Theorem elzn0s 28572

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 28558	. 2 ⊢ (𝐴 ∈ ℤs ↔ ∃𝑛 ∈ ℕs ∃𝑚 ∈ ℕs 𝐴 = (𝑛 -s 𝑚))
2		nnno 28498	. . . . . . 7 ⊢ (𝑛 ∈ ℕs → 𝑛 ∈ No )
3		nnno 28498	. . . . . . 7 ⊢ (𝑚 ∈ ℕs → 𝑚 ∈ No )
4		subscl 28236	. . . . . . 7 ⊢ ((𝑛 ∈ No ∧ 𝑚 ∈ No ) → (𝑛 -s 𝑚) ∈ No )
5	2, 3, 4	syl2an 607	. . . . . 6 ⊢ ((𝑛 ∈ ℕs ∧ 𝑚 ∈ ℕs) → (𝑛 -s 𝑚) ∈ No )
6		lestric 27913	. . . . . . . 8 ⊢ ((𝑚 ∈ No ∧ 𝑛 ∈ No ) → (𝑚 ≤s 𝑛 ∨ 𝑛 ≤s 𝑚))
7	3, 2, 6	syl2anr 608	. . . . . . 7 ⊢ ((𝑛 ∈ ℕs ∧ 𝑚 ∈ ℕs) → (𝑚 ≤s 𝑛 ∨ 𝑛 ≤s 𝑚))
8		nnn0s 28501	. . . . . . . . 9 ⊢ (𝑚 ∈ ℕs → 𝑚 ∈ ℕ0s)
9		nnn0s 28501	. . . . . . . . 9 ⊢ (𝑛 ∈ ℕs → 𝑛 ∈ ℕ0s)
10		n0subs 28537	. . . . . . . . 9 ⊢ ((𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s) → (𝑚 ≤s 𝑛 ↔ (𝑛 -s 𝑚) ∈ ℕ0s))
11	8, 9, 10	syl2anr 608	. . . . . . . 8 ⊢ ((𝑛 ∈ ℕs ∧ 𝑚 ∈ ℕs) → (𝑚 ≤s 𝑛 ↔ (𝑛 -s 𝑚) ∈ ℕ0s))
12		n0subs 28537	. . . . . . . . . 10 ⊢ ((𝑛 ∈ ℕ0s ∧ 𝑚 ∈ ℕ0s) → (𝑛 ≤s 𝑚 ↔ (𝑚 -s 𝑛) ∈ ℕ0s))
13	9, 8, 12	syl2an 607	. . . . . . . . 9 ⊢ ((𝑛 ∈ ℕs ∧ 𝑚 ∈ ℕs) → (𝑛 ≤s 𝑚 ↔ (𝑚 -s 𝑛) ∈ ℕ0s))
14	2	adantr 485	. . . . . . . . . . 11 ⊢ ((𝑛 ∈ ℕs ∧ 𝑚 ∈ ℕs) → 𝑛 ∈ No )
15	3	adantl 486	. . . . . . . . . . 11 ⊢ ((𝑛 ∈ ℕs ∧ 𝑚 ∈ ℕs) → 𝑚 ∈ No )
16	14, 15	negsubsdi2d 28254	. . . . . . . . . 10 ⊢ ((𝑛 ∈ ℕs ∧ 𝑚 ∈ ℕs) → ( -us ‘(𝑛 -s 𝑚)) = (𝑚 -s 𝑛))
17	16	eleq1d 2848	. . . . . . . . 9 ⊢ ((𝑛 ∈ ℕs ∧ 𝑚 ∈ ℕs) → (( -us ‘(𝑛 -s 𝑚)) ∈ ℕ0s ↔ (𝑚 -s 𝑛) ∈ ℕ0s))
18	13, 17	bitr4d 285	. . . . . . . 8 ⊢ ((𝑛 ∈ ℕs ∧ 𝑚 ∈ ℕs) → (𝑛 ≤s 𝑚 ↔ ( -us ‘(𝑛 -s 𝑚)) ∈ ℕ0s))
19	11, 18	orbi12d 931	. . . . . . 7 ⊢ ((𝑛 ∈ ℕs ∧ 𝑚 ∈ ℕs) → ((𝑚 ≤s 𝑛 ∨ 𝑛 ≤s 𝑚) ↔ ((𝑛 -s 𝑚) ∈ ℕ0s ∨ ( -us ‘(𝑛 -s 𝑚)) ∈ ℕ0s)))
20	7, 19	mpbid 235	. . . . . 6 ⊢ ((𝑛 ∈ ℕs ∧ 𝑚 ∈ ℕs) → ((𝑛 -s 𝑚) ∈ ℕ0s ∨ ( -us ‘(𝑛 -s 𝑚)) ∈ ℕ0s))
21	5, 20	jca 520	. . . . 5 ⊢ ((𝑛 ∈ ℕs ∧ 𝑚 ∈ ℕs) → ((𝑛 -s 𝑚) ∈ No ∧ ((𝑛 -s 𝑚) ∈ ℕ0s ∨ ( -us ‘(𝑛 -s 𝑚)) ∈ ℕ0s)))
22		eleq1 2851	. . . . . 6 ⊢ (𝐴 = (𝑛 -s 𝑚) → (𝐴 ∈ No ↔ (𝑛 -s 𝑚) ∈ No ))
23		eleq1 2851	. . . . . . 7 ⊢ (𝐴 = (𝑛 -s 𝑚) → (𝐴 ∈ ℕ0s ↔ (𝑛 -s 𝑚) ∈ ℕ0s))
24		fveq2 6883	. . . . . . . 8 ⊢ (𝐴 = (𝑛 -s 𝑚) → ( -us ‘𝐴) = ( -us ‘(𝑛 -s 𝑚)))
25	24	eleq1d 2848	. . . . . . 7 ⊢ (𝐴 = (𝑛 -s 𝑚) → (( -us ‘𝐴) ∈ ℕ0s ↔ ( -us ‘(𝑛 -s 𝑚)) ∈ ℕ0s))
26	23, 25	orbi12d 931	. . . . . 6 ⊢ (𝐴 = (𝑛 -s 𝑚) → ((𝐴 ∈ ℕ0s ∨ ( -us ‘𝐴) ∈ ℕ0s) ↔ ((𝑛 -s 𝑚) ∈ ℕ0s ∨ ( -us ‘(𝑛 -s 𝑚)) ∈ ℕ0s)))
