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Theorem prslem 14380
Description: Lemma for prsref 14381 and prstr 14382. (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 14379 . . 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 451 . 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 4209 . . . . 5  |-  ( ( x  =  X  /\  x  =  X )  ->  ( x  .<_  x  <->  X  .<_  X ) )
65anidms 627 . . . 4  |-  ( x  =  X  ->  (
x  .<_  x  <->  X  .<_  X ) )
7 breq1 4207 . . . . . 6  |-  ( x  =  X  ->  (
x  .<_  y  <->  X  .<_  y ) )
87anbi1d 686 . . . . 5  |-  ( x  =  X  ->  (
( x  .<_  y  /\  y  .<_  z )  <->  ( X  .<_  y  /\  y  .<_  z ) ) )
9 breq1 4207 . . . . 5  |-  ( x  =  X  ->  (
x  .<_  z  <->  X  .<_  z ) )
108, 9imbi12d 312 . . . 4  |-  ( x  =  X  ->  (
( ( x  .<_  y  /\  y  .<_  z )  ->  x  .<_  z )  <-> 
( ( X  .<_  y  /\  y  .<_  z )  ->  X  .<_  z ) ) )
116, 10anbi12d 692 . . 3  |-  ( x  =  X  ->  (
( x  .<_  x  /\  ( ( x  .<_  y  /\  y  .<_  z )  ->  x  .<_  z ) )  <->  ( X  .<_  X  /\  ( ( X 
.<_  y  /\  y  .<_  z )  ->  X  .<_  z ) ) ) )
12 breq2 4208 . . . . . 6  |-  ( y  =  Y  ->  ( X  .<_  y  <->  X  .<_  Y ) )
13 breq1 4207 . . . . . 6  |-  ( y  =  Y  ->  (
y  .<_  z  <->  Y  .<_  z ) )
1412, 13anbi12d 692 . . . . 5  |-  ( y  =  Y  ->  (
( X  .<_  y  /\  y  .<_  z )  <->  ( X  .<_  Y  /\  Y  .<_  z ) ) )
1514imbi1d 309 . . . 4  |-  ( y  =  Y  ->  (
( ( X  .<_  y  /\  y  .<_  z )  ->  X  .<_  z )  <-> 
( ( X  .<_  Y  /\  Y  .<_  z )  ->  X  .<_  z ) ) )
1615anbi2d 685 . . 3  |-  ( y  =  Y  ->  (
( X  .<_  X  /\  ( ( X  .<_  y  /\  y  .<_  z )  ->  X  .<_  z ) )  <->  ( X  .<_  X  /\  ( ( X 
.<_  Y  /\  Y  .<_  z )  ->  X  .<_  z ) ) ) )
17 breq2 4208 . . . . . 6  |-  ( z  =  Z  ->  ( Y  .<_  z  <->  Y  .<_  Z ) )
1817anbi2d 685 . . . . 5  |-  ( z  =  Z  ->  (
( X  .<_  Y  /\  Y  .<_  z )  <->  ( X  .<_  Y  /\  Y  .<_  Z ) ) )
19 breq2 4208 . . . . 5  |-  ( z  =  Z  ->  ( X  .<_  z  <->  X  .<_  Z ) )
2018, 19imbi12d 312 . . . 4  |-  ( z  =  Z  ->  (
( ( X  .<_  Y  /\  Y  .<_  z )  ->  X  .<_  z )  <-> 
( ( X  .<_  Y  /\  Y  .<_  Z )  ->  X  .<_  Z ) ) )
2120anbi2d 685 . . 3  |-  ( z  =  Z  ->  (
( X  .<_  X  /\  ( ( X  .<_  Y  /\  Y  .<_  z )  ->  X  .<_  z ) )  <->  ( X  .<_  X  /\  ( ( X 
.<_  Y  /\  Y  .<_  Z )  ->  X  .<_  Z ) ) ) )
2211, 16, 21rspc3v 3053 . 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 456 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 177    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725   A.wral 2697   _Vcvv 2948   class class class wbr 4204   ` cfv 5446   Basecbs 13461   lecple 13528    Preset cpreset 14375
This theorem is referenced by:  prsref  14381  prstr  14382
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-nul 4330
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-iota 5410  df-fv 5454  df-preset 14377
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