Step |
Hyp |
Ref |
Expression |
1 |
|
line2ylem.i |
|- I = { 1 , 2 } |
2 |
|
line2ylem.p |
|- P = ( RR ^m I ) |
3 |
|
ianor |
|- ( -. ( ( A =/= 0 /\ B = 0 ) /\ C = 0 ) <-> ( -. ( A =/= 0 /\ B = 0 ) \/ -. C = 0 ) ) |
4 |
|
df-ne |
|- ( C =/= 0 <-> -. C = 0 ) |
5 |
|
0re |
|- 0 e. RR |
6 |
1 2
|
prelrrx2 |
|- ( ( 0 e. RR /\ 0 e. RR ) -> { <. 1 , 0 >. , <. 2 , 0 >. } e. P ) |
7 |
5 5 6
|
mp2an |
|- { <. 1 , 0 >. , <. 2 , 0 >. } e. P |
8 |
|
eqneqall |
|- ( C = 0 -> ( C =/= 0 -> -. 0 = 0 ) ) |
9 |
8
|
com12 |
|- ( C =/= 0 -> ( C = 0 -> -. 0 = 0 ) ) |
10 |
|
eqid |
|- 0 = 0 |
11 |
10
|
pm2.24i |
|- ( -. 0 = 0 -> C = 0 ) |
12 |
9 11
|
impbid1 |
|- ( C =/= 0 -> ( C = 0 <-> -. 0 = 0 ) ) |
13 |
12
|
adantl |
|- ( ( ( A e. RR /\ B e. RR /\ C e. RR ) /\ C =/= 0 ) -> ( C = 0 <-> -. 0 = 0 ) ) |
14 |
|
xor3 |
|- ( -. ( C = 0 <-> 0 = 0 ) <-> ( C = 0 <-> -. 0 = 0 ) ) |
15 |
13 14
|
sylibr |
|- ( ( ( A e. RR /\ B e. RR /\ C e. RR ) /\ C =/= 0 ) -> -. ( C = 0 <-> 0 = 0 ) ) |
16 |
|
simp1 |
|- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> A e. RR ) |
17 |
16
|
recnd |
|- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> A e. CC ) |
18 |
17
|
mul01d |
|- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( A x. 0 ) = 0 ) |
19 |
|
simp2 |
|- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> B e. RR ) |
20 |
19
|
recnd |
|- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> B e. CC ) |
21 |
20
|
mul01d |
|- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( B x. 0 ) = 0 ) |
22 |
18 21
|
oveq12d |
|- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( ( A x. 0 ) + ( B x. 0 ) ) = ( 0 + 0 ) ) |
23 |
|
00id |
|- ( 0 + 0 ) = 0 |
24 |
22 23
|
eqtrdi |
|- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( ( A x. 0 ) + ( B x. 0 ) ) = 0 ) |
25 |
24
|
eqeq1d |
|- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( ( ( A x. 0 ) + ( B x. 0 ) ) = C <-> 0 = C ) ) |
26 |
|
eqcom |
|- ( 0 = C <-> C = 0 ) |
27 |
25 26
|
bitrdi |
|- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( ( ( A x. 0 ) + ( B x. 0 ) ) = C <-> C = 0 ) ) |
28 |
27
|
adantr |
|- ( ( ( A e. RR /\ B e. RR /\ C e. RR ) /\ C =/= 0 ) -> ( ( ( A x. 0 ) + ( B x. 0 ) ) = C <-> C = 0 ) ) |
29 |
28
|
bibi1d |
