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Theorem lmconst 21286
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 1131 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑃𝑋𝑀 ∈ ℤ) → 𝑃𝑋)
2 simp3 1132 . . . . . 6 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑃𝑋𝑀 ∈ ℤ) → 𝑀 ∈ ℤ)
3 uzid 11908 . . . . . 6 (𝑀 ∈ ℤ → 𝑀 ∈ (ℤ𝑀))
42, 3syl 17 . . . . 5 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑃𝑋𝑀 ∈ ℤ) → 𝑀 ∈ (ℤ𝑀))
5 lmconst.2 . . . . 5 𝑍 = (ℤ𝑀)
64, 5syl6eleqr 2861 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑃𝑋𝑀 ∈ ℤ) → 𝑀𝑍)
7 idd 24 . . . . 5 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑃𝑋𝑀 ∈ ℤ) ∧ 𝑘 ∈ (ℤ𝑀)) → (𝑃𝑢𝑃𝑢))
87ralrimdva 3118 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑃𝑋𝑀 ∈ ℤ) → (𝑃𝑢 → ∀𝑘 ∈ (ℤ𝑀)𝑃𝑢))
9 fveq2 6333 . . . . . 6 (𝑗 = 𝑀 → (ℤ𝑗) = (ℤ𝑀))
109raleqdv 3293 . . . . 5 (𝑗 = 𝑀 → (∀𝑘 ∈ (ℤ𝑗)𝑃𝑢 ↔ ∀𝑘 ∈ (ℤ𝑀)𝑃𝑢))
1110rspcev 3460 . . . 4 ((𝑀𝑍 ∧ ∀𝑘 ∈ (ℤ𝑀)𝑃𝑢) → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)𝑃𝑢)
126, 8, 11syl6an 663 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑃𝑋𝑀 ∈ ℤ) → (𝑃𝑢 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)𝑃𝑢))
1312ralrimivw 3116 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑃𝑋𝑀 ∈ ℤ) → ∀𝑢𝐽 (𝑃𝑢 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)𝑃𝑢))
14 simp1 1130 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑃𝑋𝑀 ∈ ℤ) → 𝐽 ∈ (TopOn‘𝑋))
15 fconst6g 6235 . . . 4 (𝑃𝑋 → (𝑍 × {𝑃}):𝑍𝑋)
161, 15syl 17 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑃𝑋𝑀 ∈ ℤ) → (𝑍 × {𝑃}):𝑍𝑋)
17 fvconst2g 6614 . . . 4 ((𝑃𝑋𝑘𝑍) → ((𝑍 × {𝑃})‘𝑘) = 𝑃)
181, 17sylan 569 . . 3 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑃𝑋𝑀 ∈ ℤ) ∧ 𝑘𝑍) → ((𝑍 × {𝑃})‘𝑘) = 𝑃)
1914, 5, 2, 16, 18lmbrf 21285 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑃𝑋𝑀 ∈ ℤ) → ((𝑍 × {𝑃})(⇝𝑡𝐽)𝑃 ↔ (𝑃𝑋 ∧ ∀𝑢𝐽 (𝑃𝑢 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)𝑃𝑢))))
201, 13, 19mpbir2and 692 1 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑃𝑋𝑀 ∈ ℤ) → (𝑍 × {𝑃})(⇝𝑡𝐽)𝑃)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382  w3a 1071   = wceq 1631  wcel 2145  wral 3061  wrex 3062  {csn 4317   class class class wbr 4787   × cxp 5248  wf 6026  cfv 6030  cz 11584  cuz 11893  TopOnctopon 20935  𝑡clm 21251
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4916  ax-nul 4924  ax-pow 4975  ax-pr 5035  ax-un 7100  ax-cnex 10198  ax-resscn 10199  ax-pre-lttri 10216  ax-pre-lttrn 10217
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3or 1072  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-nel 3047  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4227  df-pw 4300  df-sn 4318  df-pr 4320  df-op 4324  df-uni 4576  df-iun 4657  df-br 4788  df-opab 4848  df-mpt 4865  df-id 5158  df-po 5171  df-so 5172  df-xp 5256  df-rel 5257  df-cnv 5258  df-co 5259  df-dm 5260  df-rn 5261  df-res 5262  df-ima 5263  df-iota 5993  df-fun 6032  df-fn 6033  df-f 6034  df-f1 6035  df-fo 6036  df-f1o 6037  df-fv 6038  df-ov 6799  df-oprab 6800  df-mpt2 6801  df-1st 7319  df-2nd 7320  df-er 7900  df-pm 8016  df-en 8114  df-dom 8115  df-sdom 8116  df-pnf 10282  df-mnf 10283  df-xr 10284  df-ltxr 10285  df-le 10286  df-neg 10475  df-z 11585  df-uz 11894  df-top 20919  df-topon 20936  df-lm 21254
This theorem is referenced by:  hlim0  28432  occllem  28502  nlelchi  29260  hmopidmchi  29350  esumcvg  30488  xlimconst  40564
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