pjhthlem2 - Hilbert Space Explorer

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Theorem pjhthlem2 31321

Description: Lemma for pjhth 31322. (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 480	. . . . 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 31161	. . . . . 6 ⊢ (𝑥 ∈ 𝐻 → 𝑥 ∈ ℋ)
5	4	ad2antrl 728	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐻 ∧ ∀𝑧 ∈ 𝐻 (norm_ℎ‘(𝐴 −_ℎ 𝑥)) ≤ (norm_ℎ‘(𝐴 −_ℎ 𝑧)))) → 𝑥 ∈ ℋ)
6		hvsubcl 30946	. . . . 5 ⊢ ((𝐴 ∈ ℋ ∧ 𝑥 ∈ ℋ) → (𝐴 −_ℎ 𝑥) ∈ ℋ)
7	2, 5, 6	syl2anc 584	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐻 ∧ ∀𝑧 ∈ 𝐻 (norm_ℎ‘(𝐴 −_ℎ 𝑥)) ≤ (norm_ℎ‘(𝐴 −_ℎ 𝑧)))) → (𝐴 −_ℎ 𝑥) ∈ ℋ)
8	2	adantr 480	. . . . . 6 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐻 ∧ ∀𝑧 ∈ 𝐻 (norm_ℎ‘(𝐴 −_ℎ 𝑥)) ≤ (norm_ℎ‘(𝐴 −_ℎ 𝑧)))) ∧ 𝑦 ∈ 𝐻) → 𝐴 ∈ ℋ)
9		simplrl 776	. . . . . 6 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐻 ∧ ∀𝑧 ∈ 𝐻 (norm_ℎ‘(𝐴 −_ℎ 𝑥)) ≤ (norm_ℎ‘(𝐴 −_ℎ 𝑧)))) ∧ 𝑦 ∈ 𝐻) → 𝑥 ∈ 𝐻)
10		simpr 484	. . . . . 6 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐻 ∧ ∀𝑧 ∈ 𝐻 (norm_ℎ‘(𝐴 −_ℎ 𝑥)) ≤ (norm_ℎ‘(𝐴 −_ℎ 𝑧)))) ∧ 𝑦 ∈ 𝐻) → 𝑦 ∈ 𝐻)
11		simplrr 777	. . . . . 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 2729	. . . . . 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 31320	. . . . 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 3125	. . . 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 31156	. . . . 5 ⊢ 𝐻 ∈ S_ℋ
16		shocel 31211	. . . . 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 583	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐻 ∧ ∀𝑧 ∈ 𝐻 (norm_ℎ‘(𝐴 −_ℎ 𝑥)) ≤ (norm_ℎ‘(𝐴 −_ℎ 𝑧)))) → (𝐴 −_ℎ 𝑥) ∈ (⊥‘𝐻))
19		hvpncan3 30971	. . . . 5 ⊢ ((𝑥 ∈ ℋ ∧ 𝐴 ∈ ℋ) → (𝑥 +_ℎ (𝐴 −_ℎ 𝑥)) = 𝐴)
20	5, 2, 19	syl2anc 584	. . . 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 2735	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐻 ∧ ∀𝑧 ∈ 𝐻 (norm_ℎ‘(𝐴 −_ℎ 𝑥)) ≤ (norm_ℎ‘(𝐴 −_ℎ 𝑧)))) → 𝐴 = (𝑥 +_ℎ (𝐴 −_ℎ 𝑥)))
22		oveq2 7395	. . . 4 ⊢ (𝑦 = (𝐴 −_ℎ 𝑥) → (𝑥 +_ℎ 𝑦) = (𝑥 +_ℎ (𝐴 −_ℎ 𝑥)))
23	22	rspceeqv 3611	. . 3 ⊢ (((𝐴 −_ℎ 𝑥) ∈ (⊥‘𝐻) ∧ 𝐴 = (𝑥 +_ℎ (𝐴 −_ℎ 𝑥))) → ∃𝑦 ∈ (⊥‘𝐻)𝐴 = (𝑥 +_ℎ 𝑦))
24	18, 21, 23	syl2anc 584	. 2 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐻 ∧ ∀𝑧 ∈ 𝐻 (norm_ℎ‘(𝐴 −_ℎ 𝑥)) ≤ (norm_ℎ‘(𝐴 −_ℎ 𝑧)))) → ∃𝑦 ∈ (⊥‘𝐻)𝐴 = (𝑥 +_ℎ 𝑦))
