Theorem nnmulscl 28215

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

Assertion

Ref	Expression
nnmulscl	⊢ ((𝐴 ∈ ℕs ∧ 𝐵 ∈ ℕs) → (𝐴 ·s 𝐵) ∈ ℕs)

Proof of Theorem nnmulscl

Step	Hyp	Ref	Expression
1		n0mulscl 28213	. . . 4 ⊢ ((𝐴 ∈ ℕ0s ∧ 𝐵 ∈ ℕ0s) → (𝐴 ·s 𝐵) ∈ ℕ0s)
2	1	ad2ant2r 747	. . 3 ⊢ (((𝐴 ∈ ℕ0s ∧ 0s <s 𝐴) ∧ (𝐵 ∈ ℕ0s ∧ 0s <s 𝐵)) → (𝐴 ·s 𝐵) ∈ ℕ0s)
3		simpll 766	. . . . 5 ⊢ (((𝐴 ∈ ℕ0s ∧ 0s <s 𝐴) ∧ (𝐵 ∈ ℕ0s ∧ 0s <s 𝐵)) → 𝐴 ∈ ℕ0s)
4	3	n0snod 28194	. . . 4 ⊢ (((𝐴 ∈ ℕ0s ∧ 0s <s 𝐴) ∧ (𝐵 ∈ ℕ0s ∧ 0s <s 𝐵)) → 𝐴 ∈ No )
5		simprl 770	. . . . 5 ⊢ (((𝐴 ∈ ℕ0s ∧ 0s <s 𝐴) ∧ (𝐵 ∈ ℕ0s ∧ 0s <s 𝐵)) → 𝐵 ∈ ℕ0s)
6	5	n0snod 28194	. . . 4 ⊢ (((𝐴 ∈ ℕ0s ∧ 0s <s 𝐴) ∧ (𝐵 ∈ ℕ0s ∧ 0s <s 𝐵)) → 𝐵 ∈ No )
7		simplr 768	. . . 4 ⊢ (((𝐴 ∈ ℕ0s ∧ 0s <s 𝐴) ∧ (𝐵 ∈ ℕ0s ∧ 0s <s 𝐵)) → 0s <s 𝐴)
8		simprr 772	. . . 4 ⊢ (((𝐴 ∈ ℕ0s ∧ 0s <s 𝐴) ∧ (𝐵 ∈ ℕ0s ∧ 0s <s 𝐵)) → 0s <s 𝐵)
9	4, 6, 7, 8	mulsgt0d 28024	. . 3 ⊢ (((𝐴 ∈ ℕ0s ∧ 0s <s 𝐴) ∧ (𝐵 ∈ ℕ0s ∧ 0s <s 𝐵)) → 0s <s (𝐴 ·s 𝐵))
10	2, 9	jca 511	. 2 ⊢ (((𝐴 ∈ ℕ0s ∧ 0s <s 𝐴) ∧ (𝐵 ∈ ℕ0s ∧ 0s <s 𝐵)) → ((𝐴 ·s 𝐵) ∈ ℕ0s ∧ 0s <s (𝐴 ·s 𝐵)))
11		elnns2 28209	. . 3 ⊢ (𝐴 ∈ ℕs ↔ (𝐴 ∈ ℕ0s ∧ 0s <s 𝐴))
12		elnns2 28209	. . 3 ⊢ (𝐵 ∈ ℕs ↔ (𝐵 ∈ ℕ0s ∧ 0s <s 𝐵))
13	11, 12	anbi12i 628	. 2 ⊢ ((𝐴 ∈ ℕs ∧ 𝐵 ∈ ℕs) ↔ ((𝐴 ∈ ℕ0s ∧ 0s <s 𝐴) ∧ (𝐵 ∈ ℕ0s ∧ 0s <s 𝐵)))
14		elnns2 28209	. 2 ⊢ ((𝐴 ·s 𝐵) ∈ ℕs ↔ ((𝐴 ·s 𝐵) ∈ ℕ0s ∧ 0s <s (𝐴 ·s 𝐵)))
15	10, 13, 14	3imtr4i 292	1 ⊢ ((𝐴 ∈ ℕs ∧ 𝐵 ∈ ℕs) → (𝐴 ·s 𝐵) ∈ ℕs)

Syntax hints:   → wi 4   ∧ wa 395   ∈ wcel 2109   class class class wbr 5102  (class class class)co 7369   <s cslt 27528   0_s c0s 27710   ·_s cmuls 27985  ℕ_0scnn0s 28182  ℕ_scnns 28183

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5229  ax-sep 5246  ax-nul 5256  ax-pow 5315  ax-pr 5382  ax-un 7691

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rmo 3351  df-reu 3352  df-rab 3403  df-v 3446  df-sbc 3751  df-csb 3860  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3931  df-nul 4293  df-if 4485  df-pw 4561  df-sn 4586  df-pr 4588  df-tp 4590  df-op 4592  df-ot 4594  df-uni 4868  df-int 4907  df-iun 4953  df-br 5103  df-opab 5165  df-mpt 5184  df-tr 5210  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-se 5585  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6262  df-ord 6323  df-on 6324  df-lim 6325  df-suc 6326  df-iota 6452  df-fun 6501  df-fn 6502  df-f 6503  df-f1 6504  df-fo 6505  df-f1o 6506  df-fv 6507  df-riota 7326  df-ov 7372  df-oprab 7373  df-mpo 7374  df-om 7823  df-1st 7947  df-2nd 7948  df-frecs 8237  df-wrecs 8268  df-recs 8317  df-rdg 8355  df-1o 8411  df-2o 8412  df-nadd 8607  df-no 27530  df-slt 27531  df-bday 27532  df-sle 27633  df-sslt 27669  df-scut 27671  df-0s 27712  df-1s 27713  df-made 27731  df-old 27732  df-left 27734  df-right 27735  df-norec 27821  df-norec2 27832  df-adds 27843  df-negs 27903  df-subs 27904  df-muls 27986  df-n0s 28184  df-nns 28185

This theorem is referenced by:  zmulscld  28261  nnexpscl  28295  remulscllem1  28327  remulscllem2  28328