Theorem stdbdbl 24422

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 1214	. . . . . 6 ⊢ ((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ* ∧ 0 < 𝑅) ∧ (𝑃 ∈ 𝑋 ∧ 𝑆 ∈ ℝ* ∧ 𝑆 ≤ 𝑅)) ∧ 𝑧 ∈ 𝑋) → 𝑅 ∈ ℝ*)
2		simpr1 1195	. . . . . . 7 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ* ∧ 0 < 𝑅) ∧ (𝑃 ∈ 𝑋 ∧ 𝑆 ∈ ℝ* ∧ 𝑆 ≤ 𝑅)) → 𝑃 ∈ 𝑋)
3	2	adantr 480	. . . . . 6 ⊢ ((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ* ∧ 0 < 𝑅) ∧ (𝑃 ∈ 𝑋 ∧ 𝑆 ∈ ℝ* ∧ 𝑆 ≤ 𝑅)) ∧ 𝑧 ∈ 𝑋) → 𝑃 ∈ 𝑋)
4		simpr 484	. . . . . 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 24419	. . . . . 6 ⊢ ((𝑅 ∈ ℝ* ∧ 𝑃 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋) → (𝑃𝐷𝑧) = if((𝑃𝐶𝑧) ≤ 𝑅, (𝑃𝐶𝑧), 𝑅))
7	1, 3, 4, 6	syl3anc 1373	. . . . 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 5105	. . . 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 1218	. . . . . . . 8 ⊢ ((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ* ∧ 0 < 𝑅) ∧ (𝑃 ∈ 𝑋 ∧ 𝑆 ∈ ℝ* ∧ 𝑆 ≤ 𝑅)) ∧ 𝑧 ∈ 𝑋) → 𝑆 ≤ 𝑅)
10	9	biantrud 531	. . . . . . 7 ⊢ ((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ* ∧ 0 < 𝑅) ∧ (𝑃 ∈ 𝑋 ∧ 𝑆 ∈ ℝ* ∧ 𝑆 ≤ 𝑅)) ∧ 𝑧 ∈ 𝑋) → (𝑆 ≤ (𝑃𝐶𝑧) ↔ (𝑆 ≤ (𝑃𝐶𝑧) ∧ 𝑆 ≤ 𝑅)))
11		simpr2 1196	. . . . . . . . 9 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ* ∧ 0 < 𝑅) ∧ (𝑃 ∈ 𝑋 ∧ 𝑆 ∈ ℝ* ∧ 𝑆 ≤ 𝑅)) → 𝑆 ∈ ℝ*)
12	11	adantr 480	. . . . . . . 8 ⊢ ((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ* ∧ 0 < 𝑅) ∧ (𝑃 ∈ 𝑋 ∧ 𝑆 ∈ ℝ* ∧ 𝑆 ≤ 𝑅)) ∧ 𝑧 ∈ 𝑋) → 𝑆 ∈ ℝ*)
13		simpl1 1192	. . . . . . . . . 10 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ* ∧ 0 < 𝑅) ∧ (𝑃 ∈ 𝑋 ∧ 𝑆 ∈ ℝ* ∧ 𝑆 ≤ 𝑅)) → 𝐶 ∈ (∞Met‘𝑋))
14	13	adantr 480	. . . . . . . . 9 ⊢ ((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ* ∧ 0 < 𝑅) ∧ (𝑃 ∈ 𝑋 ∧ 𝑆 ∈ ℝ* ∧ 𝑆 ≤ 𝑅)) ∧ 𝑧 ∈ 𝑋) → 𝐶 ∈ (∞Met‘𝑋))
15		xmetcl 24236	. . . . . . . . 9 ⊢ ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋) → (𝑃𝐶𝑧) ∈ ℝ*)
16	14, 3, 4, 15	syl3anc 1373	. . . . . . . 8 ⊢ ((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ* ∧ 0 < 𝑅) ∧ (𝑃 ∈ 𝑋 ∧ 𝑆 ∈ ℝ* ∧ 𝑆 ≤ 𝑅)) ∧ 𝑧 ∈ 𝑋) → (𝑃𝐶𝑧) ∈ ℝ*)
17		xrlemin 13105	. . . . . . . 8 ⊢ ((𝑆 ∈ ℝ* ∧ (𝑃𝐶𝑧) ∈ ℝ* ∧ 𝑅 ∈ ℝ*) → (𝑆 ≤ if((𝑃𝐶𝑧) ≤ 𝑅, (𝑃𝐶𝑧), 𝑅) ↔ (𝑆 ≤ (𝑃𝐶𝑧) ∧ 𝑆 ≤ 𝑅)))
18	12, 16, 1, 17	syl3anc 1373	. . . . . . 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 282	. . . . . 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 11201	. . . . . 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 4525	. . . . . 6 ⊢ ((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ* ∧ 0 < 𝑅) ∧ (𝑃 ∈ 𝑋 ∧ 𝑆 ∈ ℝ* ∧ 𝑆 ≤ 𝑅)) ∧ 𝑧 ∈ 𝑋) → if((𝑃𝐶𝑧) ≤ 𝑅, (𝑃𝐶𝑧), 𝑅) ∈ ℝ*)
24		xrltnle 11201	. . . . . 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 282	. . 3 ⊢ ((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ* ∧ 0 < 𝑅) ∧ (𝑃 ∈ 𝑋 ∧ 𝑆 ∈ ℝ* ∧ 𝑆 ≤ 𝑅)) ∧ 𝑧 ∈ 𝑋) → ((𝑃𝐷𝑧) < 𝑆 ↔ (𝑃𝐶𝑧) < 𝑆))
28	27	rabbidva 3403	. 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 24420	. . . 4 ⊢ ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ* ∧ 0 < 𝑅) → 𝐷 ∈ (∞Met‘𝑋))
30	29	adantr 480	. . 3 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ* ∧ 0 < 𝑅) ∧ (𝑃 ∈ 𝑋 ∧ 𝑆 ∈ ℝ* ∧ 𝑆 ≤ 𝑅)) → 𝐷 ∈ (∞Met‘𝑋))
31		blval 24291	. . 3 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑆 ∈ ℝ*) → (𝑃(ball‘𝐷)𝑆) = {𝑧 ∈ 𝑋 ∣ (𝑃𝐷𝑧) < 𝑆})
32	30, 2, 11, 31	syl3anc 1373	. 2 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ* ∧ 0 < 𝑅) ∧ (𝑃 ∈ 𝑋 ∧ 𝑆 ∈ ℝ* ∧ 𝑆 ≤ 𝑅)) → (𝑃(ball‘𝐷)𝑆) = {𝑧 ∈ 𝑋 ∣ (𝑃𝐷𝑧) < 𝑆})
33		blval 24291	. . 3 ⊢ ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑆 ∈ ℝ*) → (𝑃(ball‘𝐶)𝑆) = {𝑧 ∈ 𝑋 ∣ (𝑃𝐶𝑧) < 𝑆})
34	13, 2, 11, 33	syl3anc 1373	. 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 2774	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 206   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109  {crab 3396  ifcif 4478   class class class wbr 5095  ‘cfv 6486  (class class class)co 7353   ∈ cmpo 7355  0cc0 11028  ℝ^*cxr 11167   < clt 11168   ≤ cle 11169  ∞Metcxmet 21265  ballcbl 21267

