Step |
Hyp |
Ref |
Expression |
1 |
|
line2.i |
|- I = { 1 , 2 } |
2 |
|
line2.e |
|- E = ( RR^ ` I ) |
3 |
|
line2.p |
|- P = ( RR ^m I ) |
4 |
|
line2.l |
|- L = ( LineM ` E ) |
5 |
|
line2.g |
|- G = { p e. P | ( ( A x. ( p ` 1 ) ) + ( B x. ( p ` 2 ) ) ) = C } |
6 |
|
line2x.x |
|- X = { <. 1 , 0 >. , <. 2 , M >. } |
7 |
|
line2x.y |
|- Y = { <. 1 , 1 >. , <. 2 , M >. } |
8 |
|
ianor |
|- ( -. ( A = 0 /\ M = ( C / B ) ) <-> ( -. A = 0 \/ -. M = ( C / B ) ) ) |
9 |
|
df-ne |
|- ( A =/= 0 <-> -. A = 0 ) |
10 |
|
df-ne |
|- ( M =/= ( C / B ) <-> -. M = ( C / B ) ) |
11 |
9 10
|
orbi12i |
|- ( ( A =/= 0 \/ M =/= ( C / B ) ) <-> ( -. A = 0 \/ -. M = ( C / B ) ) ) |
12 |
8 11
|
bitr4i |
|- ( -. ( A = 0 /\ M = ( C / B ) ) <-> ( A =/= 0 \/ M =/= ( C / B ) ) ) |
13 |
|
0red |
|- ( ( M =/= ( C / B ) /\ ( ( A e. RR /\ ( B e. RR /\ B =/= 0 ) /\ C e. RR ) /\ M e. RR ) ) -> 0 e. RR ) |
14 |
|
simp3 |
|- ( ( A e. RR /\ ( B e. RR /\ B =/= 0 ) /\ C e. RR ) -> C e. RR ) |
15 |
14
|
adantr |
|- ( ( ( A e. RR /\ ( B e. RR /\ B =/= 0 ) /\ C e. RR ) /\ M e. RR ) -> C e. RR ) |
16 |
|
simpl |
|- ( ( B e. RR /\ B =/= 0 ) -> B e. RR ) |
17 |
16
|
3ad2ant2 |
|- ( ( A e. RR /\ ( B e. RR /\ B =/= 0 ) /\ C e. RR ) -> B e. RR ) |
18 |
17
|
adantr |
|- ( ( ( A e. RR /\ ( B e. RR /\ B =/= 0 ) /\ C e. RR ) /\ M e. RR ) -> B e. RR ) |
19 |
|
simp2r |
|- ( ( A e. RR /\ ( B e. RR /\ B =/= 0 ) /\ C e. RR ) -> B =/= 0 ) |
20 |
19
|
adantr |
|- ( ( ( A e. RR /\ ( B e. RR /\ B =/= 0 ) /\ C e. RR ) /\ M e. RR ) -> B =/= 0 ) |
21 |
15 18 20
|
redivcld |
|- ( ( ( A e. RR /\ ( B e. RR /\ B =/= 0 ) /\ C e. RR ) /\ M e. RR ) -> ( C / B ) e. RR ) |
22 |
21
|
adantl |
|- ( ( M =/= ( C / B ) /\ ( ( A e. RR /\ ( B e. RR /\ B =/= 0 ) /\ C e. RR ) /\ M e. RR ) ) -> ( C / B ) e. RR ) |
23 |
1 3
|
prelrrx2 |
|- ( ( 0 e. RR /\ ( C / B ) e. RR ) -> { <. 1 , 0 >. , <. 2 , ( C / B ) >. } e. P ) |
24 |
13 22 23
|
syl2anc |
|- ( ( M =/= ( C / B ) /\ ( ( A e. RR /\ ( B e. RR /\ B =/= 0 ) /\ C e. RR ) /\ M e. RR ) ) -> { <. 1 , 0 >. , <. 2 , ( C / B ) >. } e. P ) |
25 |
|
id |
|- ( M =/= ( C / B ) -> M =/= ( C / B ) ) |
26 |
25
|
necomd |
|- ( M =/= ( C / B ) -> ( C / B ) =/= M ) |
27 |
26
|
neneqd |
|- ( M =/= ( C / B ) -> -. ( C / B ) = M ) |
28 |
27
|
a1d |
|- ( M =/= ( C / B ) -> ( C = C -> -. ( C / B ) = M ) ) |
29 |
|
eqidd |
|- ( -. ( C / B ) = M -> C = C ) |
30 |
29
|
a1i |
|- ( M =/= ( C / B ) -> ( -. ( C / B ) = M -> C = C ) ) |
31 |
28 30
|
impbid |
|- ( M =/= ( C / B ) -> ( C = C <-> -. ( C / B ) = M ) ) |
32 |
|
xor3 |
|- ( -. ( C = C <-> ( C / B ) = M ) <-> ( C = C <-> -. ( C / B ) = M ) ) |
33 |
31 32
|
sylibr |
|- ( M =/= ( C / B ) -> -. ( C = C <-> ( C / B ) = M ) ) |
34 |
33
|
adantr |
