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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 5303 | . . . 4 ⊢ (𝐼 × {(ℂfld ↾s ℝ)}) = (𝑘 ∈ 𝐼 ↦ (ℂfld ↾s ℝ)) | |
2 | 1 | oveq2i 6804 | . . 3 ⊢ ((Scalar‘ℂfld)Xs(𝐼 × {(ℂfld ↾s ℝ)})) = ((Scalar‘ℂfld)Xs(𝑘 ∈ 𝐼 ↦ (ℂfld ↾s ℝ))) |
3 | eqid 2771 | . . 3 ⊢ (Base‘((Scalar‘ℂfld)Xs(𝐼 × {(ℂfld ↾s ℝ)}))) = (Base‘((Scalar‘ℂfld)Xs(𝐼 × {(ℂfld ↾s ℝ)}))) | |
4 | ax-resscn 10195 | . . . 4 ⊢ ℝ ⊆ ℂ | |
5 | eqid 2771 | . . . . 5 ⊢ (ℂfld ↾s ℝ) = (ℂfld ↾s ℝ) | |
6 | cnfldbas 19965 | . . . . 5 ⊢ ℂ = (Base‘ℂfld) | |
7 | 5, 6 | ressbas2 16138 | . . . 4 ⊢ (ℝ ⊆ ℂ → ℝ = (Base‘(ℂfld ↾s ℝ))) |
8 | 4, 7 | ax-mp 5 | . . 3 ⊢ ℝ = (Base‘(ℂfld ↾s ℝ)) |
9 | reex 10229 | . . . . 5 ⊢ ℝ ∈ V | |
10 | cnfldds 19971 | . . . . . 6 ⊢ (abs ∘ − ) = (dist‘ℂfld) | |
11 | 5, 10 | ressds 16281 | . . . . 5 ⊢ (ℝ ∈ V → (abs ∘ − ) = (dist‘(ℂfld ↾s ℝ))) |
12 | 9, 11 | ax-mp 5 | . . . 4 ⊢ (abs ∘ − ) = (dist‘(ℂfld ↾s ℝ)) |
13 | 12 | reseq1i 5530 | . . 3 ⊢ ((abs ∘ − ) ↾ (ℝ × ℝ)) = ((dist‘(ℂfld ↾s ℝ)) ↾ (ℝ × ℝ)) |
14 | eqid 2771 | . . 3 ⊢ (dist‘((Scalar‘ℂfld)Xs(𝐼 × {(ℂfld ↾s ℝ)}))) = (dist‘((Scalar‘ℂfld)Xs(𝐼 × {(ℂfld ↾s ℝ)}))) | |
15 | fvexd 6344 | . . 3 ⊢ (𝐼 ∈ Fin → (Scalar‘ℂfld) ∈ V) | |
16 | id 22 | . . 3 ⊢ (𝐼 ∈ Fin → 𝐼 ∈ Fin) | |
17 | ovex 6823 | . . . 4 ⊢ (ℂfld ↾s ℝ) ∈ V | |
18 | 17 | a1i 11 | . . 3 ⊢ ((𝐼 ∈ Fin ∧ 𝑘 ∈ 𝐼) → (ℂfld ↾s ℝ) ∈ V) |
19 | eqid 2771 | . . . . 5 ⊢ ((abs ∘ − ) ↾ (ℝ × ℝ)) = ((abs ∘ − ) ↾ (ℝ × ℝ)) | |
20 | 19 | remet 22813 | . . . 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 22395 | . 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 2771 | . . . . . . . 8 ⊢ (Scalar‘ℂfld) = (Scalar‘ℂfld) | |
26 | 5, 25 | resssca 16239 | . . . . . . 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 16354 | . . . . 5 ⊢ (((ℂfld ↾s ℝ) ∈ V ∧ 𝐼 ∈ Fin) → 𝑌 = ((Scalar‘ℂfld)Xs(𝐼 × {(ℂfld ↾s ℝ)}))) |
29 | 17, 28 | mpan 670 | . . . 4 ⊢ (𝐼 ∈ Fin → 𝑌 = ((Scalar‘ℂfld)Xs(𝐼 × {(ℂfld ↾s ℝ)}))) |
30 | 29 | fveq2d 6336 | . . 3 ⊢ (𝐼 ∈ Fin → (dist‘𝑌) = (dist‘((Scalar‘ℂfld)Xs(𝐼 × {(ℂfld ↾s ℝ)})))) |
31 | 23, 30 | syl5eq 2817 | . 2 ⊢ (𝐼 ∈ Fin → 𝐷 = (dist‘((Scalar‘ℂfld)Xs(𝐼 × {(ℂfld ↾s ℝ)})))) |
32 | rrnequiv.1 | . . . 4 ⊢ 𝑋 = (ℝ ↑𝑚 𝐼) | |
33 | 24, 8 | pwsbas 16355 | . . . . . 6 ⊢ (((ℂfld ↾s ℝ) ∈ V ∧ 𝐼 ∈ Fin) → (ℝ ↑𝑚 𝐼) = (Base‘𝑌)) |
34 | 17, 33 | mpan 670 | . . . . 5 ⊢ (𝐼 ∈ Fin → (ℝ ↑𝑚 𝐼) = (Base‘𝑌)) |
35 | 29 | fveq2d 6336 | . . . . 5 ⊢ (𝐼 ∈ Fin → (Base‘𝑌) = (Base‘((Scalar‘ℂfld)Xs(𝐼 × {(ℂfld ↾s ℝ)})))) |
36 | 34, 35 | eqtrd 2805 | . . . 4 ⊢ (𝐼 ∈ Fin → (ℝ ↑𝑚 𝐼) = (Base‘((Scalar‘ℂfld)Xs(𝐼 × {(ℂfld ↾s ℝ)})))) |
37 | 32, 36 | syl5eq 2817 | . . 3 ⊢ (𝐼 ∈ Fin → 𝑋 = (Base‘((Scalar‘ℂfld)Xs(𝐼 × {(ℂfld ↾s ℝ)})))) |
38 | 37 | fveq2d 6336 | . 2 ⊢ (𝐼 ∈ Fin → (Met‘𝑋) = (Met‘(Base‘((Scalar‘ℂfld)Xs(𝐼 × {(ℂfld ↾s ℝ)}))))) |
