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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 5085 | . . . 4 ⊢ (𝐼 × {(ℂfld ↾s ℝ)}) = (𝑘 ∈ 𝐼 ↦ (ℂfld ↾s ℝ)) | |
2 | 1 | oveq2i 6560 | . . 3 ⊢ ((Scalar‘ℂfld)Xs(𝐼 × {(ℂfld ↾s ℝ)})) = ((Scalar‘ℂfld)Xs(𝑘 ∈ 𝐼 ↦ (ℂfld ↾s ℝ))) |
3 | eqid 2610 | . . 3 ⊢ (Base‘((Scalar‘ℂfld)Xs(𝐼 × {(ℂfld ↾s ℝ)}))) = (Base‘((Scalar‘ℂfld)Xs(𝐼 × {(ℂfld ↾s ℝ)}))) | |
4 | ax-resscn 9872 | . . . 4 ⊢ ℝ ⊆ ℂ | |
5 | eqid 2610 | . . . . 5 ⊢ (ℂfld ↾s ℝ) = (ℂfld ↾s ℝ) | |
6 | cnfldbas 19571 | . . . . 5 ⊢ ℂ = (Base‘ℂfld) | |
7 | 5, 6 | ressbas2 15758 | . . . 4 ⊢ (ℝ ⊆ ℂ → ℝ = (Base‘(ℂfld ↾s ℝ))) |
8 | 4, 7 | ax-mp 5 | . . 3 ⊢ ℝ = (Base‘(ℂfld ↾s ℝ)) |
9 | reex 9906 | . . . . 5 ⊢ ℝ ∈ V | |
10 | cnfldds 19577 | . . . . . 6 ⊢ (abs ∘ − ) = (dist‘ℂfld) | |
11 | 5, 10 | ressds 15896 | . . . . 5 ⊢ (ℝ ∈ V → (abs ∘ − ) = (dist‘(ℂfld ↾s ℝ))) |
12 | 9, 11 | ax-mp 5 | . . . 4 ⊢ (abs ∘ − ) = (dist‘(ℂfld ↾s ℝ)) |
13 | 12 | reseq1i 5313 | . . 3 ⊢ ((abs ∘ − ) ↾ (ℝ × ℝ)) = ((dist‘(ℂfld ↾s ℝ)) ↾ (ℝ × ℝ)) |
14 | eqid 2610 | . . 3 ⊢ (dist‘((Scalar‘ℂfld)Xs(𝐼 × {(ℂfld ↾s ℝ)}))) = (dist‘((Scalar‘ℂfld)Xs(𝐼 × {(ℂfld ↾s ℝ)}))) | |
15 | fvex 6113 | . . . 4 ⊢ (Scalar‘ℂfld) ∈ V | |
16 | 15 | a1i 11 | . . 3 ⊢ (𝐼 ∈ Fin → (Scalar‘ℂfld) ∈ V) |
17 | id 22 | . . 3 ⊢ (𝐼 ∈ Fin → 𝐼 ∈ Fin) | |
18 | ovex 6577 | . . . 4 ⊢ (ℂfld ↾s ℝ) ∈ V | |
19 | 18 | a1i 11 | . . 3 ⊢ ((𝐼 ∈ Fin ∧ 𝑘 ∈ 𝐼) → (ℂfld ↾s ℝ) ∈ V) |
20 | eqid 2610 | . . . . 5 ⊢ ((abs ∘ − ) ↾ (ℝ × ℝ)) = ((abs ∘ − ) ↾ (ℝ × ℝ)) | |
21 | 20 | remet 22401 | . . . 4 ⊢ ((abs ∘ − ) ↾ (ℝ × ℝ)) ∈ (Met‘ℝ) |
22 | 21 | a1i 11 | . . 3 ⊢ ((𝐼 ∈ Fin ∧ 𝑘 ∈ 𝐼) → ((abs ∘ − ) ↾ (ℝ × ℝ)) ∈ (Met‘ℝ)) |
23 | 2, 3, 8, 13, 14, 16, 17, 19, 22 | prdsmet 21985 | . 2 ⊢ (𝐼 ∈ Fin → (dist‘((Scalar‘ℂfld)Xs(𝐼 × {(ℂfld ↾s ℝ)}))) ∈ (Met‘(Base‘((Scalar‘ℂfld)Xs(𝐼 × {(ℂfld ↾s ℝ)}))))) |
24 | rrnequiv.d | . . 3 ⊢ 𝐷 = (dist‘𝑌) | |
25 | rrnequiv.y | . . . . . 6 ⊢ 𝑌 = ((ℂfld ↾s ℝ) ↑s 𝐼) | |
26 | eqid 2610 | . . . . . . . 8 ⊢ (Scalar‘ℂfld) = (Scalar‘ℂfld) | |
27 | 5, 26 | resssca 15854 | . . . . . . 7 ⊢ (ℝ ∈ V → (Scalar‘ℂfld) = (Scalar‘(ℂfld ↾s ℝ))) |
28 | 9, 27 | ax-mp 5 | . . . . . 6 ⊢ (Scalar‘ℂfld) = (Scalar‘(ℂfld ↾s ℝ)) |
29 | 25, 28 | pwsval 15969 | . . . . 5 ⊢ (((ℂfld ↾s ℝ) ∈ V ∧ 𝐼 ∈ Fin) → 𝑌 = ((Scalar‘ℂfld)Xs(𝐼 × {(ℂfld ↾s ℝ)}))) |
30 | 18, 29 | mpan 702 | . . . 4 ⊢ (𝐼 ∈ Fin → 𝑌 = ((Scalar‘ℂfld)Xs(𝐼 × {(ℂfld ↾s ℝ)}))) |
