Theorem expadds 28639

Description: Sum of exponents law for surreals. (Contributed by Scott Fenton, 7-Nov-2025.)

Assertion

Ref	Expression
expadds	⊢ ((𝐴 ∈ No ∧ 𝑀 ∈ ℕ0s ∧ 𝑁 ∈ ℕ0s) → (𝐴↑s(𝑀 +s 𝑁)) = ((𝐴↑s𝑀) ·s (𝐴↑s𝑁)))

Proof of Theorem expadds

Dummy variables 𝑗 𝑘 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		oveq2 7420	. . . . . . 7 ⊢ (𝑗 = 0s → (𝑀 +s 𝑗) = (𝑀 +s 0s ))
2	1	oveq2d 7428	. . . . . 6 ⊢ (𝑗 = 0s → (𝐴↑s(𝑀 +s 𝑗)) = (𝐴↑s(𝑀 +s 0s )))
3		oveq2 7420	. . . . . . 7 ⊢ (𝑗 = 0s → (𝐴↑s𝑗) = (𝐴↑s 0s ))
4	3	oveq2d 7428	. . . . . 6 ⊢ (𝑗 = 0s → ((𝐴↑s𝑀) ·s (𝐴↑s𝑗)) = ((𝐴↑s𝑀) ·s (𝐴↑s 0s )))
5	2, 4	eqeq12d 2778	. . . . 5 ⊢ (𝑗 = 0s → ((𝐴↑s(𝑀 +s 𝑗)) = ((𝐴↑s𝑀) ·s (𝐴↑s𝑗)) ↔ (𝐴↑s(𝑀 +s 0s )) = ((𝐴↑s𝑀) ·s (𝐴↑s 0s ))))
6	5	imbi2d 343	. . . 4 ⊢ (𝑗 = 0s → (((𝐴 ∈ No ∧ 𝑀 ∈ ℕ0s) → (𝐴↑s(𝑀 +s 𝑗)) = ((𝐴↑s𝑀) ·s (𝐴↑s𝑗))) ↔ ((𝐴 ∈ No ∧ 𝑀 ∈ ℕ0s) → (𝐴↑s(𝑀 +s 0s )) = ((𝐴↑s𝑀) ·s (𝐴↑s 0s )))))
7		oveq2 7420	. . . . . . 7 ⊢ (𝑗 = 𝑘 → (𝑀 +s 𝑗) = (𝑀 +s 𝑘))
8	7	oveq2d 7428	. . . . . 6 ⊢ (𝑗 = 𝑘 → (𝐴↑s(𝑀 +s 𝑗)) = (𝐴↑s(𝑀 +s 𝑘)))
9		oveq2 7420	. . . . . . 7 ⊢ (𝑗 = 𝑘 → (𝐴↑s𝑗) = (𝐴↑s𝑘))
10	9	oveq2d 7428	. . . . . 6 ⊢ (𝑗 = 𝑘 → ((𝐴↑s𝑀) ·s (𝐴↑s𝑗)) = ((𝐴↑s𝑀) ·s (𝐴↑s𝑘)))
11	8, 10	eqeq12d 2778	. . . . 5 ⊢ (𝑗 = 𝑘 → ((𝐴↑s(𝑀 +s 𝑗)) = ((𝐴↑s𝑀) ·s (𝐴↑s𝑗)) ↔ (𝐴↑s(𝑀 +s 𝑘)) = ((𝐴↑s𝑀) ·s (𝐴↑s𝑘))))
12	11	imbi2d 343	. . . 4 ⊢ (𝑗 = 𝑘 → (((𝐴 ∈ No ∧ 𝑀 ∈ ℕ0s) → (𝐴↑s(𝑀 +s 𝑗)) = ((𝐴↑s𝑀) ·s (𝐴↑s𝑗))) ↔ ((𝐴 ∈ No ∧ 𝑀 ∈ ℕ0s) → (𝐴↑s(𝑀 +s 𝑘)) = ((𝐴↑s𝑀) ·s (𝐴↑s𝑘)))))
13		oveq2 7420	. . . . . . 7 ⊢ (𝑗 = (𝑘 +s 1s ) → (𝑀 +s 𝑗) = (𝑀 +s (𝑘 +s 1s )))
