Theorem absmuls 28182

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 28073	. . . . . . 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 28085	. . . . . 6 ⊢ ((((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 0s ≤s 𝐴) ∧ 0s ≤s 𝐵) → 0s ≤s (𝐴 ·s 𝐵))
8		abssid 28179	. . . . . 6 ⊢ (((𝐴 ·s 𝐵) ∈ No ∧ 0s ≤s (𝐴 ·s 𝐵)) → (abss‘(𝐴 ·s 𝐵)) = (𝐴 ·s 𝐵))
9	2, 7, 8	syl2an2r 685	. . . . 5 ⊢ ((((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 0s ≤s 𝐴) ∧ 0s ≤s 𝐵) → (abss‘(𝐴 ·s 𝐵)) = (𝐴 ·s 𝐵))
10		abssid 28179	. . . . . . 7 ⊢ ((𝐵 ∈ No ∧ 0s ≤s 𝐵) → (abss‘𝐵) = 𝐵)
11	10	ad4ant24 754	. . . . . 6 ⊢ ((((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 0s ≤s 𝐴) ∧ 0s ≤s 𝐵) → (abss‘𝐵) = 𝐵)
12	11	oveq2d 7362	. . . . 5 ⊢ ((((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 0s ≤s 𝐴) ∧ 0s ≤s 𝐵) → (𝐴 ·s (abss‘𝐵)) = (𝐴 ·s 𝐵))
13	9, 12	eqtr4d 2769	. . . 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 28100	. . . . 5 ⊢ ((((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 0s ≤s 𝐴) ∧ 𝐵 ≤s 0s ) → (𝐴 ·s ( -us ‘𝐵)) = ( -us ‘(𝐴 ·s 𝐵)))
17		abssnid 28181	. . . . . . 7 ⊢ ((𝐵 ∈ No ∧ 𝐵 ≤s 0s ) → (abss‘𝐵) = ( -us ‘𝐵))
18	17	ad4ant24 754	. . . . . 6 ⊢ ((((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 0s ≤s 𝐴) ∧ 𝐵 ≤s 0s ) → (abss‘𝐵) = ( -us ‘𝐵))
19	18	oveq2d 7362	. . . . 5 ⊢ ((((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 0s ≤s 𝐴) ∧ 𝐵 ≤s 0s ) → (𝐴 ·s (abss‘𝐵)) = (𝐴 ·s ( -us ‘𝐵)))
20		negs0s 27968	. . . . . . . 8 ⊢ ( -us ‘ 0s ) = 0s
21	15	negscld 27979	. . . . . . . . . 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 27770	. . . . . . . . . . . . . 14 ⊢ 0s ∈ No
25	24	a1i 11	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 0s ≤s 𝐴) ∧ 𝐵 ≤s 0s ) → 0s ∈ No )
26	15, 25	slenegd 27990	. . . . . . . . . . . 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 5125	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 0s ≤s 𝐴) ∧ 𝐵 ≤s 0s ) → 0s ≤s ( -us ‘𝐵))
29	14, 21, 22, 28	mulsge0d 28085	. . . . . . . . 9 ⊢ ((((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 0s ≤s 𝐴) ∧ 𝐵 ≤s 0s ) → 0s ≤s (𝐴 ·s ( -us ‘𝐵)))
30	29, 16	breqtrd 5115	. . . . . . . 8 ⊢ ((((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 0s ≤s 𝐴) ∧ 𝐵 ≤s 0s ) → 0s ≤s ( -us ‘(𝐴 ·s 𝐵)))
31	20, 30	eqbrtrid 5124	. . . . . . 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 27990	. . . . . . 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 28181	. . . . . 6 ⊢ (((𝐴 ·s 𝐵) ∈ No ∧ (𝐴 ·s 𝐵) ≤s 0s ) → (abss‘(𝐴 ·s 𝐵)) = ( -us ‘(𝐴 ·s 𝐵)))
36	2, 34, 35	syl2an2r 685	. . . . 5 ⊢ ((((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 0s ≤s 𝐴) ∧ 𝐵 ≤s 0s ) → (abss‘(𝐴 ·s 𝐵)) = ( -us ‘(𝐴 ·s 𝐵)))
37	16, 19, 36	3eqtr4rd 2777	. . . 4 ⊢ ((((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 0s ≤s 𝐴) ∧ 𝐵 ≤s 0s ) → (abss‘(𝐴 ·s 𝐵)) = (𝐴 ·s (abss‘𝐵)))
38		sletric 27703	. . . . . 6 ⊢ (( 0s ∈ No ∧ 𝐵 ∈ No ) → ( 0s ≤s 𝐵 ∨ 𝐵 ≤s 0s ))
39	24, 38	mpan 690	. . . . 5 ⊢ (𝐵 ∈ No → ( 0s ≤s 𝐵 ∨ 𝐵 ≤s 0s ))
40	39	ad2antlr 727	. . . 4 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 0s ≤s 𝐴) → ( 0s ≤s 𝐵 ∨ 𝐵 ≤s 0s ))
41	13, 37, 40	mpjaodan 960	. . 3 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 0s ≤s 𝐴) → (abss‘(𝐴 ·s 𝐵)) = (𝐴 ·s (abss‘𝐵)))
42		abssid 28179	. . . . 5 ⊢ ((𝐴 ∈ No ∧ 0s ≤s 𝐴) → (abss‘𝐴) = 𝐴)
43	42	adantlr 715	. . . 4 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 0s ≤s 𝐴) → (abss‘𝐴) = 𝐴)
44	43	oveq1d 7361	. . 3 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 0s ≤s 𝐴) → ((abss‘𝐴) ·s (abss‘𝐵)) = (𝐴 ·s (abss‘𝐵)))
45	41, 44	eqtr4d 2769	. 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 28099	. . . . 5 ⊢ ((((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 𝐴 ≤s 0s ) ∧ 0s ≤s 𝐵) → (( -us ‘𝐴) ·s 𝐵) = ( -us ‘(𝐴 ·s 𝐵)))
