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Theorem stdbdbl 23044
Description: The standard bounded metric corresponding to 𝐶 generates the same balls as 𝐶 for radii less than 𝑅. (Contributed by Mario Carneiro, 26-Aug-2015.)
Hypothesis
Ref Expression
stdbdmet.1 𝐷 = (𝑥𝑋, 𝑦𝑋 ↦ if((𝑥𝐶𝑦) ≤ 𝑅, (𝑥𝐶𝑦), 𝑅))
Assertion
Ref Expression
stdbdbl (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ* ∧ 0 < 𝑅) ∧ (𝑃𝑋𝑆 ∈ ℝ*𝑆𝑅)) → (𝑃(ball‘𝐷)𝑆) = (𝑃(ball‘𝐶)𝑆))
Distinct variable groups:   𝑥,𝑦,𝐶   𝑥,𝑃,𝑦   𝑥,𝑅,𝑦   𝑥,𝑋,𝑦
Allowed substitution hints:   𝐷(𝑥,𝑦)   𝑆(𝑥,𝑦)

Proof of Theorem stdbdbl
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 simpll2 1207 . . . . . 6 ((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ* ∧ 0 < 𝑅) ∧ (𝑃𝑋𝑆 ∈ ℝ*𝑆𝑅)) ∧ 𝑧𝑋) → 𝑅 ∈ ℝ*)
2 simpr1 1188 . . . . . . 7 (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ* ∧ 0 < 𝑅) ∧ (𝑃𝑋𝑆 ∈ ℝ*𝑆𝑅)) → 𝑃𝑋)
32adantr 481 . . . . . 6 ((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ* ∧ 0 < 𝑅) ∧ (𝑃𝑋𝑆 ∈ ℝ*𝑆𝑅)) ∧ 𝑧𝑋) → 𝑃𝑋)
4 simpr 485 . . . . . 6 ((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ* ∧ 0 < 𝑅) ∧ (𝑃𝑋𝑆 ∈ ℝ*𝑆𝑅)) ∧ 𝑧𝑋) → 𝑧𝑋)
5 stdbdmet.1 . . . . . . 7 𝐷 = (𝑥𝑋, 𝑦𝑋 ↦ if((𝑥𝐶𝑦) ≤ 𝑅, (𝑥𝐶𝑦), 𝑅))
65stdbdmetval 23041 . . . . . 6 ((𝑅 ∈ ℝ*𝑃𝑋𝑧𝑋) → (𝑃𝐷𝑧) = if((𝑃𝐶𝑧) ≤ 𝑅, (𝑃𝐶𝑧), 𝑅))
71, 3, 4, 6syl3anc 1365 . . . . 5 ((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ* ∧ 0 < 𝑅) ∧ (𝑃𝑋𝑆 ∈ ℝ*𝑆𝑅)) ∧ 𝑧𝑋) → (𝑃𝐷𝑧) = if((𝑃𝐶𝑧) ≤ 𝑅, (𝑃𝐶𝑧), 𝑅))
87breq1d 5072 . . . 4 ((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ* ∧ 0 < 𝑅) ∧ (𝑃𝑋𝑆 ∈ ℝ*𝑆𝑅)) ∧ 𝑧𝑋) → ((𝑃𝐷𝑧) < 𝑆 ↔ if((𝑃𝐶𝑧) ≤ 𝑅, (𝑃𝐶𝑧), 𝑅) < 𝑆))
9 simplr3 1211 . . . . . . . 8 ((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ* ∧ 0 < 𝑅) ∧ (𝑃𝑋𝑆 ∈ ℝ*𝑆𝑅)) ∧ 𝑧𝑋) → 𝑆𝑅)
109biantrud 532 . . . . . . 7 ((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ* ∧ 0 < 𝑅) ∧ (𝑃𝑋𝑆 ∈ ℝ*𝑆𝑅)) ∧ 𝑧𝑋) → (𝑆 ≤ (𝑃𝐶𝑧) ↔ (𝑆 ≤ (𝑃𝐶𝑧) ∧ 𝑆𝑅)))
11 simpr2 1189 . . . . . . . . 9 (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ* ∧ 0 < 𝑅) ∧ (𝑃𝑋𝑆 ∈ ℝ*𝑆𝑅)) → 𝑆 ∈ ℝ*)
1211adantr 481 . . . . . . . 8 ((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ* ∧ 0 < 𝑅) ∧ (𝑃𝑋𝑆 ∈ ℝ*𝑆𝑅)) ∧ 𝑧𝑋) → 𝑆 ∈ ℝ*)
13 simpl1 1185 . . . . . . . . . 10 (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ* ∧ 0 < 𝑅) ∧ (𝑃𝑋𝑆 ∈ ℝ*𝑆𝑅)) → 𝐶 ∈ (∞Met‘𝑋))
1413adantr 481 . . . . . . . . 9 ((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ* ∧ 0 < 𝑅) ∧ (𝑃𝑋𝑆 ∈ ℝ*𝑆𝑅)) ∧ 𝑧𝑋) → 𝐶 ∈ (∞Met‘𝑋))
15 xmetcl 22858 . . . . . . . . 9 ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑃𝑋𝑧𝑋) → (𝑃𝐶𝑧) ∈ ℝ*)
