Metamath Proof Explorer


Theorem fvline

Description: Calculate the value of the Line function. (Contributed by Scott Fenton, 25-Oct-2013) (Revised by Mario Carneiro, 19-Apr-2014)

Ref Expression
Assertion fvline N A 𝔼 N B 𝔼 N A B A Line B = x | x Colinear A B

Proof

Step Hyp Ref Expression
1 eqid A B Colinear -1 = A B Colinear -1
2 fveq2 n = N 𝔼 n = 𝔼 N
3 2 eleq2d n = N A 𝔼 n A 𝔼 N
4 2 eleq2d n = N B 𝔼 n B 𝔼 N
5 3 4 3anbi12d n = N A 𝔼 n B 𝔼 n A B A 𝔼 N B 𝔼 N A B
6 5 anbi1d n = N A 𝔼 n B 𝔼 n A B A B Colinear -1 = A B Colinear -1 A 𝔼 N B 𝔼 N A B A B Colinear -1 = A B Colinear -1
7 6 rspcev N A 𝔼 N B 𝔼 N A B A B Colinear -1 = A B Colinear -1 n A 𝔼 n B 𝔼 n A B A B Colinear -1 = A B Colinear -1
8 1 7 mpanr2 N A 𝔼 N B 𝔼 N A B n A 𝔼 n B 𝔼 n A B A B Colinear -1 = A B Colinear -1
9 simpr1 N A 𝔼 N B 𝔼 N A B A 𝔼 N
10 simpr2 N A 𝔼 N B 𝔼 N A B B 𝔼 N
11 colinearex Colinear V
12 11 cnvex Colinear -1 V
13 ecexg Colinear -1 V A B Colinear -1 V
14 12 13 ax-mp A B Colinear -1 V
15 eleq1 a = A a 𝔼 n A 𝔼 n
16 neeq1 a = A a b A b
17 15 16 3anbi13d a = A a 𝔼 n b 𝔼 n a b A 𝔼 n b 𝔼 n A b
18 opeq1 a = A a b = A b
19 18 eceq1d a = A a b Colinear -1 = A b Colinear -1
20 19 eqeq2d a = A l = a b Colinear -1 l = A b Colinear -1
21 17 20 anbi12d a = A a 𝔼 n b 𝔼 n a b l = a b Colinear -1 A 𝔼 n b 𝔼 n A b l = A b Colinear -1
22 21 rexbidv a = A n a 𝔼 n b 𝔼 n a b l = a b Colinear -1 n A 𝔼 n b 𝔼 n A b l = A b Colinear -1
23 eleq1 b = B b 𝔼 n B 𝔼 n
24 neeq2 b = B A b A B
25 23 24 3anbi23d b = B A 𝔼 n b 𝔼 n A b A 𝔼 n B 𝔼 n A B
26 opeq2 b = B A b = A B
27 26 eceq1d b = B A b Colinear -1 = A B Colinear -1
28 27 eqeq2d b = B l = A b Colinear -1 l = A B Colinear -1
29 25 28 anbi12d b = B A 𝔼 n b 𝔼 n A b l = A b Colinear -1 A 𝔼 n B 𝔼 n A B l = A B Colinear -1
30 29 rexbidv b = B n A 𝔼 n b 𝔼 n A b l = A b Colinear -1 n A 𝔼 n B 𝔼 n A B l = A B Colinear -1
31 eqeq1 l = A B Colinear -1 l = A B Colinear -1 A B Colinear -1 = A B Colinear -1
32 31 anbi2d l = A B Colinear -1 A 𝔼 n B 𝔼 n A B l = A B Colinear -1 A 𝔼 n B 𝔼 n A B A B Colinear -1 = A B Colinear -1
33 32 rexbidv l = A B Colinear -1 n A 𝔼 n B 𝔼 n A B l = A B Colinear -1 n A 𝔼 n B 𝔼 n A B A B Colinear -1 = A B Colinear -1
34 22 30 33 eloprabg A 𝔼 N B 𝔼 N A B Colinear -1 V A B A B Colinear -1 a b l | n a 𝔼 n b 𝔼 n a b l = a b Colinear -1 n A 𝔼 n B 𝔼 n A B A B Colinear -1 = A B Colinear -1
35 14 34 mp3an3 A 𝔼 N B 𝔼 N A B A B Colinear -1 a b l | n a 𝔼 n b 𝔼 n a b l = a b Colinear -1 n A 𝔼 n B 𝔼 n A B A B Colinear -1 = A B Colinear -1
36 9 10 35 syl2anc N A 𝔼 N B 𝔼 N A B A B A B Colinear -1 a b l | n a 𝔼 n b 𝔼 n a b l = a b Colinear -1 n A 𝔼 n B 𝔼 n A B A B Colinear -1 = A B Colinear -1
37 8 36 mpbird N A 𝔼 N B 𝔼 N A B A B A B Colinear -1 a b l | n a 𝔼 n b 𝔼 n a b l = a b Colinear -1
38 df-ov A Line B = Line A B
39 df-br A B Line A B Colinear -1 A B A B Colinear -1 Line
40 df-line2 Line = a b l | n a 𝔼 n b 𝔼 n a b l = a b Colinear -1
41 40 eleq2i A B A B Colinear -1 Line A B A B Colinear -1 a b l | n a 𝔼 n b 𝔼 n a b l = a b Colinear -1
42 39 41 bitri A B Line A B Colinear -1 A B A B Colinear -1 a b l | n a 𝔼 n b 𝔼 n a b l = a b Colinear -1
43 funline Fun Line
44 funbrfv Fun Line A B Line A B Colinear -1 Line A B = A B Colinear -1
45 43 44 ax-mp A B Line A B Colinear -1 Line A B = A B Colinear -1
46 42 45 sylbir A B A B Colinear -1 a b l | n a 𝔼 n b 𝔼 n a b l = a b Colinear -1 Line A B = A B Colinear -1
47 38 46 syl5eq A B A B Colinear -1 a b l | n a 𝔼 n b 𝔼 n a b l = a b Colinear -1 A Line B = A B Colinear -1
48 37 47 syl N A 𝔼 N B 𝔼 N A B A Line B = A B Colinear -1
49 opex A B V
50 dfec2 A B V A B Colinear -1 = x | A B Colinear -1 x
51 49 50 ax-mp A B Colinear -1 = x | A B Colinear -1 x
52 vex x V
53 49 52 brcnv A B Colinear -1 x x Colinear A B
54 53 abbii x | A B Colinear -1 x = x | x Colinear A B
55 51 54 eqtri A B Colinear -1 = x | x Colinear A B
56 48 55 eqtrdi N A 𝔼 N B 𝔼 N A B A Line B = x | x Colinear A B