Theorem stdbdbl 23673

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 = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ if((𝑥𝐶𝑦) ≤ 𝑅, (𝑥𝐶𝑦), 𝑅))

Assertion

Ref	Expression
stdbdbl	⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ* ∧ 0 < 𝑅) ∧ (𝑃 ∈ 𝑋 ∧ 𝑆 ∈ ℝ* ∧ 𝑆 ≤ 𝑅)) → (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.

Step	Hyp	Ref	Expression
1		simpll2 1212	. . . . . 6 ⊢ ((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ* ∧ 0 < 𝑅) ∧ (𝑃 ∈ 𝑋 ∧ 𝑆 ∈ ℝ* ∧ 𝑆 ≤ 𝑅)) ∧ 𝑧 ∈ 𝑋) → 𝑅 ∈ ℝ*)
2		simpr1 1193	. . . . . . 7 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ* ∧ 0 < 𝑅) ∧ (𝑃 ∈ 𝑋 ∧ 𝑆 ∈ ℝ* ∧ 𝑆 ≤ 𝑅)) → 𝑃 ∈ 𝑋)
3	2	adantr 481	. . . . . 6 ⊢ ((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ* ∧ 0 < 𝑅) ∧ (𝑃 ∈ 𝑋 ∧ 𝑆 ∈ ℝ* ∧ 𝑆 ≤ 𝑅)) ∧ 𝑧 ∈ 𝑋) → 𝑃 ∈ 𝑋)
4		simpr 485	. . . . . 6 ⊢ ((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ* ∧ 0 < 𝑅) ∧ (𝑃 ∈ 𝑋 ∧ 𝑆 ∈ ℝ* ∧ 𝑆 ≤ 𝑅)) ∧ 𝑧 ∈ 𝑋) → 𝑧 ∈ 𝑋)
5		stdbdmet.1	. . . . . . 7 ⊢ 𝐷 = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ if((𝑥𝐶𝑦) ≤ 𝑅, (𝑥𝐶𝑦), 𝑅))
6	5	stdbdmetval 23670	. . . . . 6 ⊢ ((𝑅 ∈ ℝ* ∧ 𝑃 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋) → (𝑃𝐷𝑧) = if((𝑃𝐶𝑧) ≤ 𝑅, (𝑃𝐶𝑧), 𝑅))
7	1, 3, 4, 6	syl3anc 1370	. . . . 5 ⊢ ((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ* ∧ 0 < 𝑅) ∧ (𝑃 ∈ 𝑋 ∧ 𝑆 ∈ ℝ* ∧ 𝑆 ≤ 𝑅)) ∧ 𝑧 ∈ 𝑋) → (𝑃𝐷𝑧) = if((𝑃𝐶𝑧) ≤ 𝑅, (𝑃𝐶𝑧), 𝑅))
8	7	breq1d 5084	. . . 4 ⊢ ((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ* ∧ 0 < 𝑅) ∧ (𝑃 ∈ 𝑋 ∧ 𝑆 ∈ ℝ* ∧ 𝑆 ≤ 𝑅)) ∧ 𝑧 ∈ 𝑋) → ((𝑃𝐷𝑧) < 𝑆 ↔ if((𝑃𝐶𝑧) ≤ 𝑅, (𝑃𝐶𝑧), 𝑅) < 𝑆))
9		simplr3 1216	. . . . . . . 8 ⊢ ((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ* ∧ 0 < 𝑅) ∧ (𝑃 ∈ 𝑋 ∧ 𝑆 ∈ ℝ* ∧ 𝑆 ≤ 𝑅)) ∧ 𝑧 ∈ 𝑋) → 𝑆 ≤ 𝑅)
10	9	biantrud 532	. . . . . . 7 ⊢ ((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ* ∧ 0 < 𝑅) ∧ (𝑃 ∈ 𝑋 ∧ 𝑆 ∈ ℝ* ∧ 𝑆 ≤ 𝑅)) ∧ 𝑧 ∈ 𝑋) → (𝑆 ≤ (𝑃𝐶𝑧) ↔ (𝑆 ≤ (𝑃𝐶𝑧) ∧ 𝑆 ≤ 𝑅)))
11		simpr2 1194	. . . . . . . . 9 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ* ∧ 0 < 𝑅) ∧ (𝑃 ∈ 𝑋 ∧ 𝑆 ∈ ℝ* ∧ 𝑆 ≤ 𝑅)) → 𝑆 ∈ ℝ*)
