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Theorem eln0zs 28386
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.
StepHypRef Expression
1 n0zs 28375 . . 3 (𝑁 ∈ ℕ0s𝑁 ∈ ℤs)
2 n0sge0 28341 . . 3 (𝑁 ∈ ℕ0s → 0s ≤s 𝑁)
31, 2jca 511 . 2 (𝑁 ∈ ℕ0s → (𝑁 ∈ ℤs ∧ 0s ≤s 𝑁))
4 elzs 28370 . . . 4 (𝑁 ∈ ℤs ↔ ∃𝑥 ∈ ℕs𝑦 ∈ ℕs 𝑁 = (𝑥 -s 𝑦))
5 nnsno 28329 . . . . . . . . . 10 (𝑥 ∈ ℕs𝑥 No )
65adantr 480 . . . . . . . . 9 ((𝑥 ∈ ℕs𝑦 ∈ ℕs) → 𝑥 No )
7 nnsno 28329 . . . . . . . . . 10 (𝑦 ∈ ℕs𝑦 No )
87adantl 481 . . . . . . . . 9 ((𝑥 ∈ ℕs𝑦 ∈ ℕs) → 𝑦 No )
96, 8subsge0d 28129 . . . . . . . 8 ((𝑥 ∈ ℕs𝑦 ∈ ℕs) → ( 0s ≤s (𝑥 -s 𝑦) ↔ 𝑦 ≤s 𝑥))
10 nnn0s 28332 . . . . . . . . 9 (𝑦 ∈ ℕs𝑦 ∈ ℕ0s)
11 nnn0s 28332 . . . . . . . . 9 (𝑥 ∈ ℕs𝑥 ∈ ℕ0s)
12 n0subs 28360 . . . . . . . . 9 ((𝑦 ∈ ℕ0s𝑥 ∈ ℕ0s) → (𝑦 ≤s 𝑥 ↔ (𝑥 -s 𝑦) ∈ ℕ0s))
1310, 11, 12syl2anr 597 . . . . . . . 8 ((𝑥 ∈ ℕs𝑦 ∈ ℕs) → (𝑦 ≤s 𝑥 ↔ (𝑥 -s 𝑦) ∈ ℕ0s))
149, 13bitrd 279 . . . . . . 7 ((𝑥 ∈ ℕs𝑦 ∈ ℕs) → ( 0s ≤s (𝑥 -s 𝑦) ↔ (𝑥 -s 𝑦) ∈ ℕ0s))
1514biimpd 229 . . . . . 6 ((𝑥 ∈ ℕs𝑦 ∈ ℕs) → ( 0s ≤s (𝑥 -s 𝑦) → (𝑥 -s 𝑦) ∈ ℕ0s))
16 breq2 5147 . . . . . . 7 (𝑁 = (𝑥 -s 𝑦) → ( 0s ≤s 𝑁 ↔ 0s ≤s (𝑥 -s 𝑦)))
17 eleq1 2829 . . . . . . 7 (𝑁 = (𝑥 -s 𝑦) → (𝑁 ∈ ℕ0s ↔ (𝑥 -s 𝑦) ∈ ℕ0s))
1816, 17imbi12d 344 . . . . . 6 (𝑁 = (𝑥 -s 𝑦) → (( 0s ≤s 𝑁𝑁 ∈ ℕ0s) ↔ ( 0s ≤s (𝑥 -s 𝑦) → (𝑥 -s 𝑦) ∈ ℕ0s)))
1915, 18syl5ibrcom 247 . . . . 5 ((𝑥 ∈ ℕs𝑦 ∈ ℕs) → (𝑁 = (𝑥 -s 𝑦) → ( 0s ≤s 𝑁𝑁 ∈ ℕ0s)))
2019rexlimivv 3201 . . . 4 (∃𝑥 ∈ ℕs𝑦 ∈ ℕs 𝑁 = (𝑥 -s 𝑦) → ( 0s ≤s 𝑁𝑁 ∈ ℕ0s))
214, 20sylbi 217 . . 3 (𝑁 ∈ ℤs → ( 0s ≤s 𝑁𝑁 ∈ ℕ0s))
2221imp 406 . 2 ((𝑁 ∈ ℤs ∧ 0s ≤s 𝑁) → 𝑁 ∈ ℕ0s)
233, 22impbii 209 1 (𝑁 ∈ ℕ0s ↔ (𝑁 ∈ ℤs ∧ 0s ≤s 𝑁))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1540  wcel 2108  wrex 3070   class class class wbr 5143  (class class class)co 7431   No csur 27684   ≤s csle 27789   0s c0s 27867   -s csubs 28052  0scnn0s 28318  scnns 28319  sczs 28364
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 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-tp 4631  df-op 4633  df-ot 4635  df-uni 4908  df-int 4947  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-se 5638  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8014  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-2o 8507  df-nadd 8704  df-no 27687  df-slt 27688  df-bday 27689  df-sle 27790  df-sslt 27826  df-scut 27828  df-0s 27869  df-1s 27870  df-made 27886  df-old 27887  df-left 27889  df-right 27890  df-norec 27971  df-norec2 27982  df-adds 27993  df-negs 28053  df-subs 28054  df-n0s 28320  df-nns 28321  df-zs 28365
This theorem is referenced by:  zn0subs  28389  peano5uzs  28390
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