pjhthlem2 - Hilbert Space Explorer

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Theorem pjhthlem2 28251

Description: Lemma for pjhth 28252. (Contributed by NM, 10-Oct-1999.) (Revised by Mario Carneiro, 15-May-2014.) (New usage is discouraged.)

**Hypotheses**
Ref	Expression
pjhth.1	⊢ 𝐻 ∈ C_ℋ
pjhth.2	⊢ (𝜑 → 𝐴 ∈ ℋ)

**Assertion**
Ref	Expression
pjhthlem2	⊢ (𝜑 → ∃𝑥 ∈ 𝐻 ∃𝑦 ∈ (⊥‘𝐻)𝐴 = (𝑥 +_ℎ 𝑦))

Distinct variable groups: 𝑥,𝑦,𝐴 𝑥,𝐻,𝑦 𝜑,𝑥,𝑦

Proof of Theorem pjhthlem2 Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		pjhth.2	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ ℋ)
2	1	adantr 481	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐻 ∧ ∀𝑧 ∈ 𝐻 (norm_ℎ‘(𝐴 −_ℎ 𝑥)) ≤ (norm_ℎ‘(𝐴 −_ℎ 𝑧)))) → 𝐴 ∈ ℋ)
3		pjhth.1	. . . . . . 7 ⊢ 𝐻 ∈ C_ℋ
4	3	cheli 28089	. . . . . 6 ⊢ (𝑥 ∈ 𝐻 → 𝑥 ∈ ℋ)
5	4	ad2antrl 764	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐻 ∧ ∀𝑧 ∈ 𝐻 (norm_ℎ‘(𝐴 −_ℎ 𝑥)) ≤ (norm_ℎ‘(𝐴 −_ℎ 𝑧)))) → 𝑥 ∈ ℋ)
6		hvsubcl 27874	. . . . 5 ⊢ ((𝐴 ∈ ℋ ∧ 𝑥 ∈ ℋ) → (𝐴 −_ℎ 𝑥) ∈ ℋ)
7	2, 5, 6	syl2anc 693	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐻 ∧ ∀𝑧 ∈ 𝐻 (norm_ℎ‘(𝐴 −_ℎ 𝑥)) ≤ (norm_ℎ‘(𝐴 −_ℎ 𝑧)))) → (𝐴 −_ℎ 𝑥) ∈ ℋ)
8	2	adantr 481	. . . . . 6 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐻 ∧ ∀𝑧 ∈ 𝐻 (norm_ℎ‘(𝐴 −_ℎ 𝑥)) ≤ (norm_ℎ‘(𝐴 −_ℎ 𝑧)))) ∧ 𝑦 ∈ 𝐻) → 𝐴 ∈ ℋ)
9		simplrl 800	. . . . . 6 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐻 ∧ ∀𝑧 ∈ 𝐻 (norm_ℎ‘(𝐴 −_ℎ 𝑥)) ≤ (norm_ℎ‘(𝐴 −_ℎ 𝑧)))) ∧ 𝑦 ∈ 𝐻) → 𝑥 ∈ 𝐻)
10		simpr 477	. . . . . 6 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐻 ∧ ∀𝑧 ∈ 𝐻 (norm_ℎ‘(𝐴 −_ℎ 𝑥)) ≤ (norm_ℎ‘(𝐴 −_ℎ 𝑧)))) ∧ 𝑦 ∈ 𝐻) → 𝑦 ∈ 𝐻)
11		simplrr 801	. . . . . 6 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐻 ∧ ∀𝑧 ∈ 𝐻 (norm_ℎ‘(𝐴 −_ℎ 𝑥)) ≤ (norm_ℎ‘(𝐴 −_ℎ 𝑧)))) ∧ 𝑦 ∈ 𝐻) → ∀𝑧 ∈ 𝐻 (norm_ℎ‘(𝐴 −_ℎ 𝑥)) ≤ (norm_ℎ‘(𝐴 −_ℎ 𝑧)))
12		eqid 2622	. . . . . 6 ⊢ (((𝐴 −_ℎ 𝑥) ·_ih 𝑦) / ((𝑦 ·_ih 𝑦) + 1)) = (((𝐴 −_ℎ 𝑥) ·_ih 𝑦) / ((𝑦 ·_ih 𝑦) + 1))
