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Mirrors > Home > HSE Home > Th. List > pjhthlem2 | Structured version Visualization version GIF version |
Description: Lemma for pjhth 27636. (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 480 | . . . . 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 27473 | . . . . . 6 ⊢ (𝑥 ∈ 𝐻 → 𝑥 ∈ ℋ) |
5 | 4 | ad2antrl 760 | . . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐻 ∧ ∀𝑧 ∈ 𝐻 (normℎ‘(𝐴 −ℎ 𝑥)) ≤ (normℎ‘(𝐴 −ℎ 𝑧)))) → 𝑥 ∈ ℋ) |
6 | hvsubcl 27258 | . . . . 5 ⊢ ((𝐴 ∈ ℋ ∧ 𝑥 ∈ ℋ) → (𝐴 −ℎ 𝑥) ∈ ℋ) | |
7 | 2, 5, 6 | syl2anc 691 | . . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐻 ∧ ∀𝑧 ∈ 𝐻 (normℎ‘(𝐴 −ℎ 𝑥)) ≤ (normℎ‘(𝐴 −ℎ 𝑧)))) → (𝐴 −ℎ 𝑥) ∈ ℋ) |
8 | 2 | adantr 480 | . . . . . 6 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐻 ∧ ∀𝑧 ∈ 𝐻 (normℎ‘(𝐴 −ℎ 𝑥)) ≤ (normℎ‘(𝐴 −ℎ 𝑧)))) ∧ 𝑦 ∈ 𝐻) → 𝐴 ∈ ℋ) |
9 | simplrl 796 | . . . . . 6 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐻 ∧ ∀𝑧 ∈ 𝐻 (normℎ‘(𝐴 −ℎ 𝑥)) ≤ (normℎ‘(𝐴 −ℎ 𝑧)))) ∧ 𝑦 ∈ 𝐻) → 𝑥 ∈ 𝐻) | |
10 | simpr 476 | . . . . . 6 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐻 ∧ ∀𝑧 ∈ 𝐻 (normℎ‘(𝐴 −ℎ 𝑥)) ≤ (normℎ‘(𝐴 −ℎ 𝑧)))) ∧ 𝑦 ∈ 𝐻) → 𝑦 ∈ 𝐻) | |
11 | simplrr 797 | . . . . . 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 2610 | . . . . . 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 27634 | . . . . 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 2949 | . . . 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 27468 | . . . . 5 ⊢ 𝐻 ∈ Sℋ |
16 | shocel 27525 | . . . . 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 695 | . . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐻 ∧ ∀𝑧 ∈ 𝐻 (normℎ‘(𝐴 −ℎ 𝑥)) ≤ (normℎ‘(𝐴 −ℎ 𝑧)))) → (𝐴 −ℎ 𝑥) ∈ (⊥‘𝐻)) |
19 | hvpncan3 27283 | . . . . 5 ⊢ ((𝑥 ∈ ℋ ∧ 𝐴 ∈ ℋ) → (𝑥 +ℎ (𝐴 −ℎ 𝑥)) = 𝐴) | |
20 | 5, 2, 19 | syl2anc 691 | . . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐻 ∧ ∀𝑧 ∈ 𝐻 (normℎ‘(𝐴 −ℎ 𝑥)) ≤ (normℎ‘(𝐴 −ℎ 𝑧)))) → (𝑥 +ℎ (𝐴 −ℎ 𝑥)) = 𝐴) |
21 | 20 | eqcomd 2616 | . . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐻 ∧ ∀𝑧 ∈ 𝐻 (normℎ‘(𝐴 −ℎ 𝑥)) ≤ (normℎ‘(𝐴 −ℎ 𝑧)))) → 𝐴 = (𝑥 +ℎ (𝐴 −ℎ 𝑥))) |
22 | oveq2 6557 | . . . . 5 ⊢ (𝑦 = (𝐴 −ℎ 𝑥) → (𝑥 +ℎ 𝑦) = (𝑥 +ℎ (𝐴 −ℎ 𝑥))) | |
23 | 22 | eqeq2d 2620 | . . . 4 ⊢ (𝑦 = (𝐴 −ℎ 𝑥) → (𝐴 = (𝑥 +ℎ 𝑦) ↔ 𝐴 = (𝑥 +ℎ (𝐴 −ℎ 𝑥)))) |
