Theorem zmulscld 28291

Description: The surreal integers are closed under multiplication. (Contributed by Scott Fenton, 20-Aug-2025.)

Hypotheses

Ref	Expression
zmulscld.1	⊢ (𝜑 → 𝐴 ∈ ℤs)
zmulscld.2	⊢ (𝜑 → 𝐵 ∈ ℤs)

Assertion

Ref	Expression
zmulscld	⊢ (𝜑 → (𝐴 ·s 𝐵) ∈ ℤs)

Proof of Theorem zmulscld

Dummy variables 𝑥 𝑦 𝑧 𝑤 𝑡 𝑢 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		zmulscld.1	. . 3 ⊢ (𝜑 → 𝐴 ∈ ℤs)
2		elzs 28278	. . 3 ⊢ (𝐴 ∈ ℤs ↔ ∃𝑥 ∈ ℕs ∃𝑦 ∈ ℕs 𝐴 = (𝑥 -s 𝑦))
3	1, 2	sylib 218	. 2 ⊢ (𝜑 → ∃𝑥 ∈ ℕs ∃𝑦 ∈ ℕs 𝐴 = (𝑥 -s 𝑦))
4		zmulscld.2	. . 3 ⊢ (𝜑 → 𝐵 ∈ ℤs)
5		elzs 28278	. . 3 ⊢ (𝐵 ∈ ℤs ↔ ∃𝑧 ∈ ℕs ∃𝑤 ∈ ℕs 𝐵 = (𝑧 -s 𝑤))
6	4, 5	sylib 218	. 2 ⊢ (𝜑 → ∃𝑧 ∈ ℕs ∃𝑤 ∈ ℕs 𝐵 = (𝑧 -s 𝑤))
7		reeanv 3210	. . . . 5 ⊢ (∃𝑦 ∈ ℕs ∃𝑤 ∈ ℕs (𝐴 = (𝑥 -s 𝑦) ∧ 𝐵 = (𝑧 -s 𝑤)) ↔ (∃𝑦 ∈ ℕs 𝐴 = (𝑥 -s 𝑦) ∧ ∃𝑤 ∈ ℕs 𝐵 = (𝑧 -s 𝑤)))
8	7	2rexbii 3110	. . . 4 ⊢ (∃𝑥 ∈ ℕs ∃𝑧 ∈ ℕs ∃𝑦 ∈ ℕs ∃𝑤 ∈ ℕs (𝐴 = (𝑥 -s 𝑦) ∧ 𝐵 = (𝑧 -s 𝑤)) ↔ ∃𝑥 ∈ ℕs ∃𝑧 ∈ ℕs (∃𝑦 ∈ ℕs 𝐴 = (𝑥 -s 𝑦) ∧ ∃𝑤 ∈ ℕs 𝐵 = (𝑧 -s 𝑤)))
9		reeanv 3210	. . . 4 ⊢ (∃𝑥 ∈ ℕs ∃𝑧 ∈ ℕs (∃𝑦 ∈ ℕs 𝐴 = (𝑥 -s 𝑦) ∧ ∃𝑤 ∈ ℕs 𝐵 = (𝑧 -s 𝑤)) ↔ (∃𝑥 ∈ ℕs ∃𝑦 ∈ ℕs 𝐴 = (𝑥 -s 𝑦) ∧ ∃𝑧 ∈ ℕs ∃𝑤 ∈ ℕs 𝐵 = (𝑧 -s 𝑤)))
10	8, 9	bitri 275	. . 3 ⊢ (∃𝑥 ∈ ℕs ∃𝑧 ∈ ℕs ∃𝑦 ∈ ℕs ∃𝑤 ∈ ℕs (𝐴 = (𝑥 -s 𝑦) ∧ 𝐵 = (𝑧 -s 𝑤)) ↔ (∃𝑥 ∈ ℕs ∃𝑦 ∈ ℕs 𝐴 = (𝑥 -s 𝑦) ∧ ∃𝑧 ∈ ℕs ∃𝑤 ∈ ℕs 𝐵 = (𝑧 -s 𝑤)))
11		nnsno 28223	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℕs → 𝑥 ∈ No )
12	11	ad2antrr 726	. . . . . . . . . 10 ⊢ (((𝑥 ∈ ℕs ∧ 𝑧 ∈ ℕs) ∧ (𝑦 ∈ ℕs ∧ 𝑤 ∈ ℕs)) → 𝑥 ∈ No )
13		nnsno 28223	. . . . . . . . . . 11 ⊢ (𝑦 ∈ ℕs → 𝑦 ∈ No )
