Theorem elzn0s 28789

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 ∨ ( -s‘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 28775	. 2 ⊢ (𝐴 ∈ ℤs ↔ ∃𝑛 ∈ ℕs ∃𝑚 ∈ ℕs 𝐴 = (𝑛 −s 𝑚))
2		nnno 28715	. . . . . . 7 ⊢ (𝑛 ∈ ℕs → 𝑛 ∈ No)
3		nnno 28715	. . . . . . 7 ⊢ (𝑚 ∈ ℕs → 𝑚 ∈ No)
4		subscl 28453	. . . . . . 7 ⊢ ((𝑛 ∈ No ∧ 𝑚 ∈ No) → (𝑛 −s 𝑚) ∈ No)
5	2, 3, 4	syl2an 608	. . . . . 6 ⊢ ((𝑛 ∈ ℕs ∧ 𝑚 ∈ ℕs) → (𝑛 −s 𝑚) ∈ No)
6		lestric 28130	. . . . . . . 8 ⊢ ((𝑚 ∈ No ∧ 𝑛 ∈ No) → (𝑚 ≤s 𝑛 ∨ 𝑛 ≤s 𝑚))
7	3, 2, 6	syl2anr 609	. . . . . . 7 ⊢ ((𝑛 ∈ ℕs ∧ 𝑚 ∈ ℕs) → (𝑚 ≤s 𝑛 ∨ 𝑛 ≤s 𝑚))
8		nnn0s 28718	. . . . . . . . 9 ⊢ (𝑚 ∈ ℕs → 𝑚 ∈ ℕ0s)
9		nnn0s 28718	. . . . . . . . 9 ⊢ (𝑛 ∈ ℕs → 𝑛 ∈ ℕ0s)
10		n0subs 28754	. . . . . . . . 9 ⊢ ((𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s) → (𝑚 ≤s 𝑛 ↔ (𝑛 −s 𝑚) ∈ ℕ0s))
11	8, 9, 10	syl2anr 609	. . . . . . . 8 ⊢ ((𝑛 ∈ ℕs ∧ 𝑚 ∈ ℕs) → (𝑚 ≤s 𝑛 ↔ (𝑛 −s 𝑚) ∈ ℕ0s))
12		n0subs 28754	. . . . . . . . . 10 ⊢ ((𝑛 ∈ ℕ0s ∧ 𝑚 ∈ ℕ0s) → (𝑛 ≤s 𝑚 ↔ (𝑚 −s 𝑛) ∈ ℕ0s))
13	9, 8, 12	syl2an 608	. . . . . . . . 9 ⊢ ((𝑛 ∈ ℕs ∧ 𝑚 ∈ ℕs) → (𝑛 ≤s 𝑚 ↔ (𝑚 −s 𝑛) ∈ ℕ0s))
14	2	adantr 486	. . . . . . . . . . 11 ⊢ ((𝑛 ∈ ℕs ∧ 𝑚 ∈ ℕs) → 𝑛 ∈ No)
15	3	adantl 487	. . . . . . . . . . 11 ⊢ ((𝑛 ∈ ℕs ∧ 𝑚 ∈ ℕs) → 𝑚 ∈ No)
16	14, 15	negsubsdi2d 28471	. . . . . . . . . 10 ⊢ ((𝑛 ∈ ℕs ∧ 𝑚 ∈ ℕs) → ( -s‘(𝑛 −s 𝑚)) = (𝑚 −s 𝑛))
17	16	eleq1d 2845	. . . . . . . . 9 ⊢ ((𝑛 ∈ ℕs ∧ 𝑚 ∈ ℕs) → (( -s‘(𝑛 −s 𝑚)) ∈ ℕ0s ↔ (𝑚 −s 𝑛) ∈ ℕ0s))
18	13, 17	bitr4d 285	. . . . . . . 8 ⊢ ((𝑛 ∈ ℕs ∧ 𝑚 ∈ ℕs) → (𝑛 ≤s 𝑚 ↔ ( -s‘(𝑛 −s 𝑚)) ∈ ℕ0s))
19	11, 18	orbi12d 932	. . . . . . 7 ⊢ ((𝑛 ∈ ℕs ∧ 𝑚 ∈ ℕs) → ((𝑚 ≤s 𝑛 ∨ 𝑛 ≤s 𝑚) ↔ ((𝑛 −s 𝑚) ∈ ℕ0s ∨ ( -s‘(𝑛 −s 𝑚)) ∈ ℕ0s)))
20	7, 19	mpbid 235	. . . . . 6 ⊢ ((𝑛 ∈ ℕs ∧ 𝑚 ∈ ℕs) → ((𝑛 −s 𝑚) ∈ ℕ0s ∨ ( -s‘(𝑛 −s 𝑚)) ∈ ℕ0s))
21	5, 20	jca 521	. . . . 5 ⊢ ((𝑛 ∈ ℕs ∧ 𝑚 ∈ ℕs) → ((𝑛 −s 𝑚) ∈ No ∧ ((𝑛 −s 𝑚) ∈ ℕ0s ∨ ( -s‘(𝑛 −s 𝑚)) ∈ ℕ0s)))
22		eleq1 2848	. . . . . 6 ⊢ (𝐴 = (𝑛 −s 𝑚) → (𝐴 ∈ No ↔ (𝑛 −s 𝑚) ∈ No))
23		eleq1 2848	. . . . . . 7 ⊢ (𝐴 = (𝑛 −s 𝑚) → (𝐴 ∈ ℕ0s ↔ (𝑛 −s 𝑚) ∈ ℕ0s))
24		fveq2 6885	. . . . . . . 8 ⊢ (𝐴 = (𝑛 −s 𝑚) → ( -s‘𝐴) = ( -s‘(𝑛 −s 𝑚)))
25	24	eleq1d 2845	. . . . . . 7 ⊢ (𝐴 = (𝑛 −s 𝑚) → (( -s‘𝐴) ∈ ℕ0s ↔ ( -s‘(𝑛 −s 𝑚)) ∈ ℕ0s))
26	23, 25	orbi12d 932	. . . . . 6 ⊢ (𝐴 = (𝑛 −s 𝑚) → ((𝐴 ∈ ℕ0s ∨ ( -s‘𝐴) ∈ ℕ0s) ↔ ((𝑛 −s 𝑚) ∈ ℕ0s ∨ ( -s‘(𝑛 −s 𝑚)) ∈ ℕ0s)))
