sgnmul - Mathbox for Thierry Arnoux

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Theorem sgnmul 32554

Description: Signum of a product. (Contributed by Thierry Arnoux, 2-Oct-2018.)

**Assertion**
Ref	Expression
sgnmul	⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (sgn‘(𝐴 · 𝐵)) = ((sgn‘𝐴) · (sgn‘𝐵)))

**Proof of Theorem sgnmul**
Step	Hyp	Ref	Expression
1		remulcl 11002	. . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 · 𝐵) ∈ ℝ)
2	1	rexrd 11071	. 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 · 𝐵) ∈ ℝ^*)
3		eqeq1 2740	. 2 ⊢ ((sgn‘(𝐴 · 𝐵)) = 0 → ((sgn‘(𝐴 · 𝐵)) = ((sgn‘𝐴) · (sgn‘𝐵)) ↔ 0 = ((sgn‘𝐴) · (sgn‘𝐵))))
4		eqeq1 2740	. 2 ⊢ ((sgn‘(𝐴 · 𝐵)) = 1 → ((sgn‘(𝐴 · 𝐵)) = ((sgn‘𝐴) · (sgn‘𝐵)) ↔ 1 = ((sgn‘𝐴) · (sgn‘𝐵))))
5		eqeq1 2740	. 2 ⊢ ((sgn‘(𝐴 · 𝐵)) = -1 → ((sgn‘(𝐴 · 𝐵)) = ((sgn‘𝐴) · (sgn‘𝐵)) ↔ -1 = ((sgn‘𝐴) · (sgn‘𝐵))))
6		fveq2 6804	. . . . . . 7 ⊢ (𝐴 = 0 → (sgn‘𝐴) = (sgn‘0))
7		sgn0 14845	. . . . . . 7 ⊢ (sgn‘0) = 0
8	6, 7	eqtrdi 2792	. . . . . 6 ⊢ (𝐴 = 0 → (sgn‘𝐴) = 0)
9	8	oveq1d 7322	. . . . 5 ⊢ (𝐴 = 0 → ((sgn‘𝐴) · (sgn‘𝐵)) = (0 · (sgn‘𝐵)))
10	9	adantl 483	. . . 4 ⊢ ((((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (𝐴 · 𝐵) = 0) ∧ 𝐴 = 0) → ((sgn‘𝐴) · (sgn‘𝐵)) = (0 · (sgn‘𝐵)))
11		sgnclre 32551	. . . . . . 7 ⊢ (𝐵 ∈ ℝ → (sgn‘𝐵) ∈ ℝ)
12	11	recnd 11049	. . . . . 6 ⊢ (𝐵 ∈ ℝ → (sgn‘𝐵) ∈ ℂ)
13	12	mul02d 11219	. . . . 5 ⊢ (𝐵 ∈ ℝ → (0 · (sgn‘𝐵)) = 0)
14	13	ad3antlr 729	. . . 4 ⊢ ((((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (𝐴 · 𝐵) = 0) ∧ 𝐴 = 0) → (0 · (sgn‘𝐵)) = 0)
15	10, 14	eqtr2d 2777	. . 3 ⊢ ((((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (𝐴 · 𝐵) = 0) ∧ 𝐴 = 0) → 0 = ((sgn‘𝐴) · (sgn‘𝐵)))
16		fveq2 6804	. . . . . . 7 ⊢ (𝐵 = 0 → (sgn‘𝐵) = (sgn‘0))
17	16, 7	eqtrdi 2792	. . . . . 6 ⊢ (𝐵 = 0 → (sgn‘𝐵) = 0)
18	17	oveq2d 7323	. . . . 5 ⊢ (𝐵 = 0 → ((sgn‘𝐴) · (sgn‘𝐵)) = ((sgn‘𝐴) · 0))
19	18	adantl 483	. . . 4 ⊢ ((((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (𝐴 · 𝐵) = 0) ∧ 𝐵 = 0) → ((sgn‘𝐴) · (sgn‘𝐵)) = ((sgn‘𝐴) · 0))
20		sgnclre 32551	. . . . . . 7 ⊢ (𝐴 ∈ ℝ → (sgn‘𝐴) ∈ ℝ)
21	20	recnd 11049	. . . . . 6 ⊢ (𝐴 ∈ ℝ → (sgn‘𝐴) ∈ ℂ)
22	21	mul01d 11220	. . . . 5 ⊢ (𝐴 ∈ ℝ → ((sgn‘𝐴) · 0) = 0)
23	22	ad3antrrr 728	. . . 4 ⊢ ((((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (𝐴 · 𝐵) = 0) ∧ 𝐵 = 0) → ((sgn‘𝐴) · 0) = 0)
24	19, 23	eqtr2d 2777	. . 3 ⊢ ((((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (𝐴 · 𝐵) = 0) ∧ 𝐵 = 0) → 0 = ((sgn‘𝐴) · (sgn‘𝐵)))
25		simpl 484	. . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → 𝐴 ∈ ℝ)
