Theorem fnn0ind 9741

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 9636	. . . 4 ⊢ (𝐾 ∈ ℕ0 ↔ (𝐾 ∈ ℤ ∧ 0 ≤ 𝐾))
2		nn0z 9643	. . . . . 6 ⊢ (𝑁 ∈ ℕ0 → 𝑁 ∈ ℤ)
3		0z 9634	. . . . . . . 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 9636	. . . . . . . . . . 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 1046	. . . . . . . . 9 ⊢ ((0 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 0 ≤ 𝑁) → 𝜓)
12		zre 9627	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 ∈ ℤ → 𝑦 ∈ ℝ)
13		zre 9627	. . . . . . . . . . . . . . . 16 ⊢ (𝑁 ∈ ℤ → 𝑁 ∈ ℝ)
14		0re 8316	. . . . . . . . . . . . . . . . 17 ⊢ 0 ∈ ℝ
15		lelttr 8404	. . . . . . . . . . . . . . . . . 18 ⊢ ((0 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝑁 ∈ ℝ) → ((0 ≤ 𝑦 ∧ 𝑦 < 𝑁) → 0 < 𝑁))
16		ltle 8403	. . . . . . . . . . . . . . . . . . 19 ⊢ ((0 ∈ ℝ ∧ 𝑁 ∈ ℝ) → (0 < 𝑁 → 0 ≤ 𝑁))
17	16	3adant2 1047	. . . . . . . . . . . . . . . . . 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 1365	. . . . . . . . . . . . . . . 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 1232	. . . . . . . . . . . 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 9636	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 ∈ ℕ0 ↔ (𝑦 ∈ ℤ ∧ 0 ≤ 𝑦))
26	25	anbi1i 462	. . . . . . . . . . . . . . . 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 1235	. . . . . . . . . . . . . . . 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 1225	. . . . . . . . . . . . 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 480	. . . . . . . . 9 ⊢ (((0 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (𝑦 ∈ ℤ ∧ 0 ≤ 𝑦 ∧ 𝑦 < 𝑁)) → (𝜒 → 𝜃))
36	4, 5, 6, 7, 11, 35	fzind 9740	. . . . . . . 8 ⊢ (((0 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (𝐾 ∈ ℤ ∧ 0 ≤ 𝐾 ∧ 𝐾 ≤ 𝑁)) → 𝜏)
37	3, 36	mpanl1 438	. . . . . . 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 1234	. . . 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 1230	1 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐾 ∈ ℕ0 ∧ 𝐾 ≤ 𝑁) → 𝜏)

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   ∧ w3a 1009   = wceq 1402   ∈ wcel 2209   class class class wbr 4125  (class class class)co 6075  ℝcr 8168  0cc0 8169  1c1 8170   + caddc 8172   < clt 8350   ≤ cle 8351  ℕ₀cn0 9542  ℤcz 9623

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-setind 4679  ax-cnex 8260  ax-resscn 8261  ax-1cn 8262  ax-1re 8263  ax-icn 8264  ax-addcl 8265  ax-addrcl 8266  ax-mulcl 8267  ax-addcom 8269  ax-addass 8271  ax-distr 8273  ax-i2m1 8274  ax-0lt1 8275  ax-0id 8277  ax-rnegex 8278  ax-cnre 8280  ax-pre-ltirr 8281  ax-pre-ltwlin 8282  ax-pre-lttrn 8283  ax-pre-ltadd 8285

This theorem depends on definitions:  df-bi 117  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-int 3966  df-br 4126  df-opab 4188  df-id 4433  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-iota 5332  df-fun 5374  df-fv 5380  df-riota 6028  df-ov 6078  df-oprab 6079  df-mpo 6080  df-pnf 8352  df-mnf 8353  df-xr 8354  df-ltxr 8355  df-le 8356  df-sub 8489  df-neg 8490  df-inn 9284  df-n0 9543  df-z 9624

This theorem is referenced by:  nn0seqcvgd  12797