Theorem pjhthmo 31132

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 654	. . . 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 3224	. . . . . 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 1209	. . . . . . . . . 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 1210	. . . . . . . . . 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 1211	. . . . . . . . . 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 775	. . . . . . . . . 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 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)))) → 𝑦 ∈ 𝐵)
8		simplrr 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)))) → 𝑧 ∈ 𝐴)
9		simprlr 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)))) → 𝑤 ∈ 𝐵)
10		simprrl 779	. . . . . . . . . . 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 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 = (𝑧 +ℎ 𝑤))
12	10, 11	eqtr3d 2770	. . . . . . . . . 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 31130	. . . . . . . . 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 493	. . . . . . . 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 419	. . . . . . 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 3208	. . . . . 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 452	. . . 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 1923	. 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 2812	. . . 4 ⊢ (𝑥 = 𝑧 → (𝑥 ∈ 𝐴 ↔ 𝑧 ∈ 𝐴))
22		oveq1 7433	. . . . . . 7 ⊢ (𝑥 = 𝑧 → (𝑥 +ℎ 𝑦) = (𝑧 +ℎ 𝑦))
23	22	eqeq2d 2739	. . . . . 6 ⊢ (𝑥 = 𝑧 → (𝐶 = (𝑥 +ℎ 𝑦) ↔ 𝐶 = (𝑧 +ℎ 𝑦)))
24	23	rexbidv 3176	. . . . 5 ⊢ (𝑥 = 𝑧 → (∃𝑦 ∈ 𝐵 𝐶 = (𝑥 +ℎ 𝑦) ↔ ∃𝑦 ∈ 𝐵 𝐶 = (𝑧 +ℎ 𝑦)))
25		oveq2 7434	. . . . . . 7 ⊢ (𝑦 = 𝑤 → (𝑧 +ℎ 𝑦) = (𝑧 +ℎ 𝑤))
26	25	eqeq2d 2739	. . . . . 6 ⊢ (𝑦 = 𝑤 → (𝐶 = (𝑧 +ℎ 𝑦) ↔ 𝐶 = (𝑧 +ℎ 𝑤)))
27	26	cbvrexvw 3233	. . . . 5 ⊢ (∃𝑦 ∈ 𝐵 𝐶 = (𝑧 +ℎ 𝑦) ↔ ∃𝑤 ∈ 𝐵 𝐶 = (𝑧 +ℎ 𝑤))
28	24, 27	bitrdi 286	. . . 4 ⊢ (𝑥 = 𝑧 → (∃𝑦 ∈ 𝐵 𝐶 = (𝑥 +ℎ 𝑦) ↔ ∃𝑤 ∈ 𝐵 𝐶 = (𝑧 +ℎ 𝑤)))
29	21, 28	anbi12d 630	. . 3 ⊢ (𝑥 = 𝑧 → ((𝑥 ∈ 𝐴 ∧ ∃𝑦 ∈ 𝐵 𝐶 = (𝑥 +ℎ 𝑦)) ↔ (𝑧 ∈ 𝐴 ∧ ∃𝑤 ∈ 𝐵 𝐶 = (𝑧 +ℎ 𝑤))))
30	29	mo4 2555	. 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 394   ∧ w3a 1084  ∀wal 1531   = wceq 1533   ∈ wcel 2098  ∃*wmo 2527  ∃wrex 3067   ∩ cin 3948  (class class class)co 7426   +_ℎ cva 30750   S_ℋ csh 30758  0_ℋc0h 30765

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2699  ax-sep 5303  ax-nul 5310  ax-pow 5369  ax-pr 5433  ax-un 7746  ax-resscn 11203  ax-1cn 11204  ax-icn 11205  ax-addcl 11206  ax-addrcl 11207  ax-mulcl 11208  ax-mulrcl 11209  ax-mulcom 11210  ax-addass 11211  ax-mulass 11212  ax-distr 11213  ax-i2m1 11214  ax-1ne0 11215  ax-1rid 11216  ax-rnegex 11217  ax-rrecex 11218  ax-cnre 11219  ax-pre-lttri 11220  ax-pre-lttrn 11221  ax-pre-ltadd 11222  ax-pre-mulgt0 11223  ax-hilex 30829  ax-hfvadd 30830  ax-hvcom 30831  ax-hvass 30832  ax-hv0cl 30833  ax-hvaddid 30834  ax-hfvmul 30835  ax-hvmulid 30836  ax-hvmulass 30837  ax-hvdistr1 30838  ax-hvdistr2 30839  ax-hvmul0 30840

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-nel 3044  df-ral 3059  df-rex 3068  df-rmo 3374  df-reu 3375  df-rab 3431  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-iun 5002  df-br 5153  df-opab 5215  df-mpt 5236  df-id 5580  df-po 5594  df-so 5595  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-f1 6558  df-fo 6559  df-f1o 6560  df-fv 6561  df-riota 7382  df-ov 7429  df-oprab 7430  df-mpo 7431  df-er 8731  df-en 8971  df-dom 8972  df-sdom 8973  df-pnf 11288  df-mnf 11289  df-xr 11290  df-ltxr 11291  df-le 11292  df-sub 11484  df-neg 11485  df-div 11910  df-hvsub 30801  df-sh 31037  df-ch0 31083

This theorem is referenced by:  pjhtheu  31224  pjpreeq  31228