Theorem lmconst 14855

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 = (ℤ≥‘𝑀)

Assertion

Ref	Expression
lmconst	⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑀 ∈ ℤ) → (𝑍 × {𝑃})(⇝𝑡‘𝐽)𝑃)

Proof of Theorem lmconst

Dummy variables 𝑗 𝑘 𝑢 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simp2 1003	. 2 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑀 ∈ ℤ) → 𝑃 ∈ 𝑋)
2		simp3 1004	. . . . . 6 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑀 ∈ ℤ) → 𝑀 ∈ ℤ)
3		uzid 9704	. . . . . 6 ⊢ (𝑀 ∈ ℤ → 𝑀 ∈ (ℤ≥‘𝑀))
4	2, 3	syl 14	. . . . 5 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑀 ∈ ℤ) → 𝑀 ∈ (ℤ≥‘𝑀))
5		lmconst.2	. . . . 5 ⊢ 𝑍 = (ℤ≥‘𝑀)
6	4, 5	eleqtrrdi 2303	. . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑀 ∈ ℤ) → 𝑀 ∈ 𝑍)
7		idd 21	. . . . 5 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑀 ∈ ℤ) ∧ 𝑘 ∈ (ℤ≥‘𝑀)) → (𝑃 ∈ 𝑢 → 𝑃 ∈ 𝑢))
8	7	ralrimdva 2590	. . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑀 ∈ ℤ) → (𝑃 ∈ 𝑢 → ∀𝑘 ∈ (ℤ≥‘𝑀)𝑃 ∈ 𝑢))
9		fveq2 5603	. . . . . 6 ⊢ (𝑗 = 𝑀 → (ℤ≥‘𝑗) = (ℤ≥‘𝑀))
10	9	raleqdv 2714	. . . . 5 ⊢ (𝑗 = 𝑀 → (∀𝑘 ∈ (ℤ≥‘𝑗)𝑃 ∈ 𝑢 ↔ ∀𝑘 ∈ (ℤ≥‘𝑀)𝑃 ∈ 𝑢))
11	10	rspcev 2887	. . . 4 ⊢ ((𝑀 ∈ 𝑍 ∧ ∀𝑘 ∈ (ℤ≥‘𝑀)𝑃 ∈ 𝑢) → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)𝑃 ∈ 𝑢)
12	6, 8, 11	syl6an 1456	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑀 ∈ ℤ) → (𝑃 ∈ 𝑢 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)𝑃 ∈ 𝑢))
13	12	ralrimivw 2584	. 2 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑀 ∈ ℤ) → ∀𝑢 ∈ 𝐽 (𝑃 ∈ 𝑢 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)𝑃 ∈ 𝑢))
14		simp1 1002	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑀 ∈ ℤ) → 𝐽 ∈ (TopOn‘𝑋))
15		fconst6g 5500	. . . 4 ⊢ (𝑃 ∈ 𝑋 → (𝑍 × {𝑃}):𝑍⟶𝑋)
16	1, 15	syl 14	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑀 ∈ ℤ) → (𝑍 × {𝑃}):𝑍⟶𝑋)
17		fvconst2g 5826	. . . 4 ⊢ ((𝑃 ∈ 𝑋 ∧ 𝑘 ∈ 𝑍) → ((𝑍 × {𝑃})‘𝑘) = 𝑃)
18	1, 17	sylan 283	. . 3 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑀 ∈ ℤ) ∧ 𝑘 ∈ 𝑍) → ((𝑍 × {𝑃})‘𝑘) = 𝑃)
19	14, 5, 2, 16, 18	lmbrf 14854	. 2 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑀 ∈ ℤ) → ((𝑍 × {𝑃})(⇝𝑡‘𝐽)𝑃 ↔ (𝑃 ∈ 𝑋 ∧ ∀𝑢 ∈ 𝐽 (𝑃 ∈ 𝑢 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)𝑃 ∈ 𝑢))))
20	1, 13, 19	mpbir2and 949	1 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑀 ∈ ℤ) → (𝑍 × {𝑃})(⇝𝑡‘𝐽)𝑃)

Syntax hints:   → wi 4   ∧ wa 104   ∧ w3a 983   = wceq 1375   ∈ wcel 2180  ∀wral 2488  ∃wrex 2489  {csn 3646   class class class wbr 4062   × cxp 4694  ⟶wf 5290  ‘cfv 5294  ℤcz 9414  ℤ_≥cuz 9690  TopOnctopon 14649  ⇝_𝑡clm 14826

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 617  ax-in2 618  ax-io 713  ax-5 1473  ax-7 1474  ax-gen 1475  ax-ie1 1519  ax-ie2 1520  ax-8 1530  ax-10 1531  ax-11 1532  ax-i12 1533  ax-bndl 1535  ax-4 1536  ax-17 1552  ax-i9 1556  ax-ial 1560  ax-i5r 1561  ax-13 2182  ax-14 2183  ax-ext 2191  ax-sep 4181  ax-pow 4237  ax-pr 4272  ax-un 4501  ax-setind 4606  ax-cnex 8058  ax-resscn 8059  ax-1cn 8060  ax-1re 8061  ax-icn 8062  ax-addcl 8063  ax-addrcl 8064  ax-mulcl 8065  ax-addcom 8067  ax-addass 8069  ax-distr 8071  ax-i2m1 8072  ax-0lt1 8073  ax-0id 8075  ax-rnegex 8076  ax-cnre 8078  ax-pre-ltirr 8079  ax-pre-ltwlin 8080  ax-pre-lttrn 8081  ax-pre-apti 8082  ax-pre-ltadd 8083

This theorem depends on definitions:  df-bi 117  df-dc 839  df-3or 984  df-3an 985  df-tru 1378  df-fal 1381  df-nf 1487  df-sb 1789  df-eu 2060  df-mo 2061  df-clab 2196  df-cleq 2202  df-clel 2205  df-nfc 2341  df-ne 2381  df-nel 2476  df-ral 2493  df-rex 2494  df-reu 2495  df-rab 2497  df-v 2781  df-sbc 3009  df-csb 3105  df-dif 3179  df-un 3181  df-in 3183  df-ss 3190  df-if 3583  df-pw 3631  df-sn 3652  df-pr 3653  df-op 3655  df-uni 3868  df-int 3903  df-iun 3946  df-br 4063  df-opab 4125  df-mpt 4126  df-id 4361  df-xp 4702  df-rel 4703  df-cnv 4704  df-co 4705  df-dm 4706  df-rn 4707  df-res 4708  df-ima 4709  df-iota 5254  df-fun 5296  df-fn 5297  df-f 5298  df-fv 5302  df-riota 5927  df-ov 5977  df-oprab 5978  df-mpo 5979  df-1st 6256  df-2nd 6257  df-pm 6768  df-pnf 8151  df-mnf 8152  df-xr 8153  df-ltxr 8154  df-le 8155  df-sub 8287  df-neg 8288  df-inn 9079  df-n0 9338  df-z 9415  df-uz 9691  df-top 14637  df-topon 14650  df-lm 14829

This theorem is referenced by: (None)