Metamath Proof Explorer


Theorem xmullem2

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

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

Proof

Step Hyp Ref Expression
1 mnfnepnf -∞ ≠ +∞
2 eqeq1 ( 𝐴 = -∞ → ( 𝐴 = +∞ ↔ -∞ = +∞ ) )
3 2 necon3bbid ( 𝐴 = -∞ → ( ¬ 𝐴 = +∞ ↔ -∞ ≠ +∞ ) )
4 1 3 mpbiri ( 𝐴 = -∞ → ¬ 𝐴 = +∞ )
5 4 con2i ( 𝐴 = +∞ → ¬ 𝐴 = -∞ )
6 5 adantl ( ( 0 < 𝐵𝐴 = +∞ ) → ¬ 𝐴 = -∞ )
7 0xr 0 ∈ ℝ*
8 nltmnf ( 0 ∈ ℝ* → ¬ 0 < -∞ )
9 7 8 ax-mp ¬ 0 < -∞
10 breq2 ( 𝐴 = -∞ → ( 0 < 𝐴 ↔ 0 < -∞ ) )
11 9 10 mtbiri ( 𝐴 = -∞ → ¬ 0 < 𝐴 )
12 11 con2i ( 0 < 𝐴 → ¬ 𝐴 = -∞ )
13 12 adantr ( ( 0 < 𝐴𝐵 = +∞ ) → ¬ 𝐴 = -∞ )
14 6 13 jaoi ( ( ( 0 < 𝐵𝐴 = +∞ ) ∨ ( 0 < 𝐴𝐵 = +∞ ) ) → ¬ 𝐴 = -∞ )
15 14 a1i ( ( 𝐴 ∈ ℝ*𝐵 ∈ ℝ* ) → ( ( ( 0 < 𝐵𝐴 = +∞ ) ∨ ( 0 < 𝐴𝐵 = +∞ ) ) → ¬ 𝐴 = -∞ ) )
16 simpr ( ( 𝐴 ∈ ℝ*𝐵 ∈ ℝ* ) → 𝐵 ∈ ℝ* )
17 xrltnsym ( ( 𝐵 ∈ ℝ* ∧ 0 ∈ ℝ* ) → ( 𝐵 < 0 → ¬ 0 < 𝐵 ) )
18 16 7 17 sylancl ( ( 𝐴 ∈ ℝ*𝐵 ∈ ℝ* ) → ( 𝐵 < 0 → ¬ 0 < 𝐵 ) )
19 18 adantrd ( ( 𝐴 ∈ ℝ*𝐵 ∈ ℝ* ) → ( ( 𝐵 < 0 ∧ 𝐴 = -∞ ) → ¬ 0 < 𝐵 ) )
20 breq2 ( 𝐵 = -∞ → ( 0 < 𝐵 ↔ 0 < -∞ ) )
21 9 20 mtbiri ( 𝐵 = -∞ → ¬ 0 < 𝐵 )
22 21 adantl ( ( 𝐴 < 0 ∧ 𝐵 = -∞ ) → ¬ 0 < 𝐵 )
23 22 a1i ( ( 𝐴 ∈ ℝ*𝐵 ∈ ℝ* ) → ( ( 𝐴 < 0 ∧ 𝐵 = -∞ ) → ¬ 0 < 𝐵 ) )
