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Theorem eln0zs 28401
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 28390 . . 3 (𝑁 ∈ ℕ0s𝑁 ∈ ℤs)
2 n0sge0 28356 . . 3 (𝑁 ∈ ℕ0s → 0s ≤s 𝑁)
31, 2jca 511 . 2 (𝑁 ∈ ℕ0s → (𝑁 ∈ ℤs ∧ 0s ≤s 𝑁))
4 elzs 28385 . . . 4 (𝑁 ∈ ℤs ↔ ∃𝑥 ∈ ℕs𝑦 ∈ ℕs 𝑁 = (𝑥 -s 𝑦))
5 nnsno 28344 . . . . . . . . . 10 (𝑥 ∈ ℕs𝑥 No )
65adantr 480 . . . . . . . . 9 ((𝑥 ∈ ℕs𝑦 ∈ ℕs) → 𝑥 No )
7 nnsno 28344 . . . . . . . . . 10 (𝑦 ∈ ℕs𝑦 No )
87adantl 481 . . . . . . . . 9 ((𝑥 ∈ ℕs𝑦 ∈ ℕs) → 𝑦 No )
96, 8subsge0d 28144 . . . . . . . 8 ((𝑥 ∈ ℕs𝑦 ∈ ℕs) → ( 0s ≤s (𝑥 -s 𝑦) ↔ 𝑦 ≤s 𝑥))
10 nnn0s 28347 . . . . . . . . 9 (𝑦 ∈ ℕs𝑦 ∈ ℕ0s)
11 nnn0s 28347 . . . . . . . . 9 (𝑥 ∈ ℕs𝑥 ∈ ℕ0s)
12 n0subs 28375 . . . . . . . . 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 5152 . . . . . . 7 (𝑁 = (𝑥 -s 𝑦) → ( 0s ≤s 𝑁 ↔ 0s ≤s (𝑥 -s 𝑦)))
17 eleq1 2827 . . . . . . 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 3199 . . . 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 1537  wcel 2106  wrex 3068   class class class wbr 5148  (class class class)co 7431   No csur 27699   ≤s csle 27804   0s c0s 27882   -s csubs 28067  0scnn0s 28333  scnns 28334  sczs 28379
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-tp 4636  df-op 4638  df-ot 4640  df-uni 4913  df-int 4952  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-se 5642  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8013  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-1o 8505  df-2o 8506  df-nadd 8703  df-no 27702  df-slt 27703  df-bday 27704  df-sle 27805  df-sslt 27841  df-scut 27843  df-0s 27884  df-1s 27885  df-made 27901  df-old 27902  df-left 27904  df-right 27905  df-norec 27986  df-norec2 27997  df-adds 28008  df-negs 28068  df-subs 28069  df-n0s 28335  df-nns 28336  df-zs 28380
This theorem is referenced by:  zn0subs  28404  peano5uzs  28405
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