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Theorem fnn0ind 9274
 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 → (𝜑𝜓))
fnn0ind.2 (𝑥 = 𝑦 → (𝜑𝜒))
fnn0ind.3 (𝑥 = (𝑦 + 1) → (𝜑𝜃))
fnn0ind.4 (𝑥 = 𝐾 → (𝜑𝜏))
fnn0ind.5 (𝑁 ∈ ℕ0𝜓)
fnn0ind.6 ((𝑁 ∈ ℕ0𝑦 ∈ ℕ0𝑦 < 𝑁) → (𝜒𝜃))
Assertion
Ref Expression
fnn0ind ((𝑁 ∈ ℕ0𝐾 ∈ ℕ0𝐾𝑁) → 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
StepHypRef Expression
1 elnn0z 9174 . . . 4 (𝐾 ∈ ℕ0 ↔ (𝐾 ∈ ℤ ∧ 0 ≤ 𝐾))
2 nn0z 9181 . . . . . 6 (𝑁 ∈ ℕ0𝑁 ∈ ℤ)
3 0z 9172 . . . . . . . 8 0 ∈ ℤ
4 fnn0ind.1 . . . . . . . . 9 (𝑥 = 0 → (𝜑𝜓))
5 fnn0ind.2 . . . . . . . . 9 (𝑥 = 𝑦 → (𝜑𝜒))
6 fnn0ind.3 . . . . . . . . 9 (𝑥 = (𝑦 + 1) → (𝜑𝜃))
7 fnn0ind.4 . . . . . . . . 9 (𝑥 = 𝐾 → (𝜑𝜏))
8 elnn0z 9174 . . . . . . . . . . 11 (𝑁 ∈ ℕ0 ↔ (𝑁 ∈ ℤ ∧ 0 ≤ 𝑁))
9 fnn0ind.5 . . . . . . . . . . 11 (𝑁 ∈ ℕ0𝜓)
108, 9sylbir 134 . . . . . . . . . 10 ((𝑁 ∈ ℤ ∧ 0 ≤ 𝑁) → 𝜓)
11103adant1 1000 . . . . . . . . 9 ((0 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 0 ≤ 𝑁) → 𝜓)
12 zre 9165 . . . . . . . . . . . . . . . 16 (𝑦 ∈ ℤ → 𝑦 ∈ ℝ)
13 zre 9165 . . . . . . . . . . . . . . . 16 (𝑁 ∈ ℤ → 𝑁 ∈ ℝ)
14 0re 7872 . . . . . . . . . . . . . . . . 17 0 ∈ ℝ
15 lelttr 7959 . . . . . . . . . . . . . . . . . 18 ((0 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝑁 ∈ ℝ) → ((0 ≤ 𝑦𝑦 < 𝑁) → 0 < 𝑁))
16 ltle 7958 . . . . . . . . . . . . . . . . . . 19 ((0 ∈ ℝ ∧ 𝑁 ∈ ℝ) → (0 < 𝑁 → 0 ≤ 𝑁))
17163adant2 1001 . . . . . . . . . . . . . . . . . 18 ((0 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝑁 ∈ ℝ) → (0 < 𝑁 → 0 ≤ 𝑁))
1815, 17syld 45 . . . . . . . . . . . . . . . . 17 ((0 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝑁 ∈ ℝ) → ((0 ≤ 𝑦𝑦 < 𝑁) → 0 ≤ 𝑁))
1914, 18mp3an1 1306 . . . . . . . . . . . . . . . 16 ((𝑦 ∈ ℝ ∧ 𝑁 ∈ ℝ) → ((0 ≤ 𝑦𝑦 < 𝑁) → 0 ≤ 𝑁))
2012, 13, 19syl2an 287 . . . . . . . . . . . . . . 15 ((𝑦 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((0 ≤ 𝑦𝑦 < 𝑁) → 0 ≤ 𝑁))
2120ex 114 . . . . . . . . . . . . . 14 (𝑦 ∈ ℤ → (𝑁 ∈ ℤ → ((0 ≤ 𝑦𝑦 < 𝑁) → 0 ≤ 𝑁)))
2221com23 78 . . . . . . . . . . . . 13 (𝑦 ∈ ℤ → ((0 ≤ 𝑦𝑦 < 𝑁) → (𝑁 ∈ ℤ → 0 ≤ 𝑁)))
23223impib 1183 . . . . . . . . . . . 12 ((𝑦 ∈ ℤ ∧ 0 ≤ 𝑦𝑦 < 𝑁) → (𝑁 ∈ ℤ → 0 ≤ 𝑁))
2423impcom 124 . . . . . . . . . . 11 ((𝑁 ∈ ℤ ∧ (𝑦 ∈ ℤ ∧ 0 ≤ 𝑦𝑦 < 𝑁)) → 0 ≤ 𝑁)
25 elnn0z 9174 . . . . . . . . . . . . . . . . 17 (𝑦 ∈ ℕ0 ↔ (𝑦 ∈ ℤ ∧ 0 ≤ 𝑦))
2625anbi1i 454 . . . . . . . . . . . . . . . 16 ((𝑦 ∈ ℕ0𝑦 < 𝑁) ↔ ((𝑦 ∈ ℤ ∧ 0 ≤ 𝑦) ∧ 𝑦 < 𝑁))
27 fnn0ind.6 . . . . . . . . . . . . . . . . 17 ((𝑁 ∈ ℕ0𝑦 ∈ ℕ0𝑦 < 𝑁) → (𝜒𝜃))
28273expb 1186 . . . . . . . . . . . . . . . 16 ((𝑁 ∈ ℕ0 ∧ (𝑦 ∈ ℕ0𝑦 < 𝑁)) → (𝜒𝜃))
