Metamath Proof Explorer

Theorem brifs

Description: Binary relation form of the inner five segment predicate. (Contributed by Scott Fenton, 26-Sep-2013)

Ref Expression
Assertion brifs 
|- ( ( ( N e. NN /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) /\ E e. ( EE ` N ) ) /\ ( F e. ( EE ` N ) /\ G e. ( EE ` N ) /\ H e. ( EE ` N ) ) ) -> ( <. <. A , B >. , <. C , D >. >. InnerFiveSeg <. <. E , F >. , <. G , H >. >. <-> ( ( B Btwn <. A , C >. /\ F Btwn <. E , G >. ) /\ ( <. A , C >. Cgr <. E , G >. /\ <. B , C >. Cgr <. F , G >. ) /\ ( <. A , D >. Cgr <. E , H >. /\ <. C , D >. Cgr <. G , H >. ) ) ) )

Proof

Step Hyp Ref Expression
1 opeq1 
 |-  ( a = A -> <. a , c >. = <. A , c >. )
2 1 breq2d 
 |-  ( a = A -> ( b Btwn <. a , c >. <-> b Btwn <. A , c >. ) )
3 2 anbi1d 
 |-  ( a = A -> ( ( b Btwn <. a , c >. /\ f Btwn <. e , g >. ) <-> ( b Btwn <. A , c >. /\ f Btwn <. e , g >. ) ) )
4 1 breq1d 
 |-  ( a = A -> ( <. a , c >. Cgr <. e , g >. <-> <. A , c >. Cgr <. e , g >. ) )
5 4 anbi1d 
 |-  ( a = A -> ( ( <. a , c >. Cgr <. e , g >. /\ <. b , c >. Cgr <. f , g >. ) <-> ( <. A , c >. Cgr <. e , g >. /\ <. b , c >. Cgr <. f , g >. ) ) )
6 opeq1 
 |-  ( a = A -> <. a , d >. = <. A , d >. )
7 6 breq1d 
 |-  ( a = A -> ( <. a , d >. Cgr <. e , h >. <-> <. A , d >. Cgr <. e , h >. ) )
8 7 anbi1d 
 |-  ( a = A -> ( ( <. a , d >. Cgr <. e , h >. /\ <. c , d >. Cgr <. g , h >. ) <-> ( <. A , d >. Cgr <. e , h >. /\ <. c , d >. Cgr <. g , h >. ) ) )
9 3 5 8 3anbi123d 
 |-  ( a = A -> ( ( ( b Btwn <. a , c >. /\ f Btwn <. e , g >. ) /\ ( <. a , c >. Cgr <. e , g >. /\ <. b , c >. Cgr <. f , g >. ) /\ ( <. a , d >. Cgr <. e , h >. /\ <. c , d >. Cgr <. g , h >. ) ) <-> ( ( b Btwn <. A , c >. /\ f Btwn <. e , g >. ) /\ ( <. A , c >. Cgr <. e , g >. /\ <. b , c >. Cgr <. f , g >. ) /\ ( <. A , d >. Cgr <. e , h >. /\ <. c , d >. Cgr <. g , h >. ) ) ) )
10 breq1 
 |-  ( b = B -> ( b Btwn <. A , c >. <-> B Btwn <. A , c >. ) )
11 10 anbi1d 
 |-  ( b = B -> ( ( b Btwn <. A , c >. /\ f Btwn <. e , g >. ) <-> ( B Btwn <. A , c >. /\ f Btwn <. e , g >. ) ) )
12 opeq1 
 |-  ( b = B -> <. b , c >. = <. B , c >. )
13 12 breq1d 
 |-  ( b = B -> ( <. b , c >. Cgr <. f , g >. <-> <. B , c >. Cgr <. f , g >. ) )
14 13 anbi2d 
 |-  ( b = B -> ( ( <. A , c >. Cgr <. e , g >. /\ <. b , c >. Cgr <. f , g >. ) <-> ( <. A , c >. Cgr <. e , g >. /\ <. B , c >. Cgr <. f , g >. ) ) )
15 11 14 3anbi12d 
 |-  ( b = B -> ( ( ( b Btwn <. A , c >. /\ f Btwn <. e , g >. ) /\ ( <. A , c >. Cgr <. e , g >. /\ <. b , c >. Cgr <. f , g >. ) /\ ( <. A , d >. Cgr <. e , h >. /\ <. c , d >. Cgr <. g , h >. ) ) <-> ( ( B Btwn <. A , c >. /\ f Btwn <. e , g >. ) /\ ( <. A , c >. Cgr <. e , g >. /\ <. B , c >. Cgr <. f , g >. ) /\ ( <. A , d >. Cgr <. e , h >. /\ <. c , d >. Cgr <. g , h >. ) ) ) )
