pjhthlem2 - Hilbert Space Explorer

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Theorem pjhthlem2 31481

Description: Lemma for pjhth 31482. (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 31321	. . . . . 6 ⊢ (𝑥 ∈ 𝐻 → 𝑥 ∈ ℋ)
5	4	ad2antrl 734	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐻 ∧ ∀𝑧 ∈ 𝐻 (norm_ℎ‘(𝐴 −_ℎ 𝑥)) ≤ (norm_ℎ‘(𝐴 −_ℎ 𝑧)))) → 𝑥 ∈ ℋ)
6		hvsubcl 31106	. . . . 5 ⊢ ((𝐴 ∈ ℋ ∧ 𝑥 ∈ ℋ) → (𝐴 −_ℎ 𝑥) ∈ ℋ)
7	2, 5, 6	syl2anc 590	. . . 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 782	. . . . . 6 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐻 ∧ ∀𝑧 ∈ 𝐻 (norm_ℎ‘(𝐴 −_ℎ 𝑥)) ≤ (norm_ℎ‘(𝐴 −_ℎ 𝑧)))) ∧ 𝑦 ∈ 𝐻) → 𝑥 ∈ 𝐻)
10		simpr 485	. . . . . 6 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐻 ∧ ∀𝑧 ∈ 𝐻 (norm_ℎ‘(𝐴 −_ℎ 𝑥)) ≤ (norm_ℎ‘(𝐴 −_ℎ 𝑧)))) ∧ 𝑦 ∈ 𝐻) → 𝑦 ∈ 𝐻)
11		simplrr 783	. . . . . 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 2739	. . . . . 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 31480	. . . . 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 3131	. . . 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 31316	. . . . 5 ⊢ 𝐻 ∈ S_ℋ
16		shocel 31371	. . . . 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 589	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐻 ∧ ∀𝑧 ∈ 𝐻 (norm_ℎ‘(𝐴 −_ℎ 𝑥)) ≤ (norm_ℎ‘(𝐴 −_ℎ 𝑧)))) → (𝐴 −_ℎ 𝑥) ∈ (⊥‘𝐻))
19		hvpncan3 31131	. . . . 5 ⊢ ((𝑥 ∈ ℋ ∧ 𝐴 ∈ ℋ) → (𝑥 +_ℎ (𝐴 −_ℎ 𝑥)) = 𝐴)
20	5, 2, 19	syl2anc 590	. . . 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 2745	. . 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 3583	. . 3 ⊢ (((𝐴 −_ℎ 𝑥) ∈ (⊥‘𝐻) ∧ 𝐴 = (𝑥 +_ℎ (𝐴 −_ℎ 𝑥))) → ∃𝑦 ∈ (⊥‘𝐻)𝐴 = (𝑥 +_ℎ 𝑦))
24	18, 21, 23	syl2anc 590	. 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 31058	. . . 4 ⊢ ℋ = (BaseSet‘⟨⟨ +_ℎ , ·_ℎ ⟩, norm_ℎ⟩)
26		eqid 2739	. . . . 5 ⊢ ⟨⟨ +_ℎ , ·_ℎ ⟩, norm_ℎ⟩ = ⟨⟨ +_ℎ , ·_ℎ ⟩, norm_ℎ⟩
27	26	hhvs 31259	. . . 4 ⊢ −_ℎ = ( −_𝑣 ‘⟨⟨ +_ℎ , ·_ℎ ⟩, norm_ℎ⟩)
28	26	hhnm 31260	. . . 4 ⊢ norm_ℎ = (norm_CV‘⟨⟨ +_ℎ , ·_ℎ ⟩, norm_ℎ⟩)
29		eqid 2739	. . . . 5 ⊢ ⟨⟨( +_ℎ ↾ (𝐻 × 𝐻)), ( ·_ℎ ↾ (ℂ × 𝐻))⟩, (norm_ℎ ↾ 𝐻)⟩ = ⟨⟨( +_ℎ ↾ (𝐻 × 𝐻)), ( ·_ℎ ↾ (ℂ × 𝐻))⟩, (norm_ℎ ↾ 𝐻)⟩
30	29, 15	hhssba 31360	. . . 4 ⊢ 𝐻 = (BaseSet‘⟨⟨( +_ℎ ↾ (𝐻 × 𝐻)), ( ·_ℎ ↾ (ℂ × 𝐻))⟩, (norm_ℎ ↾ 𝐻)⟩)
31	26	hhph 31267	. . . . 5 ⊢ ⟨⟨ +_ℎ , ·_ℎ ⟩, norm_ℎ⟩ ∈ CPreHil_OLD
32	31	a1i 11	. . . 4 ⊢ (𝜑 → ⟨⟨ +_ℎ , ·_ℎ ⟩, norm_ℎ⟩ ∈ CPreHil_OLD)
33	26, 29	hhsst 31355	. . . . . . 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 31368	. . . . . 6 ⊢ ⟨⟨( +_ℎ ↾ (𝐻 × 𝐻)), ( ·_ℎ ↾ (ℂ × 𝐻))⟩, (norm_ℎ ↾ 𝐻)⟩ ∈ CBan
36		elin 3899	. . . . . 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 717	. . . . 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 30973	. . 3 ⊢ (𝜑 → ∃!𝑥 ∈ 𝐻 ∀𝑧 ∈ 𝐻 (norm_ℎ‘(𝐴 −_ℎ 𝑥)) ≤ (norm_ℎ‘(𝐴 −_ℎ 𝑧)))
40		reurex 3348	. . 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 3155	1 ⊢ (𝜑 → ∃𝑥 ∈ 𝐻 ∃𝑦 ∈ (⊥‘𝐻)𝐴 = (𝑥 +_ℎ 𝑦))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 207 ∧ wa 396 = wceq 1547 ∈ wcel 2119 ∀wral 3053 ∃wrex 3063 ∃!wreu 3342 ∩ cin 3882 ⟨cop 4561 class class class wbr 5072 × cxp 5616 ↾ cres 5620 ‘cfv 6485 (class class class)co 7356 ℂcc 11027 0cc0 11029 1c1 11030 + caddc 11032 ≤ cle 11171 / cdiv 11798 SubSpcss 30810 CPreHil_OLDccphlo 30901 CBanccbn 30951 ℋchba 31008 +_ℎ cva 31009 ·_ℎ csm 31010 ·_ih csp 31011 norm_ℎcno 31012 −_ℎ cmv 31014 S_ℋ csh 31017 C_ℋ cch 31018 ⊥cort 31019

