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 13243

Description: Lemma for xmulneg1 13247. (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 11269	. . . . . . . . . . . 12 ⊢ -∞ ≠ +∞
2		eqeq1 2736	. . . . . . . . . . . . 13 ⊢ (𝐴 = -∞ → (𝐴 = +∞ ↔ -∞ = +∞))
3	2	necon3bbid 2978	. . . . . . . . . . . 12 ⊢ (𝐴 = -∞ → (¬ 𝐴 = +∞ ↔ -∞ ≠ +∞))
4	1, 3	mpbiri 257	. . . . . . . . . . 11 ⊢ (𝐴 = -∞ → ¬ 𝐴 = +∞)
5	4	con2i 139	. . . . . . . . . 10 ⊢ (𝐴 = +∞ → ¬ 𝐴 = -∞)
6	5	adantl 482	. . . . . . . . 9 ⊢ ((0 < 𝐵 ∧ 𝐴 = +∞) → ¬ 𝐴 = -∞)
7		0xr 11260	. . . . . . . . . . . . 13 ⊢ 0 ∈ ℝ^*
8		nltmnf 13108	. . . . . . . . . . . . 13 ⊢ (0 ∈ ℝ^* → ¬ 0 < -∞)
9	7, 8	ax-mp 5	. . . . . . . . . . . 12 ⊢ ¬ 0 < -∞
10		breq2 5152	. . . . . . . . . . . 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 481	. . . . . . . . 9 ⊢ ((0 < 𝐴 ∧ 𝐵 = +∞) → ¬ 𝐴 = -∞)
14	6, 13	jaoi 855	. . . . . . . 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 485	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) → 𝐵 ∈ ℝ^)
17		xrltnsym 13115	. . . . . . . . . 10 ⊢ ((𝐵 ∈ ℝ^* ∧ 0 ∈ ℝ^*) → (𝐵 < 0 → ¬ 0 < 𝐵))
18	16, 7, 17	sylancl 586	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → (𝐵 < 0 → ¬ 0 < 𝐵))
19	18	adantrd 492	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → ((𝐵 < 0 ∧ 𝐴 = -∞) → ¬ 0 < 𝐵))
20		breq2 5152	. . . . . . . . . . 11 ⊢ (𝐵 = -∞ → (0 < 𝐵 ↔ 0 < -∞))
21	9, 20	mtbiri 326	. . . . . . . . . 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 857	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → (((𝐵 < 0 ∧ 𝐴 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = -∞)) → ¬ 0 < 𝐵))
25	15, 24	orim12d 963	. . . . . 6 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → ((((0 < 𝐵 ∧ 𝐴 = +∞) ∨ (0 < 𝐴 ∧ 𝐵 = +∞)) ∨ ((𝐵 < 0 ∧ 𝐴 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = -∞))) → (¬ 𝐴 = -∞ ∨ ¬ 0 < 𝐵)))
26		ianor 980	. . . . . . 7 ⊢ (¬ (0 < 𝐵 ∧ 𝐴 = -∞) ↔ (¬ 0 < 𝐵 ∨ ¬ 𝐴 = -∞))
27		orcom 868	. . . . . . 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 492	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → ((0 < 𝐵 ∧ 𝐴 = +∞) → ¬ 𝐵 < 0))
32		pnfnlt 13107	. . . . . . . . . . 11 ⊢ (0 ∈ ℝ^* → ¬ +∞ < 0)
33	7, 32	ax-mp 5	. . . . . . . . . 10 ⊢ ¬ +∞ < 0
34		simpr 485	. . . . . . . . . . 11 ⊢ ((0 < 𝐴 ∧ 𝐵 = +∞) → 𝐵 = +∞)
35	34	breq1d 5158	. . . . . . . . . 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 857	. . . . . . 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 491	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → ((𝐵 < 0 ∧ 𝐴 = -∞) → ¬ 𝐴 = +∞))
41		breq1 5151	. . . . . . . . . . . 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 481	. . . . . . . . 9 ⊢ ((𝐴 < 0 ∧ 𝐵 = -∞) → ¬ 𝐴 = +∞)
45	44	a1i 11	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → ((𝐴 < 0 ∧ 𝐵 = -∞) → ¬ 𝐴 = +∞))
46	40, 45	jaod 857	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → (((𝐵 < 0 ∧ 𝐴 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = -∞)) → ¬ 𝐴 = +∞))
47	38, 46	orim12d 963	. . . . . 6 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → ((((0 < 𝐵 ∧ 𝐴 = +∞) ∨ (0 < 𝐴 ∧ 𝐵 = +∞)) ∨ ((𝐵 < 0 ∧ 𝐴 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = -∞))) → (¬ 𝐵 < 0 ∨ ¬ 𝐴 = +∞)))
48		ianor 980	. . . . . 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 513	. . . 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 982	. . . 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 481	. . . . . . . . 9 ⊢ ((0 < 𝐵 ∧ 𝐴 = +∞) → ¬ 𝐵 = -∞)
