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Theorem prslem 15113
Description: Lemma for prsref 15114 and prstr 15115. (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 15112 . . 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 464 . 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 4309 . . . . 5  |-  ( ( x  =  X  /\  x  =  X )  ->  ( x  .<_  x  <->  X  .<_  X ) )
65anidms 645 . . . 4  |-  ( x  =  X  ->  (
x  .<_  x  <->  X  .<_  X ) )
7 breq1 4307 . . . . . 6  |-  ( x  =  X  ->  (
x  .<_  y  <->  X  .<_  y ) )
87anbi1d 704 . . . . 5  |-  ( x  =  X  ->  (
( x  .<_  y  /\  y  .<_  z )  <->  ( X  .<_  y  /\  y  .<_  z ) ) )
9 breq1 4307 . . . . 5  |-  ( x  =  X  ->  (
x  .<_  z  <->  X  .<_  z ) )
108, 9imbi12d 320 . . . 4  |-  ( x  =  X  ->  (
( ( x  .<_  y  /\  y  .<_  z )  ->  x  .<_  z )  <-> 
( ( X  .<_  y  /\  y  .<_  z )  ->  X  .<_  z ) ) )
116, 10anbi12d 710 . . 3  |-  ( x  =  X  ->  (
( x  .<_  x  /\  ( ( x  .<_  y  /\  y  .<_  z )  ->  x  .<_  z ) )  <->  ( X  .<_  X  /\  ( ( X 
.<_  y  /\  y  .<_  z )  ->  X  .<_  z ) ) ) )
12 breq2 4308 . . . . . 6  |-  ( y  =  Y  ->  ( X  .<_  y  <->  X  .<_  Y ) )
13 breq1 4307 . . . . . 6  |-  ( y  =  Y  ->  (
y  .<_  z  <->  Y  .<_  z ) )
1412, 13anbi12d 710 . . . . 5  |-  ( y  =  Y  ->  (
( X  .<_  y  /\  y  .<_  z )  <->  ( X  .<_  Y  /\  Y  .<_  z ) ) )
1514imbi1d 317 . . . 4  |-  ( y  =  Y  ->  (
( ( X  .<_  y  /\  y  .<_  z )  ->  X  .<_  z )  <-> 
( ( X  .<_  Y  /\  Y  .<_  z )  ->  X  .<_  z ) ) )
1615anbi2d 703 . . 3  |-  ( y  =  Y  ->  (
( X  .<_  X  /\  ( ( X  .<_  y  /\  y  .<_  z )  ->  X  .<_  z ) )  <->  ( X  .<_  X  /\  ( ( X 
.<_  Y  /\  Y  .<_  z )  ->  X  .<_  z ) ) ) )
17 breq2 4308 . . . . . 6  |-  ( z  =  Z  ->  ( Y  .<_  z  <->  Y  .<_  Z ) )
1817anbi2d 703 . . . . 5  |-  ( z  =  Z  ->  (
( X  .<_  Y  /\  Y  .<_  z )  <->  ( X  .<_  Y  /\  Y  .<_  Z ) ) )
19 breq2 4308 . . . . 5  |-  ( z  =  Z  ->  ( X  .<_  z  <->  X  .<_  Z ) )
2018, 19imbi12d 320 . . . 4  |-  ( z  =  Z  ->  (
( ( X  .<_  Y  /\  Y  .<_  z )  ->  X  .<_  z )  <-> 
( ( X  .<_  Y  /\  Y  .<_  Z )  ->  X  .<_  Z ) ) )
2120anbi2d 703 . . 3  |-  ( z  =  Z  ->  (
( X  .<_  X  /\  ( ( X  .<_  Y  /\  Y  .<_  z )  ->  X  .<_  z ) )  <->  ( X  .<_  X  /\  ( ( X 
.<_  Y  /\  Y  .<_  Z )  ->  X  .<_  Z ) ) ) )
2211, 16, 21rspc3v 3094 . 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 469 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 setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756   A.wral 2727   _Vcvv 2984   class class class wbr 4304   ` cfv 5430   Basecbs 14186   lecple 14257    Preset cpreset 15108
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-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-nul 4433
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-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2620  df-ral 2732  df-rex 2733  df-rab 2736  df-v 2986  df-sbc 3199  df-dif 3343  df-un 3345  df-in 3347  df-ss 3354  df-nul 3650  df-if 3804  df-sn 3890  df-pr 3892  df-op 3896  df-uni 4104  df-br 4305  df-iota 5393  df-fv 5438  df-preset 15110
This theorem is referenced by:  prsref  15114  prstr  15115
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