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1974 ax-7 2015 ax-8 2121 ax-9 2129 ax-10 2152 ax-11 2168 ax-12 2189 ax-ext 2711 ax-rep 5199 ax-sep 5218 ax-nul 5228 ax-pow 5294 ax-pr 5362 ax-un 7678 ax-inf2 9553 ax-cc 10348 ax-cnex 11085 ax-resscn 11086 ax-1cn 11087 ax-icn 11088 ax-addcl 11089 ax-addrcl 11090 ax-mulcl 11091 ax-mulrcl 11092 ax-mulcom 11093 ax-addass 11094 ax-mulass 11095 ax-distr 11096 ax-i2m1 11097 ax-1ne0 11098 ax-1rid 11099 ax-rnegex 11100 ax-rrecex 11101 ax-cnre 11102 ax-pre-lttri 11103 ax-pre-lttrn 11104 ax-pre-ltadd 11105 ax-pre-mulgt0 11106 ax-pre-sup 11107 ax-addf 11108 ax-mulf 11109 ax-hilex 31088 ax-hfvadd 31089 ax-hvcom 31090 ax-hvass 31091 ax-hv0cl 31092 ax-hvaddid 31093 ax-hfvmul 31094 ax-hvmulid 31095 ax-hvmulass 31096 ax-hvdistr1 31097 ax-hvdistr2 31098 ax-hvmul0 31099 ax-hfi 31168 ax-his1 31171 ax-his2 31172 ax-his3 31173 ax-his4 31174 ax-hcompl 31291

This theorem depends on definitions: df-bi 208 df-an 397 df-or 854 df-3or 1093 df-3an 1094 df-tru 1550 df-fal 1560 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2543 df-eu 2573 df-clab 2718 df-cleq 2731 df-clel 2814 df-nfc 2888 df-ne 2935 df-nel 3039 df-ral 3054 df-rex 3064 df-rmo 3344 df-reu 3345 df-rab 3392 df-v 3433 df-sbc 3724 df-csb 3832 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3903 df-nul 4262 df-if 4455 df-pw 4531 df-sn 4556 df-pr 4558 df-op 4562 df-uni 4839 df-int 4878 df-iun 4923 df-iin 4924 df-br 5073 df-opab 5135 df-mpt 5154 df-tr 5180 df-id 5513 df-eprel 5518 df-po 5526 df-so 5527 df-fr 5571 df-se 5572 df-we 5573 df-xp 5624 df-rel 5625 df-cnv 5626 df-co 5627 df-dm 5628 df-rn 5629 df-res 5630 df-ima 5631 df-pred 6252 df-ord 6313 df-on 6314 df-lim 6315 df-suc 6316 df-iota 6441 df-fun 6487 df-fn 6488 df-f 6489 df-f1 6490 df-fo 6491 df-f1o 6492 df-fv 6493 df-isom 6494 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 8765 df-pm 8766 df-en 8884 df-dom 8885 df-sdom 8886 df-fin 8887 df-fi 9314 df-sup 9345 df-inf 9346 df-oi 9415 df-card 9854 df-acn 9857 df-pnf 11172 df-mnf 11173 df-xr 11174 df-ltxr 11175 df-le 11176 df-sub 11370 df-neg 11371 df-div 11799 df-nn 12166 df-2 12235 df-3 12236 df-4 12237 df-n0 12429 df-z 12516 df-uz 12780 df-q 12890 df-rp 12934 df-xneg 13054 df-xadd 13055 df-xmul 13056 df-ico 13295 df-icc 13296 df-fz 13453 df-fl 13742 df-seq 13955 df-exp 14015 df-cj 15052 df-re 15053 df-im 15054 df-sqrt 15188 df-abs 15189 df-clim 15441 df-rlim 15442 df-rest 17376 df-topgen 17397 df-psmet 21339 df-xmet 21340 df-met 21341 df-bl 21342 df-mopn 21343 df-fbas 21344 df-fg 21345 df-top 22877 df-topon 22894 df-bases 22929 df-cld 23002 df-ntr 23003 df-cls 23004 df-nei 23081 df-lm 23212 df-haus 23298 df-fil 23829 df-fm 23921 df-flim 23922 df-flf 23923 df-cfil 25240 df-cau 25241 df-cmet 25242 df-grpo 30582 df-gid 30583 df-ginv 30584 df-gdiv 30585 df-ablo 30634 df-vc 30648 df-nv 30681 df-va 30684 df-ba 30685 df-sm 30686 df-0v 30687 df-vs 30688 df-nmcv 30689 df-ims 30690 df-ssp 30811 df-ph 30902 df-cbn 30952 df-hnorm 31057 df-hba 31058 df-hvsub 31060 df-hlim 31061 df-hcau 31062 df-sh 31296 df-ch 31310 df-oc 31341 df-ch0 31342

This theorem is referenced by: pjhth 31482 omlsii 31492

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