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Theorem lmconst 12563
 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 983 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑃𝑋𝑀 ∈ ℤ) → 𝑃𝑋)
2 simp3 984 . . . . . 6 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑃𝑋𝑀 ∈ ℤ) → 𝑀 ∈ ℤ)
3 uzid 9432 . . . . . 6 (𝑀 ∈ ℤ → 𝑀 ∈ (ℤ𝑀))
42, 3syl 14 . . . . 5 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑃𝑋𝑀 ∈ ℤ) → 𝑀 ∈ (ℤ𝑀))
5 lmconst.2 . . . . 5 𝑍 = (ℤ𝑀)
64, 5eleqtrrdi 2248 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑃𝑋𝑀 ∈ ℤ) → 𝑀𝑍)
7 idd 21 . . . . 5 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑃𝑋𝑀 ∈ ℤ) ∧ 𝑘 ∈ (ℤ𝑀)) → (𝑃𝑢𝑃𝑢))
87ralrimdva 2534 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑃𝑋𝑀 ∈ ℤ) → (𝑃𝑢 → ∀𝑘 ∈ (ℤ𝑀)𝑃𝑢))
9 fveq2 5461 . . . . . 6 (𝑗 = 𝑀 → (ℤ𝑗) = (ℤ𝑀))
109raleqdv 2655 . . . . 5 (𝑗 = 𝑀 → (∀𝑘 ∈ (ℤ𝑗)𝑃𝑢 ↔ ∀𝑘 ∈ (ℤ𝑀)𝑃𝑢))
1110rspcev 2813 . . . 4 ((𝑀𝑍 ∧ ∀𝑘 ∈ (ℤ𝑀)𝑃𝑢) → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)𝑃𝑢)
126, 8, 11syl6an 1411 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑃𝑋𝑀 ∈ ℤ) → (𝑃𝑢 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)𝑃𝑢))
1312ralrimivw 2528 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑃𝑋𝑀 ∈ ℤ) → ∀𝑢𝐽 (𝑃𝑢 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)𝑃𝑢))
14 simp1 982 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑃𝑋𝑀 ∈ ℤ) → 𝐽 ∈ (TopOn‘𝑋))
15 fconst6g 5361 . . . 4 (𝑃𝑋 → (𝑍 × {𝑃}):𝑍𝑋)
161, 15syl 14 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑃𝑋𝑀 ∈ ℤ) → (𝑍 × {𝑃}):𝑍𝑋)
17 fvconst2g 5674 . . . 4 ((𝑃𝑋𝑘𝑍) → ((𝑍 × {𝑃})‘𝑘) = 𝑃)
181, 17sylan 281 . . 3 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑃𝑋𝑀 ∈ ℤ) ∧ 𝑘𝑍) → ((𝑍 × {𝑃})‘𝑘) = 𝑃)
1914, 5, 2, 16, 18lmbrf 12562 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑃𝑋𝑀 ∈ ℤ) → ((𝑍 × {𝑃})(⇝𝑡𝐽)𝑃 ↔ (𝑃𝑋 ∧ ∀𝑢𝐽 (𝑃𝑢 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)𝑃𝑢))))
201, 13, 19mpbir2and 929 1 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑃𝑋𝑀 ∈ ℤ) → (𝑍 × {𝑃})(⇝𝑡𝐽)𝑃)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 103   ∧ w3a 963   = wceq 1332   ∈ wcel 2125  ∀wral 2432  ∃wrex 2433  {csn 3556   class class class wbr 3961   × cxp 4577  ⟶wf 5159  ‘cfv 5163  ℤcz 9146  ℤ≥cuz 9418  TopOnctopon 12355  ⇝𝑡clm 12534 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1481  ax-10 1482  ax-11 1483  ax-i12 1484  ax-bndl 1486  ax-4 1487  ax-17 1503  ax-i9 1507  ax-ial 1511  ax-i5r 1512  ax-13 2127  ax-14 2128  ax-ext 2136  ax-sep 4078  ax-pow 4130  ax-pr 4164  ax-un 4388  ax-setind 4490  ax-cnex 7802  ax-resscn 7803  ax-1cn 7804  ax-1re 7805  ax-icn 7806  ax-addcl 7807  ax-addrcl 7808  ax-mulcl 7809  ax-addcom 7811  ax-addass 7813  ax-distr 7815  ax-i2m1 7816  ax-0lt1 7817  ax-0id 7819  ax-rnegex 7820  ax-cnre 7822  ax-pre-ltirr 7823  ax-pre-ltwlin 7824  ax-pre-lttrn 7825  ax-pre-apti 7826  ax-pre-ltadd 7827 This theorem depends on definitions:  df-bi 116  df-dc 821  df-3or 964  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1740  df-eu 2006  df-mo 2007  df-clab 2141  df-cleq 2147  df-clel 2150  df-nfc 2285  df-ne 2325  df-nel 2420  df-ral 2437  df-rex 2438  df-reu 2439  df-rab 2441  df-v 2711  df-sbc 2934  df-csb 3028  df-dif 3100  df-un 3102  df-in 3104  df-ss 3111  df-if 3502  df-pw 3541  df-sn 3562  df-pr 3563  df-op 3565  df-uni 3769  df-int 3804  df-iun 3847  df-br 3962  df-opab 4022  df-mpt 4023  df-id 4248  df-xp 4585  df-rel 4586  df-cnv 4587  df-co 4588  df-dm 4589  df-rn 4590  df-res 4591  df-ima 4592  df-iota 5128  df-fun 5165  df-fn 5166  df-f 5167  df-fv 5171  df-riota 5770  df-ov 5817  df-oprab 5818  df-mpo 5819  df-1st 6078  df-2nd 6079  df-pm 6585  df-pnf 7893  df-mnf 7894  df-xr 7895  df-ltxr 7896  df-le 7897  df-sub 8027  df-neg 8028  df-inn 8813  df-n0 9070  df-z 9147  df-uz 9419  df-top 12343  df-topon 12356  df-lm 12537 This theorem is referenced by: (None)
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