Theorem pjhthmo 31373

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 657	. . . 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 3209	. . . . . 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 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)))) → 𝐴 ∈ Sℋ )
4		simpll2 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)))) → 𝐵 ∈ Sℋ )
5		simpll3 1216	. . . . . . . . . 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 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)))) → 𝑥 ∈ 𝐴)
7		simprll 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)))) → 𝑦 ∈ 𝐵)
8		simplrr 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)))) → 𝑧 ∈ 𝐴)
9		simprlr 780	. . . . . . . . . 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 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 = (𝑥 +ℎ 𝑦))
11		simprrr 782	. . . . . . . . . . 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 2773	. . . . . . . . . 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 31371	. . . . . . . . 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 494	. . . . . . . 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 420	. . . . . . 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 3193	. . . . . 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 243	. . . . 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 453	. . . 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 243	. . 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 1930	. 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 2819	. . . 4 ⊢ (𝑥 = 𝑧 → (𝑥 ∈ 𝐴 ↔ 𝑧 ∈ 𝐴))
22		oveq1 7374	. . . . . . 7 ⊢ (𝑥 = 𝑧 → (𝑥 +ℎ 𝑦) = (𝑧 +ℎ 𝑦))
23	22	eqeq2d 2747	. . . . . 6 ⊢ (𝑥 = 𝑧 → (𝐶 = (𝑥 +ℎ 𝑦) ↔ 𝐶 = (𝑧 +ℎ 𝑦)))
24	23	rexbidv 3161	. . . . 5 ⊢ (𝑥 = 𝑧 → (∃𝑦 ∈ 𝐵 𝐶 = (𝑥 +ℎ 𝑦) ↔ ∃𝑦 ∈ 𝐵 𝐶 = (𝑧 +ℎ 𝑦)))
25		oveq2 7375	. . . . . . 7 ⊢ (𝑦 = 𝑤 → (𝑧 +ℎ 𝑦) = (𝑧 +ℎ 𝑤))
26	25	eqeq2d 2747	. . . . . 6 ⊢ (𝑦 = 𝑤 → (𝐶 = (𝑧 +ℎ 𝑦) ↔ 𝐶 = (𝑧 +ℎ 𝑤)))
27	26	cbvrexvw 3216	. . . . 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 633	. . 3 ⊢ (𝑥 = 𝑧 → ((𝑥 ∈ 𝐴 ∧ ∃𝑦 ∈ 𝐵 𝐶 = (𝑥 +ℎ 𝑦)) ↔ (𝑧 ∈ 𝐴 ∧ ∃𝑤 ∈ 𝐵 𝐶 = (𝑧 +ℎ 𝑤))))
30	29	mo4 2566	. 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 234	1 ⊢ ((𝐴 ∈ Sℋ ∧ 𝐵 ∈ Sℋ ∧ (𝐴 ∩ 𝐵) = 0ℋ) → ∃*𝑥(𝑥 ∈ 𝐴 ∧ ∃𝑦 ∈ 𝐵 𝐶 = (𝑥 +ℎ 𝑦)))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1087  ∀wal 1540   = wceq 1542   ∈ wcel 2114  ∃*wmo 2537  ∃wrex 3061   ∩ cin 3888  (class class class)co 7367   +_ℎ cva 30991   S_ℋ csh 30999  0_ℋc0h 31006

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-sep 5231  ax-nul 5241  ax-pow 5307  ax-pr 5375  ax-un 7689  ax-resscn 11095  ax-1cn 11096  ax-icn 11097  ax-addcl 11098  ax-addrcl 11099  ax-mulcl 11100  ax-mulrcl 11101  ax-mulcom 11102  ax-addass 11103  ax-mulass 11104  ax-distr 11105  ax-i2m1 11106  ax-1ne0 11107  ax-1rid 11108  ax-rnegex 11109  ax-rrecex 11110  ax-cnre 11111  ax-pre-lttri 11112  ax-pre-lttrn 11113  ax-pre-ltadd 11114  ax-pre-mulgt0 11115  ax-hilex 31070  ax-hfvadd 31071  ax-hvcom 31072  ax-hvass 31073  ax-hv0cl 31074  ax-hvaddid 31075  ax-hfvmul 31076  ax-hvmulid 31077  ax-hvmulass 31078  ax-hvdistr1 31079  ax-hvdistr2 31080  ax-hvmul0 31081

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-nel 3037  df-ral 3052  df-rex 3062  df-rmo 3342  df-reu 3343  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-iun 4935  df-br 5086  df-opab 5148  df-mpt 5167  df-id 5526  df-po 5539  df-so 5540  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-f1 6503  df-fo 6504  df-f1o 6505  df-fv 6506  df-riota 7324  df-ov 7370  df-oprab 7371  df-mpo 7372  df-er 8643  df-en 8894  df-dom 8895  df-sdom 8896  df-pnf 11181  df-mnf 11182  df-xr 11183  df-ltxr 11184  df-le 11185  df-sub 11379  df-neg 11380  df-div 11808  df-hvsub 31042  df-sh 31278  df-ch0 31324

This theorem is referenced by:  pjhtheu  31465  pjpreeq  31469