Theorem absmuls 28687

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 28577	. . . . . . 7 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No) → (𝐴 ·s 𝐵) ∈ No)
2	1	adantr 486	. . . . . 6 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No) ∧ 0s ≤s 𝐴) → (𝐴 ·s 𝐵) ∈ No)
3		simplll 787	. . . . . . 7 ⊢ ((((𝐴 ∈ No ∧ 𝐵 ∈ No) ∧ 0s ≤s 𝐴) ∧ 0s ≤s 𝐵) → 𝐴 ∈ No)
4		simpllr 788	. . . . . . 7 ⊢ ((((𝐴 ∈ No ∧ 𝐵 ∈ No) ∧ 0s ≤s 𝐴) ∧ 0s ≤s 𝐵) → 𝐵 ∈ No)
5		simplr 781	. . . . . . 7 ⊢ ((((𝐴 ∈ No ∧ 𝐵 ∈ No) ∧ 0s ≤s 𝐴) ∧ 0s ≤s 𝐵) → 0s ≤s 𝐴)
6		simpr 490	. . . . . . 7 ⊢ ((((𝐴 ∈ No ∧ 𝐵 ∈ No) ∧ 0s ≤s 𝐴) ∧ 0s ≤s 𝐵) → 0s ≤s 𝐵)
7	3, 4, 5, 6	mulsge0d 28589	. . . . . 6 ⊢ ((((𝐴 ∈ No ∧ 𝐵 ∈ No) ∧ 0s ≤s 𝐴) ∧ 0s ≤s 𝐵) → 0s ≤s (𝐴 ·s 𝐵))
8		abssid 28684	. . . . . 6 ⊢ (((𝐴 ·s 𝐵) ∈ No ∧ 0s ≤s (𝐴 ·s 𝐵)) → (abss‘(𝐴 ·s 𝐵)) = (𝐴 ·s 𝐵))
9	2, 7, 8	syl2an2r 698	. . . . 5 ⊢ ((((𝐴 ∈ No ∧ 𝐵 ∈ No) ∧ 0s ≤s 𝐴) ∧ 0s ≤s 𝐵) → (abss‘(𝐴 ·s 𝐵)) = (𝐴 ·s 𝐵))
10		abssid 28684	. . . . . . 7 ⊢ ((𝐵 ∈ No ∧ 0s ≤s 𝐵) → (abss‘𝐵) = 𝐵)
11	10	ad4ant24 767	. . . . . 6 ⊢ ((((𝐴 ∈ No ∧ 𝐵 ∈ No) ∧ 0s ≤s 𝐴) ∧ 0s ≤s 𝐵) → (abss‘𝐵) = 𝐵)
12	11	oveq2d 7467	. . . . 5 ⊢ ((((𝐴 ∈ No ∧ 𝐵 ∈ No) ∧ 0s ≤s 𝐴) ∧ 0s ≤s 𝐵) → (𝐴 ·s (abss‘𝐵)) = (𝐴 ·s 𝐵))
13	9, 12	eqtr4d 2801	. . . 4 ⊢ ((((𝐴 ∈ No ∧ 𝐵 ∈ No) ∧ 0s ≤s 𝐴) ∧ 0s ≤s 𝐵) → (abss‘(𝐴 ·s 𝐵)) = (𝐴 ·s (abss‘𝐵)))
14		simplll 787	. . . . . 6 ⊢ ((((𝐴 ∈ No ∧ 𝐵 ∈ No) ∧ 0s ≤s 𝐴) ∧ 𝐵 ≤s 0s ) → 𝐴 ∈ No)
15		simpllr 788	. . . . . 6 ⊢ ((((𝐴 ∈ No ∧ 𝐵 ∈ No) ∧ 0s ≤s 𝐴) ∧ 𝐵 ≤s 0s ) → 𝐵 ∈ No)
16	14, 15	mulnegs2d 28604	. . . . 5 ⊢ ((((𝐴 ∈ No ∧ 𝐵 ∈ No) ∧ 0s ≤s 𝐴) ∧ 𝐵 ≤s 0s ) → (𝐴 ·s ( -s‘𝐵)) = ( -s‘(𝐴 ·s 𝐵)))
17		abssnid 28686	. . . . . . 7 ⊢ ((𝐵 ∈ No ∧ 𝐵 ≤s 0s ) → (abss‘𝐵) = ( -s‘𝐵))
18	17	ad4ant24 767	. . . . . 6 ⊢ ((((𝐴 ∈ No ∧ 𝐵 ∈ No) ∧ 0s ≤s 𝐴) ∧ 𝐵 ≤s 0s ) → (abss‘𝐵) = ( -s‘𝐵))
19	18	oveq2d 7467	. . . . 5 ⊢ ((((𝐴 ∈ No ∧ 𝐵 ∈ No) ∧ 0s ≤s 𝐴) ∧ 𝐵 ≤s 0s ) → (𝐴 ·s (abss‘𝐵)) = (𝐴 ·s ( -s‘𝐵)))
20		neg0s 28469	. . . . . . . 8 ⊢ ( -s‘ 0s ) = 0s
