Theorem elzn0s 28338

Description: A surreal integer is a surreal that is a non-negative integer or whose negative is a non-negative integer. (Contributed by Scott Fenton, 26-May-2025.)

Assertion

Ref	Expression
elzn0s	⊢ (𝐴 ∈ ℤs ↔ (𝐴 ∈ No ∧ (𝐴 ∈ ℕ0s ∨ ( -us ‘𝐴) ∈ ℕ0s)))

Proof of Theorem elzn0s

Dummy variables 𝑛 𝑚 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		elzs 28324	. 2 ⊢ (𝐴 ∈ ℤs ↔ ∃𝑛 ∈ ℕs ∃𝑚 ∈ ℕs 𝐴 = (𝑛 -s 𝑚))
2		nnsno 28269	. . . . . . 7 ⊢ (𝑛 ∈ ℕs → 𝑛 ∈ No )
3		nnsno 28269	. . . . . . 7 ⊢ (𝑚 ∈ ℕs → 𝑚 ∈ No )
4		subscl 28018	. . . . . . 7 ⊢ ((𝑛 ∈ No ∧ 𝑚 ∈ No ) → (𝑛 -s 𝑚) ∈ No )
5	2, 3, 4	syl2an 596	. . . . . 6 ⊢ ((𝑛 ∈ ℕs ∧ 𝑚 ∈ ℕs) → (𝑛 -s 𝑚) ∈ No )
6		sletric 27728	. . . . . . . 8 ⊢ ((𝑚 ∈ No ∧ 𝑛 ∈ No ) → (𝑚 ≤s 𝑛 ∨ 𝑛 ≤s 𝑚))
7	3, 2, 6	syl2anr 597	. . . . . . 7 ⊢ ((𝑛 ∈ ℕs ∧ 𝑚 ∈ ℕs) → (𝑚 ≤s 𝑛 ∨ 𝑛 ≤s 𝑚))
8		nnn0s 28272	. . . . . . . . 9 ⊢ (𝑚 ∈ ℕs → 𝑚 ∈ ℕ0s)
9		nnn0s 28272	. . . . . . . . 9 ⊢ (𝑛 ∈ ℕs → 𝑛 ∈ ℕ0s)
10		n0subs 28305	. . . . . . . . 9 ⊢ ((𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s) → (𝑚 ≤s 𝑛 ↔ (𝑛 -s 𝑚) ∈ ℕ0s))
11	8, 9, 10	syl2anr 597	. . . . . . . 8 ⊢ ((𝑛 ∈ ℕs ∧ 𝑚 ∈ ℕs) → (𝑚 ≤s 𝑛 ↔ (𝑛 -s 𝑚) ∈ ℕ0s))
12		n0subs 28305	. . . . . . . . . 10 ⊢ ((𝑛 ∈ ℕ0s ∧ 𝑚 ∈ ℕ0s) → (𝑛 ≤s 𝑚 ↔ (𝑚 -s 𝑛) ∈ ℕ0s))
13	9, 8, 12	syl2an 596	. . . . . . . . 9 ⊢ ((𝑛 ∈ ℕs ∧ 𝑚 ∈ ℕs) → (𝑛 ≤s 𝑚 ↔ (𝑚 -s 𝑛) ∈ ℕ0s))
14	2	adantr 480	. . . . . . . . . . 11 ⊢ ((𝑛 ∈ ℕs ∧ 𝑚 ∈ ℕs) → 𝑛 ∈ No )
15	3	adantl 481	. . . . . . . . . . 11 ⊢ ((𝑛 ∈ ℕs ∧ 𝑚 ∈ ℕs) → 𝑚 ∈ No )
16	14, 15	negsubsdi2d 28036	. . . . . . . . . 10 ⊢ ((𝑛 ∈ ℕs ∧ 𝑚 ∈ ℕs) → ( -us ‘(𝑛 -s 𝑚)) = (𝑚 -s 𝑛))
17	16	eleq1d 2819	. . . . . . . . 9 ⊢ ((𝑛 ∈ ℕs ∧ 𝑚 ∈ ℕs) → (( -us ‘(𝑛 -s 𝑚)) ∈ ℕ0s ↔ (𝑚 -s 𝑛) ∈ ℕ0s))
18	13, 17	bitr4d 282	. . . . . . . 8 ⊢ ((𝑛 ∈ ℕs ∧ 𝑚 ∈ ℕs) → (𝑛 ≤s 𝑚 ↔ ( -us ‘(𝑛 -s 𝑚)) ∈ ℕ0s))
