Theorem fnn0ind 9442

Description: Induction on the integers from 0 to 𝑁 inclusive . The first four hypotheses give us the substitution instances we need; the last two are the basis and the induction step. (Contributed by Paul Chapman, 31-Mar-2011.)

Hypotheses

Ref	Expression
fnn0ind.1	⊢ (𝑥 = 0 → (𝜑 ↔ 𝜓))
fnn0ind.2	⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜒))
fnn0ind.3	⊢ (𝑥 = (𝑦 + 1) → (𝜑 ↔ 𝜃))
fnn0ind.4	⊢ (𝑥 = 𝐾 → (𝜑 ↔ 𝜏))
fnn0ind.5	⊢ (𝑁 ∈ ℕ0 → 𝜓)
fnn0ind.6	⊢ ((𝑁 ∈ ℕ0 ∧ 𝑦 ∈ ℕ0 ∧ 𝑦 < 𝑁) → (𝜒 → 𝜃))

Assertion

Ref	Expression
fnn0ind	⊢ ((𝑁 ∈ ℕ0 ∧ 𝐾 ∈ ℕ0 ∧ 𝐾 ≤ 𝑁) → 𝜏)

Distinct variable groups:   𝑥,𝐾   𝑥,𝑁,𝑦   𝜒,𝑥   𝜑,𝑦   𝜓,𝑥   𝜏,𝑥   𝜃,𝑥

Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑦)   𝜒(𝑦)   𝜃(𝑦)   𝜏(𝑦)   𝐾(𝑦)

