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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 ) ) ) )