24 | 23 | rspcev 3282 | . . 3 ⊢ (((𝐴 −ℎ 𝑥) ∈ (⊥‘𝐻) ∧ 𝐴 = (𝑥 +ℎ (𝐴 −ℎ 𝑥))) → ∃𝑦 ∈ (⊥‘𝐻)𝐴 = (𝑥 +ℎ 𝑦)) |
25 | 18, 21, 24 | syl2anc 691 | . 2 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐻 ∧ ∀𝑧 ∈ 𝐻 (normℎ‘(𝐴 −ℎ 𝑥)) ≤ (normℎ‘(𝐴 −ℎ 𝑧)))) → ∃𝑦 ∈ (⊥‘𝐻)𝐴 = (𝑥 +ℎ 𝑦)) |
26 | df-hba 27210 | . . . 4 ⊢ ℋ = (BaseSet‘〈〈 +ℎ , ·ℎ 〉, normℎ〉) | |
27 | eqid 2610 | . . . . 5 ⊢ 〈〈 +ℎ , ·ℎ 〉, normℎ〉 = 〈〈 +ℎ , ·ℎ 〉, normℎ〉 | |
28 | 27 | hhvs 27411 | . . . 4 ⊢ −ℎ = ( −𝑣 ‘〈〈 +ℎ , ·ℎ 〉, normℎ〉) |
29 | 27 | hhnm 27412 | . . . 4 ⊢ normℎ = (normCV‘〈〈 +ℎ , ·ℎ 〉, normℎ〉) |
30 | eqid 2610 | . . . . 5 ⊢ 〈〈( +ℎ ↾ (𝐻 × 𝐻)), ( ·ℎ ↾ (ℂ × 𝐻))〉, (normℎ ↾ 𝐻)〉 = 〈〈( +ℎ ↾ (𝐻 × 𝐻)), ( ·ℎ ↾ (ℂ × 𝐻))〉, (normℎ ↾ 𝐻)〉 | |
31 | 30, 15 | hhssba 27512 | . . . 4 ⊢ 𝐻 = (BaseSet‘〈〈( +ℎ ↾ (𝐻 × 𝐻)), ( ·ℎ ↾ (ℂ × 𝐻))〉, (normℎ ↾ 𝐻)〉) |
32 | 27 | hhph 27419 | . . . . 5 ⊢ 〈〈 +ℎ , ·ℎ 〉, normℎ〉 ∈ CPreHilOLD |
33 | 32 | a1i 11 | . . . 4 ⊢ (𝜑 → 〈〈 +ℎ , ·ℎ 〉, normℎ〉 ∈ CPreHilOLD) |
34 | 27, 30 | hhsst 27507 | . . . . . . 7 ⊢ (𝐻 ∈ Sℋ → 〈〈( +ℎ ↾ (𝐻 × 𝐻)), ( ·ℎ ↾ (ℂ × 𝐻))〉, (normℎ ↾ 𝐻)〉 ∈ (SubSp‘〈〈 +ℎ , ·ℎ 〉, normℎ〉)) |
35 | 15, 34 | ax-mp 5 | . . . . . 6 ⊢ 〈〈( +ℎ ↾ (𝐻 × 𝐻)), ( ·ℎ ↾ (ℂ × 𝐻))〉, (normℎ ↾ 𝐻)〉 ∈ (SubSp‘〈〈 +ℎ , ·ℎ 〉, normℎ〉) |
36 | 30, 3 | hhssbn 27521 | . . . . . 6 ⊢ 〈〈( +ℎ ↾ (𝐻 × 𝐻)), ( ·ℎ ↾ (ℂ × 𝐻))〉, (normℎ ↾ 𝐻)〉 ∈ CBan |
37 | elin 3758 | . . . . . 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)) | |
38 | 35, 36, 37 | mpbir2an 957 | . . . . 5 ⊢ 〈〈( +ℎ ↾ (𝐻 × 𝐻)), ( ·ℎ ↾ (ℂ × 𝐻))〉, (normℎ ↾ 𝐻)〉 ∈ ((SubSp‘〈〈 +ℎ , ·ℎ 〉, normℎ〉) ∩ CBan) |
39 | 38 | a1i 11 | . . . 4 ⊢ (𝜑 → 〈〈( +ℎ ↾ (𝐻 × 𝐻)), ( ·ℎ ↾ (ℂ × 𝐻))〉, (normℎ ↾ 𝐻)〉 ∈ ((SubSp‘〈〈 +ℎ , ·ℎ 〉, normℎ〉) ∩ CBan)) |
40 | 26, 28, 29, 31, 33, 39, 1 | minveco 27124 | . . 3 ⊢ (𝜑 → ∃!𝑥 ∈ 𝐻 ∀𝑧 ∈ 𝐻 (normℎ‘(𝐴 −ℎ 𝑥)) ≤ (normℎ‘(𝐴 −ℎ 𝑧))) |
41 | reurex 3137 | . . 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))) | |
42 | 40, 41 | syl 17 | . 2 ⊢ (𝜑 → ∃𝑥 ∈ 𝐻 ∀𝑧 ∈ 𝐻 (normℎ‘(𝐴 −ℎ 𝑥)) ≤ (normℎ‘(𝐴 −ℎ 𝑧))) |
