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Theorem ntrivcvgap0 11330
 Description: A product that converges to a value apart from zero converges non-trivially. (Contributed by Scott Fenton, 18-Dec-2017.)
Hypotheses
Ref Expression
ntrivcvgn0.1 𝑍 = (ℤ𝑀)
ntrivcvgn0.2 (𝜑𝑀 ∈ ℤ)
ntrivcvgn0.3 (𝜑 → seq𝑀( · , 𝐹) ⇝ 𝑋)
ntrivcvgap0.4 (𝜑𝑋 # 0)
Assertion
Ref Expression
ntrivcvgap0 (𝜑 → ∃𝑛𝑍𝑦(𝑦 # 0 ∧ seq𝑛( · , 𝐹) ⇝ 𝑦))
Distinct variable groups:   𝑛,𝐹,𝑦   𝑛,𝑀,𝑦   𝑦,𝑋   𝑛,𝑍
Allowed substitution hints:   𝜑(𝑦,𝑛)   𝑋(𝑛)   𝑍(𝑦)

Proof of Theorem ntrivcvgap0
StepHypRef Expression
1 ntrivcvgn0.2 . . . 4 (𝜑𝑀 ∈ ℤ)
2 uzid 9352 . . . 4 (𝑀 ∈ ℤ → 𝑀 ∈ (ℤ𝑀))
31, 2syl 14 . . 3 (𝜑𝑀 ∈ (ℤ𝑀))
4 ntrivcvgn0.1 . . 3 𝑍 = (ℤ𝑀)
53, 4eleqtrrdi 2233 . 2 (𝜑𝑀𝑍)
6 ntrivcvgn0.3 . . . 4 (𝜑 → seq𝑀( · , 𝐹) ⇝ 𝑋)
7 climrel 11061 . . . . 5 Rel ⇝
87brrelex2i 4583 . . . 4 (seq𝑀( · , 𝐹) ⇝ 𝑋𝑋 ∈ V)
96, 8syl 14 . . 3 (𝜑𝑋 ∈ V)
10 ntrivcvgap0.4 . . . 4 (𝜑𝑋 # 0)
1110, 6jca 304 . . 3 (𝜑 → (𝑋 # 0 ∧ seq𝑀( · , 𝐹) ⇝ 𝑋))
12 breq1 3932 . . . 4 (𝑦 = 𝑋 → (𝑦 # 0 ↔ 𝑋 # 0))
13 breq2 3933 . . . 4 (𝑦 = 𝑋 → (seq𝑀( · , 𝐹) ⇝ 𝑦 ↔ seq𝑀( · , 𝐹) ⇝ 𝑋))
1412, 13anbi12d 464 . . 3 (𝑦 = 𝑋 → ((𝑦 # 0 ∧ seq𝑀( · , 𝐹) ⇝ 𝑦) ↔ (𝑋 # 0 ∧ seq𝑀( · , 𝐹) ⇝ 𝑋)))
159, 11, 14elabd 2829 . 2 (𝜑 → ∃𝑦(𝑦 # 0 ∧ seq𝑀( · , 𝐹) ⇝ 𝑦))
16 seqeq1 10233 . . . . . 6 (𝑛 = 𝑀 → seq𝑛( · , 𝐹) = seq𝑀( · , 𝐹))
1716breq1d 3939 . . . . 5 (𝑛 = 𝑀 → (seq𝑛( · , 𝐹) ⇝ 𝑦 ↔ seq𝑀( · , 𝐹) ⇝ 𝑦))
1817anbi2d 459 . . . 4 (𝑛 = 𝑀 → ((𝑦 # 0 ∧ seq𝑛( · , 𝐹) ⇝ 𝑦) ↔ (𝑦 # 0 ∧ seq𝑀( · , 𝐹) ⇝ 𝑦)))
1918exbidv 1797 . . 3 (𝑛 = 𝑀 → (∃𝑦(𝑦 # 0 ∧ seq𝑛( · , 𝐹) ⇝ 𝑦) ↔ ∃𝑦(𝑦 # 0 ∧ seq𝑀( · , 𝐹) ⇝ 𝑦)))
2019rspcev 2789 . 2 ((𝑀𝑍 ∧ ∃𝑦(𝑦 # 0 ∧ seq𝑀( · , 𝐹) ⇝ 𝑦)) → ∃𝑛𝑍𝑦(𝑦 # 0 ∧ seq𝑛( · , 𝐹) ⇝ 𝑦))
215, 15, 20syl2anc 408 1 (𝜑 → ∃𝑛𝑍𝑦(𝑦 # 0 ∧ seq𝑛( · , 𝐹) ⇝ 𝑦))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 103   = wceq 1331  ∃wex 1468   ∈ wcel 1480  ∃wrex 2417  Vcvv 2686   class class class wbr 3929  ‘cfv 5123  0cc0 7632   · cmul 7637   # cap 8355  ℤcz 9066  ℤ≥cuz 9338  seqcseq 10230   ⇝ cli 11059 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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-cnex 7723  ax-resscn 7724  ax-pre-ltirr 7744 This theorem depends on definitions:  df-bi 116  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-nel 2404  df-ral 2421  df-rex 2422  df-rab 2425  df-v 2688  df-sbc 2910  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-br 3930  df-opab 3990  df-mpt 3991  df-id 4215  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-iota 5088  df-fun 5125  df-fv 5131  df-ov 5777  df-oprab 5778  df-mpo 5779  df-recs 6202  df-frec 6288  df-pnf 7814  df-mnf 7815  df-xr 7816  df-ltxr 7817  df-le 7818  df-neg 7948  df-z 9067  df-uz 9339  df-seqfrec 10231  df-clim 11060 This theorem is referenced by: (None)
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