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Theorem joinval2lem 15174
Description: Lemma for joinval2 15175 and joineu 15176. (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 4292 . . 3  |-  ( y  =  X  ->  (
y  .<_  x  <->  X  .<_  x ) )
2 breq1 4292 . . 3  |-  ( y  =  Y  ->  (
y  .<_  x  <->  Y  .<_  x ) )
31, 2ralprg 3922 . 2  |-  ( ( X  e.  B  /\  Y  e.  B )  ->  ( A. y  e. 
{ X ,  Y } y  .<_  x  <->  ( X  .<_  x  /\  Y  .<_  x ) ) )
4 breq1 4292 . . . . 5  |-  ( y  =  X  ->  (
y  .<_  z  <->  X  .<_  z ) )
5 breq1 4292 . . . . 5  |-  ( y  =  Y  ->  (
y  .<_  z  <->  Y  .<_  z ) )
64, 5ralprg 3922 . . . 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 2733 . 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 705 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 1364    e. wcel 1761   A.wral 2713   {cpr 3876   class class class wbr 4289   ` cfv 5415   Basecbs 14170   lecple 14241   joincjn 15110
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ral 2718  df-rab 2722  df-v 2972  df-sbc 3184  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-nul 3635  df-if 3789  df-sn 3875  df-pr 3877  df-op 3881  df-br 4290
This theorem is referenced by:  joinval2  15175  joineu  15176
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