Theorem nnaddscl 28612

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

Assertion

Ref	Expression
nnaddscl	⊢ ((𝐴 ∈ ℕs ∧ 𝐵 ∈ ℕs) → (𝐴 +s 𝐵) ∈ ℕs)

Proof of Theorem nnaddscl

Step	Hyp	Ref	Expression
1		n0addscl 28610	. . . 4 ⊢ ((𝐴 ∈ ℕ0s ∧ 𝐵 ∈ ℕ0s) → (𝐴 +s 𝐵) ∈ ℕ0s)
2	1	ad2ant2r 760	. . 3 ⊢ (((𝐴 ∈ ℕ0s ∧ 0s <s 𝐴) ∧ (𝐵 ∈ ℕ0s ∧ 0s <s 𝐵)) → (𝐴 +s 𝐵) ∈ ℕ0s)
3		simpll 779	. . . . 5 ⊢ (((𝐴 ∈ ℕ0s ∧ 0s <s 𝐴) ∧ (𝐵 ∈ ℕ0s ∧ 0s <s 𝐵)) → 𝐴 ∈ ℕ0s)
4	3	n0nod 28591	. . . 4 ⊢ (((𝐴 ∈ ℕ0s ∧ 0s <s 𝐴) ∧ (𝐵 ∈ ℕ0s ∧ 0s <s 𝐵)) → 𝐴 ∈ No )
5		simprl 783	. . . . 5 ⊢ (((𝐴 ∈ ℕ0s ∧ 0s <s 𝐴) ∧ (𝐵 ∈ ℕ0s ∧ 0s <s 𝐵)) → 𝐵 ∈ ℕ0s)
6	5	n0nod 28591	. . . 4 ⊢ (((𝐴 ∈ ℕ0s ∧ 0s <s 𝐴) ∧ (𝐵 ∈ ℕ0s ∧ 0s <s 𝐵)) → 𝐵 ∈ No )
7		simplr 781	. . . 4 ⊢ (((𝐴 ∈ ℕ0s ∧ 0s <s 𝐴) ∧ (𝐵 ∈ ℕ0s ∧ 0s <s 𝐵)) → 0s <s 𝐴)
8		simprr 785	. . . 4 ⊢ (((𝐴 ∈ ℕ0s ∧ 0s <s 𝐴) ∧ (𝐵 ∈ ℕ0s ∧ 0s <s 𝐵)) → 0s <s 𝐵)
9	4, 6, 7, 8	addsgt0d 28280	. . 3 ⊢ (((𝐴 ∈ ℕ0s ∧ 0s <s 𝐴) ∧ (𝐵 ∈ ℕ0s ∧ 0s <s 𝐵)) → 0s <s (𝐴 +s 𝐵))
10	2, 9	jca 521	. 2 ⊢ (((𝐴 ∈ ℕ0s ∧ 0s <s 𝐴) ∧ (𝐵 ∈ ℕ0s ∧ 0s <s 𝐵)) → ((𝐴 +s 𝐵) ∈ ℕ0s ∧ 0s <s (𝐴 +s 𝐵)))
11		elnns2 28607	. . 3 ⊢ (𝐴 ∈ ℕs ↔ (𝐴 ∈ ℕ0s ∧ 0s <s 𝐴))
12		elnns2 28607	. . 3 ⊢ (𝐵 ∈ ℕs ↔ (𝐵 ∈ ℕ0s ∧ 0s <s 𝐵))
13	11, 12	anbi12i 640	. 2 ⊢ ((𝐴 ∈ ℕs ∧ 𝐵 ∈ ℕs) ↔ ((𝐴 ∈ ℕ0s ∧ 0s <s 𝐴) ∧ (𝐵 ∈ ℕ0s ∧ 0s <s 𝐵)))
14		elnns2 28607	. 2 ⊢ ((𝐴 +s 𝐵) ∈ ℕs ↔ ((𝐴 +s 𝐵) ∈ ℕ0s ∧ 0s <s (𝐴 +s 𝐵)))
15	10, 13, 14	3imtr4i 295	1 ⊢ ((𝐴 ∈ ℕs ∧ 𝐵 ∈ ℕs) → (𝐴 +s 𝐵) ∈ ℕs)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∈ wcel 2145   class class class wbr 5103  (class class class)co 7414   <s clts 27878   0_s c0s 28071   +_s cadds 28225  ℕ_0scn0s 28578  ℕ_scnns 28579

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-rep 5232  ax-sep 5251  ax-nul 5263  ax-pow 5330  ax-pr 5398  ax-un 7737

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  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 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-tp 4589  df-op 4591  df-ot 4593  df-uni 4868  df-int 4908  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-se 5609  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-pred 6299  df-ord 6360  df-on 6361  df-lim 6362  df-suc 6363  df-iota 6489  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-riota 7371  df-ov 7417  df-oprab 7418  df-mpo 7419  df-om 7864  df-1st 7987  df-2nd 7988  df-frecs 8281  df-wrecs 8312  df-recs 8361  df-rdg 8400  df-1o 8456  df-2o 8457  df-nadd 8655  df-no 27880  df-lts 27881  df-bday 27882  df-les 27982  df-slts 28024  df-cuts 28026  df-0s 28073  df-1s 28074  df-made 28093  df-old 28094  df-left 28096  df-right 28097  df-norec2 28215  df-adds 28226  df-n0s 28580  df-nns 28581

This theorem is used by:  peano2nns  28616  zaddscl  28660  zmulscld  28663  readdscl  28765