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Mirrors > Home > MPE Home > Th. List > minvecolem1 | Structured version Visualization version GIF version |
Description: Lemma for minveco 28312. 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.) (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))) |
Ref | Expression |
---|---|
minvecolem1 | ⊢ (𝜑 → (𝑅 ⊆ ℝ ∧ 𝑅 ≠ ∅ ∧ ∀𝑤 ∈ 𝑅 0 ≤ 𝑤)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | minveco.r | . . 3 ⊢ 𝑅 = ran (𝑦 ∈ 𝑌 ↦ (𝑁‘(𝐴𝑀𝑦))) | |
2 | minveco.u | . . . . . . . 8 ⊢ (𝜑 → 𝑈 ∈ CPreHilOLD) | |
3 | phnv 28241 | . . . . . . . 8 ⊢ (𝑈 ∈ CPreHilOLD → 𝑈 ∈ NrmCVec) | |
4 | 2, 3 | syl 17 | . . . . . . 7 ⊢ (𝜑 → 𝑈 ∈ NrmCVec) |
5 | 4 | adantr 474 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑌) → 𝑈 ∈ NrmCVec) |
6 | minveco.a | . . . . . . . 8 ⊢ (𝜑 → 𝐴 ∈ 𝑋) | |
7 | 6 | adantr 474 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑌) → 𝐴 ∈ 𝑋) |
8 | minveco.w | . . . . . . . . . . 11 ⊢ (𝜑 → 𝑊 ∈ ((SubSp‘𝑈) ∩ CBan)) | |
9 | elin 4019 | . . . . . . . . . . 11 ⊢ (𝑊 ∈ ((SubSp‘𝑈) ∩ CBan) ↔ (𝑊 ∈ (SubSp‘𝑈) ∧ 𝑊 ∈ CBan)) | |
10 | 8, 9 | sylib 210 | . . . . . . . . . 10 ⊢ (𝜑 → (𝑊 ∈ (SubSp‘𝑈) ∧ 𝑊 ∈ CBan)) |
11 | 10 | simpld 490 | . . . . . . . . 9 ⊢ (𝜑 → 𝑊 ∈ (SubSp‘𝑈)) |
12 | minveco.x | . . . . . . . . . 10 ⊢ 𝑋 = (BaseSet‘𝑈) | |
13 | minveco.y | . . . . . . . . . 10 ⊢ 𝑌 = (BaseSet‘𝑊) | |
14 | eqid 2778 | . . . . . . . . . 10 ⊢ (SubSp‘𝑈) = (SubSp‘𝑈) | |
15 | 12, 13, 14 | sspba 28154 | . . . . . . . . 9 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ (SubSp‘𝑈)) → 𝑌 ⊆ 𝑋) |
16 | 4, 11, 15 | syl2anc 579 | . . . . . . . 8 ⊢ (𝜑 → 𝑌 ⊆ 𝑋) |
17 | 16 | sselda 3821 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑌) → 𝑦 ∈ 𝑋) |
18 | minveco.m | . . . . . . . 8 ⊢ 𝑀 = ( −𝑣 ‘𝑈) | |
19 | 12, 18 | nvmcl 28073 | . . . . . . 7 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → (𝐴𝑀𝑦) ∈ 𝑋) |
20 | 5, 7, 17, 19 | syl3anc 1439 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑌) → (𝐴𝑀𝑦) ∈ 𝑋) |
21 | minveco.n | . . . . . . 7 ⊢ 𝑁 = (normCV‘𝑈) | |
22 | 12, 21 | nvcl 28088 | . . . . . 6 ⊢ ((𝑈 ∈ NrmCVec ∧ (𝐴𝑀𝑦) ∈ 𝑋) → (𝑁‘(𝐴𝑀𝑦)) ∈ ℝ) |
23 | 5, 20, 22 | syl2anc 579 | . . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑌) → (𝑁‘(𝐴𝑀𝑦)) ∈ ℝ) |
24 | 23 | fmpttd 6649 | . . . 4 ⊢ (𝜑 → (𝑦 ∈ 𝑌 ↦ (𝑁‘(𝐴𝑀𝑦))):𝑌⟶ℝ) |