19	11, 18	orbi12d 918	. . . . . . 7 ⊢ ((𝑛 ∈ ℕs ∧ 𝑚 ∈ ℕs) → ((𝑚 ≤s 𝑛 ∨ 𝑛 ≤s 𝑚) ↔ ((𝑛 -s 𝑚) ∈ ℕ0s ∨ ( -us ‘(𝑛 -s 𝑚)) ∈ ℕ0s)))
20	7, 19	mpbid 232	. . . . . 6 ⊢ ((𝑛 ∈ ℕs ∧ 𝑚 ∈ ℕs) → ((𝑛 -s 𝑚) ∈ ℕ0s ∨ ( -us ‘(𝑛 -s 𝑚)) ∈ ℕ0s))
21	5, 20	jca 511	. . . . 5 ⊢ ((𝑛 ∈ ℕs ∧ 𝑚 ∈ ℕs) → ((𝑛 -s 𝑚) ∈ No ∧ ((𝑛 -s 𝑚) ∈ ℕ0s ∨ ( -us ‘(𝑛 -s 𝑚)) ∈ ℕ0s)))
22		eleq1 2822	. . . . . 6 ⊢ (𝐴 = (𝑛 -s 𝑚) → (𝐴 ∈ No ↔ (𝑛 -s 𝑚) ∈ No ))
23		eleq1 2822	. . . . . . 7 ⊢ (𝐴 = (𝑛 -s 𝑚) → (𝐴 ∈ ℕ0s ↔ (𝑛 -s 𝑚) ∈ ℕ0s))
24		fveq2 6876	. . . . . . . 8 ⊢ (𝐴 = (𝑛 -s 𝑚) → ( -us ‘𝐴) = ( -us ‘(𝑛 -s 𝑚)))
25	24	eleq1d 2819	. . . . . . 7 ⊢ (𝐴 = (𝑛 -s 𝑚) → (( -us ‘𝐴) ∈ ℕ0s ↔ ( -us ‘(𝑛 -s 𝑚)) ∈ ℕ0s))
26	23, 25	orbi12d 918	. . . . . 6 ⊢ (𝐴 = (𝑛 -s 𝑚) → ((𝐴 ∈ ℕ0s ∨ ( -us ‘𝐴) ∈ ℕ0s) ↔ ((𝑛 -s 𝑚) ∈ ℕ0s ∨ ( -us ‘(𝑛 -s 𝑚)) ∈ ℕ0s)))
27	22, 26	anbi12d 632	. . . . 5 ⊢ (𝐴 = (𝑛 -s 𝑚) → ((𝐴 ∈ No ∧ (𝐴 ∈ ℕ0s ∨ ( -us ‘𝐴) ∈ ℕ0s)) ↔ ((𝑛 -s 𝑚) ∈ No ∧ ((𝑛 -s 𝑚) ∈ ℕ0s ∨ ( -us ‘(𝑛 -s 𝑚)) ∈ ℕ0s))))
28	21, 27	syl5ibrcom 247	. . . 4 ⊢ ((𝑛 ∈ ℕs ∧ 𝑚 ∈ ℕs) → (𝐴 = (𝑛 -s 𝑚) → (𝐴 ∈ No ∧ (𝐴 ∈ ℕ0s ∨ ( -us ‘𝐴) ∈ ℕ0s))))
29	28	rexlimivv 3186	. . 3 ⊢ (∃𝑛 ∈ ℕs ∃𝑚 ∈ ℕs 𝐴 = (𝑛 -s 𝑚) → (𝐴 ∈ No ∧ (𝐴 ∈ ℕ0s ∨ ( -us ‘𝐴) ∈ ℕ0s)))
30		n0p1nns 28312	. . . . . 6 ⊢ (𝐴 ∈ ℕ0s → (𝐴 +s 1s ) ∈ ℕs)
31		1nns 28293	. . . . . . 7 ⊢ 1s ∈ ℕs
32	31	a1i 11	. . . . . 6 ⊢ (𝐴 ∈ ℕ0s → 1s ∈ ℕs)
33		n0sno 28268	. . . . . . . 8 ⊢ (𝐴 ∈ ℕ0s → 𝐴 ∈ No )
34		1sno 27791	. . . . . . . 8 ⊢ 1s ∈ No
35		pncans 28028	. . . . . . . 8 ⊢ ((𝐴 ∈ No ∧ 1s ∈ No ) → ((𝐴 +s 1s ) -s 1s ) = 𝐴)
36	33, 34, 35	sylancl 586	. . . . . . 7 ⊢ (𝐴 ∈ ℕ0s → ((𝐴 +s 1s ) -s 1s ) = 𝐴)
37	36	eqcomd 2741	. . . . . 6 ⊢ (𝐴 ∈ ℕ0s → 𝐴 = ((𝐴 +s 1s ) -s 1s ))
38		rspceov 7454	. . . . . 6 ⊢ (((𝐴 +s 1s ) ∈ ℕs ∧ 1s ∈ ℕs ∧ 𝐴 = ((𝐴 +s 1s ) -s 1s )) → ∃𝑛 ∈ ℕs ∃𝑚 ∈ ℕs 𝐴 = (𝑛 -s 𝑚))
