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Theorem lmmbr 23776
 Description: Express the binary relation "sequence 𝐹 converges to point 𝑃 " in a metric space. Definition 1.4-1 of [Kreyszig] p. 25. The condition 𝐹 ⊆ (ℂ × 𝑋) allows us to use objects more general than sequences when convenient; see the comment in df-lm 21753. (Contributed by NM, 7-Dec-2006.) (Revised by Mario Carneiro, 1-May-2014.)
Hypotheses
Ref Expression
lmmbr.2 𝐽 = (MetOpen‘𝐷)
lmmbr.3 (𝜑𝐷 ∈ (∞Met‘𝑋))
Assertion
Ref Expression
lmmbr (𝜑 → (𝐹(⇝𝑡𝐽)𝑃 ↔ (𝐹 ∈ (𝑋pm ℂ) ∧ 𝑃𝑋 ∧ ∀𝑥 ∈ ℝ+𝑦 ∈ ran ℤ(𝐹𝑦):𝑦⟶(𝑃(ball‘𝐷)𝑥))))
Distinct variable groups:   𝑥,𝑦,𝐷   𝑥,𝐹,𝑦   𝑥,𝑃,𝑦   𝑥,𝑋,𝑦   𝑥,𝐽,𝑦   𝜑,𝑥
Allowed substitution hint:   𝜑(𝑦)

Proof of Theorem lmmbr
Dummy variable 𝑢 is distinct from all other variables.
StepHypRef Expression
1 lmmbr.3 . . . 4 (𝜑𝐷 ∈ (∞Met‘𝑋))
2 lmmbr.2 . . . . 5 𝐽 = (MetOpen‘𝐷)
32mopntopon 22964 . . . 4 (𝐷 ∈ (∞Met‘𝑋) → 𝐽 ∈ (TopOn‘𝑋))
41, 3syl 17 . . 3 (𝜑𝐽 ∈ (TopOn‘𝑋))
54lmbr 21782 . 2 (𝜑 → (𝐹(⇝𝑡𝐽)𝑃 ↔ (𝐹 ∈ (𝑋pm ℂ) ∧ 𝑃𝑋 ∧ ∀𝑢𝐽 (𝑃𝑢 → ∃𝑦 ∈ ran ℤ(𝐹𝑦):𝑦𝑢))))
6 rpxr 12391 . . . . . . . . . . . 12 (𝑥 ∈ ℝ+𝑥 ∈ ℝ*)
72blopn 23025 . . . . . . . . . . . 12 ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃𝑋𝑥 ∈ ℝ*) → (𝑃(ball‘𝐷)𝑥) ∈ 𝐽)
86, 7syl3an3 1159 . . . . . . . . . . 11 ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃𝑋𝑥 ∈ ℝ+) → (𝑃(ball‘𝐷)𝑥) ∈ 𝐽)
9 blcntr 22938 . . . . . . . . . . 11 ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃𝑋𝑥 ∈ ℝ+) → 𝑃 ∈ (𝑃(ball‘𝐷)𝑥))
