Theorem expadds 28698

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 7424	. . . . . . 7 ⊢ (𝑗 = 0s → (𝑀 +s 𝑗) = (𝑀 +s 0s ))
2	1	oveq2d 7432	. . . . . 6 ⊢ (𝑗 = 0s → (𝐴↑s(𝑀 +s 𝑗)) = (𝐴↑s(𝑀 +s 0s )))
3		oveq2 7424	. . . . . . 7 ⊢ (𝑗 = 0s → (𝐴↑s𝑗) = (𝐴↑s 0s ))
4	3	oveq2d 7432	. . . . . 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 7424	. . . . . . 7 ⊢ (𝑗 = 𝑘 → (𝑀 +s 𝑗) = (𝑀 +s 𝑘))
8	7	oveq2d 7432	. . . . . 6 ⊢ (𝑗 = 𝑘 → (𝐴↑s(𝑀 +s 𝑗)) = (𝐴↑s(𝑀 +s 𝑘)))
9		oveq2 7424	. . . . . . 7 ⊢ (𝑗 = 𝑘 → (𝐴↑s𝑗) = (𝐴↑s𝑘))
10	9	oveq2d 7432	. . . . . 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 7424	. . . . . . 7 ⊢ (𝑗 = (𝑘 +s 1s ) → (𝑀 +s 𝑗) = (𝑀 +s (𝑘 +s 1s )))
14	13	oveq2d 7432	. . . . . 6 ⊢ (𝑗 = (𝑘 +s 1s ) → (𝐴↑s(𝑀 +s 𝑗)) = (𝐴↑s(𝑀 +s (𝑘 +s 1s ))))
15		oveq2 7424	. . . . . . 7 ⊢ (𝑗 = (𝑘 +s 1s ) → (𝐴↑s𝑗) = (𝐴↑s(𝑘 +s 1s )))
16	15	oveq2d 7432	. . . . . 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 7424	. . . . . . 7 ⊢ (𝑗 = 𝑁 → (𝑀 +s 𝑗) = (𝑀 +s 𝑁))
20	19	oveq2d 7432	. . . . . 6 ⊢ (𝑗 = 𝑁 → (𝐴↑s(𝑀 +s 𝑗)) = (𝐴↑s(𝑀 +s 𝑁)))
21		oveq2 7424	. . . . . . 7 ⊢ (𝑗 = 𝑁 → (𝐴↑s𝑗) = (𝐴↑s𝑁))
22	21	oveq2d 7432	. . . . . 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 28694	. . . . . 6 ⊢ ((𝐴 ∈ No ∧ 𝑀 ∈ ℕ0s) → (𝐴↑s𝑀) ∈ No )
26	25	mulsridd 28377	. . . . 5 ⊢ ((𝐴 ∈ No ∧ 𝑀 ∈ ℕ0s) → ((𝐴↑s𝑀) ·s 1s ) = (𝐴↑s𝑀))
27		exps0 28690	. . . . . . 7 ⊢ (𝐴 ∈ No → (𝐴↑s 0s ) = 1s )
28	27	oveq2d 7432	. . . . . 6 ⊢ (𝐴 ∈ No → ((𝐴↑s𝑀) ·s (𝐴↑s 0s )) = ((𝐴↑s𝑀) ·s 1s ))
29	28	adantr 486	. . . . 5 ⊢ ((𝐴 ∈ No ∧ 𝑀 ∈ ℕ0s) → ((𝐴↑s𝑀) ·s (𝐴↑s 0s )) = ((𝐴↑s𝑀) ·s 1s ))
30		n0no 28586	. . . . . . . 8 ⊢ (𝑀 ∈ ℕ0s → 𝑀 ∈ No )
31	30	adantl 487	. . . . . . 7 ⊢ ((𝐴 ∈ No ∧ 𝑀 ∈ ℕ0s) → 𝑀 ∈ No )
32	31	addsridd 28228	. . . . . 6 ⊢ ((𝐴 ∈ No ∧ 𝑀 ∈ ℕ0s) → (𝑀 +s 0s ) = 𝑀)
33	32	oveq2d 7432	. . . . 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 785	. . . . . . . . 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 7431	. . . . . . . 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 486	. . . . . . . . . 10 ⊢ (((𝐴 ∈ No ∧ 𝑀 ∈ ℕ0s) ∧ (𝐴↑s(𝑀 +s 𝑘)) = ((𝐴↑s𝑀) ·s (𝐴↑s𝑘))) → (𝐴↑s𝑀) ∈ No )
38	37	adantl 487	. . . . . . . . 9 ⊢ ((𝑘 ∈ ℕ0s ∧ ((𝐴 ∈ No ∧ 𝑀 ∈ ℕ0s) ∧ (𝐴↑s(𝑀 +s 𝑘)) = ((𝐴↑s𝑀) ·s (𝐴↑s𝑘)))) → (𝐴↑s𝑀) ∈ No )
39		simprll 791	. . . . . . . . . 10 ⊢ ((𝑘 ∈ ℕ0s ∧ ((𝐴 ∈ No ∧ 𝑀 ∈ ℕ0s) ∧ (𝐴↑s(𝑀 +s 𝑘)) = ((𝐴↑s𝑀) ·s (𝐴↑s𝑘)))) → 𝐴 ∈ No )
40		simpl 488	. . . . . . . . . 10 ⊢ ((𝑘 ∈ ℕ0s ∧ ((𝐴 ∈ No ∧ 𝑀 ∈ ℕ0s) ∧ (𝐴↑s(𝑀 +s 𝑘)) = ((𝐴↑s𝑀) ·s (𝐴↑s𝑘)))) → 𝑘 ∈ ℕ0s)
41		expscl 28694	. . . . . . . . . 10 ⊢ ((𝐴 ∈ No ∧ 𝑘 ∈ ℕ0s) → (𝐴↑s𝑘) ∈ No )
42	39, 40, 41	syl2anc 596	. . . . . . . . 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 28430	. . . . . . . 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 792	. . . . . . . . . . 11 ⊢ ((𝑘 ∈ ℕ0s ∧ ((𝐴 ∈ No ∧ 𝑀 ∈ ℕ0s) ∧ (𝐴↑s(𝑀 +s 𝑘)) = ((𝐴↑s𝑀) ·s (𝐴↑s𝑘)))) → 𝑀 ∈ ℕ0s)
46	45	n0nod 28588	. . . . . . . . . 10 ⊢ ((𝑘 ∈ ℕ0s ∧ ((𝐴 ∈ No ∧ 𝑀 ∈ ℕ0s) ∧ (𝐴↑s(𝑀 +s 𝑘)) = ((𝐴↑s𝑀) ·s (𝐴↑s𝑘)))) → 𝑀 ∈ No )
47	40	n0nod 28588	. . . . . . . . . 10 ⊢ ((𝑘 ∈ ℕ0s ∧ ((𝐴 ∈ No ∧ 𝑀 ∈ ℕ0s) ∧ (𝐴↑s(𝑀 +s 𝑘)) = ((𝐴↑s𝑀) ·s (𝐴↑s𝑘)))) → 𝑘 ∈ No )
48		1no 28073	. . . . . . . . . . 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 28269	. . . . . . . . 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 7432	. . . . . . . 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 28607	. . . . . . . . . 10 ⊢ ((𝑀 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s) → (𝑀 +s 𝑘) ∈ ℕ0s)
53	45, 40, 52	syl2anc 596	. . . . . . . . 9 ⊢ ((𝑘 ∈ ℕ0s ∧ ((𝐴 ∈ No ∧ 𝑀 ∈ ℕ0s) ∧ (𝐴↑s(𝑀 +s 𝑘)) = ((𝐴↑s𝑀) ·s (𝐴↑s𝑘)))) → (𝑀 +s 𝑘) ∈ ℕ0s)
54		expsp1 28692	. . . . . . . . 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 596	. . . . . . . 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 28692	. . . . . . . . 9 ⊢ ((𝐴 ∈ No ∧ 𝑘 ∈ ℕ0s) → (𝐴↑s(𝑘 +s 1s )) = ((𝐴↑s𝑘) ·s 𝐴))
58	39, 40, 57	syl2anc 596	. . . . . . . 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 7432	. . . . . . 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 426	. . . . 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 28596	. . 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 1135	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 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145  (class class class)co 7416   No csur 27874   0_s c0s 28068   1_s c1s 28069   +_s cadds 28222   ·_s cmuls 28369  ℕ_0scn0s 28575  ↑_scexps 28675

