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Theorem xmullem2 12095

Description: Lemma for xmulneg1 12099. (Contributed by Mario Carneiro, 20-Aug-2015.)

**Assertion**
Ref	Expression
xmullem2	⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → ((((0 < 𝐵 ∧ 𝐴 = +∞) ∨ (𝐵 < 0 ∧ 𝐴 = -∞)) ∨ ((0 < 𝐴 ∧ 𝐵 = +∞) ∨ (𝐴 < 0 ∧ 𝐵 = -∞))) → ¬ (((0 < 𝐵 ∧ 𝐴 = -∞) ∨ (𝐵 < 0 ∧ 𝐴 = +∞)) ∨ ((0 < 𝐴 ∧ 𝐵 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = +∞)))))

**Proof of Theorem xmullem2**
Step	Hyp	Ref	Expression
1		mnfnepnf 10095	. . . . . . . . . . . 12 ⊢ -∞ ≠ +∞
2		eqeq1 2626	. . . . . . . . . . . . 13 ⊢ (𝐴 = -∞ → (𝐴 = +∞ ↔ -∞ = +∞))
3	2	necon3bbid 2831	. . . . . . . . . . . 12 ⊢ (𝐴 = -∞ → (¬ 𝐴 = +∞ ↔ -∞ ≠ +∞))
4	1, 3	mpbiri 248	. . . . . . . . . . 11 ⊢ (𝐴 = -∞ → ¬ 𝐴 = +∞)
5	4	con2i 134	. . . . . . . . . 10 ⊢ (𝐴 = +∞ → ¬ 𝐴 = -∞)
6	5	adantl 482	. . . . . . . . 9 ⊢ ((0 < 𝐵 ∧ 𝐴 = +∞) → ¬ 𝐴 = -∞)
7		0xr 10086	. . . . . . . . . . . . 13 ⊢ 0 ∈ ℝ^*
8		nltmnf 11963	. . . . . . . . . . . . 13 ⊢ (0 ∈ ℝ^* → ¬ 0 < -∞)
9	7, 8	ax-mp 5	. . . . . . . . . . . 12 ⊢ ¬ 0 < -∞
10		breq2 4657	. . . . . . . . . . . 12 ⊢ (𝐴 = -∞ → (0 < 𝐴 ↔ 0 < -∞))
11	9, 10	mtbiri 317	. . . . . . . . . . 11 ⊢ (𝐴 = -∞ → ¬ 0 < 𝐴)
12	11	con2i 134	. . . . . . . . . 10 ⊢ (0 < 𝐴 → ¬ 𝐴 = -∞)
13	12	adantr 481	. . . . . . . . 9 ⊢ ((0 < 𝐴 ∧ 𝐵 = +∞) → ¬ 𝐴 = -∞)
14	6, 13	jaoi 394	. . . . . . . 8 ⊢ (((0 < 𝐵 ∧ 𝐴 = +∞) ∨ (0 < 𝐴 ∧ 𝐵 = +∞)) → ¬ 𝐴 = -∞)
15	14	a1i 11	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → (((0 < 𝐵 ∧ 𝐴 = +∞) ∨ (0 < 𝐴 ∧ 𝐵 = +∞)) → ¬ 𝐴 = -∞))
16		simpr 477	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) → 𝐵 ∈ ℝ^)
17		xrltnsym 11970	. . . . . . . . . 10 ⊢ ((𝐵 ∈ ℝ^* ∧ 0 ∈ ℝ^*) → (𝐵 < 0 → ¬ 0 < 𝐵))
18	16, 7, 17	sylancl 694	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → (𝐵 < 0 → ¬ 0 < 𝐵))
19	18	adantrd 484	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → ((𝐵 < 0 ∧ 𝐴 = -∞) → ¬ 0 < 𝐵))
20		breq2 4657	. . . . . . . . . . 11 ⊢ (𝐵 = -∞ → (0 < 𝐵 ↔ 0 < -∞))
21	9, 20	mtbiri 317	. . . . . . . . . 10 ⊢ (𝐵 = -∞ → ¬ 0 < 𝐵)
22	21	adantl 482	. . . . . . . . 9 ⊢ ((𝐴 < 0 ∧ 𝐵 = -∞) → ¬ 0 < 𝐵)
23	22	a1i 11	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → ((𝐴 < 0 ∧ 𝐵 = -∞) → ¬ 0 < 𝐵))
24	19, 23	jaod 395	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → (((𝐵 < 0 ∧ 𝐴 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = -∞)) → ¬ 0 < 𝐵))
