ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  ntrivcvgap0 GIF version

Theorem ntrivcvgap0 11499
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 9488 . . . 4 (𝑀 ∈ ℤ → 𝑀 ∈ (ℤ𝑀))
31, 2syl 14 . . 3 (𝜑𝑀 ∈ (ℤ𝑀))
4 ntrivcvgn0.1 . . 3 𝑍 = (ℤ𝑀)
53, 4eleqtrrdi 2264 . 2 (𝜑𝑀𝑍)
6 ntrivcvgn0.3 . . . 4 (𝜑 → seq𝑀( · , 𝐹) ⇝ 𝑋)
7 climrel 11230 . . . . 5 Rel ⇝
87brrelex2i 4653 . . . 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 3990 . . . 4 (𝑦 = 𝑋 → (𝑦 # 0 ↔ 𝑋 # 0))
13 breq2 3991 . . . 4 (𝑦 = 𝑋 → (seq𝑀( · , 𝐹) ⇝ 𝑦 ↔ seq𝑀( · , 𝐹) ⇝ 𝑋))
1412, 13anbi12d 470 . . 3 (𝑦 = 𝑋 → ((𝑦 # 0 ∧ seq𝑀( · , 𝐹) ⇝ 𝑦) ↔ (𝑋 # 0 ∧ seq𝑀( · , 𝐹) ⇝ 𝑋)))
159, 11, 14elabd 2875 . 2 (𝜑 → ∃𝑦(𝑦 # 0 ∧ seq𝑀( · , 𝐹) ⇝ 𝑦))
16 seqeq1 10391 . . . . . 6 (𝑛 = 𝑀 → seq𝑛( · , 𝐹) = seq𝑀( · , 𝐹))
1716breq1d 3997 . . . . 5 (𝑛 = 𝑀 → (seq𝑛( · , 𝐹) ⇝ 𝑦 ↔ seq𝑀( · , 𝐹) ⇝ 𝑦))
1817anbi2d 461 . . . 4 (𝑛 = 𝑀 → ((𝑦 # 0 ∧ seq𝑛( · , 𝐹) ⇝ 𝑦) ↔ (𝑦 # 0 ∧ seq𝑀( · , 𝐹) ⇝ 𝑦)))
1918exbidv 1818 . . 3 (𝑛 = 𝑀 → (∃𝑦(𝑦 # 0 ∧ seq𝑛( · , 𝐹) ⇝ 𝑦) ↔ ∃𝑦(𝑦 # 0 ∧ seq𝑀( · , 𝐹) ⇝ 𝑦)))
2019rspcev 2834 . 2 ((𝑀𝑍 ∧ ∃𝑦(𝑦 # 0 ∧ seq𝑀( · , 𝐹) ⇝ 𝑦)) → ∃𝑛𝑍𝑦(𝑦 # 0 ∧ seq𝑛( · , 𝐹) ⇝ 𝑦))
215, 15, 20syl2anc 409 1 (𝜑 → ∃𝑛𝑍𝑦(𝑦 # 0 ∧ seq𝑛( · , 𝐹) ⇝ 𝑦))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103   = wceq 1348  wex 1485  wcel 2141  wrex 2449  Vcvv 2730   class class class wbr 3987  cfv 5196  0cc0 7761   · cmul 7766   # cap 8487  cz 9199  cuz 9474  seqcseq 10388  cli 11228
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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4105  ax-pow 4158  ax-pr 4192  ax-un 4416  ax-setind 4519  ax-cnex 7852  ax-resscn 7853  ax-pre-ltirr 7873
This theorem depends on definitions:  df-bi 116  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-nel 2436  df-ral 2453  df-rex 2454  df-rab 2457  df-v 2732  df-sbc 2956  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-pw 3566  df-sn 3587  df-pr 3588  df-op 3590  df-uni 3795  df-br 3988  df-opab 4049  df-mpt 4050  df-id 4276  df-xp 4615  df-rel 4616  df-cnv 4617  df-co 4618  df-dm 4619  df-rn 4620  df-res 4621  df-iota 5158  df-fun 5198  df-fv 5204  df-ov 5853  df-oprab 5854  df-mpo 5855  df-recs 6281  df-frec 6367  df-pnf 7943  df-mnf 7944  df-xr 7945  df-ltxr 7946  df-le 7947  df-neg 8080  df-z 9200  df-uz 9475  df-seqfrec 10389  df-clim 11229
This theorem is referenced by:  zprodap0  11531
  Copyright terms: Public domain W3C validator