Theorem n0addscl 28425

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 7399	. . . . 5 ⊢ (𝑛 = 0s → (𝐴 +s 𝑛) = (𝐴 +s 0s ))
2	1	eleq1d 2846	. . . 4 ⊢ (𝑛 = 0s → ((𝐴 +s 𝑛) ∈ ℕ0s ↔ (𝐴 +s 0s ) ∈ ℕ0s))
3	2	imbi2d 342	. . 3 ⊢ (𝑛 = 0s → ((𝐴 ∈ ℕ0s → (𝐴 +s 𝑛) ∈ ℕ0s) ↔ (𝐴 ∈ ℕ0s → (𝐴 +s 0s ) ∈ ℕ0s)))
4		oveq2 7399	. . . . 5 ⊢ (𝑛 = 𝑚 → (𝐴 +s 𝑛) = (𝐴 +s 𝑚))
5	4	eleq1d 2846	. . . 4 ⊢ (𝑛 = 𝑚 → ((𝐴 +s 𝑛) ∈ ℕ0s ↔ (𝐴 +s 𝑚) ∈ ℕ0s))
6	5	imbi2d 342	. . 3 ⊢ (𝑛 = 𝑚 → ((𝐴 ∈ ℕ0s → (𝐴 +s 𝑛) ∈ ℕ0s) ↔ (𝐴 ∈ ℕ0s → (𝐴 +s 𝑚) ∈ ℕ0s)))
7		oveq2 7399	. . . . 5 ⊢ (𝑛 = (𝑚 +s 1s ) → (𝐴 +s 𝑛) = (𝐴 +s (𝑚 +s 1s )))
8	7	eleq1d 2846	. . . 4 ⊢ (𝑛 = (𝑚 +s 1s ) → ((𝐴 +s 𝑛) ∈ ℕ0s ↔ (𝐴 +s (𝑚 +s 1s )) ∈ ℕ0s))
9	8	imbi2d 342	. . 3 ⊢ (𝑛 = (𝑚 +s 1s ) → ((𝐴 ∈ ℕ0s → (𝐴 +s 𝑛) ∈ ℕ0s) ↔ (𝐴 ∈ ℕ0s → (𝐴 +s (𝑚 +s 1s )) ∈ ℕ0s)))
10		oveq2 7399	. . . . 5 ⊢ (𝑛 = 𝐵 → (𝐴 +s 𝑛) = (𝐴 +s 𝐵))
11	10	eleq1d 2846	. . . 4 ⊢ (𝑛 = 𝐵 → ((𝐴 +s 𝑛) ∈ ℕ0s ↔ (𝐴 +s 𝐵) ∈ ℕ0s))
12	11	imbi2d 342	. . 3 ⊢ (𝑛 = 𝐵 → ((𝐴 ∈ ℕ0s → (𝐴 +s 𝑛) ∈ ℕ0s) ↔ (𝐴 ∈ ℕ0s → (𝐴 +s 𝐵) ∈ ℕ0s)))
13		n0no 28404	. . . . 5 ⊢ (𝐴 ∈ ℕ0s → 𝐴 ∈ No )
14	13	addsridd 28046	. . . 4 ⊢ (𝐴 ∈ ℕ0s → (𝐴 +s 0s ) = 𝐴)
15		id 22	. . . 4 ⊢ (𝐴 ∈ ℕ0s → 𝐴 ∈ ℕ0s)
16	14, 15	eqeltrd 2861	. . 3 ⊢ (𝐴 ∈ ℕ0s → (𝐴 +s 0s ) ∈ ℕ0s)
17	13	adantr 484	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℕ0s ∧ 𝑚 ∈ ℕ0s) → 𝐴 ∈ No )
18	17	adantr 484	. . . . . . . 8 ⊢ (((𝐴 ∈ ℕ0s ∧ 𝑚 ∈ ℕ0s) ∧ (𝐴 +s 𝑚) ∈ ℕ0s) → 𝐴 ∈ No )
19		n0no 28404	. . . . . . . . . 10 ⊢ (𝑚 ∈ ℕ0s → 𝑚 ∈ No )
20	19	adantl 485	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℕ0s ∧ 𝑚 ∈ ℕ0s) → 𝑚 ∈ No )
21	20	adantr 484	. . . . . . . 8 ⊢ (((𝐴 ∈ ℕ0s ∧ 𝑚 ∈ ℕ0s) ∧ (𝐴 +s 𝑚) ∈ ℕ0s) → 𝑚 ∈ No )
22		1no 27891	. . . . . . . . 9 ⊢ 1s ∈ No
23	22	a1i 11	. . . . . . . 8 ⊢ (((𝐴 ∈ ℕ0s ∧ 𝑚 ∈ ℕ0s) ∧ (𝐴 +s 𝑚) ∈ ℕ0s) → 1s ∈ No )
24	18, 21, 23	addsassd 28087	. . . . . . 7 ⊢ (((𝐴 ∈ ℕ0s ∧ 𝑚 ∈ ℕ0s) ∧ (𝐴 +s 𝑚) ∈ ℕ0s) → ((𝐴 +s 𝑚) +s 1s ) = (𝐴 +s (𝑚 +s 1s )))
25		peano2n0s 28411	. . . . . . . 8 ⊢ ((𝐴 +s 𝑚) ∈ ℕ0s → ((𝐴 +s 𝑚) +s 1s ) ∈ ℕ0s)
26	25	adantl 485	. . . . . . 7 ⊢ (((𝐴 ∈ ℕ0s ∧ 𝑚 ∈ ℕ0s) ∧ (𝐴 +s 𝑚) ∈ ℕ0s) → ((𝐴 +s 𝑚) +s 1s ) ∈ ℕ0s)
27	24, 26	eqeltrrd 2862	. . . . . 6 ⊢ (((𝐴 ∈ ℕ0s ∧ 𝑚 ∈ ℕ0s) ∧ (𝐴 +s 𝑚) ∈ ℕ0s) → (𝐴 +s (𝑚 +s 1s )) ∈ ℕ0s)
28	27	ex 416	. . . . 5 ⊢ ((𝐴 ∈ ℕ0s ∧ 𝑚 ∈ ℕ0s) → ((𝐴 +s 𝑚) ∈ ℕ0s → (𝐴 +s (𝑚 +s 1s )) ∈ ℕ0s))
29	28	expcom 417	. . . 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 28414	. 2 ⊢ (𝐵 ∈ ℕ0s → (𝐴 ∈ ℕ0s → (𝐴 +s 𝐵) ∈ ℕ0s))
32	31	impcom 411	1 ⊢ ((𝐴 ∈ ℕ0s ∧ 𝐵 ∈ ℕ0s) → (𝐴 +s 𝐵) ∈ ℕ0s)

