Theorem eln0zs 28816

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 28805	. . 3 ⊢ (𝑁 ∈ ℕ0s → 𝑁 ∈ ℤs)
2		n0sge0 28754	. . 3 ⊢ (𝑁 ∈ ℕ0s → 0s ≤s 𝑁)
3	1, 2	jca 521	. 2 ⊢ (𝑁 ∈ ℕ0s → (𝑁 ∈ ℤs ∧ 0s ≤s 𝑁))
4		elzs 28800	. . . 4 ⊢ (𝑁 ∈ ℤs ↔ ∃𝑥 ∈ ℕs ∃𝑦 ∈ ℕs 𝑁 = (𝑥 −s 𝑦))
5		nnno 28740	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℕs → 𝑥 ∈ No)
6	5	adantr 486	. . . . . . . . 9 ⊢ ((𝑥 ∈ ℕs ∧ 𝑦 ∈ ℕs) → 𝑥 ∈ No)
7		nnno 28740	. . . . . . . . . 10 ⊢ (𝑦 ∈ ℕs → 𝑦 ∈ No)
8	7	adantl 487	. . . . . . . . 9 ⊢ ((𝑥 ∈ ℕs ∧ 𝑦 ∈ ℕs) → 𝑦 ∈ No)
9	6, 8	subsge0d 28516	. . . . . . . 8 ⊢ ((𝑥 ∈ ℕs ∧ 𝑦 ∈ ℕs) → ( 0s ≤s (𝑥 −s 𝑦) ↔ 𝑦 ≤s 𝑥))
10		nnn0s 28743	. . . . . . . . 9 ⊢ (𝑦 ∈ ℕs → 𝑦 ∈ ℕ0s)
11		nnn0s 28743	. . . . . . . . 9 ⊢ (𝑥 ∈ ℕs → 𝑥 ∈ ℕ0s)
12		n0subs 28779	. . . . . . . . 9 ⊢ ((𝑦 ∈ ℕ0s ∧ 𝑥 ∈ ℕ0s) → (𝑦 ≤s 𝑥 ↔ (𝑥 −s 𝑦) ∈ ℕ0s))
13	10, 11, 12	syl2anr 609	. . . . . . . 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 5110	. . . . . . 7 ⊢ (𝑁 = (𝑥 −s 𝑦) → ( 0s ≤s 𝑁 ↔ 0s ≤s (𝑥 −s 𝑦)))
17		eleq1 2852	. . . . . . 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 3208	. . . 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 412	. 2 ⊢ ((𝑁 ∈ ℤs ∧ 0s ≤s 𝑁) → 𝑁 ∈ ℕ0s)
23	3, 22	impbii 212	1 ⊢ (𝑁 ∈ ℕ0s ↔ (𝑁 ∈ ℤs ∧ 0s ≤s 𝑁))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2147  ∃wrex 3090   class class class wbr 5106  (class class class)co 7443  Nocsur 28027   ≤s cles 28131   0_s c0s 28221   −_s csubs 28436  ℕ_0scn0s 28728  ℕ_scnns 28729  ℤ_sczs 28794

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 5235  ax-sep 5252  ax-nul 5263  ax-pow 5331  ax-pr 5395  ax-un 7775

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 5171  df-mpt 5190  df-tr 5216  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-se 5609  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-pred 6322  df-ord 6383  df-on 6384  df-lim 6385  df-suc 6386  df-iota 6512  df-fun 6560  df-fn 6561  df-f 6562  df-f1 6563  df-fo 6564  df-f1o 6565  df-fv 6566  df-riota 7400  df-ov 7446  df-oprab 7447  df-mpo 7448  df-om 7902  df-1st 8025  df-2nd 8026  df-frecs 8327  df-wrecs 8358  df-recs 8407  df-rdg 8446  df-1o 8504  df-2o 8505  df-nadd 8703  df-no 28030  df-lts 28031  df-bday 28032  df-les 28132  df-slts 28174  df-cuts 28176  df-0s 28223  df-1s 28224  df-made 28243  df-old 28244  df-left 28246  df-right 28247  df-norec 28354  df-norec2 28365  df-adds 28376  df-negs 28437  df-subs 28438  df-n0s 28730  df-nns 28731  df-zs 28795

This theorem is used by:  zn0subs  28819  peano5uzs  28820  z12sge0  28899