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Theorem lmconst 14176
Description: A constant sequence converges to its value. (Contributed by NM, 8-Nov-2007.) (Revised by Mario Carneiro, 14-Nov-2013.)
Hypothesis
Ref Expression
lmconst.2 𝑍 = (ℤ𝑀)
Assertion
Ref Expression
lmconst ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑃𝑋𝑀 ∈ ℤ) → (𝑍 × {𝑃})(⇝𝑡𝐽)𝑃)

Proof of Theorem lmconst
Dummy variables 𝑗 𝑘 𝑢 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simp2 1000 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑃𝑋𝑀 ∈ ℤ) → 𝑃𝑋)
2 simp3 1001 . . . . . 6 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑃𝑋𝑀 ∈ ℤ) → 𝑀 ∈ ℤ)
3 uzid 9572 . . . . . 6 (𝑀 ∈ ℤ → 𝑀 ∈ (ℤ𝑀))
42, 3syl 14 . . . . 5 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑃𝑋𝑀 ∈ ℤ) → 𝑀 ∈ (ℤ𝑀))
5 lmconst.2 . . . . 5 𝑍 = (ℤ𝑀)
64, 5eleqtrrdi 2283 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑃𝑋𝑀 ∈ ℤ) → 𝑀𝑍)
7 idd 21 . . . . 5 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑃𝑋𝑀 ∈ ℤ) ∧ 𝑘 ∈ (ℤ𝑀)) → (𝑃𝑢𝑃𝑢))
87ralrimdva 2570 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑃𝑋𝑀 ∈ ℤ) → (𝑃𝑢 → ∀𝑘 ∈ (ℤ𝑀)𝑃𝑢))
9 fveq2 5534 . . . . . 6 (𝑗 = 𝑀 → (ℤ𝑗) = (ℤ𝑀))
109raleqdv 2692 . . . . 5 (𝑗 = 𝑀 → (∀𝑘 ∈ (ℤ𝑗)𝑃𝑢 ↔ ∀𝑘 ∈ (ℤ𝑀)𝑃𝑢))
1110rspcev 2856 . . . 4 ((𝑀𝑍 ∧ ∀𝑘 ∈ (ℤ𝑀)𝑃𝑢) → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)𝑃𝑢)
126, 8, 11syl6an 1445 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑃𝑋𝑀 ∈ ℤ) → (𝑃𝑢 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)𝑃𝑢))
1312ralrimivw 2564 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑃𝑋𝑀 ∈ ℤ) → ∀𝑢𝐽 (𝑃𝑢 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)𝑃𝑢))
14 simp1 999 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑃𝑋𝑀 ∈ ℤ) → 𝐽 ∈ (TopOn‘𝑋))
15 fconst6g 5433 . . . 4 (𝑃𝑋 → (𝑍 × {𝑃}):𝑍𝑋)
161, 15syl 14 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑃𝑋𝑀 ∈ ℤ) → (𝑍 × {𝑃}):𝑍𝑋)
17 fvconst2g 5751 . . . 4 ((𝑃𝑋𝑘𝑍) → ((𝑍 × {𝑃})‘𝑘) = 𝑃)
181, 17sylan 283 . . 3 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑃𝑋𝑀 ∈ ℤ) ∧ 𝑘𝑍) → ((𝑍 × {𝑃})‘𝑘) = 𝑃)
1914, 5, 2, 16, 18lmbrf 14175 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑃𝑋𝑀 ∈ ℤ) → ((𝑍 × {𝑃})(⇝𝑡𝐽)𝑃 ↔ (𝑃𝑋 ∧ ∀𝑢𝐽 (𝑃𝑢 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)𝑃𝑢))))
201, 13, 19mpbir2and 946 1 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑃𝑋𝑀 ∈ ℤ) → (𝑍 × {𝑃})(⇝𝑡𝐽)𝑃)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  w3a 980   = wceq 1364  wcel 2160  wral 2468  wrex 2469  {csn 3607   class class class wbr 4018   × cxp 4642  wf 5231  cfv 5235  cz 9283  cuz 9558  TopOnctopon 13970  𝑡clm 14147
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4192  ax-pr 4227  ax-un 4451  ax-setind 4554  ax-cnex 7932  ax-resscn 7933  ax-1cn 7934  ax-1re 7935  ax-icn 7936  ax-addcl 7937  ax-addrcl 7938  ax-mulcl 7939  ax-addcom 7941  ax-addass 7943  ax-distr 7945  ax-i2m1 7946  ax-0lt1 7947  ax-0id 7949  ax-rnegex 7950  ax-cnre 7952  ax-pre-ltirr 7953  ax-pre-ltwlin 7954  ax-pre-lttrn 7955  ax-pre-apti 7956  ax-pre-ltadd 7957
This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-nel 2456  df-ral 2473  df-rex 2474  df-reu 2475  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-if 3550  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-int 3860  df-iun 3903  df-br 4019  df-opab 4080  df-mpt 4081  df-id 4311  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-rn 4655  df-res 4656  df-ima 4657  df-iota 5196  df-fun 5237  df-fn 5238  df-f 5239  df-fv 5243  df-riota 5852  df-ov 5899  df-oprab 5900  df-mpo 5901  df-1st 6165  df-2nd 6166  df-pm 6677  df-pnf 8024  df-mnf 8025  df-xr 8026  df-ltxr 8027  df-le 8028  df-sub 8160  df-neg 8161  df-inn 8950  df-n0 9207  df-z 9284  df-uz 9559  df-top 13958  df-topon 13971  df-lm 14150
This theorem is referenced by: (None)
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