Theorem fvline 32251

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

Assertion

Ref	Expression
fvline	|- ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ A =/= B ) ) -> ( ALine B ) = { x | x Colinear <. A , B >. } )

Distinct variable groups:   x, A   x, B

Allowed substitution hint:   N( x)

Proof of Theorem fvline

Dummy variables a b l n are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2622	. . . . 5 |- [ <. A , B >. ] `' Colinear = [ <. A , B >. ] `' Colinear
2		fveq2 6191	. . . . . . . . 9 |- ( n = N -> ( EE ` n ) = ( EE ` N ) )
3	2	eleq2d 2687	. . . . . . . 8 |- ( n = N -> ( A e. ( EE ` n ) <-> A e. ( EE ` N ) ) )
4	2	eleq2d 2687	. . . . . . . 8 |- ( n = N -> ( B e. ( EE ` n ) <-> B e. ( EE ` N ) ) )
5	3, 4	3anbi12d 1400	. . . . . . 7 |- ( n = N -> ( ( A e. ( EE ` n ) /\ B e. ( EE ` n ) /\ A =/= B ) <-> ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ A =/= B ) ) )
6	5	anbi1d 741	. . . . . 6 |- ( n = N -> ( ( ( A e. ( EE ` n ) /\ B e. ( EE ` n ) /\ A =/= B ) /\ [ <. A , B >. ] `' Colinear = [ <. A , B >. ] `' Colinear ) <-> ( ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ A =/= B ) /\ [ <. A , B >. ] `' Colinear = [ <. A , B >. ] `' Colinear ) ) )
7	6	rspcev 3309	. . . . 5 |- ( ( N e. NN /\ ( ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ A =/= B ) /\ [ <. A , B >. ] `' Colinear = [ <. A , B >. ] `' Colinear ) ) -> E. n e. NN ( ( A e. ( EE ` n ) /\ B e. ( EE ` n ) /\ A =/= B ) /\ [ <. A , B >. ] `' Colinear = [ <. A , B >. ] `' Colinear ) )
8	1, 7	mpanr2 720	. . . 4 |- ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ A =/= B ) ) -> E. n e. NN ( ( A e. ( EE ` n ) /\ B e. ( EE ` n ) /\ A =/= B ) /\ [ <. A , B >. ] `' Colinear = [ <. A , B >. ] `' Colinear ) )
9		simpr1 1067	. . . . 5 |- ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ A =/= B ) ) -> A e. ( EE ` N ) )
10		simpr2 1068	. . . . 5 |- ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ A =/= B ) ) -> B e. ( EE ` N ) )
11		colinearex 32167	. . . . . . . 8 |- Colinear e. _V
12	11	cnvex 7113	. . . . . . 7 |- `' Colinear e. _V
13		ecexg 7746	. . . . . . 7 |- ( `' Colinear e. _V -> [ <. A , B >. ] `' Colinear e. _V )
14	12, 13	ax-mp 5	. . . . . 6 |- [ <. A , B >. ] `' Colinear e. _V
15		eleq1 2689	. . . . . . . . . 10 |- ( a = A -> ( a e. ( EE ` n ) <-> A e. ( EE ` n ) ) )
16		neeq1 2856	. . . . . . . . . 10 |- ( a = A -> ( a =/= b <-> A =/= b ) )
17	15, 16	3anbi13d 1401	. . . . . . . . 9 |- ( a = A -> ( ( a e. ( EE ` n ) /\ b e. ( EE ` n ) /\ a =/= b ) <-> ( A e. ( EE ` n ) /\ b e. ( EE ` n ) /\ A =/= b ) ) )
