Theorem peano5uzs 28329

Description: Peano's inductive postulate for upper surreal integers. (Contributed by Scott Fenton, 25-Jul-2025.)

Hypotheses

Ref	Expression
peano5uzs.1	⊢ (𝜑 → 𝑁 ∈ ℤs)
peano5uzs.2	⊢ (𝜑 → 𝑁 ∈ 𝐴)
peano5uzs.3	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝑥 +s 1s ) ∈ 𝐴)

Assertion

Ref	Expression
peano5uzs	⊢ (𝜑 → {𝑘 ∈ ℤs ∣ 𝑁 ≤s 𝑘} ⊆ 𝐴)

Distinct variable groups:   𝑘,𝑁,𝑥   𝜑,𝑥,𝑘   𝑥,𝐴

Allowed substitution hint:   𝐴(𝑘)

Proof of Theorem peano5uzs

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

Step	Hyp	Ref	Expression
1		breq2 5097	. . . 4 ⊢ (𝑘 = 𝑛 → (𝑁 ≤s 𝑘 ↔ 𝑁 ≤s 𝑛))
2	1	elrab 3643	. . 3 ⊢ (𝑛 ∈ {𝑘 ∈ ℤs ∣ 𝑁 ≤s 𝑘} ↔ (𝑛 ∈ ℤs ∧ 𝑁 ≤s 𝑛))
3		zno 28307	. . . . . . 7 ⊢ (𝑛 ∈ ℤs → 𝑛 ∈ No )
4	3	adantr 480	. . . . . 6 ⊢ ((𝑛 ∈ ℤs ∧ 𝑁 ≤s 𝑛) → 𝑛 ∈ No )
5		peano5uzs.1	. . . . . . 7 ⊢ (𝜑 → 𝑁 ∈ ℤs)
6	5	znod 28308	. . . . . 6 ⊢ (𝜑 → 𝑁 ∈ No )
7		npcans 28016	. . . . . 6 ⊢ ((𝑛 ∈ No ∧ 𝑁 ∈ No ) → ((𝑛 -s 𝑁) +s 𝑁) = 𝑛)
8	4, 6, 7	syl2anr 597	. . . . 5 ⊢ ((𝜑 ∧ (𝑛 ∈ ℤs ∧ 𝑁 ≤s 𝑛)) → ((𝑛 -s 𝑁) +s 𝑁) = 𝑛)
9		simprl 770	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑛 ∈ ℤs ∧ 𝑁 ≤s 𝑛)) → 𝑛 ∈ ℤs)
10	5	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑛 ∈ ℤs ∧ 𝑁 ≤s 𝑛)) → 𝑁 ∈ ℤs)
11	9, 10	zsubscld 28321	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑛 ∈ ℤs ∧ 𝑁 ≤s 𝑛)) → (𝑛 -s 𝑁) ∈ ℤs)
12	3	adantl 481	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑛 ∈ ℤs) → 𝑛 ∈ No )
13	6	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑛 ∈ ℤs) → 𝑁 ∈ No )
14	12, 13	subsge0d 28040	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ ℤs) → ( 0s ≤s (𝑛 -s 𝑁) ↔ 𝑁 ≤s 𝑛))
15	14	biimpar 477	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑛 ∈ ℤs) ∧ 𝑁 ≤s 𝑛) → 0s ≤s (𝑛 -s 𝑁))
16	15	anasss 466	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑛 ∈ ℤs ∧ 𝑁 ≤s 𝑛)) → 0s ≤s (𝑛 -s 𝑁))
17	11, 16	jca 511	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑛 ∈ ℤs ∧ 𝑁 ≤s 𝑛)) → ((𝑛 -s 𝑁) ∈ ℤs ∧ 0s ≤s (𝑛 -s 𝑁)))
18		eln0zs 28325	. . . . . . . . 9 ⊢ ((𝑛 -s 𝑁) ∈ ℕ0s ↔ ((𝑛 -s 𝑁) ∈ ℤs ∧ 0s ≤s (𝑛 -s 𝑁)))
19	17, 18	sylibr 234	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑛 ∈ ℤs ∧ 𝑁 ≤s 𝑛)) → (𝑛 -s 𝑁) ∈ ℕ0s)
20	19	ex 412	. . . . . . 7 ⊢ (𝜑 → ((𝑛 ∈ ℤs ∧ 𝑁 ≤s 𝑛) → (𝑛 -s 𝑁) ∈ ℕ0s))
21		oveq1 7359	. . . . . . . . . . 11 ⊢ (𝑧 = 0s → (𝑧 +s 𝑁) = ( 0s +s 𝑁))
22	21	eleq1d 2818	. . . . . . . . . 10 ⊢ (𝑧 = 0s → ((𝑧 +s 𝑁) ∈ 𝐴 ↔ ( 0s +s 𝑁) ∈ 𝐴))
23	22	imbi2d 340	. . . . . . . . 9 ⊢ (𝑧 = 0s → ((𝜑 → (𝑧 +s 𝑁) ∈ 𝐴) ↔ (𝜑 → ( 0s +s 𝑁) ∈ 𝐴)))
24		oveq1 7359	. . . . . . . . . . 11 ⊢ (𝑧 = 𝑦 → (𝑧 +s 𝑁) = (𝑦 +s 𝑁))
25	24	eleq1d 2818	. . . . . . . . . 10 ⊢ (𝑧 = 𝑦 → ((𝑧 +s 𝑁) ∈ 𝐴 ↔ (𝑦 +s 𝑁) ∈ 𝐴))
26	25	imbi2d 340	. . . . . . . . 9 ⊢ (𝑧 = 𝑦 → ((𝜑 → (𝑧 +s 𝑁) ∈ 𝐴) ↔ (𝜑 → (𝑦 +s 𝑁) ∈ 𝐴)))
27		oveq1 7359	. . . . . . . . . . 11 ⊢ (𝑧 = (𝑦 +s 1s ) → (𝑧 +s 𝑁) = ((𝑦 +s 1s ) +s 𝑁))
28	27	eleq1d 2818	. . . . . . . . . 10 ⊢ (𝑧 = (𝑦 +s 1s ) → ((𝑧 +s 𝑁) ∈ 𝐴 ↔ ((𝑦 +s 1s ) +s 𝑁) ∈ 𝐴))
29	28	imbi2d 340	. . . . . . . . 9 ⊢ (𝑧 = (𝑦 +s 1s ) → ((𝜑 → (𝑧 +s 𝑁) ∈ 𝐴) ↔ (𝜑 → ((𝑦 +s 1s ) +s 𝑁) ∈ 𝐴)))
30		oveq1 7359	. . . . . . . . . . 11 ⊢ (𝑧 = (𝑛 -s 𝑁) → (𝑧 +s 𝑁) = ((𝑛 -s 𝑁) +s 𝑁))