26	25	recnd 11049	. . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → 𝐴 ∈ ℂ)
27		simpr 486	. . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → 𝐵 ∈ ℝ)
28	27	recnd 11049	. . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → 𝐵 ∈ ℂ)
29	26, 28	mul0ord 11671	. . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((𝐴 · 𝐵) = 0 ↔ (𝐴 = 0 ∨ 𝐵 = 0)))
30	29	biimpa 478	. . 3 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (𝐴 · 𝐵) = 0) → (𝐴 = 0 ∨ 𝐵 = 0))
31	15, 24, 30	mpjaodan 957	. 2 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (𝐴 · 𝐵) = 0) → 0 = ((sgn‘𝐴) · (sgn‘𝐵)))
32		simpll 765	. . . 4 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ 0 < (𝐴 · 𝐵)) → 𝐴 ∈ ℝ)
33	32	rexrd 11071	. . 3 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ 0 < (𝐴 · 𝐵)) → 𝐴 ∈ ℝ^*)
34		oveq1 7314	. . . 4 ⊢ ((sgn‘𝐴) = 0 → ((sgn‘𝐴) · (sgn‘𝐵)) = (0 · (sgn‘𝐵)))
35	34	eqeq2d 2747	. . 3 ⊢ ((sgn‘𝐴) = 0 → (1 = ((sgn‘𝐴) · (sgn‘𝐵)) ↔ 1 = (0 · (sgn‘𝐵))))
36		oveq1 7314	. . . 4 ⊢ ((sgn‘𝐴) = 1 → ((sgn‘𝐴) · (sgn‘𝐵)) = (1 · (sgn‘𝐵)))
37	36	eqeq2d 2747	. . 3 ⊢ ((sgn‘𝐴) = 1 → (1 = ((sgn‘𝐴) · (sgn‘𝐵)) ↔ 1 = (1 · (sgn‘𝐵))))
38		oveq1 7314	. . . 4 ⊢ ((sgn‘𝐴) = -1 → ((sgn‘𝐴) · (sgn‘𝐵)) = (-1 · (sgn‘𝐵)))
39	38	eqeq2d 2747	. . 3 ⊢ ((sgn‘𝐴) = -1 → (1 = ((sgn‘𝐴) · (sgn‘𝐵)) ↔ 1 = (-1 · (sgn‘𝐵))))
40		simpr 486	. . . 4 ⊢ ((((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ 0 < (𝐴 · 𝐵)) ∧ 𝐴 = 0) → 𝐴 = 0)
41	26	adantr 482	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ 0 < (𝐴 · 𝐵)) → 𝐴 ∈ ℂ)
42	28	adantr 482	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ 0 < (𝐴 · 𝐵)) → 𝐵 ∈ ℂ)
43		simpr 486	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ 0 < (𝐴 · 𝐵)) → 0 < (𝐴 · 𝐵))
44	43	gt0ne0d 11585	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ 0 < (𝐴 · 𝐵)) → (𝐴 · 𝐵) ≠ 0)
45	41, 42, 44	mulne0bad 11676	. . . . . 6 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ 0 < (𝐴 · 𝐵)) → 𝐴 ≠ 0)
46	45	neneqd 2946	. . . . 5 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ 0 < (𝐴 · 𝐵)) → ¬ 𝐴 = 0)
47	46	adantr 482	. . . 4 ⊢ ((((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ 0 < (𝐴 · 𝐵)) ∧ 𝐴 = 0) → ¬ 𝐴 = 0)
48	40, 47	pm2.21dd 194	. . 3 ⊢ ((((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ 0 < (𝐴 · 𝐵)) ∧ 𝐴 = 0) → 1 = (0 · (sgn‘𝐵)))
49	27	ad2antrr 724	. . . . . . 7 ⊢ ((((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ 0 < (𝐴 · 𝐵)) ∧ 0 < 𝐴) → 𝐵 ∈ ℝ)
50	49	rexrd 11071	. . . . . 6 ⊢ ((((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ 0 < (𝐴 · 𝐵)) ∧ 0 < 𝐴) → 𝐵 ∈ ℝ^*)
51		simpll 765	. . . . . . 7 ⊢ ((((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ 0 < (𝐴 · 𝐵)) ∧ 0 < 𝐴) → (𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ))