25 | 24 | frnd 6298 | . . 3 ⊢ (𝜑 → ran (𝑦 ∈ 𝑌 ↦ (𝑁‘(𝐴𝑀𝑦))) ⊆ ℝ) |
26 | 1, 25 | syl5eqss 3868 | . 2 ⊢ (𝜑 → 𝑅 ⊆ ℝ) |
27 | 10 | simprd 491 | . . . . . 6 ⊢ (𝜑 → 𝑊 ∈ CBan) |
28 | bnnv 28294 | . . . . . 6 ⊢ (𝑊 ∈ CBan → 𝑊 ∈ NrmCVec) | |
29 | eqid 2778 | . . . . . . 7 ⊢ (0vec‘𝑊) = (0vec‘𝑊) | |
30 | 13, 29 | nvzcl 28061 | . . . . . 6 ⊢ (𝑊 ∈ NrmCVec → (0vec‘𝑊) ∈ 𝑌) |
31 | 27, 28, 30 | 3syl 18 | . . . . 5 ⊢ (𝜑 → (0vec‘𝑊) ∈ 𝑌) |
32 | fvex 6459 | . . . . . 6 ⊢ (𝑁‘(𝐴𝑀𝑦)) ∈ V | |
33 | eqid 2778 | . . . . . 6 ⊢ (𝑦 ∈ 𝑌 ↦ (𝑁‘(𝐴𝑀𝑦))) = (𝑦 ∈ 𝑌 ↦ (𝑁‘(𝐴𝑀𝑦))) | |
34 | 32, 33 | dmmpti 6269 | . . . . 5 ⊢ dom (𝑦 ∈ 𝑌 ↦ (𝑁‘(𝐴𝑀𝑦))) = 𝑌 |
35 | 31, 34 | syl6eleqr 2870 | . . . 4 ⊢ (𝜑 → (0vec‘𝑊) ∈ dom (𝑦 ∈ 𝑌 ↦ (𝑁‘(𝐴𝑀𝑦)))) |
36 | 35 | ne0d 4150 | . . 3 ⊢ (𝜑 → dom (𝑦 ∈ 𝑌 ↦ (𝑁‘(𝐴𝑀𝑦))) ≠ ∅) |
37 | dm0rn0 5587 | . . . . 5 ⊢ (dom (𝑦 ∈ 𝑌 ↦ (𝑁‘(𝐴𝑀𝑦))) = ∅ ↔ ran (𝑦 ∈ 𝑌 ↦ (𝑁‘(𝐴𝑀𝑦))) = ∅) | |
38 | 1 | eqeq1i 2783 | . . . . 5 ⊢ (𝑅 = ∅ ↔ ran (𝑦 ∈ 𝑌 ↦ (𝑁‘(𝐴𝑀𝑦))) = ∅) |
39 | 37, 38 | bitr4i 270 | . . . 4 ⊢ (dom (𝑦 ∈ 𝑌 ↦ (𝑁‘(𝐴𝑀𝑦))) = ∅ ↔ 𝑅 = ∅) |
40 | 39 | necon3bii 3021 | . . 3 ⊢ (dom (𝑦 ∈ 𝑌 ↦ (𝑁‘(𝐴𝑀𝑦))) ≠ ∅ ↔ 𝑅 ≠ ∅) |
41 | 36, 40 | sylib 210 | . 2 ⊢ (𝜑 → 𝑅 ≠ ∅) |
42 | 12, 21 | nvge0 28100 | . . . . . 6 ⊢ ((𝑈 ∈ NrmCVec ∧ (𝐴𝑀𝑦) ∈ 𝑋) → 0 ≤ (𝑁‘(𝐴𝑀𝑦))) |
43 | 5, 20, 42 | syl2anc 579 | . . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑌) → 0 ≤ (𝑁‘(𝐴𝑀𝑦))) |
44 | 43 | ralrimiva 3148 | . . . 4 ⊢ (𝜑 → ∀𝑦 ∈ 𝑌 0 ≤ (𝑁‘(𝐴𝑀𝑦))) |
45 | 32 | rgenw 3106 | . . . . 5 ⊢ ∀𝑦 ∈ 𝑌 (𝑁‘(𝐴𝑀𝑦)) ∈ V |
46 | breq2 4890 | . . . . . 6 ⊢ (𝑤 = (𝑁‘(𝐴𝑀𝑦)) → (0 ≤ 𝑤 ↔ 0 ≤ (𝑁‘(𝐴𝑀𝑦)))) | |
47 | 33, 46 | ralrnmpt 6632 | . . . . 5 ⊢ (∀𝑦 ∈ 𝑌 (𝑁‘(𝐴𝑀𝑦)) ∈ V → (∀𝑤 ∈ ran (𝑦 ∈ 𝑌 ↦ (𝑁‘(𝐴𝑀𝑦)))0 ≤ 𝑤 ↔ ∀𝑦 ∈ 𝑌 0 ≤ (𝑁‘(𝐴𝑀𝑦)))) |
48 | 45, 47 | ax-mp 5 | . . . 4 ⊢ (∀𝑤 ∈ ran (𝑦 ∈ 𝑌 ↦ (𝑁‘(𝐴𝑀𝑦)))0 ≤ 𝑤 ↔ ∀𝑦 ∈ 𝑌 0 ≤ (𝑁‘(𝐴𝑀𝑦))) |
49 | 44, 48 | sylibr 226 | . . 3 ⊢ (𝜑 → ∀𝑤 ∈ ran (𝑦 ∈ 𝑌 ↦ (𝑁‘(𝐴𝑀𝑦)))0 ≤ 𝑤) |
50 | 1 | raleqi 3338 | . . 3 ⊢ (∀𝑤 ∈ 𝑅 0 ≤ 𝑤 ↔ ∀𝑤 ∈ ran (𝑦 ∈ 𝑌 ↦ (𝑁‘(𝐴𝑀𝑦)))0 ≤ 𝑤) |
51 | 49, 50 | sylibr 226 | . 2 ⊢ (𝜑 → ∀𝑤 ∈ 𝑅 0 ≤ 𝑤) |