14	13	ad2antrl 728	. . . . . . . . . 10 ⊢ (((𝑥 ∈ ℕs ∧ 𝑧 ∈ ℕs) ∧ (𝑦 ∈ ℕs ∧ 𝑤 ∈ ℕs)) → 𝑦 ∈ No )
15	12, 14	subscld 27973	. . . . . . . . 9 ⊢ (((𝑥 ∈ ℕs ∧ 𝑧 ∈ ℕs) ∧ (𝑦 ∈ ℕs ∧ 𝑤 ∈ ℕs)) → (𝑥 -s 𝑦) ∈ No )
16		nnsno 28223	. . . . . . . . . 10 ⊢ (𝑧 ∈ ℕs → 𝑧 ∈ No )
17	16	ad2antlr 727	. . . . . . . . 9 ⊢ (((𝑥 ∈ ℕs ∧ 𝑧 ∈ ℕs) ∧ (𝑦 ∈ ℕs ∧ 𝑤 ∈ ℕs)) → 𝑧 ∈ No )
18		nnsno 28223	. . . . . . . . . 10 ⊢ (𝑤 ∈ ℕs → 𝑤 ∈ No )
19	18	ad2antll 729	. . . . . . . . 9 ⊢ (((𝑥 ∈ ℕs ∧ 𝑧 ∈ ℕs) ∧ (𝑦 ∈ ℕs ∧ 𝑤 ∈ ℕs)) → 𝑤 ∈ No )
20	15, 17, 19	subsdid 28067	. . . . . . . 8 ⊢ (((𝑥 ∈ ℕs ∧ 𝑧 ∈ ℕs) ∧ (𝑦 ∈ ℕs ∧ 𝑤 ∈ ℕs)) → ((𝑥 -s 𝑦) ·s (𝑧 -s 𝑤)) = (((𝑥 -s 𝑦) ·s 𝑧) -s ((𝑥 -s 𝑦) ·s 𝑤)))
21		nnmulscl 28245	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ ℕs ∧ 𝑧 ∈ ℕs) → (𝑥 ·s 𝑧) ∈ ℕs)
22	21	adantr 480	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ ℕs ∧ 𝑧 ∈ ℕs) ∧ (𝑦 ∈ ℕs ∧ 𝑤 ∈ ℕs)) → (𝑥 ·s 𝑧) ∈ ℕs)
23	22	nnsnod 28225	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ ℕs ∧ 𝑧 ∈ ℕs) ∧ (𝑦 ∈ ℕs ∧ 𝑤 ∈ ℕs)) → (𝑥 ·s 𝑧) ∈ No )
24		simprl 770	. . . . . . . . . . . . 13 ⊢ (((𝑥 ∈ ℕs ∧ 𝑧 ∈ ℕs) ∧ (𝑦 ∈ ℕs ∧ 𝑤 ∈ ℕs)) → 𝑦 ∈ ℕs)
25		simplr 768	. . . . . . . . . . . . 13 ⊢ (((𝑥 ∈ ℕs ∧ 𝑧 ∈ ℕs) ∧ (𝑦 ∈ ℕs ∧ 𝑤 ∈ ℕs)) → 𝑧 ∈ ℕs)
26		nnmulscl 28245	. . . . . . . . . . . . 13 ⊢ ((𝑦 ∈ ℕs ∧ 𝑧 ∈ ℕs) → (𝑦 ·s 𝑧) ∈ ℕs)
27	24, 25, 26	syl2anc 584	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ ℕs ∧ 𝑧 ∈ ℕs) ∧ (𝑦 ∈ ℕs ∧ 𝑤 ∈ ℕs)) → (𝑦 ·s 𝑧) ∈ ℕs)
28	27	nnsnod 28225	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ ℕs ∧ 𝑧 ∈ ℕs) ∧ (𝑦 ∈ ℕs ∧ 𝑤 ∈ ℕs)) → (𝑦 ·s 𝑧) ∈ No )