21	15	negscld 28480	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ No ∧ 𝐵 ∈ No) ∧ 0s ≤s 𝐴) ∧ 𝐵 ≤s 0s ) → ( -s‘𝐵) ∈ No)
22		simplr 781	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ No ∧ 𝐵 ∈ No) ∧ 0s ≤s 𝐴) ∧ 𝐵 ≤s 0s ) → 0s ≤s 𝐴)
23		simpr 490	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ No ∧ 𝐵 ∈ No) ∧ 0s ≤s 𝐴) ∧ 𝐵 ≤s 0s ) → 𝐵 ≤s 0s )
24		0no 28252	. . . . . . . . . . . . . 14 ⊢ 0s ∈ No
25	24	a1i 11	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ∈ No ∧ 𝐵 ∈ No) ∧ 0s ≤s 𝐴) ∧ 𝐵 ≤s 0s ) → 0s ∈ No)
26	15, 25	lenegsd 28491	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ No ∧ 𝐵 ∈ No) ∧ 0s ≤s 𝐴) ∧ 𝐵 ≤s 0s ) → (𝐵 ≤s 0s ↔ ( -s‘ 0s ) ≤s ( -s‘𝐵)))
27	23, 26	mpbid 235	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ No ∧ 𝐵 ∈ No) ∧ 0s ≤s 𝐴) ∧ 𝐵 ≤s 0s ) → ( -s‘ 0s ) ≤s ( -s‘𝐵))
28	20, 27	eqbrtrrid 5147	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ No ∧ 𝐵 ∈ No) ∧ 0s ≤s 𝐴) ∧ 𝐵 ≤s 0s ) → 0s ≤s ( -s‘𝐵))
29	14, 21, 22, 28	mulsge0d 28589	. . . . . . . . 9 ⊢ ((((𝐴 ∈ No ∧ 𝐵 ∈ No) ∧ 0s ≤s 𝐴) ∧ 𝐵 ≤s 0s ) → 0s ≤s (𝐴 ·s ( -s‘𝐵)))
30	29, 16	breqtrd 5137	. . . . . . . 8 ⊢ ((((𝐴 ∈ No ∧ 𝐵 ∈ No) ∧ 0s ≤s 𝐴) ∧ 𝐵 ≤s 0s ) → 0s ≤s ( -s‘(𝐴 ·s 𝐵)))
31	20, 30	eqbrtrid 5146	. . . . . . 7 ⊢ ((((𝐴 ∈ No ∧ 𝐵 ∈ No) ∧ 0s ≤s 𝐴) ∧ 𝐵 ≤s 0s ) → ( -s‘ 0s ) ≤s ( -s‘(𝐴 ·s 𝐵)))
32	2	adantr 486	. . . . . . . 8 ⊢ ((((𝐴 ∈ No ∧ 𝐵 ∈ No) ∧ 0s ≤s 𝐴) ∧ 𝐵 ≤s 0s ) → (𝐴 ·s 𝐵) ∈ No)
33	32, 25	lenegsd 28491	. . . . . . 7 ⊢ ((((𝐴 ∈ No ∧ 𝐵 ∈ No) ∧ 0s ≤s 𝐴) ∧ 𝐵 ≤s 0s ) → ((𝐴 ·s 𝐵) ≤s 0s ↔ ( -s‘ 0s ) ≤s ( -s‘(𝐴 ·s 𝐵))))
34	31, 33	mpbird 260	. . . . . 6 ⊢ ((((𝐴 ∈ No ∧ 𝐵 ∈ No) ∧ 0s ≤s 𝐴) ∧ 𝐵 ≤s 0s ) → (𝐴 ·s 𝐵) ≤s 0s )
35		abssnid 28686	. . . . . 6 ⊢ (((𝐴 ·s 𝐵) ∈ No ∧ (𝐴 ·s 𝐵) ≤s 0s ) → (abss‘(𝐴 ·s 𝐵)) = ( -s‘(𝐴 ·s 𝐵)))
36	2, 34, 35	syl2an2r 698	. . . . 5 ⊢ ((((𝐴 ∈ No ∧ 𝐵 ∈ No) ∧ 0s ≤s 𝐴) ∧ 𝐵 ≤s 0s ) → (abss‘(𝐴 ·s 𝐵)) = ( -s‘(𝐴 ·s 𝐵)))
37	16, 19, 36	3eqtr4rd 2809	. . . 4 ⊢ ((((𝐴 ∈ No ∧ 𝐵 ∈ No) ∧ 0s ≤s 𝐴) ∧ 𝐵 ≤s 0s ) → (abss‘(𝐴 ·s 𝐵)) = (𝐴 ·s (abss‘𝐵)))
