Theorem pjhthmo 28161

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 865	. . . 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 3107	. . . . . 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 1100	. . . . . . . . . 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 1101	. . . . . . . . . 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 1102	. . . . . . . . . 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 800	. . . . . . . . . 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 802	. . . . . . . . . 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 801	. . . . . . . . . 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 803	. . . . . . . . . 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 804	. . . . . . . . . . 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 805	. . . . . . . . . . 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 2658	. . . . . . . . . 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 28159	. . . . . . . . 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 475	. . . . . . . 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 631	. . . . . . 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 3037	. . . . . 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	syl5bir 233	. . . . 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 629	. . . 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	syl5bir 233	. . 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 1856	. 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		eleq1 2689	. . . 4 ⊢ (𝑥 = 𝑧 → (𝑥 ∈ 𝐴 ↔ 𝑧 ∈ 𝐴))
22		oveq1 6657	. . . . . . 7 ⊢ (𝑥 = 𝑧 → (𝑥 +ℎ 𝑦) = (𝑧 +ℎ 𝑦))
23	22	eqeq2d 2632	. . . . . 6 ⊢ (𝑥 = 𝑧 → (𝐶 = (𝑥 +ℎ 𝑦) ↔ 𝐶 = (𝑧 +ℎ 𝑦)))
24	23	rexbidv 3052	. . . . 5 ⊢ (𝑥 = 𝑧 → (∃𝑦 ∈ 𝐵 𝐶 = (𝑥 +ℎ 𝑦) ↔ ∃𝑦 ∈ 𝐵 𝐶 = (𝑧 +ℎ 𝑦)))
25		oveq2 6658	. . . . . . 7 ⊢ (𝑦 = 𝑤 → (𝑧 +ℎ 𝑦) = (𝑧 +ℎ 𝑤))
26	25	eqeq2d 2632	. . . . . 6 ⊢ (𝑦 = 𝑤 → (𝐶 = (𝑧 +ℎ 𝑦) ↔ 𝐶 = (𝑧 +ℎ 𝑤)))
27	26	cbvrexv 3172	. . . . 5 ⊢ (∃𝑦 ∈ 𝐵 𝐶 = (𝑧 +ℎ 𝑦) ↔ ∃𝑤 ∈ 𝐵 𝐶 = (𝑧 +ℎ 𝑤))
28	24, 27	syl6bb 276	. . . 4 ⊢ (𝑥 = 𝑧 → (∃𝑦 ∈ 𝐵 𝐶 = (𝑥 +ℎ 𝑦) ↔ ∃𝑤 ∈ 𝐵 𝐶 = (𝑧 +ℎ 𝑤)))
29	21, 28	anbi12d 747	. . 3 ⊢ (𝑥 = 𝑧 → ((𝑥 ∈ 𝐴 ∧ ∃𝑦 ∈ 𝐵 𝐶 = (𝑥 +ℎ 𝑦)) ↔ (𝑧 ∈ 𝐴 ∧ ∃𝑤 ∈ 𝐵 𝐶 = (𝑧 +ℎ 𝑤))))
30	29	mo4 2517	. 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 224	1 ⊢ ((𝐴 ∈ Sℋ ∧ 𝐵 ∈ Sℋ ∧ (𝐴 ∩ 𝐵) = 0ℋ) → ∃*𝑥(𝑥 ∈ 𝐴 ∧ ∃𝑦 ∈ 𝐵 𝐶 = (𝑥 +ℎ 𝑦)))

Syntax hints:   → wi 4   ∧ wa 384   ∧ w3a 1037  ∀wal 1481   = wceq 1483   ∈ wcel 1990  ∃*wmo 2471  ∃wrex 2913   ∩ cin 3573  (class class class)co 6650   +_ℎ cva 27777   S_ℋ csh 27785  0_ℋc0h 27792

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-hilex 27856  ax-hfvadd 27857  ax-hvcom 27858  ax-hvass 27859  ax-hv0cl 27860  ax-hvaddid 27861  ax-hfvmul 27862  ax-hvmulid 27863  ax-hvmulass 27864  ax-hvdistr1 27865  ax-hvdistr2 27866  ax-hvmul0 27867

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-po 5035  df-so 5036  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-hvsub 27828  df-sh 28064  df-ch0 28110

This theorem is referenced by:  pjhtheu  28253  pjpreeq  28257