52		0red 11024	. . . . . . . 8 ⊢ ((((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ 0 < (𝐴 · 𝐵)) ∧ 0 < 𝐴) → 0 ∈ ℝ)
53		simplll 773	. . . . . . . 8 ⊢ ((((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ 0 < (𝐴 · 𝐵)) ∧ 0 < 𝐴) → 𝐴 ∈ ℝ)
54		simpr 486	. . . . . . . 8 ⊢ ((((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ 0 < (𝐴 · 𝐵)) ∧ 0 < 𝐴) → 0 < 𝐴)
55	52, 53, 54	ltled 11169	. . . . . . 7 ⊢ ((((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ 0 < (𝐴 · 𝐵)) ∧ 0 < 𝐴) → 0 ≤ 𝐴)
56		simplr 767	. . . . . . 7 ⊢ ((((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ 0 < (𝐴 · 𝐵)) ∧ 0 < 𝐴) → 0 < (𝐴 · 𝐵))
57		prodgt0 11868	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (0 ≤ 𝐴 ∧ 0 < (𝐴 · 𝐵))) → 0 < 𝐵)
58	51, 55, 56, 57	syl12anc 835	. . . . . 6 ⊢ ((((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ 0 < (𝐴 · 𝐵)) ∧ 0 < 𝐴) → 0 < 𝐵)
59		sgnp 14846	. . . . . 6 ⊢ ((𝐵 ∈ ℝ^* ∧ 0 < 𝐵) → (sgn‘𝐵) = 1)
60	50, 58, 59	syl2anc 585	. . . . 5 ⊢ ((((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ 0 < (𝐴 · 𝐵)) ∧ 0 < 𝐴) → (sgn‘𝐵) = 1)
61	60	oveq2d 7323	. . . 4 ⊢ ((((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ 0 < (𝐴 · 𝐵)) ∧ 0 < 𝐴) → (1 · (sgn‘𝐵)) = (1 · 1))
62		1t1e1 12181	. . . 4 ⊢ (1 · 1) = 1
63	61, 62	eqtr2di 2793	. . 3 ⊢ ((((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ 0 < (𝐴 · 𝐵)) ∧ 0 < 𝐴) → 1 = (1 · (sgn‘𝐵)))
64	27	ad2antrr 724	. . . . . . 7 ⊢ ((((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ 0 < (𝐴 · 𝐵)) ∧ 𝐴 < 0) → 𝐵 ∈ ℝ)
65	64	rexrd 11071	. . . . . 6 ⊢ ((((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ 0 < (𝐴 · 𝐵)) ∧ 𝐴 < 0) → 𝐵 ∈ ℝ^*)
66		simplll 773	. . . . . . . . 9 ⊢ ((((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ 0 < (𝐴 · 𝐵)) ∧ 𝐴 < 0) → 𝐴 ∈ ℝ)
67	66	renegcld 11448	. . . . . . . 8 ⊢ ((((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ 0 < (𝐴 · 𝐵)) ∧ 𝐴 < 0) → -𝐴 ∈ ℝ)
68	64	renegcld 11448	. . . . . . . 8 ⊢ ((((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ 0 < (𝐴 · 𝐵)) ∧ 𝐴 < 0) → -𝐵 ∈ ℝ)
69		0red 11024	. . . . . . . . 9 ⊢ ((((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ 0 < (𝐴 · 𝐵)) ∧ 𝐴 < 0) → 0 ∈ ℝ)
70		simpr 486	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ 0 < (𝐴 · 𝐵)) ∧ 𝐴 < 0) → 𝐴 < 0)
71	25	lt0neg1d 11590	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 < 0 ↔ 0 < -𝐴))
72	71	ad2antrr 724	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ 0 < (𝐴 · 𝐵)) ∧ 𝐴 < 0) → (𝐴 < 0 ↔ 0 < -𝐴))
