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Mirrors > Home > MPE Home > Th. List > xrsds | Structured version Visualization version GIF version |
Description: The metric of the extended real number structure. (Contributed by Mario Carneiro, 20-Aug-2015.) |
Ref | Expression |
---|---|
xrsds.d | ⊢ 𝐷 = (dist‘ℝ*𝑠) |
Ref | Expression |
---|---|
xrsds | ⊢ 𝐷 = (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ if(𝑥 ≤ 𝑦, (𝑦 +𝑒 -𝑒𝑥), (𝑥 +𝑒 -𝑒𝑦))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | xrsds.d | . 2 ⊢ 𝐷 = (dist‘ℝ*𝑠) | |
2 | id 22 | . . . . . . . 8 ⊢ (𝑦 ∈ ℝ* → 𝑦 ∈ ℝ*) | |
3 | xnegcl 12356 | . . . . . . . 8 ⊢ (𝑥 ∈ ℝ* → -𝑒𝑥 ∈ ℝ*) | |
4 | xaddcl 12382 | . . . . . . . 8 ⊢ ((𝑦 ∈ ℝ* ∧ -𝑒𝑥 ∈ ℝ*) → (𝑦 +𝑒 -𝑒𝑥) ∈ ℝ*) | |
5 | 2, 3, 4 | syl2anr 590 | . . . . . . 7 ⊢ ((𝑥 ∈ ℝ* ∧ 𝑦 ∈ ℝ*) → (𝑦 +𝑒 -𝑒𝑥) ∈ ℝ*) |
6 | xnegcl 12356 | . . . . . . . 8 ⊢ (𝑦 ∈ ℝ* → -𝑒𝑦 ∈ ℝ*) | |
7 | xaddcl 12382 | . . . . . . . 8 ⊢ ((𝑥 ∈ ℝ* ∧ -𝑒𝑦 ∈ ℝ*) → (𝑥 +𝑒 -𝑒𝑦) ∈ ℝ*) | |
8 | 6, 7 | sylan2 586 | . . . . . . 7 ⊢ ((𝑥 ∈ ℝ* ∧ 𝑦 ∈ ℝ*) → (𝑥 +𝑒 -𝑒𝑦) ∈ ℝ*) |
9 | 5, 8 | ifcld 4352 | . . . . . 6 ⊢ ((𝑥 ∈ ℝ* ∧ 𝑦 ∈ ℝ*) → if(𝑥 ≤ 𝑦, (𝑦 +𝑒 -𝑒𝑥), (𝑥 +𝑒 -𝑒𝑦)) ∈ ℝ*) |
10 | 9 | rgen2a 3159 | . . . . 5 ⊢ ∀𝑥 ∈ ℝ* ∀𝑦 ∈ ℝ* if(𝑥 ≤ 𝑦, (𝑦 +𝑒 -𝑒𝑥), (𝑥 +𝑒 -𝑒𝑦)) ∈ ℝ* |
11 | eqid 2778 | . . . . . 6 ⊢ (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ if(𝑥 ≤ 𝑦, (𝑦 +𝑒 -𝑒𝑥), (𝑥 +𝑒 -𝑒𝑦))) = (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ if(𝑥 ≤ 𝑦, (𝑦 +𝑒 -𝑒𝑥), (𝑥 +𝑒 -𝑒𝑦))) | |
12 | 11 | fmpt2 7517 | . . . . 5 ⊢ (∀𝑥 ∈ ℝ* ∀𝑦 ∈ ℝ* if(𝑥 ≤ 𝑦, (𝑦 +𝑒 -𝑒𝑥), (𝑥 +𝑒 -𝑒𝑦)) ∈ ℝ* ↔ (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ if(𝑥 ≤ 𝑦, (𝑦 +𝑒 -𝑒𝑥), (𝑥 +𝑒 -𝑒𝑦))):(ℝ* × ℝ*)⟶ℝ*) |
13 | 10, 12 | mpbi 222 | . . . 4 ⊢ (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ if(𝑥 ≤ 𝑦, (𝑦 +𝑒 -𝑒𝑥), (𝑥 +𝑒 -𝑒𝑦))):(ℝ* × ℝ*)⟶ℝ* |
14 | xrex 12134 | . . . . 5 ⊢ ℝ* ∈ V | |
15 | 14, 14 | xpex 7240 | . . . 4 ⊢ (ℝ* × ℝ*) ∈ V |
16 | fex2 7400 | . . . 4 ⊢ (((𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ if(𝑥 ≤ 𝑦, (𝑦 +𝑒 -𝑒𝑥), (𝑥 +𝑒 -𝑒𝑦))):(ℝ* × ℝ*)⟶ℝ* ∧ (ℝ* × ℝ*) ∈ V ∧ ℝ* ∈ V) → (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ if(𝑥 ≤ 𝑦, (𝑦 +𝑒 -𝑒𝑥), (𝑥 +𝑒 -𝑒𝑦))) ∈ V) | |
17 | 13, 15, 14, 16 | mp3an 1534 | . . 3 ⊢ (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ if(𝑥 ≤ 𝑦, (𝑦 +𝑒 -𝑒𝑥), (𝑥 +𝑒 -𝑒𝑦))) ∈ V |
18 | df-xrs 16548 | . . . 4 ⊢ ℝ*𝑠 = ({〈(Base‘ndx), ℝ*〉, 〈(+g‘ndx), +𝑒 〉, 〈(.r‘ndx), ·e 〉} ∪ {〈(TopSet‘ndx), (ordTop‘ ≤ )〉, 〈(le‘ndx), ≤ 〉, 〈(dist‘ndx), (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ if(𝑥 ≤ 𝑦, (𝑦 +𝑒 -𝑒𝑥), (𝑥 +𝑒 -𝑒𝑦)))〉}) | |