13	3, 8, 9, 10, 11, 12	pjhthlem1 28250	. . . . 5 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐻 ∧ ∀𝑧 ∈ 𝐻 (norm_ℎ‘(𝐴 −_ℎ 𝑥)) ≤ (norm_ℎ‘(𝐴 −_ℎ 𝑧)))) ∧ 𝑦 ∈ 𝐻) → ((𝐴 −_ℎ 𝑥) ·_ih 𝑦) = 0)
14	13	ralrimiva 2966	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐻 ∧ ∀𝑧 ∈ 𝐻 (norm_ℎ‘(𝐴 −_ℎ 𝑥)) ≤ (norm_ℎ‘(𝐴 −_ℎ 𝑧)))) → ∀𝑦 ∈ 𝐻 ((𝐴 −_ℎ 𝑥) ·_ih 𝑦) = 0)
15	3	chshii 28084	. . . . 5 ⊢ 𝐻 ∈ S_ℋ
16		shocel 28141	. . . . 5 ⊢ (𝐻 ∈ S_ℋ → ((𝐴 −_ℎ 𝑥) ∈ (⊥‘𝐻) ↔ ((𝐴 −_ℎ 𝑥) ∈ ℋ ∧ ∀𝑦 ∈ 𝐻 ((𝐴 −_ℎ 𝑥) ·_ih 𝑦) = 0)))
17	15, 16	ax-mp 5	. . . 4 ⊢ ((𝐴 −_ℎ 𝑥) ∈ (⊥‘𝐻) ↔ ((𝐴 −_ℎ 𝑥) ∈ ℋ ∧ ∀𝑦 ∈ 𝐻 ((𝐴 −_ℎ 𝑥) ·_ih 𝑦) = 0))
18	7, 14, 17	sylanbrc 698	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐻 ∧ ∀𝑧 ∈ 𝐻 (norm_ℎ‘(𝐴 −_ℎ 𝑥)) ≤ (norm_ℎ‘(𝐴 −_ℎ 𝑧)))) → (𝐴 −_ℎ 𝑥) ∈ (⊥‘𝐻))
19		hvpncan3 27899	. . . . 5 ⊢ ((𝑥 ∈ ℋ ∧ 𝐴 ∈ ℋ) → (𝑥 +_ℎ (𝐴 −_ℎ 𝑥)) = 𝐴)
20	5, 2, 19	syl2anc 693	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐻 ∧ ∀𝑧 ∈ 𝐻 (norm_ℎ‘(𝐴 −_ℎ 𝑥)) ≤ (norm_ℎ‘(𝐴 −_ℎ 𝑧)))) → (𝑥 +_ℎ (𝐴 −_ℎ 𝑥)) = 𝐴)
21	20	eqcomd 2628	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐻 ∧ ∀𝑧 ∈ 𝐻 (norm_ℎ‘(𝐴 −_ℎ 𝑥)) ≤ (norm_ℎ‘(𝐴 −_ℎ 𝑧)))) → 𝐴 = (𝑥 +_ℎ (𝐴 −_ℎ 𝑥)))
22		oveq2 6658	. . . . 5 ⊢ (𝑦 = (𝐴 −_ℎ 𝑥) → (𝑥 +_ℎ 𝑦) = (𝑥 +_ℎ (𝐴 −_ℎ 𝑥)))
23	22	eqeq2d 2632	. . . 4 ⊢ (𝑦 = (𝐴 −_ℎ 𝑥) → (𝐴 = (𝑥 +_ℎ 𝑦) ↔ 𝐴 = (𝑥 +_ℎ (𝐴 −_ℎ 𝑥))))
24	23	rspcev 3309	. . 3 ⊢ (((𝐴 −_ℎ 𝑥) ∈ (⊥‘𝐻) ∧ 𝐴 = (𝑥 +_ℎ (𝐴 −_ℎ 𝑥))) → ∃𝑦 ∈ (⊥‘𝐻)𝐴 = (𝑥 +_ℎ 𝑦))
25	18, 21, 24	syl2anc 693	. 2 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐻 ∧ ∀𝑧 ∈ 𝐻 (norm_ℎ‘(𝐴 −_ℎ 𝑥)) ≤ (norm_ℎ‘(𝐴 −_ℎ 𝑧)))) → ∃𝑦 ∈ (⊥‘𝐻)𝐴 = (𝑥 +_ℎ 𝑦))
26		df-hba 27826	. . . 4 ⊢ ℋ = (BaseSet‘⟨⟨ +_ℎ , ·_ℎ ⟩, norm_ℎ⟩)
27		eqid 2622	. . . . 5 ⊢ ⟨⟨ +_ℎ , ·_ℎ ⟩, norm_ℎ⟩ = ⟨⟨ +_ℎ , ·_ℎ ⟩, norm_ℎ⟩
28	27	hhvs 28027	. . . 4 ⊢ −_ℎ = ( −_𝑣 ‘⟨⟨ +_ℎ , ·_ℎ ⟩, norm_ℎ⟩)
29	27	hhnm 28028	. . . 4 ⊢ norm_ℎ = (norm_CV‘⟨⟨ +_ℎ , ·_ℎ ⟩, norm_ℎ⟩)
30		eqid 2622	. . . . 5 ⊢ ⟨⟨( +_ℎ ↾ (𝐻 × 𝐻)), ( ·_ℎ ↾ (ℂ × 𝐻))⟩, (norm_ℎ ↾ 𝐻)⟩ = ⟨⟨( +_ℎ ↾ (𝐻 × 𝐻)), ( ·_ℎ ↾ (ℂ × 𝐻))⟩, (norm_ℎ ↾ 𝐻)⟩
