Theorem stdbdbl 24500

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 1220	. . . . . 6 ⊢ ((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ* ∧ 0 < 𝑅) ∧ (𝑃 ∈ 𝑋 ∧ 𝑆 ∈ ℝ* ∧ 𝑆 ≤ 𝑅)) ∧ 𝑧 ∈ 𝑋) → 𝑅 ∈ ℝ*)
2		simpr1 1201	. . . . . . 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 24497	. . . . . 6 ⊢ ((𝑅 ∈ ℝ* ∧ 𝑃 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋) → (𝑃𝐷𝑧) = if((𝑃𝐶𝑧) ≤ 𝑅, (𝑃𝐶𝑧), 𝑅))
7	1, 3, 4, 6	syl3anc 1379	. . . . 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 5082	. . . 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 1224	. . . . . . . 8 ⊢ ((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ* ∧ 0 < 𝑅) ∧ (𝑃 ∈ 𝑋 ∧ 𝑆 ∈ ℝ* ∧ 𝑆 ≤ 𝑅)) ∧ 𝑧 ∈ 𝑋) → 𝑆 ≤ 𝑅)
10	9	biantrud 536	. . . . . . 7 ⊢ ((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ* ∧ 0 < 𝑅) ∧ (𝑃 ∈ 𝑋 ∧ 𝑆 ∈ ℝ* ∧ 𝑆 ≤ 𝑅)) ∧ 𝑧 ∈ 𝑋) → (𝑆 ≤ (𝑃𝐶𝑧) ↔ (𝑆 ≤ (𝑃𝐶𝑧) ∧ 𝑆 ≤ 𝑅)))
11		simpr2 1202	. . . . . . . . 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 1198	. . . . . . . . . 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 24314	. . . . . . . . 9 ⊢ ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋) → (𝑃𝐶𝑧) ∈ ℝ*)
16	14, 3, 4, 15	syl3anc 1379	. . . . . . . 8 ⊢ ((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ* ∧ 0 < 𝑅) ∧ (𝑃 ∈ 𝑋 ∧ 𝑆 ∈ ℝ* ∧ 𝑆 ≤ 𝑅)) ∧ 𝑧 ∈ 𝑋) → (𝑃𝐶𝑧) ∈ ℝ*)
17		xrlemin 13127	. . . . . . . 8 ⊢ ((𝑆 ∈ ℝ* ∧ (𝑃𝐶𝑧) ∈ ℝ* ∧ 𝑅 ∈ ℝ*) → (𝑆 ≤ if((𝑃𝐶𝑧) ≤ 𝑅, (𝑃𝐶𝑧), 𝑅) ↔ (𝑆 ≤ (𝑃𝐶𝑧) ∧ 𝑆 ≤ 𝑅)))
18	12, 16, 1, 17	syl3anc 1379	. . . . . . 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 283	. . . . . 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 319	. . . . 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 11203	. . . . . 6 ⊢ (((𝑃𝐶𝑧) ∈ ℝ* ∧ 𝑆 ∈ ℝ*) → ((𝑃𝐶𝑧) < 𝑆 ↔ ¬ 𝑆 ≤ (𝑃𝐶𝑧)))
22	16, 12, 21	syl2anc 590	. . . . 5 ⊢ ((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ* ∧ 0 < 𝑅) ∧ (𝑃 ∈ 𝑋 ∧ 𝑆 ∈ ℝ* ∧ 𝑆 ≤ 𝑅)) ∧ 𝑧 ∈ 𝑋) → ((𝑃𝐶𝑧) < 𝑆 ↔ ¬ 𝑆 ≤ (𝑃𝐶𝑧)))
23	16, 1	ifcld 4501	. . . . . 6 ⊢ ((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ* ∧ 0 < 𝑅) ∧ (𝑃 ∈ 𝑋 ∧ 𝑆 ∈ ℝ* ∧ 𝑆 ≤ 𝑅)) ∧ 𝑧 ∈ 𝑋) → if((𝑃𝐶𝑧) ≤ 𝑅, (𝑃𝐶𝑧), 𝑅) ∈ ℝ*)
24		xrltnle 11203	. . . . . 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 590	. . . . 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 312	. . . 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 283	. . 3 ⊢ ((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ* ∧ 0 < 𝑅) ∧ (𝑃 ∈ 𝑋 ∧ 𝑆 ∈ ℝ* ∧ 𝑆 ≤ 𝑅)) ∧ 𝑧 ∈ 𝑋) → ((𝑃𝐷𝑧) < 𝑆 ↔ (𝑃𝐶𝑧) < 𝑆))
28	27	rabbidva 3397	. 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 24498	. . . 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 24369	. . 3 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑆 ∈ ℝ*) → (𝑃(ball‘𝐷)𝑆) = {𝑧 ∈ 𝑋 ∣ (𝑃𝐷𝑧) < 𝑆})
32	30, 2, 11, 31	syl3anc 1379	. 2 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ* ∧ 0 < 𝑅) ∧ (𝑃 ∈ 𝑋 ∧ 𝑆 ∈ ℝ* ∧ 𝑆 ≤ 𝑅)) → (𝑃(ball‘𝐷)𝑆) = {𝑧 ∈ 𝑋 ∣ (𝑃𝐷𝑧) < 𝑆})
33		blval 24369	. . 3 ⊢ ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑆 ∈ ℝ*) → (𝑃(ball‘𝐶)𝑆) = {𝑧 ∈ 𝑋 ∣ (𝑃𝐶𝑧) < 𝑆})
34	13, 2, 11, 33	syl3anc 1379	. 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 2784	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 207   ∧ wa 396   ∧ w3a 1092   = wceq 1547   ∈ wcel 2119  {crab 3391  ifcif 4454   class class class wbr 5072  ‘cfv 6485  (class class class)co 7356   ∈ cmpo 7358  0cc0 11029  ℝ^*cxr 11169   < clt 11170   ≤ cle 11171  ∞Metcxmet 21332  ballcbl 21334

