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Mirrors > Home > MPE Home > Th. List > minveclem1 | Structured version Visualization version GIF version |
Description: Lemma for minvec 24600. 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 | . . . . . . 7 ⊢ (𝜑 → 𝑈 ∈ ℂPreHil) | |
3 | cphngp 24337 | . . . . . . 7 ⊢ (𝑈 ∈ ℂPreHil → 𝑈 ∈ NrmGrp) | |
4 | 2, 3 | syl 17 | . . . . . 6 ⊢ (𝜑 → 𝑈 ∈ NrmGrp) |
5 | cphlmod 24338 | . . . . . . . . 9 ⊢ (𝑈 ∈ ℂPreHil → 𝑈 ∈ LMod) | |
6 | 2, 5 | syl 17 | . . . . . . . 8 ⊢ (𝜑 → 𝑈 ∈ LMod) |
7 | 6 | adantr 481 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑌) → 𝑈 ∈ LMod) |
8 | minvec.a | . . . . . . . 8 ⊢ (𝜑 → 𝐴 ∈ 𝑋) | |
9 | 8 | adantr 481 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑌) → 𝐴 ∈ 𝑋) |
10 | minvec.y | . . . . . . . . 9 ⊢ (𝜑 → 𝑌 ∈ (LSubSp‘𝑈)) | |
11 | minvec.x | . . . . . . . . . 10 ⊢ 𝑋 = (Base‘𝑈) | |
12 | eqid 2738 | . . . . . . . . . 10 ⊢ (LSubSp‘𝑈) = (LSubSp‘𝑈) | |
13 | 11, 12 | lssss 20198 | . . . . . . . . 9 ⊢ (𝑌 ∈ (LSubSp‘𝑈) → 𝑌 ⊆ 𝑋) |
14 | 10, 13 | syl 17 | . . . . . . . 8 ⊢ (𝜑 → 𝑌 ⊆ 𝑋) |
15 | 14 | sselda 3921 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑌) → 𝑦 ∈ 𝑋) |
16 | minvec.m | . . . . . . . 8 ⊢ − = (-g‘𝑈) | |
17 | 11, 16 | lmodvsubcl 20168 | . . . . . . 7 ⊢ ((𝑈 ∈ LMod ∧ 𝐴 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → (𝐴 − 𝑦) ∈ 𝑋) |
18 | 7, 9, 15, 17 | syl3anc 1370 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑌) → (𝐴 − 𝑦) ∈ 𝑋) |
19 | minvec.n | . . . . . . 7 ⊢ 𝑁 = (norm‘𝑈) | |
20 | 11, 19 | nmcl 23772 | . . . . . 6 ⊢ ((𝑈 ∈ NrmGrp ∧ (𝐴 − 𝑦) ∈ 𝑋) → (𝑁‘(𝐴 − 𝑦)) ∈ ℝ) |
21 | 4, 18, 20 | syl2an2r 682 | . . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑌) → (𝑁‘(𝐴 − 𝑦)) ∈ ℝ) |
22 | 21 | fmpttd 6989 | . . . 4 ⊢ (𝜑 → (𝑦 ∈ 𝑌 ↦ (𝑁‘(𝐴 − 𝑦))):𝑌⟶ℝ) |
23 | 22 | frnd 6608 | . . 3 ⊢ (𝜑 → ran (𝑦 ∈ 𝑌 ↦ (𝑁‘(𝐴 − 𝑦))) ⊆ ℝ) |
24 | 1, 23 | eqsstrid 3969 | . 2 ⊢ (𝜑 → 𝑅 ⊆ ℝ) |
25 | 12 | lssn0 20202 | . . . 4 ⊢ (𝑌 ∈ (LSubSp‘𝑈) → 𝑌 ≠ ∅) |