|- ( ( ( A e. RR /\ B e. RR /\ C e. RR ) /\ C =/= 0 ) -> ( ( ( ( A x. 0 ) + ( B x. 0 ) ) = C <-> 0 = 0 ) <-> ( C = 0 <-> 0 = 0 ) ) ) |
30 |
15 29
|
mtbird |
|- ( ( ( A e. RR /\ B e. RR /\ C e. RR ) /\ C =/= 0 ) -> -. ( ( ( A x. 0 ) + ( B x. 0 ) ) = C <-> 0 = 0 ) ) |
31 |
|
fveq1 |
|- ( p = { <. 1 , 0 >. , <. 2 , 0 >. } -> ( p ` 1 ) = ( { <. 1 , 0 >. , <. 2 , 0 >. } ` 1 ) ) |
32 |
|
1ex |
|- 1 e. _V |
33 |
|
c0ex |
|- 0 e. _V |
34 |
|
1ne2 |
|- 1 =/= 2 |
35 |
|
fvpr1g |
|- ( ( 1 e. _V /\ 0 e. _V /\ 1 =/= 2 ) -> ( { <. 1 , 0 >. , <. 2 , 0 >. } ` 1 ) = 0 ) |
36 |
32 33 34 35
|
mp3an |
|- ( { <. 1 , 0 >. , <. 2 , 0 >. } ` 1 ) = 0 |
37 |
31 36
|
eqtrdi |
|- ( p = { <. 1 , 0 >. , <. 2 , 0 >. } -> ( p ` 1 ) = 0 ) |
38 |
37
|
oveq2d |
|- ( p = { <. 1 , 0 >. , <. 2 , 0 >. } -> ( A x. ( p ` 1 ) ) = ( A x. 0 ) ) |
39 |
|
fveq1 |
|- ( p = { <. 1 , 0 >. , <. 2 , 0 >. } -> ( p ` 2 ) = ( { <. 1 , 0 >. , <. 2 , 0 >. } ` 2 ) ) |
40 |
|
2ex |
|- 2 e. _V |
41 |
|
fvpr2g |
|- ( ( 2 e. _V /\ 0 e. _V /\ 1 =/= 2 ) -> ( { <. 1 , 0 >. , <. 2 , 0 >. } ` 2 ) = 0 ) |
42 |
40 33 34 41
|
mp3an |
|- ( { <. 1 , 0 >. , <. 2 , 0 >. } ` 2 ) = 0 |
43 |
39 42
|
eqtrdi |
|- ( p = { <. 1 , 0 >. , <. 2 , 0 >. } -> ( p ` 2 ) = 0 ) |
44 |
43
|
oveq2d |
|- ( p = { <. 1 , 0 >. , <. 2 , 0 >. } -> ( B x. ( p ` 2 ) ) = ( B x. 0 ) ) |
45 |
38 44
|
oveq12d |
|- ( p = { <. 1 , 0 >. , <. 2 , 0 >. } -> ( ( A x. ( p ` 1 ) ) + ( B x. ( p ` 2 ) ) ) = ( ( A x. 0 ) + ( B x. 0 ) ) ) |
46 |
45
|
eqeq1d |
|- ( p = { <. 1 , 0 >. , <. 2 , 0 >. } -> ( ( ( A x. ( p ` 1 ) ) + ( B x. ( p ` 2 ) ) ) = C <-> ( ( A x. 0 ) + ( B x. 0 ) ) = C ) ) |
47 |
37
|
eqeq1d |
|- ( p = { <. 1 , 0 >. , <. 2 , 0 >. } -> ( ( p ` 1 ) = 0 <-> 0 = 0 ) ) |
48 |
46 47
|
bibi12d |
|- ( p = { <. 1 , 0 >. , <. 2 , 0 >. } -> ( ( ( ( A x. ( p ` 1 ) ) + ( B x. ( p ` 2 ) ) ) = C <-> ( p ` 1 ) = 0 ) <-> ( ( ( A x. 0 ) + ( B x. 0 ) ) = C <-> 0 = 0 ) ) ) |
49 |
48
|
notbid |
|- ( p = { <. 1 , 0 >. , <. 2 , 0 >. } -> ( -. ( ( ( A x. ( p ` 1 ) ) + ( B x. ( p ` 2 ) ) ) = C <-> ( p ` 1 ) = 0 ) <-> -. ( ( ( A x. 0 ) + ( B x. 0 ) ) = C <-> 0 = 0 ) ) ) |
50 |
49
|
rspcev |
|- ( ( { <. 1 , 0 >. , <. 2 , 0 >. } e. P /\ -. ( ( ( A x. 0 ) + ( B x. 0 ) ) = C <-> 0 = 0 ) ) -> E. p e. P -. ( ( ( A x. ( p ` 1 ) ) + ( B x. ( p ` 2 ) ) ) = C <-> ( p ` 1 ) = 0 ) ) |
