Theorem peano5nni 12165

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 ∧ ∀𝑥 ∈ 𝐴 (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.

Step	Hyp	Ref	Expression
1		df-nn 12163	. . 3 ⊢ ℕ = (rec((𝑛 ∈ V ↦ (𝑛 + 1)), 1) “ ω)
2		df-ima 5644	. . 3 ⊢ (rec((𝑛 ∈ V ↦ (𝑛 + 1)), 1) “ ω) = ran (rec((𝑛 ∈ V ↦ (𝑛 + 1)), 1) ↾ ω)
3	1, 2	eqtri 2752	. 2 ⊢ ℕ = ran (rec((𝑛 ∈ V ↦ (𝑛 + 1)), 1) ↾ ω)
4		frfnom 8380	. . . . 5 ⊢ (rec((𝑛 ∈ V ↦ (𝑛 + 1)), 1) ↾ ω) Fn ω
5	4	a1i 11	. . . 4 ⊢ ((1 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑥 + 1) ∈ 𝐴) → (rec((𝑛 ∈ V ↦ (𝑛 + 1)), 1) ↾ ω) Fn ω)
6		fveq2 6840	. . . . . . . 8 ⊢ (𝑦 = ∅ → ((rec((𝑛 ∈ V ↦ (𝑛 + 1)), 1) ↾ ω)‘𝑦) = ((rec((𝑛 ∈ V ↦ (𝑛 + 1)), 1) ↾ ω)‘∅))
7	6	eleq1d 2813	. . . . . . 7 ⊢ (𝑦 = ∅ → (((rec((𝑛 ∈ V ↦ (𝑛 + 1)), 1) ↾ ω)‘𝑦) ∈ 𝐴 ↔ ((rec((𝑛 ∈ V ↦ (𝑛 + 1)), 1) ↾ ω)‘∅) ∈ 𝐴))
8		fveq2 6840	. . . . . . . 8 ⊢ (𝑦 = 𝑧 → ((rec((𝑛 ∈ V ↦ (𝑛 + 1)), 1) ↾ ω)‘𝑦) = ((rec((𝑛 ∈ V ↦ (𝑛 + 1)), 1) ↾ ω)‘𝑧))
9	8	eleq1d 2813	. . . . . . 7 ⊢ (𝑦 = 𝑧 → (((rec((𝑛 ∈ V ↦ (𝑛 + 1)), 1) ↾ ω)‘𝑦) ∈ 𝐴 ↔ ((rec((𝑛 ∈ V ↦ (𝑛 + 1)), 1) ↾ ω)‘𝑧) ∈ 𝐴))
10		fveq2 6840	. . . . . . . 8 ⊢ (𝑦 = suc 𝑧 → ((rec((𝑛 ∈ V ↦ (𝑛 + 1)), 1) ↾ ω)‘𝑦) = ((rec((𝑛 ∈ V ↦ (𝑛 + 1)), 1) ↾ ω)‘suc 𝑧))
11	10	eleq1d 2813	. . . . . . 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 11102	. . . . . . . . 9 ⊢ 1 ∈ ℂ
13		fr0g 8381	. . . . . . . . 9 ⊢ (1 ∈ ℂ → ((rec((𝑛 ∈ V ↦ (𝑛 + 1)), 1) ↾ ω)‘∅) = 1)
14	12, 13	ax-mp 5	. . . . . . . 8 ⊢ ((rec((𝑛 ∈ V ↦ (𝑛 + 1)), 1) ↾ ω)‘∅) = 1
15		simpl 482	. . . . . . . 8 ⊢ ((1 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑥 + 1) ∈ 𝐴) → 1 ∈ 𝐴)
16	14, 15	eqeltrid 2832	. . . . . . 7 ⊢ ((1 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑥 + 1) ∈ 𝐴) → ((rec((𝑛 ∈ V ↦ (𝑛 + 1)), 1) ↾ ω)‘∅) ∈ 𝐴)
17		oveq1 7376	. . . . . . . . . . . 12 ⊢ (𝑥 = ((rec((𝑛 ∈ V ↦ (𝑛 + 1)), 1) ↾ ω)‘𝑧) → (𝑥 + 1) = (((rec((𝑛 ∈ V ↦ (𝑛 + 1)), 1) ↾ ω)‘𝑧) + 1))
18	17	eleq1d 2813	. . . . . . . . . . 11 ⊢ (𝑥 = ((rec((𝑛 ∈ V ↦ (𝑛 + 1)), 1) ↾ ω)‘𝑧) → ((𝑥 + 1) ∈ 𝐴 ↔ (((rec((𝑛 ∈ V ↦ (𝑛 + 1)), 1) ↾ ω)‘𝑧) + 1) ∈ 𝐴))
19	18	rspccv 3582	. . . . . . . . . 10 ⊢ (∀𝑥 ∈ 𝐴 (𝑥 + 1) ∈ 𝐴 → (((rec((𝑛 ∈ V ↦ (𝑛 + 1)), 1) ↾ ω)‘𝑧) ∈ 𝐴 → (((rec((𝑛 ∈ V ↦ (𝑛 + 1)), 1) ↾ ω)‘𝑧) + 1) ∈ 𝐴))
20	19	ad2antlr 727	. . . . . . . . 9 ⊢ (((1 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 (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 7402	. . . . . . . . . . . 12 ⊢ (((rec((𝑛 ∈ V ↦ (𝑛 + 1)), 1) ↾ ω)‘𝑧) + 1) ∈ V
22		eqid 2729	. . . . . . . . . . . . 13 ⊢ (rec((𝑛 ∈ V ↦ (𝑛 + 1)), 1) ↾ ω) = (rec((𝑛 ∈ V ↦ (𝑛 + 1)), 1) ↾ ω)
23		oveq1 7376	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝑛 → (𝑦 + 1) = (𝑛 + 1))
24		oveq1 7376	. . . . . . . . . . . . 13 ⊢ (𝑦 = ((rec((𝑛 ∈ V ↦ (𝑛 + 1)), 1) ↾ ω)‘𝑧) → (𝑦 + 1) = (((rec((𝑛 ∈ V ↦ (𝑛 + 1)), 1) ↾ ω)‘𝑧) + 1))
25	22, 23, 24	frsucmpt2 8385	. . . . . . . . . . . 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))