31	30	eleq1d 2818	. . . . . . . . . 10 ⊢ (𝑧 = (𝑛 -s 𝑁) → ((𝑧 +s 𝑁) ∈ 𝐴 ↔ ((𝑛 -s 𝑁) +s 𝑁) ∈ 𝐴))
32	31	imbi2d 340	. . . . . . . . 9 ⊢ (𝑧 = (𝑛 -s 𝑁) → ((𝜑 → (𝑧 +s 𝑁) ∈ 𝐴) ↔ (𝜑 → ((𝑛 -s 𝑁) +s 𝑁) ∈ 𝐴)))
33		addslid 27912	. . . . . . . . . . 11 ⊢ (𝑁 ∈ No → ( 0s +s 𝑁) = 𝑁)
34	6, 33	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → ( 0s +s 𝑁) = 𝑁)
35		peano5uzs.2	. . . . . . . . . 10 ⊢ (𝜑 → 𝑁 ∈ 𝐴)
36	34, 35	eqeltrd 2833	. . . . . . . . 9 ⊢ (𝜑 → ( 0s +s 𝑁) ∈ 𝐴)
37		peano5uzs.3	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝑥 +s 1s ) ∈ 𝐴)
38	37	ralrimiva 3125	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 (𝑥 +s 1s ) ∈ 𝐴)
39		oveq1 7359	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = (𝑦 +s 𝑁) → (𝑥 +s 1s ) = ((𝑦 +s 𝑁) +s 1s ))
40	39	eleq1d 2818	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = (𝑦 +s 𝑁) → ((𝑥 +s 1s ) ∈ 𝐴 ↔ ((𝑦 +s 𝑁) +s 1s ) ∈ 𝐴))
41	40	rspccv 3570	. . . . . . . . . . . . . 14 ⊢ (∀𝑥 ∈ 𝐴 (𝑥 +s 1s ) ∈ 𝐴 → ((𝑦 +s 𝑁) ∈ 𝐴 → ((𝑦 +s 𝑁) +s 1s ) ∈ 𝐴))
42	38, 41	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((𝑦 +s 𝑁) ∈ 𝐴 → ((𝑦 +s 𝑁) +s 1s ) ∈ 𝐴))
43	42	adantl 481	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ ℕ0s ∧ 𝜑) → ((𝑦 +s 𝑁) ∈ 𝐴 → ((𝑦 +s 𝑁) +s 1s ) ∈ 𝐴))
44		n0sno 28253	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ∈ ℕ0s → 𝑦 ∈ No )
45	44	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((𝑦 ∈ ℕ0s ∧ 𝜑) → 𝑦 ∈ No )
46		1sno 27772	. . . . . . . . . . . . . . 15 ⊢ 1s ∈ No
47	46	a1i 11	. . . . . . . . . . . . . 14 ⊢ ((𝑦 ∈ ℕ0s ∧ 𝜑) → 1s ∈ No )
48	6	adantl 481	. . . . . . . . . . . . . 14 ⊢ ((𝑦 ∈ ℕ0s ∧ 𝜑) → 𝑁 ∈ No )
49	45, 47, 48	adds32d 27951	. . . . . . . . . . . . 13 ⊢ ((𝑦 ∈ ℕ0s ∧ 𝜑) → ((𝑦 +s 1s ) +s 𝑁) = ((𝑦 +s 𝑁) +s 1s ))
50	49	eleq1d 2818	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ ℕ0s ∧ 𝜑) → (((𝑦 +s 1s ) +s 𝑁) ∈ 𝐴 ↔ ((𝑦 +s 𝑁) +s 1s ) ∈ 𝐴))
51	43, 50	sylibrd 259	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ ℕ0s ∧ 𝜑) → ((𝑦 +s 𝑁) ∈ 𝐴 → ((𝑦 +s 1s ) +s 𝑁) ∈ 𝐴))
52	51	ex 412	. . . . . . . . . 10 ⊢ (𝑦 ∈ ℕ0s → (𝜑 → ((𝑦 +s 𝑁) ∈ 𝐴 → ((𝑦 +s 1s ) +s 𝑁) ∈ 𝐴)))
53	52	a2d 29	. . . . . . . . 9 ⊢ (𝑦 ∈ ℕ0s → ((𝜑 → (𝑦 +s 𝑁) ∈ 𝐴) → (𝜑 → ((𝑦 +s 1s ) +s 𝑁) ∈ 𝐴)))
54	23, 26, 29, 32, 36, 53	n0sind 28262	. . . . . . . 8 ⊢ ((𝑛 -s 𝑁) ∈ ℕ0s → (𝜑 → ((𝑛 -s 𝑁) +s 𝑁) ∈ 𝐴))
55	54	com12 32	. . . . . . 7 ⊢ (𝜑 → ((𝑛 -s 𝑁) ∈ ℕ0s → ((𝑛 -s 𝑁) +s 𝑁) ∈ 𝐴))
56	20, 55	syld 47	. . . . . 6 ⊢ (𝜑 → ((𝑛 ∈ ℤs ∧ 𝑁 ≤s 𝑛) → ((𝑛 -s 𝑁) +s 𝑁) ∈ 𝐴))
57	56	imp 406	. . . . 5 ⊢ ((𝜑 ∧ (𝑛 ∈ ℤs ∧ 𝑁 ≤s 𝑛)) → ((𝑛 -s 𝑁) +s 𝑁) ∈ 𝐴)
58	8, 57	eqeltrrd 2834	. . . 4 ⊢ ((𝜑 ∧ (𝑛 ∈ ℤs ∧ 𝑁 ≤s 𝑛)) → 𝑛 ∈ 𝐴)
59	58	ex 412	. . 3 ⊢ (𝜑 → ((𝑛 ∈ ℤs ∧ 𝑁 ≤s 𝑛) → 𝑛 ∈ 𝐴))
60	2, 59	biimtrid 242	. 2 ⊢ (𝜑 → (𝑛 ∈ {𝑘 ∈ ℤs ∣ 𝑁 ≤s 𝑘} → 𝑛 ∈ 𝐴))
61	60	ssrdv 3936	1 ⊢ (𝜑 → {𝑘 ∈ ℤs ∣ 𝑁 ≤s 𝑘} ⊆ 𝐴)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2113  ∀wral 3048  {crab 3396   ⊆ wss 3898   class class class wbr 5093  (class class class)co 7352   No csur 27579   ≤s csle 27684   0_s c0s 27767   1_s c1s 27768   +_s cadds 27903   -_s csubs 27963  ℕ_0scnn0s 28243  ℤ_sczs 28303

