Theorem n0mulscl 28130

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 7410	. . . . 5 ⊢ (𝑛 = 0s → (𝐴 ·s 𝑛) = (𝐴 ·s 0s ))
2	1	eleq1d 2810	. . . 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 7410	. . . . 5 ⊢ (𝑛 = 𝑚 → (𝐴 ·s 𝑛) = (𝐴 ·s 𝑚))
5	4	eleq1d 2810	. . . 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 7410	. . . . 5 ⊢ (𝑛 = (𝑚 +s 1s ) → (𝐴 ·s 𝑛) = (𝐴 ·s (𝑚 +s 1s )))
8	7	eleq1d 2810	. . . 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 7410	. . . . 5 ⊢ (𝑛 = 𝐵 → (𝐴 ·s 𝑛) = (𝐴 ·s 𝐵))
11	10	eleq1d 2810	. . . 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 28114	. . . . 5 ⊢ (𝐴 ∈ ℕ0s → 𝐴 ∈ No )
14		muls01 27931	. . . . 5 ⊢ (𝐴 ∈ No → (𝐴 ·s 0s ) = 0s )
15	13, 14	syl 17	. . . 4 ⊢ (𝐴 ∈ ℕ0s → (𝐴 ·s 0s ) = 0s )
16		0n0s 28118	. . . 4 ⊢ 0s ∈ ℕ0s
17	15, 16	eqeltrdi 2833	. . 3 ⊢ (𝐴 ∈ ℕ0s → (𝐴 ·s 0s ) ∈ ℕ0s)
18	13	ad2antrr 723	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℕ0s ∧ 𝑚 ∈ ℕ0s) ∧ (𝐴 ·s 𝑚) ∈ ℕ0s) → 𝐴 ∈ No )
19		n0sno 28114	. . . . . . . . . 10 ⊢ (𝑚 ∈ ℕ0s → 𝑚 ∈ No )
20	19	ad2antlr 724	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℕ0s ∧ 𝑚 ∈ ℕ0s) ∧ (𝐴 ·s 𝑚) ∈ ℕ0s) → 𝑚 ∈ No )
21		1sno 27679	. . . . . . . . . 10 ⊢ 1s ∈ No
22	21	a1i 11	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℕ0s ∧ 𝑚 ∈ ℕ0s) ∧ (𝐴 ·s 𝑚) ∈ ℕ0s) → 1s ∈ No )
23	18, 20, 22	addsdid 27975	. . . . . . . 8 ⊢ (((𝐴 ∈ ℕ0s ∧ 𝑚 ∈ ℕ0s) ∧ (𝐴 ·s 𝑚) ∈ ℕ0s) → (𝐴 ·s (𝑚 +s 1s )) = ((𝐴 ·s 𝑚) +s (𝐴 ·s 1s )))
24	13	mulsridd 27933	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℕ0s → (𝐴 ·s 1s ) = 𝐴)
25	24	oveq2d 7418	. . . . . . . . 9 ⊢ (𝐴 ∈ ℕ0s → ((𝐴 ·s 𝑚) +s (𝐴 ·s 1s )) = ((𝐴 ·s 𝑚) +s 𝐴))
26	25	ad2antrr 723	. . . . . . . 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 28129	. . . . . . . . 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 712	. . . . . . 7 ⊢ (((𝐴 ∈ ℕ0s ∧ 𝑚 ∈ ℕ0s) ∧ (𝐴 ·s 𝑚) ∈ ℕ0s) → ((𝐴 ·s 𝑚) +s 𝐴) ∈ ℕ0s)
31	27, 30	eqeltrd 2825	. . . . . 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 28121	. 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 1533   ∈ wcel 2098  (class class class)co 7402   No csur 27492   0_s c0s 27674   1_s c1s 27675   +_s cadds 27795   ·_s cmuls 27925  ℕ_0scnn0s 28104

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-rep 5276  ax-sep 5290  ax-nul 5297  ax-pow 5354  ax-pr 5418  ax-un 7719

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rex 3063  df-rmo 3368  df-reu 3369  df-rab 3425  df-v 3468  df-sbc 3771  df-csb 3887  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-pss 3960  df-nul 4316  df-if 4522  df-pw 4597  df-sn 4622  df-pr 4624  df-tp 4626  df-op 4628  df-ot 4630  df-uni 4901  df-int 4942  df-iun 4990  df-br 5140  df-opab 5202  df-mpt 5223  df-tr 5257  df-id 5565  df-eprel 5571  df-po 5579  df-so 5580  df-fr 5622  df-se 5623  df-we 5624  df-xp 5673  df-rel 5674  df-cnv 5675  df-co 5676  df-dm 5677  df-rn 5678  df-res 5679  df-ima 5680  df-pred 6291  df-ord 6358  df-on 6359  df-lim 6360  df-suc 6361  df-iota 6486  df-fun 6536  df-fn 6537  df-f 6538  df-f1 6539  df-fo 6540  df-f1o 6541  df-fv 6542  df-riota 7358  df-ov 7405  df-oprab 7406  df-mpo 7407  df-om 7850  df-1st 7969  df-2nd 7970  df-frecs 8262  df-wrecs 8293  df-recs 8367  df-rdg 8406  df-1o 8462  df-2o 8463  df-nadd 8662  df-no 27495  df-slt 27496  df-bday 27497  df-sle 27597  df-sslt 27633  df-scut 27635  df-0s 27676  df-1s 27677  df-made 27693  df-old 27694  df-left 27696  df-right 27697  df-norec 27774  df-norec2 27785  df-adds 27796  df-negs 27853  df-subs 27854  df-muls 27926  df-n0s 28106

This theorem is referenced by:  nnmulscl  28132