31	30, 15	hhssba 28128	. . . 4 ⊢ 𝐻 = (BaseSet‘⟨⟨( +_ℎ ↾ (𝐻 × 𝐻)), ( ·_ℎ ↾ (ℂ × 𝐻))⟩, (norm_ℎ ↾ 𝐻)⟩)
32	27	hhph 28035	. . . . 5 ⊢ ⟨⟨ +_ℎ , ·_ℎ ⟩, norm_ℎ⟩ ∈ CPreHil_OLD
33	32	a1i 11	. . . 4 ⊢ (𝜑 → ⟨⟨ +_ℎ , ·_ℎ ⟩, norm_ℎ⟩ ∈ CPreHil_OLD)
34	27, 30	hhsst 28123	. . . . . . 7 ⊢ (𝐻 ∈ S_ℋ → ⟨⟨( +_ℎ ↾ (𝐻 × 𝐻)), ( ·_ℎ ↾ (ℂ × 𝐻))⟩, (norm_ℎ ↾ 𝐻)⟩ ∈ (SubSp‘⟨⟨ +_ℎ , ·_ℎ ⟩, norm_ℎ⟩))
35	15, 34	ax-mp 5	. . . . . 6 ⊢ ⟨⟨( +_ℎ ↾ (𝐻 × 𝐻)), ( ·_ℎ ↾ (ℂ × 𝐻))⟩, (norm_ℎ ↾ 𝐻)⟩ ∈ (SubSp‘⟨⟨ +_ℎ , ·_ℎ ⟩, norm_ℎ⟩)
36	30, 3	hhssbn 28137	. . . . . 6 ⊢ ⟨⟨( +_ℎ ↾ (𝐻 × 𝐻)), ( ·_ℎ ↾ (ℂ × 𝐻))⟩, (norm_ℎ ↾ 𝐻)⟩ ∈ CBan
37		elin 3796	. . . . . 6 ⊢ (⟨⟨( +_ℎ ↾ (𝐻 × 𝐻)), ( ·_ℎ ↾ (ℂ × 𝐻))⟩, (norm_ℎ ↾ 𝐻)⟩ ∈ ((SubSp‘⟨⟨ +_ℎ , ·_ℎ ⟩, norm_ℎ⟩) ∩ CBan) ↔ (⟨⟨( +_ℎ ↾ (𝐻 × 𝐻)), ( ·_ℎ ↾ (ℂ × 𝐻))⟩, (norm_ℎ ↾ 𝐻)⟩ ∈ (SubSp‘⟨⟨ +_ℎ , ·_ℎ ⟩, norm_ℎ⟩) ∧ ⟨⟨( +_ℎ ↾ (𝐻 × 𝐻)), ( ·_ℎ ↾ (ℂ × 𝐻))⟩, (norm_ℎ ↾ 𝐻)⟩ ∈ CBan))
38	35, 36, 37	mpbir2an 955	. . . . 5 ⊢ ⟨⟨( +_ℎ ↾ (𝐻 × 𝐻)), ( ·_ℎ ↾ (ℂ × 𝐻))⟩, (norm_ℎ ↾ 𝐻)⟩ ∈ ((SubSp‘⟨⟨ +_ℎ , ·_ℎ ⟩, norm_ℎ⟩) ∩ CBan)
39	38	a1i 11	. . . 4 ⊢ (𝜑 → ⟨⟨( +_ℎ ↾ (𝐻 × 𝐻)), ( ·_ℎ ↾ (ℂ × 𝐻))⟩, (norm_ℎ ↾ 𝐻)⟩ ∈ ((SubSp‘⟨⟨ +_ℎ , ·_ℎ ⟩, norm_ℎ⟩) ∩ CBan))
40	26, 28, 29, 31, 33, 39, 1	minveco 27740	. . 3 ⊢ (𝜑 → ∃!𝑥 ∈ 𝐻 ∀𝑧 ∈ 𝐻 (norm_ℎ‘(𝐴 −_ℎ 𝑥)) ≤ (norm_ℎ‘(𝐴 −_ℎ 𝑧)))
41		reurex 3160	. . 3 ⊢ (∃!𝑥 ∈ 𝐻 ∀𝑧 ∈ 𝐻 (norm_ℎ‘(𝐴 −_ℎ 𝑥)) ≤ (norm_ℎ‘(𝐴 −_ℎ 𝑧)) → ∃𝑥 ∈ 𝐻 ∀𝑧 ∈ 𝐻 (norm_ℎ‘(𝐴 −_ℎ 𝑥)) ≤ (norm_ℎ‘(𝐴 −_ℎ 𝑧)))
42	40, 41	syl 17	. 2 ⊢ (𝜑 → ∃𝑥 ∈ 𝐻 ∀𝑧 ∈ 𝐻 (norm_ℎ‘(𝐴 −_ℎ 𝑥)) ≤ (norm_ℎ‘(𝐴 −_ℎ 𝑧)))
43	25, 42	reximddv 3018	1 ⊢ (𝜑 → ∃𝑥 ∈ 𝐻 ∃𝑦 ∈ (⊥‘𝐻)𝐴 = (𝑥 +_ℎ 𝑦))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 196 ∧ wa 384 = wceq 1483 ∈ wcel 1990 ∀wral 2912 ∃wrex 2913 ∃!wreu 2914 ∩ cin 3573 ⟨cop 4183 class class class wbr 4653 × cxp 5112 ↾ cres 5116 ‘cfv 5888 (class class class)co 6650 ℂcc 9934 0cc0 9936 1c1 9937 + caddc 9939 ≤ cle 10075 / cdiv 10684 SubSpcss 27576 CPreHil_OLDccphlo 27667 CBanccbn 27718 ℋchil 27776 +_ℎ cva 27777 ·_ℎ csm 27778 ·_ih csp 27779 norm_ℎcno 27780 −_ℎ cmv 27782 S_ℋ csh 27785 C_ℋ cch 27786 ⊥cort 27787