31 | 30 | fveq2d 6107 | . . 3 ⊢ (𝐼 ∈ Fin → (dist‘𝑌) = (dist‘((Scalar‘ℂfld)Xs(𝐼 × {(ℂfld ↾s ℝ)})))) |
32 | 24, 31 | syl5eq 2656 | . 2 ⊢ (𝐼 ∈ Fin → 𝐷 = (dist‘((Scalar‘ℂfld)Xs(𝐼 × {(ℂfld ↾s ℝ)})))) |
33 | rrnequiv.1 | . . . 4 ⊢ 𝑋 = (ℝ ↑𝑚 𝐼) | |
34 | 25, 8 | pwsbas 15970 | . . . . . 6 ⊢ (((ℂfld ↾s ℝ) ∈ V ∧ 𝐼 ∈ Fin) → (ℝ ↑𝑚 𝐼) = (Base‘𝑌)) |
35 | 18, 34 | mpan 702 | . . . . 5 ⊢ (𝐼 ∈ Fin → (ℝ ↑𝑚 𝐼) = (Base‘𝑌)) |
36 | 30 | fveq2d 6107 | . . . . 5 ⊢ (𝐼 ∈ Fin → (Base‘𝑌) = (Base‘((Scalar‘ℂfld)Xs(𝐼 × {(ℂfld ↾s ℝ)})))) |
37 | 35, 36 | eqtrd 2644 | . . . 4 ⊢ (𝐼 ∈ Fin → (ℝ ↑𝑚 𝐼) = (Base‘((Scalar‘ℂfld)Xs(𝐼 × {(ℂfld ↾s ℝ)})))) |
38 | 33, 37 | syl5eq 2656 | . . 3 ⊢ (𝐼 ∈ Fin → 𝑋 = (Base‘((Scalar‘ℂfld)Xs(𝐼 × {(ℂfld ↾s ℝ)})))) |
39 | 38 | fveq2d 6107 | . 2 ⊢ (𝐼 ∈ Fin → (Met‘𝑋) = (Met‘(Base‘((Scalar‘ℂfld)Xs(𝐼 × {(ℂfld ↾s ℝ)}))))) |
40 | 23, 32, 39 | 3eltr4d 2703 | 1 ⊢ (𝐼 ∈ Fin → 𝐷 ∈ (Met‘𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1475 ∈ wcel 1977 Vcvv 3173 ⊆ wss 3540 {csn 4125 ↦ cmpt 4643 × cxp 5036 ↾ cres 5040 ∘ ccom 5042 ‘cfv 5804 (class class class)co 6549 ↑𝑚 cmap 7744 Fincfn 7841 ℂcc 9813 ℝcr 9814 − cmin 10145 abscabs 13822 Basecbs 15695 ↾s cress 15696 Scalarcsca 15771 distcds 15777 Xscprds 15929 ↑s cpws 15930 Metcme 19553 ℂfldccnfld 19567 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 ax-pre-sup 9893 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-oadd 7451 df-er 7629 df-map 7746 df-ixp 7795 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-sup 8231 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-div 10564 df-nn 10898 df-2 10956 df-3 10957 df-4 10958 df-5 10959 df-6 10960 df-7 10961 df-8 10962 df-9 10963 df-n0 11170 df-z 11255 df-dec 11370 df-uz 11564 df-rp 11709 df-xneg 11822 df-xadd 11823 df-xmul 11824 df-icc 12053 df-fz 12198 df-seq 12664 df-exp 12723 df-cj 13687 df-re 13688 df-im 13689 df-sqrt 13823 df-abs 13824 df-struct 15697 df-ndx 15698 df-slot 15699 df-base 15700 df-sets 15701 df-ress 15702 df-plusg 15781 df-mulr 15782 df-starv 15783 df-sca 15784 df-vsca 15785 df-ip 15786 df-tset 15787 df-ple 15788 df-ds 15791 df-unif 15792 df-hom 15793 df-cco 15794 df-prds 15931 df-pws 15933 df-xmet 19560 df-met 19561 df-cnfld 19568 |
This theorem is referenced by: rrnequiv 32804 rrntotbnd 32805 |
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