Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > minvecolem1 | Structured version Visualization version GIF version |
Description: Lemma for minveco 27124. 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 27053 | . . . . . . . 8 ⊢ (𝑈 ∈ CPreHilOLD → 𝑈 ∈ NrmCVec) | |
4 | 2, 3 | syl 17 | . . . . . . 7 ⊢ (𝜑 → 𝑈 ∈ NrmCVec) |
5 | 4 | adantr 480 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑌) → 𝑈 ∈ NrmCVec) |
6 | minveco.a | . . . . . . . 8 ⊢ (𝜑 → 𝐴 ∈ 𝑋) | |
7 | 6 | adantr 480 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑌) → 𝐴 ∈ 𝑋) |
8 | minveco.w | . . . . . . . . . . 11 ⊢ (𝜑 → 𝑊 ∈ ((SubSp‘𝑈) ∩ CBan)) | |
9 | elin 3758 | . . . . . . . . . . 11 ⊢ (𝑊 ∈ ((SubSp‘𝑈) ∩ CBan) ↔ (𝑊 ∈ (SubSp‘𝑈) ∧ 𝑊 ∈ CBan)) | |
10 | 8, 9 | sylib 207 | . . . . . . . . . 10 ⊢ (𝜑 → (𝑊 ∈ (SubSp‘𝑈) ∧ 𝑊 ∈ CBan)) |
11 | 10 | simpld 474 | . . . . . . . . 9 ⊢ (𝜑 → 𝑊 ∈ (SubSp‘𝑈)) |
12 | minveco.x | . . . . . . . . . 10 ⊢ 𝑋 = (BaseSet‘𝑈) | |
13 | minveco.y | . . . . . . . . . 10 ⊢ 𝑌 = (BaseSet‘𝑊) | |
14 | eqid 2610 | . . . . . . . . . 10 ⊢ (SubSp‘𝑈) = (SubSp‘𝑈) | |
15 | 12, 13, 14 | sspba 26966 | . . . . . . . . 9 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ (SubSp‘𝑈)) → 𝑌 ⊆ 𝑋) |
16 | 4, 11, 15 | syl2anc 691 | . . . . . . . 8 ⊢ (𝜑 → 𝑌 ⊆ 𝑋) |
17 | 16 | sselda 3568 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑌) → 𝑦 ∈ 𝑋) |
18 | minveco.m | . . . . . . . 8 ⊢ 𝑀 = ( −𝑣 ‘𝑈) | |
19 | 12, 18 | nvmcl 26885 | . . . . . . 7 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → (𝐴𝑀𝑦) ∈ 𝑋) |
20 | 5, 7, 17, 19 | syl3anc 1318 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑌) → (𝐴𝑀𝑦) ∈ 𝑋) |
21 | minveco.n | . . . . . . 7 ⊢ 𝑁 = (normCV‘𝑈) | |
22 | 12, 21 | nvcl 26900 | . . . . . 6 ⊢ ((𝑈 ∈ NrmCVec ∧ (𝐴𝑀𝑦) ∈ 𝑋) → (𝑁‘(𝐴𝑀𝑦)) ∈ ℝ) |
23 | 5, 20, 22 | syl2anc 691 | . . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑌) → (𝑁‘(𝐴𝑀𝑦)) ∈ ℝ) |
