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Theorem elzn0s 28384
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 ∨ ( -us𝐴) ∈ ℕ0s)))

Proof of Theorem elzn0s
Dummy variables 𝑛 𝑚 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elzs 28370 . 2 (𝐴 ∈ ℤs ↔ ∃𝑛 ∈ ℕs𝑚 ∈ ℕs 𝐴 = (𝑛 -s 𝑚))
2 nnsno 28329 . . . . . . 7 (𝑛 ∈ ℕs𝑛 No )
3 nnsno 28329 . . . . . . 7 (𝑚 ∈ ℕs𝑚 No )
4 subscl 28092 . . . . . . 7 ((𝑛 No 𝑚 No ) → (𝑛 -s 𝑚) ∈ No )
52, 3, 4syl2an 596 . . . . . 6 ((𝑛 ∈ ℕs𝑚 ∈ ℕs) → (𝑛 -s 𝑚) ∈ No )
6 sletric 27809 . . . . . . . 8 ((𝑚 No 𝑛 No ) → (𝑚 ≤s 𝑛𝑛 ≤s 𝑚))
73, 2, 6syl2anr 597 . . . . . . 7 ((𝑛 ∈ ℕs𝑚 ∈ ℕs) → (𝑚 ≤s 𝑛𝑛 ≤s 𝑚))
8 nnn0s 28332 . . . . . . . . 9 (𝑚 ∈ ℕs𝑚 ∈ ℕ0s)
9 nnn0s 28332 . . . . . . . . 9 (𝑛 ∈ ℕs𝑛 ∈ ℕ0s)
10 n0subs 28360 . . . . . . . . 9 ((𝑚 ∈ ℕ0s𝑛 ∈ ℕ0s) → (𝑚 ≤s 𝑛 ↔ (𝑛 -s 𝑚) ∈ ℕ0s))
118, 9, 10syl2anr 597 . . . . . . . 8 ((𝑛 ∈ ℕs𝑚 ∈ ℕs) → (𝑚 ≤s 𝑛 ↔ (𝑛 -s 𝑚) ∈ ℕ0s))
12 n0subs 28360 . . . . . . . . . 10 ((𝑛 ∈ ℕ0s𝑚 ∈ ℕ0s) → (𝑛 ≤s 𝑚 ↔ (𝑚 -s 𝑛) ∈ ℕ0s))
139, 8, 12syl2an 596 . . . . . . . . 9 ((𝑛 ∈ ℕs𝑚 ∈ ℕs) → (𝑛 ≤s 𝑚 ↔ (𝑚 -s 𝑛) ∈ ℕ0s))
142adantr 480 . . . . . . . . . . 11 ((𝑛 ∈ ℕs𝑚 ∈ ℕs) → 𝑛 No )
153adantl 481 . . . . . . . . . . 11 ((𝑛 ∈ ℕs𝑚 ∈ ℕs) → 𝑚 No )
1614, 15negsubsdi2d 28110 . . . . . . . . . 10 ((𝑛 ∈ ℕs𝑚 ∈ ℕs) → ( -us ‘(𝑛 -s 𝑚)) = (𝑚 -s 𝑛))
1716eleq1d 2826 . . . . . . . . 9 ((𝑛 ∈ ℕs𝑚 ∈ ℕs) → (( -us ‘(𝑛 -s 𝑚)) ∈ ℕ0s ↔ (𝑚 -s 𝑛) ∈ ℕ0s))
1813, 17bitr4d 282 . . . . . . . 8 ((𝑛 ∈ ℕs𝑚 ∈ ℕs) → (𝑛 ≤s 𝑚 ↔ ( -us ‘(𝑛 -s 𝑚)) ∈ ℕ0s))
1911, 18orbi12d 919 . . . . . . 7 ((𝑛 ∈ ℕs𝑚 ∈ ℕs) → ((𝑚 ≤s 𝑛𝑛 ≤s 𝑚) ↔ ((𝑛 -s 𝑚) ∈ ℕ0s ∨ ( -us ‘(𝑛 -s 𝑚)) ∈ ℕ0s)))
207, 19mpbid 232 . . . . . 6 ((𝑛 ∈ ℕs𝑚 ∈ ℕs) → ((𝑛 -s 𝑚) ∈ ℕ0s ∨ ( -us ‘(𝑛 -s 𝑚)) ∈ ℕ0s))
215, 20jca 511 . . . . 5 ((𝑛 ∈ ℕs𝑚 ∈ ℕs) → ((𝑛 -s 𝑚) ∈ No ∧ ((𝑛 -s 𝑚) ∈ ℕ0s ∨ ( -us ‘(𝑛 -s 𝑚)) ∈ ℕ0s)))
22 eleq1 2829 . . . . . 6 (𝐴 = (𝑛 -s 𝑚) → (𝐴 No ↔ (𝑛 -s 𝑚) ∈ No ))
23 eleq1 2829 . . . . . . 7 (𝐴 = (𝑛 -s 𝑚) → (𝐴 ∈ ℕ0s ↔ (𝑛 -s 𝑚) ∈ ℕ0s))
