pjhthlem2 - Hilbert Space Explorer

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Theorem pjhthlem2 31416

Description: Lemma for pjhth 31417. (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 31256	. . . . . 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 31041	. . . . 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 2734	. . . . . 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 31415	. . . . 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 3126	. . . 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 31251	. . . . 5 ⊢ 𝐻 ∈ S_ℋ
16		shocel 31306	. . . . 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 31066	. . . . 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 2740	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐻 ∧ ∀𝑧 ∈ 𝐻 (norm_ℎ‘(𝐴 −_ℎ 𝑥)) ≤ (norm_ℎ‘(𝐴 −_ℎ 𝑧)))) → 𝐴 = (𝑥 +_ℎ (𝐴 −_ℎ 𝑥)))
22		oveq2 7364	. . . 4 ⊢ (𝑦 = (𝐴 −_ℎ 𝑥) → (𝑥 +_ℎ 𝑦) = (𝑥 +_ℎ (𝐴 −_ℎ 𝑥)))
23	22	rspceeqv 3597	. . 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 30993	. . . 4 ⊢ ℋ = (BaseSet‘⟨⟨ +_ℎ , ·_ℎ ⟩, norm_ℎ⟩)
26		eqid 2734	. . . . 5 ⊢ ⟨⟨ +_ℎ , ·_ℎ ⟩, norm_ℎ⟩ = ⟨⟨ +_ℎ , ·_ℎ ⟩, norm_ℎ⟩
27	26	hhvs 31194	. . . 4 ⊢ −_ℎ = ( −_𝑣 ‘⟨⟨ +_ℎ , ·_ℎ ⟩, norm_ℎ⟩)
28	26	hhnm 31195	. . . 4 ⊢ norm_ℎ = (norm_CV‘⟨⟨ +_ℎ , ·_ℎ ⟩, norm_ℎ⟩)
29		eqid 2734	. . . . 5 ⊢ ⟨⟨( +_ℎ ↾ (𝐻 × 𝐻)), ( ·_ℎ ↾ (ℂ × 𝐻))⟩, (norm_ℎ ↾ 𝐻)⟩ = ⟨⟨( +_ℎ ↾ (𝐻 × 𝐻)), ( ·_ℎ ↾ (ℂ × 𝐻))⟩, (norm_ℎ ↾ 𝐻)⟩
30	29, 15	hhssba 31295	. . . 4 ⊢ 𝐻 = (BaseSet‘⟨⟨( +_ℎ ↾ (𝐻 × 𝐻)), ( ·_ℎ ↾ (ℂ × 𝐻))⟩, (norm_ℎ ↾ 𝐻)⟩)
31	26	hhph 31202	. . . . 5 ⊢ ⟨⟨ +_ℎ , ·_ℎ ⟩, norm_ℎ⟩ ∈ CPreHil_OLD
32	31	a1i 11	. . . 4 ⊢ (𝜑 → ⟨⟨ +_ℎ , ·_ℎ ⟩, norm_ℎ⟩ ∈ CPreHil_OLD)
33	26, 29	hhsst 31290	. . . . . . 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 31303	. . . . . 6 ⊢ ⟨⟨( +_ℎ ↾ (𝐻 × 𝐻)), ( ·_ℎ ↾ (ℂ × 𝐻))⟩, (norm_ℎ ↾ 𝐻)⟩ ∈ CBan
36		elin 3915	. . . . . 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 30908	. . 3 ⊢ (𝜑 → ∃!𝑥 ∈ 𝐻 ∀𝑧 ∈ 𝐻 (norm_ℎ‘(𝐴 −_ℎ 𝑥)) ≤ (norm_ℎ‘(𝐴 −_ℎ 𝑧)))
40		reurex 3352	. . 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 3150	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 1541 ∈ wcel 2113 ∀wral 3049 ∃wrex 3058 ∃!wreu 3346 ∩ cin 3898 ⟨cop 4584 class class class wbr 5096 × cxp 5620 ↾ cres 5624 ‘cfv 6490 (class class class)co 7356 ℂcc 11022 0cc0 11024 1c1 11025 + caddc 11027 ≤ cle 11165 / cdiv 11792 SubSpcss 30745 CPreHil_OLDccphlo 30836 CBanccbn 30886 ℋchba 30943 +_ℎ cva 30944 ·_ℎ csm 30945 ·_ih csp 30946 norm_ℎcno 30947 −_ℎ cmv 30949 S_ℋ csh 30952 C_ℋ cch 30953 ⊥cort 30954

