Theorem n0mulscl 28311

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 7424	. . . . 5 ⊢ (𝑛 = 0s → (𝐴 ·s 𝑛) = (𝐴 ·s 0s ))
2	1	eleq1d 2811	. . . 4 ⊢ (𝑛 = 0s → ((𝐴 ·s 𝑛) ∈ ℕ0s ↔ (𝐴 ·s 0s ) ∈ ℕ0s))
3	2	imbi2d 339	. . 3 ⊢ (𝑛 = 0s → ((𝐴 ∈ ℕ0s → (𝐴 ·s 𝑛) ∈ ℕ0s) ↔ (𝐴 ∈ ℕ0s → (𝐴 ·s 0s ) ∈ ℕ0s)))
4		oveq2 7424	. . . . 5 ⊢ (𝑛 = 𝑚 → (𝐴 ·s 𝑛) = (𝐴 ·s 𝑚))
5	4	eleq1d 2811	. . . 4 ⊢ (𝑛 = 𝑚 → ((𝐴 ·s 𝑛) ∈ ℕ0s ↔ (𝐴 ·s 𝑚) ∈ ℕ0s))
6	5	imbi2d 339	. . 3 ⊢ (𝑛 = 𝑚 → ((𝐴 ∈ ℕ0s → (𝐴 ·s 𝑛) ∈ ℕ0s) ↔ (𝐴 ∈ ℕ0s → (𝐴 ·s 𝑚) ∈ ℕ0s)))
7		oveq2 7424	. . . . 5 ⊢ (𝑛 = (𝑚 +s 1s ) → (𝐴 ·s 𝑛) = (𝐴 ·s (𝑚 +s 1s )))
8	7	eleq1d 2811	. . . 4 ⊢ (𝑛 = (𝑚 +s 1s ) → ((𝐴 ·s 𝑛) ∈ ℕ0s ↔ (𝐴 ·s (𝑚 +s 1s )) ∈ ℕ0s))
9	8	imbi2d 339	. . 3 ⊢ (𝑛 = (𝑚 +s 1s ) → ((𝐴 ∈ ℕ0s → (𝐴 ·s 𝑛) ∈ ℕ0s) ↔ (𝐴 ∈ ℕ0s → (𝐴 ·s (𝑚 +s 1s )) ∈ ℕ0s)))
10		oveq2 7424	. . . . 5 ⊢ (𝑛 = 𝐵 → (𝐴 ·s 𝑛) = (𝐴 ·s 𝐵))
11	10	eleq1d 2811	. . . 4 ⊢ (𝑛 = 𝐵 → ((𝐴 ·s 𝑛) ∈ ℕ0s ↔ (𝐴 ·s 𝐵) ∈ ℕ0s))
12	11	imbi2d 339	. . 3 ⊢ (𝑛 = 𝐵 → ((𝐴 ∈ ℕ0s → (𝐴 ·s 𝑛) ∈ ℕ0s) ↔ (𝐴 ∈ ℕ0s → (𝐴 ·s 𝐵) ∈ ℕ0s)))
13		n0sno 28293	. . . . 5 ⊢ (𝐴 ∈ ℕ0s → 𝐴 ∈ No )
14		muls01 28110	. . . . 5 ⊢ (𝐴 ∈ No → (𝐴 ·s 0s ) = 0s )
15	13, 14	syl 17	. . . 4 ⊢ (𝐴 ∈ ℕ0s → (𝐴 ·s 0s ) = 0s )
16		0n0s 28299	. . . 4 ⊢ 0s ∈ ℕ0s
17	15, 16	eqeltrdi 2834	. . 3 ⊢ (𝐴 ∈ ℕ0s → (𝐴 ·s 0s ) ∈ ℕ0s)
18	13	ad2antrr 724	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℕ0s ∧ 𝑚 ∈ ℕ0s) ∧ (𝐴 ·s 𝑚) ∈ ℕ0s) → 𝐴 ∈ No )
19		n0sno 28293	. . . . . . . . . 10 ⊢ (𝑚 ∈ ℕ0s → 𝑚 ∈ No )
20	19	ad2antlr 725	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℕ0s ∧ 𝑚 ∈ ℕ0s) ∧ (𝐴 ·s 𝑚) ∈ ℕ0s) → 𝑚 ∈ No )
21		1sno 27854	. . . . . . . . . 10 ⊢ 1s ∈ No
22	21	a1i 11	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℕ0s ∧ 𝑚 ∈ ℕ0s) ∧ (𝐴 ·s 𝑚) ∈ ℕ0s) → 1s ∈ No )
23	18, 20, 22	addsdid 28154	. . . . . . . 8 ⊢ (((𝐴 ∈ ℕ0s ∧ 𝑚 ∈ ℕ0s) ∧ (𝐴 ·s 𝑚) ∈ ℕ0s) → (𝐴 ·s (𝑚 +s 1s )) = ((𝐴 ·s 𝑚) +s (𝐴 ·s 1s )))
24	13	mulsridd 28112	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℕ0s → (𝐴 ·s 1s ) = 𝐴)
25	24	oveq2d 7432	. . . . . . . . 9 ⊢ (𝐴 ∈ ℕ0s → ((𝐴 ·s 𝑚) +s (𝐴 ·s 1s )) = ((𝐴 ·s 𝑚) +s 𝐴))
26	25	ad2antrr 724	. . . . . . . 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 2766	. . . . . . 7 ⊢ (((𝐴 ∈ ℕ0s ∧ 𝑚 ∈ ℕ0s) ∧ (𝐴 ·s 𝑚) ∈ ℕ0s) → (𝐴 ·s (𝑚 +s 1s )) = ((𝐴 ·s 𝑚) +s 𝐴))
28		n0addscl 28310	. . . . . . . . 9 ⊢ (((𝐴 ·s 𝑚) ∈ ℕ0s ∧ 𝐴 ∈ ℕ0s) → ((𝐴 ·s 𝑚) +s 𝐴) ∈ ℕ0s)
29	28	ancoms 457	. . . . . . . 8 ⊢ ((𝐴 ∈ ℕ0s ∧ (𝐴 ·s 𝑚) ∈ ℕ0s) → ((𝐴 ·s 𝑚) +s 𝐴) ∈ ℕ0s)
30	29	adantlr 713	. . . . . . 7 ⊢ (((𝐴 ∈ ℕ0s ∧ 𝑚 ∈ ℕ0s) ∧ (𝐴 ·s 𝑚) ∈ ℕ0s) → ((𝐴 ·s 𝑚) +s 𝐴) ∈ ℕ0s)
31	27, 30	eqeltrd 2826	. . . . . 6 ⊢ (((𝐴 ∈ ℕ0s ∧ 𝑚 ∈ ℕ0s) ∧ (𝐴 ·s 𝑚) ∈ ℕ0s) → (𝐴 ·s (𝑚 +s 1s )) ∈ ℕ0s)
32	31	ex 411	. . . . 5 ⊢ ((𝐴 ∈ ℕ0s ∧ 𝑚 ∈ ℕ0s) → ((𝐴 ·s 𝑚) ∈ ℕ0s → (𝐴 ·s (𝑚 +s 1s )) ∈ ℕ0s))
33	32	expcom 412	. . . 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 28302	. 2 ⊢ (𝐵 ∈ ℕ0s → (𝐴 ∈ ℕ0s → (𝐴 ·s 𝐵) ∈ ℕ0s))
36	35	impcom 406	1 ⊢ ((𝐴 ∈ ℕ0s ∧ 𝐵 ∈ ℕ0s) → (𝐴 ·s 𝐵) ∈ ℕ0s)

