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Mirrors > Home > MPE Home > Th. List > xrsdsre | Structured version Visualization version GIF version |
Description: The metric on the extended reals coincides with the usual metric on the reals. (Contributed by Mario Carneiro, 4-Sep-2015.) |
Ref | Expression |
---|---|
xrsxmet.1 | ⊢ 𝐷 = (dist‘ℝ*𝑠) |
Ref | Expression |
---|---|
xrsdsre | ⊢ (𝐷 ↾ (ℝ × ℝ)) = ((abs ∘ − ) ↾ (ℝ × ℝ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | xrsxmet.1 | . . . . 5 ⊢ 𝐷 = (dist‘ℝ*𝑠) | |
2 | 1 | xrsdsreval 20191 | . . . 4 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (𝑥𝐷𝑦) = (abs‘(𝑥 − 𝑦))) |
3 | ovres 7079 | . . . 4 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (𝑥(𝐷 ↾ (ℝ × ℝ))𝑦) = (𝑥𝐷𝑦)) | |
4 | eqid 2778 | . . . . 5 ⊢ ((abs ∘ − ) ↾ (ℝ × ℝ)) = ((abs ∘ − ) ↾ (ℝ × ℝ)) | |
5 | 4 | remetdval 23004 | . . . 4 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (𝑥((abs ∘ − ) ↾ (ℝ × ℝ))𝑦) = (abs‘(𝑥 − 𝑦))) |
6 | 2, 3, 5 | 3eqtr4d 2824 | . . 3 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (𝑥(𝐷 ↾ (ℝ × ℝ))𝑦) = (𝑥((abs ∘ − ) ↾ (ℝ × ℝ))𝑦)) |
7 | 6 | rgen2a 3159 | . 2 ⊢ ∀𝑥 ∈ ℝ ∀𝑦 ∈ ℝ (𝑥(𝐷 ↾ (ℝ × ℝ))𝑦) = (𝑥((abs ∘ − ) ↾ (ℝ × ℝ))𝑦) |
8 | 1 | xrsxmet 23024 | . . . . 5 ⊢ 𝐷 ∈ (∞Met‘ℝ*) |
9 | xmetf 22546 | . . . . 5 ⊢ (𝐷 ∈ (∞Met‘ℝ*) → 𝐷:(ℝ* × ℝ*)⟶ℝ*) | |
10 | ffn 6293 | . . . . 5 ⊢ (𝐷:(ℝ* × ℝ*)⟶ℝ* → 𝐷 Fn (ℝ* × ℝ*)) | |
11 | 8, 9, 10 | mp2b 10 | . . . 4 ⊢ 𝐷 Fn (ℝ* × ℝ*) |
12 | rexpssxrxp 10423 | . . . 4 ⊢ (ℝ × ℝ) ⊆ (ℝ* × ℝ*) | |
13 | fnssres 6252 | . . . 4 ⊢ ((𝐷 Fn (ℝ* × ℝ*) ∧ (ℝ × ℝ) ⊆ (ℝ* × ℝ*)) → (𝐷 ↾ (ℝ × ℝ)) Fn (ℝ × ℝ)) | |
14 | 11, 12, 13 | mp2an 682 | . . 3 ⊢ (𝐷 ↾ (ℝ × ℝ)) Fn (ℝ × ℝ) |
15 | cnmet 22987 | . . . . 5 ⊢ (abs ∘ − ) ∈ (Met‘ℂ) | |
16 | metf 22547 | . . . . 5 ⊢ ((abs ∘ − ) ∈ (Met‘ℂ) → (abs ∘ − ):(ℂ × ℂ)⟶ℝ) | |
17 | ffn 6293 | . . . . 5 ⊢ ((abs ∘ − ):(ℂ × ℂ)⟶ℝ → (abs ∘ − ) Fn (ℂ × ℂ)) | |
18 | 15, 16, 17 | mp2b 10 | . . . 4 ⊢ (abs ∘ − ) Fn (ℂ × ℂ) |
19 | ax-resscn 10331 | . . . . 5 ⊢ ℝ ⊆ ℂ | |
20 | xpss12 5372 | . . . . 5 ⊢ ((ℝ ⊆ ℂ ∧ ℝ ⊆ ℂ) → (ℝ × ℝ) ⊆ (ℂ × ℂ)) | |
21 | 19, 19, 20 | mp2an 682 | . . . 4 ⊢ (ℝ × ℝ) ⊆ (ℂ × ℂ) |
22 | fnssres 6252 | . . . 4 ⊢ (((abs ∘ − ) Fn (ℂ × ℂ) ∧ (ℝ × ℝ) ⊆ (ℂ × ℂ)) → ((abs ∘ − ) ↾ (ℝ × ℝ)) Fn (ℝ × ℝ)) | |
23 | 18, 21, 22 | mp2an 682 | . . 3 ⊢ ((abs ∘ − ) ↾ (ℝ × ℝ)) Fn (ℝ × ℝ) |
24 | eqfnov2 7046 | . . 3 ⊢ (((𝐷 ↾ (ℝ × ℝ)) Fn (ℝ × ℝ) ∧ ((abs ∘ − ) ↾ (ℝ × ℝ)) Fn (ℝ × ℝ)) → ((𝐷 ↾ (ℝ × ℝ)) = ((abs ∘ − ) ↾ (ℝ × ℝ)) ↔ ∀𝑥 ∈ ℝ ∀𝑦 ∈ ℝ (𝑥(𝐷 ↾ (ℝ × ℝ))𝑦) = (𝑥((abs ∘ − ) ↾ (ℝ × ℝ))𝑦))) | |