52 | 26, 41, 51 | 3jca 1119 | 1 ⊢ (𝜑 → (𝑅 ⊆ ℝ ∧ 𝑅 ≠ ∅ ∧ ∀𝑤 ∈ 𝑅 0 ≤ 𝑤)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∧ wa 386 ∧ w3a 1071 = wceq 1601 ∈ wcel 2107 ≠ wne 2969 ∀wral 3090 Vcvv 3398 ∩ cin 3791 ⊆ wss 3792 ∅c0 4141 class class class wbr 4886 ↦ cmpt 4965 dom cdm 5355 ran crn 5356 ‘cfv 6135 (class class class)co 6922 ℝcr 10271 0cc0 10272 ≤ cle 10412 MetOpencmopn 20132 NrmCVeccnv 28011 BaseSetcba 28013 0veccn0v 28015 −𝑣 cnsb 28016 normCVcnmcv 28017 IndMetcims 28018 SubSpcss 28148 CPreHilOLDccphlo 28239 CBanccbn 28290 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-rep 5006 ax-sep 5017 ax-nul 5025 ax-pow 5077 ax-pr 5138 ax-un 7226 ax-cnex 10328 ax-resscn 10329 ax-1cn 10330 ax-icn 10331 ax-addcl 10332 ax-addrcl 10333 ax-mulcl 10334 ax-mulrcl 10335 ax-mulcom 10336 ax-addass 10337 ax-mulass 10338 ax-distr 10339 ax-i2m1 10340 ax-1ne0 10341 ax-1rid 10342 ax-rnegex 10343 ax-rrecex 10344 ax-cnre 10345 ax-pre-lttri 10346 ax-pre-lttrn 10347 ax-pre-ltadd 10348 ax-pre-mulgt0 10349 ax-pre-sup 10350 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-nel 3076 df-ral 3095 df-rex 3096 df-reu 3097 df-rmo 3098 df-rab 3099 df-v 3400 df-sbc 3653 df-csb 3752 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-pss 3808 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-tp 4403 df-op 4405 df-uni 4672 df-iun 4755 df-br 4887 df-opab 4949 df-mpt 4966 df-tr 4988 df-id 5261 df-eprel 5266 df-po 5274 df-so 5275 df-fr 5314 df-we 5316 df-xp 5361 df-rel 5362 df-cnv 5363 df-co 5364 df-dm 5365 df-rn 5366 df-res 5367 df-ima 5368 df-pred 5933 df-ord 5979 df-on 5980 df-lim 5981 df-suc 5982 df-iota 6099 df-fun 6137 df-fn 6138 df-f 6139 df-f1 6140 df-fo 6141 df-f1o 6142 df-fv 6143 df-riota 6883 df-ov 6925 df-oprab 6926 df-mpt2 6927 df-om 7344 df-1st 7445 df-2nd 7446 df-wrecs 7689 df-recs 7751 df-rdg 7789 df-er 8026 df-en 8242 df-dom 8243 df-sdom 8244 df-sup 8636 df-pnf 10413 df-mnf 10414 df-xr 10415 df-ltxr 10416 df-le 10417 df-sub 10608 df-neg 10609 df-div 11033 df-nn 11375 df-2 11438 df-3 11439 df-n0 11643 df-z 11729 df-uz 11993 df-rp 12138 df-seq 13120 df-exp 13179 df-cj 14246 df-re 14247 df-im 14248 df-sqrt 14382 df-abs 14383 df-grpo 27920 df-gid 27921 df-ginv 27922 df-gdiv 27923 df-ablo 27972 df-vc 27986 df-nv 28019 df-va 28022 df-ba 28023 df-sm 28024 df-0v 28025 df-vs 28026 df-nmcv 28027 df-ssp 28149 df-ph 28240 df-cbn 28291 |
This theorem is referenced by: minvecolem2 28303 minvecolem3 28304 minvecolem4c 28307 minvecolem4 28308 minvecolem5 28309 minvecolem6 28310 |
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