Theorem absmuls 28639

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 28529	. . . . . . 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 28541	. . . . . 6 ⊢ ((((𝐴 ∈ No ∧ 𝐵 ∈ No) ∧ 0s ≤s 𝐴) ∧ 0s ≤s 𝐵) → 0s ≤s (𝐴 ·s 𝐵))
8		abssid 28636	. . . . . 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 28636	. . . . . . 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 7440	. . . . 5 ⊢ ((((𝐴 ∈ No ∧ 𝐵 ∈ No) ∧ 0s ≤s 𝐴) ∧ 0s ≤s 𝐵) → (𝐴 ·s (abss‘𝐵)) = (𝐴 ·s 𝐵))
13	9, 12	eqtr4d 2798	. . . 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 28556	. . . . 5 ⊢ ((((𝐴 ∈ No ∧ 𝐵 ∈ No) ∧ 0s ≤s 𝐴) ∧ 𝐵 ≤s 0s ) → (𝐴 ·s ( -s‘𝐵)) = ( -s‘(𝐴 ·s 𝐵)))
17		abssnid 28638	. . . . . . 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 7440	. . . . 5 ⊢ ((((𝐴 ∈ No ∧ 𝐵 ∈ No) ∧ 0s ≤s 𝐴) ∧ 𝐵 ≤s 0s ) → (𝐴 ·s (abss‘𝐵)) = (𝐴 ·s ( -s‘𝐵)))
20		neg0s 28421	. . . . . . . 8 ⊢ ( -s‘ 0s ) = 0s
21	15	negscld 28432	. . . . . . . . . 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 28204	. . . . . . . . . . . . . 14 ⊢ 0s ∈ No
25	24	a1i 11	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ∈ No ∧ 𝐵 ∈ No) ∧ 0s ≤s 𝐴) ∧ 𝐵 ≤s 0s ) → 0s ∈ No)
26	15, 25	lenegsd 28443	. . . . . . . . . . . 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 5140	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ No ∧ 𝐵 ∈ No) ∧ 0s ≤s 𝐴) ∧ 𝐵 ≤s 0s ) → 0s ≤s ( -s‘𝐵))
29	14, 21, 22, 28	mulsge0d 28541	. . . . . . . . 9 ⊢ ((((𝐴 ∈ No ∧ 𝐵 ∈ No) ∧ 0s ≤s 𝐴) ∧ 𝐵 ≤s 0s ) → 0s ≤s (𝐴 ·s ( -s‘𝐵)))
30	29, 16	breqtrd 5130	. . . . . . . 8 ⊢ ((((𝐴 ∈ No ∧ 𝐵 ∈ No) ∧ 0s ≤s 𝐴) ∧ 𝐵 ≤s 0s ) → 0s ≤s ( -s‘(𝐴 ·s 𝐵)))
31	20, 30	eqbrtrid 5139	. . . . . . 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 28443	. . . . . . 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 28638	. . . . . 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 2806	. . . 4 ⊢ ((((𝐴 ∈ No ∧ 𝐵 ∈ No) ∧ 0s ≤s 𝐴) ∧ 𝐵 ≤s 0s ) → (abss‘(𝐴 ·s 𝐵)) = (𝐴 ·s (abss‘𝐵)))
38		lestric 28134	. . . . . 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 28636	. . . . 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 7439	. . 3 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No) ∧ 0s ≤s 𝐴) → ((abss‘𝐴) ·s (abss‘𝐵)) = (𝐴 ·s (abss‘𝐵)))
45	41, 44	eqtr4d 2798	. 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 28555	. . . . 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 7440	. . . . 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 28432	. . . . . . . . . 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 28443	. . . . . . . . . . . 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 5140	. . . . . . . . . 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 28541	. . . . . . . . 9 ⊢ ((((𝐴 ∈ No ∧ 𝐵 ∈ No) ∧ 𝐴 ≤s 0s ) ∧ 0s ≤s 𝐵) → 0s ≤s (( -s‘𝐴) ·s 𝐵))
60	59, 48	breqtrd 5130	. . . . . . . 8 ⊢ ((((𝐴 ∈ No ∧ 𝐵 ∈ No) ∧ 𝐴 ≤s 0s ) ∧ 0s ≤s 𝐵) → 0s ≤s ( -s‘(𝐴 ·s 𝐵)))