26	21, 25	mpan2 691	. . . . . . . . . . 11 ⊢ (𝑧 ∈ ω → ((rec((𝑛 ∈ V ↦ (𝑛 + 1)), 1) ↾ ω)‘suc 𝑧) = (((rec((𝑛 ∈ V ↦ (𝑛 + 1)), 1) ↾ ω)‘𝑧) + 1))
27	26	eleq1d 2813	. . . . . . . . . 10 ⊢ (𝑧 ∈ ω → (((rec((𝑛 ∈ V ↦ (𝑛 + 1)), 1) ↾ ω)‘suc 𝑧) ∈ 𝐴 ↔ (((rec((𝑛 ∈ V ↦ (𝑛 + 1)), 1) ↾ ω)‘𝑧) + 1) ∈ 𝐴))
28	27	adantl 481	. . . . . . . . 9 ⊢ (((1 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 (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))
29	20, 28	sylibrd 259	. . . . . . . 8 ⊢ (((1 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑥 + 1) ∈ 𝐴) ∧ 𝑧 ∈ ω) → (((rec((𝑛 ∈ V ↦ (𝑛 + 1)), 1) ↾ ω)‘𝑧) ∈ 𝐴 → ((rec((𝑛 ∈ V ↦ (𝑛 + 1)), 1) ↾ ω)‘suc 𝑧) ∈ 𝐴))
30	29	expcom 413	. . . . . . 7 ⊢ (𝑧 ∈ ω → ((1 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑥 + 1) ∈ 𝐴) → (((rec((𝑛 ∈ V ↦ (𝑛 + 1)), 1) ↾ ω)‘𝑧) ∈ 𝐴 → ((rec((𝑛 ∈ V ↦ (𝑛 + 1)), 1) ↾ ω)‘suc 𝑧) ∈ 𝐴)))
31	7, 9, 11, 16, 30	finds2 7854	. . . . . 6 ⊢ (𝑦 ∈ ω → ((1 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑥 + 1) ∈ 𝐴) → ((rec((𝑛 ∈ V ↦ (𝑛 + 1)), 1) ↾ ω)‘𝑦) ∈ 𝐴))
32	31	com12 32	. . . . 5 ⊢ ((1 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑥 + 1) ∈ 𝐴) → (𝑦 ∈ ω → ((rec((𝑛 ∈ V ↦ (𝑛 + 1)), 1) ↾ ω)‘𝑦) ∈ 𝐴))
33	32	ralrimiv 3124	. . . 4 ⊢ ((1 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑥 + 1) ∈ 𝐴) → ∀𝑦 ∈ ω ((rec((𝑛 ∈ V ↦ (𝑛 + 1)), 1) ↾ ω)‘𝑦) ∈ 𝐴)
34		ffnfv 7073	. . . 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))
35	5, 33, 34	sylanbrc 583	. . 3 ⊢ ((1 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑥 + 1) ∈ 𝐴) → (rec((𝑛 ∈ V ↦ (𝑛 + 1)), 1) ↾ ω):ω⟶𝐴)
36	35	frnd 6678	. 2 ⊢ ((1 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑥 + 1) ∈ 𝐴) → ran (rec((𝑛 ∈ V ↦ (𝑛 + 1)), 1) ↾ ω) ⊆ 𝐴)
37	3, 36	eqsstrid 3982	1 ⊢ ((1 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑥 + 1) ∈ 𝐴) → ℕ ⊆ 𝐴)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109  ∀wral 3044  Vcvv 3444   ⊆ wss 3911  ∅c0 4292   ↦ cmpt 5183  ran crn 5632   ↾ cres 5633   “ cima 5634  suc csuc 6322   Fn wfn 6494  ⟶wf 6495  ‘cfv 6499  (class class class)co 7369  ωcom 7822  reccrdg 8354  ℂcc 11042  1c1 11045   + caddc 11047  ℕcn 12162

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5246  ax-nul 5256  ax-pr 5382  ax-un 7691  ax-1cn 11102

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-reu 3352  df-rab 3403  df-v 3446  df-sbc 3751  df-csb 3860  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3931  df-nul 4293  df-if 4485  df-pw 4561  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-iun 4953  df-br 5103  df-opab 5165  df-mpt 5184  df-tr 5210  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6262  df-ord 6323  df-on 6324  df-lim 6325  df-suc 6326  df-iota 6452  df-fun 6501  df-fn 6502  df-f 6503  df-f1 6504  df-fo 6505  df-f1o 6506  df-fv 6507  df-ov 7372  df-om 7823  df-2nd 7948  df-frecs 8237  df-wrecs 8268  df-recs 8317  df-rdg 8355  df-nn 12163

This theorem is referenced by:  nnssre  12166  nnsscn  12167  dfnn2  12175  nnind  12180  nnindf  32717