1614, 3, 4, 15syl3anc 1365 . . . . . . . 8 ((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ* ∧ 0 < 𝑅) ∧ (𝑃𝑋𝑆 ∈ ℝ*𝑆𝑅)) ∧ 𝑧𝑋) → (𝑃𝐶𝑧) ∈ ℝ*)
17 xrlemin 12570 . . . . . . . 8 ((𝑆 ∈ ℝ* ∧ (𝑃𝐶𝑧) ∈ ℝ*𝑅 ∈ ℝ*) → (𝑆 ≤ if((𝑃𝐶𝑧) ≤ 𝑅, (𝑃𝐶𝑧), 𝑅) ↔ (𝑆 ≤ (𝑃𝐶𝑧) ∧ 𝑆𝑅)))
1812, 16, 1, 17syl3anc 1365 . . . . . . 7 ((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ* ∧ 0 < 𝑅) ∧ (𝑃𝑋𝑆 ∈ ℝ*𝑆𝑅)) ∧ 𝑧𝑋) → (𝑆 ≤ if((𝑃𝐶𝑧) ≤ 𝑅, (𝑃𝐶𝑧), 𝑅) ↔ (𝑆 ≤ (𝑃𝐶𝑧) ∧ 𝑆𝑅)))
1910, 18bitr4d 283 . . . . . 6 ((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ* ∧ 0 < 𝑅) ∧ (𝑃𝑋𝑆 ∈ ℝ*𝑆𝑅)) ∧ 𝑧𝑋) → (𝑆 ≤ (𝑃𝐶𝑧) ↔ 𝑆 ≤ if((𝑃𝐶𝑧) ≤ 𝑅, (𝑃𝐶𝑧), 𝑅)))
2019notbid 319 . . . . 5 ((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ* ∧ 0 < 𝑅) ∧ (𝑃𝑋𝑆 ∈ ℝ*𝑆𝑅)) ∧ 𝑧𝑋) → (¬ 𝑆 ≤ (𝑃𝐶𝑧) ↔ ¬ 𝑆 ≤ if((𝑃𝐶𝑧) ≤ 𝑅, (𝑃𝐶𝑧), 𝑅)))
21 xrltnle 10700 . . . . . 6 (((𝑃𝐶𝑧) ∈ ℝ*𝑆 ∈ ℝ*) → ((𝑃𝐶𝑧) < 𝑆 ↔ ¬ 𝑆 ≤ (𝑃𝐶𝑧)))
2216, 12, 21syl2anc 584 . . . . 5 ((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ* ∧ 0 < 𝑅) ∧ (𝑃𝑋𝑆 ∈ ℝ*𝑆𝑅)) ∧ 𝑧𝑋) → ((𝑃𝐶𝑧) < 𝑆 ↔ ¬ 𝑆 ≤ (𝑃𝐶𝑧)))
2316, 1ifcld 4514 . . . . . 6 ((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ* ∧ 0 < 𝑅) ∧ (𝑃𝑋𝑆 ∈ ℝ*𝑆𝑅)) ∧ 𝑧𝑋) → if((𝑃𝐶𝑧) ≤ 𝑅, (𝑃𝐶𝑧), 𝑅) ∈ ℝ*)
24 xrltnle 10700 . . . . . 6 ((if((𝑃𝐶𝑧) ≤ 𝑅, (𝑃𝐶𝑧), 𝑅) ∈ ℝ*𝑆 ∈ ℝ*) → (if((𝑃𝐶𝑧) ≤ 𝑅, (𝑃𝐶𝑧), 𝑅) < 𝑆 ↔ ¬ 𝑆 ≤ if((𝑃𝐶𝑧) ≤ 𝑅, (𝑃𝐶𝑧), 𝑅)))
2523, 12, 24syl2anc 584 . . . . 5 ((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ* ∧ 0 < 𝑅) ∧ (𝑃𝑋𝑆 ∈ ℝ*𝑆𝑅)) ∧ 𝑧𝑋) → (if((𝑃𝐶𝑧) ≤ 𝑅, (𝑃𝐶𝑧), 𝑅) < 𝑆 ↔ ¬ 𝑆 ≤ if((𝑃𝐶𝑧) ≤ 𝑅, (𝑃𝐶𝑧), 𝑅)))
2620, 22, 253bitr4d 312 . . . 4 ((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ* ∧ 0 < 𝑅) ∧ (𝑃𝑋𝑆 ∈ ℝ*𝑆𝑅)) ∧ 𝑧𝑋) → ((𝑃𝐶𝑧) < 𝑆 ↔ if((𝑃𝐶𝑧) ≤ 𝑅, (𝑃𝐶𝑧), 𝑅) < 𝑆))
278, 26bitr4d 283 . . 3 ((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ* ∧ 0 < 𝑅) ∧ (𝑃𝑋𝑆 ∈ ℝ*𝑆𝑅)) ∧ 𝑧𝑋) → ((𝑃𝐷𝑧) < 𝑆 ↔ (𝑃𝐶𝑧) < 𝑆))
2827rabbidva 3483 . 2 (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ* ∧ 0 < 𝑅) ∧ (𝑃𝑋𝑆 ∈ ℝ*𝑆𝑅)) → {𝑧𝑋 ∣ (𝑃𝐷𝑧) < 𝑆} = {𝑧𝑋 ∣ (𝑃𝐶𝑧) < 𝑆})
295stdbdxmet 23042 . . . 4 ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ* ∧ 0 < 𝑅) → 𝐷 ∈ (∞Met‘𝑋))
3029adantr 481 . . 3 (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ* ∧ 0 < 𝑅) ∧ (𝑃𝑋𝑆 ∈ ℝ*𝑆𝑅)) → 𝐷 ∈ (∞Met‘𝑋))
31 blval 22913 . . 3 ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃𝑋𝑆 ∈ ℝ*) → (𝑃(ball‘𝐷)𝑆) = {𝑧𝑋 ∣ (𝑃𝐷𝑧) < 𝑆})
3230, 2, 11, 31syl3anc 1365 . 2 (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ* ∧ 0 < 𝑅) ∧ (𝑃𝑋𝑆 ∈ ℝ*𝑆𝑅)) → (𝑃(ball‘𝐷)𝑆) = {𝑧𝑋 ∣ (𝑃𝐷𝑧) < 𝑆})