Syntax hints:   → wi 4   ∧ wa 399   = wceq 1559   ∈ wcel 2141  (class class class)co 7391   No csur 27692   0_s c0s 27886   1_s c1s 27887   +_s cadds 28040  ℕ_0scn0s 28393

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-rep 5224  ax-sep 5243  ax-nul 5253  ax-pow 5319  ax-pr 5387  ax-un 7713

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1098  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3076  df-rex 3086  df-rmo 3366  df-reu 3367  df-rab 3414  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4284  df-if 4478  df-pw 4554  df-sn 4580  df-pr 4582  df-tp 4584  df-op 4586  df-ot 4588  df-uni 4863  df-int 4903  df-iun 4948  df-br 5098  df-opab 5160  df-mpt 5179  df-tr 5205  df-id 5538  df-eprel 5543  df-po 5551  df-so 5552  df-fr 5596  df-se 5597  df-we 5598  df-xp 5649  df-rel 5650  df-cnv 5651  df-co 5652  df-dm 5653  df-rn 5654  df-res 5655  df-ima 5656  df-pred 6283  df-ord 6344  df-on 6345  df-lim 6346  df-suc 6347  df-iota 6472  df-fun 6518  df-fn 6519  df-f 6520  df-f1 6521  df-fo 6522  df-f1o 6523  df-fv 6524  df-riota 7348  df-ov 7394  df-oprab 7395  df-mpo 7396  df-om 7842  df-1st 7965  df-2nd 7966  df-frecs 8256  df-wrecs 8287  df-recs 8336  df-rdg 8375  df-1o 8431  df-2o 8432  df-nadd 8630  df-no 27695  df-lts 27696  df-bday 27697  df-les 27797  df-slts 27839  df-cuts 27841  df-0s 27888  df-1s 27889  df-made 27908  df-old 27909  df-left 27911  df-right 27912  df-norec2 28030  df-adds 28041  df-n0s 28395

This theorem is referenced by:  n0mulscl  28426  nnaddscl  28427  expadds  28516  pw2divscan4d  28525  addhalfcut  28540  bdaypw2n0bndlem  28544  bdaypw2bnd  28546  bdayfinbndlem1  28548  z12bdaylem2  28552  z12addscl  28558  z12sge0  28564