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Mirrors > Home > MPE Home > Th. List > Mathboxes > repwsmet | Structured version Visualization version GIF version |
Description: The supremum metric on ℝ↑𝐼 is a metric. (Contributed by Jeff Madsen, 15-Sep-2015.) |
Ref | Expression |
---|---|
rrnequiv.y | ⊢ 𝑌 = ((ℂfld ↾s ℝ) ↑s 𝐼) |
rrnequiv.d | ⊢ 𝐷 = (dist‘𝑌) |
rrnequiv.1 | ⊢ 𝑋 = (ℝ ↑𝑚 𝐼) |
Ref | Expression |
---|---|
repwsmet | ⊢ (𝐼 ∈ Fin → 𝐷 ∈ (Met‘𝑋)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fconstmpt 5163 | . . . 4 ⊢ (𝐼 × {(ℂfld ↾s ℝ)}) = (𝑘 ∈ 𝐼 ↦ (ℂfld ↾s ℝ)) | |
2 | 1 | oveq2i 6661 | . . 3 ⊢ ((Scalar‘ℂfld)Xs(𝐼 × {(ℂfld ↾s ℝ)})) = ((Scalar‘ℂfld)Xs(𝑘 ∈ 𝐼 ↦ (ℂfld ↾s ℝ))) |
3 | eqid 2622 | . . 3 ⊢ (Base‘((Scalar‘ℂfld)Xs(𝐼 × {(ℂfld ↾s ℝ)}))) = (Base‘((Scalar‘ℂfld)Xs(𝐼 × {(ℂfld ↾s ℝ)}))) | |
4 | ax-resscn 9993 | . . . 4 ⊢ ℝ ⊆ ℂ | |
5 | eqid 2622 | . . . . 5 ⊢ (ℂfld ↾s ℝ) = (ℂfld ↾s ℝ) | |
6 | cnfldbas 19750 | . . . . 5 ⊢ ℂ = (Base‘ℂfld) | |
7 | 5, 6 | ressbas2 15931 | . . . 4 ⊢ (ℝ ⊆ ℂ → ℝ = (Base‘(ℂfld ↾s ℝ))) |
8 | 4, 7 | ax-mp 5 | . . 3 ⊢ ℝ = (Base‘(ℂfld ↾s ℝ)) |
9 | reex 10027 | . . . . 5 ⊢ ℝ ∈ V | |
10 | cnfldds 19756 | . . . . . 6 ⊢ (abs ∘ − ) = (dist‘ℂfld) | |
11 | 5, 10 | ressds 16073 | . . . . 5 ⊢ (ℝ ∈ V → (abs ∘ − ) = (dist‘(ℂfld ↾s ℝ))) |
12 | 9, 11 | ax-mp 5 | . . . 4 ⊢ (abs ∘ − ) = (dist‘(ℂfld ↾s ℝ)) |
13 | 12 | reseq1i 5392 | . . 3 ⊢ ((abs ∘ − ) ↾ (ℝ × ℝ)) = ((dist‘(ℂfld ↾s ℝ)) ↾ (ℝ × ℝ)) |
14 | eqid 2622 | . . 3 ⊢ (dist‘((Scalar‘ℂfld)Xs(𝐼 × {(ℂfld ↾s ℝ)}))) = (dist‘((Scalar‘ℂfld)Xs(𝐼 × {(ℂfld ↾s ℝ)}))) | |
15 | fvexd 6203 | . . 3 ⊢ (𝐼 ∈ Fin → (Scalar‘ℂfld) ∈ V) | |
16 | id 22 | . . 3 ⊢ (𝐼 ∈ Fin → 𝐼 ∈ Fin) | |
17 | ovex 6678 | . . . 4 ⊢ (ℂfld ↾s ℝ) ∈ V | |
18 | 17 | a1i 11 | . . 3 ⊢ ((𝐼 ∈ Fin ∧ 𝑘 ∈ 𝐼) → (ℂfld ↾s ℝ) ∈ V) |
19 | eqid 2622 | . . . . 5 ⊢ ((abs ∘ − ) ↾ (ℝ × ℝ)) = ((abs ∘ − ) ↾ (ℝ × ℝ)) | |
20 | 19 | remet 22593 | . . . 4 ⊢ ((abs ∘ − ) ↾ (ℝ × ℝ)) ∈ (Met‘ℝ) |
21 | 20 | a1i 11 | . . 3 ⊢ ((𝐼 ∈ Fin ∧ 𝑘 ∈ 𝐼) → ((abs ∘ − ) ↾ (ℝ × ℝ)) ∈ (Met‘ℝ)) |
22 | 2, 3, 8, 13, 14, 15, 16, 18, 21 | prdsmet 22175 | . 2 ⊢ (𝐼 ∈ Fin → (dist‘((Scalar‘ℂfld)Xs(𝐼 × {(ℂfld ↾s ℝ)}))) ∈ (Met‘(Base‘((Scalar‘ℂfld)Xs(𝐼 × {(ℂfld ↾s ℝ)}))))) |
23 | rrnequiv.d | . . 3 ⊢ 𝐷 = (dist‘𝑌) | |
24 | rrnequiv.y | . . . . . 6 ⊢ 𝑌 = ((ℂfld ↾s ℝ) ↑s 𝐼) | |
25 | eqid 2622 | . . . . . . . 8 ⊢ (Scalar‘ℂfld) = (Scalar‘ℂfld) | |
26 | 5, 25 | resssca 16031 | . . . . . . 7 ⊢ (ℝ ∈ V → (Scalar‘ℂfld) = (Scalar‘(ℂfld ↾s ℝ))) |
27 | 9, 26 | ax-mp 5 | . . . . . 6 ⊢ (Scalar‘ℂfld) = (Scalar‘(ℂfld ↾s ℝ)) |
