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Theorem nn0ind-raph 11515
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 11332 . 2 (𝐴 ∈ ℕ0 ↔ (𝐴 ∈ ℕ ∨ 𝐴 = 0))
2 dfsbcq2 3471 . . . 4 (𝑧 = 1 → ([𝑧 / 𝑥]𝜑[1 / 𝑥]𝜑))
3 nfv 1883 . . . . 5 𝑥𝜒
4 nn0ind-raph.2 . . . . 5 (𝑥 = 𝑦 → (𝜑𝜒))
53, 4sbhypf 3284 . . . 4 (𝑧 = 𝑦 → ([𝑧 / 𝑥]𝜑𝜒))
6 nfv 1883 . . . . 5 𝑥𝜃
7 nn0ind-raph.3 . . . . 5 (𝑥 = (𝑦 + 1) → (𝜑𝜃))
86, 7sbhypf 3284 . . . 4 (𝑧 = (𝑦 + 1) → ([𝑧 / 𝑥]𝜑𝜃))
9 nfv 1883 . . . . 5 𝑥𝜏
10 nn0ind-raph.4 . . . . 5 (𝑥 = 𝐴 → (𝜑𝜏))
119, 10sbhypf 3284 . . . 4 (𝑧 = 𝐴 → ([𝑧 / 𝑥]𝜑𝜏))
12 nfsbc1v 3488 . . . . 5 𝑥[1 / 𝑥]𝜑
13 1ex 10073 . . . . 5 1 ∈ V
14 c0ex 10072 . . . . . . 7 0 ∈ V
15 0nn0 11345 . . . . . . . . . . . 12 0 ∈ ℕ0
16 eleq1a 2725 . . . . . . . . . . . 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 248 . . . . . . . . . . . . . 14 (𝑥 = 0 → 𝜑)
21 eqeq2 2662 . . . . . . . . . . . . . . . 16 (𝑦 = 0 → (𝑥 = 𝑦𝑥 = 0))
2221, 4syl6bir 244 . . . . . . . . . . . . . . 15 (𝑦 = 0 → (𝑥 = 0 → (𝜑𝜒)))
2322pm5.74d 262 . . . . . . . . . . . . . 14 (𝑦 = 0 → ((𝑥 = 0 → 𝜑) ↔ (𝑥 = 0 → 𝜒)))
2420, 23mpbii 223 . . . . . . . . . . . . 13 (𝑦 = 0 → (𝑥 = 0 → 𝜒))
2524com12 32 . . . . . . . . . . . 12 (𝑥 = 0 → (𝑦 = 0 → 𝜒))
2614, 25vtocle 3313 . . . . . . . . . . 11 (𝑦 = 0 → 𝜒)
27 nn0ind-raph.6 . . . . . . . . . . 11 (𝑦 ∈ ℕ0 → (𝜒𝜃))
2817, 26, 27sylc 65 . . . . . . . . . 10 (𝑦 = 0 → 𝜃)
2928adantr 480 . . . . . . . . 9 ((𝑦 = 0 ∧ 𝑥 = 1) → 𝜃)
30 oveq1 6697 . . . . . . . . . . . . 13 (𝑦 = 0 → (𝑦 + 1) = (0 + 1))
31 0p1e1 11170 . . . . . . . . . . . . 13 (0 + 1) = 1
3230, 31syl6eq 2701 . . . . . . . . . . . 12 (𝑦 = 0 → (𝑦 + 1) = 1)
3332eqeq2d 2661 . . . . . . . . . . 11 (𝑦 = 0 → (𝑥 = (𝑦 + 1) ↔ 𝑥 = 1))
3433, 7syl6bir 244 . . . . . . . . . 10 (𝑦 = 0 → (𝑥 = 1 → (𝜑𝜃)))
3534imp 444 . . . . . . . . 9 ((𝑦 = 0 ∧ 𝑥 = 1) → (𝜑𝜃))
3629, 35mpbird 247 . . . . . . . 8 ((𝑦 = 0 ∧ 𝑥 = 1) → 𝜑)
3736ex 449 . . . . . . 7 (𝑦 = 0 → (𝑥 = 1 → 𝜑))
3814, 37vtocle 3313 . . . . . 6 (𝑥 = 1 → 𝜑)
39 sbceq1a 3479 . . . . . 6 (𝑥 = 1 → (𝜑[1 / 𝑥]𝜑))
4038, 39mpbid 222 . . . . 5 (𝑥 = 1 → [1 / 𝑥]𝜑)
4112, 13, 40vtoclef 3312 . . . 4 [1 / 𝑥]𝜑
42 nnnn0 11337 . . . . 5 (𝑦 ∈ ℕ → 𝑦 ∈ ℕ0)
4342, 27syl 17 . . . 4 (𝑦 ∈ ℕ → (𝜒𝜃))
442, 5, 8, 11, 41, 43nnind 11076 . . 3 (𝐴 ∈ ℕ → 𝜏)
45 nfv 1883 . . . . 5 𝑥(0 = 𝐴𝜏)
46 eqeq1 2655 . . . . . 6 (𝑥 = 0 → (𝑥 = 𝐴 ↔ 0 = 𝐴))
4719bicomd 213 . . . . . . . . 9 (𝑥 = 0 → (𝜓𝜑))
4847, 10sylan9bb 736 . . . . . . . 8 ((𝑥 = 0 ∧ 𝑥 = 𝐴) → (𝜓𝜏))
4918, 48mpbii 223 . . . . . . 7 ((𝑥 = 0 ∧ 𝑥 = 𝐴) → 𝜏)
5049ex 449 . . . . . 6 (𝑥 = 0 → (𝑥 = 𝐴𝜏))
5146, 50sylbird 250 . . . . 5 (𝑥 = 0 → (0 = 𝐴𝜏))
5245, 14, 51vtoclef 3312 . . . 4 (0 = 𝐴𝜏)
5352eqcoms 2659 . . 3 (𝐴 = 0 → 𝜏)
5444, 53jaoi 393 . 2 ((𝐴 ∈ ℕ ∨ 𝐴 = 0) → 𝜏)
551, 54sylbi 207 1 (𝐴 ∈ ℕ0𝜏)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wo 382  wa 383   = wceq 1523  [wsb 1937  wcel 2030  [wsbc 3468  (class class class)co 6690  0cc0 9974  1c1 9975   + caddc 9977  cn 11058  0cn0 11330
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-resscn 10031  ax-1cn 10032  ax-icn 10033  ax-addcl 10034  ax-addrcl 10035  ax-mulcl 10036  ax-mulrcl 10037  ax-mulcom 10038  ax-addass 10039  ax-mulass 10040  ax-distr 10041  ax-i2m1 10042  ax-1ne0 10043  ax-1rid 10044  ax-rnegex 10045  ax-rrecex 10046  ax-cnre 10047  ax-pre-lttri 10048  ax-pre-lttrn 10049  ax-pre-ltadd 10050
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-ov 6693  df-om 7108  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-er 7787  df-en 7998  df-dom 7999  df-sdom 8000  df-pnf 10114  df-mnf 10115  df-ltxr 10117  df-nn 11059  df-n0 11331
This theorem is referenced by: (None)
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