Theorem brifs 32150

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

Assertion

Ref

Expression

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 of Theorem brifs

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	1	breq1d 4663	. . . 4 |- ( a = A -> ( <. a , c >.Cgr <. e , g >. <-> <. A , c >.Cgr <. e , g >. ) )
5	4	anbi1d 741	. . 3 |- ( a = A -> ( ( <. a , c >.Cgr <. e , g >. /\ <. b , c >.Cgr <. f , g >. ) <-> ( <. A , c >.Cgr <. e , g >. /\ <. b , c >.Cgr <. f , g >. ) ) )
6		opeq1 4402	. . . . 5 |- ( a = A -> <. a , d >. = <. A , d >. )
7	6	breq1d 4663	. . . 4 |- ( a = A -> ( <. a , d >.Cgr <. e , h >. <-> <. A , d >.Cgr <. e , h >. ) )
8	7	anbi1d 741	. . 3 |- ( 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 1399	. 2 |- ( 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 4656	. . . 4 |- ( b = B -> ( b Btwn <. A , c >. <-> B Btwn <. A , c >. ) )
11	10	anbi1d 741	. . 3 |- ( b = B -> ( ( b Btwn <. A , c >. /\ f Btwn <. e , g >. ) <-> ( B Btwn <. A , c >. /\ f Btwn <. e , g >. ) ) )
12		opeq1 4402	. . . . 5 |- ( b = B -> <. b , c >. = <. B , c >. )
13	12	breq1d 4663	. . . 4 |- ( b = B -> ( <. b , c >.Cgr <. f , g >. <-> <. B , c >.Cgr <. f , g >. ) )
14	13	anbi2d 740	. . 3 |- ( 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 1400	. 2 |- ( 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 4403	. . . . 5 |- ( c = C -> <. A , c >. = <. A , C >. )
17	16	breq2d 4665	. . . 4 |- ( c = C -> ( B Btwn <. A , c >. <-> B Btwn <. A , C >. ) )
18	17	anbi1d 741	. . 3 |- ( c = C -> ( ( B Btwn <. A , c >. /\ f Btwn <. e , g >. ) <-> ( B Btwn <. A , C >. /\ f Btwn <. e , g >. ) ) )
19	16	breq1d 4663	. . . 4 |- ( c = C -> ( <. A , c >.Cgr <. e , g >. <-> <. A , C >.Cgr <. e , g >. ) )
20		opeq2 4403	. . . . 5 |- ( c = C -> <. B , c >. = <. B , C >. )
21	20	breq1d 4663	. . . 4 |- ( c = C -> ( <. B , c >.Cgr <. f , g >. <-> <. B , C >.Cgr <. f , g >. ) )
22	19, 21	anbi12d 747	. . 3 |- ( c = C -> ( ( <. A , c >.Cgr <. e , g >. /\ <. B , c >.Cgr <. f , g >. ) <-> ( <. A , C >.Cgr <. e , g >. /\ <. B , C >.Cgr <. f , g >. ) ) )
23		opeq1 4402	. . . . 5 |- ( c = C -> <. c , d >. = <. C , d >. )
24	23	breq1d 4663	. . . 4 |- ( c = C -> ( <. c , d >.Cgr <. g , h >. <-> <. C , d >.Cgr <. g , h >. ) )
25	24	anbi2d 740	. . 3 |- ( 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 1399	. 2 |- ( 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 4403	. . . . 5 |- ( d = D -> <. A , d >. = <. A , D >. )
28	27	breq1d 4663	. . . 4 |- ( d = D -> ( <. A , d >.Cgr <. e , h >. <-> <. A , D >.Cgr <. e , h >. ) )
29		opeq2 4403	. . . . 5 |- ( d = D -> <. C , d >. = <. C , D >. )
30	29	breq1d 4663	. . . 4 |- ( d = D -> ( <. C , d >.Cgr <. g , h >. <-> <. C , D >.Cgr <. g , h >. ) )
31	28, 30	anbi12d 747	. . 3 |- ( 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 1405	. 2 |- ( 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 4402	. . . . 5 |- ( e = E -> <. e , g >. = <. E , g >. )
34	33	breq2d 4665	. . . 4 |- ( e = E -> ( f Btwn <. e , g >. <-> f Btwn <. E , g >. ) )
35	34	anbi2d 740	. . 3 |- ( e = E -> ( ( B Btwn <. A , C >. /\ f Btwn <. e , g >. ) <-> ( B Btwn <. A , C >. /\ f Btwn <. E , g >. ) ) )