24 19 23 jaod ( ( 𝐴 ∈ ℝ*𝐵 ∈ ℝ* ) → ( ( ( 𝐵 < 0 ∧ 𝐴 = -∞ ) ∨ ( 𝐴 < 0 ∧ 𝐵 = -∞ ) ) → ¬ 0 < 𝐵 ) )
25 15 24 orim12d ( ( 𝐴 ∈ ℝ*𝐵 ∈ ℝ* ) → ( ( ( ( 0 < 𝐵𝐴 = +∞ ) ∨ ( 0 < 𝐴𝐵 = +∞ ) ) ∨ ( ( 𝐵 < 0 ∧ 𝐴 = -∞ ) ∨ ( 𝐴 < 0 ∧ 𝐵 = -∞ ) ) ) → ( ¬ 𝐴 = -∞ ∨ ¬ 0 < 𝐵 ) ) )
26 ianor ( ¬ ( 0 < 𝐵𝐴 = -∞ ) ↔ ( ¬ 0 < 𝐵 ∨ ¬ 𝐴 = -∞ ) )
27 orcom ( ( ¬ 0 < 𝐵 ∨ ¬ 𝐴 = -∞ ) ↔ ( ¬ 𝐴 = -∞ ∨ ¬ 0 < 𝐵 ) )
28 26 27 bitri ( ¬ ( 0 < 𝐵𝐴 = -∞ ) ↔ ( ¬ 𝐴 = -∞ ∨ ¬ 0 < 𝐵 ) )
29 25 28 syl6ibr ( ( 𝐴 ∈ ℝ*𝐵 ∈ ℝ* ) → ( ( ( ( 0 < 𝐵𝐴 = +∞ ) ∨ ( 0 < 𝐴𝐵 = +∞ ) ) ∨ ( ( 𝐵 < 0 ∧ 𝐴 = -∞ ) ∨ ( 𝐴 < 0 ∧ 𝐵 = -∞ ) ) ) → ¬ ( 0 < 𝐵𝐴 = -∞ ) ) )
30 18 con2d ( ( 𝐴 ∈ ℝ*𝐵 ∈ ℝ* ) → ( 0 < 𝐵 → ¬ 𝐵 < 0 ) )
31 30 adantrd ( ( 𝐴 ∈ ℝ*𝐵 ∈ ℝ* ) → ( ( 0 < 𝐵𝐴 = +∞ ) → ¬ 𝐵 < 0 ) )
32 pnfnlt ( 0 ∈ ℝ* → ¬ +∞ < 0 )
33 7 32 ax-mp ¬ +∞ < 0
34 simpr ( ( 0 < 𝐴𝐵 = +∞ ) → 𝐵 = +∞ )
35 34 breq1d ( ( 0 < 𝐴𝐵 = +∞ ) → ( 𝐵 < 0 ↔ +∞ < 0 ) )
36 33 35 mtbiri ( ( 0 < 𝐴𝐵 = +∞ ) → ¬ 𝐵 < 0 )
37 36 a1i ( ( 𝐴 ∈ ℝ*𝐵 ∈ ℝ* ) → ( ( 0 < 𝐴𝐵 = +∞ ) → ¬ 𝐵 < 0 ) )
38 31 37 jaod ( ( 𝐴 ∈ ℝ*𝐵 ∈ ℝ* ) → ( ( ( 0 < 𝐵𝐴 = +∞ ) ∨ ( 0 < 𝐴𝐵 = +∞ ) ) → ¬ 𝐵 < 0 ) )
39 4 a1i ( ( 𝐴 ∈ ℝ*𝐵 ∈ ℝ* ) → ( 𝐴 = -∞ → ¬ 𝐴 = +∞ ) )
40 39 adantld ( ( 𝐴 ∈ ℝ*𝐵 ∈ ℝ* ) → ( ( 𝐵 < 0 ∧ 𝐴 = -∞ ) → ¬ 𝐴 = +∞ ) )
41 breq1 ( 𝐴 = +∞ → ( 𝐴 < 0 ↔ +∞ < 0 ) )
42 33 41 mtbiri ( 𝐴 = +∞ → ¬ 𝐴 < 0 )