14	13	oveq2d 7428	. . . . . 6 ⊢ (𝑗 = (𝑘 +s 1s ) → (𝐴↑s(𝑀 +s 𝑗)) = (𝐴↑s(𝑀 +s (𝑘 +s 1s ))))
15		oveq2 7420	. . . . . . 7 ⊢ (𝑗 = (𝑘 +s 1s ) → (𝐴↑s𝑗) = (𝐴↑s(𝑘 +s 1s )))
16	15	oveq2d 7428	. . . . . 6 ⊢ (𝑗 = (𝑘 +s 1s ) → ((𝐴↑s𝑀) ·s (𝐴↑s𝑗)) = ((𝐴↑s𝑀) ·s (𝐴↑s(𝑘 +s 1s ))))
17	14, 16	eqeq12d 2778	. . . . 5 ⊢ (𝑗 = (𝑘 +s 1s ) → ((𝐴↑s(𝑀 +s 𝑗)) = ((𝐴↑s𝑀) ·s (𝐴↑s𝑗)) ↔ (𝐴↑s(𝑀 +s (𝑘 +s 1s ))) = ((𝐴↑s𝑀) ·s (𝐴↑s(𝑘 +s 1s )))))
18	17	imbi2d 343	. . . 4 ⊢ (𝑗 = (𝑘 +s 1s ) → (((𝐴 ∈ No ∧ 𝑀 ∈ ℕ0s) → (𝐴↑s(𝑀 +s 𝑗)) = ((𝐴↑s𝑀) ·s (𝐴↑s𝑗))) ↔ ((𝐴 ∈ No ∧ 𝑀 ∈ ℕ0s) → (𝐴↑s(𝑀 +s (𝑘 +s 1s ))) = ((𝐴↑s𝑀) ·s (𝐴↑s(𝑘 +s 1s ))))))
19		oveq2 7420	. . . . . . 7 ⊢ (𝑗 = 𝑁 → (𝑀 +s 𝑗) = (𝑀 +s 𝑁))
20	19	oveq2d 7428	. . . . . 6 ⊢ (𝑗 = 𝑁 → (𝐴↑s(𝑀 +s 𝑗)) = (𝐴↑s(𝑀 +s 𝑁)))
21		oveq2 7420	. . . . . . 7 ⊢ (𝑗 = 𝑁 → (𝐴↑s𝑗) = (𝐴↑s𝑁))
22	21	oveq2d 7428	. . . . . 6 ⊢ (𝑗 = 𝑁 → ((𝐴↑s𝑀) ·s (𝐴↑s𝑗)) = ((𝐴↑s𝑀) ·s (𝐴↑s𝑁)))
23	20, 22	eqeq12d 2778	. . . . 5 ⊢ (𝑗 = 𝑁 → ((𝐴↑s(𝑀 +s 𝑗)) = ((𝐴↑s𝑀) ·s (𝐴↑s𝑗)) ↔ (𝐴↑s(𝑀 +s 𝑁)) = ((𝐴↑s𝑀) ·s (𝐴↑s𝑁))))
24	23	imbi2d 343	. . . 4 ⊢ (𝑗 = 𝑁 → (((𝐴 ∈ No ∧ 𝑀 ∈ ℕ0s) → (𝐴↑s(𝑀 +s 𝑗)) = ((𝐴↑s𝑀) ·s (𝐴↑s𝑗))) ↔ ((𝐴 ∈ No ∧ 𝑀 ∈ ℕ0s) → (𝐴↑s(𝑀 +s 𝑁)) = ((𝐴↑s𝑀) ·s (𝐴↑s𝑁)))))
25		expscl 28635	. . . . . 6 ⊢ ((𝐴 ∈ No ∧ 𝑀 ∈ ℕ0s) → (𝐴↑s𝑀) ∈ No )
26	25	mulsridd 28318	. . . . 5 ⊢ ((𝐴 ∈ No ∧ 𝑀 ∈ ℕ0s) → ((𝐴↑s𝑀) ·s 1s ) = (𝐴↑s𝑀))
27		exps0 28631	. . . . . . 7 ⊢ (𝐴 ∈ No → (𝐴↑s 0s ) = 1s )
28	27	oveq2d 7428	. . . . . 6 ⊢ (𝐴 ∈ No → ((𝐴↑s𝑀) ·s (𝐴↑s 0s )) = ((𝐴↑s𝑀) ·s 1s ))
29	28	adantr 485	. . . . 5 ⊢ ((𝐴 ∈ No ∧ 𝑀 ∈ ℕ0s) → ((𝐴↑s𝑀) ·s (𝐴↑s 0s )) = ((𝐴↑s𝑀) ·s 1s ))
30		n0no 28527	. . . . . . . 8 ⊢ (𝑀 ∈ ℕ0s → 𝑀 ∈ No )