10 eleq2 2905 . . . . . . . . . . . . . 14 (𝑢 = (𝑃(ball‘𝐷)𝑥) → (𝑃𝑢𝑃 ∈ (𝑃(ball‘𝐷)𝑥)))
11 feq3 6493 . . . . . . . . . . . . . . 15 (𝑢 = (𝑃(ball‘𝐷)𝑥) → ((𝐹𝑦):𝑦𝑢 ↔ (𝐹𝑦):𝑦⟶(𝑃(ball‘𝐷)𝑥)))
1211rexbidv 3301 . . . . . . . . . . . . . 14 (𝑢 = (𝑃(ball‘𝐷)𝑥) → (∃𝑦 ∈ ran ℤ(𝐹𝑦):𝑦𝑢 ↔ ∃𝑦 ∈ ran ℤ(𝐹𝑦):𝑦⟶(𝑃(ball‘𝐷)𝑥)))
1310, 12imbi12d 346 . . . . . . . . . . . . 13 (𝑢 = (𝑃(ball‘𝐷)𝑥) → ((𝑃𝑢 → ∃𝑦 ∈ ran ℤ(𝐹𝑦):𝑦𝑢) ↔ (𝑃 ∈ (𝑃(ball‘𝐷)𝑥) → ∃𝑦 ∈ ran ℤ(𝐹𝑦):𝑦⟶(𝑃(ball‘𝐷)𝑥))))
1413rspcva 3624 . . . . . . . . . . . 12 (((𝑃(ball‘𝐷)𝑥) ∈ 𝐽 ∧ ∀𝑢𝐽 (𝑃𝑢 → ∃𝑦 ∈ ran ℤ(𝐹𝑦):𝑦𝑢)) → (𝑃 ∈ (𝑃(ball‘𝐷)𝑥) → ∃𝑦 ∈ ran ℤ(𝐹𝑦):𝑦⟶(𝑃(ball‘𝐷)𝑥)))
1514impancom 452 . . . . . . . . . . 11 (((𝑃(ball‘𝐷)𝑥) ∈ 𝐽𝑃 ∈ (𝑃(ball‘𝐷)𝑥)) → (∀𝑢𝐽 (𝑃𝑢 → ∃𝑦 ∈ ran ℤ(𝐹𝑦):𝑦𝑢) → ∃𝑦 ∈ ran ℤ(𝐹𝑦):𝑦⟶(𝑃(ball‘𝐷)𝑥)))
168, 9, 15syl2anc 584 . . . . . . . . . 10 ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃𝑋𝑥 ∈ ℝ+) → (∀𝑢𝐽 (𝑃𝑢 → ∃𝑦 ∈ ran ℤ(𝐹𝑦):𝑦𝑢) → ∃𝑦 ∈ ran ℤ(𝐹𝑦):𝑦⟶(𝑃(ball‘𝐷)𝑥)))
17163expa 1112 . . . . . . . . 9 (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃𝑋) ∧ 𝑥 ∈ ℝ+) → (∀𝑢𝐽 (𝑃𝑢 → ∃𝑦 ∈ ran ℤ(𝐹𝑦):𝑦𝑢) → ∃𝑦 ∈ ran ℤ(𝐹𝑦):𝑦⟶(𝑃(ball‘𝐷)𝑥)))
1817adantlrl 716 . . . . . . . 8 (((𝐷 ∈ (∞Met‘𝑋) ∧ (𝐹 ∈ (𝑋pm ℂ) ∧ 𝑃𝑋)) ∧ 𝑥 ∈ ℝ+) → (∀𝑢𝐽 (𝑃𝑢 → ∃𝑦 ∈ ran ℤ(𝐹𝑦):𝑦𝑢) → ∃𝑦 ∈ ran ℤ(𝐹𝑦):𝑦⟶(𝑃(ball‘𝐷)𝑥)))
1918impancom 452 . . . . . . 7 (((𝐷 ∈ (∞Met‘𝑋) ∧ (𝐹 ∈ (𝑋pm ℂ) ∧ 𝑃𝑋)) ∧ ∀𝑢𝐽 (𝑃𝑢 → ∃𝑦 ∈ ran ℤ(𝐹𝑦):𝑦𝑢)) → (𝑥 ∈ ℝ+ → ∃𝑦 ∈ ran ℤ(𝐹𝑦):𝑦⟶(𝑃(ball‘𝐷)𝑥)))
2019ralrimiv 3185 . . . . . 6 (((𝐷 ∈ (∞Met‘𝑋) ∧ (𝐹 ∈ (𝑋pm ℂ) ∧ 𝑃𝑋)) ∧ ∀𝑢𝐽 (𝑃𝑢 → ∃𝑦 ∈ ran ℤ(𝐹𝑦):𝑦𝑢)) → ∀𝑥 ∈ ℝ+𝑦 ∈ ran ℤ(𝐹𝑦):𝑦⟶(𝑃(ball‘𝐷)𝑥))
