Theorem n0mulscl 28242

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

Assertion

Ref	Expression
n0mulscl	⊢ ((𝐴 ∈ ℕ0s ∧ 𝐵 ∈ ℕ0s) → (𝐴 ·s 𝐵) ∈ ℕ0s)

Proof of Theorem n0mulscl

Dummy variables 𝑛 𝑚 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		oveq2 7357	. . . . 5 ⊢ (𝑛 = 0s → (𝐴 ·s 𝑛) = (𝐴 ·s 0s ))
2	1	eleq1d 2813	. . . 4 ⊢ (𝑛 = 0s → ((𝐴 ·s 𝑛) ∈ ℕ0s ↔ (𝐴 ·s 0s ) ∈ ℕ0s))
3	2	imbi2d 340	. . 3 ⊢ (𝑛 = 0s → ((𝐴 ∈ ℕ0s → (𝐴 ·s 𝑛) ∈ ℕ0s) ↔ (𝐴 ∈ ℕ0s → (𝐴 ·s 0s ) ∈ ℕ0s)))
4		oveq2 7357	. . . . 5 ⊢ (𝑛 = 𝑚 → (𝐴 ·s 𝑛) = (𝐴 ·s 𝑚))
5	4	eleq1d 2813	. . . 4 ⊢ (𝑛 = 𝑚 → ((𝐴 ·s 𝑛) ∈ ℕ0s ↔ (𝐴 ·s 𝑚) ∈ ℕ0s))
6	5	imbi2d 340	. . 3 ⊢ (𝑛 = 𝑚 → ((𝐴 ∈ ℕ0s → (𝐴 ·s 𝑛) ∈ ℕ0s) ↔ (𝐴 ∈ ℕ0s → (𝐴 ·s 𝑚) ∈ ℕ0s)))
7		oveq2 7357	. . . . 5 ⊢ (𝑛 = (𝑚 +s 1s ) → (𝐴 ·s 𝑛) = (𝐴 ·s (𝑚 +s 1s )))
8	7	eleq1d 2813	. . . 4 ⊢ (𝑛 = (𝑚 +s 1s ) → ((𝐴 ·s 𝑛) ∈ ℕ0s ↔ (𝐴 ·s (𝑚 +s 1s )) ∈ ℕ0s))
9	8	imbi2d 340	. . 3 ⊢ (𝑛 = (𝑚 +s 1s ) → ((𝐴 ∈ ℕ0s → (𝐴 ·s 𝑛) ∈ ℕ0s) ↔ (𝐴 ∈ ℕ0s → (𝐴 ·s (𝑚 +s 1s )) ∈ ℕ0s)))
10		oveq2 7357	. . . . 5 ⊢ (𝑛 = 𝐵 → (𝐴 ·s 𝑛) = (𝐴 ·s 𝐵))
11	10	eleq1d 2813	. . . 4 ⊢ (𝑛 = 𝐵 → ((𝐴 ·s 𝑛) ∈ ℕ0s ↔ (𝐴 ·s 𝐵) ∈ ℕ0s))
12	11	imbi2d 340	. . 3 ⊢ (𝑛 = 𝐵 → ((𝐴 ∈ ℕ0s → (𝐴 ·s 𝑛) ∈ ℕ0s) ↔ (𝐴 ∈ ℕ0s → (𝐴 ·s 𝐵) ∈ ℕ0s)))
13		n0sno 28221	. . . . 5 ⊢ (𝐴 ∈ ℕ0s → 𝐴 ∈ No )
14		muls01 28020	. . . . 5 ⊢ (𝐴 ∈ No → (𝐴 ·s 0s ) = 0s )
15	13, 14	syl 17	. . . 4 ⊢ (𝐴 ∈ ℕ0s → (𝐴 ·s 0s ) = 0s )
16		0n0s 28227	. . . 4 ⊢ 0s ∈ ℕ0s
17	15, 16	eqeltrdi 2836	. . 3 ⊢ (𝐴 ∈ ℕ0s → (𝐴 ·s 0s ) ∈ ℕ0s)
18	13	ad2antrr 726	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℕ0s ∧ 𝑚 ∈ ℕ0s) ∧ (𝐴 ·s 𝑚) ∈ ℕ0s) → 𝐴 ∈ No )
19		n0sno 28221	. . . . . . . . . 10 ⊢ (𝑚 ∈ ℕ0s → 𝑚 ∈ No )
20	19	ad2antlr 727	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℕ0s ∧ 𝑚 ∈ ℕ0s) ∧ (𝐴 ·s 𝑚) ∈ ℕ0s) → 𝑚 ∈ No )
21		1sno 27741	. . . . . . . . . 10 ⊢ 1s ∈ No
22	21	a1i 11	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℕ0s ∧ 𝑚 ∈ ℕ0s) ∧ (𝐴 ·s 𝑚) ∈ ℕ0s) → 1s ∈ No )
23	18, 20, 22	addsdid 28064	. . . . . . . 8 ⊢ (((𝐴 ∈ ℕ0s ∧ 𝑚 ∈ ℕ0s) ∧ (𝐴 ·s 𝑚) ∈ ℕ0s) → (𝐴 ·s (𝑚 +s 1s )) = ((𝐴 ·s 𝑚) +s (𝐴 ·s 1s )))
24	13	mulsridd 28022	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℕ0s → (𝐴 ·s 1s ) = 𝐴)
25	24	oveq2d 7365	. . . . . . . . 9 ⊢ (𝐴 ∈ ℕ0s → ((𝐴 ·s 𝑚) +s (𝐴 ·s 1s )) = ((𝐴 ·s 𝑚) +s 𝐴))
26	25	ad2antrr 726	. . . . . . . 8 ⊢ (((𝐴 ∈ ℕ0s ∧ 𝑚 ∈ ℕ0s) ∧ (𝐴 ·s 𝑚) ∈ ℕ0s) → ((𝐴 ·s 𝑚) +s (𝐴 ·s 1s )) = ((𝐴 ·s 𝑚) +s 𝐴))
27	23, 26	eqtrd 2764	. . . . . . 7 ⊢ (((𝐴 ∈ ℕ0s ∧ 𝑚 ∈ ℕ0s) ∧ (𝐴 ·s 𝑚) ∈ ℕ0s) → (𝐴 ·s (𝑚 +s 1s )) = ((𝐴 ·s 𝑚) +s 𝐴))
28		n0addscl 28241	. . . . . . . . 9 ⊢ (((𝐴 ·s 𝑚) ∈ ℕ0s ∧ 𝐴 ∈ ℕ0s) → ((𝐴 ·s 𝑚) +s 𝐴) ∈ ℕ0s)
29	28	ancoms 458	. . . . . . . 8 ⊢ ((𝐴 ∈ ℕ0s ∧ (𝐴 ·s 𝑚) ∈ ℕ0s) → ((𝐴 ·s 𝑚) +s 𝐴) ∈ ℕ0s)
30	29	adantlr 715	. . . . . . 7 ⊢ (((𝐴 ∈ ℕ0s ∧ 𝑚 ∈ ℕ0s) ∧ (𝐴 ·s 𝑚) ∈ ℕ0s) → ((𝐴 ·s 𝑚) +s 𝐴) ∈ ℕ0s)
31	27, 30	eqeltrd 2828	. . . . . 6 ⊢ (((𝐴 ∈ ℕ0s ∧ 𝑚 ∈ ℕ0s) ∧ (𝐴 ·s 𝑚) ∈ ℕ0s) → (𝐴 ·s (𝑚 +s 1s )) ∈ ℕ0s)
32	31	ex 412	. . . . 5 ⊢ ((𝐴 ∈ ℕ0s ∧ 𝑚 ∈ ℕ0s) → ((𝐴 ·s 𝑚) ∈ ℕ0s → (𝐴 ·s (𝑚 +s 1s )) ∈ ℕ0s))
33	32	expcom 413	. . . 4 ⊢ (𝑚 ∈ ℕ0s → (𝐴 ∈ ℕ0s → ((𝐴 ·s 𝑚) ∈ ℕ0s → (𝐴 ·s (𝑚 +s 1s )) ∈ ℕ0s)))
34	33	a2d 29	. . 3 ⊢ (𝑚 ∈ ℕ0s → ((𝐴 ∈ ℕ0s → (𝐴 ·s 𝑚) ∈ ℕ0s) → (𝐴 ∈ ℕ0s → (𝐴 ·s (𝑚 +s 1s )) ∈ ℕ0s)))
35	3, 6, 9, 12, 17, 34	n0sind 28230	. 2 ⊢ (𝐵 ∈ ℕ0s → (𝐴 ∈ ℕ0s → (𝐴 ·s 𝐵) ∈ ℕ0s))
36	35	impcom 407	1 ⊢ ((𝐴 ∈ ℕ0s ∧ 𝐵 ∈ ℕ0s) → (𝐴 ·s 𝐵) ∈ ℕ0s)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109  (class class class)co 7349   No csur 27549   0_s c0s 27736   1_s c1s 27737   +_s cadds 27871   ·_s cmuls 28014  ℕ_0scnn0s 28211