12	11	adantr 481	. . . . . . . 8 ⊢ ((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ* ∧ 0 < 𝑅) ∧ (𝑃 ∈ 𝑋 ∧ 𝑆 ∈ ℝ* ∧ 𝑆 ≤ 𝑅)) ∧ 𝑧 ∈ 𝑋) → 𝑆 ∈ ℝ*)
13		simpl1 1190	. . . . . . . . . 10 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ* ∧ 0 < 𝑅) ∧ (𝑃 ∈ 𝑋 ∧ 𝑆 ∈ ℝ* ∧ 𝑆 ≤ 𝑅)) → 𝐶 ∈ (∞Met‘𝑋))
14	13	adantr 481	. . . . . . . . 9 ⊢ ((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ* ∧ 0 < 𝑅) ∧ (𝑃 ∈ 𝑋 ∧ 𝑆 ∈ ℝ* ∧ 𝑆 ≤ 𝑅)) ∧ 𝑧 ∈ 𝑋) → 𝐶 ∈ (∞Met‘𝑋))
15		xmetcl 23484	. . . . . . . . 9 ⊢ ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋) → (𝑃𝐶𝑧) ∈ ℝ*)
16	14, 3, 4, 15	syl3anc 1370	. . . . . . . 8 ⊢ ((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ* ∧ 0 < 𝑅) ∧ (𝑃 ∈ 𝑋 ∧ 𝑆 ∈ ℝ* ∧ 𝑆 ≤ 𝑅)) ∧ 𝑧 ∈ 𝑋) → (𝑃𝐶𝑧) ∈ ℝ*)
17		xrlemin 12918	. . . . . . . 8 ⊢ ((𝑆 ∈ ℝ* ∧ (𝑃𝐶𝑧) ∈ ℝ* ∧ 𝑅 ∈ ℝ*) → (𝑆 ≤ if((𝑃𝐶𝑧) ≤ 𝑅, (𝑃𝐶𝑧), 𝑅) ↔ (𝑆 ≤ (𝑃𝐶𝑧) ∧ 𝑆 ≤ 𝑅)))
18	12, 16, 1, 17	syl3anc 1370	. . . . . . 7 ⊢ ((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ* ∧ 0 < 𝑅) ∧ (𝑃 ∈ 𝑋 ∧ 𝑆 ∈ ℝ* ∧ 𝑆 ≤ 𝑅)) ∧ 𝑧 ∈ 𝑋) → (𝑆 ≤ if((𝑃𝐶𝑧) ≤ 𝑅, (𝑃𝐶𝑧), 𝑅) ↔ (𝑆 ≤ (𝑃𝐶𝑧) ∧ 𝑆 ≤ 𝑅)))
19	10, 18	bitr4d 281	. . . . . 6 ⊢ ((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ* ∧ 0 < 𝑅) ∧ (𝑃 ∈ 𝑋 ∧ 𝑆 ∈ ℝ* ∧ 𝑆 ≤ 𝑅)) ∧ 𝑧 ∈ 𝑋) → (𝑆 ≤ (𝑃𝐶𝑧) ↔ 𝑆 ≤ if((𝑃𝐶𝑧) ≤ 𝑅, (𝑃𝐶𝑧), 𝑅)))
20	19	notbid 318	. . . . 5 ⊢ ((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ* ∧ 0 < 𝑅) ∧ (𝑃 ∈ 𝑋 ∧ 𝑆 ∈ ℝ* ∧ 𝑆 ≤ 𝑅)) ∧ 𝑧 ∈ 𝑋) → (¬ 𝑆 ≤ (𝑃𝐶𝑧) ↔ ¬ 𝑆 ≤ if((𝑃𝐶𝑧) ≤ 𝑅, (𝑃𝐶𝑧), 𝑅)))
21		xrltnle 11042	. . . . . 6 ⊢ (((𝑃𝐶𝑧) ∈ ℝ* ∧ 𝑆 ∈ ℝ*) → ((𝑃𝐶𝑧) < 𝑆 ↔ ¬ 𝑆 ≤ (𝑃𝐶𝑧)))
22	16, 12, 21	syl2anc 584	. . . . 5 ⊢ ((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ* ∧ 0 < 𝑅) ∧ (𝑃 ∈ 𝑋 ∧ 𝑆 ∈ ℝ* ∧ 𝑆 ≤ 𝑅)) ∧ 𝑧 ∈ 𝑋) → ((𝑃𝐶𝑧) < 𝑆 ↔ ¬ 𝑆 ≤ (𝑃𝐶𝑧)))
23	16, 1	ifcld 4505	. . . . . 6 ⊢ ((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ* ∧ 0 < 𝑅) ∧ (𝑃 ∈ 𝑋 ∧ 𝑆 ∈ ℝ* ∧ 𝑆 ≤ 𝑅)) ∧ 𝑧 ∈ 𝑋) → if((𝑃𝐶𝑧) ≤ 𝑅, (𝑃𝐶𝑧), 𝑅) ∈ ℝ*)
24		xrltnle 11042	. . . . . 6 ⊢ ((if((𝑃𝐶𝑧) ≤ 𝑅, (𝑃𝐶𝑧), 𝑅) ∈ ℝ* ∧ 𝑆 ∈ ℝ*) → (if((𝑃𝐶𝑧) ≤ 𝑅, (𝑃𝐶𝑧), 𝑅) < 𝑆 ↔ ¬ 𝑆 ≤ if((𝑃𝐶𝑧) ≤ 𝑅, (𝑃𝐶𝑧), 𝑅)))
