Theorem lmconst 23182

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 1137	. 2 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑀 ∈ ℤ) → 𝑃 ∈ 𝑋)
2		simp3 1138	. . . . . 6 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑀 ∈ ℤ) → 𝑀 ∈ ℤ)
3		uzid 12753	. . . . . 6 ⊢ (𝑀 ∈ ℤ → 𝑀 ∈ (ℤ≥‘𝑀))
4	2, 3	syl 17	. . . . 5 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑀 ∈ ℤ) → 𝑀 ∈ (ℤ≥‘𝑀))
5		lmconst.2	. . . . 5 ⊢ 𝑍 = (ℤ≥‘𝑀)
6	4, 5	eleqtrrdi 2842	. . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑀 ∈ ℤ) → 𝑀 ∈ 𝑍)
7		idd 24	. . . . 5 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑀 ∈ ℤ) ∧ 𝑘 ∈ (ℤ≥‘𝑀)) → (𝑃 ∈ 𝑢 → 𝑃 ∈ 𝑢))
8	7	ralrimdva 3132	. . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑀 ∈ ℤ) → (𝑃 ∈ 𝑢 → ∀𝑘 ∈ (ℤ≥‘𝑀)𝑃 ∈ 𝑢))
9		fveq2 6828	. . . . . 6 ⊢ (𝑗 = 𝑀 → (ℤ≥‘𝑗) = (ℤ≥‘𝑀))
10	9	raleqdv 3292	. . . . 5 ⊢ (𝑗 = 𝑀 → (∀𝑘 ∈ (ℤ≥‘𝑗)𝑃 ∈ 𝑢 ↔ ∀𝑘 ∈ (ℤ≥‘𝑀)𝑃 ∈ 𝑢))
11	10	rspcev 3572	. . . 4 ⊢ ((𝑀 ∈ 𝑍 ∧ ∀𝑘 ∈ (ℤ≥‘𝑀)𝑃 ∈ 𝑢) → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)𝑃 ∈ 𝑢)
12	6, 8, 11	syl6an 684	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑀 ∈ ℤ) → (𝑃 ∈ 𝑢 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)𝑃 ∈ 𝑢))
13	12	ralrimivw 3128	. 2 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑀 ∈ ℤ) → ∀𝑢 ∈ 𝐽 (𝑃 ∈ 𝑢 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)𝑃 ∈ 𝑢))
14		simp1 1136	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑀 ∈ ℤ) → 𝐽 ∈ (TopOn‘𝑋))
15		fconst6g 6718	. . . 4 ⊢ (𝑃 ∈ 𝑋 → (𝑍 × {𝑃}):𝑍⟶𝑋)
16	1, 15	syl 17	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑀 ∈ ℤ) → (𝑍 × {𝑃}):𝑍⟶𝑋)
17		fvconst2g 7142	. . . 4 ⊢ ((𝑃 ∈ 𝑋 ∧ 𝑘 ∈ 𝑍) → ((𝑍 × {𝑃})‘𝑘) = 𝑃)
18	1, 17	sylan 580	. . 3 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑀 ∈ ℤ) ∧ 𝑘 ∈ 𝑍) → ((𝑍 × {𝑃})‘𝑘) = 𝑃)
19	14, 5, 2, 16, 18	lmbrf 23181	. 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 713	1 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑀 ∈ ℤ) → (𝑍 × {𝑃})(⇝𝑡‘𝐽)𝑃)

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1541   ∈ wcel 2111  ∀wral 3047  ∃wrex 3056  {csn 4575   class class class wbr 5093   × cxp 5617  ⟶wf 6483  ‘cfv 6487  ℤcz 12474  ℤ_≥cuz 12738  TopOnctopon 22831  ⇝_𝑡clm 23147

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5236  ax-nul 5246  ax-pow 5305  ax-pr 5372  ax-un 7674  ax-cnex 11068  ax-resscn 11069  ax-pre-lttri 11086  ax-pre-lttrn 11087

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-nel 3033  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-sbc 3737  df-csb 3846  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4283  df-if 4475  df-pw 4551  df-sn 4576  df-pr 4578  df-op 4582  df-uni 4859  df-iun 4943  df-br 5094  df-opab 5156  df-mpt 5175  df-id 5514  df-po 5527  df-so 5528  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-iota 6443  df-fun 6489  df-fn 6490  df-f 6491  df-f1 6492  df-fo 6493  df-f1o 6494  df-fv 6495  df-ov 7355  df-oprab 7356  df-mpo 7357  df-1st 7927  df-2nd 7928  df-er 8628  df-pm 8759  df-en 8876  df-dom 8877  df-sdom 8878  df-pnf 11154  df-mnf 11155  df-xr 11156  df-ltxr 11157  df-le 11158  df-neg 11353  df-z 12475  df-uz 12739  df-top 22815  df-topon 22832  df-lm 23150

This theorem is referenced by:  hlim0  31222  occllem  31290  nlelchi  32048  hmopidmchi  32138  esumcvg  34106  xlimconst  45928