31	30	adantl 486	. . . . . . 7 ⊢ ((𝐴 ∈ No ∧ 𝑀 ∈ ℕ0s) → 𝑀 ∈ No )
32	31	addsridd 28169	. . . . . 6 ⊢ ((𝐴 ∈ No ∧ 𝑀 ∈ ℕ0s) → (𝑀 +s 0s ) = 𝑀)
33	32	oveq2d 7428	. . . . 5 ⊢ ((𝐴 ∈ No ∧ 𝑀 ∈ ℕ0s) → (𝐴↑s(𝑀 +s 0s )) = (𝐴↑s𝑀))
34	26, 29, 33	3eqtr4rd 2808	. . . 4 ⊢ ((𝐴 ∈ No ∧ 𝑀 ∈ ℕ0s) → (𝐴↑s(𝑀 +s 0s )) = ((𝐴↑s𝑀) ·s (𝐴↑s 0s )))
35		simprr 784	. . . . . . . . 9 ⊢ ((𝑘 ∈ ℕ0s ∧ ((𝐴 ∈ No ∧ 𝑀 ∈ ℕ0s) ∧ (𝐴↑s(𝑀 +s 𝑘)) = ((𝐴↑s𝑀) ·s (𝐴↑s𝑘)))) → (𝐴↑s(𝑀 +s 𝑘)) = ((𝐴↑s𝑀) ·s (𝐴↑s𝑘)))
36	35	oveq1d 7427	. . . . . . . 8 ⊢ ((𝑘 ∈ ℕ0s ∧ ((𝐴 ∈ No ∧ 𝑀 ∈ ℕ0s) ∧ (𝐴↑s(𝑀 +s 𝑘)) = ((𝐴↑s𝑀) ·s (𝐴↑s𝑘)))) → ((𝐴↑s(𝑀 +s 𝑘)) ·s 𝐴) = (((𝐴↑s𝑀) ·s (𝐴↑s𝑘)) ·s 𝐴))
37	25	adantr 485	. . . . . . . . . 10 ⊢ (((𝐴 ∈ No ∧ 𝑀 ∈ ℕ0s) ∧ (𝐴↑s(𝑀 +s 𝑘)) = ((𝐴↑s𝑀) ·s (𝐴↑s𝑘))) → (𝐴↑s𝑀) ∈ No )
38	37	adantl 486	. . . . . . . . 9 ⊢ ((𝑘 ∈ ℕ0s ∧ ((𝐴 ∈ No ∧ 𝑀 ∈ ℕ0s) ∧ (𝐴↑s(𝑀 +s 𝑘)) = ((𝐴↑s𝑀) ·s (𝐴↑s𝑘)))) → (𝐴↑s𝑀) ∈ No )
39		simprll 790	. . . . . . . . . 10 ⊢ ((𝑘 ∈ ℕ0s ∧ ((𝐴 ∈ No ∧ 𝑀 ∈ ℕ0s) ∧ (𝐴↑s(𝑀 +s 𝑘)) = ((𝐴↑s𝑀) ·s (𝐴↑s𝑘)))) → 𝐴 ∈ No )
40		simpl 487	. . . . . . . . . 10 ⊢ ((𝑘 ∈ ℕ0s ∧ ((𝐴 ∈ No ∧ 𝑀 ∈ ℕ0s) ∧ (𝐴↑s(𝑀 +s 𝑘)) = ((𝐴↑s𝑀) ·s (𝐴↑s𝑘)))) → 𝑘 ∈ ℕ0s)
41		expscl 28635	. . . . . . . . . 10 ⊢ ((𝐴 ∈ No ∧ 𝑘 ∈ ℕ0s) → (𝐴↑s𝑘) ∈ No )
42	39, 40, 41	syl2anc 595	. . . . . . . . 9 ⊢ ((𝑘 ∈ ℕ0s ∧ ((𝐴 ∈ No ∧ 𝑀 ∈ ℕ0s) ∧ (𝐴↑s(𝑀 +s 𝑘)) = ((𝐴↑s𝑀) ·s (𝐴↑s𝑘)))) → (𝐴↑s𝑘) ∈ No )
43	38, 42, 39	mulsassd 28371	. . . . . . . 8 ⊢ ((𝑘 ∈ ℕ0s ∧ ((𝐴 ∈ No ∧ 𝑀 ∈ ℕ0s) ∧ (𝐴↑s(𝑀 +s 𝑘)) = ((𝐴↑s𝑀) ·s (𝐴↑s𝑘)))) → (((𝐴↑s𝑀) ·s (𝐴↑s𝑘)) ·s 𝐴) = ((𝐴↑s𝑀) ·s ((𝐴↑s𝑘) ·s 𝐴)))