25	23, 12, 24	syl2anc 584	. . . . 5 ⊢ ((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ* ∧ 0 < 𝑅) ∧ (𝑃 ∈ 𝑋 ∧ 𝑆 ∈ ℝ* ∧ 𝑆 ≤ 𝑅)) ∧ 𝑧 ∈ 𝑋) → (if((𝑃𝐶𝑧) ≤ 𝑅, (𝑃𝐶𝑧), 𝑅) < 𝑆 ↔ ¬ 𝑆 ≤ if((𝑃𝐶𝑧) ≤ 𝑅, (𝑃𝐶𝑧), 𝑅)))
26	20, 22, 25	3bitr4d 311	. . . 4 ⊢ ((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ* ∧ 0 < 𝑅) ∧ (𝑃 ∈ 𝑋 ∧ 𝑆 ∈ ℝ* ∧ 𝑆 ≤ 𝑅)) ∧ 𝑧 ∈ 𝑋) → ((𝑃𝐶𝑧) < 𝑆 ↔ if((𝑃𝐶𝑧) ≤ 𝑅, (𝑃𝐶𝑧), 𝑅) < 𝑆))
27	8, 26	bitr4d 281	. . 3 ⊢ ((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ* ∧ 0 < 𝑅) ∧ (𝑃 ∈ 𝑋 ∧ 𝑆 ∈ ℝ* ∧ 𝑆 ≤ 𝑅)) ∧ 𝑧 ∈ 𝑋) → ((𝑃𝐷𝑧) < 𝑆 ↔ (𝑃𝐶𝑧) < 𝑆))
28	27	rabbidva 3413	. 2 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ* ∧ 0 < 𝑅) ∧ (𝑃 ∈ 𝑋 ∧ 𝑆 ∈ ℝ* ∧ 𝑆 ≤ 𝑅)) → {𝑧 ∈ 𝑋 ∣ (𝑃𝐷𝑧) < 𝑆} = {𝑧 ∈ 𝑋 ∣ (𝑃𝐶𝑧) < 𝑆})
29	5	stdbdxmet 23671	. . . 4 ⊢ ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ* ∧ 0 < 𝑅) → 𝐷 ∈ (∞Met‘𝑋))
30	29	adantr 481	. . 3 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ* ∧ 0 < 𝑅) ∧ (𝑃 ∈ 𝑋 ∧ 𝑆 ∈ ℝ* ∧ 𝑆 ≤ 𝑅)) → 𝐷 ∈ (∞Met‘𝑋))
31		blval 23539	. . 3 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑆 ∈ ℝ*) → (𝑃(ball‘𝐷)𝑆) = {𝑧 ∈ 𝑋 ∣ (𝑃𝐷𝑧) < 𝑆})
32	30, 2, 11, 31	syl3anc 1370	. 2 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ* ∧ 0 < 𝑅) ∧ (𝑃 ∈ 𝑋 ∧ 𝑆 ∈ ℝ* ∧ 𝑆 ≤ 𝑅)) → (𝑃(ball‘𝐷)𝑆) = {𝑧 ∈ 𝑋 ∣ (𝑃𝐷𝑧) < 𝑆})
33		blval 23539	. . 3 ⊢ ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑆 ∈ ℝ*) → (𝑃(ball‘𝐶)𝑆) = {𝑧 ∈ 𝑋 ∣ (𝑃𝐶𝑧) < 𝑆})
34	13, 2, 11, 33	syl3anc 1370	. 2 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ* ∧ 0 < 𝑅) ∧ (𝑃 ∈ 𝑋 ∧ 𝑆 ∈ ℝ* ∧ 𝑆 ≤ 𝑅)) → (𝑃(ball‘𝐶)𝑆) = {𝑧 ∈ 𝑋 ∣ (𝑃𝐶𝑧) < 𝑆})
35	28, 32, 34	3eqtr4d 2788	1 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ* ∧ 0 < 𝑅) ∧ (𝑃 ∈ 𝑋 ∧ 𝑆 ∈ ℝ* ∧ 𝑆 ≤ 𝑅)) → (𝑃(ball‘𝐷)𝑆) = (𝑃(ball‘𝐶)𝑆))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 396   ∧ w3a 1086   = wceq 1539   ∈ wcel 2106  {crab 3068  ifcif 4459   class class class wbr 5074  ‘cfv 6433  (class class class)co 7275   ∈ cmpo 7277  0cc0 10871  ℝ^*cxr 11008   < clt 11009   ≤ cle 11010  ∞Metcxmet 20582  ballcbl 20584