55	54	a1i 11	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → ((0 < 𝐵 ∧ 𝐴 = +∞) → ¬ 𝐵 = -∞))
56		pnfnemnf 11268	. . . . . . . . . . 11 ⊢ +∞ ≠ -∞
57		eqeq1 2736	. . . . . . . . . . . 12 ⊢ (𝐵 = +∞ → (𝐵 = -∞ ↔ +∞ = -∞))
58	57	necon3bbid 2978	. . . . . . . . . . 11 ⊢ (𝐵 = +∞ → (¬ 𝐵 = -∞ ↔ +∞ ≠ -∞))
59	56, 58	mpbiri 257	. . . . . . . . . 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 857	. . . . . . 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 483	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) → 𝐴 ∈ ℝ^)
66		xrltnsym 13115	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ^* ∧ 0 ∈ ℝ^*) → (𝐴 < 0 → ¬ 0 < 𝐴))
67	65, 7, 66	sylancl 586	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → (𝐴 < 0 → ¬ 0 < 𝐴))
68	67	adantrd 492	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → ((𝐴 < 0 ∧ 𝐵 = -∞) → ¬ 0 < 𝐴))
69	64, 68	jaod 857	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → (((𝐵 < 0 ∧ 𝐴 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = -∞)) → ¬ 0 < 𝐴))
70	62, 69	orim12d 963	. . . . . 6 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → ((((0 < 𝐵 ∧ 𝐴 = +∞) ∨ (0 < 𝐴 ∧ 𝐵 = +∞)) ∨ ((𝐵 < 0 ∧ 𝐴 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = -∞))) → (¬ 𝐵 = -∞ ∨ ¬ 0 < 𝐴)))
71		ianor 980	. . . . . . 7 ⊢ (¬ (0 < 𝐴 ∧ 𝐵 = -∞) ↔ (¬ 0 < 𝐴 ∨ ¬ 𝐵 = -∞))
72		orcom 868	. . . . . . 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 482	. . . . . . . . 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 492	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → ((0 < 𝐴 ∧ 𝐵 = +∞) → ¬ 𝐴 < 0))
79	76, 78	jaod 857	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → (((0 < 𝐵 ∧ 𝐴 = +∞) ∨ (0 < 𝐴 ∧ 𝐵 = +∞)) → ¬ 𝐴 < 0))
80		breq1 5151	. . . . . . . . . . . 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 481	. . . . . . . . 9 ⊢ ((𝐵 < 0 ∧ 𝐴 = -∞) → ¬ 𝐵 = +∞)
84	59	con2i 139	. . . . . . . . . 10 ⊢ (𝐵 = -∞ → ¬ 𝐵 = +∞)
85	84	adantl 482	. . . . . . . . 9 ⊢ ((𝐴 < 0 ∧ 𝐵 = -∞) → ¬ 𝐵 = +∞)
86	83, 85	jaoi 855	. . . . . . . 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 963	. . . . . 6 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → ((((0 < 𝐵 ∧ 𝐴 = +∞) ∨ (0 < 𝐴 ∧ 𝐵 = +∞)) ∨ ((𝐵 < 0 ∧ 𝐴 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = -∞))) → (¬ 𝐴 < 0 ∨ ¬ 𝐵 = +∞)))
89		ianor 980	. . . . . 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 513	. . . 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 982	. . . 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 513	. 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 925	. 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 982	. 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 396 ∨ wo 845 = wceq 1541 ∈ wcel 2106 ≠ wne 2940 class class class wbr 5148 0cc0 11109 +∞cpnf 11244 -∞cmnf 11245 ℝ^*cxr 11246 < clt 11247

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7724 ax-cnex 11165 ax-resscn 11166 ax-1cn 11167 ax-addrcl 11170 ax-rnegex 11180 ax-cnre 11182 ax-pre-lttri 11183 ax-pre-lttrn 11184

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rab 3433 df-v 3476 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5574 df-po 5588 df-so 5589 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-er 8702 df-en 8939 df-dom 8940 df-sdom 8941 df-pnf 11249 df-mnf 11250 df-xr 11251 df-ltxr 11252

This theorem is referenced by: xmulneg1 13247

W3C validator