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Theorem prslem 14065
Description: Lemma for prsref 14066 and prstr 14067. (Contributed by Mario Carneiro, 1-Feb-2015.)
Hypotheses
Ref Expression
isprs.b  |-  B  =  ( Base `  K
)
isprs.l  |-  .<_  =  ( le `  K )
Assertion
Ref Expression
prslem  |-  ( ( K  e.  Preset  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )
)  ->  ( X  .<_  X  /\  ( ( X  .<_  Y  /\  Y  .<_  Z )  ->  X  .<_  Z ) ) )

Proof of Theorem prslem
Dummy variables  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 isprs.b . . . 4  |-  B  =  ( Base `  K
)
2 isprs.l . . . 4  |-  .<_  =  ( le `  K )
31, 2isprs 14064 . . 3  |-  ( K  e.  Preset 
<->  ( K  e.  _V  /\ 
A. x  e.  B  A. y  e.  B  A. z  e.  B  ( x  .<_  x  /\  ( ( x  .<_  y  /\  y  .<_  z )  ->  x  .<_  z ) ) ) )
43simprbi 450 . 2  |-  ( K  e.  Preset  ->  A. x  e.  B  A. y  e.  B  A. z  e.  B  ( x  .<_  x  /\  ( ( x  .<_  y  /\  y  .<_  z )  ->  x  .<_  z ) ) )
5 breq12 4028 . . . . 5  |-  ( ( x  =  X  /\  x  =  X )  ->  ( x  .<_  x  <->  X  .<_  X ) )
65anidms 626 . . . 4  |-  ( x  =  X  ->  (
x  .<_  x  <->  X  .<_  X ) )
7 breq1 4026 . . . . . 6  |-  ( x  =  X  ->  (
x  .<_  y  <->  X  .<_  y ) )
87anbi1d 685 . . . . 5  |-  ( x  =  X  ->  (
( x  .<_  y  /\  y  .<_  z )  <->  ( X  .<_  y  /\  y  .<_  z ) ) )
9 breq1 4026 . . . . 5  |-  ( x  =  X  ->  (
x  .<_  z  <->  X  .<_  z ) )
108, 9imbi12d 311 . . . 4  |-  ( x  =  X  ->  (
( ( x  .<_  y  /\  y  .<_  z )  ->  x  .<_  z )  <-> 
( ( X  .<_  y  /\  y  .<_  z )  ->  X  .<_  z ) ) )
116, 10anbi12d 691 . . 3  |-  ( x  =  X  ->  (
( x  .<_  x  /\  ( ( x  .<_  y  /\  y  .<_  z )  ->  x  .<_  z ) )  <->  ( X  .<_  X  /\  ( ( X 
.<_  y  /\  y  .<_  z )  ->  X  .<_  z ) ) ) )
12 breq2 4027 . . . . . 6  |-  ( y  =  Y  ->  ( X  .<_  y  <->  X  .<_  Y ) )
13 breq1 4026 . . . . . 6  |-  ( y  =  Y  ->  (
y  .<_  z  <->  Y  .<_  z ) )
1412, 13anbi12d 691 . . . . 5  |-  ( y  =  Y  ->  (
( X  .<_  y  /\  y  .<_  z )  <->  ( X  .<_  Y  /\  Y  .<_  z ) ) )
1514imbi1d 308 . . . 4  |-  ( y  =  Y  ->  (
( ( X  .<_  y  /\  y  .<_  z )  ->  X  .<_  z )  <-> 
( ( X  .<_  Y  /\  Y  .<_  z )  ->  X  .<_  z ) ) )
1615anbi2d 684 . . 3  |-  ( y  =  Y  ->  (
( X  .<_  X  /\  ( ( X  .<_  y  /\  y  .<_  z )  ->  X  .<_  z ) )  <->  ( X  .<_  X  /\  ( ( X 
.<_  Y  /\  Y  .<_  z )  ->  X  .<_  z ) ) ) )
17 breq2 4027 . . . . . 6  |-  ( z  =  Z  ->  ( Y  .<_  z  <->  Y  .<_  Z ) )
1817anbi2d 684 . . . . 5  |-  ( z  =  Z  ->  (
( X  .<_  Y  /\  Y  .<_  z )  <->  ( X  .<_  Y  /\  Y  .<_  Z ) ) )
19 breq2 4027 . . . . 5  |-  ( z  =  Z  ->  ( X  .<_  z  <->  X  .<_  Z ) )
2018, 19imbi12d 311 . . . 4  |-  ( z  =  Z  ->  (
( ( X  .<_  Y  /\  Y  .<_  z )  ->  X  .<_  z )  <-> 
( ( X  .<_  Y  /\  Y  .<_  Z )  ->  X  .<_  Z ) ) )
2120anbi2d 684 . . 3  |-  ( z  =  Z  ->  (
( X  .<_  X  /\  ( ( X  .<_  Y  /\  Y  .<_  z )  ->  X  .<_  z ) )  <->  ( X  .<_  X  /\  ( ( X 
.<_  Y  /\  Y  .<_  Z )  ->  X  .<_  Z ) ) ) )
2211, 16, 21rspc3v 2893 . 2  |-  ( ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )  ->  ( A. x  e.  B  A. y  e.  B  A. z  e.  B  ( x  .<_  x  /\  ( ( x 
.<_  y  /\  y  .<_  z )  ->  x  .<_  z ) )  -> 
( X  .<_  X  /\  ( ( X  .<_  Y  /\  Y  .<_  Z )  ->  X  .<_  Z ) ) ) )
234, 22mpan9 455 1  |-  ( ( K  e.  Preset  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )
)  ->  ( X  .<_  X  /\  ( ( X  .<_  Y  /\  Y  .<_  Z )  ->  X  .<_  Z ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   A.wral 2543   _Vcvv 2788   class class class wbr 4023   ` cfv 5255   Basecbs 13148   lecple 13215    Preset cpreset 14060
This theorem is referenced by:  prsref  14066  prstr  14067
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-nul 4149
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-iota 5219  df-fv 5263  df-preset 14062
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