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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 21408 | . . . 4 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (𝑥𝐷𝑦) = (abs‘(𝑥 − 𝑦))) |
3 | ovres 7592 | . . . 4 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (𝑥(𝐷 ↾ (ℝ × ℝ))𝑦) = (𝑥𝐷𝑦)) | |
4 | eqid 2726 | . . . . 5 ⊢ ((abs ∘ − ) ↾ (ℝ × ℝ)) = ((abs ∘ − ) ↾ (ℝ × ℝ)) | |
5 | 4 | remetdval 24796 | . . . 4 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (𝑥((abs ∘ − ) ↾ (ℝ × ℝ))𝑦) = (abs‘(𝑥 − 𝑦))) |
6 | 2, 3, 5 | 3eqtr4d 2776 | . . 3 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (𝑥(𝐷 ↾ (ℝ × ℝ))𝑦) = (𝑥((abs ∘ − ) ↾ (ℝ × ℝ))𝑦)) |
7 | 6 | rgen2 3188 | . 2 ⊢ ∀𝑥 ∈ ℝ ∀𝑦 ∈ ℝ (𝑥(𝐷 ↾ (ℝ × ℝ))𝑦) = (𝑥((abs ∘ − ) ↾ (ℝ × ℝ))𝑦) |
8 | 1 | xrsxmet 24816 | . . . . 5 ⊢ 𝐷 ∈ (∞Met‘ℝ*) |
9 | xmetf 24326 | . . . . 5 ⊢ (𝐷 ∈ (∞Met‘ℝ*) → 𝐷:(ℝ* × ℝ*)⟶ℝ*) | |
10 | ffn 6728 | . . . . 5 ⊢ (𝐷:(ℝ* × ℝ*)⟶ℝ* → 𝐷 Fn (ℝ* × ℝ*)) | |
11 | 8, 9, 10 | mp2b 10 | . . . 4 ⊢ 𝐷 Fn (ℝ* × ℝ*) |
12 | rexpssxrxp 11309 | . . . 4 ⊢ (ℝ × ℝ) ⊆ (ℝ* × ℝ*) | |
13 | fnssres 6684 | . . . 4 ⊢ ((𝐷 Fn (ℝ* × ℝ*) ∧ (ℝ × ℝ) ⊆ (ℝ* × ℝ*)) → (𝐷 ↾ (ℝ × ℝ)) Fn (ℝ × ℝ)) | |
14 | 11, 12, 13 | mp2an 690 | . . 3 ⊢ (𝐷 ↾ (ℝ × ℝ)) Fn (ℝ × ℝ) |
15 | cnmet 24779 | . . . . 5 ⊢ (abs ∘ − ) ∈ (Met‘ℂ) | |
16 | metf 24327 | . . . . 5 ⊢ ((abs ∘ − ) ∈ (Met‘ℂ) → (abs ∘ − ):(ℂ × ℂ)⟶ℝ) | |
17 | ffn 6728 | . . . . 5 ⊢ ((abs ∘ − ):(ℂ × ℂ)⟶ℝ → (abs ∘ − ) Fn (ℂ × ℂ)) | |
18 | 15, 16, 17 | mp2b 10 | . . . 4 ⊢ (abs ∘ − ) Fn (ℂ × ℂ) |
19 | ax-resscn 11215 | . . . . 5 ⊢ ℝ ⊆ ℂ | |
20 | xpss12 5697 | . . . . 5 ⊢ ((ℝ ⊆ ℂ ∧ ℝ ⊆ ℂ) → (ℝ × ℝ) ⊆ (ℂ × ℂ)) | |
21 | 19, 19, 20 | mp2an 690 | . . . 4 ⊢ (ℝ × ℝ) ⊆ (ℂ × ℂ) |
22 | fnssres 6684 | . . . 4 ⊢ (((abs ∘ − ) Fn (ℂ × ℂ) ∧ (ℝ × ℝ) ⊆ (ℂ × ℂ)) → ((abs ∘ − ) ↾ (ℝ × ℝ)) Fn (ℝ × ℝ)) | |
23 | 18, 21, 22 | mp2an 690 | . . 3 ⊢ ((abs ∘ − ) ↾ (ℝ × ℝ)) Fn (ℝ × ℝ) |
24 | eqfnov2 7556 | . . 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 690 | . 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 230 | 1 ⊢ (𝐷 ↾ (ℝ × ℝ)) = ((abs ∘ − ) ↾ (ℝ × ℝ)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∧ wa 394 = wceq 1534 ∈ wcel 2099 ∀wral 3051 ⊆ wss 3947 × cxp 5680 ↾ cres 5684 ∘ ccom 5686 Fn wfn 6549 ⟶wf 6550 ‘cfv 6554 (class class class)co 7424 ℂcc 11156 ℝcr 11157 ℝ*cxr 11297 − cmin 11494 abscabs 15239 distcds 17275 ℝ*𝑠cxrs 17515 ∞Metcxmet 21328 Metcmet 21329 |
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-sep 5304 ax-nul 5311 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-cnex 11214 ax-resscn 11215 ax-1cn 11216 ax-icn 11217 ax-addcl 11218 ax-addrcl 11219 ax-mulcl 11220 ax-mulrcl 11221 ax-mulcom 11222 ax-addass 11223 ax-mulass 11224 ax-distr 11225 ax-i2m1 11226 ax-1ne0 11227 ax-1rid 11228 ax-rnegex 11229 ax-rrecex 11230 ax-cnre 11231 ax-pre-lttri 11232 ax-pre-lttrn 11233 ax-pre-ltadd 11234 ax-pre-mulgt0 11235 ax-pre-sup 11236 |
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 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3967 df-nul 4326 df-if 4534 df-pw 4609 df-sn 4634 df-pr 4636 df-tp 4638 df-op 4640 df-uni 4914 df-iun 5003 df-br 5154 df-opab 5216 df-mpt 5237 df-tr 5271 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6312 df-ord 6379 df-on 6380 df-lim 6381 df-suc 6382 df-iota 6506 df-fun 6556 df-fn 6557 df-f 6558 df-f1 6559 df-fo 6560 df-f1o 6561 df-fv 6562 df-riota 7380 df-ov 7427 df-oprab 7428 df-mpo 7429 df-om 7877 df-1st 8003 df-2nd 8004 df-frecs 8296 df-wrecs 8327 df-recs 8401 df-rdg 8440 df-1o 8496 df-er 8734 df-map 8857 df-en 8975 df-dom 8976 df-sdom 8977 df-fin 8978 df-sup 9485 df-pnf 11300 df-mnf 11301 df-xr 11302 df-ltxr 11303 df-le 11304 df-sub 11496 df-neg 11497 df-div 11922 df-nn 12265 df-2 12327 df-3 12328 df-4 12329 df-5 12330 df-6 12331 df-7 12332 df-8 12333 df-9 12334 df-n0 12525 df-z 12611 df-dec 12730 df-uz 12875 df-rp 13029 df-xneg 13146 df-xadd 13147 df-icc 13385 df-fz 13539 df-seq 14022 df-exp 14082 df-cj 15104 df-re 15105 df-im 15106 df-sqrt 15240 df-abs 15241 df-struct 17149 df-slot 17184 df-ndx 17196 df-base 17214 df-plusg 17279 df-mulr 17280 df-tset 17285 df-ple 17286 df-ds 17288 df-xrs 17517 df-xmet 21336 df-met 21337 |
This theorem is referenced by: xrsmopn 24819 metdscn2 24864 |
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