Theorem absmuls 28334

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 28224	. . . . . . 7 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (𝐴 ·s 𝐵) ∈ No )
2	1	adantr 484	. . . . . 6 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 0s ≤s 𝐴) → (𝐴 ·s 𝐵) ∈ No )
3		simplll 784	. . . . . . 7 ⊢ ((((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 0s ≤s 𝐴) ∧ 0s ≤s 𝐵) → 𝐴 ∈ No )
4		simpllr 785	. . . . . . 7 ⊢ ((((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 0s ≤s 𝐴) ∧ 0s ≤s 𝐵) → 𝐵 ∈ No )
5		simplr 778	. . . . . . 7 ⊢ ((((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 0s ≤s 𝐴) ∧ 0s ≤s 𝐵) → 0s ≤s 𝐴)
6		simpr 488	. . . . . . 7 ⊢ ((((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 0s ≤s 𝐴) ∧ 0s ≤s 𝐵) → 0s ≤s 𝐵)
7	3, 4, 5, 6	mulsge0d 28236	. . . . . 6 ⊢ ((((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 0s ≤s 𝐴) ∧ 0s ≤s 𝐵) → 0s ≤s (𝐴 ·s 𝐵))
8		abssid 28331	. . . . . 6 ⊢ (((𝐴 ·s 𝐵) ∈ No ∧ 0s ≤s (𝐴 ·s 𝐵)) → (abss‘(𝐴 ·s 𝐵)) = (𝐴 ·s 𝐵))
9	2, 7, 8	syl2an2r 695	. . . . 5 ⊢ ((((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 0s ≤s 𝐴) ∧ 0s ≤s 𝐵) → (abss‘(𝐴 ·s 𝐵)) = (𝐴 ·s 𝐵))
10		abssid 28331	. . . . . . 7 ⊢ ((𝐵 ∈ No ∧ 0s ≤s 𝐵) → (abss‘𝐵) = 𝐵)
11	10	ad4ant24 764	. . . . . 6 ⊢ ((((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 0s ≤s 𝐴) ∧ 0s ≤s 𝐵) → (abss‘𝐵) = 𝐵)
12	11	oveq2d 7412	. . . . 5 ⊢ ((((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 0s ≤s 𝐴) ∧ 0s ≤s 𝐵) → (𝐴 ·s (abss‘𝐵)) = (𝐴 ·s 𝐵))
13	9, 12	eqtr4d 2800	. . . 4 ⊢ ((((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 0s ≤s 𝐴) ∧ 0s ≤s 𝐵) → (abss‘(𝐴 ·s 𝐵)) = (𝐴 ·s (abss‘𝐵)))
14		simplll 784	. . . . . 6 ⊢ ((((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 0s ≤s 𝐴) ∧ 𝐵 ≤s 0s ) → 𝐴 ∈ No )
15		simpllr 785	. . . . . 6 ⊢ ((((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 0s ≤s 𝐴) ∧ 𝐵 ≤s 0s ) → 𝐵 ∈ No )
16	14, 15	mulnegs2d 28251	. . . . 5 ⊢ ((((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 0s ≤s 𝐴) ∧ 𝐵 ≤s 0s ) → (𝐴 ·s ( -us ‘𝐵)) = ( -us ‘(𝐴 ·s 𝐵)))
17		abssnid 28333	. . . . . . 7 ⊢ ((𝐵 ∈ No ∧ 𝐵 ≤s 0s ) → (abss‘𝐵) = ( -us ‘𝐵))
18	17	ad4ant24 764	. . . . . 6 ⊢ ((((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 0s ≤s 𝐴) ∧ 𝐵 ≤s 0s ) → (abss‘𝐵) = ( -us ‘𝐵))
19	18	oveq2d 7412	. . . . 5 ⊢ ((((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 0s ≤s 𝐴) ∧ 𝐵 ≤s 0s ) → (𝐴 ·s (abss‘𝐵)) = (𝐴 ·s ( -us ‘𝐵)))
20		neg0s 28116	. . . . . . . 8 ⊢ ( -us ‘ 0s ) = 0s
21	15	negscld 28127	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 0s ≤s 𝐴) ∧ 𝐵 ≤s 0s ) → ( -us ‘𝐵) ∈ No )
22		simplr 778	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 0s ≤s 𝐴) ∧ 𝐵 ≤s 0s ) → 0s ≤s 𝐴)