18		opeq1 4402	. . . . . . . . . . 11 |- ( a = A -> <. a , b >. = <. A , b >. )
19	18	eceq1d 7783	. . . . . . . . . 10 |- ( a = A -> [ <. a , b >. ] `' Colinear = [ <. A , b >. ] `' Colinear )
20	19	eqeq2d 2632	. . . . . . . . 9 |- ( a = A -> ( l = [ <. a , b >. ] `' Colinear <-> l = [ <. A , b >. ] `' Colinear ) )
21	17, 20	anbi12d 747	. . . . . . . 8 |- ( a = A -> ( ( ( a e. ( EE ` n ) /\ b e. ( EE ` n ) /\ a =/= b ) /\ l = [ <. a , b >. ] `' Colinear ) <-> ( ( A e. ( EE ` n ) /\ b e. ( EE ` n ) /\ A =/= b ) /\ l = [ <. A , b >. ] `' Colinear ) ) )
22	21	rexbidv 3052	. . . . . . 7 |- ( a = A -> ( E. n e. NN ( ( a e. ( EE ` n ) /\ b e. ( EE ` n ) /\ a =/= b ) /\ l = [ <. a , b >. ] `' Colinear ) <-> E. n e. NN ( ( A e. ( EE ` n ) /\ b e. ( EE ` n ) /\ A =/= b ) /\ l = [ <. A , b >. ] `' Colinear ) ) )
23		eleq1 2689	. . . . . . . . . 10 |- ( b = B -> ( b e. ( EE ` n ) <-> B e. ( EE ` n ) ) )
24		neeq2 2857	. . . . . . . . . 10 |- ( b = B -> ( A =/= b <-> A =/= B ) )
25	23, 24	3anbi23d 1402	. . . . . . . . 9 |- ( b = B -> ( ( A e. ( EE ` n ) /\ b e. ( EE ` n ) /\ A =/= b ) <-> ( A e. ( EE ` n ) /\ B e. ( EE ` n ) /\ A =/= B ) ) )
26		opeq2 4403	. . . . . . . . . . 11 |- ( b = B -> <. A , b >. = <. A , B >. )
27	26	eceq1d 7783	. . . . . . . . . 10 |- ( b = B -> [ <. A , b >. ] `' Colinear = [ <. A , B >. ] `' Colinear )
28	27	eqeq2d 2632	. . . . . . . . 9 |- ( b = B -> ( l = [ <. A , b >. ] `' Colinear <-> l = [ <. A , B >. ] `' Colinear ) )
29	25, 28	anbi12d 747	. . . . . . . 8 |- ( b = B -> ( ( ( A e. ( EE ` n ) /\ b e. ( EE ` n ) /\ A =/= b ) /\ l = [ <. A , b >. ] `' Colinear ) <-> ( ( A e. ( EE ` n ) /\ B e. ( EE ` n ) /\ A =/= B ) /\ l = [ <. A , B >. ] `' Colinear ) ) )
30	29	rexbidv 3052	. . . . . . 7 |- ( b = B -> ( E. n e. NN ( ( A e. ( EE ` n ) /\ b e. ( EE ` n ) /\ A =/= b ) /\ l = [ <. A , b >. ] `' Colinear ) <-> E. n e. NN ( ( A e. ( EE ` n ) /\ B e. ( EE ` n ) /\ A =/= B ) /\ l = [ <. A , B >. ] `' Colinear ) ) )
31		eqeq1 2626	. . . . . . . . 9 |- ( l = [ <. A , B >. ] `' Colinear -> ( l = [ <. A , B >. ] `' Colinear <-> [ <. A , B >. ] `' Colinear = [ <. A , B >. ] `' Colinear ) )
32	31	anbi2d 740	. . . . . . . 8 |- ( l = [ <. A , B >. ] `' Colinear -> ( ( ( A e. ( EE ` n ) /\ B e. ( EE ` n ) /\ A =/= B ) /\ l = [ <. A , B >. ] `' Colinear ) <-> ( ( A e. ( EE ` n ) /\ B e. ( EE ` n ) /\ A =/= B ) /\ [ <. A , B >. ] `' Colinear = [ <. A , B >. ] `' Colinear ) ) )
33	32	rexbidv 3052	. . . . . . 7 |- ( l = [ <. A , B >. ] `' Colinear -> ( E. n e. NN ( ( A e. ( EE ` n ) /\ B e. ( EE ` n ) /\ A =/= B ) /\ l = [ <. A , B >. ] `' Colinear ) <-> E. n e. NN ( ( A e. ( EE ` n ) /\ B e. ( EE ` n ) /\ A =/= B ) /\ [ <. A , B >. ] `' Colinear = [ <. A , B >. ] `' Colinear ) ) )
