| Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
| Mirrors > Home > MPE Home > Th. List > minveclem1 | Structured version Visualization version GIF version | ||
| Description: Lemma for minvec 25470. 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 25207 | . . . . . . 7 ⊢ (𝑈 ∈ ℂPreHil → 𝑈 ∈ NrmGrp) | |
| 4 | 2, 3 | syl 17 | . . . . . 6 ⊢ (𝜑 → 𝑈 ∈ NrmGrp) |
| 5 | cphlmod 25208 | . . . . . . . . 9 ⊢ (𝑈 ∈ ℂPreHil → 𝑈 ∈ LMod) | |
| 6 | 2, 5 | syl 17 | . . . . . . . 8 ⊢ (𝜑 → 𝑈 ∈ LMod) |
| 7 | 6 | adantr 480 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑌) → 𝑈 ∈ LMod) |
| 8 | minvec.a | . . . . . . . 8 ⊢ (𝜑 → 𝐴 ∈ 𝑋) | |
| 9 | 8 | adantr 480 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑌) → 𝐴 ∈ 𝑋) |
| 10 | minvec.y | . . . . . . . . 9 ⊢ (𝜑 → 𝑌 ∈ (LSubSp‘𝑈)) | |
| 11 | minvec.x | . . . . . . . . . 10 ⊢ 𝑋 = (Base‘𝑈) | |
| 12 | eqid 2737 | . . . . . . . . . 10 ⊢ (LSubSp‘𝑈) = (LSubSp‘𝑈) | |
| 13 | 11, 12 | lssss 20934 | . . . . . . . . 9 ⊢ (𝑌 ∈ (LSubSp‘𝑈) → 𝑌 ⊆ 𝑋) |
| 14 | 10, 13 | syl 17 | . . . . . . . 8 ⊢ (𝜑 → 𝑌 ⊆ 𝑋) |
| 15 | 14 | sselda 3983 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑌) → 𝑦 ∈ 𝑋) |
| 16 | minvec.m | . . . . . . . 8 ⊢ − = (-g‘𝑈) | |
| 17 | 11, 16 | lmodvsubcl 20905 | . . . . . . 7 ⊢ ((𝑈 ∈ LMod ∧ 𝐴 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → (𝐴 − 𝑦) ∈ 𝑋) |
| 18 | 7, 9, 15, 17 | syl3anc 1373 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑌) → (𝐴 − 𝑦) ∈ 𝑋) |
| 19 | minvec.n | . . . . . . 7 ⊢ 𝑁 = (norm‘𝑈) | |
| 20 | 11, 19 | nmcl 24629 | . . . . . 6 ⊢ ((𝑈 ∈ NrmGrp ∧ (𝐴 − 𝑦) ∈ 𝑋) → (𝑁‘(𝐴 − 𝑦)) ∈ ℝ) |
| 21 | 4, 18, 20 | syl2an2r 685 | . . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑌) → (𝑁‘(𝐴 − 𝑦)) ∈ ℝ) |
| 22 | 21 | fmpttd 7135 | . . . 4 ⊢ (𝜑 → (𝑦 ∈ 𝑌 ↦ (𝑁‘(𝐴 − 𝑦))):𝑌⟶ℝ) |
| 23 | 22 | frnd 6744 | . . 3 ⊢ (𝜑 → ran (𝑦 ∈ 𝑌 ↦ (𝑁‘(𝐴 − 𝑦))) ⊆ ℝ) |
| 24 | 1, 23 | eqsstrid 4022 | . 2 ⊢ (𝜑 → 𝑅 ⊆ ℝ) |
| 25 | 12 | lssn0 20938 | . . . 4 ⊢ (𝑌 ∈ (LSubSp‘𝑈) → 𝑌 ≠ ∅) |
| 26 | 10, 25 | syl 17 | . . 3 ⊢ (𝜑 → 𝑌 ≠ ∅) |
| 27 | 1 | eqeq1i 2742 | . . . . 5 ⊢ (𝑅 = ∅ ↔ ran (𝑦 ∈ 𝑌 ↦ (𝑁‘(𝐴 − 𝑦))) = ∅) |
| 28 | dm0rn0 5935 | . . . . 5 ⊢ (dom (𝑦 ∈ 𝑌 ↦ (𝑁‘(𝐴 − 𝑦))) = ∅ ↔ ran (𝑦 ∈ 𝑌 ↦ (𝑁‘(𝐴 − 𝑦))) = ∅) | |
| 29 | fvex 6919 | . . . . . . 7 ⊢ (𝑁‘(𝐴 − 𝑦)) ∈ V | |
| 30 | eqid 2737 | . . . . . . 7 ⊢ (𝑦 ∈ 𝑌 ↦ (𝑁‘(𝐴 − 𝑦))) = (𝑦 ∈ 𝑌 ↦ (𝑁‘(𝐴 − 𝑦))) | |
| 31 | 29, 30 | dmmpti 6712 | . . . . . 6 ⊢ dom (𝑦 ∈ 𝑌 ↦ (𝑁‘(𝐴 − 𝑦))) = 𝑌 |
