Theorem eln0zs 28406

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 28395	. . 3 ⊢ (𝑁 ∈ ℕ0s → 𝑁 ∈ ℤs)
2		n0sge0 28344	. . 3 ⊢ (𝑁 ∈ ℕ0s → 0s ≤s 𝑁)
3	1, 2	jca 511	. 2 ⊢ (𝑁 ∈ ℕ0s → (𝑁 ∈ ℤs ∧ 0s ≤s 𝑁))
4		elzs 28390	. . . 4 ⊢ (𝑁 ∈ ℤs ↔ ∃𝑥 ∈ ℕs ∃𝑦 ∈ ℕs 𝑁 = (𝑥 -s 𝑦))
5		nnno 28330	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℕs → 𝑥 ∈ No )
6	5	adantr 480	. . . . . . . . 9 ⊢ ((𝑥 ∈ ℕs ∧ 𝑦 ∈ ℕs) → 𝑥 ∈ No )
7		nnno 28330	. . . . . . . . . 10 ⊢ (𝑦 ∈ ℕs → 𝑦 ∈ No )
8	7	adantl 481	. . . . . . . . 9 ⊢ ((𝑥 ∈ ℕs ∧ 𝑦 ∈ ℕs) → 𝑦 ∈ No )
9	6, 8	subsge0d 28106	. . . . . . . 8 ⊢ ((𝑥 ∈ ℕs ∧ 𝑦 ∈ ℕs) → ( 0s ≤s (𝑥 -s 𝑦) ↔ 𝑦 ≤s 𝑥))
10		nnn0s 28333	. . . . . . . . 9 ⊢ (𝑦 ∈ ℕs → 𝑦 ∈ ℕ0s)
11		nnn0s 28333	. . . . . . . . 9 ⊢ (𝑥 ∈ ℕs → 𝑥 ∈ ℕ0s)
12		n0subs 28369	. . . . . . . . 9 ⊢ ((𝑦 ∈ ℕ0s ∧ 𝑥 ∈ ℕ0s) → (𝑦 ≤s 𝑥 ↔ (𝑥 -s 𝑦) ∈ ℕ0s))
13	10, 11, 12	syl2anr 598	. . . . . . . 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 5090	. . . . . . 7 ⊢ (𝑁 = (𝑥 -s 𝑦) → ( 0s ≤s 𝑁 ↔ 0s ≤s (𝑥 -s 𝑦)))
17		eleq1 2825	. . . . . . 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 3180	. . . 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 1542   ∈ wcel 2114  ∃wrex 3062   class class class wbr 5086  (class class class)co 7360   No csur 27617   ≤s cles 27722   0_s c0s 27811   -_s csubs 28026  ℕ_0scn0s 28318  ℕ_scnns 28319  ℤ_sczs 28384

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5302  ax-pr 5370  ax-un 7682

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rmo 3343  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-tp 4573  df-op 4575  df-ot 4577  df-uni 4852  df-int 4891  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5519  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5577  df-se 5578  df-we 5579  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-pred 6259  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-riota 7317  df-ov 7363  df-oprab 7364  df-mpo 7365  df-om 7811  df-1st 7935  df-2nd 7936  df-frecs 8224  df-wrecs 8255  df-recs 8304  df-rdg 8342  df-1o 8398  df-2o 8399  df-nadd 8595  df-no 27620  df-lts 27621  df-bday 27622  df-les 27723  df-slts 27764  df-cuts 27766  df-0s 27813  df-1s 27814  df-made 27833  df-old 27834  df-left 27836  df-right 27837  df-norec 27944  df-norec2 27955  df-adds 27966  df-negs 28027  df-subs 28028  df-n0s 28320  df-nns 28321  df-zs 28385

This theorem is referenced by:  zn0subs  28409  peano5uzs  28410  z12sge0  28489