73	70, 72	mpbid 231	. . . . . . . . 9 ⊢ ((((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ 0 < (𝐴 · 𝐵)) ∧ 𝐴 < 0) → 0 < -𝐴)
74	69, 67, 73	ltled 11169	. . . . . . . 8 ⊢ ((((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ 0 < (𝐴 · 𝐵)) ∧ 𝐴 < 0) → 0 ≤ -𝐴)
75		simplr 767	. . . . . . . . 9 ⊢ ((((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ 0 < (𝐴 · 𝐵)) ∧ 𝐴 < 0) → 0 < (𝐴 · 𝐵))
76	26	ad2antrr 724	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ 0 < (𝐴 · 𝐵)) ∧ 𝐴 < 0) → 𝐴 ∈ ℂ)
77	28	ad2antrr 724	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ 0 < (𝐴 · 𝐵)) ∧ 𝐴 < 0) → 𝐵 ∈ ℂ)
78	76, 77	mul2negd 11476	. . . . . . . . 9 ⊢ ((((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ 0 < (𝐴 · 𝐵)) ∧ 𝐴 < 0) → (-𝐴 · -𝐵) = (𝐴 · 𝐵))
79	75, 78	breqtrrd 5109	. . . . . . . 8 ⊢ ((((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ 0 < (𝐴 · 𝐵)) ∧ 𝐴 < 0) → 0 < (-𝐴 · -𝐵))
80		prodgt0 11868	. . . . . . . 8 ⊢ (((-𝐴 ∈ ℝ ∧ -𝐵 ∈ ℝ) ∧ (0 ≤ -𝐴 ∧ 0 < (-𝐴 · -𝐵))) → 0 < -𝐵)
81	67, 68, 74, 79, 80	syl22anc 837	. . . . . . 7 ⊢ ((((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ 0 < (𝐴 · 𝐵)) ∧ 𝐴 < 0) → 0 < -𝐵)
82	27	lt0neg1d 11590	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐵 < 0 ↔ 0 < -𝐵))
83	82	ad2antrr 724	. . . . . . 7 ⊢ ((((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ 0 < (𝐴 · 𝐵)) ∧ 𝐴 < 0) → (𝐵 < 0 ↔ 0 < -𝐵))
84	81, 83	mpbird 257	. . . . . 6 ⊢ ((((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ 0 < (𝐴 · 𝐵)) ∧ 𝐴 < 0) → 𝐵 < 0)
85		sgnn 14850	. . . . . 6 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐵 < 0) → (sgn‘𝐵) = -1)
86	65, 84, 85	syl2anc 585	. . . . 5 ⊢ ((((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ 0 < (𝐴 · 𝐵)) ∧ 𝐴 < 0) → (sgn‘𝐵) = -1)
87	86	oveq2d 7323	. . . 4 ⊢ ((((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ 0 < (𝐴 · 𝐵)) ∧ 𝐴 < 0) → (-1 · (sgn‘𝐵)) = (-1 · -1))
88		neg1mulneg1e1 12232	. . . 4 ⊢ (-1 · -1) = 1
89	87, 88	eqtr2di 2793	. . 3 ⊢ ((((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ 0 < (𝐴 · 𝐵)) ∧ 𝐴 < 0) → 1 = (-1 · (sgn‘𝐵)))
90	33, 35, 37, 39, 48, 63, 89	sgn3da 32553	. 2 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ 0 < (𝐴 · 𝐵)) → 1 = ((sgn‘𝐴) · (sgn‘𝐵)))
91		simpll 765	. . . 4 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (𝐴 · 𝐵) < 0) → 𝐴 ∈ ℝ)
92	91	rexrd 11071	. . 3 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (𝐴 · 𝐵) < 0) → 𝐴 ∈ ℝ^*)
93	34	eqeq2d 2747	. . 3 ⊢ ((sgn‘𝐴) = 0 → (-1 = ((sgn‘𝐴) · (sgn‘𝐵)) ↔ -1 = (0 · (sgn‘𝐵))))
94	36	eqeq2d 2747	. . 3 ⊢ ((sgn‘𝐴) = 1 → (-1 = ((sgn‘𝐴) · (sgn‘𝐵)) ↔ -1 = (1 · (sgn‘𝐵))))
95	38	eqeq2d 2747	. . 3 ⊢ ((sgn‘𝐴) = -1 → (-1 = ((sgn‘𝐴) · (sgn‘𝐵)) ↔ -1 = (-1 · (sgn‘𝐵))))
