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Theorem meetfval 15519
Description: Value of meet function for a poset. (Contributed by NM, 12-Sep-2011.) (Revised by NM, 9-Sep-2018.) TODO: prove meetfval2 15520 first to reduce net proof size (existence part)?
Hypotheses
Ref Expression
meetfval.u  |-  G  =  ( glb `  K
)
meetfval.m  |-  ./\  =  ( meet `  K )
Assertion
Ref Expression
meetfval  |-  ( K  e.  V  ->  ./\  =  { <. <. x ,  y
>. ,  z >.  |  { x ,  y } G z } )
Distinct variable groups:    x, y,
z, K    z, G
Allowed substitution hints:    G( x, y)    ./\ (
x, y, z)    V( x, y, z)

Proof of Theorem meetfval
Dummy variable  p is distinct from all other variables.
StepHypRef Expression
1 elex 3104 . 2  |-  ( K  e.  V  ->  K  e.  _V )
2 meetfval.m . . 3  |-  ./\  =  ( meet `  K )
3 fvex 5866 . . . . . . 7  |-  ( Base `  K )  e.  _V
4 moeq 3261 . . . . . . . 8  |-  E* z 
z  =  ( G `
 { x ,  y } )
54a1i 11 . . . . . . 7  |-  ( ( x  e.  ( Base `  K )  /\  y  e.  ( Base `  K
) )  ->  E* z  z  =  ( G `  { x ,  y } ) )
6 eqid 2443 . . . . . . 7  |-  { <. <.
x ,  y >. ,  z >.  |  ( ( x  e.  (
Base `  K )  /\  y  e.  ( Base `  K ) )  /\  z  =  ( G `  { x ,  y } ) ) }  =  { <. <. x ,  y
>. ,  z >.  |  ( ( x  e.  ( Base `  K
)  /\  y  e.  ( Base `  K )
)  /\  z  =  ( G `  { x ,  y } ) ) }
73, 3, 5, 6oprabex 6773 . . . . . 6  |-  { <. <.
x ,  y >. ,  z >.  |  ( ( x  e.  (
Base `  K )  /\  y  e.  ( Base `  K ) )  /\  z  =  ( G `  { x ,  y } ) ) }  e.  _V
87a1i 11 . . . . 5  |-  ( K  e.  _V  ->  { <. <.
x ,  y >. ,  z >.  |  ( ( x  e.  (
Base `  K )  /\  y  e.  ( Base `  K ) )  /\  z  =  ( G `  { x ,  y } ) ) }  e.  _V )
9 meetfval.u . . . . . . . . . . . 12  |-  G  =  ( glb `  K
)
109glbfun 15497 . . . . . . . . . . 11  |-  Fun  G
11 funbrfv2b 5902 . . . . . . . . . . 11  |-  ( Fun 
G  ->  ( {
x ,  y } G z  <->  ( {
x ,  y }  e.  dom  G  /\  ( G `  { x ,  y } )  =  z ) ) )
1210, 11ax-mp 5 . . . . . . . . . 10  |-  ( { x ,  y } G z  <->  ( {
x ,  y }  e.  dom  G  /\  ( G `  { x ,  y } )  =  z ) )
13 eqid 2443 . . . . . . . . . . . . . 14  |-  ( Base `  K )  =  (
Base `  K )
14 eqid 2443 . . . . . . . . . . . . . 14  |-  ( le
`  K )  =  ( le `  K
)
15 simpl 457 . . . . . . . . . . . . . 14  |-  ( ( K  e.  _V  /\  { x ,  y }  e.  dom  G )  ->  K  e.  _V )
16 simpr 461 . . . . . . . . . . . . . 14  |-  ( ( K  e.  _V  /\  { x ,  y }  e.  dom  G )  ->  { x ,  y }  e.  dom  G )
1713, 14, 9, 15, 16glbelss 15499 . . . . . . . . . . . . 13  |-  ( ( K  e.  _V  /\  { x ,  y }  e.  dom  G )  ->  { x ,  y }  C_  ( Base `  K ) )
1817ex 434 . . . . . . . . . . . 12  |-  ( K  e.  _V  ->  ( { x ,  y }  e.  dom  G  ->  { x ,  y }  C_  ( Base `  K ) ) )
19 vex 3098 . . . . . . . . . . . . 13  |-  x  e. 
_V
20 vex 3098 . . . . . . . . . . . . 13  |-  y  e. 
