MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  joinval2lem Structured version   Unicode version

Theorem joinval2lem 15764
Description: Lemma for joinval2 15765 and joineu 15766. (Contributed by NM, 12-Sep-2018.) TODO: combine this through joineu into joinlem?
Hypotheses
Ref Expression
joinval2.b  |-  B  =  ( Base `  K
)
joinval2.l  |-  .<_  =  ( le `  K )
joinval2.j  |-  .\/  =  ( join `  K )
joinval2.k  |-  ( ph  ->  K  e.  V )
joinval2.x  |-  ( ph  ->  X  e.  B )
joinval2.y  |-  ( ph  ->  Y  e.  B )
Assertion
Ref Expression
joinval2lem  |-  ( ( X  e.  B  /\  Y  e.  B )  ->  ( ( A. y  e.  { X ,  Y } y  .<_  x  /\  A. z  e.  B  ( A. y  e.  { X ,  Y }
y  .<_  z  ->  x  .<_  z ) )  <->  ( ( X  .<_  x  /\  Y  .<_  x )  /\  A. z  e.  B  (
( X  .<_  z  /\  Y  .<_  z )  ->  x  .<_  z ) ) ) )
Distinct variable groups:    x, z, B    x,  .\/ , z    x, y, K, z    y,  .<_    x, X, y, z    x, Y, y, z
Allowed substitution hints:    ph( x, y, z)    B( y)    .\/ ( y)    .<_ ( x, z)    V( x, y, z)

Proof of Theorem joinval2lem
StepHypRef Expression
1 breq1 4459 . . 3  |-  ( y  =  X  ->  (
y  .<_  x  <->  X  .<_  x ) )
2 breq1 4459 . . 3  |-  ( y  =  Y  ->  (
y  .<_  x  <->  Y  .<_  x ) )
31, 2ralprg 4081 . 2  |-  ( ( X  e.  B  /\  Y  e.  B )  ->  ( A. y  e. 
{ X ,  Y } y  .<_  x  <->  ( X  .<_  x  /\  Y  .<_  x ) ) )
4 breq1 4459 . . . . 5  |-  ( y  =  X  ->  (
y  .<_  z  <->  X  .<_  z ) )
5 breq1 4459 . . . . 5  |-  ( y  =  Y  ->  (
y  .<_  z  <->  Y  .<_  z ) )
64, 5ralprg 4081 . . . 4  |-  ( ( X  e.  B  /\  Y  e.  B )  ->  ( A. y  e. 
{ X ,  Y } y  .<_  z  <->  ( X  .<_  z  /\  Y  .<_  z ) ) )
76imbi1d 317 . . 3  |-  ( ( X  e.  B  /\  Y  e.  B )  ->  ( ( A. y  e.  { X ,  Y } y  .<_  z  ->  x  .<_  z )  <->  ( ( X  .<_  z  /\  Y  .<_  z )  ->  x  .<_  z ) ) )
87ralbidv 2896 . 2  |-  ( ( X  e.  B  /\  Y  e.  B )  ->  ( A. z  e.  B  ( A. y  e.  { X ,  Y } y  .<_  z  ->  x  .<_  z )  <->  A. z  e.  B  ( ( X  .<_  z  /\  Y  .<_  z )  ->  x  .<_  z ) ) )
93, 8anbi12d 710 1  |-  ( ( X  e.  B  /\  Y  e.  B )  ->  ( ( A. y  e.  { X ,  Y } y  .<_  x  /\  A. z  e.  B  ( A. y  e.  { X ,  Y }
y  .<_  z  ->  x  .<_  z ) )  <->  ( ( X  .<_  x  /\  Y  .<_  x )  /\  A. z  e.  B  (
( X  .<_  z  /\  Y  .<_  z )  ->  x  .<_  z ) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1395    e. wcel 1819   A.wral 2807   {cpr 4034   class class class wbr 4456   ` cfv 5594   Basecbs 14643   lecple 14718   joincjn 15699
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ral 2812  df-rab 2816  df-v 3111  df-sbc 3328  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-sn 4033  df-pr 4035  df-op 4039  df-br 4457
This theorem is referenced by:  joinval2  15765  joineu  15766
  Copyright terms: Public domain W3C validator