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Mirrors > Home > HSE Home > Th. List > pjhthlem2 | Structured version Visualization version GIF version |
Description: Lemma for pjhth 31275. (Contributed by NM, 10-Oct-1999.) (Revised by Mario Carneiro, 15-May-2014.) (New usage is discouraged.) |
Ref | Expression |
---|---|
pjhth.1 | ⊢ 𝐻 ∈ Cℋ |
pjhth.2 | ⊢ (𝜑 → 𝐴 ∈ ℋ) |
Ref | Expression |
---|---|
pjhthlem2 | ⊢ (𝜑 → ∃𝑥 ∈ 𝐻 ∃𝑦 ∈ (⊥‘𝐻)𝐴 = (𝑥 +ℎ 𝑦)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pjhth.2 | . . . . . 6 ⊢ (𝜑 → 𝐴 ∈ ℋ) | |
2 | 1 | adantr 479 | . . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐻 ∧ ∀𝑧 ∈ 𝐻 (normℎ‘(𝐴 −ℎ 𝑥)) ≤ (normℎ‘(𝐴 −ℎ 𝑧)))) → 𝐴 ∈ ℋ) |
3 | pjhth.1 | . . . . . . 7 ⊢ 𝐻 ∈ Cℋ | |
4 | 3 | cheli 31114 | . . . . . 6 ⊢ (𝑥 ∈ 𝐻 → 𝑥 ∈ ℋ) |
5 | 4 | ad2antrl 726 | . . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐻 ∧ ∀𝑧 ∈ 𝐻 (normℎ‘(𝐴 −ℎ 𝑥)) ≤ (normℎ‘(𝐴 −ℎ 𝑧)))) → 𝑥 ∈ ℋ) |
6 | hvsubcl 30899 | . . . . 5 ⊢ ((𝐴 ∈ ℋ ∧ 𝑥 ∈ ℋ) → (𝐴 −ℎ 𝑥) ∈ ℋ) | |
7 | 2, 5, 6 | syl2anc 582 | . . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐻 ∧ ∀𝑧 ∈ 𝐻 (normℎ‘(𝐴 −ℎ 𝑥)) ≤ (normℎ‘(𝐴 −ℎ 𝑧)))) → (𝐴 −ℎ 𝑥) ∈ ℋ) |
8 | 2 | adantr 479 | . . . . . 6 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐻 ∧ ∀𝑧 ∈ 𝐻 (normℎ‘(𝐴 −ℎ 𝑥)) ≤ (normℎ‘(𝐴 −ℎ 𝑧)))) ∧ 𝑦 ∈ 𝐻) → 𝐴 ∈ ℋ) |
9 | simplrl 775 | . . . . . 6 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐻 ∧ ∀𝑧 ∈ 𝐻 (normℎ‘(𝐴 −ℎ 𝑥)) ≤ (normℎ‘(𝐴 −ℎ 𝑧)))) ∧ 𝑦 ∈ 𝐻) → 𝑥 ∈ 𝐻) | |
10 | simpr 483 | . . . . . 6 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐻 ∧ ∀𝑧 ∈ 𝐻 (normℎ‘(𝐴 −ℎ 𝑥)) ≤ (normℎ‘(𝐴 −ℎ 𝑧)))) ∧ 𝑦 ∈ 𝐻) → 𝑦 ∈ 𝐻) | |
11 | simplrr 776 | . . . . . 6 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐻 ∧ ∀𝑧 ∈ 𝐻 (normℎ‘(𝐴 −ℎ 𝑥)) ≤ (normℎ‘(𝐴 −ℎ 𝑧)))) ∧ 𝑦 ∈ 𝐻) → ∀𝑧 ∈ 𝐻 (normℎ‘(𝐴 −ℎ 𝑥)) ≤ (normℎ‘(𝐴 −ℎ 𝑧))) | |
12 | eqid 2725 | . . . . . 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 31273 | . . . . 5 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐻 ∧ ∀𝑧 ∈ 𝐻 (normℎ‘(𝐴 −ℎ 𝑥)) ≤ (normℎ‘(𝐴 −ℎ 𝑧)))) ∧ 𝑦 ∈ 𝐻) → ((𝐴 −ℎ 𝑥) ·ih 𝑦) = 0) |
14 | 13 | ralrimiva 3135 | . . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐻 ∧ ∀𝑧 ∈ 𝐻 (normℎ‘(𝐴 −ℎ 𝑥)) ≤ (normℎ‘(𝐴 −ℎ 𝑧)))) → ∀𝑦 ∈ 𝐻 ((𝐴 −ℎ 𝑥) ·ih 𝑦) = 0) |
