Metamath Proof Explorer


Theorem legid

Description: Reflexivity of the less-than relationship. Proposition 5.7 of Schwabhauser p. 42. (Contributed by Thierry Arnoux, 27-Jun-2019)

Ref Expression
Hypotheses legval.p
|- P = ( Base ` G )
legval.d
|- .- = ( dist ` G )
legval.i
|- I = ( Itv ` G )
legval.l
|- .<_ = ( leG ` G )
legval.g
|- ( ph -> G e. TarskiG )
legid.a
|- ( ph -> A e. P )
legid.b
|- ( ph -> B e. P )
Assertion legid
|- ( ph -> ( A .- B ) .<_ ( A .- B ) )

Proof

Step Hyp Ref Expression
1 legval.p
 |-  P = ( Base ` G )
2 legval.d
 |-  .- = ( dist ` G )
3 legval.i
 |-  I = ( Itv ` G )
4 legval.l
 |-  .<_ = ( leG ` G )
5 legval.g
 |-  ( ph -> G e. TarskiG )
6 legid.a
 |-  ( ph -> A e. P )
7 legid.b
 |-  ( ph -> B e. P )
8 1 2 3 5 6 7 tgbtwntriv2
 |-  ( ph -> B e. ( A I B ) )
9 eqidd
 |-  ( ph -> ( A .- B ) = ( A .- B ) )
10 eleq1
 |-  ( x = B -> ( x e. ( A I B ) <-> B e. ( A I B ) ) )
11 oveq2
 |-  ( x = B -> ( A .- x ) = ( A .- B ) )
12 11 eqeq2d
 |-  ( x = B -> ( ( A .- B ) = ( A .- x ) <-> ( A .- B ) = ( A .- B ) ) )
13 10 12 anbi12d
 |-  ( x = B -> ( ( x e. ( A I B ) /\ ( A .- B ) = ( A .- x ) ) <-> ( B e. ( A I B ) /\ ( A .- B ) = ( A .- B ) ) ) )
14 13 rspcev
 |-  ( ( B e. P /\ ( B e. ( A I B ) /\ ( A .- B ) = ( A .- B ) ) ) -> E. x e. P ( x e. ( A I B ) /\ ( A .- B ) = ( A .- x ) ) )
15 7 8 9 14 syl12anc
 |-  ( ph -> E. x e. P ( x e. ( A I B ) /\ ( A .- B ) = ( A .- x ) ) )
16 1 2 3 4 5 6 7 6 7 legov
 |-  ( ph -> ( ( A .- B ) .<_ ( A .- B ) <-> E. x e. P ( x e. ( A I B ) /\ ( A .- B ) = ( A .- x ) ) ) )
17 15 16 mpbird
 |-  ( ph -> ( A .- B ) .<_ ( A .- B ) )