Theorem pjhthmo 30555

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	⊢ ((𝐴 ∈ Sℋ ∧ 𝐵 ∈ Sℋ ∧ (𝐴 ∩ 𝐵) = 0ℋ) → ∃*𝑥(𝑥 ∈ 𝐴 ∧ ∃𝑦 ∈ 𝐵 𝐶 = (𝑥 +ℎ 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.

Step	Hyp	Ref	Expression
1		an4 655	. . . 4 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴) ∧ (∃𝑦 ∈ 𝐵 𝐶 = (𝑥 +ℎ 𝑦) ∧ ∃𝑤 ∈ 𝐵 𝐶 = (𝑧 +ℎ 𝑤))) ↔ ((𝑥 ∈ 𝐴 ∧ ∃𝑦 ∈ 𝐵 𝐶 = (𝑥 +ℎ 𝑦)) ∧ (𝑧 ∈ 𝐴 ∧ ∃𝑤 ∈ 𝐵 𝐶 = (𝑧 +ℎ 𝑤))))
2		reeanv 3227	. . . . . 6 ⊢ (∃𝑦 ∈ 𝐵 ∃𝑤 ∈ 𝐵 (𝐶 = (𝑥 +ℎ 𝑦) ∧ 𝐶 = (𝑧 +ℎ 𝑤)) ↔ (∃𝑦 ∈ 𝐵 𝐶 = (𝑥 +ℎ 𝑦) ∧ ∃𝑤 ∈ 𝐵 𝐶 = (𝑧 +ℎ 𝑤)))
3		simpll1 1213	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ Sℋ ∧ 𝐵 ∈ Sℋ ∧ (𝐴 ∩ 𝐵) = 0ℋ) ∧ (𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) ∧ ((𝑦 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵) ∧ (𝐶 = (𝑥 +ℎ 𝑦) ∧ 𝐶 = (𝑧 +ℎ 𝑤)))) → 𝐴 ∈ Sℋ )
4		simpll2 1214	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ Sℋ ∧ 𝐵 ∈ Sℋ ∧ (𝐴 ∩ 𝐵) = 0ℋ) ∧ (𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) ∧ ((𝑦 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵) ∧ (𝐶 = (𝑥 +ℎ 𝑦) ∧ 𝐶 = (𝑧 +ℎ 𝑤)))) → 𝐵 ∈ Sℋ )
5		simpll3 1215	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ Sℋ ∧ 𝐵 ∈ Sℋ ∧ (𝐴 ∩ 𝐵) = 0ℋ) ∧ (𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) ∧ ((𝑦 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵) ∧ (𝐶 = (𝑥 +ℎ 𝑦) ∧ 𝐶 = (𝑧 +ℎ 𝑤)))) → (𝐴 ∩ 𝐵) = 0ℋ)
6		simplrl 776	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ Sℋ ∧ 𝐵 ∈ Sℋ ∧ (𝐴 ∩ 𝐵) = 0ℋ) ∧ (𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) ∧ ((𝑦 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵) ∧ (𝐶 = (𝑥 +ℎ 𝑦) ∧ 𝐶 = (𝑧 +ℎ 𝑤)))) → 𝑥 ∈ 𝐴)
7		simprll 778	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ Sℋ ∧ 𝐵 ∈ Sℋ ∧ (𝐴 ∩ 𝐵) = 0ℋ) ∧ (𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) ∧ ((𝑦 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵) ∧ (𝐶 = (𝑥 +ℎ 𝑦) ∧ 𝐶 = (𝑧 +ℎ 𝑤)))) → 𝑦 ∈ 𝐵)
8		simplrr 777	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ Sℋ ∧ 𝐵 ∈ Sℋ ∧ (𝐴 ∩ 𝐵) = 0ℋ) ∧ (𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) ∧ ((𝑦 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵) ∧ (𝐶 = (𝑥 +ℎ 𝑦) ∧ 𝐶 = (𝑧 +ℎ 𝑤)))) → 𝑧 ∈ 𝐴)
9		simprlr 779	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ Sℋ ∧ 𝐵 ∈ Sℋ ∧ (𝐴 ∩ 𝐵) = 0ℋ) ∧ (𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) ∧ ((𝑦 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵) ∧ (𝐶 = (𝑥 +ℎ 𝑦) ∧ 𝐶 = (𝑧 +ℎ 𝑤)))) → 𝑤 ∈ 𝐵)
10		simprrl 780	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ Sℋ ∧ 𝐵 ∈ Sℋ ∧ (𝐴 ∩ 𝐵) = 0ℋ) ∧ (𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) ∧ ((𝑦 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵) ∧ (𝐶 = (𝑥 +ℎ 𝑦) ∧ 𝐶 = (𝑧 +ℎ 𝑤)))) → 𝐶 = (𝑥 +ℎ 𝑦))
11		simprrr 781	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ Sℋ ∧ 𝐵 ∈ Sℋ ∧ (𝐴 ∩ 𝐵) = 0ℋ) ∧ (𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) ∧ ((𝑦 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵) ∧ (𝐶 = (𝑥 +ℎ 𝑦) ∧ 𝐶 = (𝑧 +ℎ 𝑤)))) → 𝐶 = (𝑧 +ℎ 𝑤))