38		lestric 28182	. . . . . 6 ⊢ (( 0s ∈ No ∧ 𝐵 ∈ No) → ( 0s ≤s 𝐵 ∨ 𝐵 ≤s 0s ))
39	24, 38	mpan 703	. . . . 5 ⊢ (𝐵 ∈ No → ( 0s ≤s 𝐵 ∨ 𝐵 ≤s 0s ))
40	39	ad2antlr 740	. . . 4 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No) ∧ 0s ≤s 𝐴) → ( 0s ≤s 𝐵 ∨ 𝐵 ≤s 0s ))
41	13, 37, 40	mpjaodan 973	. . 3 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No) ∧ 0s ≤s 𝐴) → (abss‘(𝐴 ·s 𝐵)) = (𝐴 ·s (abss‘𝐵)))
42		abssid 28684	. . . . 5 ⊢ ((𝐴 ∈ No ∧ 0s ≤s 𝐴) → (abss‘𝐴) = 𝐴)
43	42	adantlr 728	. . . 4 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No) ∧ 0s ≤s 𝐴) → (abss‘𝐴) = 𝐴)
44	43	oveq1d 7466	. . 3 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No) ∧ 0s ≤s 𝐴) → ((abss‘𝐴) ·s (abss‘𝐵)) = (𝐴 ·s (abss‘𝐵)))
45	41, 44	eqtr4d 2801	. 2 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No) ∧ 0s ≤s 𝐴) → (abss‘(𝐴 ·s 𝐵)) = ((abss‘𝐴) ·s (abss‘𝐵)))
46		simplll 787	. . . . . 6 ⊢ ((((𝐴 ∈ No ∧ 𝐵 ∈ No) ∧ 𝐴 ≤s 0s ) ∧ 0s ≤s 𝐵) → 𝐴 ∈ No)
47		simpllr 788	. . . . . 6 ⊢ ((((𝐴 ∈ No ∧ 𝐵 ∈ No) ∧ 𝐴 ≤s 0s ) ∧ 0s ≤s 𝐵) → 𝐵 ∈ No)
48	46, 47	mulnegs1d 28603	. . . . 5 ⊢ ((((𝐴 ∈ No ∧ 𝐵 ∈ No) ∧ 𝐴 ≤s 0s ) ∧ 0s ≤s 𝐵) → (( -s‘𝐴) ·s 𝐵) = ( -s‘(𝐴 ·s 𝐵)))
49	10	ad4ant24 767	. . . . . 6 ⊢ ((((𝐴 ∈ No ∧ 𝐵 ∈ No) ∧ 𝐴 ≤s 0s ) ∧ 0s ≤s 𝐵) → (abss‘𝐵) = 𝐵)
50	49	oveq2d 7467	. . . . 5 ⊢ ((((𝐴 ∈ No ∧ 𝐵 ∈ No) ∧ 𝐴 ≤s 0s ) ∧ 0s ≤s 𝐵) → (( -s‘𝐴) ·s (abss‘𝐵)) = (( -s‘𝐴) ·s 𝐵))
51	1	adantr 486	. . . . . 6 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No) ∧ 𝐴 ≤s 0s ) → (𝐴 ·s 𝐵) ∈ No)
52	46	negscld 28480	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ No ∧ 𝐵 ∈ No) ∧ 𝐴 ≤s 0s ) ∧ 0s ≤s 𝐵) → ( -s‘𝐴) ∈ No)
53		simplr 781	. . . . . . . . . . . 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 28491	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ No ∧ 𝐵 ∈ No) ∧ 𝐴 ≤s 0s ) ∧ 0s ≤s 𝐵) → (𝐴 ≤s 0s ↔ ( -s‘ 0s ) ≤s ( -s‘𝐴)))
56	53, 55	mpbid 235	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ No ∧ 𝐵 ∈ No) ∧ 𝐴 ≤s 0s ) ∧ 0s ≤s 𝐵) → ( -s‘ 0s ) ≤s ( -s‘𝐴))
