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Theorem peano5nni 11235
Description: Peano's inductive postulate. Theorem I.36 (principle of mathematical induction) of [Apostol] p. 34. (Contributed by NM, 10-Jan-1997.) (Revised by Mario Carneiro, 17-Nov-2014.)
Assertion
Ref Expression
peano5nni ((1 ∈ 𝐴 ∧ ∀𝑥𝐴 (𝑥 + 1) ∈ 𝐴) → ℕ ⊆ 𝐴)
Distinct variable group:   𝑥,𝐴

Proof of Theorem peano5nni
Dummy variables 𝑛 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-nn 11233 . . 3 ℕ = (rec((𝑛 ∈ V ↦ (𝑛 + 1)), 1) “ ω)
2 df-ima 5279 . . 3 (rec((𝑛 ∈ V ↦ (𝑛 + 1)), 1) “ ω) = ran (rec((𝑛 ∈ V ↦ (𝑛 + 1)), 1) ↾ ω)
31, 2eqtri 2782 . 2 ℕ = ran (rec((𝑛 ∈ V ↦ (𝑛 + 1)), 1) ↾ ω)
4 frfnom 7700 . . . . 5 (rec((𝑛 ∈ V ↦ (𝑛 + 1)), 1) ↾ ω) Fn ω
54a1i 11 . . . 4 ((1 ∈ 𝐴 ∧ ∀𝑥𝐴 (𝑥 + 1) ∈ 𝐴) → (rec((𝑛 ∈ V ↦ (𝑛 + 1)), 1) ↾ ω) Fn ω)
6 fveq2 6353 . . . . . . . 8 (𝑦 = ∅ → ((rec((𝑛 ∈ V ↦ (𝑛 + 1)), 1) ↾ ω)‘𝑦) = ((rec((𝑛 ∈ V ↦ (𝑛 + 1)), 1) ↾ ω)‘∅))
76eleq1d 2824 . . . . . . 7 (𝑦 = ∅ → (((rec((𝑛 ∈ V ↦ (𝑛 + 1)), 1) ↾ ω)‘𝑦) ∈ 𝐴 ↔ ((rec((𝑛 ∈ V ↦ (𝑛 + 1)), 1) ↾ ω)‘∅) ∈ 𝐴))
8 fveq2 6353 . . . . . . . 8 (𝑦 = 𝑧 → ((rec((𝑛 ∈ V ↦ (𝑛 + 1)), 1) ↾ ω)‘𝑦) = ((rec((𝑛 ∈ V ↦ (𝑛 + 1)), 1) ↾ ω)‘𝑧))
98eleq1d 2824 . . . . . . 7 (𝑦 = 𝑧 → (((rec((𝑛 ∈ V ↦ (𝑛 + 1)), 1) ↾ ω)‘𝑦) ∈ 𝐴 ↔ ((rec((𝑛 ∈ V ↦ (𝑛 + 1)), 1) ↾ ω)‘𝑧) ∈ 𝐴))
10 fveq2 6353 . . . . . . . 8 (𝑦 = suc 𝑧 → ((rec((𝑛 ∈ V ↦ (𝑛 + 1)), 1) ↾ ω)‘𝑦) = ((rec((𝑛 ∈ V ↦ (𝑛 + 1)), 1) ↾ ω)‘suc 𝑧))
1110eleq1d 2824 . . . . . . 7 (𝑦 = suc 𝑧 → (((rec((𝑛 ∈ V ↦ (𝑛 + 1)), 1) ↾ ω)‘𝑦) ∈ 𝐴 ↔ ((rec((𝑛 ∈ V ↦ (𝑛 + 1)), 1) ↾ ω)‘suc 𝑧) ∈ 𝐴))
12 ax-1cn 10206 . . . . . . . . 9 1 ∈ ℂ
13 fr0g 7701 . . . . . . . . 9 (1 ∈ ℂ → ((rec((𝑛 ∈ V ↦ (𝑛 + 1)), 1) ↾ ω)‘∅) = 1)
1412, 13ax-mp 5 . . . . . . . 8 ((rec((𝑛 ∈ V ↦ (𝑛 + 1)), 1) ↾ ω)‘∅) = 1
15 simpl 474 . . . . . . . 8 ((1 ∈ 𝐴 ∧ ∀𝑥𝐴 (𝑥 + 1) ∈ 𝐴) → 1 ∈ 𝐴)
1614, 15syl5eqel 2843 . . . . . . 7 ((1 ∈ 𝐴 ∧ ∀𝑥𝐴 (𝑥 + 1) ∈ 𝐴) → ((rec((𝑛 ∈ V ↦ (𝑛 + 1)), 1) ↾ ω)‘∅) ∈ 𝐴)
