Theorem expadds 28427

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 7375	. . . . . . 7 ⊢ (𝑗 = 0s → (𝑀 +s 𝑗) = (𝑀 +s 0s ))
2	1	oveq2d 7383	. . . . . 6 ⊢ (𝑗 = 0s → (𝐴↑s(𝑀 +s 𝑗)) = (𝐴↑s(𝑀 +s 0s )))
3		oveq2 7375	. . . . . . 7 ⊢ (𝑗 = 0s → (𝐴↑s𝑗) = (𝐴↑s 0s ))
4	3	oveq2d 7383	. . . . . 6 ⊢ (𝑗 = 0s → ((𝐴↑s𝑀) ·s (𝐴↑s𝑗)) = ((𝐴↑s𝑀) ·s (𝐴↑s 0s )))
5	2, 4	eqeq12d 2753	. . . . 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 340	. . . 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 7375	. . . . . . 7 ⊢ (𝑗 = 𝑘 → (𝑀 +s 𝑗) = (𝑀 +s 𝑘))
8	7	oveq2d 7383	. . . . . 6 ⊢ (𝑗 = 𝑘 → (𝐴↑s(𝑀 +s 𝑗)) = (𝐴↑s(𝑀 +s 𝑘)))
9		oveq2 7375	. . . . . . 7 ⊢ (𝑗 = 𝑘 → (𝐴↑s𝑗) = (𝐴↑s𝑘))
10	9	oveq2d 7383	. . . . . 6 ⊢ (𝑗 = 𝑘 → ((𝐴↑s𝑀) ·s (𝐴↑s𝑗)) = ((𝐴↑s𝑀) ·s (𝐴↑s𝑘)))
11	8, 10	eqeq12d 2753	. . . . 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 340	. . . 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 7375	. . . . . . 7 ⊢ (𝑗 = (𝑘 +s 1s ) → (𝑀 +s 𝑗) = (𝑀 +s (𝑘 +s 1s )))
14	13	oveq2d 7383	. . . . . 6 ⊢ (𝑗 = (𝑘 +s 1s ) → (𝐴↑s(𝑀 +s 𝑗)) = (𝐴↑s(𝑀 +s (𝑘 +s 1s ))))
15		oveq2 7375	. . . . . . 7 ⊢ (𝑗 = (𝑘 +s 1s ) → (𝐴↑s𝑗) = (𝐴↑s(𝑘 +s 1s )))
16	15	oveq2d 7383	. . . . . 6 ⊢ (𝑗 = (𝑘 +s 1s ) → ((𝐴↑s𝑀) ·s (𝐴↑s𝑗)) = ((𝐴↑s𝑀) ·s (𝐴↑s(𝑘 +s 1s ))))
17	14, 16	eqeq12d 2753	. . . . 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 340	. . . 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 7375	. . . . . . 7 ⊢ (𝑗 = 𝑁 → (𝑀 +s 𝑗) = (𝑀 +s 𝑁))
20	19	oveq2d 7383	. . . . . 6 ⊢ (𝑗 = 𝑁 → (𝐴↑s(𝑀 +s 𝑗)) = (𝐴↑s(𝑀 +s 𝑁)))
21		oveq2 7375	. . . . . . 7 ⊢ (𝑗 = 𝑁 → (𝐴↑s𝑗) = (𝐴↑s𝑁))
22	21	oveq2d 7383	. . . . . 6 ⊢ (𝑗 = 𝑁 → ((𝐴↑s𝑀) ·s (𝐴↑s𝑗)) = ((𝐴↑s𝑀) ·s (𝐴↑s𝑁)))
23	20, 22	eqeq12d 2753	. . . . 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 340	. . . 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 28423	. . . . . 6 ⊢ ((𝐴 ∈ No ∧ 𝑀 ∈ ℕ0s) → (𝐴↑s𝑀) ∈ No )
26	25	mulsridd 28106	. . . . 5 ⊢ ((𝐴 ∈ No ∧ 𝑀 ∈ ℕ0s) → ((𝐴↑s𝑀) ·s 1s ) = (𝐴↑s𝑀))
27		exps0 28419	. . . . . . 7 ⊢ (𝐴 ∈ No → (𝐴↑s 0s ) = 1s )
28	27	oveq2d 7383	. . . . . 6 ⊢ (𝐴 ∈ No → ((𝐴↑s𝑀) ·s (𝐴↑s 0s )) = ((𝐴↑s𝑀) ·s 1s ))
29	28	adantr 480	. . . . 5 ⊢ ((𝐴 ∈ No ∧ 𝑀 ∈ ℕ0s) → ((𝐴↑s𝑀) ·s (𝐴↑s 0s )) = ((𝐴↑s𝑀) ·s 1s ))
30		n0no 28315	. . . . . . . 8 ⊢ (𝑀 ∈ ℕ0s → 𝑀 ∈ No )