19 | 18 | odrngds 16458 | . . 3 ⊢ ((𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ if(𝑥 ≤ 𝑦, (𝑦 +𝑒 -𝑒𝑥), (𝑥 +𝑒 -𝑒𝑦))) ∈ V → (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ if(𝑥 ≤ 𝑦, (𝑦 +𝑒 -𝑒𝑥), (𝑥 +𝑒 -𝑒𝑦))) = (dist‘ℝ*𝑠)) |
20 | 17, 19 | ax-mp 5 | . 2 ⊢ (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ if(𝑥 ≤ 𝑦, (𝑦 +𝑒 -𝑒𝑥), (𝑥 +𝑒 -𝑒𝑦))) = (dist‘ℝ*𝑠) |
21 | 1, 20 | eqtr4i 2805 | 1 ⊢ 𝐷 = (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ if(𝑥 ≤ 𝑦, (𝑦 +𝑒 -𝑒𝑥), (𝑥 +𝑒 -𝑒𝑦))) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 386 = wceq 1601 ∈ wcel 2107 ∀wral 3090 Vcvv 3398 ifcif 4307 class class class wbr 4886 × cxp 5353 ⟶wf 6131 ‘cfv 6135 (class class class)co 6922 ↦ cmpt2 6924 ℝ*cxr 10410 ≤ cle 10412 -𝑒cxne 12254 +𝑒 cxad 12255 ·e cxmu 12256 distcds 16347 ordTopcordt 16545 ℝ*𝑠cxrs 16546 |
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 5017 ax-nul 5025 ax-pow 5077 ax-pr 5138 ax-un 7226 ax-cnex 10328 ax-resscn 10329 ax-1cn 10330 ax-icn 10331 ax-addcl 10332 ax-addrcl 10333 ax-mulcl 10334 ax-mulrcl 10335 ax-mulcom 10336 ax-addass 10337 ax-mulass 10338 ax-distr 10339 ax-i2m1 10340 ax-1ne0 10341 ax-1rid 10342 ax-rnegex 10343 ax-rrecex 10344 ax-cnre 10345 ax-pre-lttri 10346 ax-pre-lttrn 10347 ax-pre-ltadd 10348 ax-pre-mulgt0 10349 |
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-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 4672 df-int 4711 df-iun 4755 df-br 4887 df-opab 4949 df-mpt 4966 df-tr 4988 df-id 5261 df-eprel 5266 df-po 5274 df-so 5275 df-fr 5314 df-we 5316 df-xp 5361 df-rel 5362 df-cnv 5363 df-co 5364 df-dm 5365 df-rn 5366 df-res 5367 df-ima 5368 df-pred 5933 df-ord 5979 df-on 5980 df-lim 5981 df-suc 5982 df-iota 6099 df-fun 6137 df-fn 6138 df-f 6139 df-f1 6140 df-fo 6141 df-f1o 6142 df-fv 6143 df-riota 6883 df-ov 6925 df-oprab 6926 df-mpt2 6927 df-om 7344 df-1st 7445 df-2nd 7446 df-wrecs 7689 df-recs 7751 df-rdg 7789 df-1o 7843 df-oadd 7847 df-er 8026 df-en 8242 df-dom 8243 df-sdom 8244 df-fin 8245 df-pnf 10413 df-mnf 10414 df-xr 10415 df-ltxr 10416 df-le 10417 df-sub 10608 df-neg 10609 df-nn 11375 df-2 11438 df-3 11439 df-4 11440 df-5 11441 df-6 11442 df-7 11443 df-8 11444 df-9 11445 df-n0 11643 df-z 11729 df-dec 11846 df-uz 11993 df-xneg 12257 df-xadd 12258 df-fz 12644 df-struct 16257 df-ndx 16258 df-slot 16259 df-base 16261 df-plusg 16351 df-mulr 16352 df-tset 16357 df-ple 16358 df-ds 16360 df-xrs 16548 |
This theorem is referenced by: xrsdsval 20186 xrsxmet 23020 |
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