17 oveq1 6821 . . . . . . . . . . . 12 (𝑥 = ((rec((𝑛 ∈ V ↦ (𝑛 + 1)), 1) ↾ ω)‘𝑧) → (𝑥 + 1) = (((rec((𝑛 ∈ V ↦ (𝑛 + 1)), 1) ↾ ω)‘𝑧) + 1))
1817eleq1d 2824 . . . . . . . . . . 11 (𝑥 = ((rec((𝑛 ∈ V ↦ (𝑛 + 1)), 1) ↾ ω)‘𝑧) → ((𝑥 + 1) ∈ 𝐴 ↔ (((rec((𝑛 ∈ V ↦ (𝑛 + 1)), 1) ↾ ω)‘𝑧) + 1) ∈ 𝐴))
1918rspccv 3446 . . . . . . . . . 10 (∀𝑥𝐴 (𝑥 + 1) ∈ 𝐴 → (((rec((𝑛 ∈ V ↦ (𝑛 + 1)), 1) ↾ ω)‘𝑧) ∈ 𝐴 → (((rec((𝑛 ∈ V ↦ (𝑛 + 1)), 1) ↾ ω)‘𝑧) + 1) ∈ 𝐴))
2019ad2antlr 765 . . . . . . . . 9 (((1 ∈ 𝐴 ∧ ∀𝑥𝐴 (𝑥 + 1) ∈ 𝐴) ∧ 𝑧 ∈ ω) → (((rec((𝑛 ∈ V ↦ (𝑛 + 1)), 1) ↾ ω)‘𝑧) ∈ 𝐴 → (((rec((𝑛 ∈ V ↦ (𝑛 + 1)), 1) ↾ ω)‘𝑧) + 1) ∈ 𝐴))
21 ovex 6842 . . . . . . . . . . . 12 (((rec((𝑛 ∈ V ↦ (𝑛 + 1)), 1) ↾ ω)‘𝑧) + 1) ∈ V
22 eqid 2760 . . . . . . . . . . . . 13 (rec((𝑛 ∈ V ↦ (𝑛 + 1)), 1) ↾ ω) = (rec((𝑛 ∈ V ↦ (𝑛 + 1)), 1) ↾ ω)
23 oveq1 6821 . . . . . . . . . . . . 13 (𝑦 = 𝑛 → (𝑦 + 1) = (𝑛 + 1))
24 oveq1 6821 . . . . . . . . . . . . 13 (𝑦 = ((rec((𝑛 ∈ V ↦ (𝑛 + 1)), 1) ↾ ω)‘𝑧) → (𝑦 + 1) = (((rec((𝑛 ∈ V ↦ (𝑛 + 1)), 1) ↾ ω)‘𝑧) + 1))
2522, 23, 24frsucmpt2 7705 . . . . . . . . . . . 12 ((𝑧 ∈ ω ∧ (((rec((𝑛 ∈ V ↦ (𝑛 + 1)), 1) ↾ ω)‘𝑧) + 1) ∈ V) → ((rec((𝑛 ∈ V ↦ (𝑛 + 1)), 1) ↾ ω)‘suc 𝑧) = (((rec((𝑛 ∈ V ↦ (𝑛 + 1)), 1) ↾ ω)‘𝑧) + 1))
2621, 25mpan2 709 . . . . . . . . . . 11 (𝑧 ∈ ω → ((rec((𝑛 ∈ V ↦ (𝑛 + 1)), 1) ↾ ω)‘suc 𝑧) = (((rec((𝑛 ∈ V ↦ (𝑛 + 1)), 1) ↾ ω)‘𝑧) + 1))
2726eleq1d 2824 . . . . . . . . . 10 (𝑧 ∈ ω → (((rec((𝑛 ∈ V ↦ (𝑛 + 1)), 1) ↾ ω)‘suc 𝑧) ∈ 𝐴 ↔ (((rec((𝑛 ∈ V ↦ (𝑛 + 1)), 1) ↾ ω)‘𝑧) + 1) ∈ 𝐴))
2827adantl 473 . . . . . . . . 9 (((1 ∈ 𝐴 ∧ ∀𝑥𝐴 (𝑥 + 1) ∈ 𝐴) ∧ 𝑧 ∈ ω) → (((rec((𝑛 ∈ V ↦ (𝑛 + 1)), 1) ↾ ω)‘suc 𝑧) ∈ 𝐴 ↔ (((rec((𝑛 ∈ V ↦ (𝑛 + 1)), 1) ↾ ω)‘𝑧) + 1) ∈ 𝐴))
2920, 28sylibrd 249 . . . . . . . 8 (((1 ∈ 𝐴 ∧ ∀𝑥𝐴 (𝑥 + 1) ∈ 𝐴) ∧ 𝑧 ∈ ω) → (((rec((𝑛 ∈ V ↦ (𝑛 + 1)), 1) ↾ ω)‘𝑧) ∈ 𝐴 → ((rec((𝑛 ∈ V ↦ (𝑛 + 1)), 1) ↾ ω)‘suc 𝑧) ∈ 𝐴))
3029expcom 450 . . . . . . 7 (𝑧 ∈ ω → ((1 ∈ 𝐴 ∧ ∀𝑥𝐴 (𝑥 + 1) ∈ 𝐴) → (((rec((𝑛 ∈ V ↦ (𝑛 + 1)), 1) ↾ ω)‘𝑧) ∈ 𝐴 → ((rec((𝑛 ∈ V ↦ (𝑛 + 1)), 1) ↾ ω)‘suc 𝑧) ∈ 𝐴)))
