Theorem nnaddscl 28732

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 28730	. . . 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 28711	. . . 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 28711	. . . 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 28400	. . 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 28727	. . 3 ⊢ (𝐴 ∈ ℕs ↔ (𝐴 ∈ ℕ0s ∧ 0s <s 𝐴))
12		elnns2 28727	. . 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 28727	. 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 7420   <s clts 27998   0_s c0s 28191   +_s cadds 28345  ℕ_0scn0s 28698  ℕ_scnns 28699

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 2733  ax-rep 5232  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7751

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 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-rmo 3366  df-reu 3367  df-rab 3414  df-v 3453  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 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 6304  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-riota 7377  df-ov 7423  df-oprab 7424  df-mpo 7425  df-om 7878  df-1st 8001  df-2nd 8002  df-frecs 8299  df-wrecs 8330  df-recs 8379  df-rdg 8418  df-1o 8476  df-2o 8477  df-nadd 8675  df-no 28000  df-lts 28001  df-bday 28002  df-les 28102  df-slts 28144  df-cuts 28146  df-0s 28193  df-1s 28194  df-made 28213  df-old 28214  df-left 28216  df-right 28217  df-norec2 28335  df-adds 28346  df-n0s 28700  df-nns 28701

This theorem is used by:  peano2nns  28736  zaddscl  28780  zmulscld  28783  readdscl  28885