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Mathbox for Jeff Madsen |
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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 5736 | . . . 4 ⊢ (𝐼 × {(ℂfld ↾s ℝ)}) = (𝑘 ∈ 𝐼 ↦ (ℂfld ↾s ℝ)) | |
2 | 1 | oveq2i 7427 | . . 3 ⊢ ((Scalar‘ℂfld)Xs(𝐼 × {(ℂfld ↾s ℝ)})) = ((Scalar‘ℂfld)Xs(𝑘 ∈ 𝐼 ↦ (ℂfld ↾s ℝ))) |
3 | eqid 2726 | . . 3 ⊢ (Base‘((Scalar‘ℂfld)Xs(𝐼 × {(ℂfld ↾s ℝ)}))) = (Base‘((Scalar‘ℂfld)Xs(𝐼 × {(ℂfld ↾s ℝ)}))) | |
4 | ax-resscn 11206 | . . . 4 ⊢ ℝ ⊆ ℂ | |
5 | eqid 2726 | . . . . 5 ⊢ (ℂfld ↾s ℝ) = (ℂfld ↾s ℝ) | |
6 | cnfldbas 21343 | . . . . 5 ⊢ ℂ = (Base‘ℂfld) | |
7 | 5, 6 | ressbas2 17246 | . . . 4 ⊢ (ℝ ⊆ ℂ → ℝ = (Base‘(ℂfld ↾s ℝ))) |
8 | 4, 7 | ax-mp 5 | . . 3 ⊢ ℝ = (Base‘(ℂfld ↾s ℝ)) |
9 | reex 11240 | . . . . 5 ⊢ ℝ ∈ V | |
10 | cnfldds 21351 | . . . . . 6 ⊢ (abs ∘ − ) = (dist‘ℂfld) | |
11 | 5, 10 | ressds 17419 | . . . . 5 ⊢ (ℝ ∈ V → (abs ∘ − ) = (dist‘(ℂfld ↾s ℝ))) |
12 | 9, 11 | ax-mp 5 | . . . 4 ⊢ (abs ∘ − ) = (dist‘(ℂfld ↾s ℝ)) |
13 | 12 | reseq1i 5977 | . . 3 ⊢ ((abs ∘ − ) ↾ (ℝ × ℝ)) = ((dist‘(ℂfld ↾s ℝ)) ↾ (ℝ × ℝ)) |
14 | eqid 2726 | . . 3 ⊢ (dist‘((Scalar‘ℂfld)Xs(𝐼 × {(ℂfld ↾s ℝ)}))) = (dist‘((Scalar‘ℂfld)Xs(𝐼 × {(ℂfld ↾s ℝ)}))) | |
15 | fvexd 6908 | . . 3 ⊢ (𝐼 ∈ Fin → (Scalar‘ℂfld) ∈ V) | |
16 | id 22 | . . 3 ⊢ (𝐼 ∈ Fin → 𝐼 ∈ Fin) | |
17 | ovex 7449 | . . . 4 ⊢ (ℂfld ↾s ℝ) ∈ V | |
18 | 17 | a1i 11 | . . 3 ⊢ ((𝐼 ∈ Fin ∧ 𝑘 ∈ 𝐼) → (ℂfld ↾s ℝ) ∈ V) |
19 | eqid 2726 | . . . . 5 ⊢ ((abs ∘ − ) ↾ (ℝ × ℝ)) = ((abs ∘ − ) ↾ (ℝ × ℝ)) | |
20 | 19 | remet 24794 | . . . 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 24364 | . 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 2726 | . . . . . . . 8 ⊢ (Scalar‘ℂfld) = (Scalar‘ℂfld) | |
26 | 5, 25 | resssca 17352 | . . . . . . 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 17496 | . . . . 5 ⊢ (((ℂfld ↾s ℝ) ∈ V ∧ 𝐼 ∈ Fin) → 𝑌 = ((Scalar‘ℂfld)Xs(𝐼 × {(ℂfld ↾s ℝ)}))) |
29 | 17, 28 | mpan 688 | . . . 4 ⊢ (𝐼 ∈ Fin → 𝑌 = ((Scalar‘ℂfld)Xs(𝐼 × {(ℂfld ↾s ℝ)}))) |
30 | 29 | fveq2d 6897 | . . 3 ⊢ (𝐼 ∈ Fin → (dist‘𝑌) = (dist‘((Scalar‘ℂfld)Xs(𝐼 × {(ℂfld ↾s ℝ)})))) |
31 | 23, 30 | eqtrid 2778 | . 2 ⊢ (𝐼 ∈ Fin → 𝐷 = (dist‘((Scalar‘ℂfld)Xs(𝐼 × {(ℂfld ↾s ℝ)})))) |
32 | rrnequiv.1 | . . . 4 ⊢ 𝑋 = (ℝ ↑m 𝐼) | |
33 | 24, 8 | pwsbas 17497 | . . . . . 6 ⊢ (((ℂfld ↾s ℝ) ∈ V ∧ 𝐼 ∈ Fin) → (ℝ ↑m 𝐼) = (Base‘𝑌)) |
34 | 17, 33 | mpan 688 | . . . . 5 ⊢ (𝐼 ∈ Fin → (ℝ ↑m 𝐼) = (Base‘𝑌)) |
35 | 29 | fveq2d 6897 | . . . . 5 ⊢ (𝐼 ∈ Fin → (Base‘𝑌) = (Base‘((Scalar‘ℂfld)Xs(𝐼 × {(ℂfld ↾s ℝ)})))) |
36 | 34, 35 | eqtrd 2766 | . . . 4 ⊢ (𝐼 ∈ Fin → (ℝ ↑m 𝐼) = (Base‘((Scalar‘ℂfld)Xs(𝐼 × {(ℂfld ↾s ℝ)})))) |
37 | 32, 36 | eqtrid 2778 | . . 3 ⊢ (𝐼 ∈ Fin → 𝑋 = (Base‘((Scalar‘ℂfld)Xs(𝐼 × {(ℂfld ↾s ℝ)})))) |