51 |
7 30 50
|
sylancr |
|- ( ( ( A e. RR /\ B e. RR /\ C e. RR ) /\ C =/= 0 ) -> E. p e. P -. ( ( ( A x. ( p ` 1 ) ) + ( B x. ( p ` 2 ) ) ) = C <-> ( p ` 1 ) = 0 ) ) |
52 |
51
|
expcom |
|- ( C =/= 0 -> ( ( A e. RR /\ B e. RR /\ C e. RR ) -> E. p e. P -. ( ( ( A x. ( p ` 1 ) ) + ( B x. ( p ` 2 ) ) ) = C <-> ( p ` 1 ) = 0 ) ) ) |
53 |
4 52
|
sylbir |
|- ( -. C = 0 -> ( ( A e. RR /\ B e. RR /\ C e. RR ) -> E. p e. P -. ( ( ( A x. ( p ` 1 ) ) + ( B x. ( p ` 2 ) ) ) = C <-> ( p ` 1 ) = 0 ) ) ) |
54 |
|
notnotb |
|- ( C = 0 <-> -. -. C = 0 ) |
55 |
|
ianor |
|- ( -. ( A =/= 0 /\ B = 0 ) <-> ( -. A =/= 0 \/ -. B = 0 ) ) |
56 |
|
df-ne |
|- ( B =/= 0 <-> -. B = 0 ) |
57 |
|
1red |
|- ( ( ( A e. RR /\ B e. RR /\ C e. RR ) /\ ( B =/= 0 /\ C = 0 ) ) -> 1 e. RR ) |
58 |
16
|
adantr |
|- ( ( ( A e. RR /\ B e. RR /\ C e. RR ) /\ ( B =/= 0 /\ C = 0 ) ) -> A e. RR ) |
59 |
58
|
renegcld |
|- ( ( ( A e. RR /\ B e. RR /\ C e. RR ) /\ ( B =/= 0 /\ C = 0 ) ) -> -u A e. RR ) |
60 |
19
|
adantr |
|- ( ( ( A e. RR /\ B e. RR /\ C e. RR ) /\ ( B =/= 0 /\ C = 0 ) ) -> B e. RR ) |
61 |
|
simprl |
|- ( ( ( A e. RR /\ B e. RR /\ C e. RR ) /\ ( B =/= 0 /\ C = 0 ) ) -> B =/= 0 ) |
62 |
59 60 61
|
redivcld |
|- ( ( ( A e. RR /\ B e. RR /\ C e. RR ) /\ ( B =/= 0 /\ C = 0 ) ) -> ( -u A / B ) e. RR ) |
63 |
1 2
|
prelrrx2 |
|- ( ( 1 e. RR /\ ( -u A / B ) e. RR ) -> { <. 1 , 1 >. , <. 2 , ( -u A / B ) >. } e. P ) |
64 |
57 62 63
|
syl2anc |
|- ( ( ( A e. RR /\ B e. RR /\ C e. RR ) /\ ( B =/= 0 /\ C = 0 ) ) -> { <. 1 , 1 >. , <. 2 , ( -u A / B ) >. } e. P ) |
65 |
|
ax-1ne0 |
|- 1 =/= 0 |
66 |
65
|
neii |
|- -. 1 = 0 |
67 |
10 66
|
2th |
|- ( 0 = 0 <-> -. 1 = 0 ) |
68 |
|
xor3 |
|- ( -. ( 0 = 0 <-> 1 = 0 ) <-> ( 0 = 0 <-> -. 1 = 0 ) ) |
69 |
67 68
|
mpbir |
|- -. ( 0 = 0 <-> 1 = 0 ) |
70 |
17
|
mulid1d |
|- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( A x. 1 ) = A ) |
71 |
70
|
adantr |
|- ( ( ( A e. RR /\ B e. RR /\ C e. RR ) /\ ( B =/= 0 /\ C = 0 ) ) -> ( A x. 1 ) = A ) |
72 |
17
|
negcld |
|- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> -u A e. CC ) |
73 |
72
|
adantr |
|- ( ( ( A e. RR /\ B e. RR /\ C e. RR ) /\ ( B =/= 0 /\ C = 0 ) ) -> -u A e. CC ) |
74 |
20
|
adantr |
|- ( ( ( A e. RR /\ B e. RR /\ C e. RR ) /\ ( B =/= 0 /\ C = 0 ) ) -> B e. CC ) |