24 fveq2 6906 . . . . . . . 8 (𝐴 = (𝑛 -s 𝑚) → ( -us𝐴) = ( -us ‘(𝑛 -s 𝑚)))
2524eleq1d 2826 . . . . . . 7 (𝐴 = (𝑛 -s 𝑚) → (( -us𝐴) ∈ ℕ0s ↔ ( -us ‘(𝑛 -s 𝑚)) ∈ ℕ0s))
2623, 25orbi12d 919 . . . . . 6 (𝐴 = (𝑛 -s 𝑚) → ((𝐴 ∈ ℕ0s ∨ ( -us𝐴) ∈ ℕ0s) ↔ ((𝑛 -s 𝑚) ∈ ℕ0s ∨ ( -us ‘(𝑛 -s 𝑚)) ∈ ℕ0s)))
2722, 26anbi12d 632 . . . . 5 (𝐴 = (𝑛 -s 𝑚) → ((𝐴 No ∧ (𝐴 ∈ ℕ0s ∨ ( -us𝐴) ∈ ℕ0s)) ↔ ((𝑛 -s 𝑚) ∈ No ∧ ((𝑛 -s 𝑚) ∈ ℕ0s ∨ ( -us ‘(𝑛 -s 𝑚)) ∈ ℕ0s))))
2821, 27syl5ibrcom 247 . . . 4 ((𝑛 ∈ ℕs𝑚 ∈ ℕs) → (𝐴 = (𝑛 -s 𝑚) → (𝐴 No ∧ (𝐴 ∈ ℕ0s ∨ ( -us𝐴) ∈ ℕ0s))))
2928rexlimivv 3201 . . 3 (∃𝑛 ∈ ℕs𝑚 ∈ ℕs 𝐴 = (𝑛 -s 𝑚) → (𝐴 No ∧ (𝐴 ∈ ℕ0s ∨ ( -us𝐴) ∈ ℕ0s)))
30 n0p1nns 28361 . . . . . 6 (𝐴 ∈ ℕ0s → (𝐴 +s 1s ) ∈ ℕs)
31 1nns 28352 . . . . . . 7 1s ∈ ℕs
3231a1i 11 . . . . . 6 (𝐴 ∈ ℕ0s → 1s ∈ ℕs)
33 n0sno 28328 . . . . . . . 8 (𝐴 ∈ ℕ0s𝐴 No )
34 1sno 27872 . . . . . . . 8 1s No
35 pncans 28102 . . . . . . . 8 ((𝐴 No ∧ 1s No ) → ((𝐴 +s 1s ) -s 1s ) = 𝐴)
3633, 34, 35sylancl 586 . . . . . . 7 (𝐴 ∈ ℕ0s → ((𝐴 +s 1s ) -s 1s ) = 𝐴)
3736eqcomd 2743 . . . . . 6 (𝐴 ∈ ℕ0s𝐴 = ((𝐴 +s 1s ) -s 1s ))
38 rspceov 7480 . . . . . 6 (((𝐴 +s 1s ) ∈ ℕs ∧ 1s ∈ ℕs𝐴 = ((𝐴 +s 1s ) -s 1s )) → ∃𝑛 ∈ ℕs𝑚 ∈ ℕs 𝐴 = (𝑛 -s 𝑚))
3930, 32, 37, 38syl3anc 1373 . . . . 5 (𝐴 ∈ ℕ0s → ∃𝑛 ∈ ℕs𝑚 ∈ ℕs 𝐴 = (𝑛 -s 𝑚))
4039adantl 481 . . . 4 ((𝐴 No 𝐴 ∈ ℕ0s) → ∃𝑛 ∈ ℕs𝑚 ∈ ℕs 𝐴 = (𝑛 -s 𝑚))
4131a1i 11 . . . . 5 ((𝐴 No ∧ ( -us𝐴) ∈ ℕ0s) → 1s ∈ ℕs)
4234a1i 11 . . . . . . . . 9 (𝐴 No → 1s No )
43 id 22 . . . . . . . . 9 (𝐴 No 𝐴 No )
4442, 43subsvald 28091 . . . . . . . 8 (𝐴 No → ( 1s -s 𝐴) = ( 1s +s ( -us𝐴)))
45 negscl 28068 . . . . . . . . 9 (𝐴 No → ( -us𝐴) ∈ No )
4642, 45addscomd 28000 . . . . . . . 8 (𝐴 No → ( 1s +s ( -us𝐴)) = (( -us𝐴) +s 1s ))
4744, 46eqtrd 2777 . . . . . . 7 (𝐴 No → ( 1s -s 𝐴) = (( -us𝐴) +s 1s ))
4847adantr 480 . . . . . 6 ((𝐴 No ∧ ( -us𝐴) ∈ ℕ0s) → ( 1s -s 𝐴) = (( -us𝐴) +s 1s ))