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 5218  ax-sep 5235  ax-nul 5245  ax-pow 5304  ax-pr 5371  ax-un 7671

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 3343  df-reu 3344  df-rab 3395  df-v 3438  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4285  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-tp 4582  df-op 4584  df-ot 4586  df-uni 4859  df-int 4897  df-iun 4943  df-br 5093  df-opab 5155  df-mpt 5174  df-tr 5200  df-id 5514  df-eprel 5519  df-po 5527  df-so 5528  df-fr 5572  df-se 5573  df-we 5574  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-pred 6249  df-ord 6310  df-on 6311  df-lim 6312  df-suc 6313  df-iota 6438  df-fun 6484  df-fn 6485  df-f 6486  df-f1 6487  df-fo 6488  df-f1o 6489  df-fv 6490  df-riota 7306  df-ov 7352  df-oprab 7353  df-mpo 7354  df-om 7800  df-1st 7924  df-2nd 7925  df-frecs 8214  df-wrecs 8245  df-recs 8294  df-rdg 8332  df-1o 8388  df-2o 8389  df-nadd 8584  df-no 27552  df-slt 27553  df-bday 27554  df-sle 27655  df-sslt 27692  df-scut 27694  df-0s 27738  df-1s 27739  df-made 27757  df-old 27758  df-left 27760  df-right 27761  df-norec 27850  df-norec2 27861  df-adds 27872  df-negs 27932  df-subs 27933  df-muls 28015  df-n0s 28213

This theorem is referenced by:  nnmulscl  28244  eucliddivs  28270  n0expscl  28324  addhalfcut  28347  zs12ge0  28360