Theorem n0addscl 28354

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 7364	. . . . 5 ⊢ (𝑛 = 0s → (𝐴 +s 𝑛) = (𝐴 +s 0s ))
2	1	eleq1d 2824	. . . 4 ⊢ (𝑛 = 0s → ((𝐴 +s 𝑛) ∈ ℕ0s ↔ (𝐴 +s 0s ) ∈ ℕ0s))
3	2	imbi2d 341	. . 3 ⊢ (𝑛 = 0s → ((𝐴 ∈ ℕ0s → (𝐴 +s 𝑛) ∈ ℕ0s) ↔ (𝐴 ∈ ℕ0s → (𝐴 +s 0s ) ∈ ℕ0s)))
4		oveq2 7364	. . . . 5 ⊢ (𝑛 = 𝑚 → (𝐴 +s 𝑛) = (𝐴 +s 𝑚))
5	4	eleq1d 2824	. . . 4 ⊢ (𝑛 = 𝑚 → ((𝐴 +s 𝑛) ∈ ℕ0s ↔ (𝐴 +s 𝑚) ∈ ℕ0s))
6	5	imbi2d 341	. . 3 ⊢ (𝑛 = 𝑚 → ((𝐴 ∈ ℕ0s → (𝐴 +s 𝑛) ∈ ℕ0s) ↔ (𝐴 ∈ ℕ0s → (𝐴 +s 𝑚) ∈ ℕ0s)))
7		oveq2 7364	. . . . 5 ⊢ (𝑛 = (𝑚 +s 1s ) → (𝐴 +s 𝑛) = (𝐴 +s (𝑚 +s 1s )))
8	7	eleq1d 2824	. . . 4 ⊢ (𝑛 = (𝑚 +s 1s ) → ((𝐴 +s 𝑛) ∈ ℕ0s ↔ (𝐴 +s (𝑚 +s 1s )) ∈ ℕ0s))
9	8	imbi2d 341	. . 3 ⊢ (𝑛 = (𝑚 +s 1s ) → ((𝐴 ∈ ℕ0s → (𝐴 +s 𝑛) ∈ ℕ0s) ↔ (𝐴 ∈ ℕ0s → (𝐴 +s (𝑚 +s 1s )) ∈ ℕ0s)))
10		oveq2 7364	. . . . 5 ⊢ (𝑛 = 𝐵 → (𝐴 +s 𝑛) = (𝐴 +s 𝐵))
11	10	eleq1d 2824	. . . 4 ⊢ (𝑛 = 𝐵 → ((𝐴 +s 𝑛) ∈ ℕ0s ↔ (𝐴 +s 𝐵) ∈ ℕ0s))
12	11	imbi2d 341	. . 3 ⊢ (𝑛 = 𝐵 → ((𝐴 ∈ ℕ0s → (𝐴 +s 𝑛) ∈ ℕ0s) ↔ (𝐴 ∈ ℕ0s → (𝐴 +s 𝐵) ∈ ℕ0s)))
13		n0no 28333	. . . . 5 ⊢ (𝐴 ∈ ℕ0s → 𝐴 ∈ No )
14	13	addsridd 27975	. . . 4 ⊢ (𝐴 ∈ ℕ0s → (𝐴 +s 0s ) = 𝐴)
15		id 22	. . . 4 ⊢ (𝐴 ∈ ℕ0s → 𝐴 ∈ ℕ0s)
16	14, 15	eqeltrd 2839	. . 3 ⊢ (𝐴 ∈ ℕ0s → (𝐴 +s 0s ) ∈ ℕ0s)
17	13	adantr 481	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℕ0s ∧ 𝑚 ∈ ℕ0s) → 𝐴 ∈ No )
18	17	adantr 481	. . . . . . . 8 ⊢ (((𝐴 ∈ ℕ0s ∧ 𝑚 ∈ ℕ0s) ∧ (𝐴 +s 𝑚) ∈ ℕ0s) → 𝐴 ∈ No )
19		n0no 28333	. . . . . . . . . 10 ⊢ (𝑚 ∈ ℕ0s → 𝑚 ∈ No )
20	19	adantl 482	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℕ0s ∧ 𝑚 ∈ ℕ0s) → 𝑚 ∈ No )
21	20	adantr 481	. . . . . . . 8 ⊢ (((𝐴 ∈ ℕ0s ∧ 𝑚 ∈ ℕ0s) ∧ (𝐴 +s 𝑚) ∈ ℕ0s) → 𝑚 ∈ No )
22		1no 27820	. . . . . . . . 9 ⊢ 1s ∈ No
23	22	a1i 11	. . . . . . . 8 ⊢ (((𝐴 ∈ ℕ0s ∧ 𝑚 ∈ ℕ0s) ∧ (𝐴 +s 𝑚) ∈ ℕ0s) → 1s ∈ No )
24	18, 21, 23	addsassd 28016	. . . . . . 7 ⊢ (((𝐴 ∈ ℕ0s ∧ 𝑚 ∈ ℕ0s) ∧ (𝐴 +s 𝑚) ∈ ℕ0s) → ((𝐴 +s 𝑚) +s 1s ) = (𝐴 +s (𝑚 +s 1s )))
25		peano2n0s 28340	. . . . . . . 8 ⊢ ((𝐴 +s 𝑚) ∈ ℕ0s → ((𝐴 +s 𝑚) +s 1s ) ∈ ℕ0s)
26	25	adantl 482	. . . . . . 7 ⊢ (((𝐴 ∈ ℕ0s ∧ 𝑚 ∈ ℕ0s) ∧ (𝐴 +s 𝑚) ∈ ℕ0s) → ((𝐴 +s 𝑚) +s 1s ) ∈ ℕ0s)
27	24, 26	eqeltrrd 2840	. . . . . 6 ⊢ (((𝐴 ∈ ℕ0s ∧ 𝑚 ∈ ℕ0s) ∧ (𝐴 +s 𝑚) ∈ ℕ0s) → (𝐴 +s (𝑚 +s 1s )) ∈ ℕ0s)
28	27	ex 413	. . . . 5 ⊢ ((𝐴 ∈ ℕ0s ∧ 𝑚 ∈ ℕ0s) → ((𝐴 +s 𝑚) ∈ ℕ0s → (𝐴 +s (𝑚 +s 1s )) ∈ ℕ0s))
29	28	expcom 414	. . . 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 28343	. 2 ⊢ (𝐵 ∈ ℕ0s → (𝐴 ∈ ℕ0s → (𝐴 +s 𝐵) ∈ ℕ0s))
32	31	impcom 408	1 ⊢ ((𝐴 ∈ ℕ0s ∧ 𝐵 ∈ ℕ0s) → (𝐴 +s 𝐵) ∈ ℕ0s)