23		simpr 488	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 0s ≤s 𝐴) ∧ 𝐵 ≤s 0s ) → 𝐵 ≤s 0s )
24		0no 27899	. . . . . . . . . . . . . 14 ⊢ 0s ∈ No
25	24	a1i 11	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 0s ≤s 𝐴) ∧ 𝐵 ≤s 0s ) → 0s ∈ No )
26	15, 25	lenegsd 28138	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 0s ≤s 𝐴) ∧ 𝐵 ≤s 0s ) → (𝐵 ≤s 0s ↔ ( -us ‘ 0s ) ≤s ( -us ‘𝐵)))
27	23, 26	mpbid 234	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 0s ≤s 𝐴) ∧ 𝐵 ≤s 0s ) → ( -us ‘ 0s ) ≤s ( -us ‘𝐵))
28	20, 27	eqbrtrrid 5136	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 0s ≤s 𝐴) ∧ 𝐵 ≤s 0s ) → 0s ≤s ( -us ‘𝐵))
29	14, 21, 22, 28	mulsge0d 28236	. . . . . . . . 9 ⊢ ((((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 0s ≤s 𝐴) ∧ 𝐵 ≤s 0s ) → 0s ≤s (𝐴 ·s ( -us ‘𝐵)))
30	29, 16	breqtrd 5126	. . . . . . . 8 ⊢ ((((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 0s ≤s 𝐴) ∧ 𝐵 ≤s 0s ) → 0s ≤s ( -us ‘(𝐴 ·s 𝐵)))
31	20, 30	eqbrtrid 5135	. . . . . . 7 ⊢ ((((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 0s ≤s 𝐴) ∧ 𝐵 ≤s 0s ) → ( -us ‘ 0s ) ≤s ( -us ‘(𝐴 ·s 𝐵)))
32	2	adantr 484	. . . . . . . 8 ⊢ ((((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 0s ≤s 𝐴) ∧ 𝐵 ≤s 0s ) → (𝐴 ·s 𝐵) ∈ No )
33	32, 25	lenegsd 28138	. . . . . . 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 259	. . . . . 6 ⊢ ((((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 0s ≤s 𝐴) ∧ 𝐵 ≤s 0s ) → (𝐴 ·s 𝐵) ≤s 0s )
35		abssnid 28333	. . . . . 6 ⊢ (((𝐴 ·s 𝐵) ∈ No ∧ (𝐴 ·s 𝐵) ≤s 0s ) → (abss‘(𝐴 ·s 𝐵)) = ( -us ‘(𝐴 ·s 𝐵)))
36	2, 34, 35	syl2an2r 695	. . . . 5 ⊢ ((((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 0s ≤s 𝐴) ∧ 𝐵 ≤s 0s ) → (abss‘(𝐴 ·s 𝐵)) = ( -us ‘(𝐴 ·s 𝐵)))
37	16, 19, 36	3eqtr4rd 2808	. . . 4 ⊢ ((((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 0s ≤s 𝐴) ∧ 𝐵 ≤s 0s ) → (abss‘(𝐴 ·s 𝐵)) = (𝐴 ·s (abss‘𝐵)))
38		lestric 27829	. . . . . 6 ⊢ (( 0s ∈ No ∧ 𝐵 ∈ No ) → ( 0s ≤s 𝐵 ∨ 𝐵 ≤s 0s ))
39	24, 38	mpan 700	. . . . 5 ⊢ (𝐵 ∈ No → ( 0s ≤s 𝐵 ∨ 𝐵 ≤s 0s ))
40	39	ad2antlr 737	. . . 4 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 0s ≤s 𝐴) → ( 0s ≤s 𝐵 ∨ 𝐵 ≤s 0s ))
41	13, 37, 40	mpjaodan 971	. . 3 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 0s ≤s 𝐴) → (abss‘(𝐴 ·s 𝐵)) = (𝐴 ·s (abss‘𝐵)))
42		abssid 28331	. . . . 5 ⊢ ((𝐴 ∈ No ∧ 0s ≤s 𝐴) → (abss‘𝐴) = 𝐴)
43	42	adantlr 725	. . . 4 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 0s ≤s 𝐴) → (abss‘𝐴) = 𝐴)
44	43	oveq1d 7411	. . 3 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 0s ≤s 𝐴) → ((abss‘𝐴) ·s (abss‘𝐵)) = (𝐴 ·s (abss‘𝐵)))
45	41, 44	eqtr4d 2800	. 2 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 0s ≤s 𝐴) → (abss‘(𝐴 ·s 𝐵)) = ((abss‘𝐴) ·s (abss‘𝐵)))