15 | 3 | chshii 31109 | . . . . 5 ⊢ 𝐻 ∈ Sℋ |
16 | shocel 31164 | . . . . 5 ⊢ (𝐻 ∈ Sℋ → ((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 581 | . . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐻 ∧ ∀𝑧 ∈ 𝐻 (normℎ‘(𝐴 −ℎ 𝑥)) ≤ (normℎ‘(𝐴 −ℎ 𝑧)))) → (𝐴 −ℎ 𝑥) ∈ (⊥‘𝐻)) |
19 | hvpncan3 30924 | . . . . 5 ⊢ ((𝑥 ∈ ℋ ∧ 𝐴 ∈ ℋ) → (𝑥 +ℎ (𝐴 −ℎ 𝑥)) = 𝐴) | |
20 | 5, 2, 19 | syl2anc 582 | . . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐻 ∧ ∀𝑧 ∈ 𝐻 (normℎ‘(𝐴 −ℎ 𝑥)) ≤ (normℎ‘(𝐴 −ℎ 𝑧)))) → (𝑥 +ℎ (𝐴 −ℎ 𝑥)) = 𝐴) |
21 | 20 | eqcomd 2731 | . . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐻 ∧ ∀𝑧 ∈ 𝐻 (normℎ‘(𝐴 −ℎ 𝑥)) ≤ (normℎ‘(𝐴 −ℎ 𝑧)))) → 𝐴 = (𝑥 +ℎ (𝐴 −ℎ 𝑥))) |
22 | oveq2 7427 | . . . 4 ⊢ (𝑦 = (𝐴 −ℎ 𝑥) → (𝑥 +ℎ 𝑦) = (𝑥 +ℎ (𝐴 −ℎ 𝑥))) | |
23 | 22 | rspceeqv 3628 | . . 3 ⊢ (((𝐴 −ℎ 𝑥) ∈ (⊥‘𝐻) ∧ 𝐴 = (𝑥 +ℎ (𝐴 −ℎ 𝑥))) → ∃𝑦 ∈ (⊥‘𝐻)𝐴 = (𝑥 +ℎ 𝑦)) |
24 | 18, 21, 23 | syl2anc 582 | . 2 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐻 ∧ ∀𝑧 ∈ 𝐻 (normℎ‘(𝐴 −ℎ 𝑥)) ≤ (normℎ‘(𝐴 −ℎ 𝑧)))) → ∃𝑦 ∈ (⊥‘𝐻)𝐴 = (𝑥 +ℎ 𝑦)) |
25 | df-hba 30851 | . . . 4 ⊢ ℋ = (BaseSet‘〈〈 +ℎ , ·ℎ 〉, normℎ〉) | |
26 | eqid 2725 | . . . . 5 ⊢ 〈〈 +ℎ , ·ℎ 〉, normℎ〉 = 〈〈 +ℎ , ·ℎ 〉, normℎ〉 | |
27 | 26 | hhvs 31052 | . . . 4 ⊢ −ℎ = ( −𝑣 ‘〈〈 +ℎ , ·ℎ 〉, normℎ〉) |
28 | 26 | hhnm 31053 | . . . 4 ⊢ normℎ = (normCV‘〈〈 +ℎ , ·ℎ 〉, normℎ〉) |
29 | eqid 2725 | . . . . 5 ⊢ 〈〈( +ℎ ↾ (𝐻 × 𝐻)), ( ·ℎ ↾ (ℂ × 𝐻))〉, (normℎ ↾ 𝐻)〉 = 〈〈( +ℎ ↾ (𝐻 × 𝐻)), ( ·ℎ ↾ (ℂ × 𝐻))〉, (normℎ ↾ 𝐻)〉 | |
30 | 29, 15 | hhssba 31153 | . . . 4 ⊢ 𝐻 = (BaseSet‘〈〈( +ℎ ↾ (𝐻 × 𝐻)), ( ·ℎ ↾ (ℂ × 𝐻))〉, (normℎ ↾ 𝐻)〉) |
31 | 26 | hhph 31060 | . . . . 5 ⊢ 〈〈 +ℎ , ·ℎ 〉, normℎ〉 ∈ CPreHilOLD |
32 | 31 | a1i 11 | . . . 4 ⊢ (𝜑 → 〈〈 +ℎ , ·ℎ 〉, normℎ〉 ∈ CPreHilOLD) |
33 | 26, 29 | hhsst 31148 | . . . . . . 7 ⊢ (𝐻 ∈ Sℋ → 〈〈( +ℎ ↾ (𝐻 × 𝐻)), ( ·ℎ ↾ (ℂ × 𝐻))〉, (normℎ ↾ 𝐻)〉 ∈ (SubSp‘〈〈 +ℎ , ·ℎ 〉, normℎ〉)) |
34 | 15, 33 | ax-mp 5 | . . . . . 6 ⊢ 〈〈( +ℎ ↾ (𝐻 × 𝐻)), ( ·ℎ ↾ (ℂ × 𝐻))〉, (normℎ ↾ 𝐻)〉 ∈ (SubSp‘〈〈 +ℎ , ·ℎ 〉, normℎ〉) |
35 | 29, 3 | hhssbnOLD 31161 | . . . . . 6 ⊢ 〈〈( +ℎ ↾ (𝐻 × 𝐻)), ( ·ℎ ↾ (ℂ × 𝐻))〉, (normℎ ↾ 𝐻)〉 ∈ CBan |