25	15, 24	orim12d 883	. . . . . 6 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → ((((0 < 𝐵 ∧ 𝐴 = +∞) ∨ (0 < 𝐴 ∧ 𝐵 = +∞)) ∨ ((𝐵 < 0 ∧ 𝐴 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = -∞))) → (¬ 𝐴 = -∞ ∨ ¬ 0 < 𝐵)))
26		ianor 509	. . . . . . 7 ⊢ (¬ (0 < 𝐵 ∧ 𝐴 = -∞) ↔ (¬ 0 < 𝐵 ∨ ¬ 𝐴 = -∞))
27		orcom 402	. . . . . . 7 ⊢ ((¬ 0 < 𝐵 ∨ ¬ 𝐴 = -∞) ↔ (¬ 𝐴 = -∞ ∨ ¬ 0 < 𝐵))
28	26, 27	bitri 264	. . . . . 6 ⊢ (¬ (0 < 𝐵 ∧ 𝐴 = -∞) ↔ (¬ 𝐴 = -∞ ∨ ¬ 0 < 𝐵))
29	25, 28	syl6ibr 242	. . . . 5 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → ((((0 < 𝐵 ∧ 𝐴 = +∞) ∨ (0 < 𝐴 ∧ 𝐵 = +∞)) ∨ ((𝐵 < 0 ∧ 𝐴 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = -∞))) → ¬ (0 < 𝐵 ∧ 𝐴 = -∞)))
30	18	con2d 129	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → (0 < 𝐵 → ¬ 𝐵 < 0))
31	30	adantrd 484	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → ((0 < 𝐵 ∧ 𝐴 = +∞) → ¬ 𝐵 < 0))
32		pnfnlt 11962	. . . . . . . . . . 11 ⊢ (0 ∈ ℝ^* → ¬ +∞ < 0)
33	7, 32	ax-mp 5	. . . . . . . . . 10 ⊢ ¬ +∞ < 0
34		simpr 477	. . . . . . . . . . 11 ⊢ ((0 < 𝐴 ∧ 𝐵 = +∞) → 𝐵 = +∞)
35	34	breq1d 4663	. . . . . . . . . 10 ⊢ ((0 < 𝐴 ∧ 𝐵 = +∞) → (𝐵 < 0 ↔ +∞ < 0))
36	33, 35	mtbiri 317	. . . . . . . . 9 ⊢ ((0 < 𝐴 ∧ 𝐵 = +∞) → ¬ 𝐵 < 0)
37	36	a1i 11	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → ((0 < 𝐴 ∧ 𝐵 = +∞) → ¬ 𝐵 < 0))
38	31, 37	jaod 395	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → (((0 < 𝐵 ∧ 𝐴 = +∞) ∨ (0 < 𝐴 ∧ 𝐵 = +∞)) → ¬ 𝐵 < 0))
39	4	a1i 11	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → (𝐴 = -∞ → ¬ 𝐴 = +∞))
40	39	adantld 483	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → ((𝐵 < 0 ∧ 𝐴 = -∞) → ¬ 𝐴 = +∞))
41		breq1 4656	. . . . . . . . . . . 12 ⊢ (𝐴 = +∞ → (𝐴 < 0 ↔ +∞ < 0))
42	33, 41	mtbiri 317	. . . . . . . . . . 11 ⊢ (𝐴 = +∞ → ¬ 𝐴 < 0)
43	42	con2i 134	. . . . . . . . . 10 ⊢ (𝐴 < 0 → ¬ 𝐴 = +∞)
44	43	adantr 481	. . . . . . . . 9 ⊢ ((𝐴 < 0 ∧ 𝐵 = -∞) → ¬ 𝐴 = +∞)
45	44	a1i 11	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → ((𝐴 < 0 ∧ 𝐵 = -∞) → ¬ 𝐴 = +∞))
46	40, 45	jaod 395	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → (((𝐵 < 0 ∧ 𝐴 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = -∞)) → ¬ 𝐴 = +∞))
47	38, 46	orim12d 883	. . . . . 6 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → ((((0 < 𝐵 ∧ 𝐴 = +∞) ∨ (0 < 𝐴 ∧ 𝐵 = +∞)) ∨ ((𝐵 < 0 ∧ 𝐴 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = -∞))) → (¬ 𝐵 < 0 ∨ ¬ 𝐴 = +∞)))