34	22, 30, 33	eloprabg 6748	. . . . . 6 |- ( ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ [ <. A , B >. ] `' Colinear e. _V ) -> ( <. <. A , B >. , [ <. A , B >. ] `' Colinear >. e. { <. <. a , b >. , l >. | E. n e. NN ( ( a e. ( EE ` n ) /\ b e. ( EE ` n ) /\ a =/= b ) /\ l = [ <. a , b >. ] `' Colinear ) } <-> E. n e. NN ( ( A e. ( EE ` n ) /\ B e. ( EE ` n ) /\ A =/= B ) /\ [ <. A , B >. ] `' Colinear = [ <. A , B >. ] `' Colinear ) ) )
35	14, 34	mp3an3 1413	. . . . 5 |- ( ( A e. ( EE ` N ) /\ B e. ( EE ` N ) ) -> ( <. <. A , B >. , [ <. A , B >. ] `' Colinear >. e. { <. <. a , b >. , l >. | E. n e. NN ( ( a e. ( EE ` n ) /\ b e. ( EE ` n ) /\ a =/= b ) /\ l = [ <. a , b >. ] `' Colinear ) } <-> E. n e. NN ( ( A e. ( EE ` n ) /\ B e. ( EE ` n ) /\ A =/= B ) /\ [ <. A , B >. ] `' Colinear = [ <. A , B >. ] `' Colinear ) ) )
36	9, 10, 35	syl2anc 693	. . . 4 |- ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ A =/= B ) ) -> ( <. <. A , B >. , [ <. A , B >. ] `' Colinear >. e. { <. <. a , b >. , l >. | E. n e. NN ( ( a e. ( EE ` n ) /\ b e. ( EE ` n ) /\ a =/= b ) /\ l = [ <. a , b >. ] `' Colinear ) } <-> E. n e. NN ( ( A e. ( EE ` n ) /\ B e. ( EE ` n ) /\ A =/= B ) /\ [ <. A , B >. ] `' Colinear = [ <. A , B >. ] `' Colinear ) ) )
37	8, 36	mpbird 247	. . 3 |- ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ A =/= B ) ) -> <. <. A , B >. , [ <. A , B >. ] `' Colinear >. e. { <. <. a , b >. , l >. | E. n e. NN ( ( a e. ( EE ` n ) /\ b e. ( EE ` n ) /\ a =/= b ) /\ l = [ <. a , b >. ] `' Colinear ) } )
38		df-ov 6653	. . . 4 |- ( ALine B ) = (Line ` <. A , B >. )
39		df-br 4654	. . . . . 6 |- ( <. A , B >.Line [ <. A , B >. ] `' Colinear <-> <. <. A , B >. , [ <. A , B >. ] `' Colinear >. e. Line )
40		df-line2 32244	. . . . . . 7 |- Line = { <. <. a , b >. , l >. | E. n e. NN ( ( a e. ( EE ` n ) /\ b e. ( EE ` n ) /\ a =/= b ) /\ l = [ <. a , b >. ] `' Colinear ) }
41	40	eleq2i 2693	. . . . . 6 |- ( <. <. A , B >. , [ <. A , B >. ] `' Colinear >. e. Line <-> <. <. A , B >. , [ <. A , B >. ] `' Colinear >. e. { <. <. a , b >. , l >. | E. n e. NN ( ( a e. ( EE ` n ) /\ b e. ( EE ` n ) /\ a =/= b ) /\ l = [ <. a , b >. ] `' Colinear ) } )
42	39, 41	bitri 264	. . . . 5 |- ( <. A , B >.Line [ <. A , B >. ] `' Colinear <-> <. <. A , B >. , [ <. A , B >. ] `' Colinear >. e. { <. <. a , b >. , l >. | E. n e. NN ( ( a e. ( EE ` n ) /\ b e. ( EE ` n ) /\ a =/= b ) /\ l = [ <. a , b >. ] `' Colinear ) } )