212mopni2 23018 . . . . . . . . . . 11 ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑢𝐽𝑃𝑢) → ∃𝑥 ∈ ℝ+ (𝑃(ball‘𝐷)𝑥) ⊆ 𝑢)
22 r19.29 3258 . . . . . . . . . . . 12 ((∀𝑥 ∈ ℝ+𝑦 ∈ ran ℤ(𝐹𝑦):𝑦⟶(𝑃(ball‘𝐷)𝑥) ∧ ∃𝑥 ∈ ℝ+ (𝑃(ball‘𝐷)𝑥) ⊆ 𝑢) → ∃𝑥 ∈ ℝ+ (∃𝑦 ∈ ran ℤ(𝐹𝑦):𝑦⟶(𝑃(ball‘𝐷)𝑥) ∧ (𝑃(ball‘𝐷)𝑥) ⊆ 𝑢))
23 fss 6523 . . . . . . . . . . . . . . . 16 (((𝐹𝑦):𝑦⟶(𝑃(ball‘𝐷)𝑥) ∧ (𝑃(ball‘𝐷)𝑥) ⊆ 𝑢) → (𝐹𝑦):𝑦𝑢)
2423expcom 414 . . . . . . . . . . . . . . 15 ((𝑃(ball‘𝐷)𝑥) ⊆ 𝑢 → ((𝐹𝑦):𝑦⟶(𝑃(ball‘𝐷)𝑥) → (𝐹𝑦):𝑦𝑢))
2524reximdv 3277 . . . . . . . . . . . . . 14 ((𝑃(ball‘𝐷)𝑥) ⊆ 𝑢 → (∃𝑦 ∈ ran ℤ(𝐹𝑦):𝑦⟶(𝑃(ball‘𝐷)𝑥) → ∃𝑦 ∈ ran ℤ(𝐹𝑦):𝑦𝑢))
2625impcom 408 . . . . . . . . . . . . 13 ((∃𝑦 ∈ ran ℤ(𝐹𝑦):𝑦⟶(𝑃(ball‘𝐷)𝑥) ∧ (𝑃(ball‘𝐷)𝑥) ⊆ 𝑢) → ∃𝑦 ∈ ran ℤ(𝐹𝑦):𝑦𝑢)
2726rexlimivw 3286 . . . . . . . . . . . 12 (∃𝑥 ∈ ℝ+ (∃𝑦 ∈ ran ℤ(𝐹𝑦):𝑦⟶(𝑃(ball‘𝐷)𝑥) ∧ (𝑃(ball‘𝐷)𝑥) ⊆ 𝑢) → ∃𝑦 ∈ ran ℤ(𝐹𝑦):𝑦𝑢)
2822, 27syl 17 . . . . . . . . . . 11 ((∀𝑥 ∈ ℝ+𝑦 ∈ ran ℤ(𝐹𝑦):𝑦⟶(𝑃(ball‘𝐷)𝑥) ∧ ∃𝑥 ∈ ℝ+ (𝑃(ball‘𝐷)𝑥) ⊆ 𝑢) → ∃𝑦 ∈ ran ℤ(𝐹𝑦):𝑦𝑢)
2921, 28sylan2 592 . . . . . . . . . 10 ((∀𝑥 ∈ ℝ+𝑦 ∈ ran ℤ(𝐹𝑦):𝑦⟶(𝑃(ball‘𝐷)𝑥) ∧ (𝐷 ∈ (∞Met‘𝑋) ∧ 𝑢𝐽𝑃𝑢)) → ∃𝑦 ∈ ran ℤ(𝐹𝑦):𝑦𝑢)
30293exp2 1348 . . . . . . . . 9 (∀𝑥 ∈ ℝ+𝑦 ∈ ran ℤ(𝐹𝑦):𝑦⟶(𝑃(ball‘𝐷)𝑥) → (𝐷 ∈ (∞Met‘𝑋) → (𝑢𝐽 → (𝑃𝑢 → ∃𝑦 ∈ ran ℤ(𝐹𝑦):𝑦𝑢))))
3130impcom 408 . . . . . . . 8 ((𝐷 ∈ (∞Met‘𝑋) ∧ ∀𝑥 ∈ ℝ+𝑦 ∈ ran ℤ(𝐹𝑦):𝑦⟶(𝑃(ball‘𝐷)𝑥)) → (𝑢𝐽 → (𝑃𝑢 → ∃𝑦 ∈ ran ℤ(𝐹𝑦):𝑦𝑢)))
3231adantlr 711 . . . . . . 7 (((𝐷 ∈ (∞Met‘𝑋) ∧ (𝐹 ∈ (𝑋pm ℂ) ∧ 𝑃𝑋)) ∧ ∀𝑥 ∈ ℝ+𝑦 ∈ ran ℤ(𝐹𝑦):𝑦⟶(𝑃(ball‘𝐷)𝑥)) → (𝑢𝐽 → (𝑃𝑢 → ∃𝑦 ∈ ran ℤ(𝐹𝑦):𝑦𝑢)))
