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Mirrors > Home > MPE Home > Th. List > resubmet | Structured version Visualization version GIF version |
Description: The subspace topology induced by a subset of the reals. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 13-Aug-2014.) |
Ref | Expression |
---|---|
resubmet.1 | ⊢ 𝑅 = (topGen‘ran (,)) |
resubmet.2 | ⊢ 𝐽 = (MetOpen‘((abs ∘ − ) ↾ (𝐴 × 𝐴))) |
Ref | Expression |
---|---|
resubmet | ⊢ (𝐴 ⊆ ℝ → 𝐽 = (𝑅 ↾t 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | resubmet.2 | . . 3 ⊢ 𝐽 = (MetOpen‘((abs ∘ − ) ↾ (𝐴 × 𝐴))) | |
2 | xpss12 5605 | . . . . . 6 ⊢ ((𝐴 ⊆ ℝ ∧ 𝐴 ⊆ ℝ) → (𝐴 × 𝐴) ⊆ (ℝ × ℝ)) | |
3 | 2 | anidms 567 | . . . . 5 ⊢ (𝐴 ⊆ ℝ → (𝐴 × 𝐴) ⊆ (ℝ × ℝ)) |
4 | 3 | resabs1d 5921 | . . . 4 ⊢ (𝐴 ⊆ ℝ → (((abs ∘ − ) ↾ (ℝ × ℝ)) ↾ (𝐴 × 𝐴)) = ((abs ∘ − ) ↾ (𝐴 × 𝐴))) |
5 | 4 | fveq2d 6775 | . . 3 ⊢ (𝐴 ⊆ ℝ → (MetOpen‘(((abs ∘ − ) ↾ (ℝ × ℝ)) ↾ (𝐴 × 𝐴))) = (MetOpen‘((abs ∘ − ) ↾ (𝐴 × 𝐴)))) |
6 | 1, 5 | eqtr4id 2799 | . 2 ⊢ (𝐴 ⊆ ℝ → 𝐽 = (MetOpen‘(((abs ∘ − ) ↾ (ℝ × ℝ)) ↾ (𝐴 × 𝐴)))) |
7 | eqid 2740 | . . . 4 ⊢ ((abs ∘ − ) ↾ (ℝ × ℝ)) = ((abs ∘ − ) ↾ (ℝ × ℝ)) | |
8 | 7 | rexmet 23952 | . . 3 ⊢ ((abs ∘ − ) ↾ (ℝ × ℝ)) ∈ (∞Met‘ℝ) |
9 | eqid 2740 | . . . 4 ⊢ (((abs ∘ − ) ↾ (ℝ × ℝ)) ↾ (𝐴 × 𝐴)) = (((abs ∘ − ) ↾ (ℝ × ℝ)) ↾ (𝐴 × 𝐴)) | |
10 | resubmet.1 | . . . . 5 ⊢ 𝑅 = (topGen‘ran (,)) | |
11 | eqid 2740 | . . . . . 6 ⊢ (MetOpen‘((abs ∘ − ) ↾ (ℝ × ℝ))) = (MetOpen‘((abs ∘ − ) ↾ (ℝ × ℝ))) | |
12 | 7, 11 | tgioo 23957 | . . . . 5 ⊢ (topGen‘ran (,)) = (MetOpen‘((abs ∘ − ) ↾ (ℝ × ℝ))) |
13 | 10, 12 | eqtri 2768 | . . . 4 ⊢ 𝑅 = (MetOpen‘((abs ∘ − ) ↾ (ℝ × ℝ))) |
14 | eqid 2740 | . . . 4 ⊢ (MetOpen‘(((abs ∘ − ) ↾ (ℝ × ℝ)) ↾ (𝐴 × 𝐴))) = (MetOpen‘(((abs ∘ − ) ↾ (ℝ × ℝ)) ↾ (𝐴 × 𝐴))) | |
15 | 9, 13, 14 | metrest 23678 | . . 3 ⊢ ((((abs ∘ − ) ↾ (ℝ × ℝ)) ∈ (∞Met‘ℝ) ∧ 𝐴 ⊆ ℝ) → (𝑅 ↾t 𝐴) = (MetOpen‘(((abs ∘ − ) ↾ (ℝ × ℝ)) ↾ (𝐴 × 𝐴)))) |
16 | 8, 15 | mpan 687 | . 2 ⊢ (𝐴 ⊆ ℝ → (𝑅 ↾t 𝐴) = (MetOpen‘(((abs ∘ − ) ↾ (ℝ × ℝ)) ↾ (𝐴 × 𝐴)))) |