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588  ax-cnex 10927  ax-resscn 10928  ax-1cn 10929  ax-icn 10930  ax-addcl 10931  ax-addrcl 10932  ax-mulcl 10933  ax-mulrcl 10934  ax-mulcom 10935  ax-addass 10936  ax-mulass 10937  ax-distr 10938  ax-i2m1 10939  ax-1ne0 10940  ax-1rid 10941  ax-rnegex 10942  ax-rrecex 10943  ax-cnre 10944  ax-pre-lttri 10945  ax-pre-lttrn 10946  ax-pre-ltadd 10947  ax-pre-mulgt0 10948

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3069  df-rex 3070  df-rmo 3071  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-po 5503  df-so 5504  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-riota 7232  df-ov 7278  df-oprab 7279  df-mpo 7280  df-1st 7831  df-2nd 7832  df-er 8498  df-map 8617  df-en 8734  df-dom 8735  df-sdom 8736  df-pnf 11011  df-mnf 11012  df-xr 11013  df-ltxr 11014  df-le 11015  df-sub 11207  df-neg 11208  df-div 11633  df-2 12036  df-rp 12731  df-xneg 12848  df-xadd 12849  df-xmul 12850  df-icc 13086  df-psmet 20589  df-xmet 20590  df-bl 20592

This theorem is referenced by:  stdbdmopn  23674  xlebnum  24128