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Mirrors > Home > MPE Home > Th. List > minveclem1 | Structured version Visualization version GIF version |
Description: Lemma for minvec 23427. The set of all distances from points of 𝑌 to 𝐴 are a nonempty set of nonnegative reals. (Contributed by Mario Carneiro, 8-May-2014.) (Revised by Mario Carneiro, 15-Oct-2015.) |
Ref | Expression |
---|---|
minvec.x | ⊢ 𝑋 = (Base‘𝑈) |
minvec.m | ⊢ − = (-g‘𝑈) |
minvec.n | ⊢ 𝑁 = (norm‘𝑈) |
minvec.u | ⊢ (𝜑 → 𝑈 ∈ ℂPreHil) |
minvec.y | ⊢ (𝜑 → 𝑌 ∈ (LSubSp‘𝑈)) |
minvec.w | ⊢ (𝜑 → (𝑈 ↾s 𝑌) ∈ CMetSp) |
minvec.a | ⊢ (𝜑 → 𝐴 ∈ 𝑋) |
minvec.j | ⊢ 𝐽 = (TopOpen‘𝑈) |
minvec.r | ⊢ 𝑅 = ran (𝑦 ∈ 𝑌 ↦ (𝑁‘(𝐴 − 𝑦))) |
Ref | Expression |
---|---|
minveclem1 | ⊢ (𝜑 → (𝑅 ⊆ ℝ ∧ 𝑅 ≠ ∅ ∧ ∀𝑤 ∈ 𝑅 0 ≤ 𝑤)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | minvec.r | . . 3 ⊢ 𝑅 = ran (𝑦 ∈ 𝑌 ↦ (𝑁‘(𝐴 − 𝑦))) | |
2 | minvec.u | . . . . . . . 8 ⊢ (𝜑 → 𝑈 ∈ ℂPreHil) | |
3 | cphngp 23193 | . . . . . . . 8 ⊢ (𝑈 ∈ ℂPreHil → 𝑈 ∈ NrmGrp) | |
4 | 2, 3 | syl 17 | . . . . . . 7 ⊢ (𝜑 → 𝑈 ∈ NrmGrp) |
5 | 4 | adantr 472 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑌) → 𝑈 ∈ NrmGrp) |
6 | cphlmod 23194 | . . . . . . . . 9 ⊢ (𝑈 ∈ ℂPreHil → 𝑈 ∈ LMod) | |
7 | 2, 6 | syl 17 | . . . . . . . 8 ⊢ (𝜑 → 𝑈 ∈ LMod) |
8 | 7 | adantr 472 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑌) → 𝑈 ∈ LMod) |
9 | minvec.a | . . . . . . . 8 ⊢ (𝜑 → 𝐴 ∈ 𝑋) | |
10 | 9 | adantr 472 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑌) → 𝐴 ∈ 𝑋) |
11 | minvec.y | . . . . . . . . 9 ⊢ (𝜑 → 𝑌 ∈ (LSubSp‘𝑈)) | |
12 | minvec.x | . . . . . . . . . 10 ⊢ 𝑋 = (Base‘𝑈) | |
13 | eqid 2760 | . . . . . . . . . 10 ⊢ (LSubSp‘𝑈) = (LSubSp‘𝑈) | |
14 | 12, 13 | lssss 19159 | . . . . . . . . 9 ⊢ (𝑌 ∈ (LSubSp‘𝑈) → 𝑌 ⊆ 𝑋) |
15 | 11, 14 | syl 17 | . . . . . . . 8 ⊢ (𝜑 → 𝑌 ⊆ 𝑋) |
16 | 15 | sselda 3744 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑌) → 𝑦 ∈ 𝑋) |
17 | minvec.m | . . . . . . . 8 ⊢ − = (-g‘𝑈) | |
18 | 12, 17 | lmodvsubcl 19130 | . . . . . . 7 ⊢ ((𝑈 ∈ LMod ∧ 𝐴 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → (𝐴 − 𝑦) ∈ 𝑋) |
19 | 8, 10, 16, 18 | syl3anc 1477 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑌) → (𝐴 − 𝑦) ∈ 𝑋) |
20 | minvec.n | . . . . . . 7 ⊢ 𝑁 = (norm‘𝑈) | |
21 | 12, 20 | nmcl 22641 | . . . . . 6 ⊢ ((𝑈 ∈ NrmGrp ∧ (𝐴 − 𝑦) ∈ 𝑋) → (𝑁‘(𝐴 − 𝑦)) ∈ ℝ) |
22 | 5, 19, 21 | syl2anc 696 | . . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑌) → (𝑁‘(𝐴 − 𝑦)) ∈ ℝ) |