33 blval 22913 . . 3 ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑃𝑋𝑆 ∈ ℝ*) → (𝑃(ball‘𝐶)𝑆) = {𝑧𝑋 ∣ (𝑃𝐶𝑧) < 𝑆})
3413, 2, 11, 33syl3anc 1365 . 2 (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ* ∧ 0 < 𝑅) ∧ (𝑃𝑋𝑆 ∈ ℝ*𝑆𝑅)) → (𝑃(ball‘𝐶)𝑆) = {𝑧𝑋 ∣ (𝑃𝐶𝑧) < 𝑆})
3528, 32, 343eqtr4d 2870 1 (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ* ∧ 0 < 𝑅) ∧ (𝑃𝑋𝑆 ∈ ℝ*𝑆𝑅)) → (𝑃(ball‘𝐷)𝑆) = (𝑃(ball‘𝐶)𝑆))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 207  wa 396  w3a 1081   = wceq 1530  wcel 2107  {crab 3146  ifcif 4469   class class class wbr 5062  cfv 6351  (class class class)co 7151  cmpo 7153  0cc0 10529  *cxr 10666   < clt 10667  cle 10668  ∞Metcxmet 20448  ballcbl 20450
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2797  ax-sep 5199  ax-nul 5206  ax-pow 5262  ax-pr 5325  ax-un 7454  ax-cnex 10585  ax-resscn 10586  ax-1cn 10587  ax-icn 10588  ax-addcl 10589  ax-addrcl 10590  ax-mulcl 10591  ax-mulrcl 10592  ax-mulcom 10593  ax-addass 10594  ax-mulass 10595  ax-distr 10596  ax-i2m1 10597  ax-1ne0 10598  ax-1rid 10599  ax-rnegex 10600  ax-rrecex 10601  ax-cnre 10602  ax-pre-lttri 10603  ax-pre-lttrn 10604  ax-pre-ltadd 10605  ax-pre-mulgt0 10606
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3or 1082  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2619  df-eu 2651  df-clab 2804  df-cleq 2818  df-clel 2897  df-nfc 2967  df-ne 3021  df-nel 3128  df-ral 3147  df-rex 3148  df-reu 3149  df-rmo 3150  df-rab 3151  df-v 3501  df-sbc 3776  df-csb 3887  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-nul 4295  df-if 4470  df-pw 4543  df-sn 4564  df-pr 4566  df-op 4570  df-uni 4837  df-iun 4918  df-br 5063  df-opab 5125  df-mpt 5143  df-id 5458  df-po 5472  df-so 5473  df-xp 5559  df-rel 5560  df-cnv 5561  df-co 5562  df-dm 5563  df-rn 5564  df-res 5565  df-ima 5566  df-iota 6311  df-fun 6353  df-fn 6354  df-f 6355  df-f1 6356  df-fo 6357  df-f1o 6358  df-fv 6359  df-riota 7109  df-ov 7154  df-oprab 7155  df-mpo 7156  df-1st 7683  df-2nd 7684  df-er 8282  df-map 8401  df-en 8502  df-dom 8503  df-sdom 8504  df-pnf 10669  df-mnf 10670  df-xr 10671  df-ltxr 10672  df-le 10673  df-sub 10864  df-neg 10865  df-div 11290  df-2 11692  df-rp 12383  df-xneg 12500  df-xadd 12501  df-xmul 12502  df-icc 12738  df-psmet 20455  df-xmet 20456  df-bl 20458
This theorem is referenced by:  stdbdmopn  23045  xlebnum  23486
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