Theorem elzn0s 28817

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 28803	. 2 ⊢ (𝐴 ∈ ℤs ↔ ∃𝑛 ∈ ℕs ∃𝑚 ∈ ℕs 𝐴 = (𝑛 −s 𝑚))
2		nnno 28743	. . . . . . 7 ⊢ (𝑛 ∈ ℕs → 𝑛 ∈ No)
3		nnno 28743	. . . . . . 7 ⊢ (𝑚 ∈ ℕs → 𝑚 ∈ No)
4		subscl 28481	. . . . . . 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 28158	. . . . . . . 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 28746	. . . . . . . . 9 ⊢ (𝑚 ∈ ℕs → 𝑚 ∈ ℕ0s)
9		nnn0s 28746	. . . . . . . . 9 ⊢ (𝑛 ∈ ℕs → 𝑛 ∈ ℕ0s)
10		n0subs 28782	. . . . . . . . 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 28782	. . . . . . . . . 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 28499	. . . . . . . . . 10 ⊢ ((𝑛 ∈ ℕs ∧ 𝑚 ∈ ℕs) → ( -s‘(𝑛 −s 𝑚)) = (𝑚 −s 𝑛))
17	16	eleq1d 2849	. . . . . . . . 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 2852	. . . . . 6 ⊢ (𝐴 = (𝑛 −s 𝑚) → (𝐴 ∈ No ↔ (𝑛 −s 𝑚) ∈ No))
23		eleq1 2852	. . . . . . 7 ⊢ (𝐴 = (𝑛 −s 𝑚) → (𝐴 ∈ ℕ0s ↔ (𝑛 −s 𝑚) ∈ ℕ0s))
24		fveq2 6911	. . . . . . . 8 ⊢ (𝐴 = (𝑛 −s 𝑚) → ( -s‘𝐴) = ( -s‘(𝑛 −s 𝑚)))
25	24	eleq1d 2849	. . . . . . 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 3208	. . 3 ⊢ (∃𝑛 ∈ ℕs ∃𝑚 ∈ ℕs 𝐴 = (𝑛 −s 𝑚) → (𝐴 ∈ No ∧ (𝐴 ∈ ℕ0s ∨ ( -s‘𝐴) ∈ ℕ0s)))
30		n0p1nns 28790	. . . . . 6 ⊢ (𝐴 ∈ ℕ0s → (𝐴 +s 1s ) ∈ ℕs)
31		1nns 28768	. . . . . . 7 ⊢ 1s ∈ ℕs
32	31	a1i 11	. . . . . 6 ⊢ (𝐴 ∈ ℕ0s → 1s ∈ ℕs)
33		n0no 28742	. . . . . . . 8 ⊢ (𝐴 ∈ ℕ0s → 𝐴 ∈ No)
34		1no 28229	. . . . . . . 8 ⊢ 1s ∈ No
35		pncans 28491	. . . . . . . 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 2769	. . . . . 6 ⊢ (𝐴 ∈ ℕ0s → 𝐴 = ((𝐴 +s 1s ) −s 1s ))
38		rspceov 7495	. . . . . 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 1399	. . . . 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 28480	. . . . . . . 8 ⊢ (𝐴 ∈ No → ( 1s −s 𝐴) = ( 1s +s ( -s‘𝐴)))
45		negscl 28455	. . . . . . . . 9 ⊢ (𝐴 ∈ No → ( -s‘𝐴) ∈ No)
46	42, 45	addscomd 28386	. . . . . . . 8 ⊢ (𝐴 ∈ No → ( 1s +s ( -s‘𝐴)) = (( -s‘𝐴) +s 1s ))
47	44, 46	eqtrd 2798	. . . . . . 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 28790	. . . . . . 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 2864	. . . . 5 ⊢ ((𝐴 ∈ No ∧ ( -s‘𝐴) ∈ ℕ0s) → ( 1s −s 𝐴) ∈ ℕs)
52	42, 43	nncansd 28516	. . . . . . 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 486	. . . . 5 ⊢ ((𝐴 ∈ No ∧ ( -s‘𝐴) ∈ ℕ0s) → 𝐴 = ( 1s −s ( 1s −s 𝐴)))
55		rspceov 7495	. . . . 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 1399	. . . 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 1570   ∈ wcel 2147  ∃wrex 3090   class class class wbr 5106  ‘cfv 6561  (class class class)co 7446  Nocsur 28030   ≤s cles 28134   1_s c1s 28225   +_s cadds 28378   -_scnegs 28438   −_s csubs 28439  ℕ_0scn0s 28731  ℕ_scnns 28732  ℤ_sczs 28797

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5236  ax-sep 5253  ax-nul 5264  ax-pow 5332  ax-pr 5396  ax-un 7778

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-tp 4593  df-op 4839  df-ot 4862  df-uni 4871  df-int 4911  df-iun 4956  df-br 5107  df-opab 5172  df-mpt 5191  df-tr 5217  df-id 5551  df-eprel 5556  df-po 5564  df-so 5565  df-fr 5609  df-se 5610  df-we 5611  df-xp 5662  df-rel 5663  df-cnv 5664  df-co 5665  df-dm 5666  df-rn 5667  df-res 5668  df-ima 5669  df-pred 6325  df-ord 6386  df-on 6387  df-lim 6388  df-suc 6389  df-iota 6515  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7403  df-ov 7449  df-oprab 7450  df-mpo 7451  df-om 7905  df-1st 8028  df-2nd 8029  df-frecs 8330  df-wrecs 8361  df-recs 8410  df-rdg 8449  df-1o 8507  df-2o 8508  df-nadd 8706  df-no 28033  df-lts 28034  df-bday 28035  df-les 28135  df-slts 28177  df-cuts 28179  df-0s 28226  df-1s 28227  df-made 28246  df-old 28247  df-left 28249  df-right 28250  df-norec 28357  df-norec2 28368  df-adds 28379  df-negs 28440  df-subs 28441  df-n0s 28733  df-nns 28734  df-zs 28798

This theorem is used by:  elzs2  28818  zsbday  28825  zcuts  28826  zcuts0  28827