Theorem expadds 28372

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 7413	. . . . . . 7 ⊢ (𝑗 = 0s → (𝑀 +s 𝑗) = (𝑀 +s 0s ))
2	1	oveq2d 7421	. . . . . 6 ⊢ (𝑗 = 0s → (𝐴↑s(𝑀 +s 𝑗)) = (𝐴↑s(𝑀 +s 0s )))
3		oveq2 7413	. . . . . . 7 ⊢ (𝑗 = 0s → (𝐴↑s𝑗) = (𝐴↑s 0s ))
4	3	oveq2d 7421	. . . . . 6 ⊢ (𝑗 = 0s → ((𝐴↑s𝑀) ·s (𝐴↑s𝑗)) = ((𝐴↑s𝑀) ·s (𝐴↑s 0s )))
5	2, 4	eqeq12d 2751	. . . . 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 7413	. . . . . . 7 ⊢ (𝑗 = 𝑘 → (𝑀 +s 𝑗) = (𝑀 +s 𝑘))
8	7	oveq2d 7421	. . . . . 6 ⊢ (𝑗 = 𝑘 → (𝐴↑s(𝑀 +s 𝑗)) = (𝐴↑s(𝑀 +s 𝑘)))
9		oveq2 7413	. . . . . . 7 ⊢ (𝑗 = 𝑘 → (𝐴↑s𝑗) = (𝐴↑s𝑘))
10	9	oveq2d 7421	. . . . . 6 ⊢ (𝑗 = 𝑘 → ((𝐴↑s𝑀) ·s (𝐴↑s𝑗)) = ((𝐴↑s𝑀) ·s (𝐴↑s𝑘)))
11	8, 10	eqeq12d 2751	. . . . 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 7413	. . . . . . 7 ⊢ (𝑗 = (𝑘 +s 1s ) → (𝑀 +s 𝑗) = (𝑀 +s (𝑘 +s 1s )))
14	13	oveq2d 7421	. . . . . 6 ⊢ (𝑗 = (𝑘 +s 1s ) → (𝐴↑s(𝑀 +s 𝑗)) = (𝐴↑s(𝑀 +s (𝑘 +s 1s ))))
15		oveq2 7413	. . . . . . 7 ⊢ (𝑗 = (𝑘 +s 1s ) → (𝐴↑s𝑗) = (𝐴↑s(𝑘 +s 1s )))
16	15	oveq2d 7421	. . . . . 6 ⊢ (𝑗 = (𝑘 +s 1s ) → ((𝐴↑s𝑀) ·s (𝐴↑s𝑗)) = ((𝐴↑s𝑀) ·s (𝐴↑s(𝑘 +s 1s ))))
17	14, 16	eqeq12d 2751	. . . . 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 7413	. . . . . . 7 ⊢ (𝑗 = 𝑁 → (𝑀 +s 𝑗) = (𝑀 +s 𝑁))
20	19	oveq2d 7421	. . . . . 6 ⊢ (𝑗 = 𝑁 → (𝐴↑s(𝑀 +s 𝑗)) = (𝐴↑s(𝑀 +s 𝑁)))
21		oveq2 7413	. . . . . . 7 ⊢ (𝑗 = 𝑁 → (𝐴↑s𝑗) = (𝐴↑s𝑁))
22	21	oveq2d 7421	. . . . . 6 ⊢ (𝑗 = 𝑁 → ((𝐴↑s𝑀) ·s (𝐴↑s𝑗)) = ((𝐴↑s𝑀) ·s (𝐴↑s𝑁)))
23	20, 22	eqeq12d 2751	. . . . 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 28369	. . . . . 6 ⊢ ((𝐴 ∈ No ∧ 𝑀 ∈ ℕ0s) → (𝐴↑s𝑀) ∈ No )
26	25	mulsridd 28069	. . . . 5 ⊢ ((𝐴 ∈ No ∧ 𝑀 ∈ ℕ0s) → ((𝐴↑s𝑀) ·s 1s ) = (𝐴↑s𝑀))
27		exps0 28365	. . . . . . 7 ⊢ (𝐴 ∈ No → (𝐴↑s 0s ) = 1s )
28	27	oveq2d 7421	. . . . . 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		n0sno 28268	. . . . . . . 8 ⊢ (𝑀 ∈ ℕ0s → 𝑀 ∈ No )
31	30	adantl 481	. . . . . . 7 ⊢ ((𝐴 ∈ No ∧ 𝑀 ∈ ℕ0s) → 𝑀 ∈ No )
32	31	addsridd 27924	. . . . . 6 ⊢ ((𝐴 ∈ No ∧ 𝑀 ∈ ℕ0s) → (𝑀 +s 0s ) = 𝑀)
33	32	oveq2d 7421	. . . . 5 ⊢ ((𝐴 ∈ No ∧ 𝑀 ∈ ℕ0s) → (𝐴↑s(𝑀 +s 0s )) = (𝐴↑s𝑀))
34	26, 29, 33	3eqtr4rd 2781	. . . 4 ⊢ ((𝐴 ∈ No ∧ 𝑀 ∈ ℕ0s) → (𝐴↑s(𝑀 +s 0s )) = ((𝐴↑s𝑀) ·s (𝐴↑s 0s )))
35		simprr 772	. . . . . . . . 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 7420	. . . . . . . 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 778	. . . . . . . . . 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 28369	. . . . . . . . . 10 ⊢ ((𝐴 ∈ No ∧ 𝑘 ∈ ℕ0s) → (𝐴↑s𝑘) ∈ No )
42	39, 40, 41	syl2anc 584	. . . . . . . . 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 28122	. . . . . . . 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 2770	. . . . . . 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 779	. . . . . . . . . . 11 ⊢ ((𝑘 ∈ ℕ0s ∧ ((𝐴 ∈ No ∧ 𝑀 ∈ ℕ0s) ∧ (𝐴↑s(𝑀 +s 𝑘)) = ((𝐴↑s𝑀) ·s (𝐴↑s𝑘)))) → 𝑀 ∈ ℕ0s)
46	45	n0snod 28270	. . . . . . . . . 10 ⊢ ((𝑘 ∈ ℕ0s ∧ ((𝐴 ∈ No ∧ 𝑀 ∈ ℕ0s) ∧ (𝐴↑s(𝑀 +s 𝑘)) = ((𝐴↑s𝑀) ·s (𝐴↑s𝑘)))) → 𝑀 ∈ No )
47	40	n0snod 28270	. . . . . . . . . 10 ⊢ ((𝑘 ∈ ℕ0s ∧ ((𝐴 ∈ No ∧ 𝑀 ∈ ℕ0s) ∧ (𝐴↑s(𝑀 +s 𝑘)) = ((𝐴↑s𝑀) ·s (𝐴↑s𝑘)))) → 𝑘 ∈ No )
48		1sno 27791	. . . . . . . . . . 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 27965	. . . . . . . . 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 7421	. . . . . . . 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 28288	. . . . . . . . . 10 ⊢ ((𝑀 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s) → (𝑀 +s 𝑘) ∈ ℕ0s)
53	45, 40, 52	syl2anc 584	. . . . . . . . 9 ⊢ ((𝑘 ∈ ℕ0s ∧ ((𝐴 ∈ No ∧ 𝑀 ∈ ℕ0s) ∧ (𝐴↑s(𝑀 +s 𝑘)) = ((𝐴↑s𝑀) ·s (𝐴↑s𝑘)))) → (𝑀 +s 𝑘) ∈ ℕ0s)
54		expsp1 28367	. . . . . . . . 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 584	. . . . . . . 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 2772	. . . . . . 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 28367	. . . . . . . . 9 ⊢ ((𝐴 ∈ No ∧ 𝑘 ∈ ℕ0s) → (𝐴↑s(𝑘 +s 1s )) = ((𝐴↑s𝑘) ·s 𝐴))
58	39, 40, 57	syl2anc 584	. . . . . . . 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 7421	. . . . . . 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 2780	. . . . . 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 28277	. . 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 1117	1 ⊢ ((𝐴 ∈ No ∧ 𝑀 ∈ ℕ0s ∧ 𝑁 ∈ ℕ0s) → (𝐴↑s(𝑀 +s 𝑁)) = ((𝐴↑s𝑀) ·s (𝐴↑s𝑁)))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2108  (class class class)co 7405   No csur 27603   0_s c0s 27786   1_s c1s 27787   +_s cadds 27918   ·_s cmuls 28061  ℕ_0scnn0s 28258  ↑_scexps 28350

