| 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 5747 | . . . 4 ⊢ (𝐼 × {(ℂfld ↾s ℝ)}) = (𝑘 ∈ 𝐼 ↦ (ℂfld ↾s ℝ)) | |
| 2 | 1 | oveq2i 7442 | . . 3 ⊢ ((Scalar‘ℂfld)Xs(𝐼 × {(ℂfld ↾s ℝ)})) = ((Scalar‘ℂfld)Xs(𝑘 ∈ 𝐼 ↦ (ℂfld ↾s ℝ))) |
| 3 | eqid 2737 | . . 3 ⊢ (Base‘((Scalar‘ℂfld)Xs(𝐼 × {(ℂfld ↾s ℝ)}))) = (Base‘((Scalar‘ℂfld)Xs(𝐼 × {(ℂfld ↾s ℝ)}))) | |
| 4 | ax-resscn 11212 | . . . 4 ⊢ ℝ ⊆ ℂ | |
| 5 | eqid 2737 | . . . . 5 ⊢ (ℂfld ↾s ℝ) = (ℂfld ↾s ℝ) | |
| 6 | cnfldbas 21368 | . . . . 5 ⊢ ℂ = (Base‘ℂfld) | |
| 7 | 5, 6 | ressbas2 17283 | . . . 4 ⊢ (ℝ ⊆ ℂ → ℝ = (Base‘(ℂfld ↾s ℝ))) |
| 8 | 4, 7 | ax-mp 5 | . . 3 ⊢ ℝ = (Base‘(ℂfld ↾s ℝ)) |
| 9 | reex 11246 | . . . . 5 ⊢ ℝ ∈ V | |
| 10 | cnfldds 21376 | . . . . . 6 ⊢ (abs ∘ − ) = (dist‘ℂfld) | |
| 11 | 5, 10 | ressds 17454 | . . . . 5 ⊢ (ℝ ∈ V → (abs ∘ − ) = (dist‘(ℂfld ↾s ℝ))) |
| 12 | 9, 11 | ax-mp 5 | . . . 4 ⊢ (abs ∘ − ) = (dist‘(ℂfld ↾s ℝ)) |
| 13 | 12 | reseq1i 5993 | . . 3 ⊢ ((abs ∘ − ) ↾ (ℝ × ℝ)) = ((dist‘(ℂfld ↾s ℝ)) ↾ (ℝ × ℝ)) |
| 14 | eqid 2737 | . . 3 ⊢ (dist‘((Scalar‘ℂfld)Xs(𝐼 × {(ℂfld ↾s ℝ)}))) = (dist‘((Scalar‘ℂfld)Xs(𝐼 × {(ℂfld ↾s ℝ)}))) | |
| 15 | fvexd 6921 | . . 3 ⊢ (𝐼 ∈ Fin → (Scalar‘ℂfld) ∈ V) | |
| 16 | id 22 | . . 3 ⊢ (𝐼 ∈ Fin → 𝐼 ∈ Fin) | |
| 17 | ovex 7464 | . . . 4 ⊢ (ℂfld ↾s ℝ) ∈ V | |
| 18 | 17 | a1i 11 | . . 3 ⊢ ((𝐼 ∈ Fin ∧ 𝑘 ∈ 𝐼) → (ℂfld ↾s ℝ) ∈ V) |
| 19 | eqid 2737 | . . . . 5 ⊢ ((abs ∘ − ) ↾ (ℝ × ℝ)) = ((abs ∘ − ) ↾ (ℝ × ℝ)) | |
| 20 | 19 | remet 24811 | . . . 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 24380 | . 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 2737 | . . . . . . . 8 ⊢ (Scalar‘ℂfld) = (Scalar‘ℂfld) | |
| 26 | 5, 25 | resssca 17387 | . . . . . . 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 17531 | . . . . 5 ⊢ (((ℂfld ↾s ℝ) ∈ V ∧ 𝐼 ∈ Fin) → 𝑌 = ((Scalar‘ℂfld)Xs(𝐼 × {(ℂfld ↾s ℝ)}))) |
| 29 | 17, 28 | mpan 690 | . . . 4 ⊢ (𝐼 ∈ Fin → 𝑌 = ((Scalar‘ℂfld)Xs(𝐼 × {(ℂfld ↾s ℝ)}))) |
| 30 | 29 | fveq2d 6910 | . . 3 ⊢ (𝐼 ∈ Fin → (dist‘𝑌) = (dist‘((Scalar‘ℂfld)Xs(𝐼 × {(ℂfld ↾s ℝ)})))) |
| 31 | 23, 30 | eqtrid 2789 | . 2 ⊢ (𝐼 ∈ Fin → 𝐷 = (dist‘((Scalar‘ℂfld)Xs(𝐼 × {(ℂfld ↾s ℝ)})))) |