36	33	breq2d 4665	. . . 4 |- ( e = E -> ( <. A , C >.Cgr <. e , g >. <-> <. A , C >.Cgr <. E , g >. ) )
37	36	anbi1d 741	. . 3 |- ( e = E -> ( ( <. A , C >.Cgr <. e , g >. /\ <. B , C >.Cgr <. f , g >. ) <-> ( <. A , C >.Cgr <. E , g >. /\ <. B , C >.Cgr <. f , g >. ) ) )
38		opeq1 4402	. . . . 5 |- ( e = E -> <. e , h >. = <. E , h >. )
39	38	breq2d 4665	. . . 4 |- ( e = E -> ( <. A , D >.Cgr <. e , h >. <-> <. A , D >.Cgr <. E , h >. ) )
40	39	anbi1d 741	. . 3 |- ( 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 1399	. 2 |- ( 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 4656	. . . 4 |- ( f = F -> ( f Btwn <. E , g >. <-> F Btwn <. E , g >. ) )
43	42	anbi2d 740	. . 3 |- ( f = F -> ( ( B Btwn <. A , C >. /\ f Btwn <. E , g >. ) <-> ( B Btwn <. A , C >. /\ F Btwn <. E , g >. ) ) )
44		opeq1 4402	. . . . 5 |- ( f = F -> <. f , g >. = <. F , g >. )
45	44	breq2d 4665	. . . 4 |- ( f = F -> ( <. B , C >.Cgr <. f , g >. <-> <. B , C >.Cgr <. F , g >. ) )
46	45	anbi2d 740	. . 3 |- ( 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 1400	. 2 |- ( 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 4403	. . . . 5 |- ( g = G -> <. E , g >. = <. E , G >. )
49	48	breq2d 4665	. . . 4 |- ( g = G -> ( F Btwn <. E , g >. <-> F Btwn <. E , G >. ) )
50	49	anbi2d 740	. . 3 |- ( g = G -> ( ( B Btwn <. A , C >. /\ F Btwn <. E , g >. ) <-> ( B Btwn <. A , C >. /\ F Btwn <. E , G >. ) ) )
51	48	breq2d 4665	. . . 4 |- ( g = G -> ( <. A , C >.Cgr <. E , g >. <-> <. A , C >.Cgr <. E , G >. ) )
52		opeq2 4403	. . . . 5 |- ( g = G -> <. F , g >. = <. F , G >. )
53	52	breq2d 4665	. . . 4 |- ( g = G -> ( <. B , C >.Cgr <. F , g >. <-> <. B , C >.Cgr <. F , G >. ) )
54	51, 53	anbi12d 747	. . 3 |- ( g = G -> ( ( <. A , C >.Cgr <. E , g >. /\ <. B , C >.Cgr <. F , g >. ) <-> ( <. A , C >.Cgr <. E , G >. /\ <. B , C >.Cgr <. F , G >. ) ) )
55		opeq1 4402	. . . . 5 |- ( g = G -> <. g , h >. = <. G , h >. )
56	55	breq2d 4665	. . . 4 |- ( g = G -> ( <. C , D >.Cgr <. g , h >. <-> <. C , D >.Cgr <. G , h >. ) )
57	56	anbi2d 740	. . 3 |- ( 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 1399	. 2 |- ( 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 4403	. . . . 5 |- ( h = H -> <. E , h >. = <. E , H >. )
60	59	breq2d 4665	. . . 4 |- ( h = H -> ( <. A , D >.Cgr <. E , h >. <-> <. A , D >.Cgr <. E , H >. ) )
61		opeq2 4403	. . . . 5 |- ( h = H -> <. G , h >. = <. G , H >. )
62	61	breq2d 4665	. . . 4 |- ( h = H -> ( <. C , D >.Cgr <. G , h >. <-> <. C , D >.Cgr <. G , H >. ) )
63	60, 62	anbi12d 747	. . 3 |- ( 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 1405	. 2 |- ( 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 6191	. 2 |- ( n = N -> ( EE ` n ) = ( EE ` N ) )
66		df-ifs 32147	. 2 |- 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 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 >. >. 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 >. ) ) ) )

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   InnerFiveSeg cifs 32142

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-ifs 32147

This theorem is referenced by:  ifscgr  32151  cgrsub  32152  btwnxfr  32163  brifs2  32185  btwnconn1lem6  32199