Theorem elzn0s 28390

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 28376	. 2 ⊢ (𝐴 ∈ ℤs ↔ ∃𝑛 ∈ ℕs ∃𝑚 ∈ ℕs 𝐴 = (𝑛 -s 𝑚))
2		nnno 28316	. . . . . . 7 ⊢ (𝑛 ∈ ℕs → 𝑛 ∈ No )
3		nnno 28316	. . . . . . 7 ⊢ (𝑚 ∈ ℕs → 𝑚 ∈ No )
4		subscl 28054	. . . . . . 7 ⊢ ((𝑛 ∈ No ∧ 𝑚 ∈ No ) → (𝑛 -s 𝑚) ∈ No )
5	2, 3, 4	syl2an 597	. . . . . 6 ⊢ ((𝑛 ∈ ℕs ∧ 𝑚 ∈ ℕs) → (𝑛 -s 𝑚) ∈ No )
6		lestric 27732	. . . . . . . 8 ⊢ ((𝑚 ∈ No ∧ 𝑛 ∈ No ) → (𝑚 ≤s 𝑛 ∨ 𝑛 ≤s 𝑚))
7	3, 2, 6	syl2anr 598	. . . . . . 7 ⊢ ((𝑛 ∈ ℕs ∧ 𝑚 ∈ ℕs) → (𝑚 ≤s 𝑛 ∨ 𝑛 ≤s 𝑚))
8		nnn0s 28319	. . . . . . . . 9 ⊢ (𝑚 ∈ ℕs → 𝑚 ∈ ℕ0s)
9		nnn0s 28319	. . . . . . . . 9 ⊢ (𝑛 ∈ ℕs → 𝑛 ∈ ℕ0s)
10		n0subs 28355	. . . . . . . . 9 ⊢ ((𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s) → (𝑚 ≤s 𝑛 ↔ (𝑛 -s 𝑚) ∈ ℕ0s))
11	8, 9, 10	syl2anr 598	. . . . . . . 8 ⊢ ((𝑛 ∈ ℕs ∧ 𝑚 ∈ ℕs) → (𝑚 ≤s 𝑛 ↔ (𝑛 -s 𝑚) ∈ ℕ0s))
12		n0subs 28355	. . . . . . . . . 10 ⊢ ((𝑛 ∈ ℕ0s ∧ 𝑚 ∈ ℕ0s) → (𝑛 ≤s 𝑚 ↔ (𝑚 -s 𝑛) ∈ ℕ0s))
13	9, 8, 12	syl2an 597	. . . . . . . . 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 28072	. . . . . . . . . 10 ⊢ ((𝑛 ∈ ℕs ∧ 𝑚 ∈ ℕs) → ( -us ‘(𝑛 -s 𝑚)) = (𝑚 -s 𝑛))
17	16	eleq1d 2821	. . . . . . . . 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 919	. . . . . . 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 2824	. . . . . 6 ⊢ (𝐴 = (𝑛 -s 𝑚) → (𝐴 ∈ No ↔ (𝑛 -s 𝑚) ∈ No ))
23		eleq1 2824	. . . . . . 7 ⊢ (𝐴 = (𝑛 -s 𝑚) → (𝐴 ∈ ℕ0s ↔ (𝑛 -s 𝑚) ∈ ℕ0s))
24		fveq2 6840	. . . . . . . 8 ⊢ (𝐴 = (𝑛 -s 𝑚) → ( -us ‘𝐴) = ( -us ‘(𝑛 -s 𝑚)))
25	24	eleq1d 2821	. . . . . . 7 ⊢ (𝐴 = (𝑛 -s 𝑚) → (( -us ‘𝐴) ∈ ℕ0s ↔ ( -us ‘(𝑛 -s 𝑚)) ∈ ℕ0s))
26	23, 25	orbi12d 919	. . . . . 6 ⊢ (𝐴 = (𝑛 -s 𝑚) → ((𝐴 ∈ ℕ0s ∨ ( -us ‘𝐴) ∈ ℕ0s) ↔ ((𝑛 -s 𝑚) ∈ ℕ0s ∨ ( -us ‘(𝑛 -s 𝑚)) ∈ ℕ0s)))
27	22, 26	anbi12d 633	. . . . 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 3179	. . 3 ⊢ (∃𝑛 ∈ ℕs ∃𝑚 ∈ ℕs 𝐴 = (𝑛 -s 𝑚) → (𝐴 ∈ No ∧ (𝐴 ∈ ℕ0s ∨ ( -us ‘𝐴) ∈ ℕ0s)))
30		n0p1nns 28363	. . . . . 6 ⊢ (𝐴 ∈ ℕ0s → (𝐴 +s 1s ) ∈ ℕs)
31		1nns 28341	. . . . . . 7 ⊢ 1s ∈ ℕs
32	31	a1i 11	. . . . . 6 ⊢ (𝐴 ∈ ℕ0s → 1s ∈ ℕs)
33		n0no 28315	. . . . . . . 8 ⊢ (𝐴 ∈ ℕ0s → 𝐴 ∈ No )
34		1no 27802	. . . . . . . 8 ⊢ 1s ∈ No
35		pncans 28064	. . . . . . . 8 ⊢ ((𝐴 ∈ No ∧ 1s ∈ No ) → ((𝐴 +s 1s ) -s 1s ) = 𝐴)
36	33, 34, 35	sylancl 587	. . . . . . 7 ⊢ (𝐴 ∈ ℕ0s → ((𝐴 +s 1s ) -s 1s ) = 𝐴)
37	36	eqcomd 2742	. . . . . 6 ⊢ (𝐴 ∈ ℕ0s → 𝐴 = ((𝐴 +s 1s ) -s 1s ))
38		rspceov 7416	. . . . . 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 1374	. . . . 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 28053	. . . . . . . 8 ⊢ (𝐴 ∈ No → ( 1s -s 𝐴) = ( 1s +s ( -us ‘𝐴)))
45		negscl 28028	. . . . . . . . 9 ⊢ (𝐴 ∈ No → ( -us ‘𝐴) ∈ No )
46	42, 45	addscomd 27959	. . . . . . . 8 ⊢ (𝐴 ∈ No → ( 1s +s ( -us ‘𝐴)) = (( -us ‘𝐴) +s 1s ))
47	44, 46	eqtrd 2771	. . . . . . 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 28363	. . . . . . 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 2836	. . . . 5 ⊢ ((𝐴 ∈ No ∧ ( -us ‘𝐴) ∈ ℕ0s) → ( 1s -s 𝐴) ∈ ℕs)
52	42, 43	nncansd 28089	. . . . . . 7 ⊢ (𝐴 ∈ No → ( 1s -s ( 1s -s 𝐴)) = 𝐴)
53	52	eqcomd 2742	. . . . . 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 7416	. . . . 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 1374	. . . 4 ⊢ ((𝐴 ∈ No ∧ ( -us ‘𝐴) ∈ ℕ0s) → ∃𝑛 ∈ ℕs ∃𝑚 ∈ ℕs 𝐴 = (𝑛 -s 𝑚))
57	40, 56	jaodan 960	. . 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 848   = wceq 1542   ∈ wcel 2114  ∃wrex 3061   class class class wbr 5085  ‘cfv 6498  (class class class)co 7367   No csur 27603   ≤s cles 27708   1_s c1s 27798   +_s cadds 27951   -u_s cnegs 28011   -_s csubs 28012  ℕ_0scn0s 28304  ℕ_scnns 28305  ℤ_sczs 28370