28 | 24, 27 | pwsval 16146 | . . . . 5 ⊢ (((ℂfld ↾s ℝ) ∈ V ∧ 𝐼 ∈ Fin) → 𝑌 = ((Scalar‘ℂfld)Xs(𝐼 × {(ℂfld ↾s ℝ)}))) |
29 | 17, 28 | mpan 706 | . . . 4 ⊢ (𝐼 ∈ Fin → 𝑌 = ((Scalar‘ℂfld)Xs(𝐼 × {(ℂfld ↾s ℝ)}))) |
30 | 29 | fveq2d 6195 | . . 3 ⊢ (𝐼 ∈ Fin → (dist‘𝑌) = (dist‘((Scalar‘ℂfld)Xs(𝐼 × {(ℂfld ↾s ℝ)})))) |
31 | 23, 30 | syl5eq 2668 | . 2 ⊢ (𝐼 ∈ Fin → 𝐷 = (dist‘((Scalar‘ℂfld)Xs(𝐼 × {(ℂfld ↾s ℝ)})))) |
32 | rrnequiv.1 | . . . 4 ⊢ 𝑋 = (ℝ ↑𝑚 𝐼) | |
33 | 24, 8 | pwsbas 16147 | . . . . . 6 ⊢ (((ℂfld ↾s ℝ) ∈ V ∧ 𝐼 ∈ Fin) → (ℝ ↑𝑚 𝐼) = (Base‘𝑌)) |
34 | 17, 33 | mpan 706 | . . . . 5 ⊢ (𝐼 ∈ Fin → (ℝ ↑𝑚 𝐼) = (Base‘𝑌)) |
35 | 29 | fveq2d 6195 | . . . . 5 ⊢ (𝐼 ∈ Fin → (Base‘𝑌) = (Base‘((Scalar‘ℂfld)Xs(𝐼 × {(ℂfld ↾s ℝ)})))) |
36 | 34, 35 | eqtrd 2656 | . . . 4 ⊢ (𝐼 ∈ Fin → (ℝ ↑𝑚 𝐼) = (Base‘((Scalar‘ℂfld)Xs(𝐼 × {(ℂfld ↾s ℝ)})))) |
37 | 32, 36 | syl5eq 2668 | . . 3 ⊢ (𝐼 ∈ Fin → 𝑋 = (Base‘((Scalar‘ℂfld)Xs(𝐼 × {(ℂfld ↾s ℝ)})))) |
38 | 37 | fveq2d 6195 | . 2 ⊢ (𝐼 ∈ Fin → (Met‘𝑋) = (Met‘(Base‘((Scalar‘ℂfld)Xs(𝐼 × {(ℂfld ↾s ℝ)}))))) |
39 | 22, 31, 38 | 3eltr4d 2716 | 1 ⊢ (𝐼 ∈ Fin → 𝐷 ∈ (Met‘𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 384 = wceq 1483 ∈ wcel 1990 Vcvv 3200 ⊆ wss 3574 {csn 4177 ↦ cmpt 4729 × cxp 5112 ↾ cres 5116 ∘ ccom 5118 ‘cfv 5888 (class class class)co 6650 ↑𝑚 cmap 7857 Fincfn 7955 ℂcc 9934 ℝcr 9935 − cmin 10266 abscabs 13974 Basecbs 15857 ↾s cress 15858 Scalarcsca 15944 distcds 15950 Xscprds 16106 ↑s cpws 16107 Metcme 19732 ℂfldccnfld 19746 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 ax-pre-sup 10014 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-oadd 7564 df-er 7742 df-map 7859 df-ixp 7909 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-sup 8348 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-div 10685 df-nn 11021 df-2 11079 df-3 11080 df-4 11081 df-5 11082 df-6 11083 df-7 11084 df-8 11085 df-9 11086 df-n0 11293 df-z 11378 df-dec 11494 df-uz 11688 df-rp 11833 df-xneg 11946 df-xadd 11947 df-xmul 11948 df-icc 12182 df-fz 12327 df-seq 12802 df-exp 12861 df-cj 13839 df-re 13840 df-im 13841 df-sqrt 13975 df-abs 13976 df-struct 15859 df-ndx 15860 df-slot 15861 df-base 15863 df-sets 15864 df-ress 15865 df-plusg 15954 df-mulr 15955 df-starv 15956 df-sca 15957 df-vsca 15958 df-ip 15959 df-tset 15960 df-ple 15961 df-ds 15964 df-unif 15965 df-hom 15966 df-cco 15967 df-prds 16108 df-pws 16110 df-xmet 19739 df-met 19740 df-cnfld 19747 |
This theorem is referenced by: rrnequiv 33634 rrntotbnd 33635 |
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