46		simplll 784	. . . . . 6 ⊢ ((((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 𝐴 ≤s 0s ) ∧ 0s ≤s 𝐵) → 𝐴 ∈ No )
47		simpllr 785	. . . . . 6 ⊢ ((((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 𝐴 ≤s 0s ) ∧ 0s ≤s 𝐵) → 𝐵 ∈ No )
48	46, 47	mulnegs1d 28250	. . . . 5 ⊢ ((((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 𝐴 ≤s 0s ) ∧ 0s ≤s 𝐵) → (( -us ‘𝐴) ·s 𝐵) = ( -us ‘(𝐴 ·s 𝐵)))
49	10	ad4ant24 764	. . . . . 6 ⊢ ((((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 𝐴 ≤s 0s ) ∧ 0s ≤s 𝐵) → (abss‘𝐵) = 𝐵)
50	49	oveq2d 7412	. . . . 5 ⊢ ((((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 𝐴 ≤s 0s ) ∧ 0s ≤s 𝐵) → (( -us ‘𝐴) ·s (abss‘𝐵)) = (( -us ‘𝐴) ·s 𝐵))
51	1	adantr 484	. . . . . 6 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 𝐴 ≤s 0s ) → (𝐴 ·s 𝐵) ∈ No )
52	46	negscld 28127	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 𝐴 ≤s 0s ) ∧ 0s ≤s 𝐵) → ( -us ‘𝐴) ∈ No )
53		simplr 778	. . . . . . . . . . . 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	lenegsd 28138	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 𝐴 ≤s 0s ) ∧ 0s ≤s 𝐵) → (𝐴 ≤s 0s ↔ ( -us ‘ 0s ) ≤s ( -us ‘𝐴)))
56	53, 55	mpbid 234	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 𝐴 ≤s 0s ) ∧ 0s ≤s 𝐵) → ( -us ‘ 0s ) ≤s ( -us ‘𝐴))
57	20, 56	eqbrtrrid 5136	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 𝐴 ≤s 0s ) ∧ 0s ≤s 𝐵) → 0s ≤s ( -us ‘𝐴))
58		simpr 488	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 𝐴 ≤s 0s ) ∧ 0s ≤s 𝐵) → 0s ≤s 𝐵)
59	52, 47, 57, 58	mulsge0d 28236	. . . . . . . . 9 ⊢ ((((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 𝐴 ≤s 0s ) ∧ 0s ≤s 𝐵) → 0s ≤s (( -us ‘𝐴) ·s 𝐵))
60	59, 48	breqtrd 5126	. . . . . . . 8 ⊢ ((((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 𝐴 ≤s 0s ) ∧ 0s ≤s 𝐵) → 0s ≤s ( -us ‘(𝐴 ·s 𝐵)))
61	20, 60	eqbrtrid 5135	. . . . . . 7 ⊢ ((((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 𝐴 ≤s 0s ) ∧ 0s ≤s 𝐵) → ( -us ‘ 0s ) ≤s ( -us ‘(𝐴 ·s 𝐵)))
62	51	adantr 484	. . . . . . . 8 ⊢ ((((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 𝐴 ≤s 0s ) ∧ 0s ≤s 𝐵) → (𝐴 ·s 𝐵) ∈ No )
63	62, 54	lenegsd 28138	. . . . . . 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 259	. . . . . 6 ⊢ ((((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 𝐴 ≤s 0s ) ∧ 0s ≤s 𝐵) → (𝐴 ·s 𝐵) ≤s 0s )
65	51, 64, 35	syl2an2r 695	. . . . 5 ⊢ ((((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 𝐴 ≤s 0s ) ∧ 0s ≤s 𝐵) → (abss‘(𝐴 ·s 𝐵)) = ( -us ‘(𝐴 ·s 𝐵)))
66	48, 50, 65	3eqtr4rd 2808	. . . 4 ⊢ ((((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 𝐴 ≤s 0s ) ∧ 0s ≤s 𝐵) → (abss‘(𝐴 ·s 𝐵)) = (( -us ‘𝐴) ·s (abss‘𝐵)))
67		simplll 784	. . . . . 6 ⊢ ((((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 𝐴 ≤s 0s ) ∧ 𝐵 ≤s 0s ) → 𝐴 ∈ No )