75 |
73 74 61
|
divcan2d |
|- ( ( ( A e. RR /\ B e. RR /\ C e. RR ) /\ ( B =/= 0 /\ C = 0 ) ) -> ( B x. ( -u A / B ) ) = -u A ) |
76 |
71 75
|
oveq12d |
|- ( ( ( A e. RR /\ B e. RR /\ C e. RR ) /\ ( B =/= 0 /\ C = 0 ) ) -> ( ( A x. 1 ) + ( B x. ( -u A / B ) ) ) = ( A + -u A ) ) |
77 |
17
|
negidd |
|- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( A + -u A ) = 0 ) |
78 |
77
|
adantr |
|- ( ( ( A e. RR /\ B e. RR /\ C e. RR ) /\ ( B =/= 0 /\ C = 0 ) ) -> ( A + -u A ) = 0 ) |
79 |
76 78
|
eqtrd |
|- ( ( ( A e. RR /\ B e. RR /\ C e. RR ) /\ ( B =/= 0 /\ C = 0 ) ) -> ( ( A x. 1 ) + ( B x. ( -u A / B ) ) ) = 0 ) |
80 |
|
simprr |
|- ( ( ( A e. RR /\ B e. RR /\ C e. RR ) /\ ( B =/= 0 /\ C = 0 ) ) -> C = 0 ) |
81 |
79 80
|
eqeq12d |
|- ( ( ( A e. RR /\ B e. RR /\ C e. RR ) /\ ( B =/= 0 /\ C = 0 ) ) -> ( ( ( A x. 1 ) + ( B x. ( -u A / B ) ) ) = C <-> 0 = 0 ) ) |
82 |
81
|
bibi1d |
|- ( ( ( A e. RR /\ B e. RR /\ C e. RR ) /\ ( B =/= 0 /\ C = 0 ) ) -> ( ( ( ( A x. 1 ) + ( B x. ( -u A / B ) ) ) = C <-> 1 = 0 ) <-> ( 0 = 0 <-> 1 = 0 ) ) ) |
83 |
69 82
|
mtbiri |
|- ( ( ( A e. RR /\ B e. RR /\ C e. RR ) /\ ( B =/= 0 /\ C = 0 ) ) -> -. ( ( ( A x. 1 ) + ( B x. ( -u A / B ) ) ) = C <-> 1 = 0 ) ) |
84 |
|
fveq1 |
|- ( p = { <. 1 , 1 >. , <. 2 , ( -u A / B ) >. } -> ( p ` 1 ) = ( { <. 1 , 1 >. , <. 2 , ( -u A / B ) >. } ` 1 ) ) |
85 |
|
fvpr1g |
|- ( ( 1 e. _V /\ 1 e. _V /\ 1 =/= 2 ) -> ( { <. 1 , 1 >. , <. 2 , ( -u A / B ) >. } ` 1 ) = 1 ) |
86 |
32 32 34 85
|
mp3an |
|- ( { <. 1 , 1 >. , <. 2 , ( -u A / B ) >. } ` 1 ) = 1 |
87 |
84 86
|
eqtrdi |
|- ( p = { <. 1 , 1 >. , <. 2 , ( -u A / B ) >. } -> ( p ` 1 ) = 1 ) |
88 |
87
|
oveq2d |
|- ( p = { <. 1 , 1 >. , <. 2 , ( -u A / B ) >. } -> ( A x. ( p ` 1 ) ) = ( A x. 1 ) ) |
89 |
|
fveq1 |
|- ( p = { <. 1 , 1 >. , <. 2 , ( -u A / B ) >. } -> ( p ` 2 ) = ( { <. 1 , 1 >. , <. 2 , ( -u A / B ) >. } ` 2 ) ) |
90 |
|
ovex |
|- ( -u A / B ) e. _V |
91 |
|
fvpr2g |
|- ( ( 2 e. _V /\ ( -u A / B ) e. _V /\ 1 =/= 2 ) -> ( { <. 1 , 1 >. , <. 2 , ( -u A / B ) >. } ` 2 ) = ( -u A / B ) ) |
92 |
40 90 34 91
|
mp3an |
|- ( { <. 1 , 1 >. , <. 2 , ( -u A / B ) >. } ` 2 ) = ( -u A / B ) |
93 |
89 92
|
eqtrdi |