3332ralrimiv 3185 . . . . . 6 (((𝐷 ∈ (∞Met‘𝑋) ∧ (𝐹 ∈ (𝑋pm ℂ) ∧ 𝑃𝑋)) ∧ ∀𝑥 ∈ ℝ+𝑦 ∈ ran ℤ(𝐹𝑦):𝑦⟶(𝑃(ball‘𝐷)𝑥)) → ∀𝑢𝐽 (𝑃𝑢 → ∃𝑦 ∈ ran ℤ(𝐹𝑦):𝑦𝑢))
3420, 33impbida 797 . . . . 5 ((𝐷 ∈ (∞Met‘𝑋) ∧ (𝐹 ∈ (𝑋pm ℂ) ∧ 𝑃𝑋)) → (∀𝑢𝐽 (𝑃𝑢 → ∃𝑦 ∈ ran ℤ(𝐹𝑦):𝑦𝑢) ↔ ∀𝑥 ∈ ℝ+𝑦 ∈ ran ℤ(𝐹𝑦):𝑦⟶(𝑃(ball‘𝐷)𝑥)))
3534pm5.32da 579 . . . 4 (𝐷 ∈ (∞Met‘𝑋) → (((𝐹 ∈ (𝑋pm ℂ) ∧ 𝑃𝑋) ∧ ∀𝑢𝐽 (𝑃𝑢 → ∃𝑦 ∈ ran ℤ(𝐹𝑦):𝑦𝑢)) ↔ ((𝐹 ∈ (𝑋pm ℂ) ∧ 𝑃𝑋) ∧ ∀𝑥 ∈ ℝ+𝑦 ∈ ran ℤ(𝐹𝑦):𝑦⟶(𝑃(ball‘𝐷)𝑥))))
36 df-3an 1083 . . . 4 ((𝐹 ∈ (𝑋pm ℂ) ∧ 𝑃𝑋 ∧ ∀𝑢𝐽 (𝑃𝑢 → ∃𝑦 ∈ ran ℤ(𝐹𝑦):𝑦𝑢)) ↔ ((𝐹 ∈ (𝑋pm ℂ) ∧ 𝑃𝑋) ∧ ∀𝑢𝐽 (𝑃𝑢 → ∃𝑦 ∈ ran ℤ(𝐹𝑦):𝑦𝑢)))
37 df-3an 1083 . . . 4 ((𝐹 ∈ (𝑋pm ℂ) ∧ 𝑃𝑋 ∧ ∀𝑥 ∈ ℝ+𝑦 ∈ ran ℤ(𝐹𝑦):𝑦⟶(𝑃(ball‘𝐷)𝑥)) ↔ ((𝐹 ∈ (𝑋pm ℂ) ∧ 𝑃𝑋) ∧ ∀𝑥 ∈ ℝ+𝑦 ∈ ran ℤ(𝐹𝑦):𝑦⟶(𝑃(ball‘𝐷)𝑥)))
3835, 36, 373bitr4g 315 . . 3 (𝐷 ∈ (∞Met‘𝑋) → ((𝐹 ∈ (𝑋pm ℂ) ∧ 𝑃𝑋 ∧ ∀𝑢𝐽 (𝑃𝑢 → ∃𝑦 ∈ ran ℤ(𝐹𝑦):𝑦𝑢)) ↔ (𝐹 ∈ (𝑋pm ℂ) ∧ 𝑃𝑋 ∧ ∀𝑥 ∈ ℝ+𝑦 ∈ ran ℤ(𝐹𝑦):𝑦⟶(𝑃(ball‘𝐷)𝑥))))
391, 38syl 17 . 2 (𝜑 → ((𝐹 ∈ (𝑋pm ℂ) ∧ 𝑃𝑋 ∧ ∀𝑢𝐽 (𝑃𝑢 → ∃𝑦 ∈ ran ℤ(𝐹𝑦):𝑦𝑢)) ↔ (𝐹 ∈ (𝑋pm ℂ) ∧ 𝑃𝑋 ∧ ∀𝑥 ∈ ℝ+𝑦 ∈ ran ℤ(𝐹𝑦):𝑦⟶(𝑃(ball‘𝐷)𝑥))))
405, 39bitrd 280 1 (𝜑 → (𝐹(⇝𝑡𝐽)𝑃 ↔ (𝐹 ∈ (𝑋pm ℂ) ∧ 𝑃𝑋 ∧ ∀𝑥 ∈ ℝ+𝑦 ∈ ran ℤ(𝐹𝑦):𝑦⟶(𝑃(ball‘𝐷)𝑥))))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 207   ∧ wa 396   ∧ w3a 1081   = wceq 1530   ∈ wcel 2106  ∀wral 3142  ∃wrex 3143   ⊆ wss 3939   class class class wbr 5062  ran crn 5554   ↾ cres 5555  ⟶wf 6347  ‘cfv 6351  (class class class)co 7151   ↑pm cpm 8400  ℂcc 10527  ℝ*cxr 10666  ℤ≥cuz 