Metamath Proof Explorer


Theorem line2x

Description: Example for a horizontal line G passing through two different points in "standard form". (Contributed by AV, 3-Feb-2023)

Ref Expression
Hypotheses line2.i I = 1 2
line2.e E = I
line2.p P = I
line2.l L = Line M E
line2.g G = p P | A p 1 + B p 2 = C
line2x.x X = 1 0 2 M
line2x.y Y = 1 1 2 M
Assertion line2x A B B 0 C M G = X L Y A = 0 M = C B

Proof

Step Hyp Ref Expression
1 line2.i I = 1 2
2 line2.e E = I
3 line2.p P = I
4 line2.l L = Line M E
5 line2.g G = p P | A p 1 + B p 2 = C
6 line2x.x X = 1 0 2 M
7 line2x.y Y = 1 1 2 M
8 5 a1i A B B 0 C M G = p P | A p 1 + B p 2 = C
9 1ex 1 V
10 2ex 2 V
11 9 10 pm3.2i 1 V 2 V
12 c0ex 0 V
13 12 jctl M 0 V M
14 1ne2 1 2
15 14 a1i M 1 2
16 fprg 1 V 2 V 0 V M 1 2 1 0 2 M : 1 2 0 M
17 0red 1 V 2 V 0
18 simpr 0 V M M
19 17 18 anim12i 1 V 2 V 0 V M 0 M
20 19 3adant3 1 V 2 V 0 V M 1 2 0 M
21 prssi 0 M 0 M
22 20 21 syl 1 V 2 V 0 V M 1 2 0 M
23 16 22 fssd 1 V 2 V 0 V M 1 2 1 0 2 M : 1 2
24 11 13 15 23 mp3an2i M 1 0 2 M : 1 2
25 1 feq2i 1 0 2 M : I 1 0 2 M : 1 2
26 24 25 sylibr M 1 0 2 M : I
27 reex V
28 prex 1 2 V
29 1 28 eqeltri I V
30 27 29 elmap 1 0 2 M I 1 0 2 M : I
31 26 30 sylibr M 1 0 2 M I
32 31 6 3 3eltr4g M X P
33 9 jctl M 1 V M
34 fprg 1 V 2 V 1 V M 1 2 1 1 2 M : 1 2 1 M
35 11 33 15 34 mp3an2i M 1 1 2 M : 1 2 1 M
36 1re 1
37 prssi 1 M 1 M
38 36 37 mpan M 1 M
39 35 38 fssd M 1 1 2 M : 1 2
40 1 feq2i 1 1 2 M : I 1 1 2 M : 1 2
41 39 40 sylibr M 1 1 2 M : I
42 27 29 pm3.2i V I V
43 elmapg V I V 1 1 2 M I 1 1 2 M : I
44 42 43 mp1i M 1 1 2 M I 1 1 2 M : I
45 41 44 mpbird M 1 1 2 M I
46 45 7 3 3eltr4g M Y P
47 opex 1 0 V
48 opex 2 M V
49 47 48 pm3.2i 1 0 V 2 M V
50 opex 1 1 V
51 50 48 pm3.2i 1 1 V 2 M V
52 49 51 pm3.2i 1 0 V 2 M V 1 1 V 2 M V
53 14 orci 1 2 0 M
54 9 12 opthne 1 0 2 M 1 2 0 M
55 53 54 mpbir 1 0 2 M
56 55 a1i M 1 0 2 M
57 0ne1 0 1
58 57 olci 1 1 0 1
59 9 12 opthne 1 0 1 1 1 1 0 1
60 58 59 mpbir 1 0 1 1
61 56 60 jctil M 1 0 1 1 1 0 2 M
62 61 orcd M 1 0 1 1 1 0 2 M 2 M 1 1 2 M 2 M
63 prneimg 1 0 V 2 M V 1 1 V 2 M V 1 0 1 1 1 0 2 M 2 M 1 1 2 M 2 M 1 0 2 M 1 1 2 M
64 52 62 63 mpsyl M 1 0 2 M 1 1 2 M
65 64 6 7 3netr4g M X Y
66 32 46 65 3jca M X P Y P X Y
67 66 adantl A B B 0 C M X P Y P X Y
68 eqid Y 1 X 1 = Y 1 X 1
69 eqid Y 2 X 2 = Y 2 X 2
70 eqid X 2 Y 1 X 1 Y 2 = X 2 Y 1 X 1 Y 2
71 1 2 3 4 68 69 70 rrx2linest X P Y P X Y X L Y = p P | Y 1 X 1 p 2 = Y 2 X 2 p 1 + X 2 Y 1 - X 1 Y 2
