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Mirrors > Home > HSE Home > Th. List > pjhthlem2 | Structured version Visualization version GIF version |
Description: Lemma for pjhth 29172. (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 483 | . . . . 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 29011 | . . . . . 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 28796 | . . . . 5 ⊢ ((𝐴 ∈ ℋ ∧ 𝑥 ∈ ℋ) → (𝐴 −ℎ 𝑥) ∈ ℋ) | |
7 | 2, 5, 6 | syl2anc 586 | . . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐻 ∧ ∀𝑧 ∈ 𝐻 (normℎ‘(𝐴 −ℎ 𝑥)) ≤ (normℎ‘(𝐴 −ℎ 𝑧)))) → (𝐴 −ℎ 𝑥) ∈ ℋ) |
8 | 2 | adantr 483 | . . . . . 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 487 | . . . . . 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 2823 | . . . . . 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 29170 | . . . . 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 3184 | . . . 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 29006 | . . . . 5 ⊢ 𝐻 ∈ Sℋ |
16 | shocel 29061 | . . . . 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 585 | . . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐻 ∧ ∀𝑧 ∈ 𝐻 (normℎ‘(𝐴 −ℎ 𝑥)) ≤ (normℎ‘(𝐴 −ℎ 𝑧)))) → (𝐴 −ℎ 𝑥) ∈ (⊥‘𝐻)) |
19 | hvpncan3 28821 | . . . . 5 ⊢ ((𝑥 ∈ ℋ ∧ 𝐴 ∈ ℋ) → (𝑥 +ℎ (𝐴 −ℎ 𝑥)) = 𝐴) | |
20 | 5, 2, 19 | syl2anc 586 | . . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐻 ∧ ∀𝑧 ∈ 𝐻 (normℎ‘(𝐴 −ℎ 𝑥)) ≤ (normℎ‘(𝐴 −ℎ 𝑧)))) → (𝑥 +ℎ (𝐴 −ℎ 𝑥)) = 𝐴) |
21 | 20 | eqcomd 2829 | . . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐻 ∧ ∀𝑧 ∈ 𝐻 (normℎ‘(𝐴 −ℎ 𝑥)) ≤ (normℎ‘(𝐴 −ℎ 𝑧)))) → 𝐴 = (𝑥 +ℎ (𝐴 −ℎ 𝑥))) |
22 | oveq2 7166 | . . . 4 ⊢ (𝑦 = (𝐴 −ℎ 𝑥) → (𝑥 +ℎ 𝑦) = (𝑥 +ℎ (𝐴 −ℎ 𝑥))) | |
23 | 22 | rspceeqv 3640 | . . 3 ⊢ (((𝐴 −ℎ 𝑥) ∈ (⊥‘𝐻) ∧ 𝐴 = (𝑥 +ℎ (𝐴 −ℎ 𝑥))) → ∃𝑦 ∈ (⊥‘𝐻)𝐴 = (𝑥 +ℎ 𝑦)) |
24 | 18, 21, 23 | syl2anc 586 | . 2 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐻 ∧ ∀𝑧 ∈ 𝐻 (normℎ‘(𝐴 −ℎ 𝑥)) ≤ (normℎ‘(𝐴 −ℎ 𝑧)))) → ∃𝑦 ∈ (⊥‘𝐻)𝐴 = (𝑥 +ℎ 𝑦)) |
25 | df-hba 28748 | . . . 4 ⊢ ℋ = (BaseSet‘〈〈 +ℎ , ·ℎ 〉, normℎ〉) | |
26 | eqid 2823 | . . . . 5 ⊢ 〈〈 +ℎ , ·ℎ 〉, normℎ〉 = 〈〈 +ℎ , ·ℎ 〉, normℎ〉 | |
27 | 26 | hhvs 28949 | . . . 4 ⊢ −ℎ = ( −𝑣 ‘〈〈 +ℎ , ·ℎ 〉, normℎ〉) |
28 | 26 | hhnm 28950 | . . . 4 ⊢ normℎ = (normCV‘〈〈 +ℎ , ·ℎ 〉, normℎ〉) |
29 | eqid 2823 | . . . . 5 ⊢ 〈〈( +ℎ ↾ (𝐻 × 𝐻)), ( ·ℎ ↾ (ℂ × 𝐻))〉, (normℎ ↾ 𝐻)〉 = 〈〈( +ℎ ↾ (𝐻 × 𝐻)), ( ·ℎ ↾ (ℂ × 𝐻))〉, (normℎ ↾ 𝐻)〉 | |
30 | 29, 15 | hhssba 29050 | . . . 4 ⊢ 𝐻 = (BaseSet‘〈〈( +ℎ ↾ (𝐻 × 𝐻)), ( ·ℎ ↾ (ℂ × 𝐻))〉, (normℎ ↾ 𝐻)〉) |
31 | 26 | hhph 28957 | . . . . 5 ⊢ 〈〈 +ℎ , ·ℎ 〉, normℎ〉 ∈ CPreHilOLD |
32 | 31 | a1i 11 | . . . 4 ⊢ (𝜑 → 〈〈 +ℎ , ·ℎ 〉, normℎ〉 ∈ CPreHilOLD) |
33 | 26, 29 | hhsst 29045 | . . . . . . 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 29058 | . . . . . 6 ⊢ 〈〈( +ℎ ↾ (𝐻 × 𝐻)), ( ·ℎ ↾ (ℂ × 𝐻))〉, (normℎ ↾ 𝐻)〉 ∈ CBan |
