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Theorem lmcvg 21805
Description: Convergence property of a converging sequence. (Contributed by Mario Carneiro, 14-Nov-2013.)
Hypotheses
Ref Expression
lmcvg.1 𝑍 = (ℤ𝑀)
lmcvg.3 (𝜑𝑃𝑈)
lmcvg.4 (𝜑𝑀 ∈ ℤ)
lmcvg.5 (𝜑𝐹(⇝𝑡𝐽)𝑃)
lmcvg.6 (𝜑𝑈𝐽)
Assertion
Ref Expression
lmcvg (𝜑 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ∈ 𝑈)
Distinct variable groups:   𝑗,𝑘,𝐹   𝑗,𝐽,𝑘   𝑃,𝑗,𝑘   𝜑,𝑗,𝑘   𝑈,𝑗,𝑘   𝑗,𝑀   𝑗,𝑍,𝑘
Allowed substitution hint:   𝑀(𝑘)

Proof of Theorem lmcvg
Dummy variable 𝑢 is distinct from all other variables.
StepHypRef Expression
1 lmcvg.3 . 2 (𝜑𝑃𝑈)
2 eleq2 2906 . . . 4 (𝑢 = 𝑈 → (𝑃𝑢𝑃𝑈))
3 eleq2 2906 . . . . 5 (𝑢 = 𝑈 → ((𝐹𝑘) ∈ 𝑢 ↔ (𝐹𝑘) ∈ 𝑈))
43rexralbidv 3306 . . . 4 (𝑢 = 𝑈 → (∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ∈ 𝑢 ↔ ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ∈ 𝑈))
52, 4imbi12d 346 . . 3 (𝑢 = 𝑈 → ((𝑃𝑢 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ∈ 𝑢) ↔ (𝑃𝑈 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ∈ 𝑈)))
6 lmcvg.5 . . . . . 6 (𝜑𝐹(⇝𝑡𝐽)𝑃)
7 lmrcl 21774 . . . . . . . . 9 (𝐹(⇝𝑡𝐽)𝑃𝐽 ∈ Top)
86, 7syl 17 . . . . . . . 8 (𝜑𝐽 ∈ Top)
9 toptopon2 21461 . . . . . . . 8 (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘ 𝐽))
108, 9sylib 219 . . . . . . 7 (𝜑𝐽 ∈ (TopOn‘ 𝐽))
11 lmcvg.1 . . . . . . 7 𝑍 = (ℤ𝑀)
12 lmcvg.4 . . . . . . 7 (𝜑𝑀 ∈ ℤ)
1310, 11, 12lmbr2 21802 . . . . . 6 (𝜑 → (𝐹(⇝𝑡𝐽)𝑃 ↔ (𝐹 ∈ ( 𝐽pm ℂ) ∧ 𝑃 𝐽 ∧ ∀𝑢𝐽 (𝑃𝑢 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹𝑘) ∈ 𝑢)))))
146, 13mpbid 233 . . . . 5 (𝜑 → (𝐹 ∈ ( 𝐽pm ℂ) ∧ 𝑃 𝐽 ∧ ∀𝑢𝐽 (𝑃𝑢 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹𝑘) ∈ 𝑢))))
1514simp3d 1138 . . . 4 (𝜑 → ∀𝑢𝐽 (𝑃𝑢 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹𝑘) ∈ 𝑢)))
16 simpr 485 . . . . . . . 8 ((𝑘 ∈ dom 𝐹 ∧ (𝐹𝑘) ∈ 𝑢) → (𝐹𝑘) ∈ 𝑢)
1716ralimi 3165 . . . . . . 7 (∀𝑘 ∈ (ℤ𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹𝑘) ∈ 𝑢) → ∀𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ∈ 𝑢)
1817reximi 3248 . . . . . 6 (∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹𝑘) ∈ 𝑢) → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ∈ 𝑢)
1918imim2i 16 . . . . 5 ((𝑃𝑢 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹𝑘) ∈ 𝑢)) → (𝑃𝑢 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ∈ 𝑢))
2019ralimi 3165 . . . 4 (∀𝑢𝐽 (𝑃𝑢 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹𝑘) ∈ 𝑢)) → ∀𝑢𝐽 (𝑃𝑢 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ∈ 𝑢))
2115, 20syl 17 . . 3 (𝜑 → ∀𝑢𝐽 (𝑃𝑢 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ∈ 𝑢))
22 lmcvg.6 . . 3 (𝜑𝑈𝐽)
235, 21, 22rspcdva 3629 . 2 (𝜑 → (𝑃𝑈 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ∈ 𝑈))
241, 23mpd 15 1 (𝜑 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ∈ 𝑈)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  w3a 1081   = wceq 1530  wcel 2107  wral 3143  wrex 3144   cuni 4837   class class class wbr 5063  dom cdm 5554  cfv 6354  (class class class)co 7150  pm cpm 8402  cc 10529  cz 11975  cuz 12237  Topctop 21436  TopOnctopon 21453  𝑡clm 21769
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798  ax-sep 5200  ax-nul 5207  ax-pow 5263  ax-pr 5326  ax-un 7455  ax-cnex 10587  ax-resscn 10588  ax-pre-lttri 10605  ax-pre-lttrn 10606
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3or 1082  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ne 3022  df-nel 3129  df-ral 3148  df-rex 3149  df-rab 3152  df-v 3502  df-sbc 3777  df-csb 3888  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-nul 4296  df-if 4471  df-pw 4544  df-sn 4565  df-pr 4567  df-op 4571  df-uni 4838  df-iun 4919  df-br 5064  df-opab 5126  df-mpt 5144  df-id 5459  df-po 5473  df-so 5474  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-iota 6313  df-fun 6356  df-fn 6357  df-f 6358  df-f1 6359  df-fo 6360  df-f1o 6361  df-fv 6362  df-ov 7153  df-oprab 7154  df-mpo 7155  df-1st 7685  df-2nd 7686  df-er 8284  df-pm 8404  df-en 8504  df-dom 8505  df-sdom 8506  df-pnf 10671  df-mnf 10672  df-xr 10673  df-ltxr 10674  df-le 10675  df-neg 10867  df-z 11976  df-uz 12238  df-top 21437  df-topon 21454  df-lm 21772
This theorem is referenced by:  lmmo  21923  1stccnp  22005  1stckgenlem  22096  iscmet3lem2  23829
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