72 67 71 syl A B B 0 C M X L Y = p P | Y 1 X 1 p 2 = Y 2 X 2 p 1 + X 2 Y 1 - X 1 Y 2
73 8 72 eqeq12d A B B 0 C M G = X L Y p P | A p 1 + B p 2 = C = p P | Y 1 X 1 p 2 = Y 2 X 2 p 1 + X 2 Y 1 - X 1 Y 2
74 rabbi p P A p 1 + B p 2 = C Y 1 X 1 p 2 = Y 2 X 2 p 1 + X 2 Y 1 - X 1 Y 2 p P | A p 1 + B p 2 = C = p P | Y 1 X 1 p 2 = Y 2 X 2 p 1 + X 2 Y 1 - X 1 Y 2
75 7 fveq1i Y 1 = 1 1 2 M 1
76 9 9 14 3pm3.2i 1 V 1 V 1 2
77 fvpr1g 1 V 1 V 1 2 1 1 2 M 1 = 1
78 76 77 mp1i M 1 1 2 M 1 = 1
79 75 78 syl5eq M Y 1 = 1
80 6 fveq1i X 1 = 1 0 2 M 1
81 9 12 14 3pm3.2i 1 V 0 V 1 2
82 fvpr1g 1 V 0 V 1 2 1 0 2 M 1 = 0
83 81 82 mp1i M 1 0 2 M 1 = 0
84 80 83 syl5eq M X 1 = 0
85 79 84 oveq12d M Y 1 X 1 = 1 0
86 1m0e1 1 0 = 1
87 85 86 eqtrdi M Y 1 X 1 = 1
88 87 oveq1d M Y 1 X 1 p 2 = 1 p 2
89 7 fveq1i Y 2 = 1 1 2 M 2
90 fvpr2g 2 V M 1 2 1 1 2 M 2 = M
91 10 14 90 mp3an13 M 1 1 2 M 2 = M
92 89 91 syl5eq M Y 2 = M
93 6 fveq1i X 2 = 1 0 2 M 2
94 fvpr2g 2 V M 1 2 1 0 2 M 2 = M
95 10 14 94 mp3an13 M 1 0 2 M 2 = M
96 93 95 syl5eq M X 2 = M
97 92 96 oveq12d M Y 2 X 2 = M M
98 recn M M
99 98 subidd M M M = 0
100 97 99 eqtrd M Y 2 X 2 = 0
101 100 oveq1d M Y 2 X 2 p 1 = 0 p 1
102 9 9 15 77 mp3an12i M 1 1 2 M 1 = 1
103 75 102 syl5eq M Y 1 = 1
104 96 103 oveq12d M X 2 Y 1 = M 1
105 ax-1rid M M 1 = M
106 104 105 eqtrd M X 2 Y 1 = M
107 9 12 15 82 mp3an12i M 1 0 2 M 1 = 0
108 80 107 syl5eq M X 1 = 0
109 108 92 oveq12d M X 1 Y 2 = 0 M
110 98 mul02d M 0 M = 0
111 109 110 eqtrd M X 1 Y 2 = 0
112 106 111 oveq12d M X 2 Y 1 X 1 Y 2 = M 0
113 98 subid1d M M 0 = M
114 112 113 eqtrd M X 2 Y 1 X 1 Y 2 = M
115 101 114 oveq12d M Y 2 X 2 p 1 + X 2 Y 1 - X 1 Y 2 = 0 p 1 + M
116 88 115 eqeq12d M Y 1 X 1 p 2 = Y 2 X 2 p 1 + X 2 Y 1 - X 1 Y 2 1 p 2 = 0 p 1 + M
117 116 adantl A B B 0 C M Y 1 X 1 p 2 = Y 2 X 2 p 1 + X 2 Y 1 - X 1 Y 2 1 p 2 = 0 p 1 + M
118 1 3 rrx2pyel p P p 2
119 118 recnd p P p 2
120 119 mulid2d p P 1 p 2 = p 2
121 1 3 rrx2pxel p P p 1
122 121 recnd p P p 1
123 122 mul02d p P 0 p 1 = 0
124 123 oveq1d p P 0 p 1 + M = 0 + M
125 120 124 eqeq12d p P 1 p 2 = 0 p 1 + M p 2 = 0 + M
126 117 125 sylan9bb A B B 0 C M p P Y 1 X 1 p 2 = Y 2 X 2 p 1 + X 2 Y 1 - X 1 Y 2 p 2 = 0 + M
127 126 bibi2d A B B 0 C M p P A p 1 + B p 2 = C Y 1 X 1 p 2 = Y 2 X 2 p 1 + X 2 Y 1 - X 1 Y 2 A p 1 + B p 2 = C p 2 = 0 + M
128 127 ralbidva A B B 0 C M p P A p 1 + B p 2 = C Y 1 X 1 p 2 = Y 2 X 2 p 1 + X 2 Y 1 - X 1 Y 2 p P A p 1 + B p 2 = C p 2 = 0 + M