29	23, 28	subscld 27973	. . . . . . . . . 10 ⊢ (((𝑥 ∈ ℕs ∧ 𝑧 ∈ ℕs) ∧ (𝑦 ∈ ℕs ∧ 𝑤 ∈ ℕs)) → ((𝑥 ·s 𝑧) -s (𝑦 ·s 𝑧)) ∈ No )
30		nnmulscl 28245	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ ℕs ∧ 𝑤 ∈ ℕs) → (𝑥 ·s 𝑤) ∈ ℕs)
31	30	ad2ant2rl 749	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ ℕs ∧ 𝑧 ∈ ℕs) ∧ (𝑦 ∈ ℕs ∧ 𝑤 ∈ ℕs)) → (𝑥 ·s 𝑤) ∈ ℕs)
32	31	nnsnod 28225	. . . . . . . . . 10 ⊢ (((𝑥 ∈ ℕs ∧ 𝑧 ∈ ℕs) ∧ (𝑦 ∈ ℕs ∧ 𝑤 ∈ ℕs)) → (𝑥 ·s 𝑤) ∈ No )
33		nnmulscl 28245	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ ℕs ∧ 𝑤 ∈ ℕs) → (𝑦 ·s 𝑤) ∈ ℕs)
34	33	adantl 481	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ ℕs ∧ 𝑧 ∈ ℕs) ∧ (𝑦 ∈ ℕs ∧ 𝑤 ∈ ℕs)) → (𝑦 ·s 𝑤) ∈ ℕs)
35	34	nnsnod 28225	. . . . . . . . . 10 ⊢ (((𝑥 ∈ ℕs ∧ 𝑧 ∈ ℕs) ∧ (𝑦 ∈ ℕs ∧ 𝑤 ∈ ℕs)) → (𝑦 ·s 𝑤) ∈ No )
36	29, 32, 35	subsubs2d 28005	. . . . . . . . 9 ⊢ (((𝑥 ∈ ℕs ∧ 𝑧 ∈ ℕs) ∧ (𝑦 ∈ ℕs ∧ 𝑤 ∈ ℕs)) → (((𝑥 ·s 𝑧) -s (𝑦 ·s 𝑧)) -s ((𝑥 ·s 𝑤) -s (𝑦 ·s 𝑤))) = (((𝑥 ·s 𝑧) -s (𝑦 ·s 𝑧)) +s ((𝑦 ·s 𝑤) -s (𝑥 ·s 𝑤))))
37	12, 14, 17	subsdird 28068	. . . . . . . . . 10 ⊢ (((𝑥 ∈ ℕs ∧ 𝑧 ∈ ℕs) ∧ (𝑦 ∈ ℕs ∧ 𝑤 ∈ ℕs)) → ((𝑥 -s 𝑦) ·s 𝑧) = ((𝑥 ·s 𝑧) -s (𝑦 ·s 𝑧)))
38	12, 14, 19	subsdird 28068	. . . . . . . . . 10 ⊢ (((𝑥 ∈ ℕs ∧ 𝑧 ∈ ℕs) ∧ (𝑦 ∈ ℕs ∧ 𝑤 ∈ ℕs)) → ((𝑥 -s 𝑦) ·s 𝑤) = ((𝑥 ·s 𝑤) -s (𝑦 ·s 𝑤)))
39	37, 38	oveq12d 7407	. . . . . . . . 9 ⊢ (((𝑥 ∈ ℕs ∧ 𝑧 ∈ ℕs) ∧ (𝑦 ∈ ℕs ∧ 𝑤 ∈ ℕs)) → (((𝑥 -s 𝑦) ·s 𝑧) -s ((𝑥 -s 𝑦) ·s 𝑤)) = (((𝑥 ·s 𝑧) -s (𝑦 ·s 𝑧)) -s ((𝑥 ·s 𝑤) -s (𝑦 ·s 𝑤))))