39	30, 32, 37, 38	syl3anc 1373	. . . . 5 ⊢ (𝐴 ∈ ℕ0s → ∃𝑛 ∈ ℕs ∃𝑚 ∈ ℕs 𝐴 = (𝑛 -s 𝑚))
40	39	adantl 481	. . . 4 ⊢ ((𝐴 ∈ No ∧ 𝐴 ∈ ℕ0s) → ∃𝑛 ∈ ℕs ∃𝑚 ∈ ℕs 𝐴 = (𝑛 -s 𝑚))
41	31	a1i 11	. . . . 5 ⊢ ((𝐴 ∈ No ∧ ( -us ‘𝐴) ∈ ℕ0s) → 1s ∈ ℕs)
42	34	a1i 11	. . . . . . . . 9 ⊢ (𝐴 ∈ No → 1s ∈ No )
43		id 22	. . . . . . . . 9 ⊢ (𝐴 ∈ No → 𝐴 ∈ No )
44	42, 43	subsvald 28017	. . . . . . . 8 ⊢ (𝐴 ∈ No → ( 1s -s 𝐴) = ( 1s +s ( -us ‘𝐴)))
45		negscl 27994	. . . . . . . . 9 ⊢ (𝐴 ∈ No → ( -us ‘𝐴) ∈ No )
46	42, 45	addscomd 27926	. . . . . . . 8 ⊢ (𝐴 ∈ No → ( 1s +s ( -us ‘𝐴)) = (( -us ‘𝐴) +s 1s ))
47	44, 46	eqtrd 2770	. . . . . . 7 ⊢ (𝐴 ∈ No → ( 1s -s 𝐴) = (( -us ‘𝐴) +s 1s ))
48	47	adantr 480	. . . . . 6 ⊢ ((𝐴 ∈ No ∧ ( -us ‘𝐴) ∈ ℕ0s) → ( 1s -s 𝐴) = (( -us ‘𝐴) +s 1s ))
49		n0p1nns 28312	. . . . . . 7 ⊢ (( -us ‘𝐴) ∈ ℕ0s → (( -us ‘𝐴) +s 1s ) ∈ ℕs)
50	49	adantl 481	. . . . . 6 ⊢ ((𝐴 ∈ No ∧ ( -us ‘𝐴) ∈ ℕ0s) → (( -us ‘𝐴) +s 1s ) ∈ ℕs)
51	48, 50	eqeltrd 2834	. . . . 5 ⊢ ((𝐴 ∈ No ∧ ( -us ‘𝐴) ∈ ℕ0s) → ( 1s -s 𝐴) ∈ ℕs)
52	42, 43	nncansd 28052	. . . . . . 7 ⊢ (𝐴 ∈ No → ( 1s -s ( 1s -s 𝐴)) = 𝐴)
53	52	eqcomd 2741	. . . . . 6 ⊢ (𝐴 ∈ No → 𝐴 = ( 1s -s ( 1s -s 𝐴)))
54	53	adantr 480	. . . . 5 ⊢ ((𝐴 ∈ No ∧ ( -us ‘𝐴) ∈ ℕ0s) → 𝐴 = ( 1s -s ( 1s -s 𝐴)))
55		rspceov 7454	. . . . 5 ⊢ (( 1s ∈ ℕs ∧ ( 1s -s 𝐴) ∈ ℕs ∧ 𝐴 = ( 1s -s ( 1s -s 𝐴))) → ∃𝑛 ∈ ℕs ∃𝑚 ∈ ℕs 𝐴 = (𝑛 -s 𝑚))
56	41, 51, 54, 55	syl3anc 1373	. . . 4 ⊢ ((𝐴 ∈ No ∧ ( -us ‘𝐴) ∈ ℕ0s) → ∃𝑛 ∈ ℕs ∃𝑚 ∈ ℕs 𝐴 = (𝑛 -s 𝑚))
57	40, 56	jaodan 959	. . 3 ⊢ ((𝐴 ∈ No ∧ (𝐴 ∈ ℕ0s ∨ ( -us ‘𝐴) ∈ ℕ0s)) → ∃𝑛 ∈ ℕs ∃𝑚 ∈ ℕs 𝐴 = (𝑛 -s 𝑚))
58	29, 57	impbii 209	. 2 ⊢ (∃𝑛 ∈ ℕs ∃𝑚 ∈ ℕs 𝐴 = (𝑛 -s 𝑚) ↔ (𝐴 ∈ No ∧ (𝐴 ∈ ℕ0s ∨ ( -us ‘𝐴) ∈ ℕ0s)))
59	1, 58	bitri 275	1 ⊢ (𝐴 ∈ ℤs ↔ (𝐴 ∈ No ∧ (𝐴 ∈ ℕ0s ∨ ( -us ‘𝐴) ∈ ℕ0s)))