36 | elin 4171 | . . . . . 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 28663 | . . 3 ⊢ (𝜑 → ∃!𝑥 ∈ 𝐻 ∀𝑧 ∈ 𝐻 (normℎ‘(𝐴 −ℎ 𝑥)) ≤ (normℎ‘(𝐴 −ℎ 𝑧))) |
40 | reurex 3433 | . . 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 3277 | 1 ⊢ (𝜑 → ∃𝑥 ∈ 𝐻 ∃𝑦 ∈ (⊥‘𝐻)𝐴 = (𝑥 +ℎ 𝑦)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ∀wral 3140 ∃wrex 3141 ∃!wreu 3142 ∩ cin 3937 〈cop 4575 class class class wbr 5068 × cxp 5555 ↾ cres 5559 ‘cfv 6357 (class class class)co 7158 ℂcc 10537 0cc0 10539 1c1 10540 + caddc 10542 ≤ cle 10678 / cdiv 11299 SubSpcss 28500 CPreHilOLDccphlo 28591 CBanccbn 28641 ℋchba 28698 +ℎ cva 28699 ·ℎ csm 28700 ·ih csp 28701 normℎcno 28702 −ℎ cmv 28704 Sℋ csh 28707 Cℋ cch 28708 ⊥cort 28709 |
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 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-inf2 9106 ax-cc 9859 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 ax-pre-sup 10617 ax-addf 10618 ax-mulf 10619 ax-hilex 28778 ax-hfvadd 28779 ax-hvcom 28780 ax-hvass 28781 ax-hv0cl 28782 ax-hvaddid 28783 ax-hfvmul 28784 ax-hvmulid 28785 ax-hvmulass 28786 ax-hvdistr1 28787 ax-hvdistr2 28788 ax-hvmul0 28789 ax-hfi 28858 ax-his1 28861 ax-his2 28862 ax-his3 28863 ax-his4 28864 ax-hcompl 28981 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-int 4879 df-iun 4923 df-iin 4924 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-se 5517 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-isom 6366 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-om 7583 df-1st 7691 df-2nd 7692 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-1o 8104 df-oadd 8108 df-omul 8109 df-er 8291 df-map 8410 df-pm 8411 df-en 8512 df-dom 8513 df-sdom 8514 df-fin 8515 df-fi 8877 df-sup 8908 df-inf 8909 df-oi 8976 df-card 9370 df-acn 9373 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-div 11300 df-nn 11641 df-2 11703 df-3 11704 df-4 11705 df-n0 11901 df-z 11985 df-uz 12247 df-q 12352 df-rp 12393 df-xneg 12510 df-xadd 12511 df-xmul 12512 df-ico 12747 df-icc 12748 df-fz 12896 df-fl 13165 df-seq 13373 df-exp 13433 df-cj 14460 df-re 14461 df-im 14462 df-sqrt 14596 df-abs 14597 df-clim 14847 df-rlim 14848 df-rest 16698 df-topgen 16719 df-psmet 20539 df-xmet 20540 df-met 20541 df-bl 20542 df-mopn 20543 df-fbas 20544 df-fg 20545 df-top 21504 df-topon 21521 df-bases 21556 df-cld 21629 df-ntr 21630 df-cls 21631 df-nei 21708 df-lm 21839 df-haus 21925 df-fil 22456 df-fm 22548 df-flim 22549 df-flf 22550 df-cfil 23860 df-cau 23861 df-cmet 23862 df-grpo 28272 df-gid 28273 df-ginv 28274 df-gdiv 28275 df-ablo 28324 df-vc 28338 df-nv 28371 df-va 28374 df-ba 28375 df-sm 28376 df-0v 28377 df-vs 28378 df-nmcv 28379 df-ims 28380 df-ssp 28501 df-ph 28592 df-cbn 28642 df-hnorm 28747 df-hba 28748 df-hvsub 28750 df-hlim 28751 df-hcau 28752 df-sh 28986 df-ch 29000 df-oc 29031 df-ch0 29032 |
This theorem is referenced by: pjhth 29172 omlsii 29182 |
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