|- ( p = { <. 1 , 1 >. , <. 2 , ( -u A / B ) >. } -> ( p ` 2 ) = ( -u A / B ) ) |
94 |
93
|
oveq2d |
|- ( p = { <. 1 , 1 >. , <. 2 , ( -u A / B ) >. } -> ( B x. ( p ` 2 ) ) = ( B x. ( -u A / B ) ) ) |
95 |
88 94
|
oveq12d |
|- ( p = { <. 1 , 1 >. , <. 2 , ( -u A / B ) >. } -> ( ( A x. ( p ` 1 ) ) + ( B x. ( p ` 2 ) ) ) = ( ( A x. 1 ) + ( B x. ( -u A / B ) ) ) ) |
96 |
95
|
eqeq1d |
|- ( p = { <. 1 , 1 >. , <. 2 , ( -u A / B ) >. } -> ( ( ( A x. ( p ` 1 ) ) + ( B x. ( p ` 2 ) ) ) = C <-> ( ( A x. 1 ) + ( B x. ( -u A / B ) ) ) = C ) ) |
97 |
87
|
eqeq1d |
|- ( p = { <. 1 , 1 >. , <. 2 , ( -u A / B ) >. } -> ( ( p ` 1 ) = 0 <-> 1 = 0 ) ) |
98 |
96 97
|
bibi12d |
|- ( p = { <. 1 , 1 >. , <. 2 , ( -u A / B ) >. } -> ( ( ( ( A x. ( p ` 1 ) ) + ( B x. ( p ` 2 ) ) ) = C <-> ( p ` 1 ) = 0 ) <-> ( ( ( A x. 1 ) + ( B x. ( -u A / B ) ) ) = C <-> 1 = 0 ) ) ) |
99 |
98
|
notbid |
|- ( p = { <. 1 , 1 >. , <. 2 , ( -u A / B ) >. } -> ( -. ( ( ( A x. ( p ` 1 ) ) + ( B x. ( p ` 2 ) ) ) = C <-> ( p ` 1 ) = 0 ) <-> -. ( ( ( A x. 1 ) + ( B x. ( -u A / B ) ) ) = C <-> 1 = 0 ) ) ) |
100 |
99
|
rspcev |
|- ( ( { <. 1 , 1 >. , <. 2 , ( -u A / B ) >. } e. P /\ -. ( ( ( A x. 1 ) + ( B x. ( -u A / B ) ) ) = C <-> 1 = 0 ) ) -> E. p e. P -. ( ( ( A x. ( p ` 1 ) ) + ( B x. ( p ` 2 ) ) ) = C <-> ( p ` 1 ) = 0 ) ) |
101 |
64 83 100
|
syl2anc |
|- ( ( ( A e. RR /\ B e. RR /\ C e. RR ) /\ ( B =/= 0 /\ C = 0 ) ) -> E. p e. P -. ( ( ( A x. ( p ` 1 ) ) + ( B x. ( p ` 2 ) ) ) = C <-> ( p ` 1 ) = 0 ) ) |
102 |
101
|
expcom |
|- ( ( B =/= 0 /\ C = 0 ) -> ( ( A e. RR /\ B e. RR /\ C e. RR ) -> E. p e. P -. ( ( ( A x. ( p ` 1 ) ) + ( B x. ( p ` 2 ) ) ) = C <-> ( p ` 1 ) = 0 ) ) ) |
103 |
102
|
ex |
|- ( B =/= 0 -> ( C = 0 -> ( ( A e. RR /\ B e. RR /\ C e. RR ) -> E. p e. P -. ( ( ( A x. ( p ` 1 ) ) + ( B x. ( p ` 2 ) ) ) = C <-> ( p ` 1 ) = 0 ) ) ) ) |
104 |
56 103
|
sylbir |
|- ( -. B = 0 -> ( C = 0 -> ( ( A e. RR /\ B e. RR /\ C e. RR ) -> E. p e. P -. ( ( ( A x. ( p ` 1 ) ) + ( B x. ( p ` 2 ) ) ) = C <-> ( p ` 1 ) = 0 ) ) ) ) |
105 |
|
notnotb |
|- ( B = 0 <-> -. -. B = 0 ) |
106 |
|
nne |
|- ( -. A =/= 0 <-> A = 0 ) |
107 |
106
|
bicomi |
|- ( A = 0 <-> -. A =/= 0 ) |
108 |
|
1re |
|- 1 e. RR |
109 |
1 2
|
prelrrx2 |
|- ( ( 1 e. RR /\ 1 e. RR ) -> { <. 1 , 1 >. , <. 2 , 1 >. } e. P ) |