96		simpr 486	. . . 4 ⊢ ((((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (𝐴 · 𝐵) < 0) ∧ 𝐴 = 0) → 𝐴 = 0)
97	26	ad2antrr 724	. . . . . 6 ⊢ ((((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (𝐴 · 𝐵) < 0) ∧ 𝐴 = 0) → 𝐴 ∈ ℂ)
98	28	ad2antrr 724	. . . . . 6 ⊢ ((((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (𝐴 · 𝐵) < 0) ∧ 𝐴 = 0) → 𝐵 ∈ ℂ)
99		simplr 767	. . . . . . 7 ⊢ ((((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (𝐴 · 𝐵) < 0) ∧ 𝐴 = 0) → (𝐴 · 𝐵) < 0)
100	99	lt0ne0d 11586	. . . . . 6 ⊢ ((((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (𝐴 · 𝐵) < 0) ∧ 𝐴 = 0) → (𝐴 · 𝐵) ≠ 0)
101	97, 98, 100	mulne0bad 11676	. . . . 5 ⊢ ((((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (𝐴 · 𝐵) < 0) ∧ 𝐴 = 0) → 𝐴 ≠ 0)
102	101	neneqd 2946	. . . 4 ⊢ ((((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (𝐴 · 𝐵) < 0) ∧ 𝐴 = 0) → ¬ 𝐴 = 0)
103	96, 102	pm2.21dd 194	. . 3 ⊢ ((((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (𝐴 · 𝐵) < 0) ∧ 𝐴 = 0) → -1 = (0 · (sgn‘𝐵)))
104	27	ad2antrr 724	. . . . . . 7 ⊢ ((((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (𝐴 · 𝐵) < 0) ∧ 0 < 𝐴) → 𝐵 ∈ ℝ)
105	104	rexrd 11071	. . . . . 6 ⊢ ((((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (𝐴 · 𝐵) < 0) ∧ 0 < 𝐴) → 𝐵 ∈ ℝ^*)
106		simplr 767	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (𝐴 · 𝐵) < 0) → 𝐵 ∈ ℝ)
107	26, 28	mulcomd 11042	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 · 𝐵) = (𝐵 · 𝐴))
108	107	breq1d 5091	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((𝐴 · 𝐵) < 0 ↔ (𝐵 · 𝐴) < 0))
109	108	biimpa 478	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (𝐴 · 𝐵) < 0) → (𝐵 · 𝐴) < 0)
110	106, 91, 109	mul2lt0rgt0 12879	. . . . . 6 ⊢ ((((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (𝐴 · 𝐵) < 0) ∧ 0 < 𝐴) → 𝐵 < 0)
111	105, 110, 85	syl2anc 585	. . . . 5 ⊢ ((((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (𝐴 · 𝐵) < 0) ∧ 0 < 𝐴) → (sgn‘𝐵) = -1)
112	111	oveq2d 7323	. . . 4 ⊢ ((((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (𝐴 · 𝐵) < 0) ∧ 0 < 𝐴) → (1 · (sgn‘𝐵)) = (1 · -1))
113		neg1cn 12133	. . . . 5 ⊢ -1 ∈ ℂ
114	113	mulid2i 11026	. . . 4 ⊢ (1 · -1) = -1
115	112, 114	eqtr2di 2793	. . 3 ⊢ ((((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (𝐴 · 𝐵) < 0) ∧ 0 < 𝐴) → -1 = (1 · (sgn‘𝐵)))
116	106	adantr 482	. . . . . . 7 ⊢ ((((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (𝐴 · 𝐵) < 0) ∧ 𝐴 < 0) → 𝐵 ∈ ℝ)
117	116	rexrd 11071	. . . . . 6 ⊢ ((((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (𝐴 · 𝐵) < 0) ∧ 𝐴 < 0) → 𝐵 ∈ ℝ^*)
