Theorem zmulscld 28290

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 28277	. . 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 28277	. . 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 3201	. . . . 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 3105	. . . 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 3201	. . . 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 28222	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℕs → 𝑥 ∈ No )
12	11	ad2antrr 726	. . . . . . . . . 10 ⊢ (((𝑥 ∈ ℕs ∧ 𝑧 ∈ ℕs) ∧ (𝑦 ∈ ℕs ∧ 𝑤 ∈ ℕs)) → 𝑥 ∈ No )
13		nnsno 28222	. . . . . . . . . . 11 ⊢ (𝑦 ∈ ℕs → 𝑦 ∈ No )
14	13	ad2antrl 728	. . . . . . . . . 10 ⊢ (((𝑥 ∈ ℕs ∧ 𝑧 ∈ ℕs) ∧ (𝑦 ∈ ℕs ∧ 𝑤 ∈ ℕs)) → 𝑦 ∈ No )
15	12, 14	subscld 27972	. . . . . . . . 9 ⊢ (((𝑥 ∈ ℕs ∧ 𝑧 ∈ ℕs) ∧ (𝑦 ∈ ℕs ∧ 𝑤 ∈ ℕs)) → (𝑥 -s 𝑦) ∈ No )
16		nnsno 28222	. . . . . . . . . 10 ⊢ (𝑧 ∈ ℕs → 𝑧 ∈ No )
17	16	ad2antlr 727	. . . . . . . . 9 ⊢ (((𝑥 ∈ ℕs ∧ 𝑧 ∈ ℕs) ∧ (𝑦 ∈ ℕs ∧ 𝑤 ∈ ℕs)) → 𝑧 ∈ No )
18		nnsno 28222	. . . . . . . . . 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 28066	. . . . . . . 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 28244	. . . . . . . . . . . . 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 28224	. . . . . . . . . . 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 28244	. . . . . . . . . . . . 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 28224	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ ℕs ∧ 𝑧 ∈ ℕs) ∧ (𝑦 ∈ ℕs ∧ 𝑤 ∈ ℕs)) → (𝑦 ·s 𝑧) ∈ No )
29	23, 28	subscld 27972	. . . . . . . . . 10 ⊢ (((𝑥 ∈ ℕs ∧ 𝑧 ∈ ℕs) ∧ (𝑦 ∈ ℕs ∧ 𝑤 ∈ ℕs)) → ((𝑥 ·s 𝑧) -s (𝑦 ·s 𝑧)) ∈ No )
30		nnmulscl 28244	. . . . . . . . . . . 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 28224	. . . . . . . . . 10 ⊢ (((𝑥 ∈ ℕs ∧ 𝑧 ∈ ℕs) ∧ (𝑦 ∈ ℕs ∧ 𝑤 ∈ ℕs)) → (𝑥 ·s 𝑤) ∈ No )
33		nnmulscl 28244	. . . . . . . . . . . 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 28224	. . . . . . . . . 10 ⊢ (((𝑥 ∈ ℕs ∧ 𝑧 ∈ ℕs) ∧ (𝑦 ∈ ℕs ∧ 𝑤 ∈ ℕs)) → (𝑦 ·s 𝑤) ∈ No )
36	29, 32, 35	subsubs2d 28004	. . . . . . . . 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 28067	. . . . . . . . . 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 28067	. . . . . . . . . 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 7367	. . . . . . . . 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 28009	. . . . . . . . 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 2774	. . . . . . . 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 2764	. . . . . . 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 28243	. . . . . . . . . 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 28243	. . . . . . . . . 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 2729	. . . . . . . . . 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 7398	. . . . . . . . . 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 28277	. . . . . . . 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 2828	. . . . . 6 ⊢ (((𝑥 ∈ ℕs ∧ 𝑧 ∈ ℕs) ∧ (𝑦 ∈ ℕs ∧ 𝑤 ∈ ℕs)) → ((𝑥 -s 𝑦) ·s (𝑧 -s 𝑤)) ∈ ℤs)
54		oveq12 7358	. . . . . . 7 ⊢ ((𝐴 = (𝑥 -s 𝑦) ∧ 𝐵 = (𝑧 -s 𝑤)) → (𝐴 ·s 𝐵) = ((𝑥 -s 𝑦) ·s (𝑧 -s 𝑤)))
55	54	eleq1d 2813	. . . . . 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 3186	. . . 4 ⊢ ((𝑥 ∈ ℕs ∧ 𝑧 ∈ ℕs) → (∃𝑦 ∈ ℕs ∃𝑤 ∈ ℕs (𝐴 = (𝑥 -s 𝑦) ∧ 𝐵 = (𝑧 -s 𝑤)) → (𝐴 ·s 𝐵) ∈ ℤs))
58	57	rexlimivv 3171	. . 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 3053  (class class class)co 7349   No csur 27549   +_s cadds 27871   -_s csubs 27931   ·_s cmuls 28014  ℕ_scnns 28212  ℤ_sczs 28271

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  df-nns 28214  df-zs 28272

This theorem is referenced by:  zsoring  28301  zexpscl  28326  pw2cutp1  28349  zs12addscl  28354  zs12bday  28361