pjhthlem2 - Hilbert Space Explorer

	Hilbert Space Explorer		< Previous Next > Nearby theorems
Mirrors > Home > HSE Home > Th. List > pjhthlem2		Structured version Visualization version GIF version

Theorem pjhthlem2 31480

Description: Lemma for pjhth 31481. (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 31320	. . . . . 6 ⊢ (𝑥 ∈ 𝐻 → 𝑥 ∈ ℋ)
5	4	ad2antrl 729	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐻 ∧ ∀𝑧 ∈ 𝐻 (norm_ℎ‘(𝐴 −_ℎ 𝑥)) ≤ (norm_ℎ‘(𝐴 −_ℎ 𝑧)))) → 𝑥 ∈ ℋ)
6		hvsubcl 31105	. . . . 5 ⊢ ((𝐴 ∈ ℋ ∧ 𝑥 ∈ ℋ) → (𝐴 −_ℎ 𝑥) ∈ ℋ)
7	2, 5, 6	syl2anc 585	. . . 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 777	. . . . . 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 778	. . . . . 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 2737	. . . . . 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 31479	. . . . 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 3130	. . . 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 31315	. . . . 5 ⊢ 𝐻 ∈ S_ℋ
16		shocel 31370	. . . . 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 584	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐻 ∧ ∀𝑧 ∈ 𝐻 (norm_ℎ‘(𝐴 −_ℎ 𝑥)) ≤ (norm_ℎ‘(𝐴 −_ℎ 𝑧)))) → (𝐴 −_ℎ 𝑥) ∈ (⊥‘𝐻))
19		hvpncan3 31130	. . . . 5 ⊢ ((𝑥 ∈ ℋ ∧ 𝐴 ∈ ℋ) → (𝑥 +_ℎ (𝐴 −_ℎ 𝑥)) = 𝐴)
20	5, 2, 19	syl2anc 585	. . . 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 2743	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐻 ∧ ∀𝑧 ∈ 𝐻 (norm_ℎ‘(𝐴 −_ℎ 𝑥)) ≤ (norm_ℎ‘(𝐴 −_ℎ 𝑧)))) → 𝐴 = (𝑥 +_ℎ (𝐴 −_ℎ 𝑥)))
22		oveq2 7376	. . . 4 ⊢ (𝑦 = (𝐴 −_ℎ 𝑥) → (𝑥 +_ℎ 𝑦) = (𝑥 +_ℎ (𝐴 −_ℎ 𝑥)))
23	22	rspceeqv 3601	. . 3 ⊢ (((𝐴 −_ℎ 𝑥) ∈ (⊥‘𝐻) ∧ 𝐴 = (𝑥 +_ℎ (𝐴 −_ℎ 𝑥))) → ∃𝑦 ∈ (⊥‘𝐻)𝐴 = (𝑥 +_ℎ 𝑦))
24	18, 21, 23	syl2anc 585	. 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 31057	. . . 4 ⊢ ℋ = (BaseSet‘⟨⟨ +_ℎ , ·_ℎ ⟩, norm_ℎ⟩)
26		eqid 2737	. . . . 5 ⊢ ⟨⟨ +_ℎ , ·_ℎ ⟩, norm_ℎ⟩ = ⟨⟨ +_ℎ , ·_ℎ ⟩, norm_ℎ⟩
27	26	hhvs 31258	. . . 4 ⊢ −_ℎ = ( −_𝑣 ‘⟨⟨ +_ℎ , ·_ℎ ⟩, norm_ℎ⟩)
28	26	hhnm 31259	. . . 4 ⊢ norm_ℎ = (norm_CV‘⟨⟨ +_ℎ , ·_ℎ ⟩, norm_ℎ⟩)
29		eqid 2737	. . . . 5 ⊢ ⟨⟨( +_ℎ ↾ (𝐻 × 𝐻)), ( ·_ℎ ↾ (ℂ × 𝐻))⟩, (norm_ℎ ↾ 𝐻)⟩ = ⟨⟨( +_ℎ ↾ (𝐻 × 𝐻)), ( ·_ℎ ↾ (ℂ × 𝐻))⟩, (norm_ℎ ↾ 𝐻)⟩
30	29, 15	hhssba 31359	. . . 4 ⊢ 𝐻 = (BaseSet‘⟨⟨( +_ℎ ↾ (𝐻 × 𝐻)), ( ·_ℎ ↾ (ℂ × 𝐻))⟩, (norm_ℎ ↾ 𝐻)⟩)
31	26	hhph 31266	. . . . 5 ⊢ ⟨⟨ +_ℎ , ·_ℎ ⟩, norm_ℎ⟩ ∈ CPreHil_OLD
32	31	a1i 11	. . . 4 ⊢ (𝜑 → ⟨⟨ +_ℎ , ·_ℎ ⟩, norm_ℎ⟩ ∈ CPreHil_OLD)
33	26, 29	hhsst 31354	. . . . . . 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 31367	. . . . . 6 ⊢ ⟨⟨( +_ℎ ↾ (𝐻 × 𝐻)), ( ·_ℎ ↾ (ℂ × 𝐻))⟩, (norm_ℎ ↾ 𝐻)⟩ ∈ CBan
36		elin 3919	. . . . . 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 712	. . . . 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 30972	. . 3 ⊢ (𝜑 → ∃!𝑥 ∈ 𝐻 ∀𝑧 ∈ 𝐻 (norm_ℎ‘(𝐴 −_ℎ 𝑥)) ≤ (norm_ℎ‘(𝐴 −_ℎ 𝑧)))
40		reurex 3356	. . 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 3154	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 1542 ∈ wcel 2114 ∀wral 3052 ∃wrex 3062 ∃!wreu 3350 ∩ cin 3902 ⟨cop 4588 class class class wbr 5100 × cxp 5630 ↾ cres 5634 ‘cfv 6500 (class class class)co 7368 ℂcc 11036 0cc0 11038 1c1 11039 + caddc 11041 ≤ cle 11179 / cdiv 11806 SubSpcss 30809 CPreHil_OLDccphlo 30900 CBanccbn 30950 ℋchba 31007 +_ℎ cva 31008 ·_ℎ csm 31009 ·_ih csp 31010 norm_ℎcno 31011 −_ℎ cmv 31013 S_ℋ csh 31016 C_ℋ cch 31017 ⊥cort 31018