57	20, 56	eqbrtrrid 5147	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ No ∧ 𝐵 ∈ No) ∧ 𝐴 ≤s 0s ) ∧ 0s ≤s 𝐵) → 0s ≤s ( -s‘𝐴))
58		simpr 490	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ No ∧ 𝐵 ∈ No) ∧ 𝐴 ≤s 0s ) ∧ 0s ≤s 𝐵) → 0s ≤s 𝐵)
59	52, 47, 57, 58	mulsge0d 28589	. . . . . . . . 9 ⊢ ((((𝐴 ∈ No ∧ 𝐵 ∈ No) ∧ 𝐴 ≤s 0s ) ∧ 0s ≤s 𝐵) → 0s ≤s (( -s‘𝐴) ·s 𝐵))
60	59, 48	breqtrd 5137	. . . . . . . 8 ⊢ ((((𝐴 ∈ No ∧ 𝐵 ∈ No) ∧ 𝐴 ≤s 0s ) ∧ 0s ≤s 𝐵) → 0s ≤s ( -s‘(𝐴 ·s 𝐵)))
61	20, 60	eqbrtrid 5146	. . . . . . 7 ⊢ ((((𝐴 ∈ No ∧ 𝐵 ∈ No) ∧ 𝐴 ≤s 0s ) ∧ 0s ≤s 𝐵) → ( -s‘ 0s ) ≤s ( -s‘(𝐴 ·s 𝐵)))
62	51	adantr 486	. . . . . . . 8 ⊢ ((((𝐴 ∈ No ∧ 𝐵 ∈ No) ∧ 𝐴 ≤s 0s ) ∧ 0s ≤s 𝐵) → (𝐴 ·s 𝐵) ∈ No)
63	62, 54	lenegsd 28491	. . . . . . 7 ⊢ ((((𝐴 ∈ No ∧ 𝐵 ∈ No) ∧ 𝐴 ≤s 0s ) ∧ 0s ≤s 𝐵) → ((𝐴 ·s 𝐵) ≤s 0s ↔ ( -s‘ 0s ) ≤s ( -s‘(𝐴 ·s 𝐵))))
64	61, 63	mpbird 260	. . . . . 6 ⊢ ((((𝐴 ∈ No ∧ 𝐵 ∈ No) ∧ 𝐴 ≤s 0s ) ∧ 0s ≤s 𝐵) → (𝐴 ·s 𝐵) ≤s 0s )
65	51, 64, 35	syl2an2r 698	. . . . 5 ⊢ ((((𝐴 ∈ No ∧ 𝐵 ∈ No) ∧ 𝐴 ≤s 0s ) ∧ 0s ≤s 𝐵) → (abss‘(𝐴 ·s 𝐵)) = ( -s‘(𝐴 ·s 𝐵)))
66	48, 50, 65	3eqtr4rd 2809	. . . 4 ⊢ ((((𝐴 ∈ No ∧ 𝐵 ∈ No) ∧ 𝐴 ≤s 0s ) ∧ 0s ≤s 𝐵) → (abss‘(𝐴 ·s 𝐵)) = (( -s‘𝐴) ·s (abss‘𝐵)))
67		simplll 787	. . . . . 6 ⊢ ((((𝐴 ∈ No ∧ 𝐵 ∈ No) ∧ 𝐴 ≤s 0s ) ∧ 𝐵 ≤s 0s ) → 𝐴 ∈ No)
68		simpllr 788	. . . . . 6 ⊢ ((((𝐴 ∈ No ∧ 𝐵 ∈ No) ∧ 𝐴 ≤s 0s ) ∧ 𝐵 ≤s 0s ) → 𝐵 ∈ No)
69	67, 68	mul2negsd 28605	. . . . 5 ⊢ ((((𝐴 ∈ No ∧ 𝐵 ∈ No) ∧ 𝐴 ≤s 0s ) ∧ 𝐵 ≤s 0s ) → (( -s‘𝐴) ·s ( -s‘𝐵)) = (𝐴 ·s 𝐵))
70	17	ad4ant24 767	. . . . . 6 ⊢ ((((𝐴 ∈ No ∧ 𝐵 ∈ No) ∧ 𝐴 ≤s 0s ) ∧ 𝐵 ≤s 0s ) → (abss‘𝐵) = ( -s‘𝐵))
71	70	oveq2d 7467	. . . . 5 ⊢ ((((𝐴 ∈ No ∧ 𝐵 ∈ No) ∧ 𝐴 ≤s 0s ) ∧ 𝐵 ≤s 0s ) → (( -s‘𝐴) ·s (abss‘𝐵)) = (( -s‘𝐴) ·s ( -s‘𝐵)))
72	67	negscld 28480	. . . . . . . 8 ⊢ ((((𝐴 ∈ No ∧ 𝐵 ∈ No) ∧ 𝐴 ≤s 0s ) ∧ 𝐵 ≤s 0s ) → ( -s‘𝐴) ∈ No)
73	68	negscld 28480	. . . . . . . 8 ⊢ ((((𝐴 ∈ No ∧ 𝐵 ∈ No) ∧ 𝐴 ≤s 0s ) ∧ 𝐵 ≤s 0s ) → ( -s‘𝐵) ∈ No)
