xmullem2 - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > xmullem2		Structured version Visualization version GIF version

Theorem xmullem2 12928

Description: Lemma for xmulneg1 12932. (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 10962	. . . . . . . . . . . 12 ⊢ -∞ ≠ +∞
2		eqeq1 2742	. . . . . . . . . . . . 13 ⊢ (𝐴 = -∞ → (𝐴 = +∞ ↔ -∞ = +∞))
3	2	necon3bbid 2980	. . . . . . . . . . . 12 ⊢ (𝐴 = -∞ → (¬ 𝐴 = +∞ ↔ -∞ ≠ +∞))
4	1, 3	mpbiri 257	. . . . . . . . . . 11 ⊢ (𝐴 = -∞ → ¬ 𝐴 = +∞)
5	4	con2i 139	. . . . . . . . . 10 ⊢ (𝐴 = +∞ → ¬ 𝐴 = -∞)
6	5	adantl 481	. . . . . . . . 9 ⊢ ((0 < 𝐵 ∧ 𝐴 = +∞) → ¬ 𝐴 = -∞)
7		0xr 10953	. . . . . . . . . . . . 13 ⊢ 0 ∈ ℝ^*
8		nltmnf 12794	. . . . . . . . . . . . 13 ⊢ (0 ∈ ℝ^* → ¬ 0 < -∞)
9	7, 8	ax-mp 5	. . . . . . . . . . . 12 ⊢ ¬ 0 < -∞
10		breq2 5074	. . . . . . . . . . . 12 ⊢ (𝐴 = -∞ → (0 < 𝐴 ↔ 0 < -∞))
11	9, 10	mtbiri 326	. . . . . . . . . . 11 ⊢ (𝐴 = -∞ → ¬ 0 < 𝐴)
12	11	con2i 139	. . . . . . . . . 10 ⊢ (0 < 𝐴 → ¬ 𝐴 = -∞)
13	12	adantr 480	. . . . . . . . 9 ⊢ ((0 < 𝐴 ∧ 𝐵 = +∞) → ¬ 𝐴 = -∞)
14	6, 13	jaoi 853	. . . . . . . 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 484	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) → 𝐵 ∈ ℝ^)
17		xrltnsym 12800	. . . . . . . . . 10 ⊢ ((𝐵 ∈ ℝ^* ∧ 0 ∈ ℝ^*) → (𝐵 < 0 → ¬ 0 < 𝐵))
18	16, 7, 17	sylancl 585	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → (𝐵 < 0 → ¬ 0 < 𝐵))
19	18	adantrd 491	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → ((𝐵 < 0 ∧ 𝐴 = -∞) → ¬ 0 < 𝐵))
20		breq2 5074	. . . . . . . . . . 11 ⊢ (𝐵 = -∞ → (0 < 𝐵 ↔ 0 < -∞))
21	9, 20	mtbiri 326	. . . . . . . . . 10 ⊢ (𝐵 = -∞ → ¬ 0 < 𝐵)
22	21	adantl 481	. . . . . . . . 9 ⊢ ((𝐴 < 0 ∧ 𝐵 = -∞) → ¬ 0 < 𝐵)
23	22	a1i 11	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → ((𝐴 < 0 ∧ 𝐵 = -∞) → ¬ 0 < 𝐵))
24	19, 23	jaod 855	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → (((𝐵 < 0 ∧ 𝐴 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = -∞)) → ¬ 0 < 𝐵))
25	15, 24	orim12d 961	. . . . . 6 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → ((((0 < 𝐵 ∧ 𝐴 = +∞) ∨ (0 < 𝐴 ∧ 𝐵 = +∞)) ∨ ((𝐵 < 0 ∧ 𝐴 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = -∞))) → (¬ 𝐴 = -∞ ∨ ¬ 0 < 𝐵)))
26		ianor 978	. . . . . . 7 ⊢ (¬ (0 < 𝐵 ∧ 𝐴 = -∞) ↔ (¬ 0 < 𝐵 ∨ ¬ 𝐴 = -∞))
27		orcom 866	. . . . . . 7 ⊢ ((¬ 0 < 𝐵 ∨ ¬ 𝐴 = -∞) ↔ (¬ 𝐴 = -∞ ∨ ¬ 0 < 𝐵))
28	26, 27	bitri 274	. . . . . 6 ⊢ (¬ (0 < 𝐵 ∧ 𝐴 = -∞) ↔ (¬ 𝐴 = -∞ ∨ ¬ 0 < 𝐵))
29	25, 28	syl6ibr 251	. . . . 5 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → ((((0 < 𝐵 ∧ 𝐴 = +∞) ∨ (0 < 𝐴 ∧ 𝐵 = +∞)) ∨ ((𝐵 < 0 ∧ 𝐴 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = -∞))) → ¬ (0 < 𝐵 ∧ 𝐴 = -∞)))
