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Theorem nn0ind-raph 9182
Description: Principle of Mathematical Induction (inference schema) on nonnegative integers. The first four hypotheses give us the substitution instances we need; the last two are the basis and the induction step. Raph Levien remarks: "This seems a bit painful. I wonder if an explicit substitution version would be easier." (Contributed by Raph Levien, 10-Apr-2004.)
Hypotheses
Ref Expression
nn0ind-raph.1 (𝑥 = 0 → (𝜑𝜓))
nn0ind-raph.2 (𝑥 = 𝑦 → (𝜑𝜒))
nn0ind-raph.3 (𝑥 = (𝑦 + 1) → (𝜑𝜃))
nn0ind-raph.4 (𝑥 = 𝐴 → (𝜑𝜏))
nn0ind-raph.5 𝜓
nn0ind-raph.6 (𝑦 ∈ ℕ0 → (𝜒𝜃))
Assertion
Ref Expression
nn0ind-raph (𝐴 ∈ ℕ0𝜏)
Distinct variable groups:   𝑥,𝑦   𝑥,𝐴   𝜓,𝑥   𝜒,𝑥   𝜃,𝑥   𝜏,𝑥   𝜑,𝑦
Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑦)   𝜒(𝑦)   𝜃(𝑦)   𝜏(𝑦)   𝐴(𝑦)

Proof of Theorem nn0ind-raph
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 elnn0 8993 . 2 (𝐴 ∈ ℕ0 ↔ (𝐴 ∈ ℕ ∨ 𝐴 = 0))
2 dfsbcq2 2912 . . . 4 (𝑧 = 1 → ([𝑧 / 𝑥]𝜑[1 / 𝑥]𝜑))
3 nfv 1508 . . . . 5 𝑥𝜒
4 nn0ind-raph.2 . . . . 5 (𝑥 = 𝑦 → (𝜑𝜒))
53, 4sbhypf 2735 . . . 4 (𝑧 = 𝑦 → ([𝑧 / 𝑥]𝜑𝜒))
6 nfv 1508 . . . . 5 𝑥𝜃
7 nn0ind-raph.3 . . . . 5 (𝑥 = (𝑦 + 1) → (𝜑𝜃))
86, 7sbhypf 2735 . . . 4 (𝑧 = (𝑦 + 1) → ([𝑧 / 𝑥]𝜑𝜃))
9 nfv 1508 . . . . 5 𝑥𝜏
10 nn0ind-raph.4 . . . . 5 (𝑥 = 𝐴 → (𝜑𝜏))
119, 10sbhypf 2735 . . . 4 (𝑧 = 𝐴 → ([𝑧 / 𝑥]𝜑𝜏))
12 nfsbc1v 2927 . . . . 5 𝑥[1 / 𝑥]𝜑
13 1ex 7775 . . . . 5 1 ∈ V
14 c0ex 7774 . . . . . . 7 0 ∈ V
15 0nn0 9006 . . . . . . . . . . . 12 0 ∈ ℕ0
16 eleq1a 2211 . . . . . . . . . . . 12 (0 ∈ ℕ0 → (𝑦 = 0 → 𝑦 ∈ ℕ0))
1715, 16ax-mp 5 . . . . . . . . . . 11 (𝑦 = 0 → 𝑦 ∈ ℕ0)
18 nn0ind-raph.5 . . . . . . . . . . . . . . 15 𝜓
19 nn0ind-raph.1 . . . . . . . . . . . . . . 15 (𝑥 = 0 → (𝜑𝜓))
2018, 19mpbiri 167 . . . . . . . . . . . . . 14 (𝑥 = 0 → 𝜑)
21 eqeq2 2149 . . . . . . . . . . . . . . . 16 (𝑦 = 0 → (𝑥 = 𝑦𝑥 = 0))
2221, 4syl6bir 163 . . . . . . . . . . . . . . 15 (𝑦 = 0 → (𝑥 = 0 → (𝜑𝜒)))
2322pm5.74d 181 . . . . . . . . . . . . . 14 (𝑦 = 0 → ((𝑥 = 0 → 𝜑) ↔ (𝑥 = 0 → 𝜒)))
2420, 23mpbii 147 . . . . . . . . . . . . 13 (𝑦 = 0 → (𝑥 = 0 → 𝜒))
2524com12 30 . . . . . . . . . . . 12 (𝑥 = 0 → (𝑦 = 0 → 𝜒))
2614, 25vtocle 2760 . . . . . . . . . . 11 (𝑦 = 0 → 𝜒)
