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Theorem lmconst 21845
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 12236 . . . . . 6 (𝑀 ∈ ℤ → 𝑀 ∈ (ℤ𝑀))
42, 3syl 17 . . . . 5 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑃𝑋𝑀 ∈ ℤ) → 𝑀 ∈ (ℤ𝑀))
5 lmconst.2 . . . . 5 𝑍 = (ℤ𝑀)
64, 5eleqtrrdi 2923 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑃𝑋𝑀 ∈ ℤ) → 𝑀𝑍)
7 idd 24 . . . . 5 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑃𝑋𝑀 ∈ ℤ) ∧ 𝑘 ∈ (ℤ𝑀)) → (𝑃𝑢𝑃𝑢))
87ralrimdva 3177 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑃𝑋𝑀 ∈ ℤ) → (𝑃𝑢 → ∀𝑘 ∈ (ℤ𝑀)𝑃𝑢))
9 fveq2 6643 . . . . . 6 (𝑗 = 𝑀 → (ℤ𝑗) = (ℤ𝑀))
109raleqdv 3396 . . . . 5 (𝑗 = 𝑀 → (∀𝑘 ∈ (ℤ𝑗)𝑃𝑢 ↔ ∀𝑘 ∈ (ℤ𝑀)𝑃𝑢))
1110rspcev 3600 . . . 4 ((𝑀𝑍 ∧ ∀𝑘 ∈ (ℤ𝑀)𝑃𝑢) → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)𝑃𝑢)
126, 8, 11syl6an 683 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑃𝑋𝑀 ∈ ℤ) → (𝑃𝑢 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)𝑃𝑢))
1312ralrimivw 3171 . 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 6541 . . . 4 (𝑃𝑋 → (𝑍 × {𝑃}):𝑍𝑋)
161, 15syl 17 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑃𝑋𝑀 ∈ ℤ) → (𝑍 × {𝑃}):𝑍𝑋)
17 fvconst2g 6937 . . . 4 ((𝑃𝑋𝑘𝑍) → ((𝑍 × {𝑃})‘𝑘) = 𝑃)
181, 17sylan 583 . . 3 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑃𝑋𝑀 ∈ ℤ) ∧ 𝑘𝑍) → ((𝑍 × {𝑃})‘𝑘) = 𝑃)
1914, 5, 2, 16, 18lmbrf 21844 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑃𝑋𝑀 ∈ ℤ) → ((𝑍 × {𝑃})(⇝𝑡𝐽)𝑃 ↔ (𝑃𝑋 ∧ ∀𝑢𝐽 (𝑃𝑢 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)𝑃𝑢))))
201, 13, 19mpbir2and 712 1 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑃𝑋𝑀 ∈ ℤ) → (𝑍 × {𝑃})(⇝𝑡𝐽)𝑃)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399  w3a 1084   = wceq 1538  wcel 2115  wral 3126  wrex 3127  {csn 4540   class class class wbr 5039   × cxp 5526  wf 6324  cfv 6328  cz 11959  cuz 12221  TopOnctopon 21494  𝑡clm 21810
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2178  ax-ext 2793  ax-sep 5176  ax-nul 5183  ax-pow 5239  ax-pr 5303  ax-un 7436  ax-cnex 10570  ax-resscn 10571  ax-pre-lttri 10588  ax-pre-lttrn 10589
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2623  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2892  df-nfc 2960  df-ne 3008  df-nel 3112  df-ral 3131  df-rex 3132  df-rab 3135  df-v 3473  df-sbc 3750  df-csb 3858  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4267  df-if 4441  df-pw 4514  df-sn 4541  df-pr 4543  df-op 4547  df-uni 4812  df-iun 4894  df-br 5040  df-opab 5102  df-mpt 5120  df-id 5433  df-po 5447  df-so 5448  df-xp 5534  df-rel 5535  df-cnv 5536  df-co 5537  df-dm 5538  df-rn 5539  df-res 5540  df-ima 5541  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-ov 7133  df-oprab 7134  df-mpo 7135  df-1st 7664  df-2nd 7665  df-er 8264  df-pm 8384  df-en 8485  df-dom 8486  df-sdom 8487  df-pnf 10654  df-mnf 10655  df-xr 10656  df-ltxr 10657  df-le 10658  df-neg 10850  df-z 11960  df-uz 12222  df-top 21478  df-topon 21495  df-lm 21813
This theorem is referenced by:  hlim0  28997  occllem  29065  nlelchi  29823  hmopidmchi  29913  esumcvg  31353  xlimconst  42286
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