Theorem lmconst 23579

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 1155	. 2 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑀 ∈ ℤ) → 𝑃 ∈ 𝑋)
2		simp3 1156	. . . . . 6 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑀 ∈ ℤ) → 𝑀 ∈ ℤ)
3		uzid 12980	. . . . . 6 ⊢ (𝑀 ∈ ℤ → 𝑀 ∈ (ℤ≥‘𝑀))
4	2, 3	syl 18	. . . . 5 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑀 ∈ ℤ) → 𝑀 ∈ (ℤ≥‘𝑀))
5		lmconst.2	. . . . 5 ⊢ 𝑍 = (ℤ≥‘𝑀)
6	4, 5	eleqtrrdi 2871	. . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑀 ∈ ℤ) → 𝑀 ∈ 𝑍)
7		idd 25	. . . . 5 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑀 ∈ ℤ) ∧ 𝑘 ∈ (ℤ≥‘𝑀)) → (𝑃 ∈ 𝑢 → 𝑃 ∈ 𝑢))
8	7	ralrimdva 3162	. . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑀 ∈ ℤ) → (𝑃 ∈ 𝑢 → ∀𝑘 ∈ (ℤ≥‘𝑀)𝑃 ∈ 𝑢))
9		fveq2 6885	. . . . . 6 ⊢ (𝑗 = 𝑀 → (ℤ≥‘𝑗) = (ℤ≥‘𝑀))
10	9	raleqdv 3319	. . . . 5 ⊢ (𝑗 = 𝑀 → (∀𝑘 ∈ (ℤ≥‘𝑗)𝑃 ∈ 𝑢 ↔ ∀𝑘 ∈ (ℤ≥‘𝑀)𝑃 ∈ 𝑢))
11	10	rspcev 3576	. . . 4 ⊢ ((𝑀 ∈ 𝑍 ∧ ∀𝑘 ∈ (ℤ≥‘𝑀)𝑃 ∈ 𝑢) → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)𝑃 ∈ 𝑢)
12	6, 8, 11	syl6an 697	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑀 ∈ ℤ) → (𝑃 ∈ 𝑢 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)𝑃 ∈ 𝑢))
13	12	ralrimivw 3158	. 2 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑀 ∈ ℤ) → ∀𝑢 ∈ 𝐽 (𝑃 ∈ 𝑢 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)𝑃 ∈ 𝑢))
14		simp1 1154	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑀 ∈ ℤ) → 𝐽 ∈ (TopOn‘𝑋))
15		fconst6g 6771	. . . 4 ⊢ (𝑃 ∈ 𝑋 → (𝑍 × {𝑃}):𝑍⟶𝑋)
16	1, 15	syl 18	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑀 ∈ ℤ) → (𝑍 × {𝑃}):𝑍⟶𝑋)
17		fvconst2g 7208	. . . 4 ⊢ ((𝑃 ∈ 𝑋 ∧ 𝑘 ∈ 𝑍) → ((𝑍 × {𝑃})‘𝑘) = 𝑃)
18	1, 17	sylan 592	. . 3 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑀 ∈ ℤ) ∧ 𝑘 ∈ 𝑍) → ((𝑍 × {𝑃})‘𝑘) = 𝑃)
19	14, 5, 2, 16, 18	lmbrf 23578	. 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 726	1 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑀 ∈ ℤ) → (𝑍 × {𝑃})(⇝𝑡‘𝐽)𝑃)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∧ w3a 1103   = wceq 1569   ∈ wcel 2144  ∀wral 3076  ∃wrex 3086  {csn 4583   class class class wbr 5102   × cxp 5648  ⟶wf 6534  ‘cfv 6538  ℤcz 12693  ℤ_≥cuz 12965  TopOnctopon 23228  ⇝_𝑡clm 23544

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7751  ax-cnex 11256  ax-resscn 11257  ax-pre-lttri 11274  ax-pre-lttrn 11275

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-nel 3062  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5545  df-po 5558  df-so 5559  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-ov 7423  df-oprab 7424  df-mpo 7425  df-1st 8001  df-2nd 8002  df-er 8717  df-pm 8850  df-en 8974  df-dom 8975  df-sdom 8976  df-pnf 11345  df-mnf 11346  df-xr 11347  df-ltxr 11348  df-le 11349  df-neg 11544  df-z 12694  df-uz 12966  df-top 23212  df-topon 23229  df-lm 23547

This theorem is used by:  hlim0  31837  occllem  31905  nlelchi  32663  hmopidmchi  32753  esumcvg  34760  xlimconst  46878