49 n0p1nns 28361 . . . . . . 7 (( -us𝐴) ∈ ℕ0s → (( -us𝐴) +s 1s ) ∈ ℕs)
5049adantl 481 . . . . . 6 ((𝐴 No ∧ ( -us𝐴) ∈ ℕ0s) → (( -us𝐴) +s 1s ) ∈ ℕs)
5148, 50eqeltrd 2841 . . . . 5 ((𝐴 No ∧ ( -us𝐴) ∈ ℕ0s) → ( 1s -s 𝐴) ∈ ℕs)
5242, 43nncansd 28126 . . . . . . 7 (𝐴 No → ( 1s -s ( 1s -s 𝐴)) = 𝐴)
5352eqcomd 2743 . . . . . 6 (𝐴 No 𝐴 = ( 1s -s ( 1s -s 𝐴)))
5453adantr 480 . . . . 5 ((𝐴 No ∧ ( -us𝐴) ∈ ℕ0s) → 𝐴 = ( 1s -s ( 1s -s 𝐴)))
55 rspceov 7480 . . . . 5 (( 1s ∈ ℕs ∧ ( 1s -s 𝐴) ∈ ℕs𝐴 = ( 1s -s ( 1s -s 𝐴))) → ∃𝑛 ∈ ℕs𝑚 ∈ ℕs 𝐴 = (𝑛 -s 𝑚))
5641, 51, 54, 55syl3anc 1373 . . . 4 ((𝐴 No ∧ ( -us𝐴) ∈ ℕ0s) → ∃𝑛 ∈ ℕs𝑚 ∈ ℕs 𝐴 = (𝑛 -s 𝑚))
5740, 56jaodan 960 . . 3 ((𝐴 No ∧ (𝐴 ∈ ℕ0s ∨ ( -us𝐴) ∈ ℕ0s)) → ∃𝑛 ∈ ℕs𝑚 ∈ ℕs 𝐴 = (𝑛 -s 𝑚))
5829, 57impbii 209 . 2 (∃𝑛 ∈ ℕs𝑚 ∈ ℕs 𝐴 = (𝑛 -s 𝑚) ↔ (𝐴 No ∧ (𝐴 ∈ ℕ0s ∨ ( -us𝐴) ∈ ℕ0s)))
591, 58bitri 275 1 (𝐴 ∈ ℤs ↔ (𝐴 No ∧ (𝐴 ∈ ℕ0s ∨ ( -us𝐴) ∈ ℕ0s)))
Colors of variables: wff setvar class
Syntax hints:  wb 206  wa 395  wo 848   = wceq 1540  wcel 2108  wrex 3070   class class class wbr 5143  cfv 6561  (class class class)co 7431   No csur 27684   ≤s csle 27789   1s c1s 27868   +s cadds 27992   -us cnegs 28051   -s csubs 28052  0scnn0s 28318  scnns 28319  sczs 28364
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-tp 4631  df-op 4633  df-ot 4635  df-uni 4908  df-int 4947  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-se 5638  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8014  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-2o 8507  df-nadd 8704  df-no 27687  df-slt 27688  df-bday 27689  df-sle 27790  df-sslt 27826  df-scut 27828  df-0s 27869  df-1s 27870  df-made 27886  df-old 27887  df-left 27889  df-right 27890  df-norec 27971  df-norec2 27982  df-adds 27993  df-negs 28053  df-subs 28054  df-n0s 28320  df-nns 28321  df-zs 28365
This theorem is referenced by:  elzs2  28385  zsbday  28392  zscut  28393
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