|- ( ( M =/= ( C / B ) /\ ( ( A e. RR /\ ( B e. RR /\ B =/= 0 ) /\ C e. RR ) /\ M e. RR ) ) -> -. ( C = C <-> ( C / B ) = M ) ) |
35 |
|
0red |
|- ( ( ( A e. RR /\ ( B e. RR /\ B =/= 0 ) /\ C e. RR ) /\ M e. RR ) -> 0 e. RR ) |
36 |
|
fv1prop |
|- ( 0 e. RR -> ( { <. 1 , 0 >. , <. 2 , ( C / B ) >. } ` 1 ) = 0 ) |
37 |
35 36
|
syl |
|- ( ( ( A e. RR /\ ( B e. RR /\ B =/= 0 ) /\ C e. RR ) /\ M e. RR ) -> ( { <. 1 , 0 >. , <. 2 , ( C / B ) >. } ` 1 ) = 0 ) |
38 |
37
|
oveq2d |
|- ( ( ( A e. RR /\ ( B e. RR /\ B =/= 0 ) /\ C e. RR ) /\ M e. RR ) -> ( A x. ( { <. 1 , 0 >. , <. 2 , ( C / B ) >. } ` 1 ) ) = ( A x. 0 ) ) |
39 |
|
recn |
|- ( A e. RR -> A e. CC ) |
40 |
39
|
mul01d |
|- ( A e. RR -> ( A x. 0 ) = 0 ) |
41 |
40
|
3ad2ant1 |
|- ( ( A e. RR /\ ( B e. RR /\ B =/= 0 ) /\ C e. RR ) -> ( A x. 0 ) = 0 ) |
42 |
41
|
adantr |
|- ( ( ( A e. RR /\ ( B e. RR /\ B =/= 0 ) /\ C e. RR ) /\ M e. RR ) -> ( A x. 0 ) = 0 ) |
43 |
38 42
|
eqtrd |
|- ( ( ( A e. RR /\ ( B e. RR /\ B =/= 0 ) /\ C e. RR ) /\ M e. RR ) -> ( A x. ( { <. 1 , 0 >. , <. 2 , ( C / B ) >. } ` 1 ) ) = 0 ) |
44 |
|
ovexd |
|- ( ( ( A e. RR /\ ( B e. RR /\ B =/= 0 ) /\ C e. RR ) /\ M e. RR ) -> ( C / B ) e. _V ) |
45 |
|
fv2prop |
|- ( ( C / B ) e. _V -> ( { <. 1 , 0 >. , <. 2 , ( C / B ) >. } ` 2 ) = ( C / B ) ) |
46 |
44 45
|
syl |
|- ( ( ( A e. RR /\ ( B e. RR /\ B =/= 0 ) /\ C e. RR ) /\ M e. RR ) -> ( { <. 1 , 0 >. , <. 2 , ( C / B ) >. } ` 2 ) = ( C / B ) ) |
47 |
46
|
oveq2d |
|- ( ( ( A e. RR /\ ( B e. RR /\ B =/= 0 ) /\ C e. RR ) /\ M e. RR ) -> ( B x. ( { <. 1 , 0 >. , <. 2 , ( C / B ) >. } ` 2 ) ) = ( B x. ( C / B ) ) ) |
48 |
14
|
recnd |
|- ( ( A e. RR /\ ( B e. RR /\ B =/= 0 ) /\ C e. RR ) -> C e. CC ) |
49 |
48
|
adantr |
|- ( ( ( A e. RR /\ ( B e. RR /\ B =/= 0 ) /\ C e. RR ) /\ M e. RR ) -> C e. CC ) |
50 |
16
|
recnd |
|- ( ( B e. RR /\ B =/= 0 ) -> B e. CC ) |
51 |
50
|
3ad2ant2 |
|- ( ( A e. RR /\ ( B e. RR /\ B =/= 0 ) /\ C e. RR ) -> B e. CC ) |
52 |
51
|
adantr |
|- ( ( ( A e. RR /\ ( B e. RR /\ B =/= 0 ) /\ C e. RR ) /\ M e. RR ) -> B e. CC ) |
53 |
49 52 20
|
divcan2d |
|- ( ( ( A e. RR /\ ( B e. RR /\ B =/= 0 ) /\ C e. RR ) /\ M e. RR ) -> ( B x. ( C / B ) ) = C ) |
54 |
47 53
|
eqtrd |
|- ( ( ( A e. RR /\ ( B e. RR /\ B =/= 0 ) /\ C e. RR ) /\ M e. RR ) -> ( B x. ( { <. 1 , 0 >. , <. 2 , ( C / B ) >. } ` 2 ) ) = C ) |
55 |
43 54
|
oveq12d |
|- ( ( ( A e. RR /\ ( B e. RR /\ B =/= 0 ) /\ C e. RR ) /\ M e. RR ) -> ( ( A x. ( { <. 1 , 0 >. , <. 2 , ( C / B ) >. } ` 1 ) ) + ( B x. ( { <. 1 , 0 >. , <. 2 , ( C / B ) >. } ` 2 ) ) ) = ( 0 + C ) ) |
56 |
55
|
adantl |
|- ( ( M =/= ( C / B ) /\ ( ( A e. RR /\ ( B e. RR /\ B =/= 0 ) /\ C e. RR ) /\ M e. RR ) ) -> ( ( A x. ( { <. 1 , 0 >. , <. 2 , ( C / B ) >. } ` 1 ) ) + ( B x. ( { <. 1 , 0 >. , <. 2 , ( C / B ) >. } ` 2 ) ) ) = ( 0 + C ) ) |