43 | 25, 42 | reximddv 3001 | 1 ⊢ (𝜑 → ∃𝑥 ∈ 𝐻 ∃𝑦 ∈ (⊥‘𝐻)𝐴 = (𝑥 +ℎ 𝑦)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ∀wral 2896 ∃wrex 2897 ∃!wreu 2898 ∩ cin 3539 〈cop 4131 class class class wbr 4583 × cxp 5036 ↾ cres 5040 ‘cfv 5804 (class class class)co 6549 ℂcc 9813 0cc0 9815 1c1 9816 + caddc 9818 ≤ cle 9954 / cdiv 10563 SubSpcss 26960 CPreHilOLDccphlo 27051 CBanccbn 27102 ℋchil 27160 +ℎ cva 27161 ·ℎ csm 27162 ·ih csp 27163 normℎcno 27164 −ℎ cmv 27166 Sℋ csh 27169 Cℋ cch 27170 ⊥cort 27171 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-inf2 8421 ax-cc 9140 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 ax-pre-sup 9893 ax-addf 9894 ax-mulf 9895 ax-hilex 27240 ax-hfvadd 27241 ax-hvcom 27242 ax-hvass 27243 ax-hv0cl 27244 ax-hvaddid 27245 ax-hfvmul 27246 ax-hvmulid 27247 ax-hvmulass 27248 ax-hvdistr1 27249 ax-hvdistr2 27250 ax-hvmul0 27251 ax-hfi 27320 ax-his1 27323 ax-his2 27324 ax-his3 27325 ax-his4 27326 ax-hcompl 27443 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-iin 4458 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-se 4998 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-isom 5813 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-oadd 7451 df-omul 7452 df-er 7629 df-map 7746 df-pm 7747 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-fi 8200 df-sup 8231 df-inf 8232 df-oi 8298 df-card 8648 df-acn 8651 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-div 10564 df-nn 10898 df-2 10956 df-3 10957 df-4 10958 df-n0 11170 df-z 11255 df-uz 11564 df-q 11665 df-rp 11709 df-xneg 11822 df-xadd 11823 df-xmul 11824 df-ico 12052 df-icc 12053 df-fz 12198 df-fl 12455 df-seq 12664 df-exp 12723 df-cj 13687 df-re 13688 df-im 13689 df-sqrt 13823 df-abs 13824 df-clim 14067 df-rlim 14068 df-rest 15906 df-topgen 15927 df-psmet 19559 df-xmet 19560 df-met 19561 df-bl 19562 df-mopn 19563 df-fbas 19564 df-fg 19565 df-top 20521 df-bases 20522 df-topon 20523 df-cld 20633 df-ntr 20634 df-cls 20635 df-nei 20712 df-lm 20843 df-haus 20929 df-fil 21460 df-fm 21552 df-flim 21553 df-flf 21554 df-cfil 22861 df-cau 22862 df-cmet 22863 df-grpo 26731 df-gid 26732 df-ginv 26733 df-gdiv 26734 df-ablo 26783 df-vc 26798 df-nv 26831 df-va 26834 df-ba 26835 df-sm 26836 df-0v 26837 df-vs 26838 df-nmcv 26839 df-ims 26840 df-ssp 26961 df-ph 27052 df-cbn 27103 df-hnorm 27209 df-hba 27210 df-hvsub 27212 df-hlim 27213 df-hcau 27214 df-sh 27448 df-ch 27462 df-oc 27493 df-ch0 27494 |
This theorem is referenced by: pjhth 27636 omlsii 27646 |
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