Theorem ntrivcvgap0 11910

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 9675	. . . 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 2300	. 2 ⊢ (𝜑 → 𝑀 ∈ 𝑍)
6		ntrivcvgn0.3	. . . 4 ⊢ (𝜑 → seq𝑀( · , 𝐹) ⇝ 𝑋)
7		climrel 11641	. . . . 5 ⊢ Rel ⇝
8	7	brrelex2i 4724	. . . 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 4051	. . . 4 ⊢ (𝑦 = 𝑋 → (𝑦 # 0 ↔ 𝑋 # 0))
13		breq2 4052	. . . 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 2920	. 2 ⊢ (𝜑 → ∃𝑦(𝑦 # 0 ∧ seq𝑀( · , 𝐹) ⇝ 𝑦))
16		seqeq1 10608	. . . . . 6 ⊢ (𝑛 = 𝑀 → seq𝑛( · , 𝐹) = seq𝑀( · , 𝐹))
17	16	breq1d 4058	. . . . 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 1849	. . 3 ⊢ (𝑛 = 𝑀 → (∃𝑦(𝑦 # 0 ∧ seq𝑛( · , 𝐹) ⇝ 𝑦) ↔ ∃𝑦(𝑦 # 0 ∧ seq𝑀( · , 𝐹) ⇝ 𝑦)))
20	19	rspcev 2879	. 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 1373  ∃wex 1516   ∈ wcel 2177  ∃wrex 2486  Vcvv 2773   class class class wbr 4048  ‘cfv 5277  0cc0 7938   · cmul 7943   # cap 8667  ℤcz 9385  ℤ_≥cuz 9661  seqcseq 10605   ⇝ cli 11639

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2179  ax-14 2180  ax-ext 2188  ax-sep 4167  ax-pow 4223  ax-pr 4258  ax-un 4485  ax-setind 4590  ax-cnex 8029  ax-resscn 8030  ax-pre-ltirr 8050

This theorem depends on definitions:  df-bi 117  df-3or 982  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ne 2378  df-nel 2473  df-ral 2490  df-rex 2491  df-rab 2494  df-v 2775  df-sbc 3001  df-dif 3170  df-un 3172  df-in 3174  df-ss 3181  df-pw 3620  df-sn 3641  df-pr 3642  df-op 3644  df-uni 3854  df-br 4049  df-opab 4111  df-mpt 4112  df-id 4345  df-xp 4686  df-rel 4687  df-cnv 4688  df-co 4689  df-dm 4690  df-rn 4691  df-res 4692  df-iota 5238  df-fun 5279  df-fv 5285  df-ov 5957  df-oprab 5958  df-mpo 5959  df-recs 6401  df-frec 6487  df-pnf 8122  df-mnf 8123  df-xr 8124  df-ltxr 8125  df-le 8126  df-neg 8259  df-z 9386  df-uz 9662  df-seqfrec 10606  df-clim 11640

This theorem is referenced by:  zprodap0  11942