23 | eqid 2760 | . . . . 5 ⊢ (𝑦 ∈ 𝑌 ↦ (𝑁‘(𝐴 − 𝑦))) = (𝑦 ∈ 𝑌 ↦ (𝑁‘(𝐴 − 𝑦))) | |
24 | 22, 23 | fmptd 6549 | . . . 4 ⊢ (𝜑 → (𝑦 ∈ 𝑌 ↦ (𝑁‘(𝐴 − 𝑦))):𝑌⟶ℝ) |
25 | frn 6214 | . . . 4 ⊢ ((𝑦 ∈ 𝑌 ↦ (𝑁‘(𝐴 − 𝑦))):𝑌⟶ℝ → ran (𝑦 ∈ 𝑌 ↦ (𝑁‘(𝐴 − 𝑦))) ⊆ ℝ) | |
26 | 24, 25 | syl 17 | . . 3 ⊢ (𝜑 → ran (𝑦 ∈ 𝑌 ↦ (𝑁‘(𝐴 − 𝑦))) ⊆ ℝ) |
27 | 1, 26 | syl5eqss 3790 | . 2 ⊢ (𝜑 → 𝑅 ⊆ ℝ) |
28 | 13 | lssn0 19163 | . . . 4 ⊢ (𝑌 ∈ (LSubSp‘𝑈) → 𝑌 ≠ ∅) |
29 | 11, 28 | syl 17 | . . 3 ⊢ (𝜑 → 𝑌 ≠ ∅) |
30 | 1 | eqeq1i 2765 | . . . . 5 ⊢ (𝑅 = ∅ ↔ ran (𝑦 ∈ 𝑌 ↦ (𝑁‘(𝐴 − 𝑦))) = ∅) |
31 | dm0rn0 5497 | . . . . 5 ⊢ (dom (𝑦 ∈ 𝑌 ↦ (𝑁‘(𝐴 − 𝑦))) = ∅ ↔ ran (𝑦 ∈ 𝑌 ↦ (𝑁‘(𝐴 − 𝑦))) = ∅) | |
32 | fvex 6363 | . . . . . . 7 ⊢ (𝑁‘(𝐴 − 𝑦)) ∈ V | |
33 | 32, 23 | dmmpti 6184 | . . . . . 6 ⊢ dom (𝑦 ∈ 𝑌 ↦ (𝑁‘(𝐴 − 𝑦))) = 𝑌 |
34 | 33 | eqeq1i 2765 | . . . . 5 ⊢ (dom (𝑦 ∈ 𝑌 ↦ (𝑁‘(𝐴 − 𝑦))) = ∅ ↔ 𝑌 = ∅) |
35 | 30, 31, 34 | 3bitr2i 288 | . . . 4 ⊢ (𝑅 = ∅ ↔ 𝑌 = ∅) |
36 | 35 | necon3bii 2984 | . . 3 ⊢ (𝑅 ≠ ∅ ↔ 𝑌 ≠ ∅) |
37 | 29, 36 | sylibr 224 | . 2 ⊢ (𝜑 → 𝑅 ≠ ∅) |
38 | 12, 20 | nmge0 22642 | . . . . . 6 ⊢ ((𝑈 ∈ NrmGrp ∧ (𝐴 − 𝑦) ∈ 𝑋) → 0 ≤ (𝑁‘(𝐴 − 𝑦))) |
39 | 5, 19, 38 | syl2anc 696 | . . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑌) → 0 ≤ (𝑁‘(𝐴 − 𝑦))) |
40 | 39 | ralrimiva 3104 | . . . 4 ⊢ (𝜑 → ∀𝑦 ∈ 𝑌 0 ≤ (𝑁‘(𝐴 − 𝑦))) |
41 | 32 | rgenw 3062 | . . . . 5 ⊢ ∀𝑦 ∈ 𝑌 (𝑁‘(𝐴 − 𝑦)) ∈ V |
42 | breq2 4808 | . . . . . 6 ⊢ (𝑤 = (𝑁‘(𝐴 − 𝑦)) → (0 ≤ 𝑤 ↔ 0 ≤ (𝑁‘(𝐴 − 𝑦)))) | |
43 | 23, 42 | ralrnmpt 6532 | . . . . 5 ⊢ (∀𝑦 ∈ 𝑌 (𝑁‘(𝐴 − 𝑦)) ∈ V → (∀𝑤 ∈ ran (𝑦 ∈ 𝑌 ↦ (𝑁‘(𝐴 − 𝑦)))0 ≤ 𝑤 ↔ ∀𝑦 ∈ 𝑌 0 ≤ (𝑁‘(𝐴 − 𝑦)))) |
44 | 41, 43 | ax-mp 5 | . . . 4 ⊢ (∀𝑤 ∈ ran (𝑦 ∈ 𝑌 ↦ (𝑁‘(𝐴 − 𝑦)))0 ≤ 𝑤 ↔ ∀𝑦 ∈ 𝑌 0 ≤ (𝑁‘(𝐴 − 𝑦))) |
45 | 40, 44 | sylibr 224 | . . 3 ⊢ (𝜑 → ∀𝑤 ∈ ran (𝑦 ∈ 𝑌 ↦ (𝑁‘(𝐴 − 𝑦)))0 ≤ 𝑤) |
46 | 1 | raleqi 3281 | . . 3 ⊢ (∀𝑤 ∈ 𝑅 0 ≤ 𝑤 ↔ ∀𝑤 ∈ ran (𝑦 ∈ 𝑌 ↦ (𝑁‘(𝐴 − 𝑦)))0 ≤ 𝑤) |
47 | 45, 46 | sylibr 224 | . 2 ⊢ (𝜑 → ∀𝑤 ∈ 𝑅 0 ≤ 𝑤) |
48 | 27, 37, 47 | 3jca 1123 | 1 ⊢ (𝜑 → (𝑅 ⊆ ℝ ∧ 𝑅 ≠ ∅ ∧ ∀𝑤 ∈ 𝑅 0 ≤ 𝑤)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∧ wa 383 ∧ w3a 1072 = wceq 1632 ∈ wcel 2139 ≠ wne 2932 ∀wral 3050 Vcvv 3340 ⊆ wss 3715 ∅c0 4058 class class class wbr 4804 ↦ cmpt 4881 dom cdm 5266 ran crn 5267 ⟶wf 6045 ‘cfv 6049 (class class class)co 6814 ℝcr 10147 0cc0 10148 ≤ cle 10287 Basecbs 16079 ↾s cress 16080 TopOpenctopn 16304 -gcsg 17645 LModclmod 19085 LSubSpclss 19154 normcnm 22602 NrmGrpcngp 22603 ℂPreHilccph 