| 32 | 31 | eqeq1i 2742 | . . . . 5 ⊢ (dom (𝑦 ∈ 𝑌 ↦ (𝑁‘(𝐴 − 𝑦))) = ∅ ↔ 𝑌 = ∅) |
| 33 | 27, 28, 32 | 3bitr2i 299 | . . . 4 ⊢ (𝑅 = ∅ ↔ 𝑌 = ∅) |
| 34 | 33 | necon3bii 2993 | . . 3 ⊢ (𝑅 ≠ ∅ ↔ 𝑌 ≠ ∅) |
| 35 | 26, 34 | sylibr 234 | . 2 ⊢ (𝜑 → 𝑅 ≠ ∅) |
| 36 | 11, 19 | nmge0 24630 | . . . . . 6 ⊢ ((𝑈 ∈ NrmGrp ∧ (𝐴 − 𝑦) ∈ 𝑋) → 0 ≤ (𝑁‘(𝐴 − 𝑦))) |
| 37 | 4, 18, 36 | syl2an2r 685 | . . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑌) → 0 ≤ (𝑁‘(𝐴 − 𝑦))) |
| 38 | 37 | ralrimiva 3146 | . . . 4 ⊢ (𝜑 → ∀𝑦 ∈ 𝑌 0 ≤ (𝑁‘(𝐴 − 𝑦))) |
| 39 | 29 | rgenw 3065 | . . . . 5 ⊢ ∀𝑦 ∈ 𝑌 (𝑁‘(𝐴 − 𝑦)) ∈ V |
| 40 | breq2 5147 | . . . . . 6 ⊢ (𝑤 = (𝑁‘(𝐴 − 𝑦)) → (0 ≤ 𝑤 ↔ 0 ≤ (𝑁‘(𝐴 − 𝑦)))) | |
| 41 | 30, 40 | ralrnmptw 7114 | . . . . 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 234 | . . 3 ⊢ (𝜑 → ∀𝑤 ∈ ran (𝑦 ∈ 𝑌 ↦ (𝑁‘(𝐴 − 𝑦)))0 ≤ 𝑤) |
| 44 | 1 | raleqi 3324 | . . 3 ⊢ (∀𝑤 ∈ 𝑅 0 ≤ 𝑤 ↔ ∀𝑤 ∈ ran (𝑦 ∈ 𝑌 ↦ (𝑁‘(𝐴 − 𝑦)))0 ≤ 𝑤) |
| 45 | 43, 44 | sylibr 234 | . 2 ⊢ (𝜑 → ∀𝑤 ∈ 𝑅 0 ≤ 𝑤) |
| 46 | 24, 35, 45 | 3jca 1129 | 1 ⊢ (𝜑 → (𝑅 ⊆ ℝ ∧ 𝑅 ≠ ∅ ∧ ∀𝑤 ∈ 𝑅 0 ≤ 𝑤)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1087 = wceq 1540 ∈ wcel 2108 ≠ wne 2940 ∀wral 3061 Vcvv 3480 ⊆ wss 3951 ∅c0 4333 class class class wbr 5143 ↦ cmpt 5225 dom cdm 5685 ran crn 5686 ‘cfv 6561 (class class class)co 7431 ℝcr 11154 0cc0 11155 ≤ cle 11296 Basecbs 17247 ↾s cress 17274 TopOpenctopn 17466 -gcsg 18953 LModclmod 20858 LSubSpclss 20929 normcnm 24589 NrmGrpcngp 24590 ℂPreHilccph 25200 CMetSpccms 25366 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 ax-pre-sup 11233 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8014 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-er 8745 df-map 8868 df-en 8986 df-dom 8987 df-sdom 8988 df-sup 9482 df-inf 9483 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-div 11921 df-nn 12267 df-2 12329 df-n0 12527 df-z 12614 df-uz 12879 df-q 12991 df-rp 13035 df-xneg 13154 df-xadd 13155 df-xmul 13156 df-0g 17486 df-topgen 17488 df-mgm 18653 df-sgrp 18732 df-mnd 18748 df-grp 18954 df-minusg 18955 df-sbg 18956 df-lmod 20860 df-lss 20930 df-psmet 21356 df-xmet 21357 df-met 21358 df-bl 21359 df-mopn 21360 df-top 22900 df-topon 22917 df-topsp 22939 df-bases 22953 df-xms 24330 df-ms 24331 df-nm 24595 df-ngp 24596 df-nlm 24599 df-cph 25202 |
| This theorem is referenced by: minveclem4c 25459 minveclem2 25460 minveclem3b 25462 minveclem4 25466 minveclem6 25468 |
| Copyright terms: Public domain | W3C validator |