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Theorem pjhthmo 28031
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 864 . . . 4 (((𝑥𝐴𝑧𝐴) ∧ (∃𝑦𝐵 𝐶 = (𝑥 + 𝑦) ∧ ∃𝑤𝐵 𝐶 = (𝑧 + 𝑤))) ↔ ((𝑥𝐴 ∧ ∃𝑦𝐵 𝐶 = (𝑥 + 𝑦)) ∧ (𝑧𝐴 ∧ ∃𝑤𝐵 𝐶 = (𝑧 + 𝑤))))
2 reeanv 3100 . . . . . 6 (∃𝑦𝐵𝑤𝐵 (𝐶 = (𝑥 + 𝑦) ∧ 𝐶 = (𝑧 + 𝑤)) ↔ (∃𝑦𝐵 𝐶 = (𝑥 + 𝑦) ∧ ∃𝑤𝐵 𝐶 = (𝑧 + 𝑤)))
3 simpll1 1098 . . . . . . . . . 10 ((((𝐴S𝐵S ∧ (𝐴𝐵) = 0) ∧ (𝑥𝐴𝑧𝐴)) ∧ ((𝑦𝐵𝑤𝐵) ∧ (𝐶 = (𝑥 + 𝑦) ∧ 𝐶 = (𝑧 + 𝑤)))) → 𝐴S )
4 simpll2 1099 . . . . . . . . . 10 ((((𝐴S𝐵S ∧ (𝐴𝐵) = 0) ∧ (𝑥𝐴𝑧𝐴)) ∧ ((𝑦𝐵𝑤𝐵) ∧ (𝐶 = (𝑥 + 𝑦) ∧ 𝐶 = (𝑧 + 𝑤)))) → 𝐵S )
5 simpll3 1100 . . . . . . . . . 10 ((((𝐴S𝐵S ∧ (𝐴𝐵) = 0) ∧ (𝑥𝐴𝑧𝐴)) ∧ ((𝑦𝐵𝑤𝐵) ∧ (𝐶 = (𝑥 + 𝑦) ∧ 𝐶 = (𝑧 + 𝑤)))) → (𝐴𝐵) = 0)
6 simplrl 799 . . . . . . . . . 10 ((((𝐴S𝐵S ∧ (𝐴𝐵) = 0) ∧ (𝑥𝐴𝑧𝐴)) ∧ ((𝑦𝐵𝑤𝐵) ∧ (𝐶 = (𝑥 + 𝑦) ∧ 𝐶 = (𝑧 + 𝑤)))) → 𝑥𝐴)
7 simprll 801 . . . . . . . . . 10 ((((𝐴S𝐵S ∧ (𝐴𝐵) = 0) ∧ (𝑥𝐴𝑧𝐴)) ∧ ((𝑦𝐵𝑤𝐵) ∧ (𝐶 = (𝑥 + 𝑦) ∧ 𝐶 = (𝑧 + 𝑤)))) → 𝑦𝐵)
8 simplrr 800 . . . . . . . . . 10 ((((𝐴S𝐵S ∧ (𝐴𝐵) = 0) ∧ (𝑥𝐴𝑧𝐴)) ∧ ((𝑦𝐵𝑤𝐵) ∧ (𝐶 = (𝑥 + 𝑦) ∧ 𝐶 = (𝑧 + 𝑤)))) → 𝑧𝐴)
9 simprlr 802 . . . . . . . . . 10 ((((𝐴S𝐵S ∧ (𝐴𝐵) = 0) ∧ (𝑥𝐴𝑧𝐴)) ∧ ((𝑦𝐵𝑤𝐵) ∧ (𝐶 = (𝑥 + 𝑦) ∧ 𝐶 = (𝑧 + 𝑤)))) → 𝑤𝐵)
10 simprrl 803 . . . . . . . . . . 11 ((((𝐴S𝐵S ∧ (𝐴𝐵) = 0) ∧ (𝑥𝐴𝑧𝐴)) ∧ ((𝑦𝐵𝑤𝐵) ∧ (𝐶 = (𝑥 + 𝑦) ∧ 𝐶 = (𝑧 + 𝑤)))) → 𝐶 = (𝑥 + 𝑦))
11 simprrr 804 . . . . . . . . . . 11 ((((𝐴S𝐵S ∧ (𝐴𝐵) = 0) ∧ (𝑥𝐴𝑧𝐴)) ∧ ((𝑦𝐵𝑤𝐵) ∧ (𝐶 = (𝑥 + 𝑦) ∧ 𝐶 = (𝑧 + 𝑤)))) → 𝐶 = (𝑧 + 𝑤))
1210, 11eqtr3d 2657 . . . . . . . . . 10 ((((𝐴S𝐵S ∧ (𝐴𝐵) = 0) ∧ (𝑥𝐴𝑧𝐴)) ∧ ((𝑦𝐵𝑤𝐵) ∧ (𝐶 = (𝑥 + 𝑦) ∧ 𝐶 = (𝑧 + 𝑤)))) → (𝑥 + 𝑦) = (𝑧 + 𝑤))
133, 4, 5, 6, 7, 8, 9, 12shuni 28029 . . . . . . . . 9 ((((𝐴S𝐵S ∧ (𝐴𝐵) = 0) ∧ (𝑥𝐴𝑧𝐴)) ∧ ((𝑦𝐵𝑤𝐵) ∧ (𝐶 = (𝑥 + 𝑦) ∧ 𝐶 = (𝑧 + 𝑤)))) → (𝑥 = 𝑧𝑦 = 𝑤))
1413simpld 475 . . . . . . . 8 ((((𝐴S𝐵S ∧ (𝐴𝐵) = 0) ∧ (𝑥𝐴𝑧𝐴)) ∧ ((𝑦𝐵𝑤𝐵) ∧ (𝐶 = (𝑥 + 𝑦) ∧ 𝐶 = (𝑧 + 𝑤)))) → 𝑥 = 𝑧)
