Theorem colinearalglem3 25788

Description: Lemma for colinearalg 25790. Translate between two forms of the colinearity condition. (Contributed by Scott Fenton, 24-Jun-2013.)

Assertion

Ref	Expression
colinearalglem3	|- ( ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) -> ( A. i e. ( 1 ... N ) A. j e. ( 1 ... N ) ( ( ( B ` i ) - ( A ` i ) ) x. ( ( C ` j ) - ( A ` j ) ) ) = ( ( ( B ` j ) - ( A ` j ) ) x. ( ( C ` i ) - ( A ` i ) ) ) <-> A. i e. ( 1 ... N ) A. j e. ( 1 ... N ) ( ( ( A ` i ) - ( C ` i ) ) x. ( ( B ` j ) - ( C ` j ) ) ) = ( ( ( A ` j ) - ( C ` j ) ) x. ( ( B ` i ) - ( C ` i ) ) ) ) )

Distinct variable groups:   A, i, j   B, i, j   C, i, j   i, N, j

Proof of Theorem colinearalglem3

Step	Hyp	Ref	Expression
1		colinearalglem2 25787	. 2 |- ( ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) -> ( A. i e. ( 1 ... N ) A. j e. ( 1 ... N ) ( ( ( B ` i ) - ( A ` i ) ) x. ( ( C ` j ) - ( A ` j ) ) ) = ( ( ( B ` j ) - ( A ` j ) ) x. ( ( C ` i ) - ( A ` i ) ) ) <-> A. i e. ( 1 ... N ) A. j e. ( 1 ... N ) ( ( ( C ` i ) - ( B ` i ) ) x. ( ( A ` j ) - ( B ` j ) ) ) = ( ( ( C ` j ) - ( B ` j ) ) x. ( ( A ` i ) - ( B ` i ) ) ) ) )
2		colinearalglem2 25787	. . 3 |- ( ( B e. ( EE ` N ) /\ C e. ( EE ` N ) /\ A e. ( EE ` N ) ) -> ( A. i e. ( 1 ... N ) A. j e. ( 1 ... N ) ( ( ( C ` i ) - ( B ` i ) ) x. ( ( A ` j ) - ( B ` j ) ) ) = ( ( ( C ` j ) - ( B ` j ) ) x. ( ( A ` i ) - ( B ` i ) ) ) <-> A. i e. ( 1 ... N ) A. j e. ( 1 ... N ) ( ( ( A ` i ) - ( C ` i ) ) x. ( ( B ` j ) - ( C ` j ) ) ) = ( ( ( A ` j ) - ( C ` j ) ) x. ( ( B ` i ) - ( C ` i ) ) ) ) )
3	2	3comr 1273	. 2 |- ( ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) -> ( A. i e. ( 1 ... N ) A. j e. ( 1 ... N ) ( ( ( C ` i ) - ( B ` i ) ) x. ( ( A ` j ) - ( B ` j ) ) ) = ( ( ( C ` j ) - ( B ` j ) ) x. ( ( A ` i ) - ( B ` i ) ) ) <-> A. i e. ( 1 ... N ) A. j e. ( 1 ... N ) ( ( ( A ` i ) - ( C ` i ) ) x. ( ( B ` j ) - ( C ` j ) ) ) = ( ( ( A ` j ) - ( C ` j ) ) x. ( ( B ` i ) - ( C ` i ) ) ) ) )
4	1, 3	bitrd 268	1 |- ( ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) -> ( A. i e. ( 1 ... N ) A. j e. ( 1 ... N ) ( ( ( B ` i ) - ( A ` i ) ) x. ( ( C ` j ) - ( A ` j ) ) ) = ( ( ( B ` j ) - ( A ` j ) ) x. ( ( C ` i ) - ( A ` i ) ) ) <-> A. i e. ( 1 ... N ) A. j e. ( 1 ... N ) ( ( ( A ` i ) - ( C ` i ) ) x. ( ( B ` j ) - ( C ` j ) ) ) = ( ( ( A ` j ) - ( C ` j ) ) x. ( ( B ` i ) - ( C ` i ) ) ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ w3a 1037   = wceq 1483   e. wcel 1990   A.wral 2912   ` cfv 5888  (class class class)co 6650   1c1 9937   x. cmul 9941   - cmin 10266   ...cfz 12326   EEcee 25768

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-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-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012

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-nel 2898  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-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-po 5035  df-so 5036  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-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-er 7742  df-map 7859  df-en 7956  df-dom 7957  df-sdom 7958  df-pnf 10076  df-mnf 10077  df-ltxr 10079  df-sub 10268  df-neg 10269  df-ee 25771

This theorem is referenced by:  colinearalg  25790