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707  ax-rep 5249  ax-sep 5266  ax-nul 5276  ax-pow 5335  ax-pr 5402  ax-un 7729

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-rmo 3359  df-reu 3360  df-rab 3416  df-v 3461  df-sbc 3766  df-csb 3875  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-pss 3946  df-nul 4309  df-if 4501  df-pw 4577  df-sn 4602  df-pr 4604  df-tp 4606  df-op 4608  df-ot 4610  df-uni 4884  df-int 4923  df-iun 4969  df-br 5120  df-opab 5182  df-mpt 5202  df-tr 5230  df-id 5548  df-eprel 5553  df-po 5561  df-so 5562  df-fr 5606  df-se 5607  df-we 5608  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667  df-pred 6290  df-ord 6355  df-on 6356  df-lim 6357  df-suc 6358  df-iota 6484  df-fun 6533  df-fn 6534  df-f 6535  df-f1 6536  df-fo 6537  df-f1o 6538  df-fv 6539  df-riota 7362  df-ov 7408  df-oprab 7409  df-mpo 7410  df-om 7862  df-1st 7988  df-2nd 7989  df-frecs 8280  df-wrecs 8311  df-recs 8385  df-rdg 8424  df-1o 8480  df-2o 8481  df-oadd 8484  df-nadd 8678  df-no 27606  df-slt 27607  df-bday 27608  df-sle 27709  df-sslt 27745  df-scut 27747  df-0s 27788  df-1s 27789  df-made 27807  df-old 27808  df-left 27810  df-right 27811  df-norec 27897  df-norec2 27908  df-adds 27919  df-negs 27979  df-subs 27980  df-muls 28062  df-seqs 28230  df-n0s 28260  df-nns 28261  df-zs 28319  df-exps 28351

This theorem is referenced by: (None)