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-sep 5218  ax-nul 5228  ax-pow 5294  ax-pr 5362  ax-un 7678  ax-cnex 11085  ax-resscn 11086  ax-1cn 11087  ax-icn 11088  ax-addcl 11089  ax-addrcl 11090  ax-mulcl 11091  ax-mulrcl 11092  ax-mulcom 11093  ax-addass 11094  ax-mulass 11095  ax-distr 11096  ax-i2m1 11097  ax-1ne0 11098  ax-1rid 11099  ax-rnegex 11100  ax-rrecex 11101  ax-cnre 11102  ax-pre-lttri 11103  ax-pre-lttrn 11104  ax-pre-ltadd 11105  ax-pre-mulgt0 11106

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3or 1093  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ne 2935  df-nel 3039  df-ral 3054  df-rex 3064  df-rmo 3344  df-reu 3345  df-rab 3392  df-v 3433  df-sbc 3724  df-csb 3832  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4262  df-if 4455  df-pw 4531  df-sn 4556  df-pr 4558  df-op 4562  df-uni 4839  df-iun 4923  df-br 5073  df-opab 5135  df-mpt 5154  df-id 5513  df-po 5526  df-so 5527  df-xp 5624  df-rel 5625  df-cnv 5626  df-co 5627  df-dm 5628  df-rn 5629  df-res 5630  df-ima 5631  df-iota 6441  df-fun 6487  df-fn 6488  df-f 6489  df-f1 6490  df-fo 6491  df-f1o 6492  df-fv 6493  df-riota 7313  df-ov 7359  df-oprab 7360  df-mpo 7361  df-1st 7931  df-2nd 7932  df-er 8633  df-map 8765  df-en 8884  df-dom 8885  df-sdom 8886  df-pnf 11172  df-mnf 11173  df-xr 11174  df-ltxr 11175  df-le 11176  df-sub 11370  df-neg 11371  df-div 11799  df-2 12235  df-rp 12934  df-xneg 13054  df-xadd 13055  df-xmul 13056  df-icc 13296  df-psmet 21339  df-xmet 21340  df-bl 21342

This theorem is referenced by:  stdbdmopn  24501  xlebnum  24950