12	10, 11	eqtr3d 2775	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ Sℋ ∧ 𝐵 ∈ Sℋ ∧ (𝐴 ∩ 𝐵) = 0ℋ) ∧ (𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) ∧ ((𝑦 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵) ∧ (𝐶 = (𝑥 +ℎ 𝑦) ∧ 𝐶 = (𝑧 +ℎ 𝑤)))) → (𝑥 +ℎ 𝑦) = (𝑧 +ℎ 𝑤))
13	3, 4, 5, 6, 7, 8, 9, 12	shuni 30553	. . . . . . . . 9 ⊢ ((((𝐴 ∈ Sℋ ∧ 𝐵 ∈ Sℋ ∧ (𝐴 ∩ 𝐵) = 0ℋ) ∧ (𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) ∧ ((𝑦 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵) ∧ (𝐶 = (𝑥 +ℎ 𝑦) ∧ 𝐶 = (𝑧 +ℎ 𝑤)))) → (𝑥 = 𝑧 ∧ 𝑦 = 𝑤))
14	13	simpld 496	. . . . . . . 8 ⊢ ((((𝐴 ∈ Sℋ ∧ 𝐵 ∈ Sℋ ∧ (𝐴 ∩ 𝐵) = 0ℋ) ∧ (𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) ∧ ((𝑦 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵) ∧ (𝐶 = (𝑥 +ℎ 𝑦) ∧ 𝐶 = (𝑧 +ℎ 𝑤)))) → 𝑥 = 𝑧)
15	14	exp32 422	. . . . . . 7 ⊢ (((𝐴 ∈ Sℋ ∧ 𝐵 ∈ Sℋ ∧ (𝐴 ∩ 𝐵) = 0ℋ) ∧ (𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) → ((𝑦 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵) → ((𝐶 = (𝑥 +ℎ 𝑦) ∧ 𝐶 = (𝑧 +ℎ 𝑤)) → 𝑥 = 𝑧)))
16	15	rexlimdvv 3211	. . . . . 6 ⊢ (((𝐴 ∈ Sℋ ∧ 𝐵 ∈ Sℋ ∧ (𝐴 ∩ 𝐵) = 0ℋ) ∧ (𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) → (∃𝑦 ∈ 𝐵 ∃𝑤 ∈ 𝐵 (𝐶 = (𝑥 +ℎ 𝑦) ∧ 𝐶 = (𝑧 +ℎ 𝑤)) → 𝑥 = 𝑧))
17	2, 16	biimtrrid 242	. . . . 5 ⊢ (((𝐴 ∈ Sℋ ∧ 𝐵 ∈ Sℋ ∧ (𝐴 ∩ 𝐵) = 0ℋ) ∧ (𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) → ((∃𝑦 ∈ 𝐵 𝐶 = (𝑥 +ℎ 𝑦) ∧ ∃𝑤 ∈ 𝐵 𝐶 = (𝑧 +ℎ 𝑤)) → 𝑥 = 𝑧))
18	17	expimpd 455	. . . 4 ⊢ ((𝐴 ∈ Sℋ ∧ 𝐵 ∈ Sℋ ∧ (𝐴 ∩ 𝐵) = 0ℋ) → (((𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴) ∧ (∃𝑦 ∈ 𝐵 𝐶 = (𝑥 +ℎ 𝑦) ∧ ∃𝑤 ∈ 𝐵 𝐶 = (𝑧 +ℎ 𝑤))) → 𝑥 = 𝑧))
19	1, 18	biimtrrid 242	. . 3 ⊢ ((𝐴 ∈ Sℋ ∧ 𝐵 ∈ Sℋ ∧ (𝐴 ∩ 𝐵) = 0ℋ) → (((𝑥 ∈ 𝐴 ∧ ∃𝑦 ∈ 𝐵 𝐶 = (𝑥 +ℎ 𝑦)) ∧ (𝑧 ∈ 𝐴 ∧ ∃𝑤 ∈ 𝐵 𝐶 = (𝑧 +ℎ 𝑤))) → 𝑥 = 𝑧))
20	19	alrimivv 1932	. 2 ⊢ ((𝐴 ∈ Sℋ ∧ 𝐵 ∈ Sℋ ∧ (𝐴 ∩ 𝐵) = 0ℋ) → ∀𝑥∀𝑧(((𝑥 ∈ 𝐴 ∧ ∃𝑦 ∈ 𝐵 𝐶 = (𝑥 +ℎ 𝑦)) ∧ (𝑧 ∈ 𝐴 ∧ ∃𝑤 ∈ 𝐵 𝐶 = (𝑧 +ℎ 𝑤))) → 𝑥 = 𝑧))
21		eleq1w 2817	. . . 4 ⊢ (𝑥 = 𝑧 → (𝑥 ∈ 𝐴 ↔ 𝑧 ∈ 𝐴))
22		oveq1 7416	. . . . . . 7 ⊢ (𝑥 = 𝑧 → (𝑥 +ℎ 𝑦) = (𝑧 +ℎ 𝑦))
23	22	eqeq2d 2744	. . . . . 6 ⊢ (𝑥 = 𝑧 → (𝐶 = (𝑥 +ℎ 𝑦) ↔ 𝐶 = (𝑧 +ℎ 𝑦)))
24	23	rexbidv 3179	. . . . 5 ⊢ (𝑥 = 𝑧 → (∃𝑦 ∈ 𝐵 𝐶 = (𝑥 +ℎ 𝑦) ↔ ∃𝑦 ∈ 𝐵 𝐶 = (𝑧 +ℎ 𝑦)))
25		oveq2 7417	. . . . . . 7 ⊢ (𝑦 = 𝑤 → (𝑧 +ℎ 𝑦) = (𝑧 +ℎ 𝑤))
26	25	eqeq2d 2744	. . . . . 6 ⊢ (𝑦 = 𝑤 → (𝐶 = (𝑧 +ℎ 𝑦) ↔ 𝐶 = (𝑧 +ℎ 𝑤)))
27	26	cbvrexvw 3236	. . . . 5 ⊢ (∃𝑦 ∈ 𝐵 𝐶 = (𝑧 +ℎ 𝑦) ↔ ∃𝑤 ∈ 𝐵 𝐶 = (𝑧 +ℎ 𝑤))
28	24, 27	bitrdi 287	. . . 4 ⊢ (𝑥 = 𝑧 → (∃𝑦 ∈ 𝐵 𝐶 = (𝑥 +ℎ 𝑦) ↔ ∃𝑤 ∈ 𝐵 𝐶 = (𝑧 +ℎ 𝑤)))
29	21, 28	anbi12d 632	. . 3 ⊢ (𝑥 = 𝑧 → ((𝑥 ∈ 𝐴 ∧ ∃𝑦 ∈ 𝐵 𝐶 = (𝑥 +ℎ 𝑦)) ↔ (𝑧 ∈ 𝐴 ∧ ∃𝑤 ∈ 𝐵 𝐶 = (𝑧 +ℎ 𝑤))))
30	29	mo4 2561	. 2 ⊢ (∃*𝑥(𝑥 ∈ 𝐴 ∧ ∃𝑦 ∈ 𝐵 𝐶 = (𝑥 +ℎ 𝑦)) ↔ ∀𝑥∀𝑧(((𝑥 ∈ 𝐴 ∧ ∃𝑦 ∈ 𝐵 𝐶 = (𝑥 +ℎ 𝑦)) ∧ (𝑧 ∈ 𝐴 ∧ ∃𝑤 ∈ 𝐵 𝐶 = (𝑧 +ℎ 𝑤))) → 𝑥 = 𝑧))
31	20, 30	sylibr 233	1 ⊢ ((𝐴 ∈ Sℋ ∧ 𝐵 ∈ Sℋ ∧ (𝐴 ∩ 𝐵) = 0ℋ) → ∃*𝑥(𝑥 ∈ 𝐴 ∧ ∃𝑦 ∈ 𝐵 𝐶 = (𝑥 +ℎ 𝑦)))

