Theorem peano5uzs 28823

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 5110	. . . 4 ⊢ (𝑘 = 𝑛 → (𝑁 ≤s 𝑘 ↔ 𝑁 ≤s 𝑛))
2	1	elrab 3648	. . 3 ⊢ (𝑛 ∈ {𝑘 ∈ ℤs ∣ 𝑁 ≤s 𝑘} ↔ (𝑛 ∈ ℤs ∧ 𝑁 ≤s 𝑛))
3		zno 28801	. . . . . . 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 28802	. . . . . 6 ⊢ (𝜑 → 𝑁 ∈ No)
7		npcans 28494	. . . . . 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 28815	. . . . . . . . . 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 28519	. . . . . . . . . . . 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 28819	. . . . . . . . 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 7453	. . . . . . . . . . 11 ⊢ (𝑧 = 0s → (𝑧 +s 𝑁) = ( 0s +s 𝑁))
22	21	eleq1d 2849	. . . . . . . . . 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 7453	. . . . . . . . . . 11 ⊢ (𝑧 = 𝑦 → (𝑧 +s 𝑁) = (𝑦 +s 𝑁))
25	24	eleq1d 2849	. . . . . . . . . 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 7453	. . . . . . . . . . 11 ⊢ (𝑧 = (𝑦 +s 1s ) → (𝑧 +s 𝑁) = ((𝑦 +s 1s ) +s 𝑁))
28	27	eleq1d 2849	. . . . . . . . . 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 7453	. . . . . . . . . . 11 ⊢ (𝑧 = (𝑛 −s 𝑁) → (𝑧 +s 𝑁) = ((𝑛 −s 𝑁) +s 𝑁))
31	30	eleq1d 2849	. . . . . . . . . 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 28387	. . . . . . . . . . 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 2864	. . . . . . . . 9 ⊢ (𝜑 → ( 0s +s 𝑁) ∈ 𝐴)
37		peano5uzs.3	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝑥 +s 1s ) ∈ 𝐴)
38	37	ralrimiva 3158	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 (𝑥 +s 1s ) ∈ 𝐴)
39		oveq1 7453	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = (𝑦 +s 𝑁) → (𝑥 +s 1s ) = ((𝑦 +s 𝑁) +s 1s ))
40	39	eleq1d 2849	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = (𝑦 +s 𝑁) → ((𝑥 +s 1s ) ∈ 𝐴 ↔ ((𝑦 +s 𝑁) +s 1s ) ∈ 𝐴))
41	40	rspccv 3577	. . . . . . . . . . . . . 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 28742	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ∈ ℕ0s → 𝑦 ∈ No)
45	44	adantr 486	. . . . . . . . . . . . . 14 ⊢ ((𝑦 ∈ ℕ0s ∧ 𝜑) → 𝑦 ∈ No)
46		1no 28229	. . . . . . . . . . . . . . 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 28426	. . . . . . . . . . . . 13 ⊢ ((𝑦 ∈ ℕ0s ∧ 𝜑) → ((𝑦 +s 1s ) +s 𝑁) = ((𝑦 +s 𝑁) +s 1s ))
50	49	eleq1d 2849	. . . . . . . . . . . 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 28752	. . . . . . . 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 2865	. . . 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 3919	1 ⊢ (𝜑 → {𝑘 ∈ ℤs ∣ 𝑁 ≤s 𝑘} ⊆ 𝐴)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2147  ∀wral 3080  {crab 3416   ⊆ wss 3900   class class class wbr 5106  (class class class)co 7446  Nocsur 28030   ≤s cles 28134   0_s c0s 28224   1_s c1s 28225   +_s cadds 28378   −_s csubs 28439  ℕ_0scn0s 28731  ℤ_sczs 28797

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5236  ax-sep 5253  ax-nul 5264  ax-pow 5332  ax-pr 5396  ax-un 7778

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-tp 4593  df-op 4839  df-ot 4862  df-uni 4871  df-int 4911  df-iun 4956  df-br 5107  df-opab 5172  df-mpt 5191  df-tr 5217  df-id 5551  df-eprel 5556  df-po 5564  df-so 5565  df-fr 5609  df-se 5610  df-we 5611  df-xp 5662  df-rel 5663  df-cnv 5664  df-co 5665  df-dm 5666  df-rn 5667  df-res 5668  df-ima 5669  df-pred 6325  df-ord 6386  df-on 6387  df-lim 6388  df-suc 6389  df-iota 6515  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7403  df-ov 7449  df-oprab 7450  df-mpo 7451  df-om 7905  df-1st 8028  df-2nd 8029  df-frecs 8330  df-wrecs 8361  df-recs 8410  df-rdg 8449  df-1o 8507  df-2o 8508  df-nadd 8706  df-no 28033  df-lts 28034  df-bday 28035  df-les 28135  df-slts 28177  df-cuts 28179  df-0s 28226  df-1s 28227  df-made 28246  df-old 28247  df-left 28249  df-right 28250  df-norec 28357  df-norec2 28368  df-adds 28379  df-negs 28440  df-subs 28441  df-n0s 28733  df-nns 28734  df-zs 28798

This theorem is used by:  uzsind  28824