31	30	adantl 481	. . . . . . 7 ⊢ ((𝐴 ∈ No ∧ 𝑀 ∈ ℕ0s) → 𝑀 ∈ No )
32	31	addsridd 27957	. . . . . 6 ⊢ ((𝐴 ∈ No ∧ 𝑀 ∈ ℕ0s) → (𝑀 +s 0s ) = 𝑀)
33	32	oveq2d 7383	. . . . 5 ⊢ ((𝐴 ∈ No ∧ 𝑀 ∈ ℕ0s) → (𝐴↑s(𝑀 +s 0s )) = (𝐴↑s𝑀))
34	26, 29, 33	3eqtr4rd 2783	. . . 4 ⊢ ((𝐴 ∈ No ∧ 𝑀 ∈ ℕ0s) → (𝐴↑s(𝑀 +s 0s )) = ((𝐴↑s𝑀) ·s (𝐴↑s 0s )))
35		simprr 773	. . . . . . . . 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 7382	. . . . . . . 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 480	. . . . . . . . . 10 ⊢ (((𝐴 ∈ No ∧ 𝑀 ∈ ℕ0s) ∧ (𝐴↑s(𝑀 +s 𝑘)) = ((𝐴↑s𝑀) ·s (𝐴↑s𝑘))) → (𝐴↑s𝑀) ∈ No )
38	37	adantl 481	. . . . . . . . 9 ⊢ ((𝑘 ∈ ℕ0s ∧ ((𝐴 ∈ No ∧ 𝑀 ∈ ℕ0s) ∧ (𝐴↑s(𝑀 +s 𝑘)) = ((𝐴↑s𝑀) ·s (𝐴↑s𝑘)))) → (𝐴↑s𝑀) ∈ No )
39		simprll 779	. . . . . . . . . 10 ⊢ ((𝑘 ∈ ℕ0s ∧ ((𝐴 ∈ No ∧ 𝑀 ∈ ℕ0s) ∧ (𝐴↑s(𝑀 +s 𝑘)) = ((𝐴↑s𝑀) ·s (𝐴↑s𝑘)))) → 𝐴 ∈ No )
40		simpl 482	. . . . . . . . . 10 ⊢ ((𝑘 ∈ ℕ0s ∧ ((𝐴 ∈ No ∧ 𝑀 ∈ ℕ0s) ∧ (𝐴↑s(𝑀 +s 𝑘)) = ((𝐴↑s𝑀) ·s (𝐴↑s𝑘)))) → 𝑘 ∈ ℕ0s)
41		expscl 28423	. . . . . . . . . 10 ⊢ ((𝐴 ∈ No ∧ 𝑘 ∈ ℕ0s) → (𝐴↑s𝑘) ∈ No )
42	39, 40, 41	syl2anc 585	. . . . . . . . 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 28159	. . . . . . . 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 2772	. . . . . . 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 780	. . . . . . . . . . 11 ⊢ ((𝑘 ∈ ℕ0s ∧ ((𝐴 ∈ No ∧ 𝑀 ∈ ℕ0s) ∧ (𝐴↑s(𝑀 +s 𝑘)) = ((𝐴↑s𝑀) ·s (𝐴↑s𝑘)))) → 𝑀 ∈ ℕ0s)
46	45	n0nod 28317	. . . . . . . . . 10 ⊢ ((𝑘 ∈ ℕ0s ∧ ((𝐴 ∈ No ∧ 𝑀 ∈ ℕ0s) ∧ (𝐴↑s(𝑀 +s 𝑘)) = ((𝐴↑s𝑀) ·s (𝐴↑s𝑘)))) → 𝑀 ∈ No )
47	40	n0nod 28317	. . . . . . . . . 10 ⊢ ((𝑘 ∈ ℕ0s ∧ ((𝐴 ∈ No ∧ 𝑀 ∈ ℕ0s) ∧ (𝐴↑s(𝑀 +s 𝑘)) = ((𝐴↑s𝑀) ·s (𝐴↑s𝑘)))) → 𝑘 ∈ No )
48		1no 27802	. . . . . . . . . . 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 27998	. . . . . . . . 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 7383	. . . . . . . 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 28336	. . . . . . . . . 10 ⊢ ((𝑀 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s) → (𝑀 +s 𝑘) ∈ ℕ0s)
53	45, 40, 52	syl2anc 585	. . . . . . . . 9 ⊢ ((𝑘 ∈ ℕ0s ∧ ((𝐴 ∈ No ∧ 𝑀 ∈ ℕ0s) ∧ (𝐴↑s(𝑀 +s 𝑘)) = ((𝐴↑s𝑀) ·s (𝐴↑s𝑘)))) → (𝑀 +s 𝑘) ∈ ℕ0s)