Syntax hints:   → wi 4   ∧ wa 397   ∧ w3a 1088  ∀wal 1540   = wceq 1542   ∈ wcel 2107  ∃*wmo 2533  ∃wrex 3071   ∩ cin 3948  (class class class)co 7409   +_ℎ cva 30173   S_ℋ csh 30181  0_ℋc0h 30188

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725  ax-resscn 11167  ax-1cn 11168  ax-icn 11169  ax-addcl 11170  ax-addrcl 11171  ax-mulcl 11172  ax-mulrcl 11173  ax-mulcom 11174  ax-addass 11175  ax-mulass 11176  ax-distr 11177  ax-i2m1 11178  ax-1ne0 11179  ax-1rid 11180  ax-rnegex 11181  ax-rrecex 11182  ax-cnre 11183  ax-pre-lttri 11184  ax-pre-lttrn 11185  ax-pre-ltadd 11186  ax-pre-mulgt0 11187  ax-hilex 30252  ax-hfvadd 30253  ax-hvcom 30254  ax-hvass 30255  ax-hv0cl 30256  ax-hvaddid 30257  ax-hfvmul 30258  ax-hvmulid 30259  ax-hvmulass 30260  ax-hvdistr1 30261  ax-hvdistr2 30262  ax-hvmul0 30263

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-rmo 3377  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-po 5589  df-so 5590  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-riota 7365  df-ov 7412  df-oprab 7413  df-mpo 7414  df-er 8703  df-en 8940  df-dom 8941  df-sdom 8942  df-pnf 11250  df-mnf 11251  df-xr 11252  df-ltxr 11253  df-le 11254  df-sub 11446  df-neg 11447  df-div 11872  df-hvsub 30224  df-sh 30460  df-ch0 30506

This theorem is referenced by:  pjhtheu  30647  pjpreeq  30651