110 |
108 108 109
|
mp2an |
|- { <. 1 , 1 >. , <. 2 , 1 >. } e. P |
111 |
|
oveq1 |
|- ( A = 0 -> ( A x. 1 ) = ( 0 x. 1 ) ) |
112 |
111
|
adantl |
|- ( ( B = 0 /\ A = 0 ) -> ( A x. 1 ) = ( 0 x. 1 ) ) |
113 |
|
ax-1cn |
|- 1 e. CC |
114 |
113
|
mul02i |
|- ( 0 x. 1 ) = 0 |
115 |
112 114
|
eqtrdi |
|- ( ( B = 0 /\ A = 0 ) -> ( A x. 1 ) = 0 ) |
116 |
|
oveq1 |
|- ( B = 0 -> ( B x. 1 ) = ( 0 x. 1 ) ) |
117 |
116
|
adantr |
|- ( ( B = 0 /\ A = 0 ) -> ( B x. 1 ) = ( 0 x. 1 ) ) |
118 |
117 114
|
eqtrdi |
|- ( ( B = 0 /\ A = 0 ) -> ( B x. 1 ) = 0 ) |
119 |
115 118
|
oveq12d |
|- ( ( B = 0 /\ A = 0 ) -> ( ( A x. 1 ) + ( B x. 1 ) ) = ( 0 + 0 ) ) |
120 |
119 23
|
eqtrdi |
|- ( ( B = 0 /\ A = 0 ) -> ( ( A x. 1 ) + ( B x. 1 ) ) = 0 ) |
121 |
|
id |
|- ( C = 0 -> C = 0 ) |
122 |
120 121
|
eqeqan12d |
|- ( ( ( B = 0 /\ A = 0 ) /\ C = 0 ) -> ( ( ( A x. 1 ) + ( B x. 1 ) ) = C <-> 0 = 0 ) ) |
123 |
122
|
bibi1d |
|- ( ( ( B = 0 /\ A = 0 ) /\ C = 0 ) -> ( ( ( ( A x. 1 ) + ( B x. 1 ) ) = C <-> 1 = 0 ) <-> ( 0 = 0 <-> 1 = 0 ) ) ) |
124 |
69 123
|
mtbiri |
|- ( ( ( B = 0 /\ A = 0 ) /\ C = 0 ) -> -. ( ( ( A x. 1 ) + ( B x. 1 ) ) = C <-> 1 = 0 ) ) |
125 |
|
fveq1 |
|- ( p = { <. 1 , 1 >. , <. 2 , 1 >. } -> ( p ` 1 ) = ( { <. 1 , 1 >. , <. 2 , 1 >. } ` 1 ) ) |
126 |
|
fvpr1g |
|- ( ( 1 e. _V /\ 1 e. _V /\ 1 =/= 2 ) -> ( { <. 1 , 1 >. , <. 2 , 1 >. } ` 1 ) = 1 ) |
127 |
32 32 34 126
|
mp3an |
|- ( { <. 1 , 1 >. , <. 2 , 1 >. } ` 1 ) = 1 |
128 |
125 127
|
eqtrdi |
|- ( p = { <. 1 , 1 >. , <. 2 , 1 >. } -> ( p ` 1 ) = 1 ) |
129 |
128
|
oveq2d |
|- ( p = { <. 1 , 1 >. , <. 2 , 1 >. } -> ( A x. ( p ` 1 ) ) = ( A x. 1 ) ) |
130 |
|
fveq1 |
|- ( p = { <. 1 , 1 >. , <. 2 , 1 >. } -> ( p ` 2 ) = ( { <. 1 , 1 >. , <. 2 , 1 >. } ` 2 ) ) |
131 |
|
fvpr2g |
|- ( ( 2 e. _V /\ 1 e. _V /\ 1 =/= 2 ) -> ( { <. 1 , 1 >. , <. 2 , 1 >. } ` 2 ) = 1 ) |
132 |
40 32 34 131
|
mp3an |
|- ( { <. 1 , 1 >. , <. 2 , 1 >. } ` 2 ) = 1 |
133 |
130 132
|
eqtrdi |
|- ( p = { <. 1 , 1 >. , <. 2 , 1 >. } -> ( p ` 2 ) = 1 ) |
134 |
133
|
oveq2d |
|- ( p = { <. 1 , 1 >. , <. 2 , 1 >. } -> ( B x. ( p ` 2 ) ) = ( B x. 1 ) ) |
135 |
129 134
|
oveq12d |
|- ( p = { <. 1 , 1 >. , <. 2 , 1 >. } -> ( ( A x. ( p ` 1 ) ) + ( B x. ( p ` 2 ) ) ) = ( ( A x. 1 ) + ( B x. 1 ) ) ) |
