Theorem n0addscl 28739

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 7432	. . . . 5 ⊢ (𝑛 = 0s → (𝐴 +s 𝑛) = (𝐴 +s 0s ))
2	1	eleq1d 2845	. . . 4 ⊢ (𝑛 = 0s → ((𝐴 +s 𝑛) ∈ ℕ0s ↔ (𝐴 +s 0s ) ∈ ℕ0s))
3	2	imbi2d 343	. . 3 ⊢ (𝑛 = 0s → ((𝐴 ∈ ℕ0s → (𝐴 +s 𝑛) ∈ ℕ0s) ↔ (𝐴 ∈ ℕ0s → (𝐴 +s 0s ) ∈ ℕ0s)))
4		oveq2 7432	. . . . 5 ⊢ (𝑛 = 𝑚 → (𝐴 +s 𝑛) = (𝐴 +s 𝑚))
5	4	eleq1d 2845	. . . 4 ⊢ (𝑛 = 𝑚 → ((𝐴 +s 𝑛) ∈ ℕ0s ↔ (𝐴 +s 𝑚) ∈ ℕ0s))
6	5	imbi2d 343	. . 3 ⊢ (𝑛 = 𝑚 → ((𝐴 ∈ ℕ0s → (𝐴 +s 𝑛) ∈ ℕ0s) ↔ (𝐴 ∈ ℕ0s → (𝐴 +s 𝑚) ∈ ℕ0s)))
7		oveq2 7432	. . . . 5 ⊢ (𝑛 = (𝑚 +s 1s ) → (𝐴 +s 𝑛) = (𝐴 +s (𝑚 +s 1s )))
8	7	eleq1d 2845	. . . 4 ⊢ (𝑛 = (𝑚 +s 1s ) → ((𝐴 +s 𝑛) ∈ ℕ0s ↔ (𝐴 +s (𝑚 +s 1s )) ∈ ℕ0s))
9	8	imbi2d 343	. . 3 ⊢ (𝑛 = (𝑚 +s 1s ) → ((𝐴 ∈ ℕ0s → (𝐴 +s 𝑛) ∈ ℕ0s) ↔ (𝐴 ∈ ℕ0s → (𝐴 +s (𝑚 +s 1s )) ∈ ℕ0s)))
10		oveq2 7432	. . . . 5 ⊢ (𝑛 = 𝐵 → (𝐴 +s 𝑛) = (𝐴 +s 𝐵))
11	10	eleq1d 2845	. . . 4 ⊢ (𝑛 = 𝐵 → ((𝐴 +s 𝑛) ∈ ℕ0s ↔ (𝐴 +s 𝐵) ∈ ℕ0s))
12	11	imbi2d 343	. . 3 ⊢ (𝑛 = 𝐵 → ((𝐴 ∈ ℕ0s → (𝐴 +s 𝑛) ∈ ℕ0s) ↔ (𝐴 ∈ ℕ0s → (𝐴 +s 𝐵) ∈ ℕ0s)))
13		n0no 28718	. . . . 5 ⊢ (𝐴 ∈ ℕ0s → 𝐴 ∈ No)
14	13	addsridd 28360	. . . 4 ⊢ (𝐴 ∈ ℕ0s → (𝐴 +s 0s ) = 𝐴)
15		id 23	. . . 4 ⊢ (𝐴 ∈ ℕ0s → 𝐴 ∈ ℕ0s)
16	14, 15	eqeltrd 2860	. . 3 ⊢ (𝐴 ∈ ℕ0s → (𝐴 +s 0s ) ∈ ℕ0s)
17	13	adantr 486	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℕ0s ∧ 𝑚 ∈ ℕ0s) → 𝐴 ∈ No)
18	17	adantr 486	. . . . . . . 8 ⊢ (((𝐴 ∈ ℕ0s ∧ 𝑚 ∈ ℕ0s) ∧ (𝐴 +s 𝑚) ∈ ℕ0s) → 𝐴 ∈ No)
19		n0no 28718	. . . . . . . . . 10 ⊢ (𝑚 ∈ ℕ0s → 𝑚 ∈ No)
20	19	adantl 487	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℕ0s ∧ 𝑚 ∈ ℕ0s) → 𝑚 ∈ No)
21	20	adantr 486	. . . . . . . 8 ⊢ (((𝐴 ∈ ℕ0s ∧ 𝑚 ∈ ℕ0s) ∧ (𝐴 +s 𝑚) ∈ ℕ0s) → 𝑚 ∈ No)
22		1no 28205	. . . . . . . . 9 ⊢ 1s ∈ No
23	22	a1i 11	. . . . . . . 8 ⊢ (((𝐴 ∈ ℕ0s ∧ 𝑚 ∈ ℕ0s) ∧ (𝐴 +s 𝑚) ∈ ℕ0s) → 1s ∈ No)
24	18, 21, 23	addsassd 28401	. . . . . . 7 ⊢ (((𝐴 ∈ ℕ0s ∧ 𝑚 ∈ ℕ0s) ∧ (𝐴 +s 𝑚) ∈ ℕ0s) → ((𝐴 +s 𝑚) +s 1s ) = (𝐴 +s (𝑚 +s 1s )))
25		peano2n0s 28725	. . . . . . . 8 ⊢ ((𝐴 +s 𝑚) ∈ ℕ0s → ((𝐴 +s 𝑚) +s 1s ) ∈ ℕ0s)
