Theorem brofs 32112

Description: Binary relation form of the outer five segment predicate. (Contributed by Scott Fenton, 21-Sep-2013.)

Assertion

Ref

Expression

brofs

|- ( ( ( 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 >. >. OuterFiveSeg <. <. E , F >. , <. G , H >. >. <-> ( ( B Btwn <. A , C >. /\ F Btwn <. E , G >. ) /\ ( <. A , B >.Cgr <. E , F >. /\ <. B , C >.Cgr <. F , G >. ) /\ ( <. A , D >.Cgr <. E , H >. /\ <. B , D >.Cgr <. F , H >. ) ) ) )

Proof of Theorem brofs

Dummy variables a b c d e f g h p q n are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		opeq1 4402	. . . . 5 |- ( a = A -> <. a , c >. = <. A , c >. )
2	1	breq2d 4665	. . . 4 |- ( a = A -> ( b Btwn <. a , c >. <-> b Btwn <. A , c >. ) )
3	2	anbi1d 741	. . 3 |- ( a = A -> ( ( b Btwn <. a , c >. /\ f Btwn <. e , g >. ) <-> ( b Btwn <. A , c >. /\ f Btwn <. e , g >. ) ) )
4		opeq1 4402	. . . . 5 |- ( a = A -> <. a , b >. = <. A , b >. )
5	4	breq1d 4663	. . . 4 |- ( a = A -> ( <. a , b >.Cgr <. e , f >. <-> <. A , b >.Cgr <. e , f >. ) )
6	5	anbi1d 741	. . 3 |- ( a = A -> ( ( <. a , b >.Cgr <. e , f >. /\ <. b , c >.Cgr <. f , g >. ) <-> ( <. A , b >.Cgr <. e , f >. /\ <. b , c >.Cgr <. f , g >. ) ) )
7		opeq1 4402	. . . . 5 |- ( a = A -> <. a , d >. = <. A , d >. )
8	7	breq1d 4663	. . . 4 |- ( a = A -> ( <. a , d >.Cgr <. e , h >. <-> <. A , d >.Cgr <. e , h >. ) )
9	8	anbi1d 741	. . 3 |- ( a = A -> ( ( <. a , d >.Cgr <. e , h >. /\ <. b , d >.Cgr <. f , h >. ) <-> ( <. A , d >.Cgr <. e , h >. /\ <. b , d >.Cgr <. f , h >. ) ) )
10	3, 6, 9	3anbi123d 1399	. 2 |- ( a = A -> ( ( ( b Btwn <. a , c >. /\ f Btwn <. e , g >. ) /\ ( <. a , b >.Cgr <. e , f >. /\ <. b , c >.Cgr <. f , g >. ) /\ ( <. a , d >.Cgr <. e , h >. /\ <. b , d >.Cgr <. f , h >. ) ) <-> ( ( b Btwn <. A , c >. /\ f Btwn <. e , g >. ) /\ ( <. A , b >.Cgr <. e , f >. /\ <. b , c >.Cgr <. f , g >. ) /\ ( <. A , d >.Cgr <. e , h >. /\ <. b , d >.Cgr <. f , h >. ) ) ) )
11		breq1 4656	. . . 4 |- ( b = B -> ( b Btwn <. A , c >. <-> B Btwn <. A , c >. ) )
12	11	anbi1d 741	. . 3 |- ( b = B -> ( ( b Btwn <. A , c >. /\ f Btwn <. e , g >. ) <-> ( B Btwn <. A , c >. /\ f Btwn <. e , g >. ) ) )
13		opeq2 4403	. . . . 5 |- ( b = B -> <. A , b >. = <. A , B >. )
14	13	breq1d 4663	. . . 4 |- ( b = B -> ( <. A , b >.Cgr <. e , f >. <-> <. A , B >.Cgr <. e , f >. ) )
15		opeq1 4402	. . . . 5 |- ( b = B -> <. b , c >. = <. B , c >. )
16	15	breq1d 4663	. . . 4 |- ( b = B -> ( <. b , c >.Cgr <. f , g >. <-> <. B , c >.Cgr <. f , g >. ) )
17	14, 16	anbi12d 747	. . 3 |- ( b = B -> ( ( <. A , b >.Cgr <. e , f >. /\ <. b , c >.Cgr <. f , g >. ) <-> ( <. A , B >.Cgr <. e , f >. /\ <. B , c >.Cgr <. f , g >. ) ) )
18		opeq1 4402	. . . . 5 |- ( b = B -> <. b , d >. = <. B , d >. )
19	18	breq1d 4663	. . . 4 |- ( b = B -> ( <. b , d >.Cgr <. f , h >. <-> <. B , d >.Cgr <. f , h >. ) )
