Theorem n0addscl 28347

Description: The non-negative surreal integers are closed under addition. (Contributed by Scott Fenton, 15-Apr-2025.)

Assertion

Ref	Expression
n0addscl	⊢ ((𝐴 ∈ ℕ0s ∧ 𝐵 ∈ ℕ0s) → (𝐴 +s 𝐵) ∈ ℕ0s)

Proof of Theorem n0addscl

Dummy variables 𝑛 𝑚 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		oveq2 7439	. . . . 5 ⊢ (𝑛 = 0s → (𝐴 +s 𝑛) = (𝐴 +s 0s ))
2	1	eleq1d 2826	. . . 4 ⊢ (𝑛 = 0s → ((𝐴 +s 𝑛) ∈ ℕ0s ↔ (𝐴 +s 0s ) ∈ ℕ0s))
3	2	imbi2d 340	. . 3 ⊢ (𝑛 = 0s → ((𝐴 ∈ ℕ0s → (𝐴 +s 𝑛) ∈ ℕ0s) ↔ (𝐴 ∈ ℕ0s → (𝐴 +s 0s ) ∈ ℕ0s)))
4		oveq2 7439	. . . . 5 ⊢ (𝑛 = 𝑚 → (𝐴 +s 𝑛) = (𝐴 +s 𝑚))
5	4	eleq1d 2826	. . . 4 ⊢ (𝑛 = 𝑚 → ((𝐴 +s 𝑛) ∈ ℕ0s ↔ (𝐴 +s 𝑚) ∈ ℕ0s))
6	5	imbi2d 340	. . 3 ⊢ (𝑛 = 𝑚 → ((𝐴 ∈ ℕ0s → (𝐴 +s 𝑛) ∈ ℕ0s) ↔ (𝐴 ∈ ℕ0s → (𝐴 +s 𝑚) ∈ ℕ0s)))
7		oveq2 7439	. . . . 5 ⊢ (𝑛 = (𝑚 +s 1s ) → (𝐴 +s 𝑛) = (𝐴 +s (𝑚 +s 1s )))
8	7	eleq1d 2826	. . . 4 ⊢ (𝑛 = (𝑚 +s 1s ) → ((𝐴 +s 𝑛) ∈ ℕ0s ↔ (𝐴 +s (𝑚 +s 1s )) ∈ ℕ0s))
9	8	imbi2d 340	. . 3 ⊢ (𝑛 = (𝑚 +s 1s ) → ((𝐴 ∈ ℕ0s → (𝐴 +s 𝑛) ∈ ℕ0s) ↔ (𝐴 ∈ ℕ0s → (𝐴 +s (𝑚 +s 1s )) ∈ ℕ0s)))
10		oveq2 7439	. . . . 5 ⊢ (𝑛 = 𝐵 → (𝐴 +s 𝑛) = (𝐴 +s 𝐵))
11	10	eleq1d 2826	. . . 4 ⊢ (𝑛 = 𝐵 → ((𝐴 +s 𝑛) ∈ ℕ0s ↔ (𝐴 +s 𝐵) ∈ ℕ0s))
12	11	imbi2d 340	. . 3 ⊢ (𝑛 = 𝐵 → ((𝐴 ∈ ℕ0s → (𝐴 +s 𝑛) ∈ ℕ0s) ↔ (𝐴 ∈ ℕ0s → (𝐴 +s 𝐵) ∈ ℕ0s)))
13		n0sno 28328	. . . . 5 ⊢ (𝐴 ∈ ℕ0s → 𝐴 ∈ No )
14	13	addsridd 27998	. . . 4 ⊢ (𝐴 ∈ ℕ0s → (𝐴 +s 0s ) = 𝐴)
15		id 22	. . . 4 ⊢ (𝐴 ∈ ℕ0s → 𝐴 ∈ ℕ0s)
16	14, 15	eqeltrd 2841	. . 3 ⊢ (𝐴 ∈ ℕ0s → (𝐴 +s 0s ) ∈ ℕ0s)
17	13	adantr 480	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℕ0s ∧ 𝑚 ∈ ℕ0s) → 𝐴 ∈ No )