49	10	ad4ant24 754	. . . . . 6 ⊢ ((((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 𝐴 ≤s 0s ) ∧ 0s ≤s 𝐵) → (abss‘𝐵) = 𝐵)
50	49	oveq2d 7362	. . . . 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 27979	. . . . . . . . . 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 27990	. . . . . . . . . . . 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 5125	. . . . . . . . . 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 28085	. . . . . . . . 9 ⊢ ((((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 𝐴 ≤s 0s ) ∧ 0s ≤s 𝐵) → 0s ≤s (( -us ‘𝐴) ·s 𝐵))
60	59, 48	breqtrd 5115	. . . . . . . 8 ⊢ ((((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 𝐴 ≤s 0s ) ∧ 0s ≤s 𝐵) → 0s ≤s ( -us ‘(𝐴 ·s 𝐵)))
61	20, 60	eqbrtrid 5124	. . . . . . 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 27990	. . . . . . 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 685	. . . . 5 ⊢ ((((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 𝐴 ≤s 0s ) ∧ 0s ≤s 𝐵) → (abss‘(𝐴 ·s 𝐵)) = ( -us ‘(𝐴 ·s 𝐵)))
66	48, 50, 65	3eqtr4rd 2777	. . . 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 28101	. . . . 5 ⊢ ((((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 𝐴 ≤s 0s ) ∧ 𝐵 ≤s 0s ) → (( -us ‘𝐴) ·s ( -us ‘𝐵)) = (𝐴 ·s 𝐵))
70	17	ad4ant24 754	. . . . . 6 ⊢ ((((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 𝐴 ≤s 0s ) ∧ 𝐵 ≤s 0s ) → (abss‘𝐵) = ( -us ‘𝐵))
71	70	oveq2d 7362	. . . . 5 ⊢ ((((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 𝐴 ≤s 0s ) ∧ 𝐵 ≤s 0s ) → (( -us ‘𝐴) ·s (abss‘𝐵)) = (( -us ‘𝐴) ·s ( -us ‘𝐵)))
72	67	negscld 27979	. . . . . . . 8 ⊢ ((((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 𝐴 ≤s 0s ) ∧ 𝐵 ≤s 0s ) → ( -us ‘𝐴) ∈ No )
73	68	negscld 27979	. . . . . . . 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 27990	. . . . . . . . . 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 5125	. . . . . . . 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 27990	. . . . . . . . . 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 5125	. . . . . . . 8 ⊢ ((((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 𝐴 ≤s 0s ) ∧ 𝐵 ≤s 0s ) → 0s ≤s ( -us ‘𝐵))
83	72, 73, 78, 82	mulsge0d 28085	. . . . . . 7 ⊢ ((((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 𝐴 ≤s 0s ) ∧ 𝐵 ≤s 0s ) → 0s ≤s (( -us ‘𝐴) ·s ( -us ‘𝐵)))
84	83, 69	breqtrd 5115	. . . . . 6 ⊢ ((((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 𝐴 ≤s 0s ) ∧ 𝐵 ≤s 0s ) → 0s ≤s (𝐴 ·s 𝐵))
85	51, 84, 8	syl2an2r 685	. . . . 5 ⊢ ((((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 𝐴 ≤s 0s ) ∧ 𝐵 ≤s 0s ) → (abss‘(𝐴 ·s 𝐵)) = (𝐴 ·s 𝐵))
86	69, 71, 85	3eqtr4rd 2777	. . . 4 ⊢ ((((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 𝐴 ≤s 0s ) ∧ 𝐵 ≤s 0s ) → (abss‘(𝐴 ·s 𝐵)) = (( -us ‘𝐴) ·s (abss‘𝐵)))
87	39	ad2antlr 727	. . . 4 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 𝐴 ≤s 0s ) → ( 0s ≤s 𝐵 ∨ 𝐵 ≤s 0s ))
88	66, 86, 87	mpjaodan 960	. . 3 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 𝐴 ≤s 0s ) → (abss‘(𝐴 ·s 𝐵)) = (( -us ‘𝐴) ·s (abss‘𝐵)))
89		abssnid 28181	. . . . 5 ⊢ ((𝐴 ∈ No ∧ 𝐴 ≤s 0s ) → (abss‘𝐴) = ( -us ‘𝐴))
90	89	oveq1d 7361	. . . 4 ⊢ ((𝐴 ∈ No ∧ 𝐴 ≤s 0s ) → ((abss‘𝐴) ·s (abss‘𝐵)) = (( -us ‘𝐴) ·s (abss‘𝐵)))
91	90	adantlr 715	. . 3 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 𝐴 ≤s 0s ) → ((abss‘𝐴) ·s (abss‘𝐵)) = (( -us ‘𝐴) ·s (abss‘𝐵)))
92	88, 91	eqtr4d 2769	. 2 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 𝐴 ≤s 0s ) → (abss‘(𝐴 ·s 𝐵)) = ((abss‘𝐴) ·s (abss‘𝐵)))
93		sletric 27703	. . . 4 ⊢ (( 0s ∈ No ∧ 𝐴 ∈ No ) → ( 0s ≤s 𝐴 ∨ 𝐴 ≤s 0s ))
94	24, 93	mpan 690	. . 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 960	1 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (abss‘(𝐴 ·s 𝐵)) = ((abss‘𝐴) ·s (abss‘𝐵)))

