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Theorem lmmbr 22782
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 20785. (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 21995 . . . 4 (𝐷 ∈ (∞Met‘𝑋) → 𝐽 ∈ (TopOn‘𝑋))
41, 3syl 17 . . 3 (𝜑𝐽 ∈ (TopOn‘𝑋))
54lmbr 20814 . 2 (𝜑 → (𝐹(⇝𝑡𝐽)𝑃 ↔ (𝐹 ∈ (𝑋pm ℂ) ∧ 𝑃𝑋 ∧ ∀𝑢𝐽 (𝑃𝑢 → ∃𝑦 ∈ ran ℤ(𝐹𝑦):𝑦𝑢))))
6 rpxr 11672 . . . . . . . . . . . 12 (𝑥 ∈ ℝ+𝑥 ∈ ℝ*)
72blopn 22056 . . . . . . . . . . . 12 ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃𝑋𝑥 ∈ ℝ*) → (𝑃(ball‘𝐷)𝑥) ∈ 𝐽)
86, 7syl3an3 1352 . . . . . . . . . . 11 ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃𝑋𝑥 ∈ ℝ+) → (𝑃(ball‘𝐷)𝑥) ∈ 𝐽)
9 blcntr 21969 . . . . . . . . . . 11 ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃𝑋𝑥 ∈ ℝ+) → 𝑃 ∈ (𝑃(ball‘𝐷)𝑥))
10 eleq2 2676 . . . . . . . . . . . . . 14 (𝑢 = (𝑃(ball‘𝐷)𝑥) → (𝑃𝑢𝑃 ∈ (𝑃(ball‘𝐷)𝑥)))
11 feq3 5927 . . . . . . . . . . . . . . 15 (𝑢 = (𝑃(ball‘𝐷)𝑥) → ((𝐹𝑦):𝑦𝑢 ↔ (𝐹𝑦):𝑦⟶(𝑃(ball‘𝐷)𝑥)))
1211rexbidv 3033 . . . . . . . . . . . . . 14 (𝑢 = (𝑃(ball‘𝐷)𝑥) → (∃𝑦 ∈ ran ℤ(𝐹𝑦):𝑦𝑢 ↔ ∃𝑦 ∈ ran ℤ(𝐹𝑦):𝑦⟶(𝑃(ball‘𝐷)𝑥)))
1310, 12imbi12d 332 . . . . . . . . . . . . 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 3279 . . . . . . . . . . . 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 454 . . . . . . . . . . 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 690 . . . . . . . . . 10 ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃𝑋𝑥 ∈ ℝ+) → (∀𝑢𝐽 (𝑃𝑢 → ∃𝑦 ∈ ran ℤ(𝐹𝑦):𝑦𝑢) → ∃𝑦 ∈ ran ℤ(𝐹𝑦):𝑦⟶(𝑃(ball‘𝐷)𝑥)))
17163expa 1256 . . . . . . . . 9 (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃𝑋) ∧ 𝑥 ∈ ℝ+) → (∀𝑢𝐽 (𝑃𝑢 → ∃𝑦 ∈ ran ℤ(𝐹𝑦):𝑦𝑢) → ∃𝑦 ∈ ran ℤ(𝐹𝑦):𝑦⟶(𝑃(ball‘𝐷)𝑥)))
1817adantlrl 751 . . . . . . . 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 454 . . . . . . 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 2947 . . . . . 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 22049 . . . . . . . . . . 11 ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑢𝐽𝑃𝑢) → ∃𝑥 ∈ ℝ+ (𝑃(ball‘𝐷)𝑥) ⊆ 𝑢)
22 r19.29 3053 . . . . . . . . . . . 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 5955 . . . . . . . . . . . . . . . 16 (((𝐹𝑦):𝑦⟶(𝑃(ball‘𝐷)𝑥) ∧ (𝑃(ball‘𝐷)𝑥) ⊆ 𝑢) → (𝐹𝑦):𝑦𝑢)
2423expcom 449 . . . . . . . . . . . . . . 15 ((𝑃(ball‘𝐷)𝑥) ⊆ 𝑢 → ((𝐹𝑦):𝑦⟶(𝑃(ball‘𝐷)𝑥) → (𝐹𝑦):𝑦𝑢))
2524reximdv 2998 . . . . . . . . . . . . . 14 ((𝑃(ball‘𝐷)𝑥) ⊆ 𝑢 → (∃𝑦 ∈ ran ℤ(𝐹𝑦):𝑦⟶(𝑃(ball‘𝐷)𝑥) → ∃𝑦 ∈ ran ℤ(𝐹𝑦):𝑦𝑢))
2625impcom 444 . . . . . . . . . . . . 13 ((∃𝑦 ∈ ran ℤ(𝐹𝑦):𝑦⟶(𝑃(ball‘𝐷)𝑥) ∧ (𝑃(ball‘𝐷)𝑥) ⊆ 𝑢) → ∃𝑦 ∈ ran ℤ(𝐹𝑦):𝑦𝑢)
2726rexlimivw 3010 . . . . . . . . . . . 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 489 . . . . . . . . . 10 ((∀𝑥 ∈ ℝ+𝑦 ∈ ran ℤ(𝐹𝑦):𝑦⟶(𝑃(ball‘𝐷)𝑥) ∧ (𝐷 ∈ (∞Met‘𝑋) ∧ 𝑢𝐽𝑃𝑢)) → ∃𝑦 ∈ ran ℤ(𝐹𝑦):𝑦𝑢)
30293exp2 1276 . . . . . . . . 9 (∀𝑥 ∈ ℝ+𝑦 ∈ ran ℤ(𝐹𝑦):𝑦⟶(𝑃(ball‘𝐷)𝑥) → (𝐷 ∈ (∞Met‘𝑋) → (𝑢𝐽 → (𝑃𝑢 → ∃𝑦 ∈ ran ℤ(𝐹𝑦):𝑦𝑢))))