27	22, 26	anbi12d 643	. . . . 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 250	. . . 4 ⊢ ((𝑛 ∈ ℕs ∧ 𝑚 ∈ ℕs) → (𝐴 = (𝑛 -s 𝑚) → (𝐴 ∈ No ∧ (𝐴 ∈ ℕ0s ∨ ( -us ‘𝐴) ∈ ℕ0s))))
29	28	rexlimivv 3207	. . 3 ⊢ (∃𝑛 ∈ ℕs ∃𝑚 ∈ ℕs 𝐴 = (𝑛 -s 𝑚) → (𝐴 ∈ No ∧ (𝐴 ∈ ℕ0s ∨ ( -us ‘𝐴) ∈ ℕ0s)))
30		n0p1nns 28545	. . . . . 6 ⊢ (𝐴 ∈ ℕ0s → (𝐴 +s 1s ) ∈ ℕs)
31		1nns 28523	. . . . . . 7 ⊢ 1s ∈ ℕs
32	31	a1i 11	. . . . . 6 ⊢ (𝐴 ∈ ℕ0s → 1s ∈ ℕs)
33		n0no 28497	. . . . . . . 8 ⊢ (𝐴 ∈ ℕ0s → 𝐴 ∈ No )
34		1no 27984	. . . . . . . 8 ⊢ 1s ∈ No
35		pncans 28246	. . . . . . . 8 ⊢ ((𝐴 ∈ No ∧ 1s ∈ No ) → ((𝐴 +s 1s ) -s 1s ) = 𝐴)
36	33, 34, 35	sylancl 597	. . . . . . 7 ⊢ (𝐴 ∈ ℕ0s → ((𝐴 +s 1s ) -s 1s ) = 𝐴)
37	36	eqcomd 2769	. . . . . 6 ⊢ (𝐴 ∈ ℕ0s → 𝐴 = ((𝐴 +s 1s ) -s 1s ))
38		rspceov 7461	. . . . . 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 1398	. . . . 5 ⊢ (𝐴 ∈ ℕ0s → ∃𝑛 ∈ ℕs ∃𝑚 ∈ ℕs 𝐴 = (𝑛 -s 𝑚))
40	39	adantl 486	. . . 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 23	. . . . . . . . 9 ⊢ (𝐴 ∈ No → 𝐴 ∈ No )
44	42, 43	subsvald 28235	. . . . . . . 8 ⊢ (𝐴 ∈ No → ( 1s -s 𝐴) = ( 1s +s ( -us ‘𝐴)))
45		negscl 28210	. . . . . . . . 9 ⊢ (𝐴 ∈ No → ( -us ‘𝐴) ∈ No )
46	42, 45	addscomd 28141	. . . . . . . 8 ⊢ (𝐴 ∈ No → ( 1s +s ( -us ‘𝐴)) = (( -us ‘𝐴) +s 1s ))
47	44, 46	eqtrd 2798	. . . . . . 7 ⊢ (𝐴 ∈ No → ( 1s -s 𝐴) = (( -us ‘𝐴) +s 1s ))
48	47	adantr 485	. . . . . 6 ⊢ ((𝐴 ∈ No ∧ ( -us ‘𝐴) ∈ ℕ0s) → ( 1s -s 𝐴) = (( -us ‘𝐴) +s 1s ))
49		n0p1nns 28545	. . . . . . 7 ⊢ (( -us ‘𝐴) ∈ ℕ0s → (( -us ‘𝐴) +s 1s ) ∈ ℕs)
50	49	adantl 486	. . . . . 6 ⊢ ((𝐴 ∈ No ∧ ( -us ‘𝐴) ∈ ℕ0s) → (( -us ‘𝐴) +s 1s ) ∈ ℕs)
51	48, 50	eqeltrd 2863	. . . . 5 ⊢ ((𝐴 ∈ No ∧ ( -us ‘𝐴) ∈ ℕ0s) → ( 1s -s 𝐴) ∈ ℕs)
52	42, 43	nncansd 28271	. . . . . . 7 ⊢ (𝐴 ∈ No → ( 1s -s ( 1s -s 𝐴)) = 𝐴)
53	52	eqcomd 2769	. . . . . 6 ⊢ (𝐴 ∈ No → 𝐴 = ( 1s -s ( 1s -s 𝐴)))
54	53	adantr 485	. . . . 5 ⊢ ((𝐴 ∈ No ∧ ( -us ‘𝐴) ∈ ℕ0s) → 𝐴 = ( 1s -s ( 1s -s 𝐴)))
55		rspceov 7461	. . . . 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 1398	. . . 4 ⊢ ((𝐴 ∈ No ∧ ( -us ‘𝐴) ∈ ℕ0s) → ∃𝑛 ∈ ℕs ∃𝑚 ∈ ℕs 𝐴 = (𝑛 -s 𝑚))
57	40, 56	jaodan 972	. . 3 ⊢ ((𝐴 ∈ No ∧ (𝐴 ∈ ℕ0s ∨ ( -us ‘𝐴) ∈ ℕ0s)) → ∃𝑛 ∈ ℕs ∃𝑚 ∈ ℕs 𝐴 = (𝑛 -s 𝑚))
58	29, 57	impbii 212	. 2 ⊢ (∃𝑛 ∈ ℕs ∃𝑚 ∈ ℕs 𝐴 = (𝑛 -s 𝑚) ↔ (𝐴 ∈ No ∧ (𝐴 ∈ ℕ0s ∨ ( -us ‘𝐴) ∈ ℕ0s)))
59	1, 58	bitri 278	1 ⊢ (𝐴 ∈ ℤs ↔ (𝐴 ∈ No ∧ (𝐴 ∈ ℕ0s ∨ ( -us ‘𝐴) ∈ ℕ0s)))

