Theorem eln0zs 28551

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 28540	. . 3 ⊢ (𝑁 ∈ ℕ0s → 𝑁 ∈ ℤs)
2		n0sge0 28489	. . 3 ⊢ (𝑁 ∈ ℕ0s → 0s ≤s 𝑁)
3	1, 2	jca 520	. 2 ⊢ (𝑁 ∈ ℕ0s → (𝑁 ∈ ℤs ∧ 0s ≤s 𝑁))
4		elzs 28535	. . . 4 ⊢ (𝑁 ∈ ℤs ↔ ∃𝑥 ∈ ℕs ∃𝑦 ∈ ℕs 𝑁 = (𝑥 -s 𝑦))
5		nnno 28475	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℕs → 𝑥 ∈ No )
6	5	adantr 485	. . . . . . . . 9 ⊢ ((𝑥 ∈ ℕs ∧ 𝑦 ∈ ℕs) → 𝑥 ∈ No )
7		nnno 28475	. . . . . . . . . 10 ⊢ (𝑦 ∈ ℕs → 𝑦 ∈ No )
8	7	adantl 486	. . . . . . . . 9 ⊢ ((𝑥 ∈ ℕs ∧ 𝑦 ∈ ℕs) → 𝑦 ∈ No )
9	6, 8	subsge0d 28251	. . . . . . . 8 ⊢ ((𝑥 ∈ ℕs ∧ 𝑦 ∈ ℕs) → ( 0s ≤s (𝑥 -s 𝑦) ↔ 𝑦 ≤s 𝑥))
10		nnn0s 28478	. . . . . . . . 9 ⊢ (𝑦 ∈ ℕs → 𝑦 ∈ ℕ0s)
11		nnn0s 28478	. . . . . . . . 9 ⊢ (𝑥 ∈ ℕs → 𝑥 ∈ ℕ0s)
12		n0subs 28514	. . . . . . . . 9 ⊢ ((𝑦 ∈ ℕ0s ∧ 𝑥 ∈ ℕ0s) → (𝑦 ≤s 𝑥 ↔ (𝑥 -s 𝑦) ∈ ℕ0s))
13	10, 11, 12	syl2anr 608	. . . . . . . 8 ⊢ ((𝑥 ∈ ℕs ∧ 𝑦 ∈ ℕs) → (𝑦 ≤s 𝑥 ↔ (𝑥 -s 𝑦) ∈ ℕ0s))
14	9, 13	bitrd 282	. . . . . . 7 ⊢ ((𝑥 ∈ ℕs ∧ 𝑦 ∈ ℕs) → ( 0s ≤s (𝑥 -s 𝑦) ↔ (𝑥 -s 𝑦) ∈ ℕ0s))
15	14	biimpd 232	. . . . . 6 ⊢ ((𝑥 ∈ ℕs ∧ 𝑦 ∈ ℕs) → ( 0s ≤s (𝑥 -s 𝑦) → (𝑥 -s 𝑦) ∈ ℕ0s))
16		breq2 5109	. . . . . . 7 ⊢ (𝑁 = (𝑥 -s 𝑦) → ( 0s ≤s 𝑁 ↔ 0s ≤s (𝑥 -s 𝑦)))
17		eleq1 2853	. . . . . . 7 ⊢ (𝑁 = (𝑥 -s 𝑦) → (𝑁 ∈ ℕ0s ↔ (𝑥 -s 𝑦) ∈ ℕ0s))
18	16, 17	imbi12d 347	. . . . . 6 ⊢ (𝑁 = (𝑥 -s 𝑦) → (( 0s ≤s 𝑁 → 𝑁 ∈ ℕ0s) ↔ ( 0s ≤s (𝑥 -s 𝑦) → (𝑥 -s 𝑦) ∈ ℕ0s)))
19	15, 18	syl5ibrcom 250	. . . . 5 ⊢ ((𝑥 ∈ ℕs ∧ 𝑦 ∈ ℕs) → (𝑁 = (𝑥 -s 𝑦) → ( 0s ≤s 𝑁 → 𝑁 ∈ ℕ0s)))
20	19	rexlimivv 3207	. . . 4 ⊢ (∃𝑥 ∈ ℕs ∃𝑦 ∈ ℕs 𝑁 = (𝑥 -s 𝑦) → ( 0s ≤s 𝑁 → 𝑁 ∈ ℕ0s))
21	4, 20	sylbi 220	. . 3 ⊢ (𝑁 ∈ ℤs → ( 0s ≤s 𝑁 → 𝑁 ∈ ℕ0s))
22	21	imp 411	. 2 ⊢ ((𝑁 ∈ ℤs ∧ 0s ≤s 𝑁) → 𝑁 ∈ ℕ0s)
23	3, 22	impbii 212	1 ⊢ (𝑁 ∈ ℕ0s ↔ (𝑁 ∈ ℤs ∧ 0s ≤s 𝑁))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1563   ∈ wcel 2145  ∃wrex 3089   class class class wbr 5105  (class class class)co 7400   No csur 27762   ≤s cles 27866   0_s c0s 27956   -_s csubs 28171  ℕ_0scn0s 28463  ℕ_scnns 28464  ℤ_sczs 28529

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2737  ax-rep 5232  ax-sep 5251  ax-nul 5261  ax-pow 5327  ax-pr 5395  ax-un 7722

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-nf 1807  df-sb 2094  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3080  df-rex 3090  df-rmo 3370  df-reu 3371  df-rab 3418  df-v 3459  df-sbc 3748  df-csb 3856  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-pss 3927  df-nul 4289  df-if 4484  df-pw 4560  df-sn 4586  df-pr 4588  df-tp 4590  df-op 4592  df-ot 4594  df-uni 4869  df-int 4909  df-iun 4954  df-br 5106  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5547  df-eprel 5552  df-po 5560  df-so 5561  df-fr 5605  df-se 5606  df-we 5607  df-xp 5658  df-rel 5659  df-cnv 5660  df-co 5661  df-dm 5662  df-rn 5663  df-res 5664  df-ima 5665  df-pred 6292  df-ord 6353  df-on 6354  df-lim 6355  df-suc 6356  df-iota 6481  df-fun 6527  df-fn 6528  df-f 6529  df-f1 6530  df-fo 6531  df-f1o 6532  df-fv 6533  df-riota 7357  df-ov 7403  df-oprab 7404  df-mpo 7405  df-om 7851  df-1st 7974  df-2nd 7975  df-frecs 8266  df-wrecs 8297  df-recs 8346  df-rdg 8385  df-1o 8441  df-2o 8442  df-nadd 8640  df-no 27765  df-lts 27766  df-bday 27767  df-les 27867  df-slts 27909  df-cuts 27911  df-0s 27958  df-1s 27959  df-made 27978  df-old 27979  df-left 27981  df-right 27982  df-norec 28089  df-norec2 28100  df-adds 28111  df-negs 28172  df-subs 28173  df-n0s 28465  df-nns 28466  df-zs 28530

This theorem is referenced by:  zn0subs  28554  peano5uzs  28555  z12sge0  28634