Syntax hints:   → wi 4   ∧ wa 394   = wceq 1534   ∈ wcel 2099  (class class class)co 7416   No csur 27666   0_s c0s 27849   1_s c1s 27850   +_s cadds 27970   ·_s cmuls 28104  ℕ_0scnn0s 28283

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 2697  ax-rep 5282  ax-sep 5296  ax-nul 5303  ax-pow 5361  ax-pr 5425  ax-un 7738

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-ral 3052  df-rex 3061  df-rmo 3364  df-reu 3365  df-rab 3420  df-v 3464  df-sbc 3776  df-csb 3892  df-dif 3949  df-un 3951  df-in 3953  df-ss 3963  df-pss 3966  df-nul 4323  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-tp 4628  df-op 4630  df-ot 4632  df-uni 4906  df-int 4947  df-iun 4995  df-br 5146  df-opab 5208  df-mpt 5229  df-tr 5263  df-id 5572  df-eprel 5578  df-po 5586  df-so 5587  df-fr 5629  df-se 5630  df-we 5631  df-xp 5680  df-rel 5681  df-cnv 5682  df-co 5683  df-dm 5684  df-rn 5685  df-res 5686  df-ima 5687  df-pred 6304  df-ord 6371  df-on 6372  df-lim 6373  df-suc 6374  df-iota 6498  df-fun 6548  df-fn 6549  df-f 6550  df-f1 6551  df-fo 6552  df-f1o 6553  df-fv 6554  df-riota 7372  df-ov 7419  df-oprab 7420  df-mpo 7421  df-om 7869  df-1st 7995  df-2nd 7996  df-frecs 8288  df-wrecs 8319  df-recs 8393  df-rdg 8432  df-1o 8488  df-2o 8489  df-nadd 8688  df-no 27669  df-slt 27670  df-bday 27671  df-sle 27772  df-sslt 27808  df-scut 27810  df-0s 27851  df-1s 27852  df-made 27868  df-old 27869  df-left 27871  df-right 27872  df-norec 27949  df-norec2 27960  df-adds 27971  df-negs 28028  df-subs 28029  df-muls 28105  df-n0s 28285

This theorem is referenced by:  nnmulscl  28313