38 | 37 | fveq2d 6897 | . 2 ⊢ (𝐼 ∈ Fin → (Met‘𝑋) = (Met‘(Base‘((Scalar‘ℂfld)Xs(𝐼 × {(ℂfld ↾s ℝ)}))))) |
39 | 22, 31, 38 | 3eltr4d 2841 | 1 ⊢ (𝐼 ∈ Fin → 𝐷 ∈ (Met‘𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 = wceq 1534 ∈ wcel 2099 Vcvv 3462 ⊆ wss 3946 {csn 4623 ↦ cmpt 5228 × cxp 5672 ↾ cres 5676 ∘ ccom 5678 ‘cfv 6546 (class class class)co 7416 ↑m cmap 8847 Fincfn 8966 ℂcc 11147 ℝcr 11148 − cmin 11485 abscabs 15234 Basecbs 17208 ↾s cress 17237 Scalarcsca 17264 distcds 17270 Xscprds 17455 ↑s cpws 17456 Metcmet 21325 ℂfldccnfld 21339 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-rep 5282 ax-sep 5296 ax-nul 5303 ax-pow 5361 ax-pr 5425 ax-un 7738 ax-cnex 11205 ax-resscn 11206 ax-1cn 11207 ax-icn 11208 ax-addcl 11209 ax-addrcl 11210 ax-mulcl 11211 ax-mulrcl 11212 ax-mulcom 11213 ax-addass 11214 ax-mulass 11215 ax-distr 11216 ax-i2m1 11217 ax-1ne0 11218 ax-1rid 11219 ax-rnegex 11220 ax-rrecex 11221 ax-cnre 11222 ax-pre-lttri 11223 ax-pre-lttrn 11224 ax-pre-ltadd 11225 ax-pre-mulgt0 11226 ax-pre-sup 11227 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3776 df-csb 3892 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-pss 3966 df-nul 4323 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-tp 4628 df-op 4630 df-uni 4906 df-iun 4995 df-br 5146 df-opab 5208 df-mpt 5229 df-tr 5263 df-id 5572 df-eprel 5578 df-po 5586 df-so 5587 df-fr 5629 df-we 5631 df-xp 5680 df-rel 5681 df-cnv 5682 df-co 5683 df-dm 5684 df-rn 5685 df-res 5686 df-ima 5687 df-pred 6304 df-ord 6371 df-on 6372 df-lim 6373 df-suc 6374 df-iota 6498 df-fun 6548 df-fn 6549 df-f 6550 df-f1 6551 df-fo 6552 df-f1o 6553 df-fv 6554 df-riota 7372 df-ov 7419 df-oprab 7420 df-mpo 7421 df-om 7869 df-1st 7995 df-2nd 7996 df-frecs 8288 df-wrecs 8319 df-recs 8393 df-rdg 8432 df-1o 8488 df-er 8726 df-map 8849 df-ixp 8919 df-en 8967 df-dom 8968 df-sdom 8969 df-fin 8970 df-sup 9478 df-pnf 11291 df-mnf 11292 df-xr 11293 df-ltxr 11294 df-le 11295 df-sub 11487 df-neg 11488 df-div 11913 df-nn 12259 df-2 12321 df-3 12322 df-4 12323 df-5 12324 df-6 12325 df-7 12326 df-8 12327 df-9 12328 df-n0 12519 df-z 12605 df-dec 12724 df-uz 12869 df-rp 13023 df-xneg 13140 df-xadd 13141 df-xmul 13142 df-icc 13379 df-fz 13533 df-seq 14016 df-exp 14076 df-cj 15099 df-re 15100 df-im 15101 df-sqrt 15235 df-abs 15236 df-struct 17144 df-sets 17161 df-slot 17179 df-ndx 17191 df-base 17209 df-ress 17238 df-plusg 17274 df-mulr 17275 df-starv 17276 df-sca 17277 df-vsca 17278 df-ip 17279 df-tset 17280 df-ple 17281 df-ds 17283 df-unif 17284 df-hom 17285 df-cco 17286 df-prds 17457 df-pws 17459 df-xmet 21332 df-met 21333 df-cnfld 21340 |
This theorem is referenced by: rrnequiv 37549 rrntotbnd 37550 |
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