129 98 addid2d M 0 + M = M
130 129 adantr M p P 0 + M = M
131 130 eqeq2d M p P p 2 = 0 + M p 2 = M
132 131 bibi2d M p P A p 1 + B p 2 = C p 2 = 0 + M A p 1 + B p 2 = C p 2 = M
133 132 ralbidva M p P A p 1 + B p 2 = C p 2 = 0 + M p P A p 1 + B p 2 = C p 2 = M
134 133 adantl A B B 0 C M p P A p 1 + B p 2 = C p 2 = 0 + M p P A p 1 + B p 2 = C p 2 = M
135 1 2 3 4 5 6 7 line2xlem A B B 0 C M p P A p 1 + B p 2 = C p 2 = M A = 0 M = C B
136 oveq1 A = 0 A p 1 = 0 p 1
137 136 adantr A = 0 M = C B A p 1 = 0 p 1
138 137 ad2antlr A B B 0 C M A = 0 M = C B p P A p 1 = 0 p 1
139 123 adantl A B B 0 C M A = 0 M = C B p P 0 p 1 = 0
140 138 139 eqtrd A B B 0 C M A = 0 M = C B p P A p 1 = 0
141 140 oveq1d A B B 0 C M A = 0 M = C B p P A p 1 + B p 2 = 0 + B p 2
142 recn B B
143 142 adantr B B 0 B
144 143 3ad2ant2 A B B 0 C B
145 144 ad3antrrr A B B 0 C M A = 0 M = C B p P B
146 119 adantl A B B 0 C M A = 0 M = C B p P p 2
147 145 146 mulcld A B B 0 C M A = 0 M = C B p P B p 2
148 147 addid2d A B B 0 C M A = 0 M = C B p P 0 + B p 2 = B p 2
149 141 148 eqtrd A B B 0 C M A = 0 M = C B p P A p 1 + B p 2 = B p 2
150 149 eqeq1d A B B 0 C M A = 0 M = C B p P A p 1 + B p 2 = C B p 2 = C
151 simp3 A B B 0 C C
152 151 recnd A B B 0 C C
153 152 ad3antrrr A B B 0 C M A = 0 M = C B p P C
154 simpl B B 0 B
155 154 recnd B B 0 B
156 155 3ad2ant2 A B B 0 C B
157 156 ad3antrrr A B B 0 C M A = 0 M = C B p P B
158 simp2r A B B 0 C B 0
159 158 ad3antrrr A B B 0 C M A = 0 M = C B p P B 0
160 153 157 146 159 divmuld A B B 0 C M A = 0 M = C B p P C B = p 2 B p 2 = C
161 simpr A = 0 M = C B M = C B
162 161 eqcomd A = 0 M = C B C B = M
163 162 ad2antlr A B B 0 C M A = 0 M = C B p P C B = M
164 163 eqeq1d A B B 0 C M A = 0 M = C B p P C B = p 2 M = p 2
165 150 160 164 3bitr2d A B B 0 C M A = 0 M = C B p P A p 1 + B p 2 = C M = p 2
166 eqcom M = p 2 p 2 = M
167 165 166 bitrdi A B B 0 C M A = 0 M = C B p P A p 1 + B p 2 = C p 2 = M
168 167 ralrimiva A B B 0 C M A = 0 M = C B p P A p 1 + B p 2 = C p 2 = M
169 168 ex A B B 0 C M A = 0 M = C B p P A p 1 + B p 2 = C p 2 = M
170 135 169 impbid A B B 0 C M p P A p 1 + B p 2 = C p 2 = M A = 0 M = C B
171 128 134 170 3bitrd A B B 0 C M p P A p 1 + B p 2 = C Y 1 X 1 p 2 = Y 2 X 2 p 1 + X 2 Y 1 - X 1 Y 2 A = 0 M = C B
172 74 171 bitr3id A B B 0 C M p P | A p 1 + B p 2 = C = p P | Y 1 X 1 p 2 = Y 2 X 2 p 1 + X 2 Y 1 - X 1 Y 2 A = 0 M = C B
173 73 172 bitrd A B B 0 C M G = X L Y A = 0 M = C B