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Theorem nn0ind-raph 12669
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 12481 . 2 (𝐴 ∈ ℕ0 ↔ (𝐴 ∈ ℕ ∨ 𝐴 = 0))
2 dfsbcq2 3780 . . . 4 (𝑧 = 1 → ([𝑧 / 𝑥]𝜑[1 / 𝑥]𝜑))
3 nfv 1916 . . . . 5 𝑥𝜒
4 nn0ind-raph.2 . . . . 5 (𝑥 = 𝑦 → (𝜑𝜒))
53, 4sbhypf 3538 . . . 4 (𝑧 = 𝑦 → ([𝑧 / 𝑥]𝜑𝜒))
6 nfv 1916 . . . . 5 𝑥𝜃
7 nn0ind-raph.3 . . . . 5 (𝑥 = (𝑦 + 1) → (𝜑𝜃))
86, 7sbhypf 3538 . . . 4 (𝑧 = (𝑦 + 1) → ([𝑧 / 𝑥]𝜑𝜃))
9 nfv 1916 . . . . 5 𝑥𝜏
10 nn0ind-raph.4 . . . . 5 (𝑥 = 𝐴 → (𝜑𝜏))
119, 10sbhypf 3538 . . . 4 (𝑧 = 𝐴 → ([𝑧 / 𝑥]𝜑𝜏))
12 nfsbc1v 3797 . . . . 5 𝑥[1 / 𝑥]𝜑
13 1ex 11217 . . . . 5 1 ∈ V
14 c0ex 11215 . . . . . . 7 0 ∈ V
15 0nn0 12494 . . . . . . . . . . . 12 0 ∈ ℕ0
16 eleq1a 2827 . . . . . . . . . . . 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 258 . . . . . . . . . . . . . 14 (𝑥 = 0 → 𝜑)
21 eqeq2 2743 . . . . . . . . . . . . . . . 16 (𝑦 = 0 → (𝑥 = 𝑦𝑥 = 0))
2221, 4syl6bir 254 . . . . . . . . . . . . . . 15 (𝑦 = 0 → (𝑥 = 0 → (𝜑𝜒)))
2322pm5.74d 273 . . . . . . . . . . . . . 14 (𝑦 = 0 → ((𝑥 = 0 → 𝜑) ↔ (𝑥 = 0 → 𝜒)))
2420, 23mpbii 232 . . . . . . . . . . . . 13 (𝑦 = 0 → (𝑥 = 0 → 𝜒))
2524com12 32 . . . . . . . . . . . 12 (𝑥 = 0 → (𝑦 = 0 → 𝜒))
2614, 25vtocle 3543 . . . . . . . . . . 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 7419 . . . . . . . . . . . . 13 (𝑦 = 0 → (𝑦 + 1) = (0 + 1))
31 0p1e1 12341 . . . . . . . . . . . . 13 (0 + 1) = 1
3230, 31eqtrdi 2787 . . . . . . . . . . . 12 (𝑦 = 0 → (𝑦 + 1) = 1)
3332eqeq2d 2742 . . . . . . . . . . 11 (𝑦 = 0 → (𝑥 = (𝑦 + 1) ↔ 𝑥 = 1))
3433, 7syl6bir 254 . . . . . . . . . 10 (𝑦 = 0 → (𝑥 = 1 → (𝜑𝜃)))
3534imp 406 . . . . . . . . 9 ((𝑦 = 0 ∧ 𝑥 = 1) → (𝜑𝜃))
3629, 35mpbird 257 . . . . . . . 8 ((𝑦 = 0 ∧ 𝑥 = 1) → 𝜑)
3736ex 412 . . . . . . 7 (𝑦 = 0 → (𝑥 = 1 → 𝜑))
3814, 37vtocle 3543 . . . . . 6 (𝑥 = 1 → 𝜑)
39 sbceq1a 3788 . . . . . 6 (𝑥 = 1 → (𝜑[1 / 𝑥]𝜑))
4038, 39mpbid 231 . . . . 5 (𝑥 = 1 → [1 / 𝑥]𝜑)
4112, 13, 40vtoclef 3550 . . . 4 [1 / 𝑥]𝜑
42 nnnn0 12486 . . . . 5 (𝑦 ∈ ℕ → 𝑦 ∈ ℕ0)
4342, 27syl 17 . . . 4 (𝑦 ∈ ℕ → (𝜒𝜃))
442, 5, 8, 11, 41, 43nnind 12237 . . 3 (𝐴 ∈ ℕ → 𝜏)
45 nfv 1916 . . . . 5 𝑥(0 = 𝐴𝜏)
46 eqeq1 2735 . . . . . 6 (𝑥 = 0 → (𝑥 = 𝐴 ↔ 0 = 𝐴))
4719bicomd 222 . . . . . . . . 9 (𝑥 = 0 → (𝜓𝜑))
4847, 10sylan9bb 509 . . . . . . . 8 ((𝑥 = 0 ∧ 𝑥 = 𝐴) → (𝜓𝜏))
4918, 48mpbii 232 . . . . . . 7 ((𝑥 = 0 ∧ 𝑥 = 𝐴) → 𝜏)
5049ex 412 . . . . . 6 (𝑥 = 0 → (𝑥 = 𝐴𝜏))
5146, 50sylbird 260 . . . . 5 (𝑥 = 0 → (0 = 𝐴𝜏))
5245, 14, 51vtoclef 3550 . . . 4 (0 = 𝐴𝜏)
5352eqcoms 2739 . . 3 (𝐴 = 0 → 𝜏)
5444, 53jaoi 854 . 2 ((𝐴 ∈ ℕ ∨ 𝐴 = 0) → 𝜏)
551, 54sylbi 216 1 (𝐴 ∈ ℕ0𝜏)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 395  wo 844   = wceq 1540  [wsb 2066  wcel 2105  [wsbc 3777  (class class class)co 7412  0cc0 11116  1c1 11117   + caddc 11119  cn 12219  0cn0 12479
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7729  ax-resscn 11173  ax-1cn 11174  ax-icn 11175  ax-addcl 11176  ax-addrcl 11177  ax-mulcl 11178  ax-mulrcl 11179  ax-mulcom 11180  ax-addass 11181  ax-mulass 11182  ax-distr 11183  ax-i2m1 11184  ax-1ne0 11185  ax-1rid 11186  ax-rnegex 11187  ax-rrecex 11188  ax-cnre 11189  ax-pre-lttri 11190  ax-pre-lttrn 11191  ax-pre-ltadd 11192
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-ov 7415  df-om 7860  df-2nd 7980  df-frecs 8272  df-wrecs 8303  df-recs 8377  df-rdg 8416  df-er 8709  df-en 8946  df-dom 8947  df-sdom 8948  df-pnf 11257  df-mnf 11258  df-ltxr 11260  df-nn 12220  df-n0 12480
This theorem is referenced by: (None)
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