298, 26, 28syl2anbr 290 . . . . . . . . . . . . . . 15 (((𝑁 ∈ ℤ ∧ 0 ≤ 𝑁) ∧ ((𝑦 ∈ ℤ ∧ 0 ≤ 𝑦) ∧ 𝑦 < 𝑁)) → (𝜒𝜃))
3029expcom 115 . . . . . . . . . . . . . 14 (((𝑦 ∈ ℤ ∧ 0 ≤ 𝑦) ∧ 𝑦 < 𝑁) → ((𝑁 ∈ ℤ ∧ 0 ≤ 𝑁) → (𝜒𝜃)))
31303impa 1177 . . . . . . . . . . . . 13 ((𝑦 ∈ ℤ ∧ 0 ≤ 𝑦𝑦 < 𝑁) → ((𝑁 ∈ ℤ ∧ 0 ≤ 𝑁) → (𝜒𝜃)))
3231expd 256 . . . . . . . . . . . 12 ((𝑦 ∈ ℤ ∧ 0 ≤ 𝑦𝑦 < 𝑁) → (𝑁 ∈ ℤ → (0 ≤ 𝑁 → (𝜒𝜃))))
3332impcom 124 . . . . . . . . . . 11 ((𝑁 ∈ ℤ ∧ (𝑦 ∈ ℤ ∧ 0 ≤ 𝑦𝑦 < 𝑁)) → (0 ≤ 𝑁 → (𝜒𝜃)))
3424, 33mpd 13 . . . . . . . . . 10 ((𝑁 ∈ ℤ ∧ (𝑦 ∈ ℤ ∧ 0 ≤ 𝑦𝑦 < 𝑁)) → (𝜒𝜃))
3534adantll 468 . . . . . . . . 9 (((0 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (𝑦 ∈ ℤ ∧ 0 ≤ 𝑦𝑦 < 𝑁)) → (𝜒𝜃))
364, 5, 6, 7, 11, 35fzind 9273 . . . . . . . 8 (((0 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (𝐾 ∈ ℤ ∧ 0 ≤ 𝐾𝐾𝑁)) → 𝜏)
373, 36mpanl1 431 . . . . . . 7 ((𝑁 ∈ ℤ ∧ (𝐾 ∈ ℤ ∧ 0 ≤ 𝐾𝐾𝑁)) → 𝜏)
3837expcom 115 . . . . . 6 ((𝐾 ∈ ℤ ∧ 0 ≤ 𝐾𝐾𝑁) → (𝑁 ∈ ℤ → 𝜏))
392, 38syl5 32 . . . . 5 ((𝐾 ∈ ℤ ∧ 0 ≤ 𝐾𝐾𝑁) → (𝑁 ∈ ℕ0𝜏))
40393expa 1185 . . . 4 (((𝐾 ∈ ℤ ∧ 0 ≤ 𝐾) ∧ 𝐾𝑁) → (𝑁 ∈ ℕ0𝜏))
411, 40sylanb 282 . . 3 ((𝐾 ∈ ℕ0𝐾𝑁) → (𝑁 ∈ ℕ0𝜏))
4241impcom 124 . 2 ((𝑁 ∈ ℕ0 ∧ (𝐾 ∈ ℕ0𝐾𝑁)) → 𝜏)
43423impb 1181 1 ((𝑁 ∈ ℕ0𝐾 ∈ ℕ0𝐾𝑁) → 𝜏)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104   ∧ w3a 963   = wceq 1335   ∈ wcel 2128   class class class wbr 3965  (class class class)co 5821  ℝcr 7725  0cc0 7726  1c1 7727   + caddc 7729   < clt 7906   ≤ cle 7907  ℕ0cn0 9084  ℤcz 9161 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-sep 4082  ax-pow 4135  ax-pr 4169  ax-un 4393  ax-setind 4495  ax-cnex 7817  ax-resscn 7818  ax-1cn 7819  ax-1re 7820  ax-icn 7821  ax-addcl 7822  ax-addrcl 7823  ax-mulcl 7824  ax-addcom 7826  ax-addass 7828  ax-distr 7830  ax-i2m1 7831  ax-0lt1 7832  ax-0id 7834  ax-rnegex 7835  ax-cnre 7837  ax-pre-ltirr 7838  ax-pre-ltwlin 7839  ax-pre-lttrn 7840  ax-pre-ltadd 7842 This theorem depends on definitions:  df-bi 116  df-3or 964  df-3an 965  df-tru 1338  df-fal 1341  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ne 2328  df-nel 2423  df-ral 2440  df-rex 2441  df-reu 2442  df-rab 2444  df-v 2714  df-sbc 2938  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-uni 3773  df-int 3808  df-br 3966  df-opab 4026  df-id 4253  df-xp 4591  df-rel 4592  df-cnv 4593  df-co 4594  df-dm 4595  df-iota 5134  df-fun 5171  df-fv 5177  df-riota 5777  df-ov 5824  df-oprab 5825  df-mpo 5826  df-pnf 7908  df-mnf 7909  df-xr 7910  df-ltxr 7911  df-le 7912  df-sub 8042  df-neg 8043  df-inn 8828  df-n0 9085  df-z 9162 This theorem is referenced by:  nn0seqcvgd  11909
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