1514exp32 630 . . . . . . 7 (((𝐴S𝐵S ∧ (𝐴𝐵) = 0) ∧ (𝑥𝐴𝑧𝐴)) → ((𝑦𝐵𝑤𝐵) → ((𝐶 = (𝑥 + 𝑦) ∧ 𝐶 = (𝑧 + 𝑤)) → 𝑥 = 𝑧)))
1615rexlimdvv 3031 . . . . . 6 (((𝐴S𝐵S ∧ (𝐴𝐵) = 0) ∧ (𝑥𝐴𝑧𝐴)) → (∃𝑦𝐵𝑤𝐵 (𝐶 = (𝑥 + 𝑦) ∧ 𝐶 = (𝑧 + 𝑤)) → 𝑥 = 𝑧))
172, 16syl5bir 233 . . . . 5 (((𝐴S𝐵S ∧ (𝐴𝐵) = 0) ∧ (𝑥𝐴𝑧𝐴)) → ((∃𝑦𝐵 𝐶 = (𝑥 + 𝑦) ∧ ∃𝑤𝐵 𝐶 = (𝑧 + 𝑤)) → 𝑥 = 𝑧))
1817expimpd 628 . . . 4 ((𝐴S𝐵S ∧ (𝐴𝐵) = 0) → (((𝑥𝐴𝑧𝐴) ∧ (∃𝑦𝐵 𝐶 = (𝑥 + 𝑦) ∧ ∃𝑤𝐵 𝐶 = (𝑧 + 𝑤))) → 𝑥 = 𝑧))
191, 18syl5bir 233 . . 3 ((𝐴S𝐵S ∧ (𝐴𝐵) = 0) → (((𝑥𝐴 ∧ ∃𝑦𝐵 𝐶 = (𝑥 + 𝑦)) ∧ (𝑧𝐴 ∧ ∃𝑤𝐵 𝐶 = (𝑧 + 𝑤))) → 𝑥 = 𝑧))
2019alrimivv 1853 . 2 ((𝐴S𝐵S ∧ (𝐴𝐵) = 0) → ∀𝑥𝑧(((𝑥𝐴 ∧ ∃𝑦𝐵 𝐶 = (𝑥 + 𝑦)) ∧ (𝑧𝐴 ∧ ∃𝑤𝐵 𝐶 = (𝑧 + 𝑤))) → 𝑥 = 𝑧))
21 eleq1 2686 . . . 4 (𝑥 = 𝑧 → (𝑥𝐴𝑧𝐴))
22 oveq1 6617 . . . . . . 7 (𝑥 = 𝑧 → (𝑥 + 𝑦) = (𝑧 + 𝑦))
2322eqeq2d 2631 . . . . . 6 (𝑥 = 𝑧 → (𝐶 = (𝑥 + 𝑦) ↔ 𝐶 = (𝑧 + 𝑦)))
2423rexbidv 3046 . . . . 5 (𝑥 = 𝑧 → (∃𝑦𝐵 𝐶 = (𝑥 + 𝑦) ↔ ∃𝑦𝐵 𝐶 = (𝑧 + 𝑦)))
25 oveq2 6618 . . . . . . 7 (𝑦 = 𝑤 → (𝑧 + 𝑦) = (𝑧 + 𝑤))
2625eqeq2d 2631 . . . . . 6 (𝑦 = 𝑤 → (𝐶 = (𝑧 + 𝑦) ↔ 𝐶 = (𝑧 + 𝑤)))
2726cbvrexv 3163 . . . . 5 (∃𝑦𝐵 𝐶 = (𝑧 + 𝑦) ↔ ∃𝑤𝐵 𝐶 = (𝑧 + 𝑤))
2824, 27syl6bb 276 . . . 4 (𝑥 = 𝑧 → (∃𝑦𝐵 𝐶 = (𝑥 + 𝑦) ↔ ∃𝑤𝐵 𝐶 = (𝑧 + 𝑤)))
2921, 28anbi12d 746 . . 3 (𝑥 = 𝑧 → ((𝑥𝐴 ∧ ∃𝑦𝐵 𝐶 = (𝑥 + 𝑦)) ↔ (𝑧𝐴 ∧ ∃𝑤𝐵 𝐶 = (𝑧 + 𝑤))))
3029mo4 2516 . 2 (∃*𝑥(𝑥𝐴 ∧ ∃𝑦𝐵 𝐶 = (𝑥 + 𝑦)) ↔ ∀𝑥𝑧(((𝑥𝐴 ∧ ∃𝑦𝐵 𝐶 = (𝑥 + 𝑦)) ∧ (𝑧𝐴 ∧ ∃𝑤𝐵 𝐶 = (𝑧 + 𝑤))) → 𝑥 = 𝑧))
3120, 30sylibr 224 1 ((𝐴S𝐵S ∧ (𝐴𝐵) = 0) → ∃*𝑥(𝑥𝐴 ∧ ∃𝑦𝐵 𝐶 = (𝑥 + 𝑦)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  w3a 1036  wal 1478   = wceq 1480  wcel 1987  ∃*wmo 2470  wrex 2908  cin 3558  (class class class)co 6610   + cva 27647   S csh 27655  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-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-hvsub 27698  df-sh 27934  df-ch0 27980
This theorem is referenced by:  pjhtheu  28123  pjpreeq  28127
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