Metamath Proof Explorer


Theorem dihjust

Description: Part of proof after Lemma N of Crawley p. 122 line 4, "the definition of phi(x) is independent of the atom q." (Contributed by NM, 2-Mar-2014)

Ref Expression
Hypotheses dihjust.b
|- B = ( Base ` K )
dihjust.l
|- .<_ = ( le ` K )
dihjust.j
|- .\/ = ( join ` K )
dihjust.m
|- ./\ = ( meet ` K )
dihjust.a
|- A = ( Atoms ` K )
dihjust.h
|- H = ( LHyp ` K )
dihjust.i
|- I = ( ( DIsoB ` K ) ` W )
dihjust.J
|- J = ( ( DIsoC ` K ) ` W )
dihjust.u
|- U = ( ( DVecH ` K ) ` W )
dihjust.s
|- .(+) = ( LSSum ` U )
Assertion dihjust
|- ( ( ( K e. HL /\ W e. H ) /\ ( ( Q e. A /\ -. Q .<_ W ) /\ ( R e. A /\ -. R .<_ W ) /\ X e. B ) /\ ( Q .\/ ( X ./\ W ) ) = ( R .\/ ( X ./\ W ) ) ) -> ( ( J ` Q ) .(+) ( I ` ( X ./\ W ) ) ) = ( ( J ` R ) .(+) ( I ` ( X ./\ W ) ) ) )

Proof

Step Hyp Ref Expression
1 dihjust.b
 |-  B = ( Base ` K )
2 dihjust.l
 |-  .<_ = ( le ` K )
3 dihjust.j
 |-  .\/ = ( join ` K )
4 dihjust.m
 |-  ./\ = ( meet ` K )
5 dihjust.a
 |-  A = ( Atoms ` K )
6 dihjust.h
 |-  H = ( LHyp ` K )
7 dihjust.i
 |-  I = ( ( DIsoB ` K ) ` W )
8 dihjust.J
 |-  J = ( ( DIsoC ` K ) ` W )
9 dihjust.u
 |-  U = ( ( DVecH ` K ) ` W )
10 dihjust.s
 |-  .(+) = ( LSSum ` U )
11 1 2 3 4 5 6 7 8 9 10 dihjustlem
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( Q e. A /\ -. Q .<_ W ) /\ ( R e. A /\ -. R .<_ W ) /\ X e. B ) /\ ( Q .\/ ( X ./\ W ) ) = ( R .\/ ( X ./\ W ) ) ) -> ( ( J ` Q ) .(+) ( I ` ( X ./\ W ) ) ) C_ ( ( J ` R ) .(+) ( I ` ( X ./\ W ) ) ) )
12 simp1
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( Q e. A /\ -. Q .<_ W ) /\ ( R e. A /\ -. R .<_ W ) /\ X e. B ) /\ ( Q .\/ ( X ./\ W ) ) = ( R .\/ ( X ./\ W ) ) ) -> ( K e. HL /\ W e. H ) )
13 simp22
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( Q e. A /\ -. Q .<_ W ) /\ ( R e. A /\ -. R .<_ W ) /\ X e. B ) /\ ( Q .\/ ( X ./\ W ) ) = ( R .\/ ( X ./\ W ) ) ) -> ( R e. A /\ -. R .<_ W ) )
14 simp21
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( Q e. A /\ -. Q .<_ W ) /\ ( R e. A /\ -. R .<_ W ) /\ X e. B ) /\ ( Q .\/ ( X ./\ W ) ) = ( R .\/ ( X ./\ W ) ) ) -> ( Q e. A /\ -. Q .<_ W ) )
15 simp23
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( Q e. A /\ -. Q .<_ W ) /\ ( R e. A /\ -. R .<_ W ) /\ X e. B ) /\ ( Q .\/ ( X ./\ W ) ) = ( R .\/ ( X ./\ W ) ) ) -> X e. B )
16 simp3
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( Q e. A /\ -. Q .<_ W ) /\ ( R e. A /\ -. R .<_ W ) /\ X e. B ) /\ ( Q .\/ ( X ./\ W ) ) = ( R .\/ ( X ./\ W ) ) ) -> ( Q .\/ ( X ./\ W ) ) = ( R .\/ ( X ./\ W ) ) )
17 16 eqcomd
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( Q e. A /\ -. Q .<_ W ) /\ ( R e. A /\ -. R .<_ W ) /\ X e. B ) /\ ( Q .\/ ( X ./\ W ) ) = ( R .\/ ( X ./\ W ) ) ) -> ( R .\/ ( X ./\ W ) ) = ( Q .\/ ( X ./\ W ) ) )
18 1 2 3 4 5 6 7 8 9 10 dihjustlem
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( R e. A /\ -. R .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ X e. B ) /\ ( R .\/ ( X ./\ W ) ) = ( Q .\/ ( X ./\ W ) ) ) -> ( ( J ` R ) .(+) ( I ` ( X ./\ W ) ) ) C_ ( ( J ` Q ) .(+) ( I ` ( X ./\ W ) ) ) )
19 12 13 14 15 17 18 syl131anc
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( Q e. A /\ -. Q .<_ W ) /\ ( R e. A /\ -. R .<_ W ) /\ X e. B ) /\ ( Q .\/ ( X ./\ W ) ) = ( R .\/ ( X ./\ W ) ) ) -> ( ( J ` R ) .(+) ( I ` ( X ./\ W ) ) ) C_ ( ( J ` Q ) .(+) ( I ` ( X ./\ W ) ) ) )
20 11 19 eqssd
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( Q e. A /\ -. Q .<_ W ) /\ ( R e. A /\ -. R .<_ W ) /\ X e. B ) /\ ( Q .\/ ( X ./\ W ) ) = ( R .\/ ( X ./\ W ) ) ) -> ( ( J ` Q ) .(+) ( I ` ( X ./\ W ) ) ) = ( ( J ` R ) .(+) ( I ` ( X ./\ W ) ) ) )