48		ianor 509	. . . . . 6 ⊢ (¬ (𝐵 < 0 ∧ 𝐴 = +∞) ↔ (¬ 𝐵 < 0 ∨ ¬ 𝐴 = +∞))
49	47, 48	syl6ibr 242	. . . . 5 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → ((((0 < 𝐵 ∧ 𝐴 = +∞) ∨ (0 < 𝐴 ∧ 𝐵 = +∞)) ∨ ((𝐵 < 0 ∧ 𝐴 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = -∞))) → ¬ (𝐵 < 0 ∧ 𝐴 = +∞)))
50	29, 49	jcad 555	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → ((((0 < 𝐵 ∧ 𝐴 = +∞) ∨ (0 < 𝐴 ∧ 𝐵 = +∞)) ∨ ((𝐵 < 0 ∧ 𝐴 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = -∞))) → (¬ (0 < 𝐵 ∧ 𝐴 = -∞) ∧ ¬ (𝐵 < 0 ∧ 𝐴 = +∞))))
51		ioran 511	. . . 4 ⊢ (¬ ((0 < 𝐵 ∧ 𝐴 = -∞) ∨ (𝐵 < 0 ∧ 𝐴 = +∞)) ↔ (¬ (0 < 𝐵 ∧ 𝐴 = -∞) ∧ ¬ (𝐵 < 0 ∧ 𝐴 = +∞)))
52	50, 51	syl6ibr 242	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → ((((0 < 𝐵 ∧ 𝐴 = +∞) ∨ (0 < 𝐴 ∧ 𝐵 = +∞)) ∨ ((𝐵 < 0 ∧ 𝐴 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = -∞))) → ¬ ((0 < 𝐵 ∧ 𝐴 = -∞) ∨ (𝐵 < 0 ∧ 𝐴 = +∞))))
53	21	con2i 134	. . . . . . . . . 10 ⊢ (0 < 𝐵 → ¬ 𝐵 = -∞)
54	53	adantr 481	. . . . . . . . 9 ⊢ ((0 < 𝐵 ∧ 𝐴 = +∞) → ¬ 𝐵 = -∞)
55	54	a1i 11	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → ((0 < 𝐵 ∧ 𝐴 = +∞) → ¬ 𝐵 = -∞))
56		pnfnemnf 10094	. . . . . . . . . . 11 ⊢ +∞ ≠ -∞
57		eqeq1 2626	. . . . . . . . . . . 12 ⊢ (𝐵 = +∞ → (𝐵 = -∞ ↔ +∞ = -∞))
58	57	necon3bbid 2831	. . . . . . . . . . 11 ⊢ (𝐵 = +∞ → (¬ 𝐵 = -∞ ↔ +∞ ≠ -∞))
59	56, 58	mpbiri 248	. . . . . . . . . 10 ⊢ (𝐵 = +∞ → ¬ 𝐵 = -∞)
60	59	adantl 482	. . . . . . . . 9 ⊢ ((0 < 𝐴 ∧ 𝐵 = +∞) → ¬ 𝐵 = -∞)
61	60	a1i 11	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → ((0 < 𝐴 ∧ 𝐵 = +∞) → ¬ 𝐵 = -∞))
62	55, 61	jaod 395	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → (((0 < 𝐵 ∧ 𝐴 = +∞) ∨ (0 < 𝐴 ∧ 𝐵 = +∞)) → ¬ 𝐵 = -∞))
63	11	adantl 482	. . . . . . . . 9 ⊢ ((𝐵 < 0 ∧ 𝐴 = -∞) → ¬ 0 < 𝐴)
64	63	a1i 11	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → ((𝐵 < 0 ∧ 𝐴 = -∞) → ¬ 0 < 𝐴))
65		simpl 473	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) → 𝐴 ∈ ℝ^)
66		xrltnsym 11970	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ^* ∧ 0 ∈ ℝ^*) → (𝐴 < 0 → ¬ 0 < 𝐴))
67	65, 7, 66	sylancl 694	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → (𝐴 < 0 → ¬ 0 < 𝐴))
68	67	adantrd 484	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → ((𝐴 < 0 ∧ 𝐵 = -∞) → ¬ 0 < 𝐴))
69	64, 68	jaod 395	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → (((𝐵 < 0 ∧ 𝐴 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = -∞)) → ¬ 0 < 𝐴))