40	23, 35, 28, 32	addsubs4d 28010	. . . . . . . . 9 ⊢ (((𝑥 ∈ ℕs ∧ 𝑧 ∈ ℕs) ∧ (𝑦 ∈ ℕs ∧ 𝑤 ∈ ℕs)) → (((𝑥 ·s 𝑧) +s (𝑦 ·s 𝑤)) -s ((𝑦 ·s 𝑧) +s (𝑥 ·s 𝑤))) = (((𝑥 ·s 𝑧) -s (𝑦 ·s 𝑧)) +s ((𝑦 ·s 𝑤) -s (𝑥 ·s 𝑤))))
41	36, 39, 40	3eqtr4d 2775	. . . . . . . 8 ⊢ (((𝑥 ∈ ℕs ∧ 𝑧 ∈ ℕs) ∧ (𝑦 ∈ ℕs ∧ 𝑤 ∈ ℕs)) → (((𝑥 -s 𝑦) ·s 𝑧) -s ((𝑥 -s 𝑦) ·s 𝑤)) = (((𝑥 ·s 𝑧) +s (𝑦 ·s 𝑤)) -s ((𝑦 ·s 𝑧) +s (𝑥 ·s 𝑤))))
42	20, 41	eqtrd 2765	. . . . . . 7 ⊢ (((𝑥 ∈ ℕs ∧ 𝑧 ∈ ℕs) ∧ (𝑦 ∈ ℕs ∧ 𝑤 ∈ ℕs)) → ((𝑥 -s 𝑦) ·s (𝑧 -s 𝑤)) = (((𝑥 ·s 𝑧) +s (𝑦 ·s 𝑤)) -s ((𝑦 ·s 𝑧) +s (𝑥 ·s 𝑤))))
43		nnaddscl 28244	. . . . . . . . . 10 ⊢ (((𝑥 ·s 𝑧) ∈ ℕs ∧ (𝑦 ·s 𝑤) ∈ ℕs) → ((𝑥 ·s 𝑧) +s (𝑦 ·s 𝑤)) ∈ ℕs)
44	22, 34, 43	syl2anc 584	. . . . . . . . 9 ⊢ (((𝑥 ∈ ℕs ∧ 𝑧 ∈ ℕs) ∧ (𝑦 ∈ ℕs ∧ 𝑤 ∈ ℕs)) → ((𝑥 ·s 𝑧) +s (𝑦 ·s 𝑤)) ∈ ℕs)
45		nnaddscl 28244	. . . . . . . . . 10 ⊢ (((𝑦 ·s 𝑧) ∈ ℕs ∧ (𝑥 ·s 𝑤) ∈ ℕs) → ((𝑦 ·s 𝑧) +s (𝑥 ·s 𝑤)) ∈ ℕs)
46	27, 31, 45	syl2anc 584	. . . . . . . . 9 ⊢ (((𝑥 ∈ ℕs ∧ 𝑧 ∈ ℕs) ∧ (𝑦 ∈ ℕs ∧ 𝑤 ∈ ℕs)) → ((𝑦 ·s 𝑧) +s (𝑥 ·s 𝑤)) ∈ ℕs)
47		eqid 2730	. . . . . . . . . 10 ⊢ (((𝑥 ·s 𝑧) +s (𝑦 ·s 𝑤)) -s ((𝑦 ·s 𝑧) +s (𝑥 ·s 𝑤))) = (((𝑥 ·s 𝑧) +s (𝑦 ·s 𝑤)) -s ((𝑦 ·s 𝑧) +s (𝑥 ·s 𝑤)))
48		rspceov 7438	. . . . . . . . . 10 ⊢ ((((𝑥 ·s 𝑧) +s (𝑦 ·s 𝑤)) ∈ ℕs ∧ ((𝑦 ·s 𝑧) +s (𝑥 ·s 𝑤)) ∈ ℕs ∧ (((𝑥 ·s 𝑧) +s (𝑦 ·s 𝑤)) -s ((𝑦 ·s 𝑧) +s (𝑥 ·s 𝑤))) = (((𝑥 ·s 𝑧) +s (𝑦 ·s 𝑤)) -s ((𝑦 ·s 𝑧) +s (𝑥 ·s 𝑤)))) → ∃𝑡 ∈ ℕs ∃𝑢 ∈ ℕs (((𝑥 ·s 𝑧) +s (𝑦 ·s 𝑤)) -s ((𝑦 ·s 𝑧) +s (𝑥 ·s 𝑤))) = (𝑡 -s 𝑢))
49	47, 48	mp3an3 1452	. . . . . . . . 9 ⊢ ((((𝑥 ·s 𝑧) +s (𝑦 ·s 𝑤)) ∈ ℕs ∧ ((𝑦 ·s 𝑧) +s (𝑥 ·s 𝑤)) ∈ ℕs) → ∃𝑡 ∈ ℕs ∃𝑢 ∈ ℕs (((𝑥 ·s 𝑧) +s (𝑦 ·s 𝑤)) -s ((𝑦 ·s 𝑧) +s (𝑥 ·s 𝑤))) = (𝑡 -s 𝑢))