26	25	adantl 487	. . . . . . 7 ⊢ (((𝐴 ∈ ℕ0s ∧ 𝑚 ∈ ℕ0s) ∧ (𝐴 +s 𝑚) ∈ ℕ0s) → ((𝐴 +s 𝑚) +s 1s ) ∈ ℕ0s)
27	24, 26	eqeltrrd 2861	. . . . . 6 ⊢ (((𝐴 ∈ ℕ0s ∧ 𝑚 ∈ ℕ0s) ∧ (𝐴 +s 𝑚) ∈ ℕ0s) → (𝐴 +s (𝑚 +s 1s )) ∈ ℕ0s)
28	27	ex 418	. . . . 5 ⊢ ((𝐴 ∈ ℕ0s ∧ 𝑚 ∈ ℕ0s) → ((𝐴 +s 𝑚) ∈ ℕ0s → (𝐴 +s (𝑚 +s 1s )) ∈ ℕ0s))
29	28	expcom 419	. . . 4 ⊢ (𝑚 ∈ ℕ0s → (𝐴 ∈ ℕ0s → ((𝐴 +s 𝑚) ∈ ℕ0s → (𝐴 +s (𝑚 +s 1s )) ∈ ℕ0s)))
30	29	a2d 30	. . 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 28728	. 2 ⊢ (𝐵 ∈ ℕ0s → (𝐴 ∈ ℕ0s → (𝐴 +s 𝐵) ∈ ℕ0s))
32	31	impcom 413	1 ⊢ ((𝐴 ∈ ℕ0s ∧ 𝐵 ∈ ℕ0s) → (𝐴 +s 𝐵) ∈ ℕ0s)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1569   ∈ wcel 2144  (class class class)co 7424  Nocsur 28006   0_s c0s 28200   1_s c1s 28201   +_s cadds 28354  ℕ_0scn0s 28707

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5327  ax-pr 5391  ax-un 7755

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3848  df-ss 3897  df-pss 4013  df-dif 4046  df-in 4090  df-un 4150  df-nul 4278  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-tp 4589  df-op 4835  df-ot 4858  df-uni 4867  df-int 4907  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5546  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-se 5605  df-we 5606  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-pred 6307  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6497  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7381  df-ov 7427  df-oprab 7428  df-mpo 7429  df-om 7882  df-1st 8005  df-2nd 8006  df-frecs 8306  df-wrecs 8337  df-recs 8386  df-rdg 8425  df-1o 8483  df-2o 8484  df-nadd 8682  df-no 28009  df-lts 28010  df-bday 28011  df-les 28111  df-slts 28153  df-cuts 28155  df-0s 28202  df-1s 28203  df-made 28222  df-old 28223  df-left 28225  df-right 28226  df-norec2 28344  df-adds 28355  df-n0s 28709

This theorem is used by:  n0mulscl  28740  nnaddscl  28741  expadds  28830  pw2divscan4d  28839  addhalfcut  28854  bdaypw2n0bndlem  28858  bdaypw2bnd  28860  bdayfinbndlem1  28862  z12bdaylem2  28866  z12addscl  28872  z12sge0  28878