Theorem eln0zs 28288

Description: Non-negative surreal integer property expressed in terms of integers. (Contributed by Scott Fenton, 25-Jul-2025.)

Assertion

Ref	Expression
eln0zs	⊢ (𝑁 ∈ ℕ0s ↔ (𝑁 ∈ ℤs ∧ 0s ≤s 𝑁))

Proof of Theorem eln0zs

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

Step	Hyp	Ref	Expression
1		n0zs 28277	. . 3 ⊢ (𝑁 ∈ ℕ0s → 𝑁 ∈ ℤs)
2		n0sge0 28230	. . 3 ⊢ (𝑁 ∈ ℕ0s → 0s ≤s 𝑁)
3	1, 2	jca 511	. 2 ⊢ (𝑁 ∈ ℕ0s → (𝑁 ∈ ℤs ∧ 0s ≤s 𝑁))
4		elzs 28272	. . . 4 ⊢ (𝑁 ∈ ℤs ↔ ∃𝑥 ∈ ℕs ∃𝑦 ∈ ℕs 𝑁 = (𝑥 -s 𝑦))
5		nnsno 28217	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℕs → 𝑥 ∈ No )
6	5	adantr 480	. . . . . . . . 9 ⊢ ((𝑥 ∈ ℕs ∧ 𝑦 ∈ ℕs) → 𝑥 ∈ No )
7		nnsno 28217	. . . . . . . . . 10 ⊢ (𝑦 ∈ ℕs → 𝑦 ∈ No )
8	7	adantl 481	. . . . . . . . 9 ⊢ ((𝑥 ∈ ℕs ∧ 𝑦 ∈ ℕs) → 𝑦 ∈ No )
9	6, 8	subsge0d 28003	. . . . . . . 8 ⊢ ((𝑥 ∈ ℕs ∧ 𝑦 ∈ ℕs) → ( 0s ≤s (𝑥 -s 𝑦) ↔ 𝑦 ≤s 𝑥))
10		nnn0s 28220	. . . . . . . . 9 ⊢ (𝑦 ∈ ℕs → 𝑦 ∈ ℕ0s)
11		nnn0s 28220	. . . . . . . . 9 ⊢ (𝑥 ∈ ℕs → 𝑥 ∈ ℕ0s)
12		n0subs 28253	. . . . . . . . 9 ⊢ ((𝑦 ∈ ℕ0s ∧ 𝑥 ∈ ℕ0s) → (𝑦 ≤s 𝑥 ↔ (𝑥 -s 𝑦) ∈ ℕ0s))
13	10, 11, 12	syl2anr 597	. . . . . . . 8 ⊢ ((𝑥 ∈ ℕs ∧ 𝑦 ∈ ℕs) → (𝑦 ≤s 𝑥 ↔ (𝑥 -s 𝑦) ∈ ℕ0s))
14	9, 13	bitrd 279	. . . . . . 7 ⊢ ((𝑥 ∈ ℕs ∧ 𝑦 ∈ ℕs) → ( 0s ≤s (𝑥 -s 𝑦) ↔ (𝑥 -s 𝑦) ∈ ℕ0s))
15	14	biimpd 229	. . . . . 6 ⊢ ((𝑥 ∈ ℕs ∧ 𝑦 ∈ ℕs) → ( 0s ≤s (𝑥 -s 𝑦) → (𝑥 -s 𝑦) ∈ ℕ0s))
16		breq2 5111	. . . . . . 7 ⊢ (𝑁 = (𝑥 -s 𝑦) → ( 0s ≤s 𝑁 ↔ 0s ≤s (𝑥 -s 𝑦)))
17		eleq1 2816	. . . . . . 7 ⊢ (𝑁 = (𝑥 -s 𝑦) → (𝑁 ∈ ℕ0s ↔ (𝑥 -s 𝑦) ∈ ℕ0s))
18	16, 17	imbi12d 344	. . . . . 6 ⊢ (𝑁 = (𝑥 -s 𝑦) → (( 0s ≤s 𝑁 → 𝑁 ∈ ℕ0s) ↔ ( 0s ≤s (𝑥 -s 𝑦) → (𝑥 -s 𝑦) ∈ ℕ0s)))
19	15, 18	syl5ibrcom 247	. . . . 5 ⊢ ((𝑥 ∈ ℕs ∧ 𝑦 ∈ ℕs) → (𝑁 = (𝑥 -s 𝑦) → ( 0s ≤s 𝑁 → 𝑁 ∈ ℕ0s)))
20	19	rexlimivv 3179	. . . 4 ⊢ (∃𝑥 ∈ ℕs ∃𝑦 ∈ ℕs 𝑁 = (𝑥 -s 𝑦) → ( 0s ≤s 𝑁 → 𝑁 ∈ ℕ0s))
21	4, 20	sylbi 217	. . 3 ⊢ (𝑁 ∈ ℤs → ( 0s ≤s 𝑁 → 𝑁 ∈ ℕ0s))
22	21	imp 406	. 2 ⊢ ((𝑁 ∈ ℤs ∧ 0s ≤s 𝑁) → 𝑁 ∈ ℕ0s)
23	3, 22	impbii 209	1 ⊢ (𝑁 ∈ ℕ0s ↔ (𝑁 ∈ ℤs ∧ 0s ≤s 𝑁))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109  ∃wrex 3053   class class class wbr 5107  (class class class)co 7387   No csur 27551   ≤s csle 27656   0_s c0s 27734   -_s csubs 27926  ℕ_0scnn0s 28206  ℕ_scnns 28207  ℤ_sczs 28266

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-rep 5234  ax-sep 5251  ax-nul 5261  ax-pow 5320  ax-pr 5387  ax-un 7711

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-rmo 3354  df-reu 3355  df-rab 3406  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3934  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-tp 4594  df-op 4596  df-ot 4598  df-uni 4872  df-int 4911  df-iun 4957  df-br 5108  df-opab 5170  df-mpt 5189  df-tr 5215  df-id 5533  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5591  df-se 5592  df-we 5593  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-pred 6274  df-ord 6335  df-on 6336  df-lim 6337  df-suc 6338  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-f1 6516  df-fo 6517  df-f1o 6518  df-fv 6519  df-riota 7344  df-ov 7390  df-oprab 7391  df-mpo 7392  df-om 7843  df-1st 7968  df-2nd 7969  df-frecs 8260  df-wrecs 8291  df-recs 8340  df-rdg 8378  df-1o 8434  df-2o 8435  df-nadd 8630  df-no 27554  df-slt 27555  df-bday 27556  df-sle 27657  df-sslt 27693  df-scut 27695  df-0s 27736  df-1s 27737  df-made 27755  df-old 27756  df-left 27758  df-right 27759  df-norec 27845  df-norec2 27856  df-adds 27867  df-negs 27927  df-subs 27928  df-n0s 28208  df-nns 28209  df-zs 28267

This theorem is referenced by:  zn0subs  28291  peano5uzs  28292  zs12ge0  28342