27 nn0ind-raph.6 . . . . . . . . . . 11 (𝑦 ∈ ℕ0 → (𝜒𝜃))
2817, 26, 27sylc 62 . . . . . . . . . 10 (𝑦 = 0 → 𝜃)
2928adantr 274 . . . . . . . . 9 ((𝑦 = 0 ∧ 𝑥 = 1) → 𝜃)
30 oveq1 5781 . . . . . . . . . . . . 13 (𝑦 = 0 → (𝑦 + 1) = (0 + 1))
31 0p1e1 8848 . . . . . . . . . . . . 13 (0 + 1) = 1
3230, 31syl6eq 2188 . . . . . . . . . . . 12 (𝑦 = 0 → (𝑦 + 1) = 1)
3332eqeq2d 2151 . . . . . . . . . . 11 (𝑦 = 0 → (𝑥 = (𝑦 + 1) ↔ 𝑥 = 1))
3433, 7syl6bir 163 . . . . . . . . . 10 (𝑦 = 0 → (𝑥 = 1 → (𝜑𝜃)))
3534imp 123 . . . . . . . . 9 ((𝑦 = 0 ∧ 𝑥 = 1) → (𝜑𝜃))
3629, 35mpbird 166 . . . . . . . 8 ((𝑦 = 0 ∧ 𝑥 = 1) → 𝜑)
3736ex 114 . . . . . . 7 (𝑦 = 0 → (𝑥 = 1 → 𝜑))
3814, 37vtocle 2760 . . . . . 6 (𝑥 = 1 → 𝜑)
39 sbceq1a 2918 . . . . . 6 (𝑥 = 1 → (𝜑[1 / 𝑥]𝜑))
4038, 39mpbid 146 . . . . 5 (𝑥 = 1 → [1 / 𝑥]𝜑)
4112, 13, 40vtoclef 2759 . . . 4 [1 / 𝑥]𝜑
42 nnnn0 8998 . . . . 5 (𝑦 ∈ ℕ → 𝑦 ∈ ℕ0)
4342, 27syl 14 . . . 4 (𝑦 ∈ ℕ → (𝜒𝜃))
442, 5, 8, 11, 41, 43nnind 8750 . . 3 (𝐴 ∈ ℕ → 𝜏)
45 nfv 1508 . . . . 5 𝑥(0 = 𝐴𝜏)
46 eqeq1 2146 . . . . . 6 (𝑥 = 0 → (𝑥 = 𝐴 ↔ 0 = 𝐴))
4719bicomd 140 . . . . . . . . 9 (𝑥 = 0 → (𝜓𝜑))
4847, 10sylan9bb 457 . . . . . . . 8 ((𝑥 = 0 ∧ 𝑥 = 𝐴) → (𝜓𝜏))
4918, 48mpbii 147 . . . . . . 7 ((𝑥 = 0 ∧ 𝑥 = 𝐴) → 𝜏)
5049ex 114 . . . . . 6 (𝑥 = 0 → (𝑥 = 𝐴𝜏))
5146, 50sylbird 169 . . . . 5 (𝑥 = 0 → (0 = 𝐴𝜏))
5245, 14, 51vtoclef 2759 . . . 4 (0 = 𝐴𝜏)
5352eqcoms 2142 . . 3 (𝐴 = 0 → 𝜏)
5444, 53jaoi 705 . 2 ((𝐴 ∈ ℕ ∨ 𝐴 = 0) → 𝜏)
551, 54sylbi 120 1 (𝐴 ∈ ℕ0𝜏)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104  wo 697   = wceq 1331  wcel 1480  [wsb 1735  [wsbc 2909  (class class class)co 5774  0cc0 7634  1c1 7635   + caddc 7637  cn 8734  0cn0 8991
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-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-cnex 7725  ax-resscn 7726  ax-1cn 7727  ax-1re 7728  ax-icn 7729  ax-addcl 7730  ax-addrcl 7731  ax-mulcl 7732  ax-addcom 7734  ax-i2m1 7739  ax-0id 7742
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-rab 2425  df-v 2688  df-sbc 2910  df-un 3075  df-in 3077  df-ss 3084  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-int 3772  df-br 3930  df-iota 5088  df-fv 5131  df-ov 5777  df-inn 8735  df-n0 8992
This theorem is referenced by: (None)
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