44	36, 43	eqtrd 2797	. . . . . . 7 ⊢ ((𝑘 ∈ ℕ0s ∧ ((𝐴 ∈ No ∧ 𝑀 ∈ ℕ0s) ∧ (𝐴↑s(𝑀 +s 𝑘)) = ((𝐴↑s𝑀) ·s (𝐴↑s𝑘)))) → ((𝐴↑s(𝑀 +s 𝑘)) ·s 𝐴) = ((𝐴↑s𝑀) ·s ((𝐴↑s𝑘) ·s 𝐴)))
45		simprlr 791	. . . . . . . . . . 11 ⊢ ((𝑘 ∈ ℕ0s ∧ ((𝐴 ∈ No ∧ 𝑀 ∈ ℕ0s) ∧ (𝐴↑s(𝑀 +s 𝑘)) = ((𝐴↑s𝑀) ·s (𝐴↑s𝑘)))) → 𝑀 ∈ ℕ0s)
46	45	n0nod 28529	. . . . . . . . . 10 ⊢ ((𝑘 ∈ ℕ0s ∧ ((𝐴 ∈ No ∧ 𝑀 ∈ ℕ0s) ∧ (𝐴↑s(𝑀 +s 𝑘)) = ((𝐴↑s𝑀) ·s (𝐴↑s𝑘)))) → 𝑀 ∈ No )
47	40	n0nod 28529	. . . . . . . . . 10 ⊢ ((𝑘 ∈ ℕ0s ∧ ((𝐴 ∈ No ∧ 𝑀 ∈ ℕ0s) ∧ (𝐴↑s(𝑀 +s 𝑘)) = ((𝐴↑s𝑀) ·s (𝐴↑s𝑘)))) → 𝑘 ∈ No )
48		1no 28014	. . . . . . . . . . 11 ⊢ 1s ∈ No
49	48	a1i 11	. . . . . . . . . 10 ⊢ ((𝑘 ∈ ℕ0s ∧ ((𝐴 ∈ No ∧ 𝑀 ∈ ℕ0s) ∧ (𝐴↑s(𝑀 +s 𝑘)) = ((𝐴↑s𝑀) ·s (𝐴↑s𝑘)))) → 1s ∈ No )
50	46, 47, 49	addsassd 28210	. . . . . . . . 9 ⊢ ((𝑘 ∈ ℕ0s ∧ ((𝐴 ∈ No ∧ 𝑀 ∈ ℕ0s) ∧ (𝐴↑s(𝑀 +s 𝑘)) = ((𝐴↑s𝑀) ·s (𝐴↑s𝑘)))) → ((𝑀 +s 𝑘) +s 1s ) = (𝑀 +s (𝑘 +s 1s )))
51	50	oveq2d 7428	. . . . . . . 8 ⊢ ((𝑘 ∈ ℕ0s ∧ ((𝐴 ∈ No ∧ 𝑀 ∈ ℕ0s) ∧ (𝐴↑s(𝑀 +s 𝑘)) = ((𝐴↑s𝑀) ·s (𝐴↑s𝑘)))) → (𝐴↑s((𝑀 +s 𝑘) +s 1s )) = (𝐴↑s(𝑀 +s (𝑘 +s 1s ))))
52		n0addscl 28548	. . . . . . . . . 10 ⊢ ((𝑀 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s) → (𝑀 +s 𝑘) ∈ ℕ0s)
53	45, 40, 52	syl2anc 595	. . . . . . . . 9 ⊢ ((𝑘 ∈ ℕ0s ∧ ((𝐴 ∈ No ∧ 𝑀 ∈ ℕ0s) ∧ (𝐴↑s(𝑀 +s 𝑘)) = ((𝐴↑s𝑀) ·s (𝐴↑s𝑘)))) → (𝑀 +s 𝑘) ∈ ℕ0s)
54		expsp1 28633	. . . . . . . . 9 ⊢ ((𝐴 ∈ No ∧ (𝑀 +s 𝑘) ∈ ℕ0s) → (𝐴↑s((𝑀 +s 𝑘) +s 1s )) = ((𝐴↑s(𝑀 +s 𝑘)) ·s 𝐴))
55	39, 53, 54	syl2anc 595	. . . . . . . 8 ⊢ ((𝑘 ∈ ℕ0s ∧ ((𝐴 ∈ No ∧ 𝑀 ∈ ℕ0s) ∧ (𝐴↑s(𝑀 +s 𝑘)) = ((𝐴↑s𝑀) ·s (𝐴↑s𝑘)))) → (𝐴↑s((𝑀 +s 𝑘) +s 1s )) = ((𝐴↑s(𝑀 +s 𝑘)) ·s 𝐴))