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1547   ∈ wcel 2119  (class class class)co 7356   No csur 27621   0_s c0s 27815   1_s c1s 27816   +_s cadds 27969  ℕ_0scn0s 28322

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-rep 5199  ax-sep 5218  ax-nul 5228  ax-pow 5294  ax-pr 5362  ax-un 7678

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3or 1093  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ne 2935  df-ral 3054  df-rex 3064  df-rmo 3344  df-reu 3345  df-rab 3392  df-v 3433  df-sbc 3724  df-csb 3832  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3903  df-nul 4262  df-if 4455  df-pw 4531  df-sn 4556  df-pr 4558  df-tp 4560  df-op 4562  df-ot 4564  df-uni 4839  df-int 4878  df-iun 4923  df-br 5073  df-opab 5135  df-mpt 5154  df-tr 5180  df-id 5513  df-eprel 5518  df-po 5526  df-so 5527  df-fr 5571  df-se 5572  df-we 5573  df-xp 5624  df-rel 5625  df-cnv 5626  df-co 5627  df-dm 5628  df-rn 5629  df-res 5630  df-ima 5631  df-pred 6252  df-ord 6313  df-on 6314  df-lim 6315  df-suc 6316  df-iota 6441  df-fun 6487  df-fn 6488  df-f 6489  df-f1 6490  df-fo 6491  df-f1o 6492  df-fv 6493  df-riota 7313  df-ov 7359  df-oprab 7360  df-mpo 7361  df-om 7807  df-1st 7931  df-2nd 7932  df-frecs 8221  df-wrecs 8252  df-recs 8301  df-rdg 8339  df-1o 8395  df-2o 8396  df-nadd 8592  df-no 27624  df-lts 27625  df-bday 27626  df-les 27727  df-slts 27768  df-cuts 27770  df-0s 27817  df-1s 27818  df-made 27837  df-old 27838  df-left 27840  df-right 27841  df-norec2 27959  df-adds 27970  df-n0s 28324

This theorem is referenced by:  n0mulscl  28355  nnaddscl  28356  expadds  28445  pw2divscan4d  28454  addhalfcut  28469  bdaypw2n0bndlem  28473  bdaypw2bnd  28475  bdayfinbndlem1  28477  z12bdaylem2  28481  z12addscl  28487  z12sge0  28493