Theorem absmuls 28286

Description: Surreal absolute value distributes over multiplication. (Contributed by Scott Fenton, 16-Apr-2025.)

Assertion

Ref	Expression
absmuls	⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (abss‘(𝐴 ·s 𝐵)) = ((abss‘𝐴) ·s (abss‘𝐵)))

Proof of Theorem absmuls

Step	Hyp	Ref	Expression
1		mulscl 28178	. . . . . . 7 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (𝐴 ·s 𝐵) ∈ No )
2	1	adantr 480	. . . . . 6 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 0s ≤s 𝐴) → (𝐴 ·s 𝐵) ∈ No )
3		simplll 774	. . . . . . 7 ⊢ ((((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 0s ≤s 𝐴) ∧ 0s ≤s 𝐵) → 𝐴 ∈ No )
4		simpllr 775	. . . . . . 7 ⊢ ((((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 0s ≤s 𝐴) ∧ 0s ≤s 𝐵) → 𝐵 ∈ No )
5		simplr 768	. . . . . . 7 ⊢ ((((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 0s ≤s 𝐴) ∧ 0s ≤s 𝐵) → 0s ≤s 𝐴)
6		simpr 484	. . . . . . 7 ⊢ ((((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 0s ≤s 𝐴) ∧ 0s ≤s 𝐵) → 0s ≤s 𝐵)
7	3, 4, 5, 6	mulsge0d 28190	. . . . . 6 ⊢ ((((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 0s ≤s 𝐴) ∧ 0s ≤s 𝐵) → 0s ≤s (𝐴 ·s 𝐵))
8		abssid 28283	. . . . . 6 ⊢ (((𝐴 ·s 𝐵) ∈ No ∧ 0s ≤s (𝐴 ·s 𝐵)) → (abss‘(𝐴 ·s 𝐵)) = (𝐴 ·s 𝐵))
9	2, 7, 8	syl2an2r 684	. . . . 5 ⊢ ((((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 0s ≤s 𝐴) ∧ 0s ≤s 𝐵) → (abss‘(𝐴 ·s 𝐵)) = (𝐴 ·s 𝐵))
10		abssid 28283	. . . . . . 7 ⊢ ((𝐵 ∈ No ∧ 0s ≤s 𝐵) → (abss‘𝐵) = 𝐵)
11	10	ad4ant24 753	. . . . . 6 ⊢ ((((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 0s ≤s 𝐴) ∧ 0s ≤s 𝐵) → (abss‘𝐵) = 𝐵)
12	11	oveq2d 7464	. . . . 5 ⊢ ((((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 0s ≤s 𝐴) ∧ 0s ≤s 𝐵) → (𝐴 ·s (abss‘𝐵)) = (𝐴 ·s 𝐵))
13	9, 12	eqtr4d 2783	. . . 4 ⊢ ((((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 0s ≤s 𝐴) ∧ 0s ≤s 𝐵) → (abss‘(𝐴 ·s 𝐵)) = (𝐴 ·s (abss‘𝐵)))
14		simplll 774	. . . . . 6 ⊢ ((((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 0s ≤s 𝐴) ∧ 𝐵 ≤s 0s ) → 𝐴 ∈ No )
15		simpllr 775	. . . . . 6 ⊢ ((((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 0s ≤s 𝐴) ∧ 𝐵 ≤s 0s ) → 𝐵 ∈ No )
16	14, 15	mulnegs2d 28205	. . . . 5 ⊢ ((((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 0s ≤s 𝐴) ∧ 𝐵 ≤s 0s ) → (𝐴 ·s ( -us ‘𝐵)) = ( -us ‘(𝐴 ·s 𝐵)))
17		abssnid 28285	. . . . . . 7 ⊢ ((𝐵 ∈ No ∧ 𝐵 ≤s 0s ) → (abss‘𝐵) = ( -us ‘𝐵))
18	17	ad4ant24 753	. . . . . 6 ⊢ ((((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 0s ≤s 𝐴) ∧ 𝐵 ≤s 0s ) → (abss‘𝐵) = ( -us ‘𝐵))
19	18	oveq2d 7464	. . . . 5 ⊢ ((((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 0s ≤s 𝐴) ∧ 𝐵 ≤s 0s ) → (𝐴 ·s (abss‘𝐵)) = (𝐴 ·s ( -us ‘𝐵)))
20		negs0s 28076	. . . . . . . 8 ⊢ ( -us ‘ 0s ) = 0s