26 | 10, 25 | syl 17 | . . 3 ⊢ (𝜑 → 𝑌 ≠ ∅) |
27 | 1 | eqeq1i 2743 | . . . . 5 ⊢ (𝑅 = ∅ ↔ ran (𝑦 ∈ 𝑌 ↦ (𝑁‘(𝐴 − 𝑦))) = ∅) |
28 | dm0rn0 5834 | . . . . 5 ⊢ (dom (𝑦 ∈ 𝑌 ↦ (𝑁‘(𝐴 − 𝑦))) = ∅ ↔ ran (𝑦 ∈ 𝑌 ↦ (𝑁‘(𝐴 − 𝑦))) = ∅) | |
29 | fvex 6787 | . . . . . . 7 ⊢ (𝑁‘(𝐴 − 𝑦)) ∈ V | |
30 | eqid 2738 | . . . . . . 7 ⊢ (𝑦 ∈ 𝑌 ↦ (𝑁‘(𝐴 − 𝑦))) = (𝑦 ∈ 𝑌 ↦ (𝑁‘(𝐴 − 𝑦))) | |
31 | 29, 30 | dmmpti 6577 | . . . . . 6 ⊢ dom (𝑦 ∈ 𝑌 ↦ (𝑁‘(𝐴 − 𝑦))) = 𝑌 |
32 | 31 | eqeq1i 2743 | . . . . 5 ⊢ (dom (𝑦 ∈ 𝑌 ↦ (𝑁‘(𝐴 − 𝑦))) = ∅ ↔ 𝑌 = ∅) |
33 | 27, 28, 32 | 3bitr2i 299 | . . . 4 ⊢ (𝑅 = ∅ ↔ 𝑌 = ∅) |
34 | 33 | necon3bii 2996 | . . 3 ⊢ (𝑅 ≠ ∅ ↔ 𝑌 ≠ ∅) |
35 | 26, 34 | sylibr 233 | . 2 ⊢ (𝜑 → 𝑅 ≠ ∅) |
36 | 11, 19 | nmge0 23773 | . . . . . 6 ⊢ ((𝑈 ∈ NrmGrp ∧ (𝐴 − 𝑦) ∈ 𝑋) → 0 ≤ (𝑁‘(𝐴 − 𝑦))) |
37 | 4, 18, 36 | syl2an2r 682 | . . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑌) → 0 ≤ (𝑁‘(𝐴 − 𝑦))) |
38 | 37 | ralrimiva 3103 | . . . 4 ⊢ (𝜑 → ∀𝑦 ∈ 𝑌 0 ≤ (𝑁‘(𝐴 − 𝑦))) |
39 | 29 | rgenw 3076 | . . . . 5 ⊢ ∀𝑦 ∈ 𝑌 (𝑁‘(𝐴 − 𝑦)) ∈ V |
40 | breq2 5078 | . . . . . 6 ⊢ (𝑤 = (𝑁‘(𝐴 − 𝑦)) → (0 ≤ 𝑤 ↔ 0 ≤ (𝑁‘(𝐴 − 𝑦)))) | |
41 | 30, 40 | ralrnmptw 6970 | . . . . 5 ⊢ (∀𝑦 ∈ 𝑌 (𝑁‘(𝐴 − 𝑦)) ∈ V → (∀𝑤 ∈ ran (𝑦 ∈ 𝑌 ↦ (𝑁‘(𝐴 − 𝑦)))0 ≤ 𝑤 ↔ ∀𝑦 ∈ 𝑌 0 ≤ (𝑁‘(𝐴 − 𝑦)))) |
42 | 39, 41 | ax-mp 5 | . . . 4 ⊢ (∀𝑤 ∈ ran (𝑦 ∈ 𝑌 ↦ (𝑁‘(𝐴 − 𝑦)))0 ≤ 𝑤 ↔ ∀𝑦 ∈ 𝑌 0 ≤ (𝑁‘(𝐴 − 𝑦))) |
43 | 38, 42 | sylibr 233 | . . 3 ⊢ (𝜑 → ∀𝑤 ∈ ran (𝑦 ∈ 𝑌 ↦ (𝑁‘(𝐴 − 𝑦)))0 ≤ 𝑤) |
44 | 1 | raleqi 3346 | . . 3 ⊢ (∀𝑤 ∈ 𝑅 0 ≤ 𝑤 ↔ ∀𝑤 ∈ ran (𝑦 ∈ 𝑌 ↦ (𝑁‘(𝐴 − 𝑦)))0 ≤ 𝑤) |
45 | 43, 44 | sylibr 233 | . 2 ⊢ (𝜑 → ∀𝑤 ∈ 𝑅 0 ≤ 𝑤) |
46 | 24, 35, 45 | 3jca 1127 | 1 ⊢ (𝜑 → (𝑅 ⊆ ℝ ∧ 𝑅 ≠ ∅ ∧ ∀𝑤 ∈ 𝑅 0 ≤ 𝑤)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 ∧ w3a 1086 = wceq 1539 ∈ wcel 2106 ≠ wne 2943 ∀wral 3064 Vcvv 3432 ⊆ wss 3887 ∅c0 4256 class class class wbr 5074 ↦ cmpt 5157 dom cdm 5589 ran crn 5590 ‘cfv 6433 (class class class)co 7275 ℝcr 10870 0cc0 10871 ≤ cle 11010 Basecbs 16912 ↾s cress 16941 TopOpenctopn 17132 -gcsg 18579 