43 42 con2i ( 𝐴 < 0 → ¬ 𝐴 = +∞ )
44 43 adantr ( ( 𝐴 < 0 ∧ 𝐵 = -∞ ) → ¬ 𝐴 = +∞ )
45 44 a1i ( ( 𝐴 ∈ ℝ*𝐵 ∈ ℝ* ) → ( ( 𝐴 < 0 ∧ 𝐵 = -∞ ) → ¬ 𝐴 = +∞ ) )
46 40 45 jaod ( ( 𝐴 ∈ ℝ*𝐵 ∈ ℝ* ) → ( ( ( 𝐵 < 0 ∧ 𝐴 = -∞ ) ∨ ( 𝐴 < 0 ∧ 𝐵 = -∞ ) ) → ¬ 𝐴 = +∞ ) )
47 38 46 orim12d ( ( 𝐴 ∈ ℝ*𝐵 ∈ ℝ* ) → ( ( ( ( 0 < 𝐵𝐴 = +∞ ) ∨ ( 0 < 𝐴𝐵 = +∞ ) ) ∨ ( ( 𝐵 < 0 ∧ 𝐴 = -∞ ) ∨ ( 𝐴 < 0 ∧ 𝐵 = -∞ ) ) ) → ( ¬ 𝐵 < 0 ∨ ¬ 𝐴 = +∞ ) ) )
48 ianor ( ¬ ( 𝐵 < 0 ∧ 𝐴 = +∞ ) ↔ ( ¬ 𝐵 < 0 ∨ ¬ 𝐴 = +∞ ) )
49 47 48 syl6ibr ( ( 𝐴 ∈ ℝ*𝐵 ∈ ℝ* ) → ( ( ( ( 0 < 𝐵𝐴 = +∞ ) ∨ ( 0 < 𝐴𝐵 = +∞ ) ) ∨ ( ( 𝐵 < 0 ∧ 𝐴 = -∞ ) ∨ ( 𝐴 < 0 ∧ 𝐵 = -∞ ) ) ) → ¬ ( 𝐵 < 0 ∧ 𝐴 = +∞ ) ) )
50 29 49 jcad ( ( 𝐴 ∈ ℝ*𝐵 ∈ ℝ* ) → ( ( ( ( 0 < 𝐵𝐴 = +∞ ) ∨ ( 0 < 𝐴𝐵 = +∞ ) ) ∨ ( ( 𝐵 < 0 ∧ 𝐴 = -∞ ) ∨ ( 𝐴 < 0 ∧ 𝐵 = -∞ ) ) ) → ( ¬ ( 0 < 𝐵𝐴 = -∞ ) ∧ ¬ ( 𝐵 < 0 ∧ 𝐴 = +∞ ) ) ) )
51 ioran ( ¬ ( ( 0 < 𝐵𝐴 = -∞ ) ∨ ( 𝐵 < 0 ∧ 𝐴 = +∞ ) ) ↔ ( ¬ ( 0 < 𝐵𝐴 = -∞ ) ∧ ¬ ( 𝐵 < 0 ∧ 𝐴 = +∞ ) ) )
52 50 51 syl6ibr ( ( 𝐴 ∈ ℝ*𝐵 ∈ ℝ* ) → ( ( ( ( 0 < 𝐵𝐴 = +∞ ) ∨ ( 0 < 𝐴𝐵 = +∞ ) ) ∨ ( ( 𝐵 < 0 ∧ 𝐴 = -∞ ) ∨ ( 𝐴 < 0 ∧ 𝐵 = -∞ ) ) ) → ¬ ( ( 0 < 𝐵𝐴 = -∞ ) ∨ ( 𝐵 < 0 ∧ 𝐴 = +∞ ) ) ) )
53 21 con2i ( 0 < 𝐵 → ¬ 𝐵 = -∞ )
54 53 adantr ( ( 0 < 𝐵𝐴 = +∞ ) → ¬ 𝐵 = -∞ )
55 54 a1i ( ( 𝐴 ∈ ℝ*𝐵 ∈ ℝ* ) → ( ( 0 < 𝐵𝐴 = +∞ ) → ¬ 𝐵 = -∞ ) )
