Theorem ntrivcvgap0 12100

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 = (ℤ≥‘𝑀)
ntrivcvgn0.2	⊢ (𝜑 → 𝑀 ∈ ℤ)
ntrivcvgn0.3	⊢ (𝜑 → seq𝑀( · , 𝐹) ⇝ 𝑋)
ntrivcvgap0.4	⊢ (𝜑 → 𝑋 # 0)

Assertion

Ref	Expression
ntrivcvgap0	⊢ (𝜑 → ∃𝑛 ∈ 𝑍 ∃𝑦(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

Step	Hyp	Ref	Expression
1		ntrivcvgn0.2	. . . 4 ⊢ (𝜑 → 𝑀 ∈ ℤ)
2		uzid 9760	. . . 4 ⊢ (𝑀 ∈ ℤ → 𝑀 ∈ (ℤ≥‘𝑀))
3	1, 2	syl 14	. . 3 ⊢ (𝜑 → 𝑀 ∈ (ℤ≥‘𝑀))
4		ntrivcvgn0.1	. . 3 ⊢ 𝑍 = (ℤ≥‘𝑀)
5	3, 4	eleqtrrdi 2323	. 2 ⊢ (𝜑 → 𝑀 ∈ 𝑍)
6		ntrivcvgn0.3	. . . 4 ⊢ (𝜑 → seq𝑀( · , 𝐹) ⇝ 𝑋)
7		climrel 11831	. . . . 5 ⊢ Rel ⇝
8	7	brrelex2i 4768	. . . 4 ⊢ (seq𝑀( · , 𝐹) ⇝ 𝑋 → 𝑋 ∈ V)
9	6, 8	syl 14	. . 3 ⊢ (𝜑 → 𝑋 ∈ V)
10		ntrivcvgap0.4	. . . 4 ⊢ (𝜑 → 𝑋 # 0)
11	10, 6	jca 306	. . 3 ⊢ (𝜑 → (𝑋 # 0 ∧ seq𝑀( · , 𝐹) ⇝ 𝑋))
12		breq1 4089	. . . 4 ⊢ (𝑦 = 𝑋 → (𝑦 # 0 ↔ 𝑋 # 0))
13		breq2 4090	. . . 4 ⊢ (𝑦 = 𝑋 → (seq𝑀( · , 𝐹) ⇝ 𝑦 ↔ seq𝑀( · , 𝐹) ⇝ 𝑋))
14	12, 13	anbi12d 473	. . 3 ⊢ (𝑦 = 𝑋 → ((𝑦 # 0 ∧ seq𝑀( · , 𝐹) ⇝ 𝑦) ↔ (𝑋 # 0 ∧ seq𝑀( · , 𝐹) ⇝ 𝑋)))
15	9, 11, 14	elabd 2949	. 2 ⊢ (𝜑 → ∃𝑦(𝑦 # 0 ∧ seq𝑀( · , 𝐹) ⇝ 𝑦))
16		seqeq1 10702	. . . . . 6 ⊢ (𝑛 = 𝑀 → seq𝑛( · , 𝐹) = seq𝑀( · , 𝐹))
17	16	breq1d 4096	. . . . 5 ⊢ (𝑛 = 𝑀 → (seq𝑛( · , 𝐹) ⇝ 𝑦 ↔ seq𝑀( · , 𝐹) ⇝ 𝑦))
18	17	anbi2d 464	. . . 4 ⊢ (𝑛 = 𝑀 → ((𝑦 # 0 ∧ seq𝑛( · , 𝐹) ⇝ 𝑦) ↔ (𝑦 # 0 ∧ seq𝑀( · , 𝐹) ⇝ 𝑦)))
19	18	exbidv 1871	. . 3 ⊢ (𝑛 = 𝑀 → (∃𝑦(𝑦 # 0 ∧ seq𝑛( · , 𝐹) ⇝ 𝑦) ↔ ∃𝑦(𝑦 # 0 ∧ seq𝑀( · , 𝐹) ⇝ 𝑦)))
20	19	rspcev 2908	. 2 ⊢ ((𝑀 ∈ 𝑍 ∧ ∃𝑦(𝑦 # 0 ∧ seq𝑀( · , 𝐹) ⇝ 𝑦)) → ∃𝑛 ∈ 𝑍 ∃𝑦(𝑦 # 0 ∧ seq𝑛( · , 𝐹) ⇝ 𝑦))
21	5, 15, 20	syl2anc 411	1 ⊢ (𝜑 → ∃𝑛 ∈ 𝑍 ∃𝑦(𝑦 # 0 ∧ seq𝑛( · , 𝐹) ⇝ 𝑦))

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1395  ∃wex 1538   ∈ wcel 2200  ∃wrex 2509  Vcvv 2800   class class class wbr 4086  ‘cfv 5324  0cc0 8022   · cmul 8027   # cap 8751  ℤcz 9469  ℤ_≥cuz 9745  seqcseq 10699   ⇝ cli 11829

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4205  ax-pow 4262  ax-pr 4297  ax-un 4528  ax-setind 4633  ax-cnex 8113  ax-resscn 8114  ax-pre-ltirr 8134

This theorem depends on definitions:  df-bi 117  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-rab 2517  df-v 2802  df-sbc 3030  df-dif 3200  df-un 3202  df-in 3204  df-ss 3211  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3892  df-br 4087  df-opab 4149  df-mpt 4150  df-id 4388  df-xp 4729  df-rel 4730  df-cnv 4731  df-co 4732  df-dm 4733  df-rn 4734  df-res 4735  df-iota 5284  df-fun 5326  df-fv 5332  df-ov 6016  df-oprab 6017  df-mpo 6018  df-recs 6466  df-frec 6552  df-pnf 8206  df-mnf 8207  df-xr 8208  df-ltxr 8209  df-le 8210  df-neg 8343  df-z 9470  df-uz 9746  df-seqfrec 10700  df-clim 11830

This theorem is referenced by:  zprodap0  12132