61	20, 60	eqbrtrid 5139	. . . . . . 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 28443	. . . . . . 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 2806	. . . 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 28557	. . . . 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 7440	. . . . 5 ⊢ ((((𝐴 ∈ No ∧ 𝐵 ∈ No) ∧ 𝐴 ≤s 0s ) ∧ 𝐵 ≤s 0s ) → (( -s‘𝐴) ·s (abss‘𝐵)) = (( -s‘𝐴) ·s ( -s‘𝐵)))
72	67	negscld 28432	. . . . . . . 8 ⊢ ((((𝐴 ∈ No ∧ 𝐵 ∈ No) ∧ 𝐴 ≤s 0s ) ∧ 𝐵 ≤s 0s ) → ( -s‘𝐴) ∈ No)
73	68	negscld 28432	. . . . . . . 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 28443	. . . . . . . . . 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 5140	. . . . . . . 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 28443	. . . . . . . . . 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 5140	. . . . . . . 8 ⊢ ((((𝐴 ∈ No ∧ 𝐵 ∈ No) ∧ 𝐴 ≤s 0s ) ∧ 𝐵 ≤s 0s ) → 0s ≤s ( -s‘𝐵))
83	72, 73, 78, 82	mulsge0d 28541	. . . . . . 7 ⊢ ((((𝐴 ∈ No ∧ 𝐵 ∈ No) ∧ 𝐴 ≤s 0s ) ∧ 𝐵 ≤s 0s ) → 0s ≤s (( -s‘𝐴) ·s ( -s‘𝐵)))
84	83, 69	breqtrd 5130	. . . . . 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 2806	. . . 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 28638	. . . . 5 ⊢ ((𝐴 ∈ No ∧ 𝐴 ≤s 0s ) → (abss‘𝐴) = ( -s‘𝐴))
90	89	oveq1d 7439	. . . 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 2798	. 2 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No) ∧ 𝐴 ≤s 0s ) → (abss‘(𝐴 ·s 𝐵)) = ((abss‘𝐴) ·s (abss‘𝐵)))
93		lestric 28134	. . . 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 1569   ∈ wcel 2144   class class class wbr 5102  ‘cfv 6543  (class class class)co 7424  Nocsur 28006   ≤s cles 28110   0_s c0s 28200   -_scnegs 28414   ·_s cmuls 28501  abs_scabss 28632

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5327  ax-pr 5391  ax-un 7755

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3848  df-ss 3897  df-pss 4013  df-dif 4046  df-in 4090  df-un 4150  df-nul 4278  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-tp 4589  df-op 4835  df-ot 4858  df-uni 4867  df-int 4907  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5546  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-se 5605  df-we 5606  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-pred 6307  df-ord 6368  df-on 6369  df-suc 6371  df-iota 6497  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7381  df-ov 7427  df-oprab 7428  df-mpo 7429  df-1st 8005  df-2nd 8006  df-frecs 8307  df-wrecs 8338  df-recs 8387  df-1o 8484  df-2o 8485  df-nadd 8683  df-no 28009  df-lts 28010  df-bday 28011  df-les 28111  df-slts 28153  df-cuts 28155  df-0s 28202  df-made 28222  df-old 28223  df-left 28225  df-right 28226  df-norec 28333  df-norec2 28344  df-adds 28355  df-negs 28416  df-subs 28417  df-muls 28502  df-abss 28633

This theorem is used by:  remulscllem2  28896