Theorem peano5uzs 28796

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 5106	. . . 4 ⊢ (𝑘 = 𝑛 → (𝑁 ≤s 𝑘 ↔ 𝑁 ≤s 𝑛))
2	1	elrab 3644	. . 3 ⊢ (𝑛 ∈ {𝑘 ∈ ℤs ∣ 𝑁 ≤s 𝑘} ↔ (𝑛 ∈ ℤs ∧ 𝑁 ≤s 𝑛))
3		zno 28774	. . . . . . 7 ⊢ (𝑛 ∈ ℤs → 𝑛 ∈ No)
4	3	adantr 486	. . . . . 6 ⊢ ((𝑛 ∈ ℤs ∧ 𝑁 ≤s 𝑛) → 𝑛 ∈ No)
5		peano5uzs.1	. . . . . . 7 ⊢ (𝜑 → 𝑁 ∈ ℤs)
6	5	znod 28775	. . . . . 6 ⊢ (𝜑 → 𝑁 ∈ No)
7		npcans 28467	. . . . . 6 ⊢ ((𝑛 ∈ No ∧ 𝑁 ∈ No) → ((𝑛 −s 𝑁) +s 𝑁) = 𝑛)
8	4, 6, 7	syl2anr 609	. . . . 5 ⊢ ((𝜑 ∧ (𝑛 ∈ ℤs ∧ 𝑁 ≤s 𝑛)) → ((𝑛 −s 𝑁) +s 𝑁) = 𝑛)
9		simprl 783	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑛 ∈ ℤs ∧ 𝑁 ≤s 𝑛)) → 𝑛 ∈ ℤs)
10	5	adantr 486	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑛 ∈ ℤs ∧ 𝑁 ≤s 𝑛)) → 𝑁 ∈ ℤs)
11	9, 10	zsubscld 28788	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑛 ∈ ℤs ∧ 𝑁 ≤s 𝑛)) → (𝑛 −s 𝑁) ∈ ℤs)
12	3	adantl 487	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑛 ∈ ℤs) → 𝑛 ∈ No)
13	6	adantr 486	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑛 ∈ ℤs) → 𝑁 ∈ No)
14	12, 13	subsge0d 28492	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ ℤs) → ( 0s ≤s (𝑛 −s 𝑁) ↔ 𝑁 ≤s 𝑛))
15	14	biimpar 483	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑛 ∈ ℤs) ∧ 𝑁 ≤s 𝑛) → 0s ≤s (𝑛 −s 𝑁))
16	15	anasss 472	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑛 ∈ ℤs ∧ 𝑁 ≤s 𝑛)) → 0s ≤s (𝑛 −s 𝑁))
17	11, 16	jca 521	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑛 ∈ ℤs ∧ 𝑁 ≤s 𝑛)) → ((𝑛 −s 𝑁) ∈ ℤs ∧ 0s ≤s (𝑛 −s 𝑁)))
18		eln0zs 28792	. . . . . . . . 9 ⊢ ((𝑛 −s 𝑁) ∈ ℕ0s ↔ ((𝑛 −s 𝑁) ∈ ℤs ∧ 0s ≤s (𝑛 −s 𝑁)))
19	17, 18	sylibr 237	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑛 ∈ ℤs ∧ 𝑁 ≤s 𝑛)) → (𝑛 −s 𝑁) ∈ ℕ0s)
20	19	ex 418	. . . . . . 7 ⊢ (𝜑 → ((𝑛 ∈ ℤs ∧ 𝑁 ≤s 𝑛) → (𝑛 −s 𝑁) ∈ ℕ0s))
21		oveq1 7428	. . . . . . . . . . 11 ⊢ (𝑧 = 0s → (𝑧 +s 𝑁) = ( 0s +s 𝑁))
22	21	eleq1d 2845	. . . . . . . . . 10 ⊢ (𝑧 = 0s → ((𝑧 +s 𝑁) ∈ 𝐴 ↔ ( 0s +s 𝑁) ∈ 𝐴))
23	22	imbi2d 343	. . . . . . . . 9 ⊢ (𝑧 = 0s → ((𝜑 → (𝑧 +s 𝑁) ∈ 𝐴) ↔ (𝜑 → ( 0s +s 𝑁) ∈ 𝐴)))
24		oveq1 7428	. . . . . . . . . . 11 ⊢ (𝑧 = 𝑦 → (𝑧 +s 𝑁) = (𝑦 +s 𝑁))
25	24	eleq1d 2845	. . . . . . . . . 10 ⊢ (𝑧 = 𝑦 → ((𝑧 +s 𝑁) ∈ 𝐴 ↔ (𝑦 +s 𝑁) ∈ 𝐴))
26	25	imbi2d 343	. . . . . . . . 9 ⊢ (𝑧 = 𝑦 → ((𝜑 → (𝑧 +s 𝑁) ∈ 𝐴) ↔ (𝜑 → (𝑦 +s 𝑁) ∈ 𝐴)))