57 |
48
|
addid2d |
|- ( ( A e. RR /\ ( B e. RR /\ B =/= 0 ) /\ C e. RR ) -> ( 0 + C ) = C ) |
58 |
57
|
adantr |
|- ( ( ( A e. RR /\ ( B e. RR /\ B =/= 0 ) /\ C e. RR ) /\ M e. RR ) -> ( 0 + C ) = C ) |
59 |
58
|
adantl |
|- ( ( M =/= ( C / B ) /\ ( ( A e. RR /\ ( B e. RR /\ B =/= 0 ) /\ C e. RR ) /\ M e. RR ) ) -> ( 0 + C ) = C ) |
60 |
56 59
|
eqtrd |
|- ( ( M =/= ( C / B ) /\ ( ( A e. RR /\ ( B e. RR /\ B =/= 0 ) /\ C e. RR ) /\ M e. RR ) ) -> ( ( A x. ( { <. 1 , 0 >. , <. 2 , ( C / B ) >. } ` 1 ) ) + ( B x. ( { <. 1 , 0 >. , <. 2 , ( C / B ) >. } ` 2 ) ) ) = C ) |
61 |
60
|
eqeq1d |
|- ( ( M =/= ( C / B ) /\ ( ( A e. RR /\ ( B e. RR /\ B =/= 0 ) /\ C e. RR ) /\ M e. RR ) ) -> ( ( ( A x. ( { <. 1 , 0 >. , <. 2 , ( C / B ) >. } ` 1 ) ) + ( B x. ( { <. 1 , 0 >. , <. 2 , ( C / B ) >. } ` 2 ) ) ) = C <-> C = C ) ) |
62 |
46
|
eqeq1d |
|- ( ( ( A e. RR /\ ( B e. RR /\ B =/= 0 ) /\ C e. RR ) /\ M e. RR ) -> ( ( { <. 1 , 0 >. , <. 2 , ( C / B ) >. } ` 2 ) = M <-> ( C / B ) = M ) ) |
63 |
62
|
adantl |
|- ( ( M =/= ( C / B ) /\ ( ( A e. RR /\ ( B e. RR /\ B =/= 0 ) /\ C e. RR ) /\ M e. RR ) ) -> ( ( { <. 1 , 0 >. , <. 2 , ( C / B ) >. } ` 2 ) = M <-> ( C / B ) = M ) ) |
64 |
61 63
|
bibi12d |
|- ( ( M =/= ( C / B ) /\ ( ( A e. RR /\ ( B e. RR /\ B =/= 0 ) /\ C e. RR ) /\ M e. RR ) ) -> ( ( ( ( A x. ( { <. 1 , 0 >. , <. 2 , ( C / B ) >. } ` 1 ) ) + ( B x. ( { <. 1 , 0 >. , <. 2 , ( C / B ) >. } ` 2 ) ) ) = C <-> ( { <. 1 , 0 >. , <. 2 , ( C / B ) >. } ` 2 ) = M ) <-> ( C = C <-> ( C / B ) = M ) ) ) |
65 |
34 64
|
mtbird |
|- ( ( M =/= ( C / B ) /\ ( ( A e. RR /\ ( B e. RR /\ B =/= 0 ) /\ C e. RR ) /\ M e. RR ) ) -> -. ( ( ( A x. ( { <. 1 , 0 >. , <. 2 , ( C / B ) >. } ` 1 ) ) + ( B x. ( { <. 1 , 0 >. , <. 2 , ( C / B ) >. } ` 2 ) ) ) = C <-> ( { <. 1 , 0 >. , <. 2 , ( C / B ) >. } ` 2 ) = M ) ) |
66 |
|
fveq1 |
|- ( p = { <. 1 , 0 >. , <. 2 , ( C / B ) >. } -> ( p ` 1 ) = ( { <. 1 , 0 >. , <. 2 , ( C / B ) >. } ` 1 ) ) |
67 |
66
|
oveq2d |
|- ( p = { <. 1 , 0 >. , <. 2 , ( C / B ) >. } -> ( A x. ( p ` 1 ) ) = ( A x. ( { <. 1 , 0 >. , <. 2 , ( C / B ) >. } ` 1 ) ) ) |
68 |
|
fveq1 |
|- ( p = { <. 1 , 0 >. , <. 2 , ( C / B ) >. } -> ( p ` 2 ) = ( { <. 1 , 0 >. , <. 2 , ( C / B ) >. } ` 2 ) ) |
69 |
68
|
oveq2d |
|- ( p = { <. 1 , 0 >. , <. 2 , ( C / B ) >. } -> ( B x. ( p ` 2 ) ) = ( B x. ( { <. 1 , 0 >. , <. 2 , ( C / B ) >. } ` 2 ) ) ) |
70 |
67 69
|
oveq12d |
|- ( p = { <. 1 , 0 >. , <. 2 , ( C / B ) >. } -> ( ( A x. ( p ` 1 ) ) + ( B x. ( p ` 2 ) ) ) = ( ( A x. ( { <. 1 , 0 >. , <. 2 , ( C / B ) >. } ` 1 ) ) + ( B x. ( { <. 1 , 0 >. , <. 2 , ( C / B ) >. } ` 2 ) ) ) ) |
71 |
70
|
eqeq1d |
|- ( p = { <. 1 , 0 >. , <. 2 , ( C / B ) >. } -> ( ( ( A x. ( p ` 1 ) ) + ( B x. ( p ` 2 ) ) ) = C <-> ( ( A x. ( { <. 1 , 0 >. , <. 2 , ( C / B ) >. } ` 1 ) ) + ( B x. ( { <. 1 , 0 >. , <. 2 , ( C / B ) >. } ` 2 ) ) ) = C ) ) |
