Theorem expadds 28851

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 7451	. . . . . . 7 ⊢ (𝑗 = 0s → (𝑀 +s 𝑗) = (𝑀 +s 0s ))
2	1	oveq2d 7459	. . . . . 6 ⊢ (𝑗 = 0s → (𝐴↑s(𝑀 +s 𝑗)) = (𝐴↑s(𝑀 +s 0s )))
3		oveq2 7451	. . . . . . 7 ⊢ (𝑗 = 0s → (𝐴↑s𝑗) = (𝐴↑s 0s ))
4	3	oveq2d 7459	. . . . . 6 ⊢ (𝑗 = 0s → ((𝐴↑s𝑀) ·s (𝐴↑s𝑗)) = ((𝐴↑s𝑀) ·s (𝐴↑s 0s )))
5	2, 4	eqeq12d 2779	. . . . 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 7451	. . . . . . 7 ⊢ (𝑗 = 𝑘 → (𝑀 +s 𝑗) = (𝑀 +s 𝑘))
8	7	oveq2d 7459	. . . . . 6 ⊢ (𝑗 = 𝑘 → (𝐴↑s(𝑀 +s 𝑗)) = (𝐴↑s(𝑀 +s 𝑘)))
9		oveq2 7451	. . . . . . 7 ⊢ (𝑗 = 𝑘 → (𝐴↑s𝑗) = (𝐴↑s𝑘))
10	9	oveq2d 7459	. . . . . 6 ⊢ (𝑗 = 𝑘 → ((𝐴↑s𝑀) ·s (𝐴↑s𝑗)) = ((𝐴↑s𝑀) ·s (𝐴↑s𝑘)))
11	8, 10	eqeq12d 2779	. . . . 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 7451	. . . . . . 7 ⊢ (𝑗 = (𝑘 +s 1s ) → (𝑀 +s 𝑗) = (𝑀 +s (𝑘 +s 1s )))
14	13	oveq2d 7459	. . . . . 6 ⊢ (𝑗 = (𝑘 +s 1s ) → (𝐴↑s(𝑀 +s 𝑗)) = (𝐴↑s(𝑀 +s (𝑘 +s 1s ))))
15		oveq2 7451	. . . . . . 7 ⊢ (𝑗 = (𝑘 +s 1s ) → (𝐴↑s𝑗) = (𝐴↑s(𝑘 +s 1s )))
16	15	oveq2d 7459	. . . . . 6 ⊢ (𝑗 = (𝑘 +s 1s ) → ((𝐴↑s𝑀) ·s (𝐴↑s𝑗)) = ((𝐴↑s𝑀) ·s (𝐴↑s(𝑘 +s 1s ))))
17	14, 16	eqeq12d 2779	. . . . 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 7451	. . . . . . 7 ⊢ (𝑗 = 𝑁 → (𝑀 +s 𝑗) = (𝑀 +s 𝑁))
20	19	oveq2d 7459	. . . . . 6 ⊢ (𝑗 = 𝑁 → (𝐴↑s(𝑀 +s 𝑗)) = (𝐴↑s(𝑀 +s 𝑁)))
21		oveq2 7451	. . . . . . 7 ⊢ (𝑗 = 𝑁 → (𝐴↑s𝑗) = (𝐴↑s𝑁))
22	21	oveq2d 7459	. . . . . 6 ⊢ (𝑗 = 𝑁 → ((𝐴↑s𝑀) ·s (𝐴↑s𝑗)) = ((𝐴↑s𝑀) ·s (𝐴↑s𝑁)))
23	20, 22	eqeq12d 2779	. . . . 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 28847	. . . . . 6 ⊢ ((𝐴 ∈ No ∧ 𝑀 ∈ ℕ0s) → (𝐴↑s𝑀) ∈ No)
26	25	mulsridd 28530	. . . . 5 ⊢ ((𝐴 ∈ No ∧ 𝑀 ∈ ℕ0s) → ((𝐴↑s𝑀) ·s 1s ) = (𝐴↑s𝑀))
27		exps0 28843	. . . . . . 7 ⊢ (𝐴 ∈ No → (𝐴↑s 0s ) = 1s )
28	27	oveq2d 7459	. . . . . 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 28739	. . . . . . . 8 ⊢ (𝑀 ∈ ℕ0s → 𝑀 ∈ No)
31	30	adantl 487	. . . . . . 7 ⊢ ((𝐴 ∈ No ∧ 𝑀 ∈ ℕ0s) → 𝑀 ∈ No)
32	31	addsridd 28381	. . . . . 6 ⊢ ((𝐴 ∈ No ∧ 𝑀 ∈ ℕ0s) → (𝑀 +s 0s ) = 𝑀)
33	32	oveq2d 7459	. . . . 5 ⊢ ((𝐴 ∈ No ∧ 𝑀 ∈ ℕ0s) → (𝐴↑s(𝑀 +s 0s )) = (𝐴↑s𝑀))
34	26, 29, 33	3eqtr4rd 2809	. . . 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 7458	. . . . . . . 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 28847	. . . . . . . . . 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 28583	. . . . . . . 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 2798	. . . . . . 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 28741	. . . . . . . . . 10 ⊢ ((𝑘 ∈ ℕ0s ∧ ((𝐴 ∈ No ∧ 𝑀 ∈ ℕ0s) ∧ (𝐴↑s(𝑀 +s 𝑘)) = ((𝐴↑s𝑀) ·s (𝐴↑s𝑘)))) → 𝑀 ∈ No)
47	40	n0nod 28741	. . . . . . . . . 10 ⊢ ((𝑘 ∈ ℕ0s ∧ ((𝐴 ∈ No ∧ 𝑀 ∈ ℕ0s) ∧ (𝐴↑s(𝑀 +s 𝑘)) = ((𝐴↑s𝑀) ·s (𝐴↑s𝑘)))) → 𝑘 ∈ No)
48		1no 28226	. . . . . . . . . . 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 28422	. . . . . . . . 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 7459	. . . . . . . 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 28760	. . . . . . . . . 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 28845	. . . . . . . . 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 2800	. . . . . . 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 28845	. . . . . . . . 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 7459	. . . . . . 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 2808	. . . . . 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 28749	. . 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 2147  (class class class)co 7443  Nocsur 28027   0_s c0s 28221   1_s c1s 28222   +_s cadds 28375   ·_s cmuls 28522  ℕ_0scn0s 28728  ↑_scexps 28828