30	18	con2d 134	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → (0 < 𝐵 → ¬ 𝐵 < 0))
31	30	adantrd 491	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → ((0 < 𝐵 ∧ 𝐴 = +∞) → ¬ 𝐵 < 0))
32		pnfnlt 12793	. . . . . . . . . . 11 ⊢ (0 ∈ ℝ^* → ¬ +∞ < 0)
33	7, 32	ax-mp 5	. . . . . . . . . 10 ⊢ ¬ +∞ < 0
34		simpr 484	. . . . . . . . . . 11 ⊢ ((0 < 𝐴 ∧ 𝐵 = +∞) → 𝐵 = +∞)
35	34	breq1d 5080	. . . . . . . . . 10 ⊢ ((0 < 𝐴 ∧ 𝐵 = +∞) → (𝐵 < 0 ↔ +∞ < 0))
36	33, 35	mtbiri 326	. . . . . . . . 9 ⊢ ((0 < 𝐴 ∧ 𝐵 = +∞) → ¬ 𝐵 < 0)
37	36	a1i 11	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → ((0 < 𝐴 ∧ 𝐵 = +∞) → ¬ 𝐵 < 0))
38	31, 37	jaod 855	. . . . . . 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 490	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → ((𝐵 < 0 ∧ 𝐴 = -∞) → ¬ 𝐴 = +∞))
41		breq1 5073	. . . . . . . . . . . 12 ⊢ (𝐴 = +∞ → (𝐴 < 0 ↔ +∞ < 0))
42	33, 41	mtbiri 326	. . . . . . . . . . 11 ⊢ (𝐴 = +∞ → ¬ 𝐴 < 0)
43	42	con2i 139	. . . . . . . . . 10 ⊢ (𝐴 < 0 → ¬ 𝐴 = +∞)
44	43	adantr 480	. . . . . . . . 9 ⊢ ((𝐴 < 0 ∧ 𝐵 = -∞) → ¬ 𝐴 = +∞)
45	44	a1i 11	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → ((𝐴 < 0 ∧ 𝐵 = -∞) → ¬ 𝐴 = +∞))
46	40, 45	jaod 855	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → (((𝐵 < 0 ∧ 𝐴 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = -∞)) → ¬ 𝐴 = +∞))
47	38, 46	orim12d 961	. . . . . 6 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → ((((0 < 𝐵 ∧ 𝐴 = +∞) ∨ (0 < 𝐴 ∧ 𝐵 = +∞)) ∨ ((𝐵 < 0 ∧ 𝐴 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = -∞))) → (¬ 𝐵 < 0 ∨ ¬ 𝐴 = +∞)))
48		ianor 978	. . . . . 6 ⊢ (¬ (𝐵 < 0 ∧ 𝐴 = +∞) ↔ (¬ 𝐵 < 0 ∨ ¬ 𝐴 = +∞))
49	47, 48	syl6ibr 251	. . . . 5 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → ((((0 < 𝐵 ∧ 𝐴 = +∞) ∨ (0 < 𝐴 ∧ 𝐵 = +∞)) ∨ ((𝐵 < 0 ∧ 𝐴 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = -∞))) → ¬ (𝐵 < 0 ∧ 𝐴 = +∞)))
50	29, 49	jcad 512	. . . 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 980	. . . 4 ⊢ (¬ ((0 < 𝐵 ∧ 𝐴 = -∞) ∨ (𝐵 < 0 ∧ 𝐴 = +∞)) ↔ (¬ (0 < 𝐵 ∧ 𝐴 = -∞) ∧ ¬ (𝐵 < 0 ∧ 𝐴 = +∞)))
52	50, 51	syl6ibr 251	. . 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 139	. . . . . . . . . 10 ⊢ (0 < 𝐵 → ¬ 𝐵 = -∞)
54	53	adantr 480	. . . . . . . . 9 ⊢ ((0 < 𝐵 ∧ 𝐴 = +∞) → ¬ 𝐵 = -∞)
55	54	a1i 11	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → ((0 < 𝐵 ∧ 𝐴 = +∞) → ¬ 𝐵 = -∞))
56		pnfnemnf 10961	. . . . . . . . . . 11 ⊢ +∞ ≠ -∞
57		eqeq1 2742	. . . . . . . . . . . 12 ⊢ (𝐵 = +∞ → (𝐵 = -∞ ↔ +∞ = -∞))