21	15	negscld 28087	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 0s ≤s 𝐴) ∧ 𝐵 ≤s 0s ) → ( -us ‘𝐵) ∈ No )
22		simplr 768	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 0s ≤s 𝐴) ∧ 𝐵 ≤s 0s ) → 0s ≤s 𝐴)
23		simpr 484	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 0s ≤s 𝐴) ∧ 𝐵 ≤s 0s ) → 𝐵 ≤s 0s )
24		0sno 27889	. . . . . . . . . . . . . 14 ⊢ 0s ∈ No
25	24	a1i 11	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 0s ≤s 𝐴) ∧ 𝐵 ≤s 0s ) → 0s ∈ No )
26	15, 25	slenegd 28098	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 0s ≤s 𝐴) ∧ 𝐵 ≤s 0s ) → (𝐵 ≤s 0s ↔ ( -us ‘ 0s ) ≤s ( -us ‘𝐵)))
27	23, 26	mpbid 232	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 0s ≤s 𝐴) ∧ 𝐵 ≤s 0s ) → ( -us ‘ 0s ) ≤s ( -us ‘𝐵))
28	20, 27	eqbrtrrid 5202	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 0s ≤s 𝐴) ∧ 𝐵 ≤s 0s ) → 0s ≤s ( -us ‘𝐵))
29	14, 21, 22, 28	mulsge0d 28190	. . . . . . . . 9 ⊢ ((((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 0s ≤s 𝐴) ∧ 𝐵 ≤s 0s ) → 0s ≤s (𝐴 ·s ( -us ‘𝐵)))
30	29, 16	breqtrd 5192	. . . . . . . 8 ⊢ ((((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 0s ≤s 𝐴) ∧ 𝐵 ≤s 0s ) → 0s ≤s ( -us ‘(𝐴 ·s 𝐵)))
31	20, 30	eqbrtrid 5201	. . . . . . 7 ⊢ ((((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 0s ≤s 𝐴) ∧ 𝐵 ≤s 0s ) → ( -us ‘ 0s ) ≤s ( -us ‘(𝐴 ·s 𝐵)))
32	2	adantr 480	. . . . . . . 8 ⊢ ((((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 0s ≤s 𝐴) ∧ 𝐵 ≤s 0s ) → (𝐴 ·s 𝐵) ∈ No )
33	32, 25	slenegd 28098	. . . . . . 7 ⊢ ((((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 0s ≤s 𝐴) ∧ 𝐵 ≤s 0s ) → ((𝐴 ·s 𝐵) ≤s 0s ↔ ( -us ‘ 0s ) ≤s ( -us ‘(𝐴 ·s 𝐵))))
34	31, 33	mpbird 257	. . . . . 6 ⊢ ((((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 0s ≤s 𝐴) ∧ 𝐵 ≤s 0s ) → (𝐴 ·s 𝐵) ≤s 0s )
35		abssnid 28285	. . . . . 6 ⊢ (((𝐴 ·s 𝐵) ∈ No ∧ (𝐴 ·s 𝐵) ≤s 0s ) → (abss‘(𝐴 ·s 𝐵)) = ( -us ‘(𝐴 ·s 𝐵)))
36	2, 34, 35	syl2an2r 684	. . . . 5 ⊢ ((((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 0s ≤s 𝐴) ∧ 𝐵 ≤s 0s ) → (abss‘(𝐴 ·s 𝐵)) = ( -us ‘(𝐴 ·s 𝐵)))
37	16, 19, 36	3eqtr4rd 2791	. . . 4 ⊢ ((((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 0s ≤s 𝐴) ∧ 𝐵 ≤s 0s ) → (abss‘(𝐴 ·s 𝐵)) = (𝐴 ·s (abss‘𝐵)))
38		sletric 27827	. . . . . 6 ⊢ (( 0s ∈ No ∧ 𝐵 ∈ No ) → ( 0s ≤s 𝐵 ∨ 𝐵 ≤s 0s ))
39	24, 38	mpan 689	. . . . 5 ⊢ (𝐵 ∈ No → ( 0s ≤s 𝐵 ∨ 𝐵 ≤s 0s ))
40	39	ad2antlr 726	. . . 4 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 0s ≤s 𝐴) → ( 0s ≤s 𝐵 ∨ 𝐵 ≤s 0s ))
41	13, 37, 40	mpjaodan 959	. . 3 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 0s ≤s 𝐴) → (abss‘(𝐴 ·s 𝐵)) = (𝐴 ·s (abss‘𝐵)))
42		abssid 28283	. . . . 5 ⊢ ((𝐴 ∈ No ∧ 0s ≤s 𝐴) → (abss‘𝐴) = 𝐴)
43	42	adantlr 714	. . . 4 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 0s ≤s 𝐴) → (abss‘𝐴) = 𝐴)