136 |
135
|
eqeq1d |
|- ( p = { <. 1 , 1 >. , <. 2 , 1 >. } -> ( ( ( A x. ( p ` 1 ) ) + ( B x. ( p ` 2 ) ) ) = C <-> ( ( A x. 1 ) + ( B x. 1 ) ) = C ) ) |
137 |
128
|
eqeq1d |
|- ( p = { <. 1 , 1 >. , <. 2 , 1 >. } -> ( ( p ` 1 ) = 0 <-> 1 = 0 ) ) |
138 |
136 137
|
bibi12d |
|- ( p = { <. 1 , 1 >. , <. 2 , 1 >. } -> ( ( ( ( A x. ( p ` 1 ) ) + ( B x. ( p ` 2 ) ) ) = C <-> ( p ` 1 ) = 0 ) <-> ( ( ( A x. 1 ) + ( B x. 1 ) ) = C <-> 1 = 0 ) ) ) |
139 |
138
|
notbid |
|- ( p = { <. 1 , 1 >. , <. 2 , 1 >. } -> ( -. ( ( ( A x. ( p ` 1 ) ) + ( B x. ( p ` 2 ) ) ) = C <-> ( p ` 1 ) = 0 ) <-> -. ( ( ( A x. 1 ) + ( B x. 1 ) ) = C <-> 1 = 0 ) ) ) |
140 |
139
|
rspcev |
|- ( ( { <. 1 , 1 >. , <. 2 , 1 >. } e. P /\ -. ( ( ( A x. 1 ) + ( B x. 1 ) ) = C <-> 1 = 0 ) ) -> E. p e. P -. ( ( ( A x. ( p ` 1 ) ) + ( B x. ( p ` 2 ) ) ) = C <-> ( p ` 1 ) = 0 ) ) |
141 |
110 124 140
|
sylancr |
|- ( ( ( B = 0 /\ A = 0 ) /\ C = 0 ) -> E. p e. P -. ( ( ( A x. ( p ` 1 ) ) + ( B x. ( p ` 2 ) ) ) = C <-> ( p ` 1 ) = 0 ) ) |
142 |
141
|
a1d |
|- ( ( ( B = 0 /\ A = 0 ) /\ C = 0 ) -> ( ( A e. RR /\ B e. RR /\ C e. RR ) -> E. p e. P -. ( ( ( A x. ( p ` 1 ) ) + ( B x. ( p ` 2 ) ) ) = C <-> ( p ` 1 ) = 0 ) ) ) |
143 |
142
|
ex |
|- ( ( B = 0 /\ A = 0 ) -> ( C = 0 -> ( ( A e. RR /\ B e. RR /\ C e. RR ) -> E. p e. P -. ( ( ( A x. ( p ` 1 ) ) + ( B x. ( p ` 2 ) ) ) = C <-> ( p ` 1 ) = 0 ) ) ) ) |
144 |
105 107 143
|
syl2anbr |
|- ( ( -. -. B = 0 /\ -. A =/= 0 ) -> ( C = 0 -> ( ( A e. RR /\ B e. RR /\ C e. RR ) -> E. p e. P -. ( ( ( A x. ( p ` 1 ) ) + ( B x. ( p ` 2 ) ) ) = C <-> ( p ` 1 ) = 0 ) ) ) ) |
145 |
104 144
|
jaoi3 |
|- ( ( -. B = 0 \/ -. A =/= 0 ) -> ( C = 0 -> ( ( A e. RR /\ B e. RR /\ C e. RR ) -> E. p e. P -. ( ( ( A x. ( p ` 1 ) ) + ( B x. ( p ` 2 ) ) ) = C <-> ( p ` 1 ) = 0 ) ) ) ) |
146 |
145
|
orcoms |
|- ( ( -. A =/= 0 \/ -. B = 0 ) -> ( C = 0 -> ( ( A e. RR /\ B e. RR /\ C e. RR ) -> E. p e. P -. ( ( ( A x. ( p ` 1 ) ) + ( B x. ( p ` 2 ) ) ) = C <-> ( p ` 1 ) = 0 ) ) ) ) |
147 |
55 146
|
sylbi |
|- ( -. ( A =/= 0 /\ B = 0 ) -> ( C = 0 -> ( ( A e. RR /\ B e. RR /\ C e. RR ) -> E. p e. P -. ( ( ( A x. ( p ` 1 ) ) + ( B x. ( p ` 2 ) ) ) = C <-> ( p ` 1 ) = 0 ) ) ) ) |
148 |
147
|
com12 |