20	19	anbi2d 740	. . 3 |- ( b = B -> ( ( <. A , d >.Cgr <. e , h >. /\ <. b , d >.Cgr <. f , h >. ) <-> ( <. A , d >.Cgr <. e , h >. /\ <. B , d >.Cgr <. f , h >. ) ) )
21	12, 17, 20	3anbi123d 1399	. 2 |- ( b = B -> ( ( ( b Btwn <. A , c >. /\ f Btwn <. e , g >. ) /\ ( <. A , b >.Cgr <. e , f >. /\ <. b , c >.Cgr <. f , g >. ) /\ ( <. A , d >.Cgr <. e , h >. /\ <. b , d >.Cgr <. f , h >. ) ) <-> ( ( B Btwn <. A , c >. /\ f Btwn <. e , g >. ) /\ ( <. A , B >.Cgr <. e , f >. /\ <. B , c >.Cgr <. f , g >. ) /\ ( <. A , d >.Cgr <. e , h >. /\ <. B , d >.Cgr <. f , h >. ) ) ) )
22		opeq2 4403	. . . . 5 |- ( c = C -> <. A , c >. = <. A , C >. )
23	22	breq2d 4665	. . . 4 |- ( c = C -> ( B Btwn <. A , c >. <-> B Btwn <. A , C >. ) )
24	23	anbi1d 741	. . 3 |- ( c = C -> ( ( B Btwn <. A , c >. /\ f Btwn <. e , g >. ) <-> ( B Btwn <. A , C >. /\ f Btwn <. e , g >. ) ) )
25		opeq2 4403	. . . . 5 |- ( c = C -> <. B , c >. = <. B , C >. )
26	25	breq1d 4663	. . . 4 |- ( c = C -> ( <. B , c >.Cgr <. f , g >. <-> <. B , C >.Cgr <. f , g >. ) )
27	26	anbi2d 740	. . 3 |- ( c = C -> ( ( <. A , B >.Cgr <. e , f >. /\ <. B , c >.Cgr <. f , g >. ) <-> ( <. A , B >.Cgr <. e , f >. /\ <. B , C >.Cgr <. f , g >. ) ) )
28	24, 27	3anbi12d 1400	. 2 |- ( c = C -> ( ( ( B Btwn <. A , c >. /\ f Btwn <. e , g >. ) /\ ( <. A , B >.Cgr <. e , f >. /\ <. B , c >.Cgr <. f , g >. ) /\ ( <. A , d >.Cgr <. e , h >. /\ <. B , d >.Cgr <. f , h >. ) ) <-> ( ( B Btwn <. A , C >. /\ f Btwn <. e , g >. ) /\ ( <. A , B >.Cgr <. e , f >. /\ <. B , C >.Cgr <. f , g >. ) /\ ( <. A , d >.Cgr <. e , h >. /\ <. B , d >.Cgr <. f , h >. ) ) ) )
29		opeq2 4403	. . . . 5 |- ( d = D -> <. A , d >. = <. A , D >. )
30	29	breq1d 4663	. . . 4 |- ( d = D -> ( <. A , d >.Cgr <. e , h >. <-> <. A , D >.Cgr <. e , h >. ) )
31		opeq2 4403	. . . . 5 |- ( d = D -> <. B , d >. = <. B , D >. )
32	31	breq1d 4663	. . . 4 |- ( d = D -> ( <. B , d >.Cgr <. f , h >. <-> <. B , D >.Cgr <. f , h >. ) )
33	30, 32	anbi12d 747	. . 3 |- ( d = D -> ( ( <. A , d >.Cgr <. e , h >. /\ <. B , d >.Cgr <. f , h >. ) <-> ( <. A , D >.Cgr <. e , h >. /\ <. B , D >.Cgr <. f , h >. ) ) )
34	33	3anbi3d 1405	. 2 |- ( d = D -> ( ( ( B Btwn <. A , C >. /\ f Btwn <. e , g >. ) /\ ( <. A , B >.Cgr <. e , f >. /\ <. B , C >.Cgr <. f , g >. ) /\ ( <. A , d >.Cgr <. e , h >. /\ <. B , d >.Cgr <. f , h >. ) ) <-> ( ( B Btwn <. A , C >. /\ f Btwn <. e , g >. ) /\ ( <. A , B >.Cgr <. e , f >. /\ <. B , C >.Cgr <. f , g >. ) /\ ( <. A , D >.Cgr <. e , h >. /\ <. B , D >.Cgr <. f , h >. ) ) ) )
35		opeq1 4402	. . . . 5 |- ( e = E -> <. e , g >. = <. E , g >. )
36	35	breq2d 4665	. . . 4 |- ( e = E -> ( f Btwn <. e , g >. <-> f Btwn <. E , g >. ) )
37	36	anbi2d 740	. . 3 |- ( e = E -> ( ( B Btwn <. A , C >. /\ f Btwn <. e , g >. ) <-> ( B Btwn <. A , C >. /\ f Btwn <. E , g >. ) ) )