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 2708  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5307  ax-pr 5375  ax-un 7689

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 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3062  df-rmo 3342  df-reu 3343  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-pss 3909  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-tp 4572  df-op 4574  df-ot 4576  df-uni 4851  df-int 4890  df-iun 4935  df-br 5086  df-opab 5148  df-mpt 5167  df-tr 5193  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-se 5585  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6265  df-ord 6326  df-on 6327  df-lim 6328  df-suc 6329  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-f1 6503  df-fo 6504  df-f1o 6505  df-fv 6506  df-riota 7324  df-ov 7370  df-oprab 7371  df-mpo 7372  df-om 7818  df-1st 7942  df-2nd 7943  df-frecs 8231  df-wrecs 8262  df-recs 8311  df-rdg 8349  df-1o 8405  df-2o 8406  df-nadd 8602  df-no 27606  df-lts 27607  df-bday 27608  df-les 27709  df-slts 27750  df-cuts 27752  df-0s 27799  df-1s 27800  df-made 27819  df-old 27820  df-left 27822  df-right 27823  df-norec 27930  df-norec2 27941  df-adds 27952  df-negs 28013  df-subs 28014  df-n0s 28306  df-nns 28307  df-zs 28371

This theorem is referenced by:  elzs2  28391  zsbday  28398  zcuts  28399  zcuts0  28400