23186 CMetSpccms 23349 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1871 ax-4 1886 ax-5 1988 ax-6 2054 ax-7 2090 ax-8 2141 ax-9 2148 ax-10 2168 ax-11 2183 ax-12 2196 ax-13 2391 ax-ext 2740 ax-rep 4923 ax-sep 4933 ax-nul 4941 ax-pow 4992 ax-pr 5055 ax-un 7115 ax-cnex 10204 ax-resscn 10205 ax-1cn 10206 ax-icn 10207 ax-addcl 10208 ax-addrcl 10209 ax-mulcl 10210 ax-mulrcl 10211 ax-mulcom 10212 ax-addass 10213 ax-mulass 10214 ax-distr 10215 ax-i2m1 10216 ax-1ne0 10217 ax-1rid 10218 ax-rnegex 10219 ax-rrecex 10220 ax-cnre 10221 ax-pre-lttri 10222 ax-pre-lttrn 10223 ax-pre-ltadd 10224 ax-pre-mulgt0 10225 ax-pre-sup 10226 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1073 df-3an 1074 df-tru 1635 df-ex 1854 df-nf 1859 df-sb 2047 df-eu 2611 df-mo 2612 df-clab 2747 df-cleq 2753 df-clel 2756 df-nfc 2891 df-ne 2933 df-nel 3036 df-ral 3055 df-rex 3056 df-reu 3057 df-rmo 3058 df-rab 3059 df-v 3342 df-sbc 3577 df-csb 3675 df-dif 3718 df-un 3720 df-in 3722 df-ss 3729 df-pss 3731 df-nul 4059 df-if 4231 df-pw 4304 df-sn 4322 df-pr 4324 df-tp 4326 df-op 4328 df-uni 4589 df-iun 4674 df-br 4805 df-opab 4865 df-mpt 4882 df-tr 4905 df-id 5174 df-eprel 5179 df-po 5187 df-so 5188 df-fr 5225 df-we 5227 df-xp 5272 df-rel 5273 df-cnv 5274 df-co 5275 df-dm 5276 df-rn 5277 df-res 5278 df-ima 5279 df-pred 5841 df-ord 5887 df-on 5888 df-lim 5889 df-suc 5890 df-iota 6012 df-fun 6051 df-fn 6052 df-f 6053 df-f1 6054 df-fo 6055 df-f1o 6056 df-fv 6057 df-riota 6775 df-ov 6817 df-oprab 6818 df-mpt2 6819 df-om 7232 df-1st 7334 df-2nd 7335 df-wrecs 7577 df-recs 7638 df-rdg 7676 df-er 7913 df-map 8027 df-en 8124 df-dom 8125 df-sdom 8126 df-sup 8515 df-inf 8516 df-pnf 10288 df-mnf 10289 df-xr 10290 df-ltxr 10291 df-le 10292 df-sub 10480 df-neg 10481 df-div 10897 df-nn 11233 df-2 11291 df-n0 11505 df-z 11590 df-uz 11900 df-q 12002 df-rp 12046 df-xneg 12159 df-xadd 12160 df-xmul 12161 df-0g 16324 df-topgen 16326 df-mgm 17463 df-sgrp 17505 df-mnd 17516 df-grp 17646 df-minusg 17647 df-sbg 17648 df-lmod 19087 df-lss 19155 df-psmet 19960 df-xmet 19961 df-met 19962 df-bl 19963 df-mopn 19964 df-top 20921 df-topon 20938 df-topsp 20959 df-bases 20972 df-xms 22346 df-ms 22347 df-nm 22608 df-ngp 22609 df-nlm 22612 df-cph 23188 |
This theorem is referenced by: minveclem4c 23416 minveclem2 23417 minveclem3b 23419 minveclem4 23423 minveclem6 23425 |
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