36 | elin 3960 | . . . . . 6 ⊢ (〈〈( +ℎ ↾ (𝐻 × 𝐻)), ( ·ℎ ↾ (ℂ × 𝐻))〉, (normℎ ↾ 𝐻)〉 ∈ ((SubSp‘〈〈 +ℎ , ·ℎ 〉, normℎ〉) ∩ CBan) ↔ (〈〈( +ℎ ↾ (𝐻 × 𝐻)), ( ·ℎ ↾ (ℂ × 𝐻))〉, (normℎ ↾ 𝐻)〉 ∈ (SubSp‘〈〈 +ℎ , ·ℎ 〉, normℎ〉) ∧ 〈〈( +ℎ ↾ (𝐻 × 𝐻)), ( ·ℎ ↾ (ℂ × 𝐻))〉, (normℎ ↾ 𝐻)〉 ∈ CBan)) | |
37 | 34, 35, 36 | mpbir2an 709 | . . . . 5 ⊢ 〈〈( +ℎ ↾ (𝐻 × 𝐻)), ( ·ℎ ↾ (ℂ × 𝐻))〉, (normℎ ↾ 𝐻)〉 ∈ ((SubSp‘〈〈 +ℎ , ·ℎ 〉, normℎ〉) ∩ CBan) |
38 | 37 | a1i 11 | . . . 4 ⊢ (𝜑 → 〈〈( +ℎ ↾ (𝐻 × 𝐻)), ( ·ℎ ↾ (ℂ × 𝐻))〉, (normℎ ↾ 𝐻)〉 ∈ ((SubSp‘〈〈 +ℎ , ·ℎ 〉, normℎ〉) ∩ CBan)) |
39 | 25, 27, 28, 30, 32, 38, 1 | minveco 30766 | . . 3 ⊢ (𝜑 → ∃!𝑥 ∈ 𝐻 ∀𝑧 ∈ 𝐻 (normℎ‘(𝐴 −ℎ 𝑥)) ≤ (normℎ‘(𝐴 −ℎ 𝑧))) |
40 | reurex 3367 | . . 3 ⊢ (∃!𝑥 ∈ 𝐻 ∀𝑧 ∈ 𝐻 (normℎ‘(𝐴 −ℎ 𝑥)) ≤ (normℎ‘(𝐴 −ℎ 𝑧)) → ∃𝑥 ∈ 𝐻 ∀𝑧 ∈ 𝐻 (normℎ‘(𝐴 −ℎ 𝑥)) ≤ (normℎ‘(𝐴 −ℎ 𝑧))) | |
41 | 39, 40 | syl 17 | . 2 ⊢ (𝜑 → ∃𝑥 ∈ 𝐻 ∀𝑧 ∈ 𝐻 (normℎ‘(𝐴 −ℎ 𝑥)) ≤ (normℎ‘(𝐴 −ℎ 𝑧))) |
42 | 24, 41 | reximddv 3160 | 1 ⊢ (𝜑 → ∃𝑥 ∈ 𝐻 ∃𝑦 ∈ (⊥‘𝐻)𝐴 = (𝑥 +ℎ 𝑦)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 394 = wceq 1533 ∈ wcel 2098 ∀wral 3050 ∃wrex 3059 ∃!wreu 3361 ∩ cin 3943 〈cop 4636 class class class wbr 5149 × cxp 5676 ↾ cres 5680 ‘cfv 6549 (class class class)co 7419 ℂcc 11138 0cc0 11140 1c1 11141 + caddc 11143 ≤ cle 11281 / cdiv 11903 SubSpcss 30603 CPreHilOLDccphlo 30694 CBanccbn 30744 ℋchba 30801 +ℎ cva 30802 ·ℎ csm 30803 ·ih csp 30804 normℎcno 30805 −ℎ cmv 30807 Sℋ csh 30810 Cℋ cch 30811 ⊥cort 30812 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5365 ax-pr 5429 ax-un 7741 ax-inf2 9666 ax-cc 10460 ax-cnex 11196 ax-resscn 11197 ax-1cn 11198 ax-icn 11199 ax-addcl 11200 ax-addrcl 11201 ax-mulcl 11202 ax-mulrcl 11203 ax-mulcom 11204 ax-addass 11205 ax-mulass 11206 ax-distr 11207 ax-i2m1 11208 ax-1ne0 11209 ax-1rid 11210 ax-rnegex 11211 ax-rrecex 11212 ax-cnre 11213 ax-pre-lttri 11214 ax-pre-lttrn 11215 ax-pre-ltadd 11216 ax-pre-mulgt0 11217 ax-pre-sup 11218 ax-addf 11219 ax-mulf 11220 ax-hilex 30881 ax-hfvadd 30882 ax-hvcom 30883 ax-hvass 30884 ax-hv0cl 30885 ax-hvaddid 30886 ax-hfvmul 30887 ax-hvmulid 30888 ax-hvmulass 30889 ax-hvdistr1 30890 ax-hvdistr2 30891 ax-hvmul0 30892 ax-hfi 30961 ax-his1 30964 ax-his2 30965 ax-his3 