72 |
68
|
eqeq1d |
|- ( p = { <. 1 , 0 >. , <. 2 , ( C / B ) >. } -> ( ( p ` 2 ) = M <-> ( { <. 1 , 0 >. , <. 2 , ( C / B ) >. } ` 2 ) = M ) ) |
73 |
71 72
|
bibi12d |
|- ( p = { <. 1 , 0 >. , <. 2 , ( C / B ) >. } -> ( ( ( ( A x. ( p ` 1 ) ) + ( B x. ( p ` 2 ) ) ) = C <-> ( p ` 2 ) = M ) <-> ( ( ( A x. ( { <. 1 , 0 >. , <. 2 , ( C / B ) >. } ` 1 ) ) + ( B x. ( { <. 1 , 0 >. , <. 2 , ( C / B ) >. } ` 2 ) ) ) = C <-> ( { <. 1 , 0 >. , <. 2 , ( C / B ) >. } ` 2 ) = M ) ) ) |
74 |
73
|
notbid |
|- ( p = { <. 1 , 0 >. , <. 2 , ( C / B ) >. } -> ( -. ( ( ( A x. ( p ` 1 ) ) + ( B x. ( p ` 2 ) ) ) = C <-> ( p ` 2 ) = M ) <-> -. ( ( ( A x. ( { <. 1 , 0 >. , <. 2 , ( C / B ) >. } ` 1 ) ) + ( B x. ( { <. 1 , 0 >. , <. 2 , ( C / B ) >. } ` 2 ) ) ) = C <-> ( { <. 1 , 0 >. , <. 2 , ( C / B ) >. } ` 2 ) = M ) ) ) |
75 |
74
|
rspcev |
|- ( ( { <. 1 , 0 >. , <. 2 , ( C / B ) >. } e. P /\ -. ( ( ( A x. ( { <. 1 , 0 >. , <. 2 , ( C / B ) >. } ` 1 ) ) + ( B x. ( { <. 1 , 0 >. , <. 2 , ( C / B ) >. } ` 2 ) ) ) = C <-> ( { <. 1 , 0 >. , <. 2 , ( C / B ) >. } ` 2 ) = M ) ) -> E. p e. P -. ( ( ( A x. ( p ` 1 ) ) + ( B x. ( p ` 2 ) ) ) = C <-> ( p ` 2 ) = M ) ) |
76 |
24 65 75
|
syl2anc |
|- ( ( M =/= ( C / B ) /\ ( ( A e. RR /\ ( B e. RR /\ B =/= 0 ) /\ C e. RR ) /\ M e. RR ) ) -> E. p e. P -. ( ( ( A x. ( p ` 1 ) ) + ( B x. ( p ` 2 ) ) ) = C <-> ( p ` 2 ) = M ) ) |
77 |
76
|
ex |
|- ( M =/= ( C / B ) -> ( ( ( A e. RR /\ ( B e. RR /\ B =/= 0 ) /\ C e. RR ) /\ M e. RR ) -> E. p e. P -. ( ( ( A x. ( p ` 1 ) ) + ( B x. ( p ` 2 ) ) ) = C <-> ( p ` 2 ) = M ) ) ) |
78 |
|
nne |
|- ( -. M =/= ( C / B ) <-> M = ( C / B ) ) |
79 |
|
1red |
|- ( ( A e. RR /\ ( B e. RR /\ B =/= 0 ) /\ C e. RR ) -> 1 e. RR ) |
80 |
14 17 19
|
redivcld |
|- ( ( A e. RR /\ ( B e. RR /\ B =/= 0 ) /\ C e. RR ) -> ( C / B ) e. RR ) |
81 |
79 80
|
jca |
|- ( ( A e. RR /\ ( B e. RR /\ B =/= 0 ) /\ C e. RR ) -> ( 1 e. RR /\ ( C / B ) e. RR ) ) |
82 |
81
|
adantr |
|- ( ( ( A e. RR /\ ( B e. RR /\ B =/= 0 ) /\ C e. RR ) /\ M e. RR ) -> ( 1 e. RR /\ ( C / B ) e. RR ) ) |
83 |
1 3
|
prelrrx2 |
|- ( ( 1 e. RR /\ ( C / B ) e. RR ) -> { <. 1 , 1 >. , <. 2 , ( C / B ) >. } e. P ) |
84 |
82 83
|
syl |
|- ( ( ( A e. RR /\ ( B e. RR /\ B =/= 0 ) /\ C e. RR ) /\ M e. RR ) -> { <. 1 , 1 >. , <. 2 , ( C / B ) >. } e. P ) |
85 |
84
|
adantl |
|- ( ( ( M = ( C / B ) /\ A =/= 0 ) /\ ( ( A e. RR /\ ( B e. RR /\ B =/= 0 ) /\ C e. RR ) /\ M e. RR ) ) -> { <. 1 , 1 >. , <. 2 , ( C / B ) >. } e. P ) |
86 |
|
eqneqall |
|- ( A = 0 -> ( A =/= 0 -> -. ( C / B ) = M ) ) |
87 |
86
|
com12 |
|- ( A =/= 0 -> ( A = 0 -> -. ( C / B ) = M ) ) |
88 |
87
|
adantl |
