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Mirrors > Home > MPE Home > Th. List > minvecolem4c | Structured version Visualization version GIF version |
Description: Lemma for minveco 27740. The infimum of the distances to 𝐴 is a real number. (Contributed by Mario Carneiro, 16-Jun-2014.) (Revised by AV, 4-Oct-2020.) (New usage is discouraged.) |
Ref | Expression |
---|---|
minveco.x | ⊢ 𝑋 = (BaseSet‘𝑈) |
minveco.m | ⊢ 𝑀 = ( −𝑣 ‘𝑈) |
minveco.n | ⊢ 𝑁 = (normCV‘𝑈) |
minveco.y | ⊢ 𝑌 = (BaseSet‘𝑊) |
minveco.u | ⊢ (𝜑 → 𝑈 ∈ CPreHilOLD) |
minveco.w | ⊢ (𝜑 → 𝑊 ∈ ((SubSp‘𝑈) ∩ CBan)) |
minveco.a | ⊢ (𝜑 → 𝐴 ∈ 𝑋) |
minveco.d | ⊢ 𝐷 = (IndMet‘𝑈) |
minveco.j | ⊢ 𝐽 = (MetOpen‘𝐷) |
minveco.r | ⊢ 𝑅 = ran (𝑦 ∈ 𝑌 ↦ (𝑁‘(𝐴𝑀𝑦))) |
minveco.s | ⊢ 𝑆 = inf(𝑅, ℝ, < ) |
minveco.f | ⊢ (𝜑 → 𝐹:ℕ⟶𝑌) |
minveco.1 | ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((𝐴𝐷(𝐹‘𝑛))↑2) ≤ ((𝑆↑2) + (1 / 𝑛))) |
Ref | Expression |
---|---|
minvecolem4c | ⊢ (𝜑 → 𝑆 ∈ ℝ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | minveco.s | . 2 ⊢ 𝑆 = inf(𝑅, ℝ, < ) | |
2 | minveco.x | . . . . 5 ⊢ 𝑋 = (BaseSet‘𝑈) | |
3 | minveco.m | . . . . 5 ⊢ 𝑀 = ( −𝑣 ‘𝑈) | |
4 | minveco.n | . . . . 5 ⊢ 𝑁 = (normCV‘𝑈) | |
5 | minveco.y | . . . . 5 ⊢ 𝑌 = (BaseSet‘𝑊) | |
6 | minveco.u | . . . . 5 ⊢ (𝜑 → 𝑈 ∈ CPreHilOLD) | |
7 | minveco.w | . . . . 5 ⊢ (𝜑 → 𝑊 ∈ ((SubSp‘𝑈) ∩ CBan)) | |
8 | minveco.a | . . . . 5 ⊢ (𝜑 → 𝐴 ∈ 𝑋) | |
9 | minveco.d | . . . . 5 ⊢ 𝐷 = (IndMet‘𝑈) | |
10 | minveco.j | . . . . 5 ⊢ 𝐽 = (MetOpen‘𝐷) | |
11 | minveco.r | . . . . 5 ⊢ 𝑅 = ran (𝑦 ∈ 𝑌 ↦ (𝑁‘(𝐴𝑀𝑦))) | |
12 | 2, 3, 4, 5, 6, 7, 8, 9, 10, 11 | minvecolem1 27730 | . . . 4 ⊢ (𝜑 → (𝑅 ⊆ ℝ ∧ 𝑅 ≠ ∅ ∧ ∀𝑤 ∈ 𝑅 0 ≤ 𝑤)) |
13 | 12 | simp1d 1073 | . . 3 ⊢ (𝜑 → 𝑅 ⊆ ℝ) |
14 | 12 | simp2d 1074 | . . 3 ⊢ (𝜑 → 𝑅 ≠ ∅) |
15 | 0re 10040 | . . . 4 ⊢ 0 ∈ ℝ | |
16 | 12 | simp3d 1075 | . . . 4 ⊢ (𝜑 → ∀𝑤 ∈ 𝑅 0 ≤ 𝑤) |
17 | breq1 4656 | . . . . . 6 ⊢ (𝑥 = 0 → (𝑥 ≤ 𝑤 ↔ 0 ≤ 𝑤)) | |
18 | 17 | ralbidv 2986 | . . . . 5 ⊢ (𝑥 = 0 → (∀𝑤 ∈ 𝑅 𝑥 ≤ 𝑤 ↔ ∀𝑤 ∈ 𝑅 0 ≤ 𝑤)) |
19 | 18 | rspcev 3309 | . . . 4 ⊢ ((0 ∈ ℝ ∧ ∀𝑤 ∈ 𝑅 0 ≤ 𝑤) → ∃𝑥 ∈ ℝ ∀𝑤 ∈ 𝑅 𝑥 ≤ 𝑤) |
20 | 15, 16, 19 | sylancr 695 | . . 3 ⊢ (𝜑 → ∃𝑥 ∈ ℝ ∀𝑤 ∈ 𝑅 𝑥 ≤ 𝑤) |
21 | infrecl 11005 | . . 3 ⊢ ((𝑅 ⊆ ℝ ∧ 𝑅 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑤 ∈ 𝑅 𝑥 ≤ 𝑤) → inf(𝑅, ℝ, < ) ∈ ℝ) | |
22 | 13, 14, 20, 21 | syl3anc 1326 | . 2 ⊢ (𝜑 → inf(𝑅, ℝ, < ) ∈ ℝ) |
23 | 1, 22 | syl5eqel 2705 | 1 ⊢ (𝜑 → 𝑆 ∈ ℝ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 384 = wceq 1483 ∈ wcel 1990 ≠ wne 2794 ∀wral 2912 ∃wrex 2913 ∩ cin 3573 ⊆ wss 3574 ∅c0 3915 class class class wbr 4653 ↦ cmpt 4729 ran crn 5115 ⟶wf 5884 ‘cfv 5888 (class class class)co 6650 infcinf 8347 ℝcr 9935 0cc0 9936 1c1 9937 + caddc 9939 < clt 10074 ≤ cle 10075 / cdiv 10684 ℕcn 11020 2c2 11070 ↑cexp 12860 MetOpencmopn 19736 BaseSetcba 27441 −𝑣 cnsb 27444 normCVcnmcv 27445 IndMetcims 27446 SubSpcss 27576 CPreHilOLDccphlo 27667 CBanccbn 27718 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 ax-pre-sup 10014 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-sup 8348 df-inf 8349 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-div 10685 df-nn 11021 df-2 11079 df-3 11080 df-n0 11293 df-z 11378 df-uz 11688 df-rp 11833 df-seq 12802 df-exp 12861 df-cj 13839 df-re 13840 df-im 13841 df-sqrt 13975 df-abs 13976 df-grpo 27347 df-gid 27348 df-ginv 27349 df-gdiv 27350 df-ablo 27399 df-vc 27414 df-nv 27447 df-va 27450 df-ba 27451 df-sm 27452 df-0v 27453 df-vs 27454 df-nmcv 27455 df-ssp 27577 df-ph 27668 df-cbn 27719 |
This theorem is referenced by: minvecolem4 27736 |
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