17 | 6, 16 | eqtr4d 2783 | 1 ⊢ (𝐴 ⊆ ℝ → 𝐽 = (𝑅 ↾t 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 ∈ wcel 2110 ⊆ wss 3892 × cxp 5588 ran crn 5591 ↾ cres 5592 ∘ ccom 5594 ‘cfv 6432 (class class class)co 7271 ℝcr 10871 − cmin 11205 (,)cioo 13078 abscabs 14943 ↾t crest 17129 topGenctg 17146 ∞Metcxmet 20580 MetOpencmopn 20585 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2015 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2711 ax-rep 5214 ax-sep 5227 ax-nul 5234 ax-pow 5292 ax-pr 5356 ax-un 7582 ax-cnex 10928 ax-resscn 10929 ax-1cn 10930 ax-icn 10931 ax-addcl 10932 ax-addrcl 10933 ax-mulcl 10934 ax-mulrcl 10935 ax-mulcom 10936 ax-addass 10937 ax-mulass 10938 ax-distr 10939 ax-i2m1 10940 ax-1ne0 10941 ax-1rid 10942 ax-rnegex 10943 ax-rrecex 10944 ax-cnre 10945 ax-pre-lttri 10946 ax-pre-lttrn 10947 ax-pre-ltadd 10948 ax-pre-mulgt0 10949 ax-pre-sup 10950 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2072 df-mo 2542 df-eu 2571 df-clab 2718 df-cleq 2732 df-clel 2818 df-nfc 2891 df-ne 2946 df-nel 3052 df-ral 3071 df-rex 3072 df-reu 3073 df-rmo 3074 df-rab 3075 df-v 3433 df-sbc 3721 df-csb 3838 df-dif 3895 df-un 3897 df-in 3899 df-ss 3909 df-pss 3911 df-nul 4263 df-if 4466 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4846 df-iun 4932 df-br 5080 df-opab 5142 df-mpt 5163 df-tr 5197 df-id 5490 df-eprel 5496 df-po 5504 df-so 5505 df-fr 5545 df-we 5547 df-xp 5596 df-rel 5597 df-cnv 5598 df-co 5599 df-dm 5600 df-rn 5601 df-res 5602 df-ima 5603 df-pred 6201 df-ord 6268 df-on 6269 df-lim 6270 df-suc 6271 df-iota 6390 df-fun 6434 df-fn 6435 df-f 6436 df-f1 6437 df-fo 6438 df-f1o 6439 df-fv 6440 df-riota 7228 df-ov 7274 df-oprab 7275 df-mpo 7276 df-om 7707 df-1st 7824 df-2nd 7825 df-frecs 8088 df-wrecs 8119 df-recs 8193 df-rdg 8232 df-er 8481 df-map 8600 df-en 8717 df-dom 8718 df-sdom 8719 df-sup 9179 df-inf 9180 df-pnf 11012 df-mnf 11013 df-xr 11014 df-ltxr 11015 df-le 11016 df-sub 11207 df-neg 11208 df-div 11633 df-nn 11974 df-2 12036 df-3 12037 df-n0 12234 df-z 12320 df-uz 12582 df-q 12688 df-rp 12730 df-xneg 12847 df-xadd 12848 df-xmul 12849 df-ioo 13082 df-seq 13720 df-exp 13781 df-cj 14808 df-re 14809 df-im 14810 df-sqrt 14944 df-abs 14945 df-rest 17131 df-topgen 17152 df-psmet 20587 df-xmet 20588 df-met 20589 df-bl 20590 df-mopn 20591 df-top 22041 df-topon 22058 df-bases 22094 |
This theorem is referenced by: dfii2 24043 icoopnst 24100 iocopnst 24101 evthicc 24621 |
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