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2182 ax-ext 2706 ax-rep 5222 ax-sep 5239 ax-nul 5249 ax-pow 5308 ax-pr 5375 ax-un 7678 ax-inf2 9548 ax-cc 10343 ax-cnex 11080 ax-resscn 11081 ax-1cn 11082 ax-icn 11083 ax-addcl 11084 ax-addrcl 11085 ax-mulcl 11086 ax-mulrcl 11087 ax-mulcom 11088 ax-addass 11089 ax-mulass 11090 ax-distr 11091 ax-i2m1 11092 ax-1ne0 11093 ax-1rid 11094 ax-rnegex 11095 ax-rrecex 11096 ax-cnre 11097 ax-pre-lttri 11098 ax-pre-lttrn 11099 ax-pre-ltadd 11100 ax-pre-mulgt0 11101 ax-pre-sup 11102 ax-addf 11103 ax-mulf 11104 ax-hilex 31023 ax-hfvadd 31024 ax-hvcom 31025 ax-hvass 31026 ax-hv0cl 31027 ax-hvaddid 31028 ax-hfvmul 31029 ax-hvmulid 31030 ax-hvmulass 31031 ax-hvdistr1 31032 ax-hvdistr2 31033 ax-hvmul0 31034 ax-hfi 31103 ax-his1 31106 ax-his2 31107 ax-his3 31108 ax-his4 31109 ax-hcompl 31226

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2537 df-eu 2567 df-clab 2713 df-cleq 2726 df-clel 2809 df-nfc 2883 df-ne 2931 df-nel 3035 df-ral 3050 df-rex 3059 df-rmo 3348 df-reu 3349 df-rab 3398 df-v 3440 df-sbc 3739 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-nul 4284 df-if 4478 df-pw 4554 df-sn 4579 df-pr 4581 df-op 4585 df-uni 4862 df-int 4901 df-iun 4946 df-iin 4947 df-br 5097 df-opab 5159 df-mpt 5178 df-tr 5204 df-id 5517 df-eprel 5522 df-po 5530 df-so 5531 df-fr 5575 df-se 5576 df-we 5577 df-xp 5628 df-rel 5629 df-cnv 5630 df-co 5631 df-dm 5632 df-rn 5633 df-res 5634 df-ima 5635 df-pred 6257 df-ord 6318 df-on 6319 df-lim 6320 df-suc 6321 df-iota 6446 df-fun 6492 df-fn 6493 df-f 6494 df-f1 6495 df-fo 6496 df-f1o 6497 df-fv 6498 df-isom 6499 df-riota 7313 df-ov 7359 df-oprab 7360 df-mpo 7361 df-om 7807 df-1st 7931 df-2nd 7932 df-frecs 8221 df-wrecs 8252 df-recs 8301 df-rdg 8339 df-1o 8395 df-2o 8396 df-oadd 8399 df-omul 8400 df-er 8633 df-map 8763 df-pm 8764 df-en 8882 df-dom 8883 df-sdom 8884 df-fin 8885 df-fi 9312 df-sup 9343 df-inf 9344 df-oi 9413 df-card 9849 df-acn 9852 df-pnf 11166 df-mnf 11167 df-xr 11168 df-ltxr 11169 df-le 11170 df-sub 11364 df-neg 11365 df-div 11793 df-nn 12144 df-2 12206 df-3 12207 df-4 12208 df-n0 12400 df-z 12487 df-uz 12750 df-q 12860 df-rp 12904 df-xneg 13024 df-xadd 13025 df-xmul 13026 df-ico 13265 df-icc 13266 df-fz 13422 df-fl 13710 df-seq 13923 df-exp 13983 df-cj 15020 df-re 15021 df-im 15022 df-sqrt 15156 df-abs 15157 df-clim 15409 df-rlim 15410 df-rest 17340 df-topgen 17361 df-psmet 21299 df-xmet 21300 df-met 21301 df-bl 21302 df-mopn 21303 df-fbas 21304 df-fg 21305 df-top 22836 df-topon 22853 df-bases 22888 df-cld 22961 df-ntr 22962 df-cls 22963 df-nei 23040 df-lm 23171 df-haus 23257 df-fil 23788 df-fm 23880 df-flim 23881 df-flf 23882 df-cfil 25209 df-cau 25210 df-cmet 25211 df-grpo 30517 df-gid 30518 df-ginv 30519 df-gdiv 30520 df-ablo 30569 df-vc 30583 df-nv 30616 df-va 30619 df-ba 30620 df-sm 30621 df-0v 30622 df-vs 30623 df-nmcv 30624 df-ims 30625 df-ssp 30746 df-ph 30837 df-cbn 30887 df-hnorm 30992 df-hba 30993 df-hvsub 30995 df-hlim 30996 df-hcau 30997 df-sh 31231 df-ch 31245 df-oc 31276 df-ch0 31277

This theorem is referenced by: pjhth 31417 omlsii 31427

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