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2734  ax-rep 5236  ax-sep 5255  ax-nul 5267  ax-pow 5334  ax-pr 5402  ax-un 7739

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  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 3367  df-reu 3368  df-rab 3415  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4283  df-if 4486  df-pw 4562  df-sn 4588  df-pr 4590  df-tp 4592  df-op 4594  df-ot 4596  df-uni 4871  df-int 4911  df-iun 4956  df-br 5108  df-opab 5172  df-mpt 5191  df-tr 5217  df-id 5554  df-eprel 5559  df-po 5567  df-so 5568  df-fr 5612  df-se 5613  df-we 5614  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-pred 6303  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-riota 7373  df-ov 7419  df-oprab 7420  df-mpo 7421  df-om 7866  df-1st 7989  df-2nd 7990  df-frecs 8283  df-wrecs 8314  df-recs 8363  df-rdg 8402  df-1o 8458  df-2o 8459  df-oadd 8462  df-nadd 8657  df-no 27877  df-lts 27878  df-bday 27879  df-les 27979  df-slts 28021  df-cuts 28023  df-0s 28070  df-1s 28071  df-made 28090  df-old 28091  df-left 28093  df-right 28094  df-norec 28201  df-norec2 28212  df-adds 28223  df-negs 28284  df-subs 28285  df-muls 28370  df-seqs 28547  df-n0s 28577  df-nns 28578  df-zs 28642  df-exps 28676

This theorem is used by:  pw2divscan4d  28707  bdayfinbndlem1  28730