Theorem n0mulscl 28363

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 7439	. . . . 5 ⊢ (𝑛 = 0s → (𝐴 ·s 𝑛) = (𝐴 ·s 0s ))
2	1	eleq1d 2824	. . . 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 7439	. . . . 5 ⊢ (𝑛 = 𝑚 → (𝐴 ·s 𝑛) = (𝐴 ·s 𝑚))
5	4	eleq1d 2824	. . . 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 7439	. . . . 5 ⊢ (𝑛 = (𝑚 +s 1s ) → (𝐴 ·s 𝑛) = (𝐴 ·s (𝑚 +s 1s )))
8	7	eleq1d 2824	. . . 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 7439	. . . . 5 ⊢ (𝑛 = 𝐵 → (𝐴 ·s 𝑛) = (𝐴 ·s 𝐵))
11	10	eleq1d 2824	. . . 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 28343	. . . . 5 ⊢ (𝐴 ∈ ℕ0s → 𝐴 ∈ No )
14		muls01 28153	. . . . 5 ⊢ (𝐴 ∈ No → (𝐴 ·s 0s ) = 0s )
15	13, 14	syl 17	. . . 4 ⊢ (𝐴 ∈ ℕ0s → (𝐴 ·s 0s ) = 0s )
16		0n0s 28349	. . . 4 ⊢ 0s ∈ ℕ0s
17	15, 16	eqeltrdi 2847	. . 3 ⊢ (𝐴 ∈ ℕ0s → (𝐴 ·s 0s ) ∈ ℕ0s)
18	13	ad2antrr 726	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℕ0s ∧ 𝑚 ∈ ℕ0s) ∧ (𝐴 ·s 𝑚) ∈ ℕ0s) → 𝐴 ∈ No )
19		n0sno 28343	. . . . . . . . . 10 ⊢ (𝑚 ∈ ℕ0s → 𝑚 ∈ No )
20	19	ad2antlr 727	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℕ0s ∧ 𝑚 ∈ ℕ0s) ∧ (𝐴 ·s 𝑚) ∈ ℕ0s) → 𝑚 ∈ No )
21		1sno 27887	. . . . . . . . . 10 ⊢ 1s ∈ No
22	21	a1i 11	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℕ0s ∧ 𝑚 ∈ ℕ0s) ∧ (𝐴 ·s 𝑚) ∈ ℕ0s) → 1s ∈ No )
23	18, 20, 22	addsdid 28197	. . . . . . . 8 ⊢ (((𝐴 ∈ ℕ0s ∧ 𝑚 ∈ ℕ0s) ∧ (𝐴 ·s 𝑚) ∈ ℕ0s) → (𝐴 ·s (𝑚 +s 1s )) = ((𝐴 ·s 𝑚) +s (𝐴 ·s 1s )))
24	13	mulsridd 28155	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℕ0s → (𝐴 ·s 1s ) = 𝐴)
25	24	oveq2d 7447	. . . . . . . . 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 2775	. . . . . . 7 ⊢ (((𝐴 ∈ ℕ0s ∧ 𝑚 ∈ ℕ0s) ∧ (𝐴 ·s 𝑚) ∈ ℕ0s) → (𝐴 ·s (𝑚 +s 1s )) = ((𝐴 ·s 𝑚) +s 𝐴))
28		n0addscl 28362	. . . . . . . . 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 2839	. . . . . 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 28352	. 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 1537   ∈ wcel 2106  (class class class)co 7431   No csur 27699   0_s c0s 27882   1_s c1s 27883   +_s cadds 28007   ·_s cmuls 28147  ℕ_0scnn0s 28333

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-tp 4636  df-op 4638  df-ot 4640  df-uni 4913  df-int 4952  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-se 5642  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8013  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-1o 8505  df-2o 8506  df-nadd 8703  df-no 27702  df-slt 27703  df-bday 27704  df-sle 27805  df-sslt 27841  df-scut 27843  df-0s 27884  df-1s 27885  df-made 27901  df-old 27902  df-left 27904  df-right 27905  df-norec 27986  df-norec2 27997  df-adds 28008  df-negs 28068  df-subs 28069  df-muls 28148  df-n0s 28335

This theorem is referenced by:  nnmulscl  28365  addhalfcut  28434