70	62, 69	orim12d 883	. . . . . 6 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → ((((0 < 𝐵 ∧ 𝐴 = +∞) ∨ (0 < 𝐴 ∧ 𝐵 = +∞)) ∨ ((𝐵 < 0 ∧ 𝐴 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = -∞))) → (¬ 𝐵 = -∞ ∨ ¬ 0 < 𝐴)))
71		ianor 509	. . . . . . 7 ⊢ (¬ (0 < 𝐴 ∧ 𝐵 = -∞) ↔ (¬ 0 < 𝐴 ∨ ¬ 𝐵 = -∞))
72		orcom 402	. . . . . . 7 ⊢ ((¬ 0 < 𝐴 ∨ ¬ 𝐵 = -∞) ↔ (¬ 𝐵 = -∞ ∨ ¬ 0 < 𝐴))
73	71, 72	bitri 264	. . . . . 6 ⊢ (¬ (0 < 𝐴 ∧ 𝐵 = -∞) ↔ (¬ 𝐵 = -∞ ∨ ¬ 0 < 𝐴))
74	70, 73	syl6ibr 242	. . . . 5 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → ((((0 < 𝐵 ∧ 𝐴 = +∞) ∨ (0 < 𝐴 ∧ 𝐵 = +∞)) ∨ ((𝐵 < 0 ∧ 𝐴 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = -∞))) → ¬ (0 < 𝐴 ∧ 𝐵 = -∞)))
75	42	adantl 482	. . . . . . . . 9 ⊢ ((0 < 𝐵 ∧ 𝐴 = +∞) → ¬ 𝐴 < 0)
76	75	a1i 11	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → ((0 < 𝐵 ∧ 𝐴 = +∞) → ¬ 𝐴 < 0))
77	67	con2d 129	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → (0 < 𝐴 → ¬ 𝐴 < 0))
78	77	adantrd 484	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → ((0 < 𝐴 ∧ 𝐵 = +∞) → ¬ 𝐴 < 0))
79	76, 78	jaod 395	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → (((0 < 𝐵 ∧ 𝐴 = +∞) ∨ (0 < 𝐴 ∧ 𝐵 = +∞)) → ¬ 𝐴 < 0))
80		breq1 4656	. . . . . . . . . . . 12 ⊢ (𝐵 = +∞ → (𝐵 < 0 ↔ +∞ < 0))
81	33, 80	mtbiri 317	. . . . . . . . . . 11 ⊢ (𝐵 = +∞ → ¬ 𝐵 < 0)
82	81	con2i 134	. . . . . . . . . 10 ⊢ (𝐵 < 0 → ¬ 𝐵 = +∞)
83	82	adantr 481	. . . . . . . . 9 ⊢ ((𝐵 < 0 ∧ 𝐴 = -∞) → ¬ 𝐵 = +∞)
84	59	con2i 134	. . . . . . . . . 10 ⊢ (𝐵 = -∞ → ¬ 𝐵 = +∞)
85	84	adantl 482	. . . . . . . . 9 ⊢ ((𝐴 < 0 ∧ 𝐵 = -∞) → ¬ 𝐵 = +∞)
86	83, 85	jaoi 394	. . . . . . . 8 ⊢ (((𝐵 < 0 ∧ 𝐴 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = -∞)) → ¬ 𝐵 = +∞)
87	86	a1i 11	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → (((𝐵 < 0 ∧ 𝐴 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = -∞)) → ¬ 𝐵 = +∞))
88	79, 87	orim12d 883	. . . . . 6 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → ((((0 < 𝐵 ∧ 𝐴 = +∞) ∨ (0 < 𝐴 ∧ 𝐵 = +∞)) ∨ ((𝐵 < 0 ∧ 𝐴 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = -∞))) → (¬ 𝐴 < 0 ∨ ¬ 𝐵 = +∞)))
89		ianor 509	. . . . . 6 ⊢ (¬ (𝐴 < 0 ∧ 𝐵 = +∞) ↔ (¬ 𝐴 < 0 ∨ ¬ 𝐵 = +∞))
90	88, 89	syl6ibr 242	. . . . 5 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → ((((0 < 𝐵 ∧ 𝐴 = +∞) ∨ (0 < 𝐴 ∧ 𝐵 = +∞)) ∨ ((𝐵 < 0 ∧ 𝐴 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = -∞))) → ¬ (𝐴 < 0 ∧ 𝐵 = +∞)))
