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Metamath Proof Explorer

Theorem meetval

Description: Meet value. Since both sides evaluate to (/) when they don't exist, for convenience we drop the { X , Y } e. dom G requirement. (Contributed by NM, 9-Sep-2018)

Ref Expression
Hypotheses meetdef.u 
|- G = ( glb ` K )
meetdef.m 
|- ./\ = ( meet ` K )
meetdef.k 
|- ( ph -> K e. V )
meetdef.x 
|- ( ph -> X e. W )
meetdef.y 
|- ( ph -> Y e. Z )
Assertion meetval 
|- ( ph -> ( X ./\ Y ) = ( G ` { X , Y } ) )

Proof

Step Hyp Ref Expression
1 meetdef.u 
 |-  G = ( glb ` K )
2 meetdef.m 
 |-  ./\ = ( meet ` K )
3 meetdef.k 
 |-  ( ph -> K e. V )
4 meetdef.x 
 |-  ( ph -> X e. W )
5 meetdef.y 
 |-  ( ph -> Y e. Z )
6 1 2 meetfval2 
 |-  ( K e. V -> ./\ = { <. <. x , y >. , z >. | ( { x , y } e. dom G /\ z = ( G ` { x , y } ) ) } )
7 3 6 syl 
 |-  ( ph -> ./\ = { <. <. x , y >. , z >. | ( { x , y } e. dom G /\ z = ( G ` { x , y } ) ) } )
8 7 oveqd 
 |-  ( ph -> ( X ./\ Y ) = ( X { <. <. x , y >. , z >. | ( { x , y } e. dom G /\ z = ( G ` { x , y } ) ) } Y ) )
9 8 adantr 
 |-  ( ( ph /\ { X , Y } e. dom G ) -> ( X ./\ Y ) = ( X { <. <. x , y >. , z >. | ( { x , y } e. dom G /\ z = ( G ` { x , y } ) ) } Y ) )
10 simpr 
 |-  ( ( ph /\ { X , Y } e. dom G ) -> { X , Y } e. dom G )
11 eqidd 
 |-  ( ( ph /\ { X , Y } e. dom G ) -> ( G ` { X , Y } ) = ( G ` { X , Y } ) )
12 fvexd 
 |-  ( ph -> ( G ` { X , Y } ) e. _V )
13 preq12 
 |-  ( ( x = X /\ y = Y ) -> { x , y } = { X , Y } )
14 13 eleq1d 
 |-  ( ( x = X /\ y = Y ) -> ( { x , y } e. dom G <-> { X , Y } e. dom G ) )
15 14 3adant3 
 |-  ( ( x = X /\ y = Y /\ z = ( G ` { X , Y } ) ) -> ( { x , y } e. dom G <-> { X , Y } e. dom G ) )
16 simp3 
 |-  ( ( x = X /\ y = Y /\ z = ( G ` { X , Y } ) ) -> z = ( G ` { X , Y } ) )
17 13 fveq2d 
 |-  ( ( x = X /\ y = Y ) -> ( G ` { x , y } ) = ( G ` { X , Y } ) )
18 17 3adant3 
 |-  ( ( x = X /\ y = Y /\ z = ( G ` { X , Y } ) ) -> ( G ` { x , y } ) = ( G ` { X , Y } ) )
19 16 18 eqeq12d 
 |-  ( ( x = X /\ y = Y /\ z = ( G ` { X , Y } ) ) -> ( z = ( G ` { x , y } ) <-> ( G ` { X , Y } ) = ( G ` { X , Y } ) ) )
20 15 19 anbi12d 
 |-  ( ( x = X /\ y = Y /\ z = ( G ` { X , Y } ) ) -> ( ( { x , y } e. dom G /\ z = ( G ` { x , y } ) ) <-> ( { X , Y } e. dom G /\ ( G ` { X , Y } ) = ( G ` { X , Y } ) ) ) )
21 moeq 
 |-  E* z z = ( G ` { x , y } )
22 21 moani 
 |-  E* z ( { x , y } e. dom G /\ z = ( G ` { x , y } ) )
23 eqid 
 |-  { <. <. x , y >. , z >. | ( { x , y } e. dom G /\ z = ( G ` { x , y } ) ) } = { <. <. x , y >. , z >. | ( { x , y } e. dom G /\ z = ( G ` { x , y } ) ) }
24 20 22 23 ovigg 
 |-  ( ( X e. W /\ Y e. Z /\ ( G ` { X , Y } ) e. _V ) -> ( ( { X , Y } e. dom G /\ ( G ` { X , Y } ) = ( G ` { X , Y } ) ) -> ( X { <. <. x , y >. , z >. | ( { x , y } e. dom G /\ z = ( G ` { x , y } ) ) } Y ) = ( G ` { X , Y } ) ) )
25 4 5 12 24 syl3anc 
 |-  ( ph -> ( ( { X , Y } e. dom G /\ ( G ` { X , Y } ) = ( G ` { X , Y } ) ) -> ( X { <. <. x , y >. , z >. | ( { x , y } e. dom G /\ z = ( G ` { x , y } ) ) } Y ) = ( G ` { X , Y } ) ) )
26 25 adantr 
 |-  ( ( ph /\ { X , Y } e. dom G ) -> ( ( { X , Y } e. dom G /\ ( G ` { X , Y } ) = ( G ` { X , Y } ) ) -> ( X { <. <. x , y >. , z >. | ( { x , y } e. dom G /\ z = ( G ` { x , y } ) ) } Y ) = ( G ` { X , Y } ) ) )
27 10 11 26 mp2and 
 |-  ( ( ph /\ { X , Y } e. dom G ) -> ( X { <. <. x , y >. , z >. | ( { x , y } e. dom G /\ z = ( G ` { x , y } ) ) } Y ) = ( G ` { X , Y } ) )
28 9 27 eqtrd 
 |-  ( ( ph /\ { X , Y } e. dom G ) -> ( X ./\ Y ) = ( G ` { X , Y } ) )
29 1 2 3 4 5 meetdef 
 |-  ( ph -> ( <. X , Y >. e. dom ./\ <-> { X , Y } e. dom G ) )
30 29 notbid 
 |-  ( ph -> ( -. <. X , Y >. e. dom ./\ <-> -. { X , Y } e. dom G ) )
31 df-ov 
 |-  ( X ./\ Y ) = ( ./\ ` <. X , Y >. )
32 ndmfv 
 |-  ( -. <. X , Y >. e. dom ./\ -> ( ./\ ` <. X , Y >. ) = (/) )
33 31 32 eqtrid 
 |-  ( -. <. X , Y >. e. dom ./\ -> ( X ./\ Y ) = (/) )
34 30 33 biimtrrdi 
 |-  ( ph -> ( -. { X , Y } e. dom G -> ( X ./\ Y ) = (/) ) )
35 34 imp 
 |-  ( ( ph /\ -. { X , Y } e. dom G ) -> ( X ./\ Y ) = (/) )
36 ndmfv 
 |-  ( -. { X , Y } e. dom G -> ( G ` { X , Y } ) = (/) )
37 36 adantl 
 |-  ( ( ph /\ -. { X , Y } e. dom G ) -> ( G ` { X , Y } ) = (/) )
38 35 37 eqtr4d 
 |-  ( ( ph /\ -. { X , Y } e. dom G ) -> ( X ./\ Y ) = ( G ` { X , Y } ) )
39 28 38 pm2.61dan 
 |-  ( ph -> ( X ./\ Y ) = ( G ` { X , Y } ) )