Metamath Proof Explorer

Theorem 5segofs

Description: Rephrase ax5seg using the outer five segment predicate. Theorem 2.10 of Schwabhauser p. 28. (Contributed by Scott Fenton, 21-Sep-2013)

Ref Expression
Assertion 5segofs 
|- ( ( ( 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 >. >. /\ A =/= B ) -> <. C , D >. Cgr <. G , H >. ) )

Proof

Step Hyp Ref Expression
1 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 >. ) ) ) )
2 1 anbi1d 
 |-  ( ( ( 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 >. >. /\ A =/= 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 >. ) ) /\ A =/= B ) ) )
3 simpr 
 |-  ( ( ( ( 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 >. ) ) /\ A =/= B ) -> A =/= B )
4 simpl1l 
 |-  ( ( ( ( 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 >. ) ) /\ A =/= B ) -> B Btwn <. A , C >. )
5 simpl1r 
 |-  ( ( ( ( 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 >. ) ) /\ A =/= B ) -> F Btwn <. E , G >. )
6 3 4 5 3jca 
 |-  ( ( ( ( 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 >. ) ) /\ A =/= B ) -> ( A =/= B /\ B Btwn <. A , C >. /\ F Btwn <. E , G >. ) )
7 simpl2 
 |-  ( ( ( ( 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 >. ) ) /\ A =/= B ) -> ( <. A , B >. Cgr <. E , F >. /\ <. B , C >. Cgr <. F , G >. ) )
8 simpl3 
 |-  ( ( ( ( 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 >. ) ) /\ A =/= B ) -> ( <. A , D >. Cgr <. E , H >. /\ <. B , D >. Cgr <. F , H >. ) )
9 6 7 8 3jca 
 |-  ( ( ( ( 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 >. ) ) /\ A =/= B ) -> ( ( A =/= 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 >. ) ) )
10 2 9 biimtrdi 
 |-  ( ( ( 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 >. >. /\ A =/= B ) -> ( ( A =/= 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 >. ) ) ) )
11 ax5seg 
 |-  ( ( ( 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 /\ 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 >. ) ) -> <. C , D >. Cgr <. G , H >. ) )
12 10 11 syld 
 |-  ( ( ( 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 >. >. /\ A =/= B ) -> <. C , D >. Cgr <. G , H >. ) )