56 pnfnemnf +∞ ≠ -∞
57 eqeq1 ( 𝐵 = +∞ → ( 𝐵 = -∞ ↔ +∞ = -∞ ) )
58 57 necon3bbid ( 𝐵 = +∞ → ( ¬ 𝐵 = -∞ ↔ +∞ ≠ -∞ ) )
59 56 58 mpbiri ( 𝐵 = +∞ → ¬ 𝐵 = -∞ )
60 59 adantl ( ( 0 < 𝐴𝐵 = +∞ ) → ¬ 𝐵 = -∞ )
61 60 a1i ( ( 𝐴 ∈ ℝ*𝐵 ∈ ℝ* ) → ( ( 0 < 𝐴𝐵 = +∞ ) → ¬ 𝐵 = -∞ ) )
62 55 61 jaod ( ( 𝐴 ∈ ℝ*𝐵 ∈ ℝ* ) → ( ( ( 0 < 𝐵𝐴 = +∞ ) ∨ ( 0 < 𝐴𝐵 = +∞ ) ) → ¬ 𝐵 = -∞ ) )
63 11 adantl ( ( 𝐵 < 0 ∧ 𝐴 = -∞ ) → ¬ 0 < 𝐴 )
64 63 a1i ( ( 𝐴 ∈ ℝ*𝐵 ∈ ℝ* ) → ( ( 𝐵 < 0 ∧ 𝐴 = -∞ ) → ¬ 0 < 𝐴 ) )
65 simpl ( ( 𝐴 ∈ ℝ*𝐵 ∈ ℝ* ) → 𝐴 ∈ ℝ* )
66 xrltnsym ( ( 𝐴 ∈ ℝ* ∧ 0 ∈ ℝ* ) → ( 𝐴 < 0 → ¬ 0 < 𝐴 ) )
67 65 7 66 sylancl ( ( 𝐴 ∈ ℝ*𝐵 ∈ ℝ* ) → ( 𝐴 < 0 → ¬ 0 < 𝐴 ) )
68 67 adantrd ( ( 𝐴 ∈ ℝ*𝐵 ∈ ℝ* ) → ( ( 𝐴 < 0 ∧ 𝐵 = -∞ ) → ¬ 0 < 𝐴 ) )
69 64 68 jaod ( ( 𝐴 ∈ ℝ*𝐵 ∈ ℝ* ) → ( ( ( 𝐵 < 0 ∧ 𝐴 = -∞ ) ∨ ( 𝐴 < 0 ∧ 𝐵 = -∞ ) ) → ¬ 0 < 𝐴 ) )
70 62 69 orim12d ( ( 𝐴 ∈ ℝ*𝐵 ∈ ℝ* ) → ( ( ( ( 0 < 𝐵𝐴 = +∞ ) ∨ ( 0 < 𝐴𝐵 = +∞ ) ) ∨ ( ( 𝐵 < 0 ∧ 𝐴 = -∞ ) ∨ ( 𝐴 < 0 ∧ 𝐵 = -∞ ) ) ) → ( ¬ 𝐵 = -∞ ∨ ¬ 0 < 𝐴 ) ) )
71 ianor ( ¬ ( 0 < 𝐴𝐵 = -∞ ) ↔ ( ¬ 0 < 𝐴 ∨ ¬ 𝐵 = -∞ ) )
72 orcom ( ( ¬ 0 < 𝐴 ∨ ¬ 𝐵 = -∞ ) ↔ ( ¬ 𝐵 = -∞ ∨ ¬ 0 < 𝐴 ) )
73 71 72 bitri ( ¬ ( 0 < 𝐴𝐵 = -∞ ) ↔ ( ¬ 𝐵 = -∞ ∨ ¬ 0 < 𝐴 ) )
74 70 73 syl6ibr ( ( 𝐴 ∈ ℝ*𝐵 ∈ ℝ* ) → ( ( ( ( 0 < 𝐵𝐴 = +∞ ) ∨ ( 0 < 𝐴𝐵 = +∞ ) ) ∨ ( ( 𝐵 < 0 ∧ 𝐴 = -∞ ) ∨ ( 𝐴 < 0 ∧ 𝐵 = -∞ ) ) ) → ¬ ( 0 < 𝐴𝐵 = -∞ ) ) )