27	22, 26	anbi12d 644	. . . . 5 ⊢ (𝐴 = (𝑛 −s 𝑚) → ((𝐴 ∈ No ∧ (𝐴 ∈ ℕ0s ∨ ( -s‘𝐴) ∈ ℕ0s)) ↔ ((𝑛 −s 𝑚) ∈ No ∧ ((𝑛 −s 𝑚) ∈ ℕ0s ∨ ( -s‘(𝑛 −s 𝑚)) ∈ ℕ0s))))
28	21, 27	syl5ibrcom 250	. . . 4 ⊢ ((𝑛 ∈ ℕs ∧ 𝑚 ∈ ℕs) → (𝐴 = (𝑛 −s 𝑚) → (𝐴 ∈ No ∧ (𝐴 ∈ ℕ0s ∨ ( -s‘𝐴) ∈ ℕ0s))))
29	28	rexlimivv 3204	. . 3 ⊢ (∃𝑛 ∈ ℕs ∃𝑚 ∈ ℕs 𝐴 = (𝑛 −s 𝑚) → (𝐴 ∈ No ∧ (𝐴 ∈ ℕ0s ∨ ( -s‘𝐴) ∈ ℕ0s)))
30		n0p1nns 28762	. . . . . 6 ⊢ (𝐴 ∈ ℕ0s → (𝐴 +s 1s ) ∈ ℕs)
31		1nns 28740	. . . . . . 7 ⊢ 1s ∈ ℕs
32	31	a1i 11	. . . . . 6 ⊢ (𝐴 ∈ ℕ0s → 1s ∈ ℕs)
33		n0no 28714	. . . . . . . 8 ⊢ (𝐴 ∈ ℕ0s → 𝐴 ∈ No)
34		1no 28201	. . . . . . . 8 ⊢ 1s ∈ No
35		pncans 28463	. . . . . . . 8 ⊢ ((𝐴 ∈ No ∧ 1s ∈ No) → ((𝐴 +s 1s ) −s 1s ) = 𝐴)
36	33, 34, 35	sylancl 598	. . . . . . 7 ⊢ (𝐴 ∈ ℕ0s → ((𝐴 +s 1s ) −s 1s ) = 𝐴)
37	36	eqcomd 2766	. . . . . 6 ⊢ (𝐴 ∈ ℕ0s → 𝐴 = ((𝐴 +s 1s ) −s 1s ))
38		rspceov 7469	. . . . . 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 487	. . . 4 ⊢ ((𝐴 ∈ No ∧ 𝐴 ∈ ℕ0s) → ∃𝑛 ∈ ℕs ∃𝑚 ∈ ℕs 𝐴 = (𝑛 −s 𝑚))
41	31	a1i 11	. . . . 5 ⊢ ((𝐴 ∈ No ∧ ( -s‘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 28452	. . . . . . . 8 ⊢ (𝐴 ∈ No → ( 1s −s 𝐴) = ( 1s +s ( -s‘𝐴)))
45		negscl 28427	. . . . . . . . 9 ⊢ (𝐴 ∈ No → ( -s‘𝐴) ∈ No)
46	42, 45	addscomd 28358	. . . . . . . 8 ⊢ (𝐴 ∈ No → ( 1s +s ( -s‘𝐴)) = (( -s‘𝐴) +s 1s ))
47	44, 46	eqtrd 2795	. . . . . . 7 ⊢ (𝐴 ∈ No → ( 1s −s 𝐴) = (( -s‘𝐴) +s 1s ))
48	47	adantr 486	. . . . . 6 ⊢ ((𝐴 ∈ No ∧ ( -s‘𝐴) ∈ ℕ0s) → ( 1s −s 𝐴) = (( -s‘𝐴) +s 1s ))
49		n0p1nns 28762	. . . . . . 7 ⊢ (( -s‘𝐴) ∈ ℕ0s → (( -s‘𝐴) +s 1s ) ∈ ℕs)
50	49	adantl 487	. . . . . 6 ⊢ ((𝐴 ∈ No ∧ ( -s‘𝐴) ∈ ℕ0s) → (( -s‘𝐴) +s 1s ) ∈ ℕs)
51	48, 50	eqeltrd 2860	. . . . 5 ⊢ ((𝐴 ∈ No ∧ ( -s‘𝐴) ∈ ℕ0s) → ( 1s −s 𝐴) ∈ ℕs)
52	42, 43	nncansd 28488	. . . . . . 7 ⊢ (𝐴 ∈ No → ( 1s −s ( 1s −s 𝐴)) = 𝐴)
53	52	eqcomd 2766	. . . . . 6 ⊢ (𝐴 ∈ No → 𝐴 = ( 1s −s ( 1s −s 𝐴)))
54	53	adantr 486	. . . . 5 ⊢ ((𝐴 ∈ No ∧ ( -s‘𝐴) ∈ ℕ0s) → 𝐴 = ( 1s −s ( 1s −s 𝐴)))
55		rspceov 7469	. . . . 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 ∧ ( -s‘𝐴) ∈ ℕ0s) → ∃𝑛 ∈ ℕs ∃𝑚 ∈ ℕs 𝐴 = (𝑛 −s 𝑚))
57	40, 56	jaodan 972	. . 3 ⊢ ((𝐴 ∈ No ∧ (𝐴 ∈ ℕ0s ∨ ( -s‘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 ∨ ( -s‘𝐴) ∈ ℕ0s)))
59	1, 58	bitri 278	1 ⊢ (𝐴 ∈ ℤs ↔ (𝐴 ∈ No ∧ (𝐴 ∈ ℕ0s ∨ ( -s‘𝐴) ∈ ℕ0s)))