58	57	necon3bbid 2980	. . . . . . . . . . 11 ⊢ (𝐵 = +∞ → (¬ 𝐵 = -∞ ↔ +∞ ≠ -∞))
59	56, 58	mpbiri 257	. . . . . . . . . 10 ⊢ (𝐵 = +∞ → ¬ 𝐵 = -∞)
60	59	adantl 481	. . . . . . . . 9 ⊢ ((0 < 𝐴 ∧ 𝐵 = +∞) → ¬ 𝐵 = -∞)
61	60	a1i 11	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → ((0 < 𝐴 ∧ 𝐵 = +∞) → ¬ 𝐵 = -∞))
62	55, 61	jaod 855	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → (((0 < 𝐵 ∧ 𝐴 = +∞) ∨ (0 < 𝐴 ∧ 𝐵 = +∞)) → ¬ 𝐵 = -∞))
63	11	adantl 481	. . . . . . . . 9 ⊢ ((𝐵 < 0 ∧ 𝐴 = -∞) → ¬ 0 < 𝐴)
64	63	a1i 11	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → ((𝐵 < 0 ∧ 𝐴 = -∞) → ¬ 0 < 𝐴))
65		simpl 482	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) → 𝐴 ∈ ℝ^)
66		xrltnsym 12800	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ^* ∧ 0 ∈ ℝ^*) → (𝐴 < 0 → ¬ 0 < 𝐴))
67	65, 7, 66	sylancl 585	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → (𝐴 < 0 → ¬ 0 < 𝐴))
68	67	adantrd 491	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → ((𝐴 < 0 ∧ 𝐵 = -∞) → ¬ 0 < 𝐴))
69	64, 68	jaod 855	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → (((𝐵 < 0 ∧ 𝐴 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = -∞)) → ¬ 0 < 𝐴))
70	62, 69	orim12d 961	. . . . . 6 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → ((((0 < 𝐵 ∧ 𝐴 = +∞) ∨ (0 < 𝐴 ∧ 𝐵 = +∞)) ∨ ((𝐵 < 0 ∧ 𝐴 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = -∞))) → (¬ 𝐵 = -∞ ∨ ¬ 0 < 𝐴)))
71		ianor 978	. . . . . . 7 ⊢ (¬ (0 < 𝐴 ∧ 𝐵 = -∞) ↔ (¬ 0 < 𝐴 ∨ ¬ 𝐵 = -∞))
72		orcom 866	. . . . . . 7 ⊢ ((¬ 0 < 𝐴 ∨ ¬ 𝐵 = -∞) ↔ (¬ 𝐵 = -∞ ∨ ¬ 0 < 𝐴))
73	71, 72	bitri 274	. . . . . 6 ⊢ (¬ (0 < 𝐴 ∧ 𝐵 = -∞) ↔ (¬ 𝐵 = -∞ ∨ ¬ 0 < 𝐴))
74	70, 73	syl6ibr 251	. . . . 5 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → ((((0 < 𝐵 ∧ 𝐴 = +∞) ∨ (0 < 𝐴 ∧ 𝐵 = +∞)) ∨ ((𝐵 < 0 ∧ 𝐴 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = -∞))) → ¬ (0 < 𝐴 ∧ 𝐵 = -∞)))
75	42	adantl 481	. . . . . . . . 9 ⊢ ((0 < 𝐵 ∧ 𝐴 = +∞) → ¬ 𝐴 < 0)
76	75	a1i 11	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → ((0 < 𝐵 ∧ 𝐴 = +∞) → ¬ 𝐴 < 0))
77	67	con2d 134	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → (0 < 𝐴 → ¬ 𝐴 < 0))
78	77	adantrd 491	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → ((0 < 𝐴 ∧ 𝐵 = +∞) → ¬ 𝐴 < 0))
79	76, 78	jaod 855	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → (((0 < 𝐵 ∧ 𝐴 = +∞) ∨ (0 < 𝐴 ∧ 𝐵 = +∞)) → ¬ 𝐴 < 0))
80		breq1 5073	. . . . . . . . . . . 12 ⊢ (𝐵 = +∞ → (𝐵 < 0 ↔ +∞ < 0))
81	33, 80	mtbiri 326	. . . . . . . . . . 11 ⊢ (𝐵 = +∞ → ¬ 𝐵 < 0)
82	81	con2i 139	. . . . . . . . . 10 ⊢ (𝐵 < 0 → ¬ 𝐵 = +∞)