|- ( ( M = ( C / B ) /\ A =/= 0 ) -> ( A = 0 -> -. ( C / B ) = M ) ) |
89 |
|
pm2.24 |
|- ( ( C / B ) = M -> ( -. ( C / B ) = M -> A = 0 ) ) |
90 |
89
|
eqcoms |
|- ( M = ( C / B ) -> ( -. ( C / B ) = M -> A = 0 ) ) |
91 |
90
|
adantr |
|- ( ( M = ( C / B ) /\ A =/= 0 ) -> ( -. ( C / B ) = M -> A = 0 ) ) |
92 |
88 91
|
impbid |
|- ( ( M = ( C / B ) /\ A =/= 0 ) -> ( A = 0 <-> -. ( C / B ) = M ) ) |
93 |
|
xor3 |
|- ( -. ( A = 0 <-> ( C / B ) = M ) <-> ( A = 0 <-> -. ( C / B ) = M ) ) |
94 |
92 93
|
sylibr |
|- ( ( M = ( C / B ) /\ A =/= 0 ) -> -. ( A = 0 <-> ( C / B ) = M ) ) |
95 |
94
|
adantr |
|- ( ( ( M = ( C / B ) /\ A =/= 0 ) /\ ( ( A e. RR /\ ( B e. RR /\ B =/= 0 ) /\ C e. RR ) /\ M e. RR ) ) -> -. ( A = 0 <-> ( C / B ) = M ) ) |
96 |
|
simprl1 |
|- ( ( ( M = ( C / B ) /\ A =/= 0 ) /\ ( ( A e. RR /\ ( B e. RR /\ B =/= 0 ) /\ C e. RR ) /\ M e. RR ) ) -> A e. RR ) |
97 |
96
|
recnd |
|- ( ( ( M = ( C / B ) /\ A =/= 0 ) /\ ( ( A e. RR /\ ( B e. RR /\ B =/= 0 ) /\ C e. RR ) /\ M e. RR ) ) -> A e. CC ) |
98 |
15
|
adantl |
|- ( ( ( M = ( C / B ) /\ A =/= 0 ) /\ ( ( A e. RR /\ ( B e. RR /\ B =/= 0 ) /\ C e. RR ) /\ M e. RR ) ) -> C e. RR ) |
99 |
98
|
recnd |
|- ( ( ( M = ( C / B ) /\ A =/= 0 ) /\ ( ( A e. RR /\ ( B e. RR /\ B =/= 0 ) /\ C e. RR ) /\ M e. RR ) ) -> C e. CC ) |
100 |
97 99
|
addcomd |
|- ( ( ( M = ( C / B ) /\ A =/= 0 ) /\ ( ( A e. RR /\ ( B e. RR /\ B =/= 0 ) /\ C e. RR ) /\ M e. RR ) ) -> ( A + C ) = ( C + A ) ) |
101 |
100
|
eqeq1d |
|- ( ( ( M = ( C / B ) /\ A =/= 0 ) /\ ( ( A e. RR /\ ( B e. RR /\ B =/= 0 ) /\ C e. RR ) /\ M e. RR ) ) -> ( ( A + C ) = C <-> ( C + A ) = C ) ) |
102 |
|
recn |
|- ( C e. RR -> C e. CC ) |
103 |
39 102
|
anim12ci |
|- ( ( A e. RR /\ C e. RR ) -> ( C e. CC /\ A e. CC ) ) |
104 |
103
|
3adant2 |
|- ( ( A e. RR /\ ( B e. RR /\ B =/= 0 ) /\ C e. RR ) -> ( C e. CC /\ A e. CC ) ) |
105 |
104
|
adantr |
|- ( ( ( A e. RR /\ ( B e. RR /\ B =/= 0 ) /\ C e. RR ) /\ M e. RR ) -> ( C e. CC /\ A e. CC ) ) |
106 |
105
|
adantl |
|- ( ( ( M = ( C / B ) /\ A =/= 0 ) /\ ( ( A e. RR /\ ( B e. RR /\ B =/= 0 ) /\ C e. RR ) /\ M e. RR ) ) -> ( C e. CC /\ A e. CC ) ) |
107 |
|
addid0 |
|- ( ( C e. CC /\ A e. CC ) -> ( ( C + A ) = C <-> A = 0 ) ) |
108 |
106 107
|
syl |
|- ( ( ( M = ( C / B ) /\ A =/= 0 ) /\ ( ( A e. RR /\ ( B e. RR /\ B =/= 0 ) /\ C e. RR ) /\ M e. RR ) ) -> ( ( C + A ) = C <-> A = 0 ) ) |
109 |
101 108
|
bitrd |
|- ( ( ( M = ( C / B ) /\ A =/= 0 ) /\ ( ( A e. RR /\ ( B e. RR /\ B =/= 0 ) /\ C e. RR ) /\ M e. RR ) ) -> ( ( A + C ) = C <-> A = 0 ) ) |
110 |
109
|
bibi1d |
|- ( ( ( M = ( C / B ) /\ A =/= 0 ) /\ ( ( A e. RR /\ ( B e. RR /\ B =/= 0 ) /\ C e. RR ) /\ M e. RR ) ) -> ( ( ( A + C ) = C <-> ( C / B ) = M ) <-> ( A = 0 <-> ( C / B ) = M ) ) ) |
111 |
95 110
|
mtbird |