43		funline 32249	. . . . . 6 |- Fun Line
44		funbrfv 6234	. . . . . 6 |- ( Fun Line -> ( <. A , B >.Line [ <. A , B >. ] `' Colinear -> (Line ` <. A , B >. ) = [ <. A , B >. ] `' Colinear ) )
45	43, 44	ax-mp 5	. . . . 5 |- ( <. A , B >.Line [ <. A , B >. ] `' Colinear -> (Line ` <. A , B >. ) = [ <. A , B >. ] `' Colinear )
46	42, 45	sylbir 225	. . . 4 |- ( <. <. A , B >. , [ <. A , B >. ] `' Colinear >. e. { <. <. a , b >. , l >. | E. n e. NN ( ( a e. ( EE ` n ) /\ b e. ( EE ` n ) /\ a =/= b ) /\ l = [ <. a , b >. ] `' Colinear ) } -> (Line ` <. A , B >. ) = [ <. A , B >. ] `' Colinear )
47	38, 46	syl5eq 2668	. . 3 |- ( <. <. A , B >. , [ <. A , B >. ] `' Colinear >. e. { <. <. a , b >. , l >. | E. n e. NN ( ( a e. ( EE ` n ) /\ b e. ( EE ` n ) /\ a =/= b ) /\ l = [ <. a , b >. ] `' Colinear ) } -> ( ALine B ) = [ <. A , B >. ] `' Colinear )
48	37, 47	syl 17	. 2 |- ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ A =/= B ) ) -> ( ALine B ) = [ <. A , B >. ] `' Colinear )
49		opex 4932	. . . 4 |- <. A , B >. e. _V
50		dfec2 7745	. . . 4 |- ( <. A , B >. e. _V -> [ <. A , B >. ] `' Colinear = { x | <. A , B >. `' Colinear x } )
51	49, 50	ax-mp 5	. . 3 |- [ <. A , B >. ] `' Colinear = { x | <. A , B >. `' Colinear x }
52		vex 3203	. . . . 5 |- x e. _V
53	49, 52	brcnv 5305	. . . 4 |- ( <. A , B >. `' Colinear x <-> x Colinear <. A , B >. )
54	53	abbii 2739	. . 3 |- { x | <. A , B >. `' Colinear x } = { x | x Colinear <. A , B >. }
55	51, 54	eqtri 2644	. 2 |- [ <. A , B >. ] `' Colinear = { x | x Colinear <. A , B >. }
56	48, 55	syl6eq 2672	1 |- ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ A =/= B ) ) -> ( ALine B ) = { x | x Colinear <. A , B >. } )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   {cab 2608   =/= wne 2794   E.wrex 2913   _Vcvv 3200   <.cop 4183   class class class wbr 4653   `'ccnv 5113   Fun wfun 5882   ` cfv 5888  (class class class)co 6650   {coprab 6651   [cec 7740   NNcn 11020   EEcee 25768   Colinear ccolin 32144  Linecline2 32241

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-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-i2m1 10004  ax-1ne0 10005  ax-rrecex 10008  ax-cnre 10009

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  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-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-om 7066  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-ec 7744  df-nn 11021  df-colinear 32146  df-line2 32244

This theorem is referenced by:  liness  32252  fvline2  32253  ellines  32259