Theorem nnaddscl 28439

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 28437	. . . 4 ⊢ ((𝐴 ∈ ℕ0s ∧ 𝐵 ∈ ℕ0s) → (𝐴 +s 𝐵) ∈ ℕ0s)
2	1	ad2ant2r 757	. . 3 ⊢ (((𝐴 ∈ ℕ0s ∧ 0s <s 𝐴) ∧ (𝐵 ∈ ℕ0s ∧ 0s <s 𝐵)) → (𝐴 +s 𝐵) ∈ ℕ0s)
3		simpll 776	. . . . 5 ⊢ (((𝐴 ∈ ℕ0s ∧ 0s <s 𝐴) ∧ (𝐵 ∈ ℕ0s ∧ 0s <s 𝐵)) → 𝐴 ∈ ℕ0s)
4	3	n0nod 28418	. . . 4 ⊢ (((𝐴 ∈ ℕ0s ∧ 0s <s 𝐴) ∧ (𝐵 ∈ ℕ0s ∧ 0s <s 𝐵)) → 𝐴 ∈ No )
5		simprl 780	. . . . 5 ⊢ (((𝐴 ∈ ℕ0s ∧ 0s <s 𝐴) ∧ (𝐵 ∈ ℕ0s ∧ 0s <s 𝐵)) → 𝐵 ∈ ℕ0s)
6	5	n0nod 28418	. . . 4 ⊢ (((𝐴 ∈ ℕ0s ∧ 0s <s 𝐴) ∧ (𝐵 ∈ ℕ0s ∧ 0s <s 𝐵)) → 𝐵 ∈ No )
7		simplr 778	. . . 4 ⊢ (((𝐴 ∈ ℕ0s ∧ 0s <s 𝐴) ∧ (𝐵 ∈ ℕ0s ∧ 0s <s 𝐵)) → 0s <s 𝐴)
8		simprr 782	. . . 4 ⊢ (((𝐴 ∈ ℕ0s ∧ 0s <s 𝐴) ∧ (𝐵 ∈ ℕ0s ∧ 0s <s 𝐵)) → 0s <s 𝐵)
9	4, 6, 7, 8	addsgt0d 28107	. . 3 ⊢ (((𝐴 ∈ ℕ0s ∧ 0s <s 𝐴) ∧ (𝐵 ∈ ℕ0s ∧ 0s <s 𝐵)) → 0s <s (𝐴 +s 𝐵))
10	2, 9	jca 519	. 2 ⊢ (((𝐴 ∈ ℕ0s ∧ 0s <s 𝐴) ∧ (𝐵 ∈ ℕ0s ∧ 0s <s 𝐵)) → ((𝐴 +s 𝐵) ∈ ℕ0s ∧ 0s <s (𝐴 +s 𝐵)))
11		elnns2 28434	. . 3 ⊢ (𝐴 ∈ ℕs ↔ (𝐴 ∈ ℕ0s ∧ 0s <s 𝐴))
12		elnns2 28434	. . 3 ⊢ (𝐵 ∈ ℕs ↔ (𝐵 ∈ ℕ0s ∧ 0s <s 𝐵))
13	11, 12	anbi12i 637	. 2 ⊢ ((𝐴 ∈ ℕs ∧ 𝐵 ∈ ℕs) ↔ ((𝐴 ∈ ℕ0s ∧ 0s <s 𝐴) ∧ (𝐵 ∈ ℕ0s ∧ 0s <s 𝐵)))
14		elnns2 28434	. 2 ⊢ ((𝐴 +s 𝐵) ∈ ℕs ↔ ((𝐴 +s 𝐵) ∈ ℕ0s ∧ 0s <s (𝐴 +s 𝐵)))
15	10, 13, 14	3imtr4i 294	1 ⊢ ((𝐴 ∈ ℕs ∧ 𝐵 ∈ ℕs) → (𝐴 +s 𝐵) ∈ ℕs)

Syntax hints:   → wi 4   ∧ wa 399   ∈ wcel 2142   class class class wbr 5100  (class class class)co 7396   <s clts 27705   0_s c0s 27898   +_s cadds 28052  ℕ_0scn0s 28405  ℕ_scnns 28406

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-rep 5227  ax-sep 5246  ax-nul 5256  ax-pow 5322  ax-pr 5390  ax-un 7718

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1099  df-3an 1100  df-tru 1563  df-fal 1573  df-ex 1800  df-nf 1804  df-sb 2091  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3077  df-rex 3087  df-rmo 3367  df-reu 3368  df-rab 3415  df-v 3456  df-sbc 3745  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-pss 3924  df-nul 4286  df-if 4481  df-pw 4557  df-sn 4583  df-pr 4585  df-tp 4587  df-op 4589  df-ot 4591  df-uni 4866  df-int 4906  df-iun 4951  df-br 5101  df-opab 5163  df-mpt 5182  df-tr 5208  df-id 5542  df-eprel 5547  df-po 5555  df-so 5556  df-fr 5600  df-se 5601  df-we 5602  df-xp 5653  df-rel 5654  df-cnv 5655  df-co 5656  df-dm 5657  df-rn 5658  df-res 5659  df-ima 5660  df-pred 6288  df-ord 6349  df-on 6350  df-lim 6351  df-suc 6352  df-iota 6477  df-fun 6523  df-fn 6524  df-f 6525  df-f1 6526  df-fo 6527  df-f1o 6528  df-fv 6529  df-riota 7353  df-ov 7399  df-oprab 7400  df-mpo 7401  df-om 7847  df-1st 7970  df-2nd 7971  df-frecs 8262  df-wrecs 8293  df-recs 8342  df-rdg 8381  df-1o 8437  df-2o 8438  df-nadd 8636  df-no 27707  df-lts 27708  df-bday 27709  df-les 27809  df-slts 27851  df-cuts 27853  df-0s 27900  df-1s 27901  df-made 27920  df-old 27921  df-left 27923  df-right 27924  df-norec2 28042  df-adds 28053  df-n0s 28407  df-nns 28408

This theorem is referenced by:  peano2nns  28443  zaddscl  28487  zmulscld  28490  readdscl  28592