12235  ℝ+crp 12382  ∞Metcxmet 20446  ballcbl 20448  MetOpencmopn 20451  TopOnctopon 21434  ⇝𝑡clm 21750 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 2108  ax-9 2116  ax-10 2137  ax-11 2152  ax-12 2167  ax-ext 2796  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  ax-pre-sup 10607 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 2615  df-eu 2649  df-clab 2803  df-cleq 2817  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-pss 3957  df-nul 4295  df-if 4470  df-pw 4543  df-sn 4564  df-pr 4566  df-tp 4568  df-op 4570  df-uni 4837  df-iun 4918  df-br 5063  df-opab 5125  df-mpt 5143  df-tr 5169  df-id 5458  df-eprel 5463  df-po 5472  df-so 5473  df-fr 5512  df-we 5514  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-pred 6145  df-ord 6191  df-on 6192  df-lim 6193  df-suc 6194  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-om 7572  df-1st 7683  df-2nd 7684  df-wrecs 7941  df-recs 8002  df-rdg 8040  df-er 8282  df-map 8401  df-en 8502  df-dom 8503  df-sdom 8504  df-sup 8898  df-inf 8899  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-nn 11631  df-2 11692  df-n0 11890  df-z 11974  df-uz 12236  df-q 12341  df-rp 12383  df-xneg 12500  df-xadd 12501  df-xmul 12502  df-topgen 16709  df-psmet 20453  df-xmet 20454  df-bl 20456  df-mopn 20457  df-top 21418  df-topon 21435  df-bases 21470  df-lm 21753 This theorem is referenced by:  lmmbr2  23777  lmcau  23831
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