|- ( ( ( M = ( C / B ) /\ A =/= 0 ) /\ ( ( A e. RR /\ ( B e. RR /\ B =/= 0 ) /\ C e. RR ) /\ M e. RR ) ) -> -. ( ( A + C ) = C <-> ( C / B ) = M ) ) |
112 |
|
1ex |
|- 1 e. _V |
113 |
112
|
a1i |
|- ( ( ( A e. RR /\ ( B e. RR /\ B =/= 0 ) /\ C e. RR ) /\ M e. RR ) -> 1 e. _V ) |
114 |
|
fv1prop |
|- ( 1 e. _V -> ( { <. 1 , 1 >. , <. 2 , ( C / B ) >. } ` 1 ) = 1 ) |
115 |
113 114
|
syl |
|- ( ( ( A e. RR /\ ( B e. RR /\ B =/= 0 ) /\ C e. RR ) /\ M e. RR ) -> ( { <. 1 , 1 >. , <. 2 , ( C / B ) >. } ` 1 ) = 1 ) |
116 |
115
|
oveq2d |
|- ( ( ( A e. RR /\ ( B e. RR /\ B =/= 0 ) /\ C e. RR ) /\ M e. RR ) -> ( A x. ( { <. 1 , 1 >. , <. 2 , ( C / B ) >. } ` 1 ) ) = ( A x. 1 ) ) |
117 |
|
ax-1rid |
|- ( A e. RR -> ( A x. 1 ) = A ) |
118 |
117
|
3ad2ant1 |
|- ( ( A e. RR /\ ( B e. RR /\ B =/= 0 ) /\ C e. RR ) -> ( A x. 1 ) = A ) |
119 |
118
|
adantr |
|- ( ( ( A e. RR /\ ( B e. RR /\ B =/= 0 ) /\ C e. RR ) /\ M e. RR ) -> ( A x. 1 ) = A ) |
120 |
116 119
|
eqtrd |
|- ( ( ( A e. RR /\ ( B e. RR /\ B =/= 0 ) /\ C e. RR ) /\ M e. RR ) -> ( A x. ( { <. 1 , 1 >. , <. 2 , ( C / B ) >. } ` 1 ) ) = A ) |
121 |
|
fv2prop |
|- ( ( C / B ) e. _V -> ( { <. 1 , 1 >. , <. 2 , ( C / B ) >. } ` 2 ) = ( C / B ) ) |
122 |
44 121
|
syl |
|- ( ( ( A e. RR /\ ( B e. RR /\ B =/= 0 ) /\ C e. RR ) /\ M e. RR ) -> ( { <. 1 , 1 >. , <. 2 , ( C / B ) >. } ` 2 ) = ( C / B ) ) |
123 |
122
|
oveq2d |
|- ( ( ( A e. RR /\ ( B e. RR /\ B =/= 0 ) /\ C e. RR ) /\ M e. RR ) -> ( B x. ( { <. 1 , 1 >. , <. 2 , ( C / B ) >. } ` 2 ) ) = ( B x. ( C / B ) ) ) |
124 |
15
|
recnd |
|- ( ( ( A e. RR /\ ( B e. RR /\ B =/= 0 ) /\ C e. RR ) /\ M e. RR ) -> C e. CC ) |
125 |
124 52 20
|
divcan2d |
|- ( ( ( A e. RR /\ ( B e. RR /\ B =/= 0 ) /\ C e. RR ) /\ M e. RR ) -> ( B x. ( C / B ) ) = C ) |
126 |
123 125
|
eqtrd |
|- ( ( ( A e. RR /\ ( B e. RR /\ B =/= 0 ) /\ C e. RR ) /\ M e. RR ) -> ( B x. ( { <. 1 , 1 >. , <. 2 , ( C / B ) >. } ` 2 ) ) = C ) |
127 |
120 126
|
oveq12d |
|- ( ( ( A e. RR /\ ( B e. RR /\ B =/= 0 ) /\ C e. RR ) /\ M e. RR ) -> ( ( A x. ( { <. 1 , 1 >. , <. 2 , ( C / B ) >. } ` 1 ) ) + ( B x. ( { <. 1 , 1 >. , <. 2 , ( C / B ) >. } ` 2 ) ) ) = ( A + C ) ) |
128 |
127
|
eqeq1d |
|- ( ( ( A e. RR /\ ( B e. RR /\ B =/= 0 ) /\ C e. RR ) /\ M e. RR ) -> ( ( ( A x. ( { <. 1 , 1 >. , <. 2 , ( C / B ) >. } ` 1 ) ) + ( B x. ( { <. 1 , 1 >. , <. 2 , ( C / B ) >. } ` 2 ) ) ) = C <-> ( A + C ) = C ) ) |
129 |
122
|
eqeq1d |
|- ( ( ( A e. RR /\ ( B e. RR /\ B =/= 0 ) /\ C e. RR ) /\ M e. RR ) -> ( ( { <. 1 , 1 >. , <. 2 , ( C / B ) >. } ` 2 ) = M <-> ( C / B ) = M ) ) |
130 |
128 129
|
bibi12d |
|- ( ( ( A e. RR /\ ( B e. RR /\ B =/= 0 ) /\ C e. RR ) /\ M e. RR ) -> ( ( ( ( A x. ( { <. 1 , 1 >. , <. 2 , ( C / B ) >. } ` 1 ) ) + ( B x. ( { <. 1 , 1 >. , <. 2 , ( C / B ) >. } ` 2 ) ) ) = C <-> ( { <. 1 , 1 >. , <. 2 , ( C / B ) >. } ` 2 ) = M ) <-> ( ( A + C ) = C <-> ( C / B ) = M ) ) ) |