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-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-inf2 8538 ax-cc 9257 ax-cnex 9992 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-pre-sup 10014 ax-addf 10015 ax-mulf 10016 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 ax-hfi 27936 ax-his1 27939 ax-his2 27940 ax-his3 27941 ax-his4 27942 ax-hcompl 28059

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-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-iin 4523 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-se 5074 df-we 5075 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-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 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-isom 5897 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-oadd 7564 df-omul 7565 df-er 7742 df-map 7859 df-pm 7860 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-fi 8317 df-sup 8348 df-inf 8349 df-oi 8415 df-card 8765 df-acn 8768 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-nn 11021 df-2 11079 df-3 11080 df-4 11081 df-n0 11293 df-z 11378 df-uz 11688 df-q 11789 df-rp 11833 df-xneg 11946 df-xadd 11947 df-xmul 11948 df-ico 12181 df-icc 12182 df-fz 12327 df-fl 12593 df-seq 12802 df-exp 12861 df-cj 13839 df-re 13840 df-im 13841 df-sqrt 13975 df-abs 13976 df-clim 14219 df-rlim 14220 df-rest 16083 df-topgen 16104 df-psmet 19738 df-xmet 19739 df-met 19740 df-bl 19741 df-mopn 19742 df-fbas 19743 df-fg 19744 df-top 20699 df-topon 20716 df-bases 20750 df-cld 20823 df-ntr 20824 df-cls 20825 df-nei 20902 df-lm 21033 df-haus 21119 df-fil 21650 df-fm 21742 df-flim 21743 df-flf 21744 df-cfil 23053 df-cau 23054 df-cmet 23055 df-grpo 27347 df-gid 27348 df-ginv 27349 df-gdiv 27350 df-ablo 27399 df-vc 27414 df-nv 27447 df-va 27450 df-ba 27451 df-sm 27452 df-0v 27453 df-vs 27454 df-nmcv 27455 df-ims 27456 df-ssp 27577 df-ph 27668 df-cbn 27719 df-hnorm 27825 df-hba 27826 df-hvsub 27828 df-hlim 27829 df-hcau 27830 df-sh 28064 df-ch 28078 df-oc 28109 df-ch0 28110

This theorem is referenced by: pjhth 28252 omlsii 28262

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