Theorem peano2n0s 27940

Description: Peano postulate: the successor of a non-negative surreal integer is a non-negative surreal integer. (Contributed by Scott Fenton, 17-Mar-2025.)

Assertion

Ref	Expression
peano2n0s	⊢ (𝐴 ∈ ℕ0s → (𝐴 +s 1s ) ∈ ℕ0s)

Proof of Theorem peano2n0s

Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		frfnom 8437	. . . 4 ⊢ (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 0s ) ↾ ω) Fn ω
2		fvelrnb 6951	. . . 4 ⊢ ((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 0s ) ↾ ω) Fn ω → (𝐴 ∈ ran (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 0s ) ↾ ω) ↔ ∃𝑦 ∈ ω ((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 0s ) ↾ ω)‘𝑦) = 𝐴))
3	1, 2	ax-mp 5	. . 3 ⊢ (𝐴 ∈ ran (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 0s ) ↾ ω) ↔ ∃𝑦 ∈ ω ((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 0s ) ↾ ω)‘𝑦) = 𝐴)
4		ovex 7444	. . . . . . 7 ⊢ (((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 0s ) ↾ ω)‘𝑦) +s 1s ) ∈ V
5		eqid 2730	. . . . . . . 8 ⊢ (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 0s ) ↾ ω) = (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 0s ) ↾ ω)
6		oveq1 7418	. . . . . . . 8 ⊢ (𝑧 = 𝑥 → (𝑧 +s 1s ) = (𝑥 +s 1s ))
7		oveq1 7418	. . . . . . . 8 ⊢ (𝑧 = ((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 0s ) ↾ ω)‘𝑦) → (𝑧 +s 1s ) = (((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 0s ) ↾ ω)‘𝑦) +s 1s ))
8	5, 6, 7	frsucmpt2 8442	. . . . . . 7 ⊢ ((𝑦 ∈ ω ∧ (((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 0s ) ↾ ω)‘𝑦) +s 1s ) ∈ V) → ((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 0s ) ↾ ω)‘suc 𝑦) = (((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 0s ) ↾ ω)‘𝑦) +s 1s ))
9	4, 8	mpan2 687	. . . . . 6 ⊢ (𝑦 ∈ ω → ((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 0s ) ↾ ω)‘suc 𝑦) = (((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 0s ) ↾ ω)‘𝑦) +s 1s ))
10		peano2 7883	. . . . . . . 8 ⊢ (𝑦 ∈ ω → suc 𝑦 ∈ ω)
11		fnfvelrn 7081	. . . . . . . 8 ⊢ (((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 0s ) ↾ ω) Fn ω ∧ suc 𝑦 ∈ ω) → ((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 0s ) ↾ ω)‘suc 𝑦) ∈ ran (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 0s ) ↾ ω))
12	1, 10, 11	sylancr 585	. . . . . . 7 ⊢ (𝑦 ∈ ω → ((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 0s ) ↾ ω)‘suc 𝑦) ∈ ran (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 0s ) ↾ ω))
13		df-n0s 27931	. . . . . . . 8 ⊢ ℕ0s = (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 0s ) “ ω)
14		df-ima 5688	. . . . . . . 8 ⊢ (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 0s ) “ ω) = ran (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 0s ) ↾ ω)
15	13, 14	eqtri 2758	. . . . . . 7 ⊢ ℕ0s = ran (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 0s ) ↾ ω)
16	12, 15	eleqtrrdi 2842	. . . . . 6 ⊢ (𝑦 ∈ ω → ((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 0s ) ↾ ω)‘suc 𝑦) ∈ ℕ0s)
17	9, 16	eqeltrrd 2832	. . . . 5 ⊢ (𝑦 ∈ ω → (((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 0s ) ↾ ω)‘𝑦) +s 1s ) ∈ ℕ0s)
18		oveq1 7418	. . . . . 6 ⊢ (((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 0s ) ↾ ω)‘𝑦) = 𝐴 → (((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 0s ) ↾ ω)‘𝑦) +s 1s ) = (𝐴 +s 1s ))
19	18	eleq1d 2816	. . . . 5 ⊢ (((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 0s ) ↾ ω)‘𝑦) = 𝐴 → ((((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 0s ) ↾ ω)‘𝑦) +s 1s ) ∈ ℕ0s ↔ (𝐴 +s 1s ) ∈ ℕ0s))
20	17, 19	syl5ibcom 244	. . . 4 ⊢ (𝑦 ∈ ω → (((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 0s ) ↾ ω)‘𝑦) = 𝐴 → (𝐴 +s 1s ) ∈ ℕ0s))
21	20	rexlimiv 3146	. . 3 ⊢ (∃𝑦 ∈ ω ((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 0s ) ↾ ω)‘𝑦) = 𝐴 → (𝐴 +s 1s ) ∈ ℕ0s)
22	3, 21	sylbi 216	. 2 ⊢ (𝐴 ∈ ran (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 0s ) ↾ ω) → (𝐴 +s 1s ) ∈ ℕ0s)
23	22, 15	eleq2s 2849	1 ⊢ (𝐴 ∈ ℕ0s → (𝐴 +s 1s ) ∈ ℕ0s)

Syntax hints:   → wi 4   ↔ wb 205   = wceq 1539   ∈ wcel 2104  ∃wrex 3068  Vcvv 3472   ↦ cmpt 5230  ran crn 5676   ↾ cres 5677   “ cima 5678  suc csuc 6365   Fn wfn 6537  ‘cfv 6542  (class class class)co 7411  ωcom 7857  reccrdg 8411   0_s c0s 27560   1_s c1s 27561   +_s cadds 27681  ℕ_0scnn0s 27929

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 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-sep 5298  ax-nul 5305  ax-pr 5426  ax-un 7727

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3375  df-rab 3431  df-v 3474  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-pred 6299  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-ov 7414  df-om 7858  df-2nd 7978  df-frecs 8268  df-wrecs 8299  df-recs 8373  df-rdg 8412  df-n0s 27931

This theorem is referenced by:  dfn0s2  27941  n0sind  27942