74		simplr 781	. . . . . . . . . 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 28491	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ No ∧ 𝐵 ∈ No) ∧ 𝐴 ≤s 0s ) ∧ 𝐵 ≤s 0s ) → (𝐴 ≤s 0s ↔ ( -s‘ 0s ) ≤s ( -s‘𝐴)))
77	74, 76	mpbid 235	. . . . . . . . 9 ⊢ ((((𝐴 ∈ No ∧ 𝐵 ∈ No) ∧ 𝐴 ≤s 0s ) ∧ 𝐵 ≤s 0s ) → ( -s‘ 0s ) ≤s ( -s‘𝐴))
78	20, 77	eqbrtrrid 5147	. . . . . . . 8 ⊢ ((((𝐴 ∈ No ∧ 𝐵 ∈ No) ∧ 𝐴 ≤s 0s ) ∧ 𝐵 ≤s 0s ) → 0s ≤s ( -s‘𝐴))
79		simpr 490	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ No ∧ 𝐵 ∈ No) ∧ 𝐴 ≤s 0s ) ∧ 𝐵 ≤s 0s ) → 𝐵 ≤s 0s )
80	68, 75	lenegsd 28491	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ No ∧ 𝐵 ∈ No) ∧ 𝐴 ≤s 0s ) ∧ 𝐵 ≤s 0s ) → (𝐵 ≤s 0s ↔ ( -s‘ 0s ) ≤s ( -s‘𝐵)))
81	79, 80	mpbid 235	. . . . . . . . 9 ⊢ ((((𝐴 ∈ No ∧ 𝐵 ∈ No) ∧ 𝐴 ≤s 0s ) ∧ 𝐵 ≤s 0s ) → ( -s‘ 0s ) ≤s ( -s‘𝐵))
82	20, 81	eqbrtrrid 5147	. . . . . . . 8 ⊢ ((((𝐴 ∈ No ∧ 𝐵 ∈ No) ∧ 𝐴 ≤s 0s ) ∧ 𝐵 ≤s 0s ) → 0s ≤s ( -s‘𝐵))
83	72, 73, 78, 82	mulsge0d 28589	. . . . . . 7 ⊢ ((((𝐴 ∈ No ∧ 𝐵 ∈ No) ∧ 𝐴 ≤s 0s ) ∧ 𝐵 ≤s 0s ) → 0s ≤s (( -s‘𝐴) ·s ( -s‘𝐵)))
84	83, 69	breqtrd 5137	. . . . . 6 ⊢ ((((𝐴 ∈ No ∧ 𝐵 ∈ No) ∧ 𝐴 ≤s 0s ) ∧ 𝐵 ≤s 0s ) → 0s ≤s (𝐴 ·s 𝐵))
85	51, 84, 8	syl2an2r 698	. . . . 5 ⊢ ((((𝐴 ∈ No ∧ 𝐵 ∈ No) ∧ 𝐴 ≤s 0s ) ∧ 𝐵 ≤s 0s ) → (abss‘(𝐴 ·s 𝐵)) = (𝐴 ·s 𝐵))
86	69, 71, 85	3eqtr4rd 2809	. . . 4 ⊢ ((((𝐴 ∈ No ∧ 𝐵 ∈ No) ∧ 𝐴 ≤s 0s ) ∧ 𝐵 ≤s 0s ) → (abss‘(𝐴 ·s 𝐵)) = (( -s‘𝐴) ·s (abss‘𝐵)))
87	39	ad2antlr 740	. . . 4 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No) ∧ 𝐴 ≤s 0s ) → ( 0s ≤s 𝐵 ∨ 𝐵 ≤s 0s ))
88	66, 86, 87	mpjaodan 973	. . 3 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No) ∧ 𝐴 ≤s 0s ) → (abss‘(𝐴 ·s 𝐵)) = (( -s‘𝐴) ·s (abss‘𝐵)))
89		abssnid 28686	. . . . 5 ⊢ ((𝐴 ∈ No ∧ 𝐴 ≤s 0s ) → (abss‘𝐴) = ( -s‘𝐴))
90	89	oveq1d 7466	. . . 4 ⊢ ((𝐴 ∈ No ∧ 𝐴 ≤s 0s ) → ((abss‘𝐴) ·s (abss‘𝐵)) = (( -s‘𝐴) ·s (abss‘𝐵)))
91	90	adantlr 728	. . 3 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No) ∧ 𝐴 ≤s 0s ) → ((abss‘𝐴) ·s (abss‘𝐵)) = (( -s‘𝐴) ·s (abss‘𝐵)))