| 32 | rrnequiv.1 | . . . 4 ⊢ 𝑋 = (ℝ ↑m 𝐼) | |
| 33 | 24, 8 | pwsbas 17532 | . . . . . 6 ⊢ (((ℂfld ↾s ℝ) ∈ V ∧ 𝐼 ∈ Fin) → (ℝ ↑m 𝐼) = (Base‘𝑌)) |
| 34 | 17, 33 | mpan 690 | . . . . 5 ⊢ (𝐼 ∈ Fin → (ℝ ↑m 𝐼) = (Base‘𝑌)) |
| 35 | 29 | fveq2d 6910 | . . . . 5 ⊢ (𝐼 ∈ Fin → (Base‘𝑌) = (Base‘((Scalar‘ℂfld)Xs(𝐼 × {(ℂfld ↾s ℝ)})))) |
| 36 | 34, 35 | eqtrd 2777 | . . . 4 ⊢ (𝐼 ∈ Fin → (ℝ ↑m 𝐼) = (Base‘((Scalar‘ℂfld)Xs(𝐼 × {(ℂfld ↾s ℝ)})))) |
| 37 | 32, 36 | eqtrid 2789 | . . 3 ⊢ (𝐼 ∈ Fin → 𝑋 = (Base‘((Scalar‘ℂfld)Xs(𝐼 × {(ℂfld ↾s ℝ)})))) |
| 38 | 37 | fveq2d 6910 | . 2 ⊢ (𝐼 ∈ Fin → (Met‘𝑋) = (Met‘(Base‘((Scalar‘ℂfld)Xs(𝐼 × {(ℂfld ↾s ℝ)}))))) |
| 39 | 22, 31, 38 | 3eltr4d 2856 | 1 ⊢ (𝐼 ∈ Fin → 𝐷 ∈ (Met‘𝑋)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 Vcvv 3480 ⊆ wss 3951 {csn 4626 ↦ cmpt 5225 × cxp 5683 ↾ cres 5687 ∘ ccom 5689 ‘cfv 6561 (class class class)co 7431 ↑m cmap 8866 Fincfn 8985 ℂcc 11153 ℝcr 11154 − cmin 11492 abscabs 15273 Basecbs 17247 ↾s cress 17274 Scalarcsca 17300 distcds 17306 Xscprds 17490 ↑s cpws 17491 Metcmet 21350 ℂfldccnfld 21364 |
| 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-rep 5279 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-tp 4631 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-1o 8506 df-er 8745 df-map 8868 df-ixp 8938 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-sup 9482 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-3 12330 df-4 12331 df-5 12332 df-6 12333 df-7 12334 df-8 12335 df-9 12336 df-n0 12527 df-z 12614 df-dec 12734 df-uz 12879 df-rp 13035 df-xneg 13154 df-xadd 13155 df-xmul 13156 df-icc 13394 df-fz 13548 df-seq 14043 df-exp 14103 df-cj 15138 df-re 15139 df-im 15140 df-sqrt 15274 df-abs 15275 df-struct 17184 df-sets 17201 df-slot 17219 df-ndx 17231 df-base 17248 df-ress 17275 df-plusg 17310 df-mulr 17311 df-starv 17312 df-sca 17313 df-vsca 17314 df-ip 17315 df-tset 17316 df-ple 17317 df-ds 17319 df-unif 17320 df-hom 17321 df-cco 17322 df-prds 17492 df-pws 17494 df-xmet 21357 df-met 21358 df-cnfld 21365 |
| This theorem is referenced by: rrnequiv 37842 rrntotbnd 37843 |
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