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Theorem pjhthmo 31334
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 3235 . . . . . 6 (∃𝑦𝐵𝑤𝐵 (𝐶 = (𝑥 + 𝑦) ∧ 𝐶 = (𝑧 + 𝑤)) ↔ (∃𝑦𝐵 𝐶 = (𝑥 + 𝑦) ∧ ∃𝑤𝐵 𝐶 = (𝑧 + 𝑤)))
3 simpll1 1212 . . . . . . . . . 10 ((((𝐴S𝐵S ∧ (𝐴𝐵) = 0) ∧ (𝑥𝐴𝑧𝐴)) ∧ ((𝑦𝐵𝑤𝐵) ∧ (𝐶 = (𝑥 + 𝑦) ∧ 𝐶 = (𝑧 + 𝑤)))) → 𝐴S )
4 simpll2 1213 . . . . . . . . . 10 ((((𝐴S𝐵S ∧ (𝐴𝐵) = 0) ∧ (𝑥𝐴𝑧𝐴)) ∧ ((𝑦𝐵𝑤𝐵) ∧ (𝐶 = (𝑥 + 𝑦) ∧ 𝐶 = (𝑧 + 𝑤)))) → 𝐵S )
5 simpll3 1214 . . . . . . . . . 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 2782 . . . . . . . . . 10 ((((𝐴S𝐵S ∧ (𝐴𝐵) = 0) ∧ (𝑥𝐴𝑧𝐴)) ∧ ((𝑦𝐵𝑤𝐵) ∧ (𝐶 = (𝑥 + 𝑦) ∧ 𝐶 = (𝑧 + 𝑤)))) → (𝑥 + 𝑦) = (𝑧 + 𝑤))
133, 4, 5, 6, 7, 8, 9, 12shuni 31332 . . . . . . . . 9 ((((𝐴S𝐵S ∧ (𝐴𝐵) = 0) ∧ (𝑥𝐴𝑧𝐴)) ∧ ((𝑦𝐵𝑤𝐵) ∧ (𝐶 = (𝑥 + 𝑦) ∧ 𝐶 = (𝑧 + 𝑤)))) → (𝑥 = 𝑧𝑦 = 𝑤))
1413simpld 494 . . . . . . . 8 ((((𝐴S𝐵S ∧ (𝐴𝐵) = 0) ∧ (𝑥𝐴𝑧𝐴)) ∧ ((𝑦𝐵𝑤𝐵) ∧ (𝐶 = (𝑥 + 𝑦) ∧ 𝐶 = (𝑧 + 𝑤)))) → 𝑥 = 𝑧)
1514exp32 420 . . . . . . 7 (((𝐴S𝐵S ∧ (𝐴𝐵) = 0) ∧ (𝑥𝐴𝑧𝐴)) → ((𝑦𝐵𝑤𝐵) → ((𝐶 = (𝑥 + 𝑦) ∧ 𝐶 = (𝑧 + 𝑤)) → 𝑥 = 𝑧)))
1615rexlimdvv 3218 . . . . . 6 (((𝐴S𝐵S ∧ (𝐴𝐵) = 0) ∧ (𝑥𝐴𝑧𝐴)) → (∃𝑦𝐵𝑤𝐵 (𝐶 = (𝑥 + 𝑦) ∧ 𝐶 = (𝑧 + 𝑤)) → 𝑥 = 𝑧))
172, 16biimtrrid 243 . . . . 5 (((𝐴S𝐵S ∧ (𝐴𝐵) = 0) ∧ (𝑥𝐴𝑧𝐴)) → ((∃𝑦𝐵 𝐶 = (𝑥 + 𝑦) ∧ ∃𝑤𝐵 𝐶 = (𝑧 + 𝑤)) → 𝑥 = 𝑧))
1817expimpd 453 . . . 4 ((𝐴S𝐵S ∧ (𝐴𝐵) = 0) → (((𝑥𝐴𝑧𝐴) ∧ (∃𝑦𝐵 𝐶 = (𝑥 + 𝑦) ∧ ∃𝑤𝐵 𝐶 = (𝑧 + 𝑤))) → 𝑥 = 𝑧))
191, 18biimtrrid 243 . . 3 ((𝐴S𝐵S ∧ (𝐴𝐵) = 0) → (((𝑥𝐴 ∧ ∃𝑦𝐵 𝐶 = (𝑥 + 𝑦)) ∧ (𝑧𝐴 ∧ ∃𝑤𝐵 𝐶 = (𝑧 + 𝑤))) → 𝑥 = 𝑧))
2019alrimivv 1927 . 2 ((𝐴S𝐵S ∧ (𝐴𝐵) = 0) → ∀𝑥𝑧(((𝑥𝐴 ∧ ∃𝑦𝐵 𝐶 = (𝑥 + 𝑦)) ∧ (𝑧𝐴 ∧ ∃𝑤𝐵 𝐶 = (𝑧 + 𝑤))) → 𝑥 = 𝑧))
21 eleq1w 2827 . . . 4 (𝑥 = 𝑧 → (𝑥𝐴𝑧𝐴))
22 oveq1 7455 . . . . . . 7 (𝑥 = 𝑧 → (𝑥 + 𝑦) = (𝑧 + 𝑦))
2322eqeq2d 2751 . . . . . 6 (𝑥 = 𝑧 → (𝐶 = (𝑥 + 𝑦) ↔ 𝐶 = (𝑧 + 𝑦)))
2423rexbidv 3185 . . . . 5 (𝑥 = 𝑧 → (∃𝑦𝐵 𝐶 = (𝑥 + 𝑦) ↔ ∃𝑦𝐵 𝐶 = (𝑧 + 𝑦)))
25 oveq2 7456 . . . . . . 7 (𝑦 = 𝑤 → (𝑧 + 𝑦) = (𝑧 + 𝑤))