25 | 14, 23, 24 | mp2an 682 | . 2 ⊢ ((𝐷 ↾ (ℝ × ℝ)) = ((abs ∘ − ) ↾ (ℝ × ℝ)) ↔ ∀𝑥 ∈ ℝ ∀𝑦 ∈ ℝ (𝑥(𝐷 ↾ (ℝ × ℝ))𝑦) = (𝑥((abs ∘ − ) ↾ (ℝ × ℝ))𝑦)) |
26 | 7, 25 | mpbir 223 | 1 ⊢ (𝐷 ↾ (ℝ × ℝ)) = ((abs ∘ − ) ↾ (ℝ × ℝ)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 198 ∧ wa 386 = wceq 1601 ∈ wcel 2107 ∀wral 3090 ⊆ wss 3792 × cxp 5355 ↾ cres 5359 ∘ ccom 5361 Fn wfn 6132 ⟶wf 6133 ‘cfv 6137 (class class class)co 6924 ℂcc 10272 ℝcr 10273 ℝ*cxr 10412 − cmin 10608 abscabs 14385 distcds 16351 ℝ*𝑠cxrs 16550 ∞Metcxmet 20131 Metcmet 20132 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-sep 5019 ax-nul 5027 ax-pow 5079 ax-pr 5140 ax-un 7228 ax-cnex 10330 ax-resscn 10331 ax-1cn 10332 ax-icn 10333 ax-addcl 10334 ax-addrcl 10335 ax-mulcl 10336 ax-mulrcl 10337 ax-mulcom 10338 ax-addass 10339 ax-mulass 10340 ax-distr 10341 ax-i2m1 10342 ax-1ne0 10343 ax-1rid 10344 ax-rnegex 10345 ax-rrecex 10346 ax-cnre 10347 ax-pre-lttri 10348 ax-pre-lttrn 10349 ax-pre-ltadd 10350 ax-pre-mulgt0 10351 ax-pre-sup 10352 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-nel 3076 df-ral 3095 df-rex 3096 df-reu 3097 df-rmo 3098 df-rab 3099 df-v 3400 df-sbc 3653 df-csb 3752 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-pss 3808 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-tp 4403 df-op 4405 df-uni 4674 df-int 4713 df-iun 4757 df-br 4889 df-opab 4951 df-mpt 4968 df-tr 4990 df-id 5263 df-eprel 5268 df-po 5276 df-so 5277 df-fr 5316 df-we 5318 df-xp 5363 df-rel 5364 df-cnv 5365 df-co 5366 df-dm 5367 df-rn 5368 df-res 5369 df-ima 5370 df-pred 5935 df-ord 5981 df-on 5982 df-lim 5983 df-suc 5984 df-iota 6101 df-fun 6139 df-fn 6140 df-f 6141 df-f1 6142 df-fo 6143 df-f1o 6144 df-fv 6145 df-riota 6885 df-ov 6927 df-oprab 6928 df-mpt2 6929 df-om 7346 df-1st 7447 df-2nd 7448 df-wrecs 7691 df-recs 7753 df-rdg 7791 df-1o 7845 df-oadd 7849 df-er 8028 df-map 8144 df-en 8244 df-dom 8245 df-sdom 8246 df-fin 8247 df-sup 8638 df-pnf 10415 df-mnf 10416 df-xr 10417 df-ltxr 10418 df-le 10419 df-sub 10610 df-neg 10611 df-div 11035 df-nn 11379 df-2 11442 df-3 11443 df-4 11444 df-5 11445 df-6 11446 df-7 11447 df-8 11448 df-9 11449 df-n0 11647 df-z 11733 df-dec 11850 df-uz 11997 df-rp 12142 df-xneg 12261 df-xadd 12262 df-icc 12498 df-fz 12648 df-seq 13124 df-exp 13183 df-cj 14250 df-re 14251 df-im 14252 df-sqrt 14386 df-abs 14387 df-struct 16261 df-ndx 16262 df-slot 16263 df-base 16265 df-plusg 16355 df-mulr 16356 df-tset 16361 df-ple 16362 df-ds 16364 df-xrs 16552 df-xmet 20139 df-met 20140 |
This theorem is referenced by: xrsmopn 23027 metdscn2 23072 |
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