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 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705  ax-rep 5219  ax-sep 5236  ax-nul 5246  ax-pow 5305  ax-pr 5372  ax-un 7674

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2882  df-ne 2930  df-ral 3049  df-rex 3058  df-rmo 3347  df-reu 3348  df-rab 3397  df-v 3439  df-sbc 3738  df-csb 3847  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-pss 3918  df-nul 4283  df-if 4475  df-pw 4551  df-sn 4576  df-pr 4578  df-tp 4580  df-op 4582  df-ot 4584  df-uni 4859  df-int 4898  df-iun 4943  df-br 5094  df-opab 5156  df-mpt 5175  df-tr 5201  df-id 5514  df-eprel 5519  df-po 5527  df-so 5528  df-fr 5572  df-se 5573  df-we 5574  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-pred 6253  df-ord 6314  df-on 6315  df-lim 6316  df-suc 6317  df-iota 6442  df-fun 6488  df-fn 6489  df-f 6490  df-f1 6491  df-fo 6492  df-f1o 6493  df-fv 6494  df-riota 7309  df-ov 7355  df-oprab 7356  df-mpo 7357  df-om 7803  df-1st 7927  df-2nd 7928  df-frecs 8217  df-wrecs 8248  df-recs 8297  df-rdg 8335  df-1o 8391  df-2o 8392  df-nadd 8587  df-no 27582  df-slt 27583  df-bday 27584  df-sle 27685  df-sslt 27722  df-scut 27724  df-0s 27769  df-1s 27770  df-made 27789  df-old 27790  df-left 27792  df-right 27793  df-norec 27882  df-norec2 27893  df-adds 27904  df-negs 27964  df-subs 27965  df-n0s 28245  df-nns 28246  df-zs 28304

This theorem is referenced by:  uzsind  28330