30966 ax-his4 30967 ax-hcompl 31084 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-nel 3036 df-ral 3051 df-rex 3060 df-rmo 3363 df-reu 3364 df-rab 3419 df-v 3463 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3964 df-nul 4323 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4910 df-int 4951 df-iun 4999 df-iin 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-se 5634 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6307 df-ord 6374 df-on 6375 df-lim 6376 df-suc 6377 df-iota 6501 df-fun 6551 df-fn 6552 df-f 6553 df-f1 6554 df-fo 6555 df-f1o 6556 df-fv 6557 df-isom 6558 df-riota 7375 df-ov 7422 df-oprab 7423 df-mpo 7424 df-om 7872 df-1st 7994 df-2nd 7995 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-1o 8487 df-oadd 8491 df-omul 8492 df-er 8725 df-map 8847 df-pm 8848 df-en 8965 df-dom 8966 df-sdom 8967 df-fin 8968 df-fi 9436 df-sup 9467 df-inf 9468 df-oi 9535 df-card 9964 df-acn 9967 df-pnf 11282 df-mnf 11283 df-xr 11284 df-ltxr 11285 df-le 11286 df-sub 11478 df-neg 11479 df-div 11904 df-nn 12246 df-2 12308 df-3 12309 df-4 12310 df-n0 12506 df-z 12592 df-uz 12856 df-q 12966 df-rp 13010 df-xneg 13127 df-xadd 13128 df-xmul 13129 df-ico 13365 df-icc 13366 df-fz 13520 df-fl 13793 df-seq 14003 df-exp 14063 df-cj 15082 df-re 15083 df-im 15084 df-sqrt 15218 df-abs 15219 df-clim 15468 df-rlim 15469 df-rest 17407 df-topgen 17428 df-psmet 21288 df-xmet 21289 df-met 21290 df-bl 21291 df-mopn 21292 df-fbas 21293 df-fg 21294 df-top 22840 df-topon 22857 df-bases 22893 df-cld 22967 df-ntr 22968 df-cls 22969 df-nei 23046 df-lm 23177 df-haus 23263 df-fil 23794 df-fm 23886 df-flim 23887 df-flf 23888 df-cfil 25227 df-cau 25228 df-cmet 25229 df-grpo 30375 df-gid 30376 df-ginv 30377 df-gdiv 30378 df-ablo 30427 df-vc 30441 df-nv 30474 df-va 30477 df-ba 30478 df-sm 30479 df-0v 30480 df-vs 30481 df-nmcv 30482 df-ims 30483 df-ssp 30604 df-ph 30695 df-cbn 30745 df-hnorm 30850 df-hba 30851 df-hvsub 30853 df-hlim 30854 df-hcau 30855 df-sh 31089 df-ch 31103 df-oc 31134 df-ch0 31135 |
This theorem is referenced by: pjhth 31275 omlsii 31285 |
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