75 42 adantl ( ( 0 < 𝐵𝐴 = +∞ ) → ¬ 𝐴 < 0 )
76 75 a1i ( ( 𝐴 ∈ ℝ*𝐵 ∈ ℝ* ) → ( ( 0 < 𝐵𝐴 = +∞ ) → ¬ 𝐴 < 0 ) )
77 67 con2d ( ( 𝐴 ∈ ℝ*𝐵 ∈ ℝ* ) → ( 0 < 𝐴 → ¬ 𝐴 < 0 ) )
78 77 adantrd ( ( 𝐴 ∈ ℝ*𝐵 ∈ ℝ* ) → ( ( 0 < 𝐴𝐵 = +∞ ) → ¬ 𝐴 < 0 ) )
79 76 78 jaod ( ( 𝐴 ∈ ℝ*𝐵 ∈ ℝ* ) → ( ( ( 0 < 𝐵𝐴 = +∞ ) ∨ ( 0 < 𝐴𝐵 = +∞ ) ) → ¬ 𝐴 < 0 ) )
80 breq1 ( 𝐵 = +∞ → ( 𝐵 < 0 ↔ +∞ < 0 ) )
81 33 80 mtbiri ( 𝐵 = +∞ → ¬ 𝐵 < 0 )
82 81 con2i ( 𝐵 < 0 → ¬ 𝐵 = +∞ )
83 82 adantr ( ( 𝐵 < 0 ∧ 𝐴 = -∞ ) → ¬ 𝐵 = +∞ )
84 59 con2i ( 𝐵 = -∞ → ¬ 𝐵 = +∞ )
85 84 adantl ( ( 𝐴 < 0 ∧ 𝐵 = -∞ ) → ¬ 𝐵 = +∞ )
86 83 85 jaoi ( ( ( 𝐵 < 0 ∧ 𝐴 = -∞ ) ∨ ( 𝐴 < 0 ∧ 𝐵 = -∞ ) ) → ¬ 𝐵 = +∞ )
87 86 a1i ( ( 𝐴 ∈ ℝ*𝐵 ∈ ℝ* ) → ( ( ( 𝐵 < 0 ∧ 𝐴 = -∞ ) ∨ ( 𝐴 < 0 ∧ 𝐵 = -∞ ) ) → ¬ 𝐵 = +∞ ) )
88 79 87 orim12d ( ( 𝐴 ∈ ℝ*𝐵 ∈ ℝ* ) → ( ( ( ( 0 < 𝐵𝐴 = +∞ ) ∨ ( 0 < 𝐴𝐵 = +∞ ) ) ∨ ( ( 𝐵 < 0 ∧ 𝐴 = -∞ ) ∨ ( 𝐴 < 0 ∧ 𝐵 = -∞ ) ) ) → ( ¬ 𝐴 < 0 ∨ ¬ 𝐵 = +∞ ) ) )
89 ianor ( ¬ ( 𝐴 < 0 ∧ 𝐵 = +∞ ) ↔ ( ¬ 𝐴 < 0 ∨ ¬ 𝐵 = +∞ ) )
90 88 89 syl6ibr ( ( 𝐴 ∈ ℝ*𝐵 ∈ ℝ* ) → ( ( ( ( 0 < 𝐵𝐴 = +∞ ) ∨ ( 0 < 𝐴𝐵 = +∞ ) ) ∨ ( ( 𝐵 < 0 ∧ 𝐴 = -∞ ) ∨ ( 𝐴 < 0 ∧ 𝐵 = -∞ ) ) ) → ¬ ( 𝐴 < 0 ∧ 𝐵 = +∞ ) ) )
