Theorem pjhthmo 31396

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 3210	. . . . . 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 2774	. . . . . . . . . 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 31394	. . . . . . . . 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 3194	. . . . . 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 2820	. . . 4 ⊢ (𝑥 = 𝑧 → (𝑥 ∈ 𝐴 ↔ 𝑧 ∈ 𝐴))
22		oveq1 7377	. . . . . . 7 ⊢ (𝑥 = 𝑧 → (𝑥 +ℎ 𝑦) = (𝑧 +ℎ 𝑦))
23	22	eqeq2d 2748	. . . . . 6 ⊢ (𝑥 = 𝑧 → (𝐶 = (𝑥 +ℎ 𝑦) ↔ 𝐶 = (𝑧 +ℎ 𝑦)))
24	23	rexbidv 3162	. . . . 5 ⊢ (𝑥 = 𝑧 → (∃𝑦 ∈ 𝐵 𝐶 = (𝑥 +ℎ 𝑦) ↔ ∃𝑦 ∈ 𝐵 𝐶 = (𝑧 +ℎ 𝑦)))
25		oveq2 7378	. . . . . . 7 ⊢ (𝑦 = 𝑤 → (𝑧 +ℎ 𝑦) = (𝑧 +ℎ 𝑤))
26	25	eqeq2d 2748	. . . . . 6 ⊢ (𝑦 = 𝑤 → (𝐶 = (𝑧 +ℎ 𝑦) ↔ 𝐶 = (𝑧 +ℎ 𝑤)))
27	26	cbvrexvw 3217	. . . . 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 2567	. 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 2538  ∃wrex 3062   ∩ cin 3902  (class class class)co 7370   +_ℎ cva 31014   S_ℋ csh 31022  0_ℋc0h 31029

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 2709  ax-sep 5245  ax-nul 5255  ax-pow 5314  ax-pr 5381  ax-un 7692  ax-resscn 11097  ax-1cn 11098  ax-icn 11099  ax-addcl 11100  ax-addrcl 11101  ax-mulcl 11102  ax-mulrcl 11103  ax-mulcom 11104  ax-addass 11105  ax-mulass 11106  ax-distr 11107  ax-i2m1 11108  ax-1ne0 11109  ax-1rid 11110  ax-rnegex 11111  ax-rrecex 11112  ax-cnre 11113  ax-pre-lttri 11114  ax-pre-lttrn 11115  ax-pre-ltadd 11116  ax-pre-mulgt0 11117  ax-hilex 31093  ax-hfvadd 31094  ax-hvcom 31095  ax-hvass 31096  ax-hv0cl 31097  ax-hvaddid 31098  ax-hfvmul 31099  ax-hvmulid 31100  ax-hvmulass 31101  ax-hvdistr1 31102  ax-hvdistr2 31103  ax-hvmul0 31104

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 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3063  df-rmo 3352  df-reu 3353  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-iun 4950  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5529  df-po 5542  df-so 5543  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-iota 6458  df-fun 6504  df-fn 6505  df-f 6506  df-f1 6507  df-fo 6508  df-f1o 6509  df-fv 6510  df-riota 7327  df-ov 7373  df-oprab 7374  df-mpo 7375  df-er 8647  df-en 8898  df-dom 8899  df-sdom 8900  df-pnf 11182  df-mnf 11183  df-xr 11184  df-ltxr 11185  df-le 11186  df-sub 11380  df-neg 11381  df-div 11809  df-hvsub 31065  df-sh 31301  df-ch0 31347

This theorem is referenced by:  pjhtheu  31488  pjpreeq  31492