Theorem eln0zs 28408

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 28397	. . 3 ⊢ (𝑁 ∈ ℕ0s → 𝑁 ∈ ℤs)
2		n0sge0 28346	. . 3 ⊢ (𝑁 ∈ ℕ0s → 0s ≤s 𝑁)
3	1, 2	jca 511	. 2 ⊢ (𝑁 ∈ ℕ0s → (𝑁 ∈ ℤs ∧ 0s ≤s 𝑁))
4		elzs 28392	. . . 4 ⊢ (𝑁 ∈ ℤs ↔ ∃𝑥 ∈ ℕs ∃𝑦 ∈ ℕs 𝑁 = (𝑥 -s 𝑦))
5		nnno 28332	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℕs → 𝑥 ∈ No )
6	5	adantr 480	. . . . . . . . 9 ⊢ ((𝑥 ∈ ℕs ∧ 𝑦 ∈ ℕs) → 𝑥 ∈ No )
7		nnno 28332	. . . . . . . . . 10 ⊢ (𝑦 ∈ ℕs → 𝑦 ∈ No )
8	7	adantl 481	. . . . . . . . 9 ⊢ ((𝑥 ∈ ℕs ∧ 𝑦 ∈ ℕs) → 𝑦 ∈ No )
9	6, 8	subsge0d 28108	. . . . . . . 8 ⊢ ((𝑥 ∈ ℕs ∧ 𝑦 ∈ ℕs) → ( 0s ≤s (𝑥 -s 𝑦) ↔ 𝑦 ≤s 𝑥))
10		nnn0s 28335	. . . . . . . . 9 ⊢ (𝑦 ∈ ℕs → 𝑦 ∈ ℕ0s)
11		nnn0s 28335	. . . . . . . . 9 ⊢ (𝑥 ∈ ℕs → 𝑥 ∈ ℕ0s)
12		n0subs 28371	. . . . . . . . 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 5104	. . . . . . 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 5100  (class class class)co 7368   No csur 27619   ≤s cles 27724   0_s c0s 27813   -_s csubs 28028  ℕ_0scn0s 28320  ℕ_scnns 28321  ℤ_sczs 28386

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 5226  ax-sep 5243  ax-nul 5253  ax-pow 5312  ax-pr 5379  ax-un 7690

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 3352  df-reu 3353  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-tp 4587  df-op 4589  df-ot 4591  df-uni 4866  df-int 4905  df-iun 4950  df-br 5101  df-opab 5163  df-mpt 5182  df-tr 5208  df-id 5527  df-eprel 5532  df-po 5540  df-so 5541  df-fr 5585  df-se 5586  df-we 5587  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-pred 6267  df-ord 6328  df-on 6329  df-lim 6330  df-suc 6331  df-iota 6456  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-riota 7325  df-ov 7371  df-oprab 7372  df-mpo 7373  df-om 7819  df-1st 7943  df-2nd 7944  df-frecs 8233  df-wrecs 8264  df-recs 8313  df-rdg 8351  df-1o 8407  df-2o 8408  df-nadd 8604  df-no 27622  df-lts 27623  df-bday 27624  df-les 27725  df-slts 27766  df-cuts 27768  df-0s 27815  df-1s 27816  df-made 27835  df-old 27836  df-left 27838  df-right 27839  df-norec 27946  df-norec2 27957  df-adds 27968  df-negs 28029  df-subs 28030  df-n0s 28322  df-nns 28323  df-zs 28387

This theorem is referenced by:  zn0subs  28411  peano5uzs  28412  z12sge0  28491