18	17	adantr 480	. . . . . . . 8 ⊢ (((𝐴 ∈ ℕ0s ∧ 𝑚 ∈ ℕ0s) ∧ (𝐴 +s 𝑚) ∈ ℕ0s) → 𝐴 ∈ No )
19		n0sno 28328	. . . . . . . . . 10 ⊢ (𝑚 ∈ ℕ0s → 𝑚 ∈ No )
20	19	adantl 481	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℕ0s ∧ 𝑚 ∈ ℕ0s) → 𝑚 ∈ No )
21	20	adantr 480	. . . . . . . 8 ⊢ (((𝐴 ∈ ℕ0s ∧ 𝑚 ∈ ℕ0s) ∧ (𝐴 +s 𝑚) ∈ ℕ0s) → 𝑚 ∈ No )
22		1sno 27872	. . . . . . . . 9 ⊢ 1s ∈ No
23	22	a1i 11	. . . . . . . 8 ⊢ (((𝐴 ∈ ℕ0s ∧ 𝑚 ∈ ℕ0s) ∧ (𝐴 +s 𝑚) ∈ ℕ0s) → 1s ∈ No )
24	18, 21, 23	addsassd 28039	. . . . . . 7 ⊢ (((𝐴 ∈ ℕ0s ∧ 𝑚 ∈ ℕ0s) ∧ (𝐴 +s 𝑚) ∈ ℕ0s) → ((𝐴 +s 𝑚) +s 1s ) = (𝐴 +s (𝑚 +s 1s )))
25		peano2n0s 28335	. . . . . . . 8 ⊢ ((𝐴 +s 𝑚) ∈ ℕ0s → ((𝐴 +s 𝑚) +s 1s ) ∈ ℕ0s)
26	25	adantl 481	. . . . . . 7 ⊢ (((𝐴 ∈ ℕ0s ∧ 𝑚 ∈ ℕ0s) ∧ (𝐴 +s 𝑚) ∈ ℕ0s) → ((𝐴 +s 𝑚) +s 1s ) ∈ ℕ0s)
27	24, 26	eqeltrrd 2842	. . . . . 6 ⊢ (((𝐴 ∈ ℕ0s ∧ 𝑚 ∈ ℕ0s) ∧ (𝐴 +s 𝑚) ∈ ℕ0s) → (𝐴 +s (𝑚 +s 1s )) ∈ ℕ0s)
28	27	ex 412	. . . . 5 ⊢ ((𝐴 ∈ ℕ0s ∧ 𝑚 ∈ ℕ0s) → ((𝐴 +s 𝑚) ∈ ℕ0s → (𝐴 +s (𝑚 +s 1s )) ∈ ℕ0s))
29	28	expcom 413	. . . 4 ⊢ (𝑚 ∈ ℕ0s → (𝐴 ∈ ℕ0s → ((𝐴 +s 𝑚) ∈ ℕ0s → (𝐴 +s (𝑚 +s 1s )) ∈ ℕ0s)))
30	29	a2d 29	. . 3 ⊢ (𝑚 ∈ ℕ0s → ((𝐴 ∈ ℕ0s → (𝐴 +s 𝑚) ∈ ℕ0s) → (𝐴 ∈ ℕ0s → (𝐴 +s (𝑚 +s 1s )) ∈ ℕ0s)))
31	3, 6, 9, 12, 16, 30	n0sind 28337	. 2 ⊢ (𝐵 ∈ ℕ0s → (𝐴 ∈ ℕ0s → (𝐴 +s 𝐵) ∈ ℕ0s))
32	31	impcom 407	1 ⊢ ((𝐴 ∈ ℕ0s ∧ 𝐵 ∈ ℕ0s) → (𝐴 +s 𝐵) ∈ ℕ0s)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2108  (class class class)co 7431   No csur 27684   0_s c0s 27867   1_s c1s 27868   +_s cadds 27992  ℕ_0scnn0s 28318

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-norec2 27982  df-adds 27993  df-n0s 28320

This theorem is referenced by:  n0mulscl  28348  nnaddscl  28349  addhalfcut  28419