44	43	oveq1d 7463	. . 3 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 0s ≤s 𝐴) → ((abss‘𝐴) ·s (abss‘𝐵)) = (𝐴 ·s (abss‘𝐵)))
45	41, 44	eqtr4d 2783	. 2 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 0s ≤s 𝐴) → (abss‘(𝐴 ·s 𝐵)) = ((abss‘𝐴) ·s (abss‘𝐵)))
46		simplll 774	. . . . . 6 ⊢ ((((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 𝐴 ≤s 0s ) ∧ 0s ≤s 𝐵) → 𝐴 ∈ No )
47		simpllr 775	. . . . . 6 ⊢ ((((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 𝐴 ≤s 0s ) ∧ 0s ≤s 𝐵) → 𝐵 ∈ No )
48	46, 47	mulnegs1d 28204	. . . . 5 ⊢ ((((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 𝐴 ≤s 0s ) ∧ 0s ≤s 𝐵) → (( -us ‘𝐴) ·s 𝐵) = ( -us ‘(𝐴 ·s 𝐵)))
49	10	ad4ant24 753	. . . . . 6 ⊢ ((((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 𝐴 ≤s 0s ) ∧ 0s ≤s 𝐵) → (abss‘𝐵) = 𝐵)
50	49	oveq2d 7464	. . . . 5 ⊢ ((((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 𝐴 ≤s 0s ) ∧ 0s ≤s 𝐵) → (( -us ‘𝐴) ·s (abss‘𝐵)) = (( -us ‘𝐴) ·s 𝐵))
51	1	adantr 480	. . . . . 6 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 𝐴 ≤s 0s ) → (𝐴 ·s 𝐵) ∈ No )
52	46	negscld 28087	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 𝐴 ≤s 0s ) ∧ 0s ≤s 𝐵) → ( -us ‘𝐴) ∈ No )
53		simplr 768	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 𝐴 ≤s 0s ) ∧ 0s ≤s 𝐵) → 𝐴 ≤s 0s )
54	24	a1i 11	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 𝐴 ≤s 0s ) ∧ 0s ≤s 𝐵) → 0s ∈ No )
55	46, 54	slenegd 28098	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 𝐴 ≤s 0s ) ∧ 0s ≤s 𝐵) → (𝐴 ≤s 0s ↔ ( -us ‘ 0s ) ≤s ( -us ‘𝐴)))
56	53, 55	mpbid 232	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 𝐴 ≤s 0s ) ∧ 0s ≤s 𝐵) → ( -us ‘ 0s ) ≤s ( -us ‘𝐴))
57	20, 56	eqbrtrrid 5202	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 𝐴 ≤s 0s ) ∧ 0s ≤s 𝐵) → 0s ≤s ( -us ‘𝐴))
58		simpr 484	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 𝐴 ≤s 0s ) ∧ 0s ≤s 𝐵) → 0s ≤s 𝐵)
59	52, 47, 57, 58	mulsge0d 28190	. . . . . . . . 9 ⊢ ((((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 𝐴 ≤s 0s ) ∧ 0s ≤s 𝐵) → 0s ≤s (( -us ‘𝐴) ·s 𝐵))
60	59, 48	breqtrd 5192	. . . . . . . 8 ⊢ ((((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 𝐴 ≤s 0s ) ∧ 0s ≤s 𝐵) → 0s ≤s ( -us ‘(𝐴 ·s 𝐵)))
61	20, 60	eqbrtrid 5201	. . . . . . 7 ⊢ ((((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 𝐴 ≤s 0s ) ∧ 0s ≤s 𝐵) → ( -us ‘ 0s ) ≤s ( -us ‘(𝐴 ·s 𝐵)))
62	51	adantr 480	. . . . . . . 8 ⊢ ((((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 𝐴 ≤s 0s ) ∧ 0s ≤s 𝐵) → (𝐴 ·s 𝐵) ∈ No )
63	62, 54	slenegd 28098	. . . . . . 7 ⊢ ((((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 𝐴 ≤s 0s ) ∧ 0s ≤s 𝐵) → ((𝐴 ·s 𝐵) ≤s 0s ↔ ( -us ‘ 0s ) ≤s ( -us ‘(𝐴 ·s 𝐵))))
64	61, 63	mpbird 257	. . . . . 6 ⊢ ((((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 𝐴 ≤s 0s ) ∧ 0s ≤s 𝐵) → (𝐴 ·s 𝐵) ≤s 0s )
