Theorem ntrivcvgap0 11714

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 9615	. . . 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 2290	. 2 ⊢ (𝜑 → 𝑀 ∈ 𝑍)
6		ntrivcvgn0.3	. . . 4 ⊢ (𝜑 → seq𝑀( · , 𝐹) ⇝ 𝑋)
7		climrel 11445	. . . . 5 ⊢ Rel ⇝
8	7	brrelex2i 4707	. . . 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 4036	. . . 4 ⊢ (𝑦 = 𝑋 → (𝑦 # 0 ↔ 𝑋 # 0))
13		breq2 4037	. . . 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 2909	. 2 ⊢ (𝜑 → ∃𝑦(𝑦 # 0 ∧ seq𝑀( · , 𝐹) ⇝ 𝑦))
16		seqeq1 10542	. . . . . 6 ⊢ (𝑛 = 𝑀 → seq𝑛( · , 𝐹) = seq𝑀( · , 𝐹))
17	16	breq1d 4043	. . . . 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 1839	. . 3 ⊢ (𝑛 = 𝑀 → (∃𝑦(𝑦 # 0 ∧ seq𝑛( · , 𝐹) ⇝ 𝑦) ↔ ∃𝑦(𝑦 # 0 ∧ seq𝑀( · , 𝐹) ⇝ 𝑦)))
20	19	rspcev 2868	. 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 1364  ∃wex 1506   ∈ wcel 2167  ∃wrex 2476  Vcvv 2763   class class class wbr 4033  ‘cfv 5258  0cc0 7879   · cmul 7884   # cap 8608  ℤcz 9326  ℤ_≥cuz 9601  seqcseq 10539   ⇝ cli 11443

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 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-sep 4151  ax-pow 4207  ax-pr 4242  ax-un 4468  ax-setind 4573  ax-cnex 7970  ax-resscn 7971  ax-pre-ltirr 7991

This theorem depends on definitions:  df-bi 117  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-nel 2463  df-ral 2480  df-rex 2481  df-rab 2484  df-v 2765  df-sbc 2990  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-br 4034  df-opab 4095  df-mpt 4096  df-id 4328  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-res 4675  df-iota 5219  df-fun 5260  df-fv 5266  df-ov 5925  df-oprab 5926  df-mpo 5927  df-recs 6363  df-frec 6449  df-pnf 8063  df-mnf 8064  df-xr 8065  df-ltxr 8066  df-le 8067  df-neg 8200  df-z 9327  df-uz 9602  df-seqfrec 10540  df-clim 11444

This theorem is referenced by:  zprodap0  11746