Proof of Theorem fnn0ind

Step	Hyp	Ref	Expression
1		elnn0z 9339	. . . 4 ⊢ (𝐾 ∈ ℕ0 ↔ (𝐾 ∈ ℤ ∧ 0 ≤ 𝐾))
2		nn0z 9346	. . . . . 6 ⊢ (𝑁 ∈ ℕ0 → 𝑁 ∈ ℤ)
3		0z 9337	. . . . . . . 8 ⊢ 0 ∈ ℤ
4		fnn0ind.1	. . . . . . . . 9 ⊢ (𝑥 = 0 → (𝜑 ↔ 𝜓))
5		fnn0ind.2	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜒))
6		fnn0ind.3	. . . . . . . . 9 ⊢ (𝑥 = (𝑦 + 1) → (𝜑 ↔ 𝜃))
7		fnn0ind.4	. . . . . . . . 9 ⊢ (𝑥 = 𝐾 → (𝜑 ↔ 𝜏))
8		elnn0z 9339	. . . . . . . . . . 11 ⊢ (𝑁 ∈ ℕ0 ↔ (𝑁 ∈ ℤ ∧ 0 ≤ 𝑁))
9		fnn0ind.5	. . . . . . . . . . 11 ⊢ (𝑁 ∈ ℕ0 → 𝜓)
10	8, 9	sylbir 135	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ℤ ∧ 0 ≤ 𝑁) → 𝜓)
11	10	3adant1 1017	. . . . . . . . 9 ⊢ ((0 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 0 ≤ 𝑁) → 𝜓)
12		zre 9330	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 ∈ ℤ → 𝑦 ∈ ℝ)
13		zre 9330	. . . . . . . . . . . . . . . 16 ⊢ (𝑁 ∈ ℤ → 𝑁 ∈ ℝ)
14		0re 8026	. . . . . . . . . . . . . . . . 17 ⊢ 0 ∈ ℝ
15		lelttr 8115	. . . . . . . . . . . . . . . . . 18 ⊢ ((0 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝑁 ∈ ℝ) → ((0 ≤ 𝑦 ∧ 𝑦 < 𝑁) → 0 < 𝑁))
16		ltle 8114	. . . . . . . . . . . . . . . . . . 19 ⊢ ((0 ∈ ℝ ∧ 𝑁 ∈ ℝ) → (0 < 𝑁 → 0 ≤ 𝑁))
17	16	3adant2 1018	. . . . . . . . . . . . . . . . . 18 ⊢ ((0 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝑁 ∈ ℝ) → (0 < 𝑁 → 0 ≤ 𝑁))
18	15, 17	syld 45	. . . . . . . . . . . . . . . . 17 ⊢ ((0 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝑁 ∈ ℝ) → ((0 ≤ 𝑦 ∧ 𝑦 < 𝑁) → 0 ≤ 𝑁))
19	14, 18	mp3an1 1335	. . . . . . . . . . . . . . . 16 ⊢ ((𝑦 ∈ ℝ ∧ 𝑁 ∈ ℝ) → ((0 ≤ 𝑦 ∧ 𝑦 < 𝑁) → 0 ≤ 𝑁))
20	12, 13, 19	syl2an 289	. . . . . . . . . . . . . . 15 ⊢ ((𝑦 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((0 ≤ 𝑦 ∧ 𝑦 < 𝑁) → 0 ≤ 𝑁))
21	20	ex 115	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ ℤ → (𝑁 ∈ ℤ → ((0 ≤ 𝑦 ∧ 𝑦 < 𝑁) → 0 ≤ 𝑁)))
22	21	com23 78	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ ℤ → ((0 ≤ 𝑦 ∧ 𝑦 < 𝑁) → (𝑁 ∈ ℤ → 0 ≤ 𝑁)))
23	22	3impib 1203	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ ℤ ∧ 0 ≤ 𝑦 ∧ 𝑦 < 𝑁) → (𝑁 ∈ ℤ → 0 ≤ 𝑁))
24	23	impcom 125	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ ℤ ∧ (𝑦 ∈ ℤ ∧ 0 ≤ 𝑦 ∧ 𝑦 < 𝑁)) → 0 ≤ 𝑁)
25		elnn0z 9339	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 ∈ ℕ0 ↔ (𝑦 ∈ ℤ ∧ 0 ≤ 𝑦))
26	25	anbi1i 458	. . . . . . . . . . . . . . . 16 ⊢ ((𝑦 ∈ ℕ0 ∧ 𝑦 < 𝑁) ↔ ((𝑦 ∈ ℤ ∧ 0 ≤ 𝑦) ∧ 𝑦 < 𝑁))
27		fnn0ind.6	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑦 ∈ ℕ0 ∧ 𝑦 < 𝑁) → (𝜒 → 𝜃))
28	27	3expb 1206	. . . . . . . . . . . . . . . 16 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑦 ∈ ℕ0 ∧ 𝑦 < 𝑁)) → (𝜒 → 𝜃))
29	8, 26, 28	syl2anbr 292	. . . . . . . . . . . . . . 15 ⊢ (((𝑁 ∈ ℤ ∧ 0 ≤ 𝑁) ∧ ((𝑦 ∈ ℤ ∧ 0 ≤ 𝑦) ∧ 𝑦 < 𝑁)) → (𝜒 → 𝜃))
30	29	expcom 116	. . . . . . . . . . . . . 14 ⊢ (((𝑦 ∈ ℤ ∧ 0 ≤ 𝑦) ∧ 𝑦 < 𝑁) → ((𝑁 ∈ ℤ ∧ 0 ≤ 𝑁) → (𝜒 → 𝜃)))
31	30	3impa 1196	. . . . . . . . . . . . 13 ⊢ ((𝑦 ∈ ℤ ∧ 0 ≤ 𝑦 ∧ 𝑦 < 𝑁) → ((𝑁 ∈ ℤ ∧ 0 ≤ 𝑁) → (𝜒 → 𝜃)))
32	31	expd 258	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ ℤ ∧ 0 ≤ 𝑦 ∧ 𝑦 < 𝑁) → (𝑁 ∈ ℤ → (0 ≤ 𝑁 → (𝜒 → 𝜃))))
33	32	impcom 125	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ ℤ ∧ (𝑦 ∈ ℤ ∧ 0 ≤ 𝑦 ∧ 𝑦 < 𝑁)) → (0 ≤ 𝑁 → (𝜒 → 𝜃)))
34	24, 33	mpd 13	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ℤ ∧ (𝑦 ∈ ℤ ∧ 0 ≤ 𝑦 ∧ 𝑦 < 𝑁)) → (𝜒 → 𝜃))
35	34	adantll 476	. . . . . . . . 9 ⊢ (((0 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (𝑦 ∈ ℤ ∧ 0 ≤ 𝑦 ∧ 𝑦 < 𝑁)) → (𝜒 → 𝜃))
36	4, 5, 6, 7, 11, 35	fzind 9441	. . . . . . . 8 ⊢ (((0 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (𝐾 ∈ ℤ ∧ 0 ≤ 𝐾 ∧ 𝐾 ≤ 𝑁)) → 𝜏)
37	3, 36	mpanl1 434	. . . . . . 7 ⊢ ((𝑁 ∈ ℤ ∧ (𝐾 ∈ ℤ ∧ 0 ≤ 𝐾 ∧ 𝐾 ≤ 𝑁)) → 𝜏)
38	37	expcom 116	. . . . . 6 ⊢ ((𝐾 ∈ ℤ ∧ 0 ≤ 𝐾 ∧ 𝐾 ≤ 𝑁) → (𝑁 ∈ ℤ → 𝜏))
39	2, 38	syl5 32	. . . . 5 ⊢ ((𝐾 ∈ ℤ ∧ 0 ≤ 𝐾 ∧ 𝐾 ≤ 𝑁) → (𝑁 ∈ ℕ0 → 𝜏))
40	39	3expa 1205	. . . 4 ⊢ (((𝐾 ∈ ℤ ∧ 0 ≤ 𝐾) ∧ 𝐾 ≤ 𝑁) → (𝑁 ∈ ℕ0 → 𝜏))
41	1, 40	sylanb 284	. . 3 ⊢ ((𝐾 ∈ ℕ0 ∧ 𝐾 ≤ 𝑁) → (𝑁 ∈ ℕ0 → 𝜏))
42	41	impcom 125	. 2 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝐾 ∈ ℕ0 ∧ 𝐾 ≤ 𝑁)) → 𝜏)
43	42	3impb 1201	1 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐾 ∈ ℕ0 ∧ 𝐾 ≤ 𝑁) → 𝜏)

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   ∧ w3a 980   = wceq 1364   ∈ wcel 2167   class class class wbr 4033  (class class class)co 5922  ℝcr 7878  0cc0 7879  1c1 7880   + caddc 7882   < clt 8061   ≤ cle 8062  ℕ₀cn0 9249  ℤcz 9326

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-1cn 7972  ax-1re 7973  ax-icn 7974  ax-addcl 7975  ax-addrcl 7976  ax-mulcl 7977  ax-addcom 7979  ax-addass 7981  ax-distr 7983  ax-i2m1 7984  ax-0lt1 7985  ax-0id 7987  ax-rnegex 7988  ax-cnre 7990  ax-pre-ltirr 7991  ax-pre-ltwlin 7992  ax-pre-lttrn 7993  ax-pre-ltadd 7995

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-reu 2482  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-int 3875  df-br 4034  df-opab 4095  df-id 4328  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-iota 5219  df-fun 5260  df-fv 5266  df-riota 5877  df-ov 5925  df-oprab 5926  df-mpo 5927  df-pnf 8063  df-mnf 8064  df-xr 8065  df-ltxr 8066  df-le 8067  df-sub 8199  df-neg 8200  df-inn 8991  df-n0 9250  df-z 9327

This theorem is referenced by:  nn0seqcvgd  12209