Theorem n0addscl 28340

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 7366	. . . . 5 ⊢ (𝑛 = 0s → (𝐴 +s 𝑛) = (𝐴 +s 0s ))
2	1	eleq1d 2821	. . . 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 7366	. . . . 5 ⊢ (𝑛 = 𝑚 → (𝐴 +s 𝑛) = (𝐴 +s 𝑚))
5	4	eleq1d 2821	. . . 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 7366	. . . . 5 ⊢ (𝑛 = (𝑚 +s 1s ) → (𝐴 +s 𝑛) = (𝐴 +s (𝑚 +s 1s )))
8	7	eleq1d 2821	. . . 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 7366	. . . . 5 ⊢ (𝑛 = 𝐵 → (𝐴 +s 𝑛) = (𝐴 +s 𝐵))
11	10	eleq1d 2821	. . . 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		n0no 28319	. . . . 5 ⊢ (𝐴 ∈ ℕ0s → 𝐴 ∈ No )
14	13	addsridd 27961	. . . 4 ⊢ (𝐴 ∈ ℕ0s → (𝐴 +s 0s ) = 𝐴)
15		id 22	. . . 4 ⊢ (𝐴 ∈ ℕ0s → 𝐴 ∈ ℕ0s)
16	14, 15	eqeltrd 2836	. . 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		n0no 28319	. . . . . . . . . 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		1no 27806	. . . . . . . . 9 ⊢ 1s ∈ No
23	22	a1i 11	. . . . . . . 8 ⊢ (((𝐴 ∈ ℕ0s ∧ 𝑚 ∈ ℕ0s) ∧ (𝐴 +s 𝑚) ∈ ℕ0s) → 1s ∈ No )
24	18, 21, 23	addsassd 28002	. . . . . . 7 ⊢ (((𝐴 ∈ ℕ0s ∧ 𝑚 ∈ ℕ0s) ∧ (𝐴 +s 𝑚) ∈ ℕ0s) → ((𝐴 +s 𝑚) +s 1s ) = (𝐴 +s (𝑚 +s 1s )))
25		peano2n0s 28326	. . . . . . . 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 2837	. . . . . 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 28329	. 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 1541   ∈ wcel 2113  (class class class)co 7358   No csur 27607   0_s c0s 27801   1_s c1s 27802   +_s cadds 27955  ℕ_0scn0s 28308

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 2184  ax-ext 2708  ax-rep 5224  ax-sep 5241  ax-nul 5251  ax-pow 5310  ax-pr 5377  ax-un 7680

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 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-rmo 3350  df-reu 3351  df-rab 3400  df-v 3442  df-sbc 3741  df-csb 3850  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-pss 3921  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-tp 4585  df-op 4587  df-ot 4589  df-uni 4864  df-int 4903  df-iun 4948  df-br 5099  df-opab 5161  df-mpt 5180  df-tr 5206  df-id 5519  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5577  df-se 5578  df-we 5579  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-pred 6259  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-riota 7315  df-ov 7361  df-oprab 7362  df-mpo 7363  df-om 7809  df-1st 7933  df-2nd 7934  df-frecs 8223  df-wrecs 8254  df-recs 8303  df-rdg 8341  df-1o 8397  df-2o 8398  df-nadd 8594  df-no 27610  df-lts 27611  df-bday 27612  df-les 27713  df-slts 27754  df-cuts 27756  df-0s 27803  df-1s 27804  df-made 27823  df-old 27824  df-left 27826  df-right 27827  df-norec2 27945  df-adds 27956  df-n0s 28310

This theorem is referenced by:  n0mulscl  28341  nnaddscl  28342  expadds  28431  pw2divscan4d  28440  addhalfcut  28455  bdaypw2n0bndlem  28459  bdaypw2bnd  28461  bdayfinbndlem1  28463  z12bdaylem2  28467  z12addscl  28473  z12sge0  28479