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-rep 5226 ax-sep 5243 ax-nul 5253 ax-pow 5312 ax-pr 5379 ax-un 7690 ax-inf2 9562 ax-cc 10357 ax-cnex 11094 ax-resscn 11095 ax-1cn 11096 ax-icn 11097 ax-addcl 11098 ax-addrcl 11099 ax-mulcl 11100 ax-mulrcl 11101 ax-mulcom 11102 ax-addass 11103 ax-mulass 11104 ax-distr 11105 ax-i2m1 11106 ax-1ne0 11107 ax-1rid 11108 ax-rnegex 11109 ax-rrecex 11110 ax-cnre 11111 ax-pre-lttri 11112 ax-pre-lttrn 11113 ax-pre-ltadd 11114 ax-pre-mulgt0 11115 ax-pre-sup 11116 ax-addf 11117 ax-mulf 11118 ax-hilex 31087 ax-hfvadd 31088 ax-hvcom 31089 ax-hvass 31090 ax-hv0cl 31091 ax-hvaddid 31092 ax-hfvmul 31093 ax-hvmulid 31094 ax-hvmulass 31095 ax-hvdistr1 31096 ax-hvdistr2 31097 ax-hvmul0 31098 ax-hfi 31167 ax-his1 31170 ax-his2 31171 ax-his3 31172 ax-his4 31173 ax-hcompl 31290

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3063 df-rmo 3352 df-reu 3353 df-rab 3402 df-v 3444 df-sbc 3743 df-csb 3852 df-dif 3906 df-un 3908 df-in 3910 df-ss 3920 df-pss 3923 df-nul 4288 df-if 4482 df-pw 4558 df-sn 4583 df-pr 4585 df-op 4589 df-uni 4866 df-int 4905 df-iun 4950 df-iin 4951 df-br 5101 df-opab 5163 df-mpt 5182 df-tr 5208 df-id 5527 df-eprel 5532 df-po 5540 df-so 5541 df-fr 5585 df-se 5586 df-we 5587 df-xp 5638 df-rel 5639 df-cnv 5640 df-co 5641 df-dm 5642 df-rn 5643 df-res 5644 df-ima 5645 df-pred 6267 df-ord 6328 df-on 6329 df-lim 6330 df-suc 6331 df-iota 6456 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-isom 6509 df-riota 7325 df-ov 7371 df-oprab 7372 df-mpo 7373 df-om 7819 df-1st 7943 df-2nd 7944 df-frecs 8233 df-wrecs 8264 df-recs 8313 df-rdg 8351 df-1o 8407 df-2o 8408 df-oadd 8411 df-omul 8412 df-er 8645 df-map 8777 df-pm 8778 df-en 8896 df-dom 8897 df-sdom 8898 df-fin 8899 df-fi 9326 df-sup 9357 df-inf 9358 df-oi 9427 df-card 9863 df-acn 9866 df-pnf 11180 df-mnf 11181 df-xr 11182 df-ltxr 11183 df-le 11184 df-sub 11378 df-neg 11379 df-div 11807 df-nn 12158 df-2 12220 df-3 12221 df-4 12222 df-n0 12414 df-z 12501 df-uz 12764 df-q 12874 df-rp 12918 df-xneg 13038 df-xadd 13039 df-xmul 13040 df-ico 13279 df-icc 13280 df-fz 13436 df-fl 13724 df-seq 13937 df-exp 13997 df-cj 15034 df-re 15035 df-im 15036 df-sqrt 15170 df-abs 15171 df-clim 15423 df-rlim 15424 df-rest 17354 df-topgen 17375 df-psmet 21313 df-xmet 21314 df-met 21315 df-bl 21316 df-mopn 21317 df-fbas 21318 df-fg 21319 df-top 22850 df-topon 22867 df-bases 22902 df-cld 22975 df-ntr 22976 df-cls 22977 df-nei 23054 df-lm 23185 df-haus 23271 df-fil 23802 df-fm 23894 df-flim 23895 df-flf 23896 df-cfil 25223 df-cau 25224 df-cmet 25225 df-grpo 30581 df-gid 30582 df-ginv 30583 df-gdiv 30584 df-ablo 30633 df-vc 30647 df-nv 30680 df-va 30683 df-ba 30684 df-sm 30685 df-0v 30686 df-vs 30687 df-nmcv 30688 df-ims 30689 df-ssp 30810 df-ph 30901 df-cbn 30951 df-hnorm 31056 df-hba 31057 df-hvsub 31059 df-hlim 31060 df-hcau 31061 df-sh 31295 df-ch 31309 df-oc 31340 df-ch0 31341

This theorem is referenced by: pjhth 31481 omlsii 31491

W3C validator