39 | 22, 31, 38 | 3eltr4d 2865 | 1 ⊢ (𝐼 ∈ Fin → 𝐷 ∈ (Met‘𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 382 = wceq 1631 ∈ wcel 2145 Vcvv 3351 ⊆ wss 3723 {csn 4316 ↦ cmpt 4863 × cxp 5247 ↾ cres 5251 ∘ ccom 5253 ‘cfv 6031 (class class class)co 6793 ↑𝑚 cmap 8009 Fincfn 8109 ℂcc 10136 ℝcr 10137 − cmin 10468 abscabs 14182 Basecbs 16064 ↾s cress 16065 Scalarcsca 16152 distcds 16158 Xscprds 16314 ↑s cpws 16315 Metcme 19947 ℂfldccnfld 19961 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-8 2147 ax-9 2154 ax-10 2174 ax-11 2190 ax-12 2203 ax-13 2408 ax-ext 2751 ax-rep 4904 ax-sep 4915 ax-nul 4923 ax-pow 4974 ax-pr 5034 ax-un 7096 ax-cnex 10194 ax-resscn 10195 ax-1cn 10196 ax-icn 10197 ax-addcl 10198 ax-addrcl 10199 ax-mulcl 10200 ax-mulrcl 10201 ax-mulcom 10202 ax-addass 10203 ax-mulass 10204 ax-distr 10205 ax-i2m1 10206 ax-1ne0 10207 ax-1rid 10208 ax-rnegex 10209 ax-rrecex 10210 ax-cnre 10211 ax-pre-lttri 10212 ax-pre-lttrn 10213 ax-pre-ltadd 10214 ax-pre-mulgt0 10215 ax-pre-sup 10216 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 837 df-3or 1072 df-3an 1073 df-tru 1634 df-ex 1853 df-nf 1858 df-sb 2050 df-eu 2622 df-mo 2623 df-clab 2758 df-cleq 2764 df-clel 2767 df-nfc 2902 df-ne 2944 df-nel 3047 df-ral 3066 df-rex 3067 df-reu 3068 df-rmo 3069 df-rab 3070 df-v 3353 df-sbc 3588 df-csb 3683 df-dif 3726 df-un 3728 df-in 3730 df-ss 3737 df-pss 3739 df-nul 4064 df-if 4226 df-pw 4299 df-sn 4317 df-pr 4319 df-tp 4321 df-op 4323 df-uni 4575 df-int 4612 df-iun 4656 df-br 4787 df-opab 4847 df-mpt 4864 df-tr 4887 df-id 5157 df-eprel 5162 df-po 5170 df-so 5171 df-fr 5208 df-we 5210 df-xp 5255 df-rel 5256 df-cnv 5257 df-co 5258 df-dm 5259 df-rn 5260 df-res 5261 df-ima 5262 df-pred 5823 df-ord 5869 df-on 5870 df-lim 5871 df-suc 5872 df-iota 5994 df-fun 6033 df-fn 6034 df-f 6035 df-f1 6036 df-fo 6037 df-f1o 6038 df-fv 6039 df-riota 6754 df-ov 6796 df-oprab 6797 df-mpt2 6798 df-om 7213 df-1st 7315 df-2nd 7316 df-wrecs 7559 df-recs 7621 df-rdg 7659 df-1o 7713 df-oadd 7717 df-er 7896 df-map 8011 df-ixp 8063 df-en 8110 df-dom 8111 df-sdom 8112 df-fin 8113 df-sup 8504 df-pnf 10278 df-mnf 10279 df-xr 10280 df-ltxr 10281 df-le 10282 df-sub 10470 df-neg 10471 df-div 10887 df-nn 11223 df-2 11281 df-3 11282 df-4 11283 df-5 11284 df-6 11285 df-7 11286 df-8 11287 df-9 11288 df-n0 11495 df-z 11580 df-dec 11696 df-uz 11889 df-rp 12036 df-xneg 12151 df-xadd 12152 df-xmul 12153 df-icc 12387 df-fz 12534 df-seq 13009 df-exp 13068 df-cj 14047 df-re 14048 df-im 14049 df-sqrt 14183 df-abs 14184 df-struct 16066 df-ndx 16067 df-slot 16068 df-base 16070 df-sets 16071 df-ress 16072 df-plusg 16162 df-mulr 16163 df-starv 16164 df-sca 16165 df-vsca 16166 df-ip 16167 df-tset 16168 df-ple 16169 df-ds 16172 df-unif 16173 df-hom 16174 df-cco 16175 df-prds 16316 df-pws 16318 df-xmet 19954 df-met 19955 df-cnfld 19962 |
This theorem is referenced by: rrnequiv 33966 rrntotbnd 33967 |
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