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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 20592 | . . . 4 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (𝑥𝐷𝑦) = (abs‘(𝑥 − 𝑦))) |
3 | ovres 7316 | . . . 4 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (𝑥(𝐷 ↾ (ℝ × ℝ))𝑦) = (𝑥𝐷𝑦)) | |
4 | eqid 2823 | . . . . 5 ⊢ ((abs ∘ − ) ↾ (ℝ × ℝ)) = ((abs ∘ − ) ↾ (ℝ × ℝ)) | |
5 | 4 | remetdval 23399 | . . . 4 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (𝑥((abs ∘ − ) ↾ (ℝ × ℝ))𝑦) = (abs‘(𝑥 − 𝑦))) |
6 | 2, 3, 5 | 3eqtr4d 2868 | . . 3 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (𝑥(𝐷 ↾ (ℝ × ℝ))𝑦) = (𝑥((abs ∘ − ) ↾ (ℝ × ℝ))𝑦)) |
7 | 6 | rgen2 3205 | . 2 ⊢ ∀𝑥 ∈ ℝ ∀𝑦 ∈ ℝ (𝑥(𝐷 ↾ (ℝ × ℝ))𝑦) = (𝑥((abs ∘ − ) ↾ (ℝ × ℝ))𝑦) |
8 | 1 | xrsxmet 23419 | . . . . 5 ⊢ 𝐷 ∈ (∞Met‘ℝ*) |
9 | xmetf 22941 | . . . . 5 ⊢ (𝐷 ∈ (∞Met‘ℝ*) → 𝐷:(ℝ* × ℝ*)⟶ℝ*) | |
10 | ffn 6516 | . . . . 5 ⊢ (𝐷:(ℝ* × ℝ*)⟶ℝ* → 𝐷 Fn (ℝ* × ℝ*)) | |
11 | 8, 9, 10 | mp2b 10 | . . . 4 ⊢ 𝐷 Fn (ℝ* × ℝ*) |
12 | rexpssxrxp 10688 | . . . 4 ⊢ (ℝ × ℝ) ⊆ (ℝ* × ℝ*) | |
13 | fnssres 6472 | . . . 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 23382 | . . . . 5 ⊢ (abs ∘ − ) ∈ (Met‘ℂ) | |
16 | metf 22942 | . . . . 5 ⊢ ((abs ∘ − ) ∈ (Met‘ℂ) → (abs ∘ − ):(ℂ × ℂ)⟶ℝ) | |
17 | ffn 6516 | . . . . 5 ⊢ ((abs ∘ − ):(ℂ × ℂ)⟶ℝ → (abs ∘ − ) Fn (ℂ × ℂ)) | |
18 | 15, 16, 17 | mp2b 10 | . . . 4 ⊢ (abs ∘ − ) Fn (ℂ × ℂ) |
19 | ax-resscn 10596 | . . . . 5 ⊢ ℝ ⊆ ℂ | |
20 | xpss12 5572 | . . . . 5 ⊢ ((ℝ ⊆ ℂ ∧ ℝ ⊆ ℂ) → (ℝ × ℝ) ⊆ (ℂ × ℂ)) | |
21 | 19, 19, 20 | mp2an 690 | . . . 4 ⊢ (ℝ × ℝ) ⊆ (ℂ × ℂ) |
22 | fnssres 6472 | . . . 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 7283 | . . 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 233 | 1 ⊢ (𝐷 ↾ (ℝ × ℝ)) = ((abs ∘ − ) ↾ (ℝ × ℝ)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ∀wral 3140 ⊆ wss 3938 × cxp 5555 ↾ cres 5559 ∘ ccom 5561 Fn wfn 6352 ⟶wf 6353 ‘cfv 6357 (class class class)co 7158 ℂcc 10537 ℝcr 10538 ℝ*cxr 10676 − cmin 10872 abscabs 14595 distcds 16576 ℝ*𝑠cxrs 16775 ∞Metcxmet 20532 Metcmet 20533 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 ax-pre-sup 10617 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-int 4879 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-om 7583 df-1st 7691 df-2nd 7692 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-1o 8104 df-oadd 8108 df-er 8291 df-map 8410 df-en 8512 df-dom 8513 df-sdom 8514 df-fin 8515 df-sup 8908 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-div 11300 df-nn 11641 df-2 11703 df-3 11704 df-4 11705 df-5 11706 df-6 11707 df-7 11708 df-8 11709 df-9 11710 df-n0 11901 df-z 11985 df-dec 12102 df-uz 12247 df-rp 12393 df-xneg 12510 df-xadd 12511 df-icc 12748 df-fz 12896 df-seq 13373 df-exp 13433 df-cj 14460 df-re 14461 df-im 14462 df-sqrt 14596 df-abs 14597 df-struct 16487 df-ndx 16488 df-slot 16489 df-base 16491 df-plusg 16580 df-mulr 16581 df-tset 16586 df-ple 16587 df-ds 16589 df-xrs 16777 df-xmet 20540 df-met 20541 |
This theorem is referenced by: xrsmopn 23422 metdscn2 23467 |
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