Theorem elzn0s 28323

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 28309	. 2 ⊢ (𝐴 ∈ ℤs ↔ ∃𝑛 ∈ ℕs ∃𝑚 ∈ ℕs 𝐴 = (𝑛 -s 𝑚))
2		nnsno 28254	. . . . . . 7 ⊢ (𝑛 ∈ ℕs → 𝑛 ∈ No )
3		nnsno 28254	. . . . . . 7 ⊢ (𝑚 ∈ ℕs → 𝑚 ∈ No )
4		subscl 28003	. . . . . . 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 27704	. . . . . . . 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 28257	. . . . . . . . 9 ⊢ (𝑚 ∈ ℕs → 𝑚 ∈ ℕ0s)
9		nnn0s 28257	. . . . . . . . 9 ⊢ (𝑛 ∈ ℕs → 𝑛 ∈ ℕ0s)
10		n0subs 28290	. . . . . . . . 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 28290	. . . . . . . . . 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 28021	. . . . . . . . . 10 ⊢ ((𝑛 ∈ ℕs ∧ 𝑚 ∈ ℕs) → ( -us ‘(𝑛 -s 𝑚)) = (𝑚 -s 𝑛))
17	16	eleq1d 2818	. . . . . . . . 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 2821	. . . . . 6 ⊢ (𝐴 = (𝑛 -s 𝑚) → (𝐴 ∈ No ↔ (𝑛 -s 𝑚) ∈ No ))
23		eleq1 2821	. . . . . . 7 ⊢ (𝐴 = (𝑛 -s 𝑚) → (𝐴 ∈ ℕ0s ↔ (𝑛 -s 𝑚) ∈ ℕ0s))
24		fveq2 6828	. . . . . . . 8 ⊢ (𝐴 = (𝑛 -s 𝑚) → ( -us ‘𝐴) = ( -us ‘(𝑛 -s 𝑚)))
25	24	eleq1d 2818	. . . . . . 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 3175	. . 3 ⊢ (∃𝑛 ∈ ℕs ∃𝑚 ∈ ℕs 𝐴 = (𝑛 -s 𝑚) → (𝐴 ∈ No ∧ (𝐴 ∈ ℕ0s ∨ ( -us ‘𝐴) ∈ ℕ0s)))
30		n0p1nns 28297	. . . . . 6 ⊢ (𝐴 ∈ ℕ0s → (𝐴 +s 1s ) ∈ ℕs)
31		1nns 28278	. . . . . . 7 ⊢ 1s ∈ ℕs
32	31	a1i 11	. . . . . 6 ⊢ (𝐴 ∈ ℕ0s → 1s ∈ ℕs)
33		n0sno 28253	. . . . . . . 8 ⊢ (𝐴 ∈ ℕ0s → 𝐴 ∈ No )
34		1sno 27772	. . . . . . . 8 ⊢ 1s ∈ No
35		pncans 28013	. . . . . . . 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 2739	. . . . . 6 ⊢ (𝐴 ∈ ℕ0s → 𝐴 = ((𝐴 +s 1s ) -s 1s ))
38		rspceov 7401	. . . . . 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 28002	. . . . . . . 8 ⊢ (𝐴 ∈ No → ( 1s -s 𝐴) = ( 1s +s ( -us ‘𝐴)))
45		negscl 27979	. . . . . . . . 9 ⊢ (𝐴 ∈ No → ( -us ‘𝐴) ∈ No )
46	42, 45	addscomd 27911	. . . . . . . 8 ⊢ (𝐴 ∈ No → ( 1s +s ( -us ‘𝐴)) = (( -us ‘𝐴) +s 1s ))
47	44, 46	eqtrd 2768	. . . . . . 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 28297	. . . . . . 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 2833	. . . . 5 ⊢ ((𝐴 ∈ No ∧ ( -us ‘𝐴) ∈ ℕ0s) → ( 1s -s 𝐴) ∈ ℕs)
52	42, 43	nncansd 28037	. . . . . . 7 ⊢ (𝐴 ∈ No → ( 1s -s ( 1s -s 𝐴)) = 𝐴)
53	52	eqcomd 2739	. . . . . 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 7401	. . . . 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 1541   ∈ wcel 2113  ∃wrex 3057   class class class wbr 5093  ‘cfv 6486  (class class class)co 7352   No csur 27579   ≤s csle 27684   1_s c1s 27768   +_s cadds 27903   -u_s cnegs 27962   -_s csubs 27963  ℕ_0scnn0s 28243  ℕ_scnns 28244  ℤ_sczs 28303

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705  ax-rep 5219  ax-sep 5236  ax-nul 5246  ax-pow 5305  ax-pr 5372  ax-un 7674

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2882  df-ne 2930  df-ral 3049  df-rex 3058  df-rmo 3347  df-reu 3348  df-rab 3397  df-v 3439  df-sbc 3738  df-csb 3847  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-pss 3918  df-nul 4283  df-if 4475  df-pw 4551  df-sn 4576  df-pr 4578  df-tp 4580  df-op 4582  df-ot 4584  df-uni 4859  df-int 4898  df-iun 4943  df-br 5094  df-opab 5156  df-mpt 5175  df-tr 5201  df-id 5514  df-eprel 5519  df-po 5527  df-so 5528  df-fr 5572  df-se 5573  df-we 5574  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-pred 6253  df-ord 6314  df-on 6315  df-lim 6316  df-suc 6317  df-iota 6442  df-fun 6488  df-fn 6489  df-f 6490  df-f1 6491  df-fo 6492  df-f1o 6493  df-fv 6494  df-riota 7309  df-ov 7355  df-oprab 7356  df-mpo 7357  df-om 7803  df-1st 7927  df-2nd 7928  df-frecs 8217  df-wrecs 8248  df-recs 8297  df-rdg 8335  df-1o 8391  df-2o 8392  df-nadd 8587  df-no 27582  df-slt 27583  df-bday 27584  df-sle 27685  df-sslt 27722  df-scut 27724  df-0s 27769  df-1s 27770  df-made 27789  df-old 27790  df-left 27792  df-right 27793  df-norec 27882  df-norec2 27893  df-adds 27904  df-negs 27964  df-subs 27965  df-n0s 28245  df-nns 28246  df-zs 28304

This theorem is referenced by:  elzs2  28324  zsbday  28331  zscut  28332