50	44, 46, 49	syl2anc 584	. . . . . . . 8 ⊢ (((𝑥 ∈ ℕs ∧ 𝑧 ∈ ℕs) ∧ (𝑦 ∈ ℕs ∧ 𝑤 ∈ ℕs)) → ∃𝑡 ∈ ℕs ∃𝑢 ∈ ℕs (((𝑥 ·s 𝑧) +s (𝑦 ·s 𝑤)) -s ((𝑦 ·s 𝑧) +s (𝑥 ·s 𝑤))) = (𝑡 -s 𝑢))
51		elzs 28278	. . . . . . . 8 ⊢ ((((𝑥 ·s 𝑧) +s (𝑦 ·s 𝑤)) -s ((𝑦 ·s 𝑧) +s (𝑥 ·s 𝑤))) ∈ ℤs ↔ ∃𝑡 ∈ ℕs ∃𝑢 ∈ ℕs (((𝑥 ·s 𝑧) +s (𝑦 ·s 𝑤)) -s ((𝑦 ·s 𝑧) +s (𝑥 ·s 𝑤))) = (𝑡 -s 𝑢))
52	50, 51	sylibr 234	. . . . . . 7 ⊢ (((𝑥 ∈ ℕs ∧ 𝑧 ∈ ℕs) ∧ (𝑦 ∈ ℕs ∧ 𝑤 ∈ ℕs)) → (((𝑥 ·s 𝑧) +s (𝑦 ·s 𝑤)) -s ((𝑦 ·s 𝑧) +s (𝑥 ·s 𝑤))) ∈ ℤs)
53	42, 52	eqeltrd 2829	. . . . . 6 ⊢ (((𝑥 ∈ ℕs ∧ 𝑧 ∈ ℕs) ∧ (𝑦 ∈ ℕs ∧ 𝑤 ∈ ℕs)) → ((𝑥 -s 𝑦) ·s (𝑧 -s 𝑤)) ∈ ℤs)
54		oveq12 7398	. . . . . . 7 ⊢ ((𝐴 = (𝑥 -s 𝑦) ∧ 𝐵 = (𝑧 -s 𝑤)) → (𝐴 ·s 𝐵) = ((𝑥 -s 𝑦) ·s (𝑧 -s 𝑤)))
55	54	eleq1d 2814	. . . . . 6 ⊢ ((𝐴 = (𝑥 -s 𝑦) ∧ 𝐵 = (𝑧 -s 𝑤)) → ((𝐴 ·s 𝐵) ∈ ℤs ↔ ((𝑥 -s 𝑦) ·s (𝑧 -s 𝑤)) ∈ ℤs))
56	53, 55	syl5ibrcom 247	. . . . 5 ⊢ (((𝑥 ∈ ℕs ∧ 𝑧 ∈ ℕs) ∧ (𝑦 ∈ ℕs ∧ 𝑤 ∈ ℕs)) → ((𝐴 = (𝑥 -s 𝑦) ∧ 𝐵 = (𝑧 -s 𝑤)) → (𝐴 ·s 𝐵) ∈ ℤs))
57	56	rexlimdvva 3195	. . . 4 ⊢ ((𝑥 ∈ ℕs ∧ 𝑧 ∈ ℕs) → (∃𝑦 ∈ ℕs ∃𝑤 ∈ ℕs (𝐴 = (𝑥 -s 𝑦) ∧ 𝐵 = (𝑧 -s 𝑤)) → (𝐴 ·s 𝐵) ∈ ℤs))
58	57	rexlimivv 3180	. . 3 ⊢ (∃𝑥 ∈ ℕs ∃𝑧 ∈ ℕs ∃𝑦 ∈ ℕs ∃𝑤 ∈ ℕs (𝐴 = (𝑥 -s 𝑦) ∧ 𝐵 = (𝑧 -s 𝑤)) → (𝐴 ·s 𝐵) ∈ ℤs)
59	10, 58	sylbir 235	. 2 ⊢ ((∃𝑥 ∈ ℕs ∃𝑦 ∈ ℕs 𝐴 = (𝑥 -s 𝑦) ∧ ∃𝑧 ∈ ℕs ∃𝑤 ∈ ℕs 𝐵 = (𝑧 -s 𝑤)) → (𝐴 ·s 𝐵) ∈ ℤs)
60	3, 6, 59	syl2anc 584	1 ⊢ (𝜑 → (𝐴 ·s 𝐵) ∈ ℤs)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109  ∃wrex 3054  (class class class)co 7389   No csur 27557   +_s cadds 27872   -_s csubs 27932   ·_s cmuls 28015  ℕ_scnns 28213  ℤ_sczs 28272