|- ( C = 0 -> ( -. ( A =/= 0 /\ B = 0 ) -> ( ( A e. RR /\ B e. RR /\ C e. RR ) -> E. p e. P -. ( ( ( A x. ( p ` 1 ) ) + ( B x. ( p ` 2 ) ) ) = C <-> ( p ` 1 ) = 0 ) ) ) ) |
149 |
54 148
|
sylbir |
|- ( -. -. C = 0 -> ( -. ( A =/= 0 /\ B = 0 ) -> ( ( A e. RR /\ B e. RR /\ C e. RR ) -> E. p e. P -. ( ( ( A x. ( p ` 1 ) ) + ( B x. ( p ` 2 ) ) ) = C <-> ( p ` 1 ) = 0 ) ) ) ) |
150 |
149
|
imp |
|- ( ( -. -. C = 0 /\ -. ( A =/= 0 /\ B = 0 ) ) -> ( ( A e. RR /\ B e. RR /\ C e. RR ) -> E. p e. P -. ( ( ( A x. ( p ` 1 ) ) + ( B x. ( p ` 2 ) ) ) = C <-> ( p ` 1 ) = 0 ) ) ) |
151 |
53 150
|
jaoi3 |
|- ( ( -. C = 0 \/ -. ( A =/= 0 /\ B = 0 ) ) -> ( ( A e. RR /\ B e. RR /\ C e. RR ) -> E. p e. P -. ( ( ( A x. ( p ` 1 ) ) + ( B x. ( p ` 2 ) ) ) = C <-> ( p ` 1 ) = 0 ) ) ) |
152 |
151
|
orcoms |
|- ( ( -. ( A =/= 0 /\ B = 0 ) \/ -. C = 0 ) -> ( ( A e. RR /\ B e. RR /\ C e. RR ) -> E. p e. P -. ( ( ( A x. ( p ` 1 ) ) + ( B x. ( p ` 2 ) ) ) = C <-> ( p ` 1 ) = 0 ) ) ) |
153 |
152
|
com12 |
|- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( ( -. ( A =/= 0 /\ B = 0 ) \/ -. C = 0 ) -> E. p e. P -. ( ( ( A x. ( p ` 1 ) ) + ( B x. ( p ` 2 ) ) ) = C <-> ( p ` 1 ) = 0 ) ) ) |
154 |
3 153
|
syl5bi |
|- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( -. ( ( A =/= 0 /\ B = 0 ) /\ C = 0 ) -> E. p e. P -. ( ( ( A x. ( p ` 1 ) ) + ( B x. ( p ` 2 ) ) ) = C <-> ( p ` 1 ) = 0 ) ) ) |
155 |
|
rexnal |
|- ( E. p e. P -. ( ( ( A x. ( p ` 1 ) ) + ( B x. ( p ` 2 ) ) ) = C <-> ( p ` 1 ) = 0 ) <-> -. A. p e. P ( ( ( A x. ( p ` 1 ) ) + ( B x. ( p ` 2 ) ) ) = C <-> ( p ` 1 ) = 0 ) ) |
156 |
154 155
|
syl6ib |
|- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( -. ( ( A =/= 0 /\ B = 0 ) /\ C = 0 ) -> -. A. p e. P ( ( ( A x. ( p ` 1 ) ) + ( B x. ( p ` 2 ) ) ) = C <-> ( p ` 1 ) = 0 ) ) ) |
157 |
156
|
con4d |
|- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( A. p e. P ( ( ( A x. ( p ` 1 ) ) + ( B x. ( p ` 2 ) ) ) = C <-> ( p ` 1 ) = 0 ) -> ( ( A =/= 0 /\ B = 0 ) /\ C = 0 ) ) ) |
158 |
|
df-3an |
|- ( ( A =/= 0 /\ B = 0 /\ C = 0 ) <-> ( ( A =/= 0 /\ B = 0 ) /\ C = 0 ) ) |
159 |
157 158
|
syl6ibr |
|- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( A. p e. P ( ( ( A x. ( p ` 1 ) ) + ( B x. ( p ` 2 ) ) ) = C <-> ( p ` 1 ) = 0 ) -> ( A =/= 0 /\ B = 0 /\ C = 0 ) ) ) |