27		oveq1 7428	. . . . . . . . . . 11 ⊢ (𝑧 = (𝑦 +s 1s ) → (𝑧 +s 𝑁) = ((𝑦 +s 1s ) +s 𝑁))
28	27	eleq1d 2845	. . . . . . . . . 10 ⊢ (𝑧 = (𝑦 +s 1s ) → ((𝑧 +s 𝑁) ∈ 𝐴 ↔ ((𝑦 +s 1s ) +s 𝑁) ∈ 𝐴))
29	28	imbi2d 343	. . . . . . . . 9 ⊢ (𝑧 = (𝑦 +s 1s ) → ((𝜑 → (𝑧 +s 𝑁) ∈ 𝐴) ↔ (𝜑 → ((𝑦 +s 1s ) +s 𝑁) ∈ 𝐴)))
30		oveq1 7428	. . . . . . . . . . 11 ⊢ (𝑧 = (𝑛 −s 𝑁) → (𝑧 +s 𝑁) = ((𝑛 −s 𝑁) +s 𝑁))
31	30	eleq1d 2845	. . . . . . . . . 10 ⊢ (𝑧 = (𝑛 −s 𝑁) → ((𝑧 +s 𝑁) ∈ 𝐴 ↔ ((𝑛 −s 𝑁) +s 𝑁) ∈ 𝐴))
32	31	imbi2d 343	. . . . . . . . 9 ⊢ (𝑧 = (𝑛 −s 𝑁) → ((𝜑 → (𝑧 +s 𝑁) ∈ 𝐴) ↔ (𝜑 → ((𝑛 −s 𝑁) +s 𝑁) ∈ 𝐴)))
33		addslid 28360	. . . . . . . . . . 11 ⊢ (𝑁 ∈ No → ( 0s +s 𝑁) = 𝑁)
34	6, 33	syl 18	. . . . . . . . . 10 ⊢ (𝜑 → ( 0s +s 𝑁) = 𝑁)
35		peano5uzs.2	. . . . . . . . . 10 ⊢ (𝜑 → 𝑁 ∈ 𝐴)
36	34, 35	eqeltrd 2860	. . . . . . . . 9 ⊢ (𝜑 → ( 0s +s 𝑁) ∈ 𝐴)
37		peano5uzs.3	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝑥 +s 1s ) ∈ 𝐴)
38	37	ralrimiva 3154	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 (𝑥 +s 1s ) ∈ 𝐴)
39		oveq1 7428	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = (𝑦 +s 𝑁) → (𝑥 +s 1s ) = ((𝑦 +s 𝑁) +s 1s ))
40	39	eleq1d 2845	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = (𝑦 +s 𝑁) → ((𝑥 +s 1s ) ∈ 𝐴 ↔ ((𝑦 +s 𝑁) +s 1s ) ∈ 𝐴))
41	40	rspccv 3573	. . . . . . . . . . . . . 14 ⊢ (∀𝑥 ∈ 𝐴 (𝑥 +s 1s ) ∈ 𝐴 → ((𝑦 +s 𝑁) ∈ 𝐴 → ((𝑦 +s 𝑁) +s 1s ) ∈ 𝐴))
42	38, 41	syl 18	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((𝑦 +s 𝑁) ∈ 𝐴 → ((𝑦 +s 𝑁) +s 1s ) ∈ 𝐴))
43	42	adantl 487	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ ℕ0s ∧ 𝜑) → ((𝑦 +s 𝑁) ∈ 𝐴 → ((𝑦 +s 𝑁) +s 1s ) ∈ 𝐴))
44		n0no 28715	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ∈ ℕ0s → 𝑦 ∈ No)
45	44	adantr 486	. . . . . . . . . . . . . 14 ⊢ ((𝑦 ∈ ℕ0s ∧ 𝜑) → 𝑦 ∈ No)
46		1no 28202	. . . . . . . . . . . . . . 15 ⊢ 1s ∈ No
47	46	a1i 11	. . . . . . . . . . . . . 14 ⊢ ((𝑦 ∈ ℕ0s ∧ 𝜑) → 1s ∈ No)
48	6	adantl 487	. . . . . . . . . . . . . 14 ⊢ ((𝑦 ∈ ℕ0s ∧ 𝜑) → 𝑁 ∈ No)
49	45, 47, 48	adds32d 28399	. . . . . . . . . . . . 13 ⊢ ((𝑦 ∈ ℕ0s ∧ 𝜑) → ((𝑦 +s 1s ) +s 𝑁) = ((𝑦 +s 𝑁) +s 1s ))
50	49	eleq1d 2845	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ ℕ0s ∧ 𝜑) → (((𝑦 +s 1s ) +s 𝑁) ∈ 𝐴 ↔ ((𝑦 +s 𝑁) +s 1s ) ∈ 𝐴))