68		simpllr 785	. . . . . 6 ⊢ ((((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 𝐴 ≤s 0s ) ∧ 𝐵 ≤s 0s ) → 𝐵 ∈ No )
69	67, 68	mul2negsd 28252	. . . . 5 ⊢ ((((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 𝐴 ≤s 0s ) ∧ 𝐵 ≤s 0s ) → (( -us ‘𝐴) ·s ( -us ‘𝐵)) = (𝐴 ·s 𝐵))
70	17	ad4ant24 764	. . . . . 6 ⊢ ((((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 𝐴 ≤s 0s ) ∧ 𝐵 ≤s 0s ) → (abss‘𝐵) = ( -us ‘𝐵))
71	70	oveq2d 7412	. . . . 5 ⊢ ((((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 𝐴 ≤s 0s ) ∧ 𝐵 ≤s 0s ) → (( -us ‘𝐴) ·s (abss‘𝐵)) = (( -us ‘𝐴) ·s ( -us ‘𝐵)))
72	67	negscld 28127	. . . . . . . 8 ⊢ ((((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 𝐴 ≤s 0s ) ∧ 𝐵 ≤s 0s ) → ( -us ‘𝐴) ∈ No )
73	68	negscld 28127	. . . . . . . 8 ⊢ ((((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 𝐴 ≤s 0s ) ∧ 𝐵 ≤s 0s ) → ( -us ‘𝐵) ∈ No )
74		simplr 778	. . . . . . . . . 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	lenegsd 28138	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 𝐴 ≤s 0s ) ∧ 𝐵 ≤s 0s ) → (𝐴 ≤s 0s ↔ ( -us ‘ 0s ) ≤s ( -us ‘𝐴)))
77	74, 76	mpbid 234	. . . . . . . . 9 ⊢ ((((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 𝐴 ≤s 0s ) ∧ 𝐵 ≤s 0s ) → ( -us ‘ 0s ) ≤s ( -us ‘𝐴))
78	20, 77	eqbrtrrid 5136	. . . . . . . 8 ⊢ ((((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 𝐴 ≤s 0s ) ∧ 𝐵 ≤s 0s ) → 0s ≤s ( -us ‘𝐴))
79		simpr 488	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 𝐴 ≤s 0s ) ∧ 𝐵 ≤s 0s ) → 𝐵 ≤s 0s )
80	68, 75	lenegsd 28138	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 𝐴 ≤s 0s ) ∧ 𝐵 ≤s 0s ) → (𝐵 ≤s 0s ↔ ( -us ‘ 0s ) ≤s ( -us ‘𝐵)))
81	79, 80	mpbid 234	. . . . . . . . 9 ⊢ ((((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 𝐴 ≤s 0s ) ∧ 𝐵 ≤s 0s ) → ( -us ‘ 0s ) ≤s ( -us ‘𝐵))
82	20, 81	eqbrtrrid 5136	. . . . . . . 8 ⊢ ((((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 𝐴 ≤s 0s ) ∧ 𝐵 ≤s 0s ) → 0s ≤s ( -us ‘𝐵))
83	72, 73, 78, 82	mulsge0d 28236	. . . . . . 7 ⊢ ((((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 𝐴 ≤s 0s ) ∧ 𝐵 ≤s 0s ) → 0s ≤s (( -us ‘𝐴) ·s ( -us ‘𝐵)))
84	83, 69	breqtrd 5126	. . . . . 6 ⊢ ((((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 𝐴 ≤s 0s ) ∧ 𝐵 ≤s 0s ) → 0s ≤s (𝐴 ·s 𝐵))
85	51, 84, 8	syl2an2r 695	. . . . 5 ⊢ ((((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 𝐴 ≤s 0s ) ∧ 𝐵 ≤s 0s ) → (abss‘(𝐴 ·s 𝐵)) = (𝐴 ·s 𝐵))
86	69, 71, 85	3eqtr4rd 2808	. . . 4 ⊢ ((((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 𝐴 ≤s 0s ) ∧ 𝐵 ≤s 0s ) → (abss‘(𝐴 ·s 𝐵)) = (( -us ‘𝐴) ·s (abss‘𝐵)))
87	39	ad2antlr 737	. . . 4 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 𝐴 ≤s 0s ) → ( 0s ≤s 𝐵 ∨ 𝐵 ≤s 0s ))
