Theorem nnmulscl 28212

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 28210	. . . 4 ⊢ ((𝐴 ∈ ℕ0s ∧ 𝐵 ∈ ℕ0s) → (𝐴 ·s 𝐵) ∈ ℕ0s)
2	1	ad2ant2r 746	. . 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 28196	. . . 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 28196	. . . 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 28044	. . 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 28208	. . 3 ⊢ (𝐴 ∈ ℕs ↔ (𝐴 ∈ ℕ0s ∧ 0s <s 𝐴))
12		elnns2 28208	. . 3 ⊢ (𝐵 ∈ ℕs ↔ (𝐵 ∈ ℕ0s ∧ 0s <s 𝐵))
13	11, 12	anbi12i 627	. 2 ⊢ ((𝐴 ∈ ℕs ∧ 𝐵 ∈ ℕs) ↔ ((𝐴 ∈ ℕ0s ∧ 0s <s 𝐴) ∧ (𝐵 ∈ ℕ0s ∧ 0s <s 𝐵)))
14		elnns2 28208	. 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 2099   class class class wbr 5148  (class class class)co 7420   <s cslt 27573   0_s c0s 27754   ·_s cmuls 28005  ℕ_0scnn0s 28184  ℕ_scnns 28185

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5365  ax-pr 5429  ax-un 7740

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3or 1086  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-rmo 3373  df-reu 3374  df-rab 3430  df-v 3473  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-tp 4634  df-op 4636  df-ot 4638  df-uni 4909  df-int 4950  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5576  df-eprel 5582  df-po 5590  df-so 5591  df-fr 5633  df-se 5634  df-we 5635  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-pred 6305  df-ord 6372  df-on 6373  df-lim 6374  df-suc 6375  df-iota 6500  df-fun 6550  df-fn 6551  df-f 6552  df-f1 6553  df-fo 6554  df-f1o 6555  df-fv 6556  df-riota 7376  df-ov 7423  df-oprab 7424  df-mpo 7425  df-om 7871  df-1st 7993  df-2nd 7994  df-frecs 8286  df-wrecs 8317  df-recs 8391  df-rdg 8430  df-1o 8486  df-2o 8487  df-nadd 8686  df-no 27575  df-slt 27576  df-bday 27577  df-sle 27677  df-sslt 27713  df-scut 27715  df-0s 27756  df-1s 27757  df-made 27773  df-old 27774  df-left 27776  df-right 27777  df-norec 27854  df-norec2 27865  df-adds 27876  df-negs 27933  df-subs 27934  df-muls 28006  df-n0s 28186  df-nns 28187

This theorem is referenced by:  remulscllem1  28227  remulscllem2  28228