25		df-hba 30898	. . . 4 ⊢ ℋ = (BaseSet‘⟨⟨ +_ℎ , ·_ℎ ⟩, norm_ℎ⟩)
26		eqid 2729	. . . . 5 ⊢ ⟨⟨ +_ℎ , ·_ℎ ⟩, norm_ℎ⟩ = ⟨⟨ +_ℎ , ·_ℎ ⟩, norm_ℎ⟩
27	26	hhvs 31099	. . . 4 ⊢ −_ℎ = ( −_𝑣 ‘⟨⟨ +_ℎ , ·_ℎ ⟩, norm_ℎ⟩)
28	26	hhnm 31100	. . . 4 ⊢ norm_ℎ = (norm_CV‘⟨⟨ +_ℎ , ·_ℎ ⟩, norm_ℎ⟩)
29		eqid 2729	. . . . 5 ⊢ ⟨⟨( +_ℎ ↾ (𝐻 × 𝐻)), ( ·_ℎ ↾ (ℂ × 𝐻))⟩, (norm_ℎ ↾ 𝐻)⟩ = ⟨⟨( +_ℎ ↾ (𝐻 × 𝐻)), ( ·_ℎ ↾ (ℂ × 𝐻))⟩, (norm_ℎ ↾ 𝐻)⟩
30	29, 15	hhssba 31200	. . . 4 ⊢ 𝐻 = (BaseSet‘⟨⟨( +_ℎ ↾ (𝐻 × 𝐻)), ( ·_ℎ ↾ (ℂ × 𝐻))⟩, (norm_ℎ ↾ 𝐻)⟩)
31	26	hhph 31107	. . . . 5 ⊢ ⟨⟨ +_ℎ , ·_ℎ ⟩, norm_ℎ⟩ ∈ CPreHil_OLD
32	31	a1i 11	. . . 4 ⊢ (𝜑 → ⟨⟨ +_ℎ , ·_ℎ ⟩, norm_ℎ⟩ ∈ CPreHil_OLD)
33	26, 29	hhsst 31195	. . . . . . 7 ⊢ (𝐻 ∈ S_ℋ → ⟨⟨( +_ℎ ↾ (𝐻 × 𝐻)), ( ·_ℎ ↾ (ℂ × 𝐻))⟩, (norm_ℎ ↾ 𝐻)⟩ ∈ (SubSp‘⟨⟨ +_ℎ , ·_ℎ ⟩, norm_ℎ⟩))
34	15, 33	ax-mp 5	. . . . . 6 ⊢ ⟨⟨( +_ℎ ↾ (𝐻 × 𝐻)), ( ·_ℎ ↾ (ℂ × 𝐻))⟩, (norm_ℎ ↾ 𝐻)⟩ ∈ (SubSp‘⟨⟨ +_ℎ , ·_ℎ ⟩, norm_ℎ⟩)
35	29, 3	hhssbnOLD 31208	. . . . . 6 ⊢ ⟨⟨( +_ℎ ↾ (𝐻 × 𝐻)), ( ·_ℎ ↾ (ℂ × 𝐻))⟩, (norm_ℎ ↾ 𝐻)⟩ ∈ CBan
36		elin 3930	. . . . . 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))
37	34, 35, 36	mpbir2an 711	. . . . 5 ⊢ ⟨⟨( +_ℎ ↾ (𝐻 × 𝐻)), ( ·_ℎ ↾ (ℂ × 𝐻))⟩, (norm_ℎ ↾ 𝐻)⟩ ∈ ((SubSp‘⟨⟨ +_ℎ , ·_ℎ ⟩, norm_ℎ⟩) ∩ CBan)
38	37	a1i 11	. . . 4 ⊢ (𝜑 → ⟨⟨( +_ℎ ↾ (𝐻 × 𝐻)), ( ·_ℎ ↾ (ℂ × 𝐻))⟩, (norm_ℎ ↾ 𝐻)⟩ ∈ ((SubSp‘⟨⟨ +_ℎ , ·_ℎ ⟩, norm_ℎ⟩) ∩ CBan))
39	25, 27, 28, 30, 32, 38, 1	minveco 30813	. . 3 ⊢ (𝜑 → ∃!𝑥 ∈ 𝐻 ∀𝑧 ∈ 𝐻 (norm_ℎ‘(𝐴 −_ℎ 𝑥)) ≤ (norm_ℎ‘(𝐴 −_ℎ 𝑧)))
40		reurex 3358	. . 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)))
41	39, 40	syl 17	. 2 ⊢ (𝜑 → ∃𝑥 ∈ 𝐻 ∀𝑧 ∈ 𝐻 (norm_ℎ‘(𝐴 −_ℎ 𝑥)) ≤ (norm_ℎ‘(𝐴 −_ℎ 𝑧)))
42	24, 41	reximddv 3149	1 ⊢ (𝜑 → ∃𝑥 ∈ 𝐻 ∃𝑦 ∈ (⊥‘𝐻)𝐴 = (𝑥 +_ℎ 𝑦))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1540 ∈ wcel 2109 ∀wral 3044 ∃wrex 3053 ∃!wreu 3352 ∩ cin 3913 ⟨cop 4595 class class class wbr 5107 × cxp 5636 ↾ cres 5640 ‘cfv 6511 (class class class)co 7387 ℂcc 11066 0cc0 11068 1c1 11069 + caddc 11071 ≤ cle 11209 / cdiv 11835 SubSpcss 30650 CPreHil_OLDccphlo 30741 CBanccbn 30791 ℋchba 30848 +_ℎ cva 30849 ·_ℎ csm 30850 ·_ih csp 30851 norm_ℎcno 30852 −_ℎ cmv 30854 S_ℋ csh 30857 C_ℋ cch 30858 ⊥cort 30859