3130impcom 444 . . . . . . . 8 ((𝐷 ∈ (∞Met‘𝑋) ∧ ∀𝑥 ∈ ℝ+𝑦 ∈ ran ℤ(𝐹𝑦):𝑦⟶(𝑃(ball‘𝐷)𝑥)) → (𝑢𝐽 → (𝑃𝑢 → ∃𝑦 ∈ ran ℤ(𝐹𝑦):𝑦𝑢)))
3231adantlr 746 . . . . . . 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 2947 . . . . . 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 872 . . . . 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 670 . . . 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 1032 . . . 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 1032 . . . 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 301 . . 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 266 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 194  wa 382  w3a 1030   = wceq 1474  wcel 1976  wral 2895  wrex 2896  wss 3539   class class class wbr 4577  ran crn 5029  cres 5030  wf 5786  cfv 5790  (class class class)co 6527  pm cpm 7722  cc 9790  *cxr 9929  cuz 11519  +crp 11664  ∞Metcxmt 19498  ballcbl 19500  MetOpencmopn 19503  TopOnctopon 20460  𝑡clm 20782
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2032  ax-13 2232  ax-ext 2589  ax-sep 4703  ax-nul 4712  ax-pow 4764  ax-pr 4828  ax-un 6824  ax-cnex 9848  ax-resscn 9849  ax-1cn 9850  ax-icn 9851  ax-addcl 9852  ax-addrcl 9853  ax-mulcl 9854  ax-mulrcl 9855  ax-mulcom 9856  ax-addass 9857  ax-mulass 9858  ax-distr 9859  ax-i2m1 9860  ax-1ne0 9861  ax-1rid 9862  ax-rnegex 9863  ax-rrecex 9864  ax-cnre 9865  ax-pre-lttri 9866  ax-pre-lttrn 9867  ax-pre-ltadd 9868  ax-pre-mulgt0 9869  ax-pre-sup 9870
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-nel 2782  df-ral 2900  df-rex 2901  df-reu 2902  df-rmo 2903  df-rab 2904  df-v 3174  df-sbc 3402  df-csb 3499  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-pss 3555  df-nul 3874  df-if 4036  df-pw 4109  df-sn 4125  df-pr 4127  df-tp 4129  df-op 4131  df-uni 4367  df-iun 4451  df-br 4578  df-opab 4638  df-mpt 4639  df-tr 4675  df-eprel 4939  df-id 4943  df-po 4949  df-so 4950  df-fr 4987  df-we 4989  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-rn 5039  df-res 5040  df-ima 5041  df-pred 5583  df-ord 5629  df-on 5630  df-lim 5631  df-suc 5632  df-iota 5754  df-fun 5792  df-fn 5793  df-f 5794  df-f1 5795  df-fo 5796  df-f1o 5797  df-fv 5798  df-riota 6489  df-ov 6530  df-oprab 6531  df-mpt2 6532  df-om 6935  df-1st 7036  df-2nd 7037  df-wrecs 7271  df-recs 7332  df-rdg 7370  df-er 7606  df-map 7723  df-en 7819  df-dom 7820  df-sdom 7821  df-sup 8208  df-inf 8209  df-pnf 9932  df-mnf 9933  df-xr 9934  df-ltxr 9935  df-le 9936  df-sub 10119  df-neg 10120  df-div 10534  df-nn 10868  df-2 10926  df-n0 11140  df-z 11211  df-uz 11520  df-q 11621  df-rp 11665  df-xneg 11778  df-xadd 11779  df-xmul 11780  df-topgen 15873  df-psmet 19505  df-xmet 19506  df-bl 19508  df-mopn 19509  df-top 20463  df-bases 20464  df-topon 20465  df-lm 20785
This theorem is referenced by:  lmmbr2  22783  lmcau  22836
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