2625eqeq2d 2751 . . . . . 6 (𝑦 = 𝑤 → (𝐶 = (𝑧 + 𝑦) ↔ 𝐶 = (𝑧 + 𝑤)))
2726cbvrexvw 3244 . . . . 5 (∃𝑦𝐵 𝐶 = (𝑧 + 𝑦) ↔ ∃𝑤𝐵 𝐶 = (𝑧 + 𝑤))
2824, 27bitrdi 287 . . . 4 (𝑥 = 𝑧 → (∃𝑦𝐵 𝐶 = (𝑥 + 𝑦) ↔ ∃𝑤𝐵 𝐶 = (𝑧 + 𝑤)))
2921, 28anbi12d 631 . . 3 (𝑥 = 𝑧 → ((𝑥𝐴 ∧ ∃𝑦𝐵 𝐶 = (𝑥 + 𝑦)) ↔ (𝑧𝐴 ∧ ∃𝑤𝐵 𝐶 = (𝑧 + 𝑤))))
3029mo4 2569 . 2 (∃*𝑥(𝑥𝐴 ∧ ∃𝑦𝐵 𝐶 = (𝑥 + 𝑦)) ↔ ∀𝑥𝑧(((𝑥𝐴 ∧ ∃𝑦𝐵 𝐶 = (𝑥 + 𝑦)) ∧ (𝑧𝐴 ∧ ∃𝑤𝐵 𝐶 = (𝑧 + 𝑤))) → 𝑥 = 𝑧))
3120, 30sylibr 234 1 ((𝐴S𝐵S ∧ (𝐴𝐵) = 0) → ∃*𝑥(𝑥𝐴 ∧ ∃𝑦𝐵 𝐶 = (𝑥 + 𝑦)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1087  wal 1535   = wceq 1537  wcel 2108  ∃*wmo 2541  wrex 3076  cin 3975  (class class class)co 7448   + cva 30952   S csh 30960  0c0h 30967
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770  ax-resscn 11241  ax-1cn 11242  ax-icn 11243  ax-addcl 11244  ax-addrcl 11245  ax-mulcl 11246  ax-mulrcl 11247  ax-mulcom 11248  ax-addass 11249  ax-mulass 11250  ax-distr 11251  ax-i2m1 11252  ax-1ne0 11253  ax-1rid 11254  ax-rnegex 11255  ax-rrecex 11256  ax-cnre 11257  ax-pre-lttri 11258  ax-pre-lttrn 11259  ax-pre-ltadd 11260  ax-pre-mulgt0 11261  ax-hilex 31031  ax-hfvadd 31032  ax-hvcom 31033  ax-hvass 31034  ax-hv0cl 31035  ax-hvaddid 31036  ax-hfvmul 31037  ax-hvmulid 31038  ax-hvmulass 31039  ax-hvdistr1 31040  ax-hvdistr2 31041  ax-hvmul0 31042
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-nel 3053  df-ral 3068  df-rex 3077  df-rmo 3388  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-po 5607  df-so 5608  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-riota 7404  df-ov 7451  df-oprab 7452  df-mpo 7453  df-er 8763  df-en 9004  df-dom 9005  df-sdom 9006  df-pnf 11326  df-mnf 11327  df-xr 11328  df-ltxr 11329  df-le 11330  df-sub 11522  df-neg 11523  df-div 11948  df-hvsub 31003  df-sh 31239  df-ch0 31285
This theorem is referenced by:  pjhtheu  31426  pjpreeq  31430
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