131 |
130
|
notbid |
|- ( ( ( A e. RR /\ ( B e. RR /\ B =/= 0 ) /\ C e. RR ) /\ M e. RR ) -> ( -. ( ( ( A x. ( { <. 1 , 1 >. , <. 2 , ( C / B ) >. } ` 1 ) ) + ( B x. ( { <. 1 , 1 >. , <. 2 , ( C / B ) >. } ` 2 ) ) ) = C <-> ( { <. 1 , 1 >. , <. 2 , ( C / B ) >. } ` 2 ) = M ) <-> -. ( ( A + C ) = C <-> ( C / B ) = M ) ) ) |
132 |
131
|
adantl |
|- ( ( ( M = ( C / B ) /\ A =/= 0 ) /\ ( ( A e. RR /\ ( B e. RR /\ B =/= 0 ) /\ C e. RR ) /\ M e. RR ) ) -> ( -. ( ( ( A x. ( { <. 1 , 1 >. , <. 2 , ( C / B ) >. } ` 1 ) ) + ( B x. ( { <. 1 , 1 >. , <. 2 , ( C / B ) >. } ` 2 ) ) ) = C <-> ( { <. 1 , 1 >. , <. 2 , ( C / B ) >. } ` 2 ) = M ) <-> -. ( ( A + C ) = C <-> ( C / B ) = M ) ) ) |
133 |
111 132
|
mpbird |
|- ( ( ( M = ( C / B ) /\ A =/= 0 ) /\ ( ( A e. RR /\ ( B e. RR /\ B =/= 0 ) /\ C e. RR ) /\ M e. RR ) ) -> -. ( ( ( A x. ( { <. 1 , 1 >. , <. 2 , ( C / B ) >. } ` 1 ) ) + ( B x. ( { <. 1 , 1 >. , <. 2 , ( C / B ) >. } ` 2 ) ) ) = C <-> ( { <. 1 , 1 >. , <. 2 , ( C / B ) >. } ` 2 ) = M ) ) |
134 |
|
fveq1 |
|- ( p = { <. 1 , 1 >. , <. 2 , ( C / B ) >. } -> ( p ` 1 ) = ( { <. 1 , 1 >. , <. 2 , ( C / B ) >. } ` 1 ) ) |
135 |
134
|
oveq2d |
|- ( p = { <. 1 , 1 >. , <. 2 , ( C / B ) >. } -> ( A x. ( p ` 1 ) ) = ( A x. ( { <. 1 , 1 >. , <. 2 , ( C / B ) >. } ` 1 ) ) ) |
136 |
|
fveq1 |
|- ( p = { <. 1 , 1 >. , <. 2 , ( C / B ) >. } -> ( p ` 2 ) = ( { <. 1 , 1 >. , <. 2 , ( C / B ) >. } ` 2 ) ) |
137 |
136
|
oveq2d |
|- ( p = { <. 1 , 1 >. , <. 2 , ( C / B ) >. } -> ( B x. ( p ` 2 ) ) = ( B x. ( { <. 1 , 1 >. , <. 2 , ( C / B ) >. } ` 2 ) ) ) |
138 |
135 137
|
oveq12d |
|- ( p = { <. 1 , 1 >. , <. 2 , ( C / B ) >. } -> ( ( A x. ( p ` 1 ) ) + ( B x. ( p ` 2 ) ) ) = ( ( A x. ( { <. 1 , 1 >. , <. 2 , ( C / B ) >. } ` 1 ) ) + ( B x. ( { <. 1 , 1 >. , <. 2 , ( C / B ) >. } ` 2 ) ) ) ) |
139 |
138
|
eqeq1d |
|- ( p = { <. 1 , 1 >. , <. 2 , ( C / B ) >. } -> ( ( ( A x. ( p ` 1 ) ) + ( B x. ( p ` 2 ) ) ) = C <-> ( ( A x. ( { <. 1 , 1 >. , <. 2 , ( C / B ) >. } ` 1 ) ) + ( B x. ( { <. 1 , 1 >. , <. 2 , ( C / B ) >. } ` 2 ) ) ) = C ) ) |
140 |
136
|
eqeq1d |
|- ( p = { <. 1 , 1 >. , <. 2 , ( C / B ) >. } -> ( ( p ` 2 ) = M <-> ( { <. 1 , 1 >. , <. 2 , ( C / B ) >. } ` 2 ) = M ) ) |
141 |
139 140
|
bibi12d |
|- ( p = { <. 1 , 1 >. , <. 2 , ( C / B ) >. } -> ( ( ( ( A x. ( p ` 1 ) ) + ( B x. ( p ` 2 ) ) ) = C <-> ( p ` 2 ) = M ) <-> ( ( ( A x. ( { <. 1 , 1 >. , <. 2 , ( C / B ) >. } ` 1 ) ) + ( B x. ( { <. 1 , 1 >. , <. 2 , ( C / B ) >. } ` 2 ) ) ) = C <-> ( { <. 1 , 1 >. , <. 2 , ( C / B ) >. } ` 2 ) = M ) ) ) |
142 |
141
|
notbid |