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-rep 5234 ax-sep 5251 ax-nul 5261 ax-pow 5320 ax-pr 5387 ax-un 7711 ax-inf2 9594 ax-cc 10388 ax-cnex 11124 ax-resscn 11125 ax-1cn 11126 ax-icn 11127 ax-addcl 11128 ax-addrcl 11129 ax-mulcl 11130 ax-mulrcl 11131 ax-mulcom 11132 ax-addass 11133 ax-mulass 11134 ax-distr 11135 ax-i2m1 11136 ax-1ne0 11137 ax-1rid 11138 ax-rnegex 11139 ax-rrecex 11140 ax-cnre 11141 ax-pre-lttri 11142 ax-pre-lttrn 11143 ax-pre-ltadd 11144 ax-pre-mulgt0 11145 ax-pre-sup 11146 ax-addf 11147 ax-mulf 11148 ax-hilex 30928 ax-hfvadd 30929 ax-hvcom 30930 ax-hvass 30931 ax-hv0cl 30932 ax-hvaddid 30933 ax-hfvmul 30934 ax-hvmulid 30935 ax-hvmulass 30936 ax-hvdistr1 30937 ax-hvdistr2 30938 ax-hvmul0 30939 ax-hfi 31008 ax-his1 31011 ax-his2 31012 ax-his3 31013 ax-his4 31014 ax-hcompl 31131

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-rmo 3354 df-reu 3355 df-rab 3406 df-v 3449 df-sbc 3754 df-csb 3863 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-pss 3934 df-nul 4297 df-if 4489 df-pw 4565 df-sn 4590 df-pr 4592 df-op 4596 df-uni 4872 df-int 4911 df-iun 4957 df-iin 4958 df-br 5108 df-opab 5170 df-mpt 5189 df-tr 5215 df-id 5533 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5591 df-se 5592 df-we 5593 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-pred 6274 df-ord 6335 df-on 6336 df-lim 6337 df-suc 6338 df-iota 6464 df-fun 6513 df-fn 6514 df-f 6515 df-f1 6516 df-fo 6517 df-f1o 6518 df-fv 6519 df-isom 6520 df-riota 7344 df-ov 7390 df-oprab 7391 df-mpo 7392 df-om 7843 df-1st 7968 df-2nd 7969 df-frecs 8260 df-wrecs 8291 df-recs 8340 df-rdg 8378 df-1o 8434 df-2o 8435 df-oadd 8438 df-omul 8439 df-er 8671 df-map 8801 df-pm 8802 df-en 8919 df-dom 8920 df-sdom 8921 df-fin 8922 df-fi 9362 df-sup 9393 df-inf 9394 df-oi 9463 df-card 9892 df-acn 9895 df-pnf 11210 df-mnf 11211 df-xr 11212 df-ltxr 11213 df-le 11214 df-sub 11407 df-neg 11408 df-div 11836 df-nn 12187 df-2 12249 df-3 12250 df-4 12251 df-n0 12443 df-z 12530 df-uz 12794 df-q 12908 df-rp 12952 df-xneg 13072 df-xadd 13073 df-xmul 13074 df-ico 13312 df-icc 13313 df-fz 13469 df-fl 13754 df-seq 13967 df-exp 14027 df-cj 15065 df-re 15066 df-im 15067 df-sqrt 15201 df-abs 15202 df-clim 15454 df-rlim 15455 df-rest 17385 df-topgen 17406 df-psmet 21256 df-xmet 21257 df-met 21258 df-bl 21259 df-mopn 21260 df-fbas 21261 df-fg 21262 df-top 22781 df-topon 22798 df-bases 22833 df-cld 22906 df-ntr 22907 df-cls 22908 df-nei 22985 df-lm 23116 df-haus 23202 df-fil 23733 df-fm 23825 df-flim 23826 df-flf 23827 df-cfil 25155 df-cau 25156 df-cmet 25157 df-grpo 30422 df-gid 30423 df-ginv 30424 df-gdiv 30425 df-ablo 30474 df-vc 30488 df-nv 30521 df-va 30524 df-ba 30525 df-sm 30526 df-0v 30527 df-vs 30528 df-nmcv 30529 df-ims 30530 df-ssp 30651 df-ph 30742 df-cbn 30792 df-hnorm 30897 df-hba 30898 df-hvsub 30900 df-hlim 30901 df-hcau 30902 df-sh 31136 df-ch 31150 df-oc 31181 df-ch0 31182

This theorem is referenced by: pjhth 31322 omlsii 31332

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