65	51, 64, 35	syl2an2r 684	. . . . 5 ⊢ ((((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 𝐴 ≤s 0s ) ∧ 0s ≤s 𝐵) → (abss‘(𝐴 ·s 𝐵)) = ( -us ‘(𝐴 ·s 𝐵)))
66	48, 50, 65	3eqtr4rd 2791	. . . 4 ⊢ ((((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 𝐴 ≤s 0s ) ∧ 0s ≤s 𝐵) → (abss‘(𝐴 ·s 𝐵)) = (( -us ‘𝐴) ·s (abss‘𝐵)))
67		simplll 774	. . . . . 6 ⊢ ((((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 𝐴 ≤s 0s ) ∧ 𝐵 ≤s 0s ) → 𝐴 ∈ No )
68		simpllr 775	. . . . . 6 ⊢ ((((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 𝐴 ≤s 0s ) ∧ 𝐵 ≤s 0s ) → 𝐵 ∈ No )
69	67, 68	mul2negsd 28206	. . . . 5 ⊢ ((((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 𝐴 ≤s 0s ) ∧ 𝐵 ≤s 0s ) → (( -us ‘𝐴) ·s ( -us ‘𝐵)) = (𝐴 ·s 𝐵))
70	17	ad4ant24 753	. . . . . 6 ⊢ ((((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 𝐴 ≤s 0s ) ∧ 𝐵 ≤s 0s ) → (abss‘𝐵) = ( -us ‘𝐵))
71	70	oveq2d 7464	. . . . 5 ⊢ ((((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 𝐴 ≤s 0s ) ∧ 𝐵 ≤s 0s ) → (( -us ‘𝐴) ·s (abss‘𝐵)) = (( -us ‘𝐴) ·s ( -us ‘𝐵)))
72	67	negscld 28087	. . . . . . . 8 ⊢ ((((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 𝐴 ≤s 0s ) ∧ 𝐵 ≤s 0s ) → ( -us ‘𝐴) ∈ No )
73	68	negscld 28087	. . . . . . . 8 ⊢ ((((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 𝐴 ≤s 0s ) ∧ 𝐵 ≤s 0s ) → ( -us ‘𝐵) ∈ No )
74		simplr 768	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 𝐴 ≤s 0s ) ∧ 𝐵 ≤s 0s ) → 𝐴 ≤s 0s )
75	24	a1i 11	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 𝐴 ≤s 0s ) ∧ 𝐵 ≤s 0s ) → 0s ∈ No )
76	67, 75	slenegd 28098	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 𝐴 ≤s 0s ) ∧ 𝐵 ≤s 0s ) → (𝐴 ≤s 0s ↔ ( -us ‘ 0s ) ≤s ( -us ‘𝐴)))
77	74, 76	mpbid 232	. . . . . . . . 9 ⊢ ((((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 𝐴 ≤s 0s ) ∧ 𝐵 ≤s 0s ) → ( -us ‘ 0s ) ≤s ( -us ‘𝐴))
78	20, 77	eqbrtrrid 5202	. . . . . . . 8 ⊢ ((((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 𝐴 ≤s 0s ) ∧ 𝐵 ≤s 0s ) → 0s ≤s ( -us ‘𝐴))
79		simpr 484	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 𝐴 ≤s 0s ) ∧ 𝐵 ≤s 0s ) → 𝐵 ≤s 0s )
80	68, 75	slenegd 28098	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 𝐴 ≤s 0s ) ∧ 𝐵 ≤s 0s ) → (𝐵 ≤s 0s ↔ ( -us ‘ 0s ) ≤s ( -us ‘𝐵)))
81	79, 80	mpbid 232	. . . . . . . . 9 ⊢ ((((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 𝐴 ≤s 0s ) ∧ 𝐵 ≤s 0s ) → ( -us ‘ 0s ) ≤s ( -us ‘𝐵))
82	20, 81	eqbrtrrid 5202	. . . . . . . 8 ⊢ ((((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 𝐴 ≤s 0s ) ∧ 𝐵 ≤s 0s ) → 0s ≤s ( -us ‘𝐵))
83	72, 73, 78, 82	mulsge0d 28190	. . . . . . 7 ⊢ ((((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 𝐴 ≤s 0s ) ∧ 𝐵 ≤s 0s ) → 0s ≤s (( -us ‘𝐴) ·s ( -us ‘𝐵)))