24 | eqid 2610 | . . . . 5 ⊢ (𝑦 ∈ 𝑌 ↦ (𝑁‘(𝐴𝑀𝑦))) = (𝑦 ∈ 𝑌 ↦ (𝑁‘(𝐴𝑀𝑦))) | |
25 | 23, 24 | fmptd 6292 | . . . 4 ⊢ (𝜑 → (𝑦 ∈ 𝑌 ↦ (𝑁‘(𝐴𝑀𝑦))):𝑌⟶ℝ) |
26 | frn 5966 | . . . 4 ⊢ ((𝑦 ∈ 𝑌 ↦ (𝑁‘(𝐴𝑀𝑦))):𝑌⟶ℝ → ran (𝑦 ∈ 𝑌 ↦ (𝑁‘(𝐴𝑀𝑦))) ⊆ ℝ) | |
27 | 25, 26 | syl 17 | . . 3 ⊢ (𝜑 → ran (𝑦 ∈ 𝑌 ↦ (𝑁‘(𝐴𝑀𝑦))) ⊆ ℝ) |
28 | 1, 27 | syl5eqss 3612 | . 2 ⊢ (𝜑 → 𝑅 ⊆ ℝ) |
29 | 10 | simprd 478 | . . . . . 6 ⊢ (𝜑 → 𝑊 ∈ CBan) |
30 | bnnv 27106 | . . . . . 6 ⊢ (𝑊 ∈ CBan → 𝑊 ∈ NrmCVec) | |
31 | eqid 2610 | . . . . . . 7 ⊢ (0vec‘𝑊) = (0vec‘𝑊) | |
32 | 13, 31 | nvzcl 26873 | . . . . . 6 ⊢ (𝑊 ∈ NrmCVec → (0vec‘𝑊) ∈ 𝑌) |
33 | 29, 30, 32 | 3syl 18 | . . . . 5 ⊢ (𝜑 → (0vec‘𝑊) ∈ 𝑌) |
34 | fvex 6113 | . . . . . 6 ⊢ (𝑁‘(𝐴𝑀𝑦)) ∈ V | |
35 | 34, 24 | dmmpti 5936 | . . . . 5 ⊢ dom (𝑦 ∈ 𝑌 ↦ (𝑁‘(𝐴𝑀𝑦))) = 𝑌 |
36 | 33, 35 | syl6eleqr 2699 | . . . 4 ⊢ (𝜑 → (0vec‘𝑊) ∈ dom (𝑦 ∈ 𝑌 ↦ (𝑁‘(𝐴𝑀𝑦)))) |
37 | ne0i 3880 | . . . 4 ⊢ ((0vec‘𝑊) ∈ dom (𝑦 ∈ 𝑌 ↦ (𝑁‘(𝐴𝑀𝑦))) → dom (𝑦 ∈ 𝑌 ↦ (𝑁‘(𝐴𝑀𝑦))) ≠ ∅) | |
38 | 36, 37 | syl 17 | . . 3 ⊢ (𝜑 → dom (𝑦 ∈ 𝑌 ↦ (𝑁‘(𝐴𝑀𝑦))) ≠ ∅) |
39 | dm0rn0 5263 | . . . . 5 ⊢ (dom (𝑦 ∈ 𝑌 ↦ (𝑁‘(𝐴𝑀𝑦))) = ∅ ↔ ran (𝑦 ∈ 𝑌 ↦ (𝑁‘(𝐴𝑀𝑦))) = ∅) | |
40 | 1 | eqeq1i 2615 | . . . . 5 ⊢ (𝑅 = ∅ ↔ ran (𝑦 ∈ 𝑌 ↦ (𝑁‘(𝐴𝑀𝑦))) = ∅) |
41 | 39, 40 | bitr4i 266 | . . . 4 ⊢ (dom (𝑦 ∈ 𝑌 ↦ (𝑁‘(𝐴𝑀𝑦))) = ∅ ↔ 𝑅 = ∅) |
42 | 41 | necon3bii 2834 | . . 3 ⊢ (dom (𝑦 ∈ 𝑌 ↦ (𝑁‘(𝐴𝑀𝑦))) ≠ ∅ ↔ 𝑅 ≠ ∅) |
43 | 38, 42 | sylib 207 | . 2 ⊢ (𝜑 → 𝑅 ≠ ∅) |
44 | 12, 21 | nvge0 26912 | . . . . . 6 ⊢ ((𝑈 ∈ NrmCVec ∧ (𝐴𝑀𝑦) ∈ 𝑋) → 0 ≤ (𝑁‘(𝐴𝑀𝑦))) |
45 | 5, 20, 44 | syl2anc 691 | . . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑌) → 0 ≤ (𝑁‘(𝐴𝑀𝑦))) |
46 | 45 | ralrimiva 2949 | . . . 4 ⊢ (𝜑 → ∀𝑦 ∈ 𝑌 0 ≤ (𝑁‘(𝐴𝑀𝑦))) |
47 | 34 | rgenw 2908 | . . . . 5 ⊢ ∀𝑦 ∈ 𝑌 (𝑁‘(𝐴𝑀𝑦)) ∈ V |
48 | breq2 4587 | . . . . . 6 ⊢ (𝑤 = (𝑁‘(𝐴𝑀𝑦)) → (0 ≤ 𝑤 ↔ 0 ≤ (𝑁‘(𝐴𝑀𝑦)))) | |