LModclmod 20123 LSubSpclss 20193 normcnm 23732 NrmGrpcngp 23733 ℂPreHilccph 24330 CMetSpccms 24496 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 ax-cnex 10927 ax-resscn 10928 ax-1cn 10929 ax-icn 10930 ax-addcl 10931 ax-addrcl 10932 ax-mulcl 10933 ax-mulrcl 10934 ax-mulcom 10935 ax-addass 10936 ax-mulass 10937 ax-distr 10938 ax-i2m1 10939 ax-1ne0 10940 ax-1rid 10941 ax-rnegex 10942 ax-rrecex 10943 ax-cnre 10944 ax-pre-lttri 10945 ax-pre-lttrn 10946 ax-pre-ltadd 10947 ax-pre-mulgt0 10948 ax-pre-sup 10949 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-rmo 3071 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-pss 3906 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-iun 4926 df-br 5075 df-opab 5137 df-mpt 5158 df-tr 5192 df-id 5489 df-eprel 5495 df-po 5503 df-so 5504 df-fr 5544 df-we 5546 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-pred 6202 df-ord 6269 df-on 6270 df-lim 6271 df-suc 6272 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-riota 7232 df-ov 7278 df-oprab 7279 df-mpo 7280 df-om 7713 df-1st 7831 df-2nd 7832 df-frecs 8097 df-wrecs 8128 df-recs 8202 df-rdg 8241 df-er 8498 df-map 8617 df-en 8734 df-dom 8735 df-sdom 8736 df-sup 9201 df-inf 9202 df-pnf 11011 df-mnf 11012 df-xr 11013 df-ltxr 11014 df-le 11015 df-sub 11207 df-neg 11208 df-div 11633 df-nn 11974 df-2 12036 df-n0 12234 df-z 12320 df-uz 12583 df-q 12689 df-rp 12731 df-xneg 12848 df-xadd 12849 df-xmul 12850 df-0g 17152 df-topgen 17154 df-mgm 18326 df-sgrp 18375 df-mnd 18386 df-grp 18580 df-minusg 18581 df-sbg 18582 df-lmod 20125 df-lss 20194 df-psmet 20589 df-xmet 20590 df-met 20591 df-bl 20592 df-mopn 20593 df-top 22043 df-topon 22060 df-topsp 22082 df-bases 22096 df-xms 23473 df-ms 23474 df-nm 23738 df-ngp 23739 df-nlm 23742 df-cph 24332 |
This theorem is referenced by: minveclem4c 24589 minveclem2 24590 minveclem3b 24592 minveclem4 24596 minveclem6 24598 |
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