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Theorem meetle 15198
Description: A meet is less than or equal to a third value iff each argument is less than or equal to the third value. (Contributed by NM, 16-Sep-2011.) (Revised by NM, 12-Sep-2018.)
Hypotheses
Ref Expression
meetle.b  |-  B  =  ( Base `  K
)
meetle.l  |-  .<_  =  ( le `  K )
meetle.m  |-  ./\  =  ( meet `  K )
meetle.k  |-  ( ph  ->  K  e.  Poset )
meetle.x  |-  ( ph  ->  X  e.  B )
meetle.y  |-  ( ph  ->  Y  e.  B )
meetle.z  |-  ( ph  ->  Z  e.  B )
meetle.e  |-  ( ph  -> 
<. X ,  Y >.  e. 
dom  ./\  )
Assertion
Ref Expression
meetle  |-  ( ph  ->  ( ( Z  .<_  X  /\  Z  .<_  Y )  <-> 
Z  .<_  ( X  ./\  Y ) ) )

Proof of Theorem meetle
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 meetle.z . . 3  |-  ( ph  ->  Z  e.  B )
2 meetle.b . . . . 5  |-  B  =  ( Base `  K
)
3 meetle.l . . . . 5  |-  .<_  =  ( le `  K )
4 meetle.m . . . . 5  |-  ./\  =  ( meet `  K )
5 meetle.k . . . . 5  |-  ( ph  ->  K  e.  Poset )
6 meetle.x . . . . 5  |-  ( ph  ->  X  e.  B )
7 meetle.y . . . . 5  |-  ( ph  ->  Y  e.  B )
8 meetle.e . . . . 5  |-  ( ph  -> 
<. X ,  Y >.  e. 
dom  ./\  )
92, 3, 4, 5, 6, 7, 8meetlem 15195 . . . 4  |-  ( ph  ->  ( ( ( X 
./\  Y )  .<_  X  /\  ( X  ./\  Y )  .<_  Y )  /\  A. z  e.  B  ( ( z  .<_  X  /\  z  .<_  Y )  ->  z  .<_  ( X 
./\  Y ) ) ) )
109simprd 463 . . 3  |-  ( ph  ->  A. z  e.  B  ( ( z  .<_  X  /\  z  .<_  Y )  ->  z  .<_  ( X 
./\  Y ) ) )
11 breq1 4295 . . . . . 6  |-  ( z  =  Z  ->  (
z  .<_  X  <->  Z  .<_  X ) )
12 breq1 4295 . . . . . 6  |-  ( z  =  Z  ->  (
z  .<_  Y  <->  Z  .<_  Y ) )
1311, 12anbi12d 710 . . . . 5  |-  ( z  =  Z  ->  (
( z  .<_  X  /\  z  .<_  Y )  <->  ( Z  .<_  X  /\  Z  .<_  Y ) ) )
14 breq1 4295 . . . . 5  |-  ( z  =  Z  ->  (
z  .<_  ( X  ./\  Y )  <->  Z  .<_  ( X 
./\  Y ) ) )
1513, 14imbi12d 320 . . . 4  |-  ( z  =  Z  ->  (
( ( z  .<_  X  /\  z  .<_  Y )  ->  z  .<_  ( X 
./\  Y ) )  <-> 
( ( Z  .<_  X  /\  Z  .<_  Y )  ->  Z  .<_  ( X 
./\  Y ) ) ) )
1615rspcva 3071 . . 3  |-  ( ( Z  e.  B  /\  A. z  e.  B  ( ( z  .<_  X  /\  z  .<_  Y )  -> 
z  .<_  ( X  ./\  Y ) ) )  -> 
( ( Z  .<_  X  /\  Z  .<_  Y )  ->  Z  .<_  ( X 
./\  Y ) ) )
171, 10, 16syl2anc 661 . 2  |-  ( ph  ->  ( ( Z  .<_  X  /\  Z  .<_  Y )  ->  Z  .<_  ( X 
./\  Y ) ) )
182, 3, 4, 5, 6, 7, 8lemeet1 15196 . . . 4  |-  ( ph  ->  ( X  ./\  Y
)  .<_  X )
192, 4, 5, 6, 7, 8meetcl 15190 . . . . 5  |-  ( ph  ->  ( X  ./\  Y
)  e.  B )
202, 3postr 15123 . . . . 5  |-  ( ( K  e.  Poset  /\  ( Z  e.  B  /\  ( X  ./\  Y )  e.  B  /\  X  e.  B ) )  -> 
( ( Z  .<_  ( X  ./\  Y )  /\  ( X  ./\  Y
)  .<_  X )  ->  Z  .<_  X ) )
215, 1, 19, 6, 20syl13anc 1220 . . . 4  |-  ( ph  ->  ( ( Z  .<_  ( X  ./\  Y )  /\  ( X  ./\  Y
)  .<_  X )  ->  Z  .<_  X ) )
2218, 21mpan2d 674 . . 3  |-  ( ph  ->  ( Z  .<_  ( X 
./\  Y )  ->  Z  .<_  X ) )
232, 3, 4, 5, 6, 7, 8lemeet2 15197 . . . 4  |-  ( ph  ->  ( X  ./\  Y
)  .<_  Y )
242, 3postr 15123 . . . . 5  |-  ( ( K  e.  Poset  /\  ( Z  e.  B  /\  ( X  ./\  Y )  e.  B  /\  Y  e.  B ) )  -> 
( ( Z  .<_  ( X  ./\  Y )  /\  ( X  ./\  Y
)  .<_  Y )  ->  Z  .<_  Y ) )
255, 1, 19, 7, 24syl13anc 1220 . . . 4  |-  ( ph  ->  ( ( Z  .<_  ( X  ./\  Y )  /\  ( X  ./\  Y
)  .<_  Y )  ->  Z  .<_  Y ) )
2623, 25mpan2d 674 . . 3  |-  ( ph  ->  ( Z  .<_  ( X 
./\  Y )  ->  Z  .<_  Y ) )
2722, 26jcad 533 . 2  |-  ( ph  ->  ( Z  .<_  ( X 
./\  Y )  -> 
( Z  .<_  X  /\  Z  .<_  Y ) ) )
2817, 27impbid 191 1  |-  ( ph  ->  ( ( Z  .<_  X  /\  Z  .<_  Y )  <-> 
Z  .<_  ( X  ./\  Y ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1369    e. wcel 1756   A.wral 2715   <.cop 3883   class class class wbr 4292   dom cdm 4840   ` cfv 5418  (class class class)co 6091   Basecbs 14174   lecple 14245   Posetcpo 15110   meetcmee 15115
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4403  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-ral 2720  df-rex 2721  df-reu 2722  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-op 3884  df-uni 4092  df-iun 4173  df-br 4293  df-opab 4351  df-mpt 4352  df-id 4636  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426  df-riota 6052  df-ov 6094  df-oprab 6095  df-poset 15116  df-glb 15145  df-meet 15147
This theorem is referenced by:  latlem12  15248
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