49 | 24, 48 | ralrnmpt 6276 | . . . . 5 ⊢ (∀𝑦 ∈ 𝑌 (𝑁‘(𝐴𝑀𝑦)) ∈ V → (∀𝑤 ∈ ran (𝑦 ∈ 𝑌 ↦ (𝑁‘(𝐴𝑀𝑦)))0 ≤ 𝑤 ↔ ∀𝑦 ∈ 𝑌 0 ≤ (𝑁‘(𝐴𝑀𝑦)))) |
50 | 47, 49 | ax-mp 5 | . . . 4 ⊢ (∀𝑤 ∈ ran (𝑦 ∈ 𝑌 ↦ (𝑁‘(𝐴𝑀𝑦)))0 ≤ 𝑤 ↔ ∀𝑦 ∈ 𝑌 0 ≤ (𝑁‘(𝐴𝑀𝑦))) |
51 | 46, 50 | sylibr 223 | . . 3 ⊢ (𝜑 → ∀𝑤 ∈ ran (𝑦 ∈ 𝑌 ↦ (𝑁‘(𝐴𝑀𝑦)))0 ≤ 𝑤) |
52 | 1 | raleqi 3119 | . . 3 ⊢ (∀𝑤 ∈ 𝑅 0 ≤ 𝑤 ↔ ∀𝑤 ∈ ran (𝑦 ∈ 𝑌 ↦ (𝑁‘(𝐴𝑀𝑦)))0 ≤ 𝑤) |
53 | 51, 52 | sylibr 223 | . 2 ⊢ (𝜑 → ∀𝑤 ∈ 𝑅 0 ≤ 𝑤) |
54 | 28, 43, 53 | 3jca 1235 | 1 ⊢ (𝜑 → (𝑅 ⊆ ℝ ∧ 𝑅 ≠ ∅ ∧ ∀𝑤 ∈ 𝑅 0 ≤ 𝑤)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 ≠ wne 2780 ∀wral 2896 Vcvv 3173 ∩ cin 3539 ⊆ wss 3540 ∅c0 3874 class class class wbr 4583 ↦ cmpt 4643 dom cdm 5038 ran crn 5039 ⟶wf 5800 ‘cfv 5804 (class class class)co 6549 ℝcr 9814 0cc0 9815 ≤ cle 9954 MetOpencmopn 19557 NrmCVeccnv 26823 BaseSetcba 26825 0veccn0v 26827 −𝑣 cnsb 26828 normCVcnmcv 26829 IndMetcims 26830 SubSpcss 26960 CPreHilOLDccphlo 27051 CBanccbn 27102 |
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-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 |
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-iun 4457 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-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-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-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-sup 8231 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-n0 11170 df-z 11255 df-uz 11564 df-rp 11709 df-seq 12664 df-exp 12723 df-cj 13687 df-re 13688 df-im 13689 df-sqrt 13823 df-abs 13824 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-ssp 26961 df-ph 27052 df-cbn 27103 |
This theorem is referenced by: minvecolem2 27115 minvecolem3 27116 minvecolem4c 27119 minvecolem4 27120 minvecolem5 27121 minvecolem6 27122 |
Copyright terms: Public domain | W3C validator |