16 opeq2 
 |-  ( c = C -> <. A , c >. = <. A , C >. )
17 16 breq2d 
 |-  ( c = C -> ( B Btwn <. A , c >. <-> B Btwn <. A , C >. ) )
18 17 anbi1d 
 |-  ( c = C -> ( ( B Btwn <. A , c >. /\ f Btwn <. e , g >. ) <-> ( B Btwn <. A , C >. /\ f Btwn <. e , g >. ) ) )
19 16 breq1d 
 |-  ( c = C -> ( <. A , c >. Cgr <. e , g >. <-> <. A , C >. Cgr <. e , g >. ) )
20 opeq2 
 |-  ( c = C -> <. B , c >. = <. B , C >. )
21 20 breq1d 
 |-  ( c = C -> ( <. B , c >. Cgr <. f , g >. <-> <. B , C >. Cgr <. f , g >. ) )
22 19 21 anbi12d 
 |-  ( c = C -> ( ( <. A , c >. Cgr <. e , g >. /\ <. B , c >. Cgr <. f , g >. ) <-> ( <. A , C >. Cgr <. e , g >. /\ <. B , C >. Cgr <. f , g >. ) ) )
23 opeq1 
 |-  ( c = C -> <. c , d >. = <. C , d >. )
24 23 breq1d 
 |-  ( c = C -> ( <. c , d >. Cgr <. g , h >. <-> <. C , d >. Cgr <. g , h >. ) )
25 24 anbi2d 
 |-  ( c = C -> ( ( <. A , d >. Cgr <. e , h >. /\ <. c , d >. Cgr <. g , h >. ) <-> ( <. A , d >. Cgr <. e , h >. /\ <. C , d >. Cgr <. g , h >. ) ) )
26 18 22 25 3anbi123d 
 |-  ( c = C -> ( ( ( B Btwn <. A , c >. /\ f Btwn <. e , g >. ) /\ ( <. A , c >. Cgr <. e , g >. /\ <. B , c >. Cgr <. f , g >. ) /\ ( <. A , d >. Cgr <. e , h >. /\ <. c , d >. Cgr <. g , h >. ) ) <-> ( ( B Btwn <. A , C >. /\ f Btwn <. e , g >. ) /\ ( <. A , C >. Cgr <. e , g >. /\ <. B , C >. Cgr <. f , g >. ) /\ ( <. A , d >. Cgr <. e , h >. /\ <. C , d >. Cgr <. g , h >. ) ) ) )
27 opeq2 
 |-  ( d = D -> <. A , d >. = <. A , D >. )
28 27 breq1d 
 |-  ( d = D -> ( <. A , d >. Cgr <. e , h >. <-> <. A , D >. Cgr <. e , h >. ) )
29 opeq2 
 |-  ( d = D -> <. C , d >. = <. C , D >. )
30 29 breq1d 
 |-  ( d = D -> ( <. C , d >. Cgr <. g , h >. <-> <. C , D >. Cgr <. g , h >. ) )
31 28 30 anbi12d 
 |-  ( d = D -> ( ( <. A , d >. Cgr <. e , h >. /\ <. C , d >. Cgr <. g , h >. ) <-> ( <. A , D >. Cgr <. e , h >. /\ <. C , D >. Cgr <. g , h >. ) ) )
32 31 3anbi3d 
 |-  ( d = D -> ( ( ( B Btwn <. A , C >. /\ f Btwn <. e , g >. ) /\ ( <. A , C >. Cgr <. e , g >. /\ <. B , C >. Cgr <. f , g >. ) /\ ( <. A , d >. Cgr <. e , h >. /\ <. C , d >. Cgr <. g , h >. ) ) <-> ( ( B Btwn <. A , C >. /\ f Btwn <. e , g >. ) /\ ( <. A , C >. Cgr <. e , g >. /\ <. B , C >. Cgr <. f , g >. ) /\ ( <. A , D >. Cgr <. e , h >. /\ <. C , D >. Cgr <. g , h >. ) ) ) )
33 opeq1 
 |-  ( e = E -> <. e , g >. = <. E , g >. )
34 33 breq2d 
 |-  ( e = E -> ( f Btwn <. e , g >. <-> f Btwn <. E , g >. ) )
35 34 anbi2d 