51	43, 50	sylibrd 262	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ ℕ0s ∧ 𝜑) → ((𝑦 +s 𝑁) ∈ 𝐴 → ((𝑦 +s 1s ) +s 𝑁) ∈ 𝐴))
52	51	ex 418	. . . . . . . . . 10 ⊢ (𝑦 ∈ ℕ0s → (𝜑 → ((𝑦 +s 𝑁) ∈ 𝐴 → ((𝑦 +s 1s ) +s 𝑁) ∈ 𝐴)))
53	52	a2d 30	. . . . . . . . 9 ⊢ (𝑦 ∈ ℕ0s → ((𝜑 → (𝑦 +s 𝑁) ∈ 𝐴) → (𝜑 → ((𝑦 +s 1s ) +s 𝑁) ∈ 𝐴)))
54	23, 26, 29, 32, 36, 53	n0sind 28725	. . . . . . . 8 ⊢ ((𝑛 −s 𝑁) ∈ ℕ0s → (𝜑 → ((𝑛 −s 𝑁) +s 𝑁) ∈ 𝐴))
55	54	com12 33	. . . . . . 7 ⊢ (𝜑 → ((𝑛 −s 𝑁) ∈ ℕ0s → ((𝑛 −s 𝑁) +s 𝑁) ∈ 𝐴))
56	20, 55	syld 48	. . . . . 6 ⊢ (𝜑 → ((𝑛 ∈ ℤs ∧ 𝑁 ≤s 𝑛) → ((𝑛 −s 𝑁) +s 𝑁) ∈ 𝐴))
57	56	imp 412	. . . . 5 ⊢ ((𝜑 ∧ (𝑛 ∈ ℤs ∧ 𝑁 ≤s 𝑛)) → ((𝑛 −s 𝑁) +s 𝑁) ∈ 𝐴)
58	8, 57	eqeltrrd 2861	. . . 4 ⊢ ((𝜑 ∧ (𝑛 ∈ ℤs ∧ 𝑁 ≤s 𝑛)) → 𝑛 ∈ 𝐴)
59	58	ex 418	. . 3 ⊢ (𝜑 → ((𝑛 ∈ ℤs ∧ 𝑁 ≤s 𝑛) → 𝑛 ∈ 𝐴))
60	2, 59	biimtrid 245	. 2 ⊢ (𝜑 → (𝑛 ∈ {𝑘 ∈ ℤs ∣ 𝑁 ≤s 𝑘} → 𝑛 ∈ 𝐴))
61	60	ssrdv 3914	1 ⊢ (𝜑 → {𝑘 ∈ ℤs ∣ 𝑁 ≤s 𝑘} ⊆ 𝐴)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1569   ∈ wcel 2144  ∀wral 3076  {crab 3412   ⊆ wss 3895   class class class wbr 5102  (class class class)co 7421  Nocsur 28003   ≤s cles 28107   0_s c0s 28197   1_s c1s 28198   +_s cadds 28351   −_s csubs 28412  ℕ_0scn0s 28704  ℤ_sczs 28770

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7752

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-pss 4014  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-tp 4589  df-op 4835  df-ot 4858  df-uni 4867  df-int 4907  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5545  df-eprel 5550  df-po 5558  df-so 5559  df-fr 5603  df-se 5604  df-we 5605  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-pred 6304  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6494  df-fun 6541  df-fn 6542  df-f 6543  df-f1 6544  df-fo 6545  df-f1o 6546  df-fv 6547  df-riota 7378  df-ov 7424  df-oprab 7425  df-mpo 7426  df-om 7879  df-1st 8002  df-2nd 8003  df-frecs 8303  df-wrecs 8334  df-recs 8383  df-rdg 8422  df-1o 8480  df-2o 8481  df-nadd 8679  df-no 28006  df-lts 28007  df-bday 28008  df-les 28108  df-slts 28150  df-cuts 28152  df-0s 28199  df-1s 28200  df-made 28219  df-old 28220  df-left 28222  df-right 28223  df-norec 28330  df-norec2 28341  df-adds 28352  df-negs 28413  df-subs 28414  df-n0s 28706  df-nns 28707  df-zs 28771

This theorem is used by:  uzsind  28797