_V
2119, 20prss 4169 . . . . . . . . . . . 12  |-  ( ( x  e.  ( Base `  K )  /\  y  e.  ( Base `  K
) )  <->  { x ,  y }  C_  ( Base `  K )
)
2218, 21syl6ibr 227 . . . . . . . . . . 11  |-  ( K  e.  _V  ->  ( { x ,  y }  e.  dom  G  ->  ( x  e.  (
Base `  K )  /\  y  e.  ( Base `  K ) ) ) )
23 eqcom 2452 . . . . . . . . . . . . 13  |-  ( ( G `  { x ,  y } )  =  z  <->  z  =  ( G `  { x ,  y } ) )
2423biimpi 194 . . . . . . . . . . . 12  |-  ( ( G `  { x ,  y } )  =  z  ->  z  =  ( G `  { x ,  y } ) )
2524a1i 11 . . . . . . . . . . 11  |-  ( K  e.  _V  ->  (
( G `  {
x ,  y } )  =  z  -> 
z  =  ( G `
 { x ,  y } ) ) )
2622, 25anim12d 563 . . . . . . . . . 10  |-  ( K  e.  _V  ->  (
( { x ,  y }  e.  dom  G  /\  ( G `  { x ,  y } )  =  z )  ->  ( (
x  e.  ( Base `  K )  /\  y  e.  ( Base `  K
) )  /\  z  =  ( G `  { x ,  y } ) ) ) )
2712, 26syl5bi 217 . . . . . . . . 9  |-  ( K  e.  _V  ->  ( { x ,  y } G z  -> 
( ( x  e.  ( Base `  K
)  /\  y  e.  ( Base `  K )
)  /\  z  =  ( G `  { x ,  y } ) ) ) )
2827alrimiv 1706 . . . . . . . 8  |-  ( K  e.  _V  ->  A. z
( { x ,  y } G z  ->  ( ( x  e.  ( Base `  K
)  /\  y  e.  ( Base `  K )
)  /\  z  =  ( G `  { x ,  y } ) ) ) )
2928alrimiv 1706 . . . . . . 7  |-  ( K  e.  _V  ->  A. y A. z ( { x ,  y } G
z  ->  ( (
x  e.  ( Base `  K )  /\  y  e.  ( Base `  K
) )  /\  z  =  ( G `  { x ,  y } ) ) ) )
3029alrimiv 1706 . . . . . 6  |-  ( K  e.  _V  ->  A. x A. y A. z ( { x ,  y } G z  -> 
( ( x  e.  ( Base `  K
)  /\  y  e.  ( Base `  K )
)  /\  z  =  ( G `  { x ,  y } ) ) ) )
31 ssoprab2 6338 . . . . . 6  |-  ( A. x A. y A. z
( { x ,  y } G z  ->  ( ( x  e.  ( Base `  K
)  /\  y  e.  ( Base `  K )
)  /\  z  =  ( G `  { x ,  y } ) ) )  ->  { <. <.
x ,  y >. ,  z >.  |  {
x ,  y } G z }  C_  {
<. <. x ,  y
>. ,  z >.  |  ( ( x  e.  ( Base `  K
)  /\  y  e.  ( Base `  K )
)  /\  z  =  ( G `  { x ,  y } ) ) } )
3230, 31syl 16 . . . . 5  |-  ( K  e.  _V  ->  { <. <.
x ,  y >. ,  z >.  |  {
x ,  y } G z }  C_  {
<. <. x ,  y
>. ,  z >.  |  ( ( x  e.  ( Base `  K
)  /\  y  e.  ( Base `  K )
)  /\  z  =  ( G `  { x ,  y } ) ) } )
338, 32ssexd 4584 . . . 4  |-  ( K  e.  _V  ->  { <. <.
x ,  y >. ,  z >.  |  {
x ,  y } G z }  e.  _V )
34 fveq2 5856 . . . . . . . 8  |-  ( p  =  K  ->  ( glb `  p )  =  ( glb `  K
) )
3534, 9syl6eqr 2502 . . . . . . 7  |-  ( p  =  K  ->  ( glb `  p )  =  G )
3635breqd 4448 . . . . . 6  |-  ( p  =  K  ->  ( { x ,  y }  ( glb `  p
) z  <->  { x ,  y } G
z ) )
3736oprabbidv 6336 . . . . 5  |-  ( p  =  K  ->  { <. <.
x ,  y >. ,  z >.  |  {
x ,  y }  ( glb `  p
) z }  =  { <. <. x ,  y
>. ,  z >.  |  { x ,  y } G z } )
38 df-meet 15481 . . . . 5  |-  meet  =  ( p  e.  _V  |->  { <. <. x ,  y
>. ,  z >.  |  { x ,  y }  ( glb `  p
) z } )
3937, 38fvmptg 5939 . . . 4  |-  ( ( K  e.  _V  /\  {
<. <. x ,  y
>. ,  z >.  |  { x ,  y } G z }  e.  _V )  -> 
( meet `  K )  =  { <. <. x ,  y
>. ,  z >.  |  { x ,  y } G z } )
4033, 39mpdan 668 . . 3  |-  ( K  e.  _V  ->  ( meet `  K )  =  { <. <. x ,  y
>. ,  z >.  |  { x ,  y } G z } )
412, 40syl5eq 2496 . 2  |-  ( K  e.  _V  ->  ./\  =  { <. <. x ,  y
>. ,  z >.  |  { x ,  y } G z } )
421, 41syl 16 1  |-  ( K  e.  V  ->  ./\  =  { <. <. x ,  y
>. ,  z >.  |  { x ,  y } G z } )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369   A.wal 1381    = wceq 1383    e. wcel 1804   E*wmo 2269   _Vcvv 3095    C_ wss 3461   {cpr 4016   class class class wbr 4437   dom cdm 4989   Fun wfun 5572   ` cfv 5578   {coprab 6282   Basecbs 14509   lecple 14581   glbcglb 15446   meetcmee 15448
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-rep 4548  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-ral 2798  df-rex 2799  df-reu 2800  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-op 4021  df-uni 4235  df-iun 4317  df-br 4438  df-opab 4496  df-mpt 4497  df-id 4785  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586  df-riota 6242  df-oprab 6285  df-glb 15479  df-meet 15481
This theorem is referenced by:  meetfval2  15520  meet0  15641  odumeet  15644  odujoin  15646
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