84	83, 69	breqtrd 5192	. . . . . 6 ⊢ ((((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 𝐴 ≤s 0s ) ∧ 𝐵 ≤s 0s ) → 0s ≤s (𝐴 ·s 𝐵))
85	51, 84, 8	syl2an2r 684	. . . . 5 ⊢ ((((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 𝐴 ≤s 0s ) ∧ 𝐵 ≤s 0s ) → (abss‘(𝐴 ·s 𝐵)) = (𝐴 ·s 𝐵))
86	69, 71, 85	3eqtr4rd 2791	. . . 4 ⊢ ((((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 𝐴 ≤s 0s ) ∧ 𝐵 ≤s 0s ) → (abss‘(𝐴 ·s 𝐵)) = (( -us ‘𝐴) ·s (abss‘𝐵)))
87	39	ad2antlr 726	. . . 4 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 𝐴 ≤s 0s ) → ( 0s ≤s 𝐵 ∨ 𝐵 ≤s 0s ))
88	66, 86, 87	mpjaodan 959	. . 3 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 𝐴 ≤s 0s ) → (abss‘(𝐴 ·s 𝐵)) = (( -us ‘𝐴) ·s (abss‘𝐵)))
89		abssnid 28285	. . . . 5 ⊢ ((𝐴 ∈ No ∧ 𝐴 ≤s 0s ) → (abss‘𝐴) = ( -us ‘𝐴))
90	89	oveq1d 7463	. . . 4 ⊢ ((𝐴 ∈ No ∧ 𝐴 ≤s 0s ) → ((abss‘𝐴) ·s (abss‘𝐵)) = (( -us ‘𝐴) ·s (abss‘𝐵)))
91	90	adantlr 714	. . 3 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 𝐴 ≤s 0s ) → ((abss‘𝐴) ·s (abss‘𝐵)) = (( -us ‘𝐴) ·s (abss‘𝐵)))
92	88, 91	eqtr4d 2783	. 2 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 𝐴 ≤s 0s ) → (abss‘(𝐴 ·s 𝐵)) = ((abss‘𝐴) ·s (abss‘𝐵)))
93		sletric 27827	. . . 4 ⊢ (( 0s ∈ No ∧ 𝐴 ∈ No ) → ( 0s ≤s 𝐴 ∨ 𝐴 ≤s 0s ))
94	24, 93	mpan 689	. . 3 ⊢ (𝐴 ∈ No → ( 0s ≤s 𝐴 ∨ 𝐴 ≤s 0s ))
95	94	adantr 480	. 2 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → ( 0s ≤s 𝐴 ∨ 𝐴 ≤s 0s ))
96	45, 92, 95	mpjaodan 959	1 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (abss‘(𝐴 ·s 𝐵)) = ((abss‘𝐴) ·s (abss‘𝐵)))

Syntax hints:   → wi 4   ∧ wa 395   ∨ wo 846   = wceq 1537   ∈ wcel 2108   class class class wbr 5166  ‘cfv 6573  (class class class)co 7448   No csur 27702   ≤s csle 27807   0_s c0s 27885   -u_s cnegs 28069   ·_s cmuls 28150  abs_scabss 28279

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rmo 3388  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-tp 4653  df-op 4655  df-ot 4657  df-uni 4932  df-int 4971  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-tr 5284  df-id 5593  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-se 5653  df-we 5654  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-pred 6332  df-ord 6398  df-on 6399  df-suc 6401  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-riota 7404  df-ov 7451  df-oprab 7452  df-mpo 7453  df-1st 8030  df-2nd 8031  df-frecs 8322  df-wrecs 8353  df-recs 8427  df-1o 8522  df-2o 8523  df-nadd 8722  df-no 27705  df-slt 27706  df-bday 27707  df-sle 27808  df-sslt 27844  df-scut 27846  df-0s 27887  df-made 27904  df-old 27905  df-left 27907  df-right 27908  df-norec 27989  df-norec2 28000  df-adds 28011  df-negs 28071  df-subs 28072  df-muls 28151  df-abss 28280

This theorem is referenced by:  remulscllem2  28451