56	51, 55	eqtr3d 2799	. . . . . . 7 ⊢ ((𝑘 ∈ ℕ0s ∧ ((𝐴 ∈ No ∧ 𝑀 ∈ ℕ0s) ∧ (𝐴↑s(𝑀 +s 𝑘)) = ((𝐴↑s𝑀) ·s (𝐴↑s𝑘)))) → (𝐴↑s(𝑀 +s (𝑘 +s 1s ))) = ((𝐴↑s(𝑀 +s 𝑘)) ·s 𝐴))
57		expsp1 28633	. . . . . . . . 9 ⊢ ((𝐴 ∈ No ∧ 𝑘 ∈ ℕ0s) → (𝐴↑s(𝑘 +s 1s )) = ((𝐴↑s𝑘) ·s 𝐴))
58	39, 40, 57	syl2anc 595	. . . . . . . 8 ⊢ ((𝑘 ∈ ℕ0s ∧ ((𝐴 ∈ No ∧ 𝑀 ∈ ℕ0s) ∧ (𝐴↑s(𝑀 +s 𝑘)) = ((𝐴↑s𝑀) ·s (𝐴↑s𝑘)))) → (𝐴↑s(𝑘 +s 1s )) = ((𝐴↑s𝑘) ·s 𝐴))
59	58	oveq2d 7428	. . . . . . 7 ⊢ ((𝑘 ∈ ℕ0s ∧ ((𝐴 ∈ No ∧ 𝑀 ∈ ℕ0s) ∧ (𝐴↑s(𝑀 +s 𝑘)) = ((𝐴↑s𝑀) ·s (𝐴↑s𝑘)))) → ((𝐴↑s𝑀) ·s (𝐴↑s(𝑘 +s 1s ))) = ((𝐴↑s𝑀) ·s ((𝐴↑s𝑘) ·s 𝐴)))
60	44, 56, 59	3eqtr4d 2807	. . . . . 6 ⊢ ((𝑘 ∈ ℕ0s ∧ ((𝐴 ∈ No ∧ 𝑀 ∈ ℕ0s) ∧ (𝐴↑s(𝑀 +s 𝑘)) = ((𝐴↑s𝑀) ·s (𝐴↑s𝑘)))) → (𝐴↑s(𝑀 +s (𝑘 +s 1s ))) = ((𝐴↑s𝑀) ·s (𝐴↑s(𝑘 +s 1s ))))
61	60	exp32 425	. . . . 5 ⊢ (𝑘 ∈ ℕ0s → ((𝐴 ∈ No ∧ 𝑀 ∈ ℕ0s) → ((𝐴↑s(𝑀 +s 𝑘)) = ((𝐴↑s𝑀) ·s (𝐴↑s𝑘)) → (𝐴↑s(𝑀 +s (𝑘 +s 1s ))) = ((𝐴↑s𝑀) ·s (𝐴↑s(𝑘 +s 1s ))))))
62	61	a2d 30	. . . 4 ⊢ (𝑘 ∈ ℕ0s → (((𝐴 ∈ No ∧ 𝑀 ∈ ℕ0s) → (𝐴↑s(𝑀 +s 𝑘)) = ((𝐴↑s𝑀) ·s (𝐴↑s𝑘))) → ((𝐴 ∈ No ∧ 𝑀 ∈ ℕ0s) → (𝐴↑s(𝑀 +s (𝑘 +s 1s ))) = ((𝐴↑s𝑀) ·s (𝐴↑s(𝑘 +s 1s ))))))
63	6, 12, 18, 24, 34, 62	n0sind 28537	. . 3 ⊢ (𝑁 ∈ ℕ0s → ((𝐴 ∈ No ∧ 𝑀 ∈ ℕ0s) → (𝐴↑s(𝑀 +s 𝑁)) = ((𝐴↑s𝑀) ·s (𝐴↑s𝑁))))
64	63	com12 33	. 2 ⊢ ((𝐴 ∈ No ∧ 𝑀 ∈ ℕ0s) → (𝑁 ∈ ℕ0s → (𝐴↑s(𝑀 +s 𝑁)) = ((𝐴↑s𝑀) ·s (𝐴↑s𝑁))))
65	64	3impia 1134	1 ⊢ ((𝐴 ∈ No ∧ 𝑀 ∈ ℕ0s ∧ 𝑁 ∈ ℕ0s) → (𝐴↑s(𝑀 +s 𝑁)) = ((𝐴↑s𝑀) ·s (𝐴↑s𝑁)))

This proof depends on syntax axioms:   → wi 4   ∧ wa 400   ∧ w3a 1102   = wceq 1569   ∈ wcel 2142  (class class class)co 7412   No csur 27815   0_s c0s 28009   1_s c1s 28010   +_s cadds 28163   ·_s cmuls 28310  ℕ_0scn0s 28516  ↑_scexps 28616