88	66, 86, 87	mpjaodan 971	. . 3 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 𝐴 ≤s 0s ) → (abss‘(𝐴 ·s 𝐵)) = (( -us ‘𝐴) ·s (abss‘𝐵)))
89		abssnid 28333	. . . . 5 ⊢ ((𝐴 ∈ No ∧ 𝐴 ≤s 0s ) → (abss‘𝐴) = ( -us ‘𝐴))
90	89	oveq1d 7411	. . . 4 ⊢ ((𝐴 ∈ No ∧ 𝐴 ≤s 0s ) → ((abss‘𝐴) ·s (abss‘𝐵)) = (( -us ‘𝐴) ·s (abss‘𝐵)))
91	90	adantlr 725	. . 3 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 𝐴 ≤s 0s ) → ((abss‘𝐴) ·s (abss‘𝐵)) = (( -us ‘𝐴) ·s (abss‘𝐵)))
92	88, 91	eqtr4d 2800	. 2 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 𝐴 ≤s 0s ) → (abss‘(𝐴 ·s 𝐵)) = ((abss‘𝐴) ·s (abss‘𝐵)))
93		lestric 27829	. . . 4 ⊢ (( 0s ∈ No ∧ 𝐴 ∈ No ) → ( 0s ≤s 𝐴 ∨ 𝐴 ≤s 0s ))
94	24, 93	mpan 700	. . 3 ⊢ (𝐴 ∈ No → ( 0s ≤s 𝐴 ∨ 𝐴 ≤s 0s ))
95	94	adantr 484	. 2 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → ( 0s ≤s 𝐴 ∨ 𝐴 ≤s 0s ))
96	45, 92, 95	mpjaodan 971	1 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (abss‘(𝐴 ·s 𝐵)) = ((abss‘𝐴) ·s (abss‘𝐵)))

Syntax hints:   → wi 4   ∧ wa 399   ∨ wo 858   = wceq 1560   ∈ wcel 2142   class class class wbr 5100  ‘cfv 6521  (class class class)co 7396   No csur 27701   ≤s cles 27805   0_s c0s 27895   -u_s cnegs 28109   ·_s cmuls 28196  abs_scabss 28327

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-rep 5227  ax-sep 5246  ax-nul 5256  ax-pow 5322  ax-pr 5390  ax-un 7718

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1099  df-3an 1100  df-tru 1563  df-fal 1573  df-ex 1800  df-nf 1804  df-sb 2091  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3077  df-rex 3087  df-rmo 3367  df-reu 3368  df-rab 3415  df-v 3456  df-sbc 3745  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-pss 3924  df-nul 4286  df-if 4481  df-pw 4557  df-sn 4583  df-pr 4585  df-tp 4587  df-op 4589  df-ot 4591  df-uni 4866  df-int 4906  df-iun 4951  df-br 5101  df-opab 5163  df-mpt 5182  df-tr 5208  df-id 5542  df-eprel 5547  df-po 5555  df-so 5556  df-fr 5600  df-se 5601  df-we 5602  df-xp 5653  df-rel 5654  df-cnv 5655  df-co 5656  df-dm 5657  df-rn 5658  df-res 5659  df-ima 5660  df-pred 6288  df-ord 6349  df-on 6350  df-suc 6352  df-iota 6477  df-fun 6523  df-fn 6524  df-f 6525  df-f1 6526  df-fo 6527  df-f1o 6528  df-fv 6529  df-riota 7353  df-ov 7399  df-oprab 7400  df-mpo 7401  df-1st 7970  df-2nd 7971  df-frecs 8262  df-wrecs 8293  df-recs 8342  df-1o 8437  df-2o 8438  df-nadd 8636  df-no 27704  df-lts 27705  df-bday 27706  df-les 27806  df-slts 27848  df-cuts 27850  df-0s 27897  df-made 27917  df-old 27918  df-left 27920  df-right 27921  df-norec 28028  df-norec2 28039  df-adds 28050  df-negs 28111  df-subs 28112  df-muls 28197  df-abss 28328

This theorem is referenced by:  remulscllem2  28591