54		expsp1 28421	. . . . . . . . 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 585	. . . . . . . 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 2774	. . . . . . 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 28421	. . . . . . . . 9 ⊢ ((𝐴 ∈ No ∧ 𝑘 ∈ ℕ0s) → (𝐴↑s(𝑘 +s 1s )) = ((𝐴↑s𝑘) ·s 𝐴))
58	39, 40, 57	syl2anc 585	. . . . . . . 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 7383	. . . . . . 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 2782	. . . . . 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 420	. . . . 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 29	. . . 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 28325	. . 3 ⊢ (𝑁 ∈ ℕ0s → ((𝐴 ∈ No ∧ 𝑀 ∈ ℕ0s) → (𝐴↑s(𝑀 +s 𝑁)) = ((𝐴↑s𝑀) ·s (𝐴↑s𝑁))))
64	63	com12 32	. 2 ⊢ ((𝐴 ∈ No ∧ 𝑀 ∈ ℕ0s) → (𝑁 ∈ ℕ0s → (𝐴↑s(𝑀 +s 𝑁)) = ((𝐴↑s𝑀) ·s (𝐴↑s𝑁))))
65	64	3impia 1118	1 ⊢ ((𝐴 ∈ No ∧ 𝑀 ∈ ℕ0s ∧ 𝑁 ∈ ℕ0s) → (𝐴↑s(𝑀 +s 𝑁)) = ((𝐴↑s𝑀) ·s (𝐴↑s𝑁)))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114  (class class class)co 7367   No csur 27603   0_s c0s 27797   1_s c1s 27798   +_s cadds 27951   ·_s cmuls 28098  ℕ_0scn0s 28304  ↑_scexps 28404

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5213  ax-sep 5232  ax-nul 5242  ax-pow 5308  ax-pr 5376  ax-un 7689

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rmo 3343  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-tp 4573  df-op 4575  df-ot 4577  df-uni 4852  df-int 4891  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-se 5585  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6266  df-ord 6327  df-on 6328  df-lim 6329  df-suc 6330  df-iota 6455  df-fun 6501  df-fn 6502  df-f 6503  df-f1 6504  df-fo 6505  df-f1o 6506  df-fv 6507  df-riota 7324  df-ov 7370  df-oprab 7371  df-mpo 7372  df-om 7818  df-1st 7942  df-2nd 7943  df-frecs 8231  df-wrecs 8262  df-recs 8311  df-rdg 8349  df-1o 8405  df-2o 8406  df-oadd 8409  df-nadd 8602  df-no 27606  df-lts 27607  df-bday 27608  df-les 27709  df-slts 27750  df-cuts 27752  df-0s 27799  df-1s 27800  df-made 27819  df-old 27820  df-left 27822  df-right 27823  df-norec 27930  df-norec2 27941  df-adds 27952  df-negs 28013  df-subs 28014  df-muls 28099  df-seqs 28276  df-n0s 28306  df-nns 28307  df-zs 28371  df-exps 28405

This theorem is referenced by:  pw2divscan4d  28436  bdayfinbndlem1  28459