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-rep 5237  ax-sep 5256  ax-nul 5268  ax-pow 5335  ax-pr 5403  ax-un 7734

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1103  df-3an 1104  df-tru 1572  df-fal 1582  df-ex 1809  df-nf 1813  df-sb 2096  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-rmo 3368  df-reu 3369  df-rab 3416  df-v 3456  df-sbc 3744  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-pss 3924  df-nul 4286  df-if 4487  df-pw 4563  df-sn 4589  df-pr 4591  df-tp 4593  df-op 4595  df-ot 4597  df-uni 4872  df-int 4912  df-iun 4957  df-br 5109  df-opab 5173  df-mpt 5192  df-tr 5218  df-id 5555  df-eprel 5560  df-po 5568  df-so 5569  df-fr 5613  df-se 5614  df-we 5615  df-xp 5666  df-rel 5667  df-cnv 5668  df-co 5669  df-dm 5670  df-rn 5671  df-res 5672  df-ima 5673  df-pred 6302  df-ord 6363  df-on 6364  df-lim 6365  df-suc 6366  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-riota 7369  df-ov 7415  df-oprab 7416  df-mpo 7417  df-om 7861  df-1st 7984  df-2nd 7985  df-frecs 8276  df-wrecs 8307  df-recs 8356  df-rdg 8395  df-1o 8451  df-2o 8452  df-oadd 8455  df-nadd 8650  df-no 27818  df-lts 27819  df-bday 27820  df-les 27920  df-slts 27962  df-cuts 27964  df-0s 28011  df-1s 28012  df-made 28031  df-old 28032  df-left 28034  df-right 28035  df-norec 28142  df-norec2 28153  df-adds 28164  df-negs 28225  df-subs 28226  df-muls 28311  df-seqs 28488  df-n0s 28518  df-nns 28519  df-zs 28583  df-exps 28617

This theorem is used by:  pw2divscan4d  28648  bdayfinbndlem1  28671