91	74, 90	jcad 555	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → ((((0 < 𝐵 ∧ 𝐴 = +∞) ∨ (0 < 𝐴 ∧ 𝐵 = +∞)) ∨ ((𝐵 < 0 ∧ 𝐴 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = -∞))) → (¬ (0 < 𝐴 ∧ 𝐵 = -∞) ∧ ¬ (𝐴 < 0 ∧ 𝐵 = +∞))))
92		ioran 511	. . . 4 ⊢ (¬ ((0 < 𝐴 ∧ 𝐵 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = +∞)) ↔ (¬ (0 < 𝐴 ∧ 𝐵 = -∞) ∧ ¬ (𝐴 < 0 ∧ 𝐵 = +∞)))
93	91, 92	syl6ibr 242	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → ((((0 < 𝐵 ∧ 𝐴 = +∞) ∨ (0 < 𝐴 ∧ 𝐵 = +∞)) ∨ ((𝐵 < 0 ∧ 𝐴 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = -∞))) → ¬ ((0 < 𝐴 ∧ 𝐵 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = +∞))))
94	52, 93	jcad 555	. 2 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → ((((0 < 𝐵 ∧ 𝐴 = +∞) ∨ (0 < 𝐴 ∧ 𝐵 = +∞)) ∨ ((𝐵 < 0 ∧ 𝐴 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = -∞))) → (¬ ((0 < 𝐵 ∧ 𝐴 = -∞) ∨ (𝐵 < 0 ∧ 𝐴 = +∞)) ∧ ¬ ((0 < 𝐴 ∧ 𝐵 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = +∞)))))
95		or4 550	. 2 ⊢ ((((0 < 𝐵 ∧ 𝐴 = +∞) ∨ (𝐵 < 0 ∧ 𝐴 = -∞)) ∨ ((0 < 𝐴 ∧ 𝐵 = +∞) ∨ (𝐴 < 0 ∧ 𝐵 = -∞))) ↔ (((0 < 𝐵 ∧ 𝐴 = +∞) ∨ (0 < 𝐴 ∧ 𝐵 = +∞)) ∨ ((𝐵 < 0 ∧ 𝐴 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = -∞))))
96		ioran 511	. 2 ⊢ (¬ (((0 < 𝐵 ∧ 𝐴 = -∞) ∨ (𝐵 < 0 ∧ 𝐴 = +∞)) ∨ ((0 < 𝐴 ∧ 𝐵 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = +∞))) ↔ (¬ ((0 < 𝐵 ∧ 𝐴 = -∞) ∨ (𝐵 < 0 ∧ 𝐴 = +∞)) ∧ ¬ ((0 < 𝐴 ∧ 𝐵 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = +∞))))
97	94, 95, 96	3imtr4g 285	1 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → ((((0 < 𝐵 ∧ 𝐴 = +∞) ∨ (𝐵 < 0 ∧ 𝐴 = -∞)) ∨ ((0 < 𝐴 ∧ 𝐵 = +∞) ∨ (𝐴 < 0 ∧ 𝐵 = -∞))) → ¬ (((0 < 𝐵 ∧ 𝐴 = -∞) ∨ (𝐵 < 0 ∧ 𝐴 = +∞)) ∨ ((0 < 𝐴 ∧ 𝐵 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = +∞)))))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∨ wo 383 ∧ wa 384 = wceq 1483 ∈ wcel 1990 ≠ wne 2794 class class class wbr 4653 0cc0 9936 +∞cpnf 10071 -∞cmnf 10072 ℝ^*cxr 10073 < clt 10074

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-i2m1 10004 ax-1ne0 10005 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-po 5035 df-so 5036 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-ov 6653 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079

This theorem is referenced by: xmulneg1 12099

W3C validator