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5238  ax-nul 5248  ax-pow 5307  ax-pr 5374  ax-un 7675  ax-cnex 11084  ax-resscn 11085  ax-1cn 11086  ax-icn 11087  ax-addcl 11088  ax-addrcl 11089  ax-mulcl 11090  ax-mulrcl 11091  ax-mulcom 11092  ax-addass 11093  ax-mulass 11094  ax-distr 11095  ax-i2m1 11096  ax-1ne0 11097  ax-1rid 11098  ax-rnegex 11099  ax-rrecex 11100  ax-cnre 11101  ax-pre-lttri 11102  ax-pre-lttrn 11103  ax-pre-ltadd 11104  ax-pre-mulgt0 11105

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-nel 3030  df-ral 3045  df-rex 3054  df-rmo 3345  df-reu 3346  df-rab 3397  df-v 3440  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4479  df-pw 4555  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4862  df-iun 4946  df-br 5096  df-opab 5158  df-mpt 5177  df-id 5518  df-po 5531  df-so 5532  df-xp 5629  df-rel 5630  df-cnv 5631  df-co 5632  df-dm 5633  df-rn 5634  df-res 5635  df-ima 5636  df-iota 6442  df-fun 6488  df-fn 6489  df-f 6490  df-f1 6491  df-fo 6492  df-f1o 6493  df-fv 6494  df-riota 7310  df-ov 7356  df-oprab 7357  df-mpo 7358  df-1st 7931  df-2nd 7932  df-er 8632  df-map 8762  df-en 8880  df-dom 8881  df-sdom 8882  df-pnf 11170  df-mnf 11171  df-xr 11172  df-ltxr 11173  df-le 11174  df-sub 11368  df-neg 11369  df-div 11797  df-2 12210  df-rp 12913  df-xneg 13033  df-xadd 13034  df-xmul 13035  df-icc 13274  df-psmet 21272  df-xmet 21273  df-bl 21275

This theorem is referenced by:  stdbdmopn  24423  xlebnum  24881