38		opeq1 4402	. . . . 5 |- ( e = E -> <. e , f >. = <. E , f >. )
39	38	breq2d 4665	. . . 4 |- ( e = E -> ( <. A , B >.Cgr <. e , f >. <-> <. A , B >.Cgr <. E , f >. ) )
40	39	anbi1d 741	. . 3 |- ( e = E -> ( ( <. A , B >.Cgr <. e , f >. /\ <. B , C >.Cgr <. f , g >. ) <-> ( <. A , B >.Cgr <. E , f >. /\ <. B , C >.Cgr <. f , g >. ) ) )
41		opeq1 4402	. . . . 5 |- ( e = E -> <. e , h >. = <. E , h >. )
42	41	breq2d 4665	. . . 4 |- ( e = E -> ( <. A , D >.Cgr <. e , h >. <-> <. A , D >.Cgr <. E , h >. ) )
43	42	anbi1d 741	. . 3 |- ( e = E -> ( ( <. A , D >.Cgr <. e , h >. /\ <. B , D >.Cgr <. f , h >. ) <-> ( <. A , D >.Cgr <. E , h >. /\ <. B , D >.Cgr <. f , h >. ) ) )
44	37, 40, 43	3anbi123d 1399	. 2 |- ( e = E -> ( ( ( B Btwn <. A , C >. /\ f Btwn <. e , g >. ) /\ ( <. A , B >.Cgr <. e , f >. /\ <. B , C >.Cgr <. f , g >. ) /\ ( <. A , D >.Cgr <. e , h >. /\ <. B , D >.Cgr <. f , h >. ) ) <-> ( ( B Btwn <. A , C >. /\ f Btwn <. E , g >. ) /\ ( <. A , B >.Cgr <. E , f >. /\ <. B , C >.Cgr <. f , g >. ) /\ ( <. A , D >.Cgr <. E , h >. /\ <. B , D >.Cgr <. f , h >. ) ) ) )
45		breq1 4656	. . . 4 |- ( f = F -> ( f Btwn <. E , g >. <-> F Btwn <. E , g >. ) )
46	45	anbi2d 740	. . 3 |- ( f = F -> ( ( B Btwn <. A , C >. /\ f Btwn <. E , g >. ) <-> ( B Btwn <. A , C >. /\ F Btwn <. E , g >. ) ) )
47		opeq2 4403	. . . . 5 |- ( f = F -> <. E , f >. = <. E , F >. )
48	47	breq2d 4665	. . . 4 |- ( f = F -> ( <. A , B >.Cgr <. E , f >. <-> <. A , B >.Cgr <. E , F >. ) )
49		opeq1 4402	. . . . 5 |- ( f = F -> <. f , g >. = <. F , g >. )
50	49	breq2d 4665	. . . 4 |- ( f = F -> ( <. B , C >.Cgr <. f , g >. <-> <. B , C >.Cgr <. F , g >. ) )
51	48, 50	anbi12d 747	. . 3 |- ( f = F -> ( ( <. A , B >.Cgr <. E , f >. /\ <. B , C >.Cgr <. f , g >. ) <-> ( <. A , B >.Cgr <. E , F >. /\ <. B , C >.Cgr <. F , g >. ) ) )
52		opeq1 4402	. . . . 5 |- ( f = F -> <. f , h >. = <. F , h >. )
53	52	breq2d 4665	. . . 4 |- ( f = F -> ( <. B , D >.Cgr <. f , h >. <-> <. B , D >.Cgr <. F , h >. ) )
54	53	anbi2d 740	. . 3 |- ( f = F -> ( ( <. A , D >.Cgr <. E , h >. /\ <. B , D >.Cgr <. f , h >. ) <-> ( <. A , D >.Cgr <. E , h >. /\ <. B , D >.Cgr <. F , h >. ) ) )
55	46, 51, 54	3anbi123d 1399	. 2 |- ( f = F -> ( ( ( B Btwn <. A , C >. /\ f Btwn <. E , g >. ) /\ ( <. A , B >.Cgr <. E , f >. /\ <. B , C >.Cgr <. f , g >. ) /\ ( <. A , D >.Cgr <. E , h >. /\ <. B , D >.Cgr <. f , h >. ) ) <-> ( ( B Btwn <. A , C >. /\ F Btwn <. E , g >. ) /\ ( <. A , B >.Cgr <. E , F >. /\ <. B , C >.Cgr <. F , g >. ) /\ ( <. A , D >.Cgr <. E , h >. /\ <. B , D >.Cgr <. F , h >. ) ) ) )
56		opeq2 4403	. . . . 5 |- ( g = G -> <. E , g >. = <. E , G >. )
57	56	breq2d 4665	. . . 4 |- ( g = G -> ( F Btwn <. E , g >. <-> F Btwn <. E , G >. ) )