Syntax hints:   ↔ wb 206   ∧ wa 395   ∨ wo 847   = wceq 1540   ∈ wcel 2108  ∃wrex 3060   class class class wbr 5119  ‘cfv 6531  (class class class)co 7405   No csur 27603   ≤s csle 27708   1_s c1s 27787   +_s cadds 27918   -u_s cnegs 27977   -_s csubs 27978  ℕ_0scnn0s 28258  ℕ_scnns 28259  ℤ_sczs 28318

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 2707  ax-rep 5249  ax-sep 5266  ax-nul 5276  ax-pow 5335  ax-pr 5402  ax-un 7729

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-rmo 3359  df-reu 3360  df-rab 3416  df-v 3461  df-sbc 3766  df-csb 3875  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-pss 3946  df-nul 4309  df-if 4501  df-pw 4577  df-sn 4602  df-pr 4604  df-tp 4606  df-op 4608  df-ot 4610  df-uni 4884  df-int 4923  df-iun 4969  df-br 5120  df-opab 5182  df-mpt 5202  df-tr 5230  df-id 5548  df-eprel 5553  df-po 5561  df-so 5562  df-fr 5606  df-se 5607  df-we 5608  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667  df-pred 6290  df-ord 6355  df-on 6356  df-lim 6357  df-suc 6358  df-iota 6484  df-fun 6533  df-fn 6534  df-f 6535  df-f1 6536  df-fo 6537  df-f1o 6538  df-fv 6539  df-riota 7362  df-ov 7408  df-oprab 7409  df-mpo 7410  df-om 7862  df-1st 7988  df-2nd 7989  df-frecs 8280  df-wrecs 8311  df-recs 8385  df-rdg 8424  df-1o 8480  df-2o 8481  df-nadd 8678  df-no 27606  df-slt 27607  df-bday 27608  df-sle 27709  df-sslt 27745  df-scut 27747  df-0s 27788  df-1s 27789  df-made 27807  df-old 27808  df-left 27810  df-right 27811  df-norec 27897  df-norec2 27908  df-adds 27919  df-negs 27979  df-subs 27980  df-n0s 28260  df-nns 28261  df-zs 28319

This theorem is referenced by:  elzs2  28339  zsbday  28346  zscut  28347