Syntax hints:   → wi 4   ∧ wa 395   ∨ wo 847   = wceq 1541   ∈ wcel 2111   class class class wbr 5089  ‘cfv 6481  (class class class)co 7346   No csur 27578   ≤s csle 27683   0_s c0s 27766   -u_s cnegs 27961   ·_s cmuls 28045  abs_scabss 28175

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-rep 5215  ax-sep 5232  ax-nul 5242  ax-pow 5301  ax-pr 5368  ax-un 7668

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-rmo 3346  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3737  df-csb 3846  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-pss 3917  df-nul 4281  df-if 4473  df-pw 4549  df-sn 4574  df-pr 4576  df-tp 4578  df-op 4580  df-ot 4582  df-uni 4857  df-int 4896  df-iun 4941  df-br 5090  df-opab 5152  df-mpt 5171  df-tr 5197  df-id 5509  df-eprel 5514  df-po 5522  df-so 5523  df-fr 5567  df-se 5568  df-we 5569  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-res 5626  df-ima 5627  df-pred 6248  df-ord 6309  df-on 6310  df-suc 6312  df-iota 6437  df-fun 6483  df-fn 6484  df-f 6485  df-f1 6486  df-fo 6487  df-f1o 6488  df-fv 6489  df-riota 7303  df-ov 7349  df-oprab 7350  df-mpo 7351  df-1st 7921  df-2nd 7922  df-frecs 8211  df-wrecs 8242  df-recs 8291  df-1o 8385  df-2o 8386  df-nadd 8581  df-no 27581  df-slt 27582  df-bday 27583  df-sle 27684  df-sslt 27721  df-scut 27723  df-0s 27768  df-made 27788  df-old 27789  df-left 27791  df-right 27792  df-norec 27881  df-norec2 27892  df-adds 27903  df-negs 27963  df-subs 27964  df-muls 28046  df-abss 28176

This theorem is referenced by:  remulscllem2  28403