((𝑥𝐴 ∧ ∃𝑦𝐵 𝐶 = (𝑥 + 𝑦)) ∧ (𝑧𝐴 ∧ ∃𝑤𝐵 𝐶 = (𝑧 + 𝑤))) → 𝑥 = 𝑧))
21 eleq1w 2841 . . . 4 (𝑥 = 𝑧 → (𝑥𝐴𝑧𝐴))
22 oveq1 6929 . . . . . . 7 (𝑥 = 𝑧 → (𝑥 + 𝑦) = (𝑧 + 𝑦))
2322eqeq2d 2787 . . . . . 6 (𝑥 = 𝑧 → (𝐶 = (𝑥 + 𝑦) ↔ 𝐶 = (𝑧 + 𝑦)))
2423rexbidv 3236 . . . . 5 (𝑥 = 𝑧 → (∃𝑦𝐵 𝐶 = (𝑥 + 𝑦) ↔ ∃𝑦𝐵 𝐶 = (𝑧 + 𝑦)))
25 oveq2 6930 . . . . . . 7 (𝑦 = 𝑤 → (𝑧 + 𝑦) = (𝑧 + 𝑤))
2625eqeq2d 2787 . . . . . 6 (𝑦 = 𝑤 → (𝐶 = (𝑧 + 𝑦) ↔ 𝐶 = (𝑧 + 𝑤)))
2726cbvrexv 3367 . . . . 5 (∃𝑦𝐵 𝐶 = (𝑧 + 𝑦) ↔ ∃𝑤𝐵 𝐶 = (𝑧 + 𝑤))
2824, 27syl6bb 279 . . . 4 (𝑥 = 𝑧 → (∃𝑦𝐵 𝐶 = (𝑥 + 𝑦) ↔ ∃𝑤𝐵 𝐶 = (𝑧 + 𝑤)))
2921, 28anbi12d 624 . . 3 (𝑥 = 𝑧 → ((𝑥𝐴 ∧ ∃𝑦𝐵 𝐶 = (𝑥 + 𝑦)) ↔ (𝑧𝐴 ∧ ∃𝑤𝐵 𝐶 = (𝑧 + 𝑤))))
3029mo4 2584 . 2 (∃*𝑥(𝑥𝐴 ∧ ∃𝑦𝐵 𝐶 = (𝑥 + 𝑦)) ↔ ∀𝑥𝑧(((𝑥𝐴 ∧ ∃𝑦𝐵 𝐶 = (𝑥 + 𝑦)) ∧ (𝑧𝐴 ∧ ∃𝑤𝐵 𝐶 = (𝑧 + 𝑤))) → 𝑥 = 𝑧))
3120, 30sylibr 226 1 ((𝐴S𝐵S ∧ (𝐴𝐵) = 0) → ∃*𝑥(𝑥𝐴 ∧ ∃𝑦𝐵 𝐶 = (𝑥 + 𝑦)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 386   ∧ w3a 1071  ∀wal 1599   = wceq 1601   ∈ wcel 2106  ∃*wmo 2548  ∃wrex 3090   ∩ cin 3790  (class class class)co 6922   +ℎ cva 28363   Sℋ csh 28371  0ℋc0h 28378 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2054  ax-8 2108  ax-9 2115  ax-10 2134  ax-11 2149  ax-12 2162  ax-13 2333  ax-ext 2753  ax-sep 5017  ax-nul 5025  ax-pow 5077  ax-pr 5138  ax-un 7226  ax-resscn 10329  ax-1cn 10330  ax-icn 10331  ax-addcl 10332  ax-addrcl 10333  ax-mulcl 10334  ax-mulrcl 10335  ax-mulcom 10336  ax-addass 10337  ax-mulass 10338  ax-distr 10339  ax-i2m1 10340  ax-1ne0 10341  ax-1rid 10342  ax-rnegex 10343  ax-rrecex 10344  ax-cnre 10345  ax-pre-lttri 10346  ax-pre-lttrn 10347  ax-pre-ltadd 10348  ax-pre-mulgt0 10349  ax-hilex 28442  ax-hfvadd 28443  ax-hvcom 28444  ax-hvass 28445  ax-hv0cl 28446  ax-hvaddid 28447  ax-hfvmul 28448  ax-hvmulid 28449  ax-hvmulass 28450  ax-hvdistr1 28451  ax-hvdistr2 28452  ax-hvmul0 28453 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3or 1072  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2550  df-eu 2586  df-clab 2763  df-cleq 2769  df-clel 2773  df-nfc 2920  df-ne 2969  df-nel 3075  df-ral 3094  df-rex 3095  df-reu 3096  df-rmo 3097  df-rab 3098  df-v 3399  df-sbc 3652  df-csb 3751  df-dif 3794  df-un 3796  df-in 3798  df-ss 3805  df-nul 4141  df-if 4307  df-pw 4380  df-sn 4398  df-pr 4400  df-op 4404  df-uni 4672  df-iun 4755  df-br 4887  df-opab 4949  df-mpt 4966  df-id 5261  df-po 5274  df-so 5275  df-xp 5361  df-rel 5362  df-cnv 5363  df-co 5364  df-dm 5365  df-rn 5366  df-res 5367  df-ima 5368  df-iota 6099  df-fun 6137  df-fn 6138  df-f 6139  df-f1 6140  df-fo 6141  df-f1o 6142  df-fv 6143  df-riota 6883  df-ov 6925  df-oprab 6926  df-mpt2 6927  df-er 8026  df-en 8242  df-dom 8243  df-sdom 8244  df-pnf 10413  df-mnf 10414  df-xr 10415  df-ltxr 10416  df-le 10417  df-sub 10608  df-neg 10609  df-div 11033  df-hvsub 28414  df-sh 28650  df-ch0 28696 This theorem is referenced by:  pjhtheu  28839  pjpreeq  28843
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