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 2702  ax-rep 5236  ax-sep 5253  ax-nul 5263  ax-pow 5322  ax-pr 5389  ax-un 7713

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 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-rmo 3356  df-reu 3357  df-rab 3409  df-v 3452  df-sbc 3756  df-csb 3865  df-dif 3919  df-un 3921  df-in 3923  df-ss 3933  df-pss 3936  df-nul 4299  df-if 4491  df-pw 4567  df-sn 4592  df-pr 4594  df-tp 4596  df-op 4598  df-ot 4600  df-uni 4874  df-int 4913  df-iun 4959  df-br 5110  df-opab 5172  df-mpt 5191  df-tr 5217  df-id 5535  df-eprel 5540  df-po 5548  df-so 5549  df-fr 5593  df-se 5594  df-we 5595  df-xp 5646  df-rel 5647  df-cnv 5648  df-co 5649  df-dm 5650  df-rn 5651  df-res 5652  df-ima 5653  df-pred 6276  df-ord 6337  df-on 6338  df-lim 6339  df-suc 6340  df-iota 6466  df-fun 6515  df-fn 6516  df-f 6517  df-f1 6518  df-fo 6519  df-f1o 6520  df-fv 6521  df-riota 7346  df-ov 7392  df-oprab 7393  df-mpo 7394  df-om 7845  df-1st 7970  df-2nd 7971  df-frecs 8262  df-wrecs 8293  df-recs 8342  df-rdg 8380  df-1o 8436  df-2o 8437  df-nadd 8632  df-no 27560  df-slt 27561  df-bday 27562  df-sle 27663  df-sslt 27699  df-scut 27701  df-0s 27742  df-1s 27743  df-made 27761  df-old 27762  df-left 27764  df-right 27765  df-norec 27851  df-norec2 27862  df-adds 27873  df-negs 27933  df-subs 27934  df-muls 28016  df-n0s 28214  df-nns 28215  df-zs 28273

This theorem is referenced by:  pw2cutp1  28342  zs12bday  28349