317, 9, 11, 16, 30finds2 7260 . . . . . 6 (𝑦 ∈ ω → ((1 ∈ 𝐴 ∧ ∀𝑥𝐴 (𝑥 + 1) ∈ 𝐴) → ((rec((𝑛 ∈ V ↦ (𝑛 + 1)), 1) ↾ ω)‘𝑦) ∈ 𝐴))
3231com12 32 . . . . 5 ((1 ∈ 𝐴 ∧ ∀𝑥𝐴 (𝑥 + 1) ∈ 𝐴) → (𝑦 ∈ ω → ((rec((𝑛 ∈ V ↦ (𝑛 + 1)), 1) ↾ ω)‘𝑦) ∈ 𝐴))
3332ralrimiv 3103 . . . 4 ((1 ∈ 𝐴 ∧ ∀𝑥𝐴 (𝑥 + 1) ∈ 𝐴) → ∀𝑦 ∈ ω ((rec((𝑛 ∈ V ↦ (𝑛 + 1)), 1) ↾ ω)‘𝑦) ∈ 𝐴)
34 ffnfv 6552 . . . 4 ((rec((𝑛 ∈ V ↦ (𝑛 + 1)), 1) ↾ ω):ω⟶𝐴 ↔ ((rec((𝑛 ∈ V ↦ (𝑛 + 1)), 1) ↾ ω) Fn ω ∧ ∀𝑦 ∈ ω ((rec((𝑛 ∈ V ↦ (𝑛 + 1)), 1) ↾ ω)‘𝑦) ∈ 𝐴))
355, 33, 34sylanbrc 701 . . 3 ((1 ∈ 𝐴 ∧ ∀𝑥𝐴 (𝑥 + 1) ∈ 𝐴) → (rec((𝑛 ∈ V ↦ (𝑛 + 1)), 1) ↾ ω):ω⟶𝐴)
36 frn 6214 . . 3 ((rec((𝑛 ∈ V ↦ (𝑛 + 1)), 1) ↾ ω):ω⟶𝐴 → ran (rec((𝑛 ∈ V ↦ (𝑛 + 1)), 1) ↾ ω) ⊆ 𝐴)
3735, 36syl 17 . 2 ((1 ∈ 𝐴 ∧ ∀𝑥𝐴 (𝑥 + 1) ∈ 𝐴) → ran (rec((𝑛 ∈ V ↦ (𝑛 + 1)), 1) ↾ ω) ⊆ 𝐴)
383, 37syl5eqss 3790 1 ((1 ∈ 𝐴 ∧ ∀𝑥𝐴 (𝑥 + 1) ∈ 𝐴) → ℕ ⊆ 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1632  wcel 2139  wral 3050  Vcvv 3340  wss 3715  c0 4058  cmpt 4881  ran crn 5267  cres 5268  cima 5269  suc csuc 5886   Fn wfn 6044  wf 6045  cfv 6049  (class class class)co 6814  ωcom 7231  reccrdg 7675  cc 10146  1c1 10149   + caddc 10151  cn 11232
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7115  ax-1cn 10206
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-reu 3057  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-pss 3731  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-tp 4326  df-op 4328  df-uni 4589  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-tr 4905  df-id 5174  df-eprel 5179  df-po 5187  df-so 5188  df-fr 5225  df-we 5227  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-pred 5841  df-ord 5887  df-on 5888  df-lim 5889  df-suc 5890  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-ov 6817  df-om 7232  df-wrecs 7577  df-recs 7638  df-rdg 7676  df-nn 11233
This theorem is referenced by:  nnssre  11236  dfnn2  11245  nnind  11250  nnindf  29895
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