91 74 90 jcad ( ( 𝐴 ∈ ℝ*𝐵 ∈ ℝ* ) → ( ( ( ( 0 < 𝐵𝐴 = +∞ ) ∨ ( 0 < 𝐴𝐵 = +∞ ) ) ∨ ( ( 𝐵 < 0 ∧ 𝐴 = -∞ ) ∨ ( 𝐴 < 0 ∧ 𝐵 = -∞ ) ) ) → ( ¬ ( 0 < 𝐴𝐵 = -∞ ) ∧ ¬ ( 𝐴 < 0 ∧ 𝐵 = +∞ ) ) ) )
92 ioran ( ¬ ( ( 0 < 𝐴𝐵 = -∞ ) ∨ ( 𝐴 < 0 ∧ 𝐵 = +∞ ) ) ↔ ( ¬ ( 0 < 𝐴𝐵 = -∞ ) ∧ ¬ ( 𝐴 < 0 ∧ 𝐵 = +∞ ) ) )
93 91 92 syl6ibr ( ( 𝐴 ∈ ℝ*𝐵 ∈ ℝ* ) → ( ( ( ( 0 < 𝐵𝐴 = +∞ ) ∨ ( 0 < 𝐴𝐵 = +∞ ) ) ∨ ( ( 𝐵 < 0 ∧ 𝐴 = -∞ ) ∨ ( 𝐴 < 0 ∧ 𝐵 = -∞ ) ) ) → ¬ ( ( 0 < 𝐴𝐵 = -∞ ) ∨ ( 𝐴 < 0 ∧ 𝐵 = +∞ ) ) ) )
94 52 93 jcad ( ( 𝐴 ∈ ℝ*𝐵 ∈ ℝ* ) → ( ( ( ( 0 < 𝐵𝐴 = +∞ ) ∨ ( 0 < 𝐴𝐵 = +∞ ) ) ∨ ( ( 𝐵 < 0 ∧ 𝐴 = -∞ ) ∨ ( 𝐴 < 0 ∧ 𝐵 = -∞ ) ) ) → ( ¬ ( ( 0 < 𝐵𝐴 = -∞ ) ∨ ( 𝐵 < 0 ∧ 𝐴 = +∞ ) ) ∧ ¬ ( ( 0 < 𝐴𝐵 = -∞ ) ∨ ( 𝐴 < 0 ∧ 𝐵 = +∞ ) ) ) ) )
95 or4 ( ( ( ( 0 < 𝐵𝐴 = +∞ ) ∨ ( 𝐵 < 0 ∧ 𝐴 = -∞ ) ) ∨ ( ( 0 < 𝐴𝐵 = +∞ ) ∨ ( 𝐴 < 0 ∧ 𝐵 = -∞ ) ) ) ↔ ( ( ( 0 < 𝐵𝐴 = +∞ ) ∨ ( 0 < 𝐴𝐵 = +∞ ) ) ∨ ( ( 𝐵 < 0 ∧ 𝐴 = -∞ ) ∨ ( 𝐴 < 0 ∧ 𝐵 = -∞ ) ) ) )
96 ioran ( ¬ ( ( ( 0 < 𝐵𝐴 = -∞ ) ∨ ( 𝐵 < 0 ∧ 𝐴 = +∞ ) ) ∨ ( ( 0 < 𝐴𝐵 = -∞ ) ∨ ( 𝐴 < 0 ∧ 𝐵 = +∞ ) ) ) ↔ ( ¬ ( ( 0 < 𝐵𝐴 = -∞ ) ∨ ( 𝐵 < 0 ∧ 𝐴 = +∞ ) ) ∧ ¬ ( ( 0 < 𝐴𝐵 = -∞ ) ∨ ( 𝐴 < 0 ∧ 𝐵 = +∞ ) ) ) )
97 94 95 96 3imtr4g ( ( 𝐴 ∈ ℝ*𝐵 ∈ ℝ* ) → ( ( ( ( 0 < 𝐵𝐴 = +∞ ) ∨ ( 𝐵 < 0 ∧ 𝐴 = -∞ ) ) ∨ ( ( 0 < 𝐴𝐵 = +∞ ) ∨ ( 𝐴 < 0 ∧ 𝐵 = -∞ ) ) ) → ¬ ( ( ( 0 < 𝐵𝐴 = -∞ ) ∨ ( 𝐵 < 0 ∧ 𝐴 = +∞ ) ) ∨ ( ( 0 < 𝐴𝐵 = -∞ ) ∨ ( 𝐴 < 0 ∧ 𝐵 = +∞ ) ) ) ) )