 |-  ( e = E -> ( ( B Btwn <. A , C >. /\ f Btwn <. e , g >. ) <-> ( B Btwn <. A , C >. /\ f Btwn <. E , g >. ) ) )
36 33 breq2d 
 |-  ( e = E -> ( <. A , C >. Cgr <. e , g >. <-> <. A , C >. Cgr <. E , g >. ) )
37 36 anbi1d 
 |-  ( e = E -> ( ( <. A , C >. Cgr <. e , g >. /\ <. B , C >. Cgr <. f , g >. ) <-> ( <. A , C >. Cgr <. E , g >. /\ <. B , C >. Cgr <. f , g >. ) ) )
38 opeq1 
 |-  ( e = E -> <. e , h >. = <. E , h >. )
39 38 breq2d 
 |-  ( e = E -> ( <. A , D >. Cgr <. e , h >. <-> <. A , D >. Cgr <. E , h >. ) )
40 39 anbi1d 
 |-  ( e = E -> ( ( <. A , D >. Cgr <. e , h >. /\ <. C , D >. Cgr <. g , h >. ) <-> ( <. A , D >. Cgr <. E , h >. /\ <. C , D >. Cgr <. g , h >. ) ) )
41 35 37 40 3anbi123d 
 |-  ( e = E -> ( ( ( B Btwn <. A , C >. /\ f Btwn <. e , g >. ) /\ ( <. A , C >. Cgr <. e , g >. /\ <. B , C >. Cgr <. f , g >. ) /\ ( <. A , D >. Cgr <. e , h >. /\ <. C , D >. Cgr <. g , h >. ) ) <-> ( ( B Btwn <. A , C >. /\ f Btwn <. E , g >. ) /\ ( <. A , C >. Cgr <. E , g >. /\ <. B , C >. Cgr <. f , g >. ) /\ ( <. A , D >. Cgr <. E , h >. /\ <. C , D >. Cgr <. g , h >. ) ) ) )
42 breq1 
 |-  ( f = F -> ( f Btwn <. E , g >. <-> F Btwn <. E , g >. ) )
43 42 anbi2d 
 |-  ( f = F -> ( ( B Btwn <. A , C >. /\ f Btwn <. E , g >. ) <-> ( B Btwn <. A , C >. /\ F Btwn <. E , g >. ) ) )
44 opeq1 
 |-  ( f = F -> <. f , g >. = <. F , g >. )
45 44 breq2d 
 |-  ( f = F -> ( <. B , C >. Cgr <. f , g >. <-> <. B , C >. Cgr <. F , g >. ) )
46 45 anbi2d 
 |-  ( f = F -> ( ( <. A , C >. Cgr <. E , g >. /\ <. B , C >. Cgr <. f , g >. ) <-> ( <. A , C >. Cgr <. E , g >. /\ <. B , C >. Cgr <. F , g >. ) ) )
47 43 46 3anbi12d 
 |-  ( f = F -> ( ( ( B Btwn <. A , C >. /\ f Btwn <. E , g >. ) /\ ( <. A , C >. Cgr <. E , g >. /\ <. B , C >. Cgr <. f , g >. ) /\ ( <. A , D >. Cgr <. E , h >. /\ <. C , D >. Cgr <. g , h >. ) ) <-> ( ( B Btwn <. A , C >. /\ F Btwn <. E , g >. ) /\ ( <. A , C >. Cgr <. E , g >. /\ <. B , C >. Cgr <. F , g >. ) /\ ( <. A , D >. Cgr <. E , h >. /\ <. C , D >. Cgr <. g , h >. ) ) ) )
48 opeq2 
 |-  ( g = G -> <. E , g >. = <. E , G >. )
49 48 breq2d 
 |-  ( g = G -> ( F Btwn <. E , g >. <-> F Btwn <. E , G >. ) )
50 49 anbi2d 
 |-  ( g = G -> ( ( B Btwn <. A , C >. /\ F Btwn <. E , g >. ) <-> ( B Btwn <. A , C >. /\ F Btwn <. E , G >. ) ) )
51 48 breq2d 
 |-  ( g = G -> ( <. A , C >. Cgr <. E , g >. <-> <. A , C >. Cgr <. E , G >. ) )
52 opeq2 
 |-  ( g = G -> <. F , g >. = <. F , G >. )
53 52 breq2d 
 |-  ( g = G -> ( <. B , C >. Cgr <. F , g >. <-> <. B , C >. Cgr <. F , G >. ) )
54 51 53 anbi12d 
 |-  ( g = G -> ( ( <. A , C >. Cgr <. E , g >. /\ <. B , C >. Cgr <. F , g >. ) <-> ( <. A , C >. Cgr <. E , G >. /\ <. B , C >. Cgr <. F , G >. ) ) )