58	57	anbi2d 740	. . 3 |- ( g = G -> ( ( B Btwn <. A , C >. /\ F Btwn <. E , g >. ) <-> ( B Btwn <. A , C >. /\ F Btwn <. E , G >. ) ) )
59		opeq2 4403	. . . . 5 |- ( g = G -> <. F , g >. = <. F , G >. )
60	59	breq2d 4665	. . . 4 |- ( g = G -> ( <. B , C >.Cgr <. F , g >. <-> <. B , C >.Cgr <. F , G >. ) )
61	60	anbi2d 740	. . 3 |- ( g = G -> ( ( <. A , B >.Cgr <. E , F >. /\ <. B , C >.Cgr <. F , g >. ) <-> ( <. A , B >.Cgr <. E , F >. /\ <. B , C >.Cgr <. F , G >. ) ) )
62	58, 61	3anbi12d 1400	. 2 |- ( g = G -> ( ( ( B Btwn <. A , C >. /\ F Btwn <. E , g >. ) /\ ( <. A , B >.Cgr <. E , F >. /\ <. B , C >.Cgr <. F , g >. ) /\ ( <. A , D >.Cgr <. E , h >. /\ <. B , D >.Cgr <. F , h >. ) ) <-> ( ( B Btwn <. A , C >. /\ F Btwn <. E , G >. ) /\ ( <. A , B >.Cgr <. E , F >. /\ <. B , C >.Cgr <. F , G >. ) /\ ( <. A , D >.Cgr <. E , h >. /\ <. B , D >.Cgr <. F , h >. ) ) ) )
63		opeq2 4403	. . . . 5 |- ( h = H -> <. E , h >. = <. E , H >. )
64	63	breq2d 4665	. . . 4 |- ( h = H -> ( <. A , D >.Cgr <. E , h >. <-> <. A , D >.Cgr <. E , H >. ) )
65		opeq2 4403	. . . . 5 |- ( h = H -> <. F , h >. = <. F , H >. )
66	65	breq2d 4665	. . . 4 |- ( h = H -> ( <. B , D >.Cgr <. F , h >. <-> <. B , D >.Cgr <. F , H >. ) )
67	64, 66	anbi12d 747	. . 3 |- ( h = H -> ( ( <. A , D >.Cgr <. E , h >. /\ <. B , D >.Cgr <. F , h >. ) <-> ( <. A , D >.Cgr <. E , H >. /\ <. B , D >.Cgr <. F , H >. ) ) )
68	67	3anbi3d 1405	. 2 |- ( h = H -> ( ( ( B Btwn <. A , C >. /\ F Btwn <. E , G >. ) /\ ( <. A , B >.Cgr <. E , F >. /\ <. B , C >.Cgr <. F , G >. ) /\ ( <. A , D >.Cgr <. E , h >. /\ <. B , D >.Cgr <. F , h >. ) ) <-> ( ( B Btwn <. A , C >. /\ F Btwn <. E , G >. ) /\ ( <. A , B >.Cgr <. E , F >. /\ <. B , C >.Cgr <. F , G >. ) /\ ( <. A , D >.Cgr <. E , H >. /\ <. B , D >.Cgr <. F , H >. ) ) ) )
69		fveq2 6191	. 2 |- ( n = N -> ( EE ` n ) = ( EE ` N ) )
70		df-ofs 32090	. 2 |- OuterFiveSeg = { <. 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 , b >.Cgr <. e , f >. /\ <. b , c >.Cgr <. f , g >. ) /\ ( <. a , d >.Cgr <. e , h >. /\ <. b , d >.Cgr <. f , h >. ) ) ) }
71	10, 21, 28, 34, 44, 55, 62, 68, 69, 70	br8 31646	1 |- ( ( ( 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 >. >. OuterFiveSeg <. <. E , F >. , <. G , H >. >. <-> ( ( B Btwn <. A , C >. /\ F Btwn <. E , G >. ) /\ ( <. A , B >.Cgr <. E , F >. /\ <. B , C >.Cgr <. F , G >. ) /\ ( <. A , D >.Cgr <. E , H >. /\ <. B , D >.Cgr <. F , H >. ) ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   <.cop 4183   class class class wbr 4653   ` cfv 5888   NNcn 11020   EEcee 25768   Btwn cbtwn 25769  Cgrccgr 25770   OuterFiveSeg cofs 32089

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-iota 5851  df-fv 5896  df-ofs 32090

This theorem is referenced by:  5segofs  32113  ofscom  32114  cgrextend  32115  segconeq  32117  ifscgr  32151  brofs2  32184