Theorem zmulscld 28321

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 28308	. . 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 28308	. . 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 3204	. . . . 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 3108	. . . 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 3204	. . . 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 28253	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℕs → 𝑥 ∈ No )
12	11	ad2antrr 726	. . . . . . . . . 10 ⊢ (((𝑥 ∈ ℕs ∧ 𝑧 ∈ ℕs) ∧ (𝑦 ∈ ℕs ∧ 𝑤 ∈ ℕs)) → 𝑥 ∈ No )
13		nnsno 28253	. . . . . . . . . . 11 ⊢ (𝑦 ∈ ℕs → 𝑦 ∈ No )
14	13	ad2antrl 728	. . . . . . . . . 10 ⊢ (((𝑥 ∈ ℕs ∧ 𝑧 ∈ ℕs) ∧ (𝑦 ∈ ℕs ∧ 𝑤 ∈ ℕs)) → 𝑦 ∈ No )
15	12, 14	subscld 28003	. . . . . . . . 9 ⊢ (((𝑥 ∈ ℕs ∧ 𝑧 ∈ ℕs) ∧ (𝑦 ∈ ℕs ∧ 𝑤 ∈ ℕs)) → (𝑥 -s 𝑦) ∈ No )
16		nnsno 28253	. . . . . . . . . 10 ⊢ (𝑧 ∈ ℕs → 𝑧 ∈ No )
17	16	ad2antlr 727	. . . . . . . . 9 ⊢ (((𝑥 ∈ ℕs ∧ 𝑧 ∈ ℕs) ∧ (𝑦 ∈ ℕs ∧ 𝑤 ∈ ℕs)) → 𝑧 ∈ No )
18		nnsno 28253	. . . . . . . . . 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 28097	. . . . . . . 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 28275	. . . . . . . . . . . . 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 28255	. . . . . . . . . . 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 28275	. . . . . . . . . . . . 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 28255	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ ℕs ∧ 𝑧 ∈ ℕs) ∧ (𝑦 ∈ ℕs ∧ 𝑤 ∈ ℕs)) → (𝑦 ·s 𝑧) ∈ No )
29	23, 28	subscld 28003	. . . . . . . . . 10 ⊢ (((𝑥 ∈ ℕs ∧ 𝑧 ∈ ℕs) ∧ (𝑦 ∈ ℕs ∧ 𝑤 ∈ ℕs)) → ((𝑥 ·s 𝑧) -s (𝑦 ·s 𝑧)) ∈ No )
30		nnmulscl 28275	. . . . . . . . . . . 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 28255	. . . . . . . . . 10 ⊢ (((𝑥 ∈ ℕs ∧ 𝑧 ∈ ℕs) ∧ (𝑦 ∈ ℕs ∧ 𝑤 ∈ ℕs)) → (𝑥 ·s 𝑤) ∈ No )
33		nnmulscl 28275	. . . . . . . . . . . 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 28255	. . . . . . . . . 10 ⊢ (((𝑥 ∈ ℕs ∧ 𝑧 ∈ ℕs) ∧ (𝑦 ∈ ℕs ∧ 𝑤 ∈ ℕs)) → (𝑦 ·s 𝑤) ∈ No )
36	29, 32, 35	subsubs2d 28035	. . . . . . . . 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 28098	. . . . . . . . . 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 28098	. . . . . . . . . 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 7364	. . . . . . . . 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 28040	. . . . . . . . 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 2776	. . . . . . . 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 2766	. . . . . . 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 28274	. . . . . . . . . 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 28274	. . . . . . . . . 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 2731	. . . . . . . . . 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 7395	. . . . . . . . . 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 28308	. . . . . . . 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 2831	. . . . . 6 ⊢ (((𝑥 ∈ ℕs ∧ 𝑧 ∈ ℕs) ∧ (𝑦 ∈ ℕs ∧ 𝑤 ∈ ℕs)) → ((𝑥 -s 𝑦) ·s (𝑧 -s 𝑤)) ∈ ℤs)
54		oveq12 7355	. . . . . . 7 ⊢ ((𝐴 = (𝑥 -s 𝑦) ∧ 𝐵 = (𝑧 -s 𝑤)) → (𝐴 ·s 𝐵) = ((𝑥 -s 𝑦) ·s (𝑧 -s 𝑤)))
55	54	eleq1d 2816	. . . . . 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 3189	. . . 4 ⊢ ((𝑥 ∈ ℕs ∧ 𝑧 ∈ ℕs) → (∃𝑦 ∈ ℕs ∃𝑤 ∈ ℕs (𝐴 = (𝑥 -s 𝑦) ∧ 𝐵 = (𝑧 -s 𝑤)) → (𝐴 ·s 𝐵) ∈ ℤs))
58	57	rexlimivv 3174	. . 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 1541   ∈ wcel 2111  ∃wrex 3056  (class class class)co 7346   No csur 27578   +_s cadds 27902   -_s csubs 27962   ·_s cmuls 28045  ℕ_scnns 28243  ℤ_sczs 28302

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-rep 5215  ax-sep 5232  ax-nul 5242  ax-pow 5301  ax-pr 5368  ax-un 7668

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-rmo 3346  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3737  df-csb 3846  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-pss 3917  df-nul 4281  df-if 4473  df-pw 4549  df-sn 4574  df-pr 4576  df-tp 4578  df-op 4580  df-ot 4582  df-uni 4857  df-int 4896  df-iun 4941  df-br 5090  df-opab 5152  df-mpt 5171  df-tr 5197  df-id 5509  df-eprel 5514  df-po 5522  df-so 5523  df-fr 5567  df-se 5568  df-we 5569  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-res 5626  df-ima 5627  df-pred 6248  df-ord 6309  df-on 6310  df-lim 6311  df-suc 6312  df-iota 6437  df-fun 6483  df-fn 6484  df-f 6485  df-f1 6486  df-fo 6487  df-f1o 6488  df-fv 6489  df-riota 7303  df-ov 7349  df-oprab 7350  df-mpo 7351  df-om 7797  df-1st 7921  df-2nd 7922  df-frecs 8211  df-wrecs 8242  df-recs 8291  df-rdg 8329  df-1o 8385  df-2o 8386  df-nadd 8581  df-no 27581  df-slt 27582  df-bday 27583  df-sle 27684  df-sslt 27721  df-scut 27723  df-0s 27768  df-1s 27769  df-made 27788  df-old 27789  df-left 27791  df-right 27792  df-norec 27881  df-norec2 27892  df-adds 27903  df-negs 27963  df-subs 27964  df-muls 28046  df-n0s 28244  df-nns 28245  df-zs 28303

This theorem is referenced by:  zsoring  28332  zexpscl  28357  pw2cutp1  28381  zs12addscl  28387  zs12bday  28394