Syntax hints:   ↔ wb 209   ∧ wa 400   ∨ wo 860   = wceq 1570   ∈ wcel 2143  ∃wrex 3089   class class class wbr 5110  ‘cfv 6538  (class class class)co 7412   No csur 27785   ≤s cles 27889   1_s c1s 27980   +_s cadds 28133   -u_s cnegs 28193   -_s csubs 28194  ℕ_0scn0s 28486  ℕ_scnns 28487  ℤ_sczs 28552

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5239  ax-sep 5258  ax-nul 5270  ax-pow 5338  ax-pr 5406  ax-un 7734

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3746  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-pss 3926  df-nul 4288  df-if 4489  df-pw 4565  df-sn 4591  df-pr 4593  df-tp 4595  df-op 4597  df-ot 4599  df-uni 4874  df-int 4914  df-iun 4959  df-br 5111  df-opab 5175  df-mpt 5194  df-tr 5220  df-id 5558  df-eprel 5563  df-po 5571  df-so 5572  df-fr 5616  df-se 5617  df-we 5618  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-pred 6304  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-riota 7369  df-ov 7415  df-oprab 7416  df-mpo 7417  df-om 7864  df-1st 7987  df-2nd 7988  df-frecs 8279  df-wrecs 8310  df-recs 8359  df-rdg 8398  df-1o 8454  df-2o 8455  df-nadd 8653  df-no 27788  df-lts 27789  df-bday 27790  df-les 27890  df-slts 27932  df-cuts 27934  df-0s 27981  df-1s 27982  df-made 28001  df-old 28002  df-left 28004  df-right 28005  df-norec 28112  df-norec2 28123  df-adds 28134  df-negs 28195  df-subs 28196  df-n0s 28488  df-nns 28489  df-zs 28553

This theorem is referenced by:  elzs2  28573  zsbday  28580  zcuts  28581  zcuts0  28582