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Theorem lmconst 23256
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 1134 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑃𝑋𝑀 ∈ ℤ) → 𝑃𝑋)
2 simp3 1135 . . . . . 6 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑃𝑋𝑀 ∈ ℤ) → 𝑀 ∈ ℤ)
3 uzid 12889 . . . . . 6 (𝑀 ∈ ℤ → 𝑀 ∈ (ℤ𝑀))
42, 3syl 17 . . . . 5 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑃𝑋𝑀 ∈ ℤ) → 𝑀 ∈ (ℤ𝑀))
5 lmconst.2 . . . . 5 𝑍 = (ℤ𝑀)
64, 5eleqtrrdi 2837 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑃𝑋𝑀 ∈ ℤ) → 𝑀𝑍)
7 idd 24 . . . . 5 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑃𝑋𝑀 ∈ ℤ) ∧ 𝑘 ∈ (ℤ𝑀)) → (𝑃𝑢𝑃𝑢))
87ralrimdva 3144 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑃𝑋𝑀 ∈ ℤ) → (𝑃𝑢 → ∀𝑘 ∈ (ℤ𝑀)𝑃𝑢))
9 fveq2 6901 . . . . . 6 (𝑗 = 𝑀 → (ℤ𝑗) = (ℤ𝑀))
109raleqdv 3315 . . . . 5 (𝑗 = 𝑀 → (∀𝑘 ∈ (ℤ𝑗)𝑃𝑢 ↔ ∀𝑘 ∈ (ℤ𝑀)𝑃𝑢))
1110rspcev 3608 . . . 4 ((𝑀𝑍 ∧ ∀𝑘 ∈ (ℤ𝑀)𝑃𝑢) → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)𝑃𝑢)
126, 8, 11syl6an 682 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑃𝑋𝑀 ∈ ℤ) → (𝑃𝑢 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)𝑃𝑢))
1312ralrimivw 3140 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑃𝑋𝑀 ∈ ℤ) → ∀𝑢𝐽 (𝑃𝑢 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)𝑃𝑢))
14 simp1 1133 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑃𝑋𝑀 ∈ ℤ) → 𝐽 ∈ (TopOn‘𝑋))
15 fconst6g 6791 . . . 4 (𝑃𝑋 → (𝑍 × {𝑃}):𝑍𝑋)
161, 15syl 17 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑃𝑋𝑀 ∈ ℤ) → (𝑍 × {𝑃}):𝑍𝑋)
17 fvconst2g 7219 . . . 4 ((𝑃𝑋𝑘𝑍) → ((𝑍 × {𝑃})‘𝑘) = 𝑃)
181, 17sylan 578 . . 3 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑃𝑋𝑀 ∈ ℤ) ∧ 𝑘𝑍) → ((𝑍 × {𝑃})‘𝑘) = 𝑃)
1914, 5, 2, 16, 18lmbrf 23255 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑃𝑋𝑀 ∈ ℤ) → ((𝑍 × {𝑃})(⇝𝑡𝐽)𝑃 ↔ (𝑃𝑋 ∧ ∀𝑢𝐽 (𝑃𝑢 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)𝑃𝑢))))
201, 13, 19mpbir2and 711 1 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑃𝑋𝑀 ∈ ℤ) → (𝑍 × {𝑃})(⇝𝑡𝐽)𝑃)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394  w3a 1084   = wceq 1534  wcel 2099  wral 3051  wrex 3060  {csn 4633   class class class wbr 5153   × cxp 5680  wf 6550  cfv 6554  cz 12610  cuz 12874  TopOnctopon 22903  𝑡clm 23221
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-sep 5304  ax-nul 5311  ax-pow 5369  ax-pr 5433  ax-un 7746  ax-cnex 11214  ax-resscn 11215  ax-pre-lttri 11232  ax-pre-lttrn 11233
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-nel 3037  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3464  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4326  df-if 4534  df-pw 4609  df-sn 4634  df-pr 4636  df-op 4640  df-uni 4914  df-iun 5003  df-br 5154  df-opab 5216  df-mpt 5237  df-id 5580  df-po 5594  df-so 5595  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-iota 6506  df-fun 6556  df-fn 6557  df-f 6558  df-f1 6559  df-fo 6560  df-f1o 6561  df-fv 6562  df-ov 7427  df-oprab 7428  df-mpo 7429  df-1st 8003  df-2nd 8004  df-er 8734  df-pm 8858  df-en 8975  df-dom 8976  df-sdom 8977  df-pnf 11300  df-mnf 11301  df-xr 11302  df-ltxr 11303  df-le 11304  df-neg 11497  df-z 12611  df-uz 12875  df-top 22887  df-topon 22904  df-lm 23224
This theorem is referenced by:  hlim0  31168  occllem  31236  nlelchi  31994  hmopidmchi  32084  esumcvg  33919  xlimconst  45446
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