55 opeq1 
 |-  ( g = G -> <. g , h >. = <. G , h >. )
56 55 breq2d 
 |-  ( g = G -> ( <. C , D >. Cgr <. g , h >. <-> <. C , D >. Cgr <. G , h >. ) )
57 56 anbi2d 
 |-  ( g = G -> ( ( <. A , D >. Cgr <. E , h >. /\ <. C , D >. Cgr <. g , h >. ) <-> ( <. A , D >. Cgr <. E , h >. /\ <. C , D >. Cgr <. G , h >. ) ) )
58 50 54 57 3anbi123d 
 |-  ( g = G -> ( ( ( B Btwn <. A , C >. /\ F Btwn <. E , g >. ) /\ ( <. A , C >. Cgr <. E , g >. /\ <. B , C >. Cgr <. F , g >. ) /\ ( <. A , D >. Cgr <. E , h >. /\ <. C , D >. Cgr <. g , h >. ) ) <-> ( ( B Btwn <. A , C >. /\ F Btwn <. E , G >. ) /\ ( <. A , C >. Cgr <. E , G >. /\ <. B , C >. Cgr <. F , G >. ) /\ ( <. A , D >. Cgr <. E , h >. /\ <. C , D >. Cgr <. G , h >. ) ) ) )
59 opeq2 
 |-  ( h = H -> <. E , h >. = <. E , H >. )
60 59 breq2d 
 |-  ( h = H -> ( <. A , D >. Cgr <. E , h >. <-> <. A , D >. Cgr <. E , H >. ) )
61 opeq2 
 |-  ( h = H -> <. G , h >. = <. G , H >. )
62 61 breq2d 
 |-  ( h = H -> ( <. C , D >. Cgr <. G , h >. <-> <. C , D >. Cgr <. G , H >. ) )
63 60 62 anbi12d 
 |-  ( h = H -> ( ( <. A , D >. Cgr <. E , h >. /\ <. C , D >. Cgr <. G , h >. ) <-> ( <. A , D >. Cgr <. E , H >. /\ <. C , D >. Cgr <. G , H >. ) ) )
64 63 3anbi3d 
 |-  ( h = H -> ( ( ( B Btwn <. A , C >. /\ F Btwn <. E , G >. ) /\ ( <. A , C >. Cgr <. E , G >. /\ <. B , C >. Cgr <. F , G >. ) /\ ( <. A , D >. Cgr <. E , h >. /\ <. C , D >. Cgr <. G , h >. ) ) <-> ( ( B Btwn <. A , C >. /\ F Btwn <. E , G >. ) /\ ( <. A , C >. Cgr <. E , G >. /\ <. B , C >. Cgr <. F , G >. ) /\ ( <. A , D >. Cgr <. E , H >. /\ <. C , D >. Cgr <. G , H >. ) ) ) )
65 fveq2 
 |-  ( n = N -> ( EE ` n ) = ( EE ` N ) )
66 df-ifs 
 |-  InnerFiveSeg = { <. p , q >. | E. n e. NN E. a e. ( EE ` n ) E. b e. ( EE ` n ) E. c e. ( EE ` n ) E. d e. ( EE ` n ) E. e e. ( EE ` n ) E. f e. ( EE ` n ) E. g e. ( EE ` n ) E. h e. ( EE ` n ) ( p = <. <. a , b >. , <. c , d >. >. /\ q = <. <. e , f >. , <. g , h >. >. /\ ( ( b Btwn <. a , c >. /\ f Btwn <. e , g >. ) /\ ( <. a , c >. Cgr <. e , g >. /\ <. b , c >. Cgr <. f , g >. ) /\ ( <. a , d >. Cgr <. e , h >. /\ <. c , d >. Cgr <. g , h >. ) ) ) }
67 9 15 26 32 41 47 58 64 65 66 br8 
 |-  ( ( ( N e. NN /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) /\ E e. ( EE ` N ) ) /\ ( F e. ( EE ` N ) /\ G e. ( EE ` N ) /\ H e. ( EE ` N ) ) ) -> ( <. <. A , B >. , <. C , D >. >. InnerFiveSeg <. <. E , F >. , <. G , H >. >. <-> ( ( B Btwn <. A , C >. /\ F Btwn <. E , G >. ) /\ ( <. A , C >. Cgr <. E , G >. /\ <. B , C >. Cgr <. F , G >. ) /\ ( <. A , D >. Cgr <. E , H >. /\ <. C , D >. Cgr <. G , H >. ) ) ) )