83	82	adantr 480	. . . . . . . . 9 ⊢ ((𝐵 < 0 ∧ 𝐴 = -∞) → ¬ 𝐵 = +∞)
84	59	con2i 139	. . . . . . . . . 10 ⊢ (𝐵 = -∞ → ¬ 𝐵 = +∞)
85	84	adantl 481	. . . . . . . . 9 ⊢ ((𝐴 < 0 ∧ 𝐵 = -∞) → ¬ 𝐵 = +∞)
86	83, 85	jaoi 853	. . . . . . . 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 961	. . . . . 6 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → ((((0 < 𝐵 ∧ 𝐴 = +∞) ∨ (0 < 𝐴 ∧ 𝐵 = +∞)) ∨ ((𝐵 < 0 ∧ 𝐴 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = -∞))) → (¬ 𝐴 < 0 ∨ ¬ 𝐵 = +∞)))
89		ianor 978	. . . . . 6 ⊢ (¬ (𝐴 < 0 ∧ 𝐵 = +∞) ↔ (¬ 𝐴 < 0 ∨ ¬ 𝐵 = +∞))
90	88, 89	syl6ibr 251	. . . . 5 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → ((((0 < 𝐵 ∧ 𝐴 = +∞) ∨ (0 < 𝐴 ∧ 𝐵 = +∞)) ∨ ((𝐵 < 0 ∧ 𝐴 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = -∞))) → ¬ (𝐴 < 0 ∧ 𝐵 = +∞)))
91	74, 90	jcad 512	. . . 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 980	. . . 4 ⊢ (¬ ((0 < 𝐴 ∧ 𝐵 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = +∞)) ↔ (¬ (0 < 𝐴 ∧ 𝐵 = -∞) ∧ ¬ (𝐴 < 0 ∧ 𝐵 = +∞)))
93	91, 92	syl6ibr 251	. . 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 512	. 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 923	. 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 980	. 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 295	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 ∧ wa 395 ∨ wo 843 = wceq 1539 ∈ wcel 2108 ≠ wne 2942 class class class wbr 5070 0cc0 10802 +∞cpnf 10937 -∞cmnf 10938 ℝ^*cxr 10939 < clt 10940

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1799 ax-4 1813 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2156 ax-12 2173 ax-ext 2709 ax-sep 5218 ax-nul 5225 ax-pow 5283 ax-pr 5347 ax-un 7566 ax-cnex 10858 ax-resscn 10859 ax-1cn 10860 ax-addrcl 10863 ax-rnegex 10873 ax-cnre 10875 ax-pre-lttri 10876 ax-pre-lttrn 10877

This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1784 df-nf 1788 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2888 df-ne 2943 df-nel 3049 df-ral 3068 df-rex 3069 df-rab 3072 df-v 3424 df-sbc 3712 df-csb 3829 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-nul 4254 df-if 4457 df-pw 4532 df-sn 4559 df-pr 4561 df-op 4565 df-uni 4837 df-br 5071 df-opab 5133 df-mpt 5154 df-id 5480 df-po 5494 df-so 5495 df-xp 5586 df-rel 5587 df-cnv 5588 df-co 5589 df-dm 5590 df-rn 5591 df-res 5592 df-ima 5593 df-iota 6376 df-fun 6420 df-fn 6421 df-f 6422 df-f1 6423 df-fo 6424 df-f1o 6425 df-fv 6426 df-er 8456 df-en 8692 df-dom 8693 df-sdom 8694 df-pnf 10942 df-mnf 10943 df-xr 10944 df-ltxr 10945

This theorem is referenced by: xmulneg1 12932

W3C validator