92	88, 91	eqtr4d 2801	. 2 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No) ∧ 𝐴 ≤s 0s ) → (abss‘(𝐴 ·s 𝐵)) = ((abss‘𝐴) ·s (abss‘𝐵)))
93		lestric 28182	. . . 4 ⊢ (( 0s ∈ No ∧ 𝐴 ∈ No) → ( 0s ≤s 𝐴 ∨ 𝐴 ≤s 0s ))
94	24, 93	mpan 703	. . 3 ⊢ (𝐴 ∈ No → ( 0s ≤s 𝐴 ∨ 𝐴 ≤s 0s ))
95	94	adantr 486	. 2 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No) → ( 0s ≤s 𝐴 ∨ 𝐴 ≤s 0s ))
96	45, 92, 95	mpjaodan 973	1 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No) → (abss‘(𝐴 ·s 𝐵)) = ((abss‘𝐴) ·s (abss‘𝐵)))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∨ wo 861   = wceq 1570   ∈ wcel 2147   class class class wbr 5108  ‘cfv 6566  (class class class)co 7451  Nocsur 28054   ≤s cles 28158   0_s c0s 28248   -_scnegs 28462   ·_s cmuls 28549  abs_scabss 28680

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 5238  ax-sep 5255  ax-nul 5266  ax-pow 5334  ax-pr 5398  ax-un 7783

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 4841  df-ot 4864  df-uni 4873  df-int 4913  df-iun 4958  df-br 5109  df-opab 5174  df-mpt 5193  df-tr 5219  df-id 5553  df-eprel 5558  df-po 5566  df-so 5567  df-fr 5611  df-se 5612  df-we 5613  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-rn 5669  df-res 5670  df-ima 5671  df-pred 6330  df-ord 6391  df-on 6392  df-suc 6394  df-iota 6520  df-fun 6568  df-fn 6569  df-f 6570  df-f1 6571  df-fo 6572  df-f1o 6573  df-fv 6574  df-riota 7408  df-ov 7454  df-oprab 7455  df-mpo 7456  df-1st 8035  df-2nd 8036  df-frecs 8337  df-wrecs 8368  df-recs 8417  df-1o 8514  df-2o 8515  df-nadd 8713  df-no 28057  df-lts 28058  df-bday 28059  df-les 28159  df-slts 28201  df-cuts 28203  df-0s 28250  df-made 28270  df-old 28271  df-left 28273  df-right 28274  df-norec 28381  df-norec2 28392  df-adds 28403  df-negs 28464  df-subs 28465  df-muls 28550  df-abss 28681

This theorem is used by:  remulscllem2  28944