118	106, 91, 109	mul2lt0rlt0 12878	. . . . . 6 ⊢ ((((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (𝐴 · 𝐵) < 0) ∧ 𝐴 < 0) → 0 < 𝐵)
119	117, 118, 59	syl2anc 585	. . . . 5 ⊢ ((((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (𝐴 · 𝐵) < 0) ∧ 𝐴 < 0) → (sgn‘𝐵) = 1)
120	119	oveq2d 7323	. . . 4 ⊢ ((((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (𝐴 · 𝐵) < 0) ∧ 𝐴 < 0) → (-1 · (sgn‘𝐵)) = (-1 · 1))
121	113	mulid1i 11025	. . . 4 ⊢ (-1 · 1) = -1
122	120, 121	eqtr2di 2793	. . 3 ⊢ ((((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (𝐴 · 𝐵) < 0) ∧ 𝐴 < 0) → -1 = (-1 · (sgn‘𝐵)))
123	92, 93, 94, 95, 103, 115, 122	sgn3da 32553	. 2 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (𝐴 · 𝐵) < 0) → -1 = ((sgn‘𝐴) · (sgn‘𝐵)))
124	2, 3, 4, 5, 31, 90, 123	sgn3da 32553	1 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (sgn‘(𝐴 · 𝐵)) = ((sgn‘𝐴) · (sgn‘𝐵)))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 397 ∨ wo 845 = wceq 1539 ∈ wcel 2104 class class class wbr 5081 ‘cfv 6458 (class class class)co 7307 ℂcc 10915 ℝcr 10916 0cc0 10917 1c1 10918 · cmul 10922 ℝ^*cxr 11054 < clt 11055 ≤ cle 11056 -cneg 11252 sgncsgn 14842

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2707 ax-sep 5232 ax-nul 5239 ax-pow 5297 ax-pr 5361 ax-un 7620 ax-cnex 10973 ax-resscn 10974 ax-1cn 10975 ax-icn 10976 ax-addcl 10977 ax-addrcl 10978 ax-mulcl 10979 ax-mulrcl 10980 ax-mulcom 10981 ax-addass 10982 ax-mulass 10983 ax-distr 10984 ax-i2m1 10985 ax-1ne0 10986 ax-1rid 10987 ax-rnegex 10988 ax-rrecex 10989 ax-cnre 10990 ax-pre-lttri 10991 ax-pre-lttrn 10992 ax-pre-ltadd 10993 ax-pre-mulgt0 10994

This theorem depends on definitions: df-bi 206 df-an 398 df-or 846 df-3or 1088 df-3an 1089 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2887 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3285 df-reu 3286 df-rab 3287 df-v 3439 df-sbc 3722 df-csb 3838 df-dif 3895 df-un 3897 df-in 3899 df-ss 3909 df-nul 4263 df-if 4466 df-pw 4541 df-sn 4566 df-pr 4568 df-tp 4570 df-op 4572 df-uni 4845 df-br 5082 df-opab 5144 df-mpt 5165 df-id 5500 df-po 5514 df-so 5515 df-xp 5606 df-rel 5607 df-cnv 5608 df-co 5609 df-dm 5610 df-rn 5611 df-res 5612 df-ima 5613 df-iota 6410 df-fun 6460 df-fn 6461 df-f 6462 df-f1 6463 df-fo 6464 df-f1o 6465 df-fv 6466 df-riota 7264 df-ov 7310 df-oprab 7311 df-mpo 7312 df-er 8529 df-en 8765 df-dom 8766 df-sdom 8767 df-pnf 11057 df-mnf 11058 df-xr 11059 df-ltxr 11060 df-le 11061 df-sub 11253 df-neg 11254 df-div 11679 df-rp 12777 df-sgn 14843

This theorem is referenced by: sgnmulrp2 32555 sgnmulsgn 32561 sgnmulsgp 32562

W3C validator