This proof depends on syntax axioms:   ↔ wb 209   ∧ wa 401   ∨ wo 861   = wceq 1569   ∈ wcel 2144  ∃wrex 3086   class class class wbr 5102  ‘cfv 6538  (class class class)co 7420  Nocsur 28002   ≤s cles 28106   1_s c1s 28197   +_s cadds 28350   -_scnegs 28410   −_s csubs 28411  ℕ_0scn0s 28703  ℕ_scnns 28704  ℤ_sczs 28769

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7751

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-pss 4014  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-tp 4589  df-op 4835  df-ot 4858  df-uni 4867  df-int 4907  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5545  df-eprel 5550  df-po 5558  df-so 5559  df-fr 5603  df-se 5604  df-we 5605  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  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 7377  df-ov 7423  df-oprab 7424  df-mpo 7425  df-om 7878  df-1st 8001  df-2nd 8002  df-frecs 8302  df-wrecs 8333  df-recs 8382  df-rdg 8421  df-1o 8479  df-2o 8480  df-nadd 8678  df-no 28005  df-lts 28006  df-bday 28007  df-les 28107  df-slts 28149  df-cuts 28151  df-0s 28198  df-1s 28199  df-made 28218  df-old 28219  df-left 28221  df-right 28222  df-norec 28329  df-norec2 28340  df-adds 28351  df-negs 28412  df-subs 28413  df-n0s 28705  df-nns 28706  df-zs 28770

This theorem is used by:  elzs2  28790  zsbday  28797  zcuts  28798  zcuts0  28799