|- ( p = { <. 1 , 1 >. , <. 2 , ( C / B ) >. } -> ( -. ( ( ( A x. ( p ` 1 ) ) + ( B x. ( p ` 2 ) ) ) = C <-> ( p ` 2 ) = M ) <-> -. ( ( ( A x. ( { <. 1 , 1 >. , <. 2 , ( C / B ) >. } ` 1 ) ) + ( B x. ( { <. 1 , 1 >. , <. 2 , ( C / B ) >. } ` 2 ) ) ) = C <-> ( { <. 1 , 1 >. , <. 2 , ( C / B ) >. } ` 2 ) = M ) ) ) |
143 |
142
|
rspcev |
|- ( ( { <. 1 , 1 >. , <. 2 , ( C / B ) >. } e. P /\ -. ( ( ( A x. ( { <. 1 , 1 >. , <. 2 , ( C / B ) >. } ` 1 ) ) + ( B x. ( { <. 1 , 1 >. , <. 2 , ( C / B ) >. } ` 2 ) ) ) = C <-> ( { <. 1 , 1 >. , <. 2 , ( C / B ) >. } ` 2 ) = M ) ) -> E. p e. P -. ( ( ( A x. ( p ` 1 ) ) + ( B x. ( p ` 2 ) ) ) = C <-> ( p ` 2 ) = M ) ) |
144 |
85 133 143
|
syl2anc |
|- ( ( ( M = ( C / B ) /\ A =/= 0 ) /\ ( ( A e. RR /\ ( B e. RR /\ B =/= 0 ) /\ C e. RR ) /\ M e. RR ) ) -> E. p e. P -. ( ( ( A x. ( p ` 1 ) ) + ( B x. ( p ` 2 ) ) ) = C <-> ( p ` 2 ) = M ) ) |
145 |
144
|
ex |
|- ( ( M = ( C / B ) /\ A =/= 0 ) -> ( ( ( A e. RR /\ ( B e. RR /\ B =/= 0 ) /\ C e. RR ) /\ M e. RR ) -> E. p e. P -. ( ( ( A x. ( p ` 1 ) ) + ( B x. ( p ` 2 ) ) ) = C <-> ( p ` 2 ) = M ) ) ) |
146 |
78 145
|
sylanb |
|- ( ( -. M =/= ( C / B ) /\ A =/= 0 ) -> ( ( ( A e. RR /\ ( B e. RR /\ B =/= 0 ) /\ C e. RR ) /\ M e. RR ) -> E. p e. P -. ( ( ( A x. ( p ` 1 ) ) + ( B x. ( p ` 2 ) ) ) = C <-> ( p ` 2 ) = M ) ) ) |
147 |
77 146
|
jaoi3 |
|- ( ( M =/= ( C / B ) \/ A =/= 0 ) -> ( ( ( A e. RR /\ ( B e. RR /\ B =/= 0 ) /\ C e. RR ) /\ M e. RR ) -> E. p e. P -. ( ( ( A x. ( p ` 1 ) ) + ( B x. ( p ` 2 ) ) ) = C <-> ( p ` 2 ) = M ) ) ) |
148 |
147
|
orcoms |
|- ( ( A =/= 0 \/ M =/= ( C / B ) ) -> ( ( ( A e. RR /\ ( B e. RR /\ B =/= 0 ) /\ C e. RR ) /\ M e. RR ) -> E. p e. P -. ( ( ( A x. ( p ` 1 ) ) + ( B x. ( p ` 2 ) ) ) = C <-> ( p ` 2 ) = M ) ) ) |
149 |
148
|
com12 |
|- ( ( ( A e. RR /\ ( B e. RR /\ B =/= 0 ) /\ C e. RR ) /\ M e. RR ) -> ( ( A =/= 0 \/ M =/= ( C / B ) ) -> E. p e. P -. ( ( ( A x. ( p ` 1 ) ) + ( B x. ( p ` 2 ) ) ) = C <-> ( p ` 2 ) = M ) ) ) |
150 |
|
rexnal |
|- ( E. p e. P -. ( ( ( A x. ( p ` 1 ) ) + ( B x. ( p ` 2 ) ) ) = C <-> ( p ` 2 ) = M ) <-> -. A. p e. P ( ( ( A x. ( p ` 1 ) ) + ( B x. ( p ` 2 ) ) ) = C <-> ( p ` 2 ) = M ) ) |
151 |
149 150
|
syl6ib |
|- ( ( ( A e. RR /\ ( B e. RR /\ B =/= 0 ) /\ C e. RR ) /\ M e. RR ) -> ( ( A =/= 0 \/ M =/= ( C / B ) ) -> -. A. p e. P ( ( ( A x. ( p ` 1 ) ) + ( B x. ( p ` 2 ) ) ) = C <-> ( p ` 2 ) = M ) ) ) |
152 |
12 151
|
syl5bi |
|- ( ( ( A e. RR /\ ( B e. RR /\ B =/= 0 ) /\ C e. RR ) /\ M e. RR ) -> ( -. ( A = 0 /\ M = ( C / B ) ) -> -. A. p e. P ( ( ( A x. ( p ` 1 ) ) + ( B x. ( p ` 2 ) ) ) = C <-> ( p ` 2 ) = M ) ) ) |
153 |
152
|
con4d |
|- ( ( ( A e. RR /\ ( B e. RR /\ B =/= 0 ) /\ C e. RR ) /\ M e. RR ) -> ( A. p e. P ( ( ( A x. ( p ` 1 ) ) + ( B x. ( p ` 2 ) ) ) = C <-> ( p ` 2 ) = M ) -> ( A = 0 /\ M = ( C / B ) ) ) ) |