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5235  ax-sep 5252  ax-nul 5263  ax-pow 5331  ax-pr 5395  ax-un 7775

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 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-tp 4593  df-op 4839  df-ot 4862  df-uni 4871  df-int 4911  df-iun 4956  df-br 5107  df-opab 5171  df-mpt 5190  df-tr 5216  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-se 5609  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-pred 6322  df-ord 6383  df-on 6384  df-lim 6385  df-suc 6386  df-iota 6512  df-fun 6560  df-fn 6561  df-f 6562  df-f1 6563  df-fo 6564  df-f1o 6565  df-fv 6566  df-riota 7400  df-ov 7446  df-oprab 7447  df-mpo 7448  df-om 7902  df-1st 8025  df-2nd 8026  df-frecs 8327  df-wrecs 8358  df-recs 8407  df-rdg 8446  df-1o 8504  df-2o 8505  df-oadd 8508  df-nadd 8703  df-no 28030  df-lts 28031  df-bday 28032  df-les 28132  df-slts 28174  df-cuts 28176  df-0s 28223  df-1s 28224  df-made 28243  df-old 28244  df-left 28246  df-right 28247  df-norec 28354  df-norec2 28365  df-adds 28376  df-negs 28437  df-subs 28438  df-muls 28523  df-seqs 28700  df-n0s 28730  df-nns 28731  df-zs 28795  df-exps 28829

This theorem is used by:  pw2divscan4d  28860  bdayfinbndlem1  28883