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Theorem cvrval2 29757
Description: Binary relation expressing  Y covers  X. Definition of covers in [Kalmbach] p. 15. (cvbr2 23739 analog.) (Contributed by NM, 16-Nov-2011.)
Hypotheses
Ref Expression
cvrletr.b  |-  B  =  ( Base `  K
)
cvrletr.l  |-  .<_  =  ( le `  K )
cvrletr.s  |-  .<  =  ( lt `  K )
cvrletr.c  |-  C  =  (  <o  `  K )
Assertion
Ref Expression
cvrval2  |-  ( ( K  e.  A  /\  X  e.  B  /\  Y  e.  B )  ->  ( X C Y  <-> 
( X  .<  Y  /\  A. z  e.  B  ( ( X  .<  z  /\  z  .<_  Y )  ->  z  =  Y ) ) ) )
Distinct variable groups:    z, A    z, B    z, K    z, X    z, Y
Allowed substitution hints:    C( z)    .< ( z)    .<_ ( z)

Proof of Theorem cvrval2
StepHypRef Expression
1 cvrletr.b . . 3  |-  B  =  ( Base `  K
)
2 cvrletr.s . . 3  |-  .<  =  ( lt `  K )
3 cvrletr.c . . 3  |-  C  =  (  <o  `  K )
41, 2, 3cvrval 29752 . 2  |-  ( ( K  e.  A  /\  X  e.  B  /\  Y  e.  B )  ->  ( X C Y  <-> 
( X  .<  Y  /\  -.  E. z  e.  B  ( X  .<  z  /\  z  .<  Y ) ) ) )
5 iman 414 . . . . . . . 8  |-  ( ( ( X  .<  z  /\  z  .<_  Y )  ->  z  =  Y )  <->  -.  ( ( X  .<  z  /\  z  .<_  Y )  /\  -.  z  =  Y )
)
6 df-ne 2569 . . . . . . . . 9  |-  ( z  =/=  Y  <->  -.  z  =  Y )
76anbi2i 676 . . . . . . . 8  |-  ( ( ( X  .<  z  /\  z  .<_  Y )  /\  z  =/=  Y
)  <->  ( ( X 
.<  z  /\  z  .<_  Y )  /\  -.  z  =  Y )
)
85, 7xchbinxr 303 . . . . . . 7  |-  ( ( ( X  .<  z  /\  z  .<_  Y )  ->  z  =  Y )  <->  -.  ( ( X  .<  z  /\  z  .<_  Y )  /\  z  =/=  Y ) )
9 cvrletr.l . . . . . . . . . . . . 13  |-  .<_  =  ( le `  K )
109, 2pltval 14372 . . . . . . . . . . . 12  |-  ( ( K  e.  A  /\  z  e.  B  /\  Y  e.  B )  ->  ( z  .<  Y  <->  ( z  .<_  Y  /\  z  =/= 
Y ) ) )
11103com23 1159 . . . . . . . . . . 11  |-  ( ( K  e.  A  /\  Y  e.  B  /\  z  e.  B )  ->  ( z  .<  Y  <->  ( z  .<_  Y  /\  z  =/= 
Y ) ) )
12113expa 1153 . . . . . . . . . 10  |-  ( ( ( K  e.  A  /\  Y  e.  B
)  /\  z  e.  B )  ->  (
z  .<  Y  <->  ( z  .<_  Y  /\  z  =/= 
Y ) ) )
1312anbi2d 685 . . . . . . . . 9  |-  ( ( ( K  e.  A  /\  Y  e.  B
)  /\  z  e.  B )  ->  (
( X  .<  z  /\  z  .<  Y )  <-> 
( X  .<  z  /\  ( z  .<_  Y  /\  z  =/=  Y ) ) ) )
14 anass 631 . . . . . . . . 9  |-  ( ( ( X  .<  z  /\  z  .<_  Y )  /\  z  =/=  Y
)  <->  ( X  .<  z  /\  ( z  .<_  Y  /\  z  =/=  Y
) ) )
1513, 14syl6rbbr 256 . . . . . . . 8  |-  ( ( ( K  e.  A  /\  Y  e.  B
)  /\  z  e.  B )  ->  (
( ( X  .<  z  /\  z  .<_  Y )  /\  z  =/=  Y
)  <->  ( X  .<  z  /\  z  .<  Y ) ) )
1615notbid 286 . . . . . . 7  |-  ( ( ( K  e.  A  /\  Y  e.  B
)  /\  z  e.  B )  ->  ( -.  ( ( X  .<  z  /\  z  .<_  Y )  /\  z  =/=  Y
)  <->  -.  ( X  .<  z  /\  z  .<  Y ) ) )
178, 16syl5bb 249 . . . . . 6  |-  ( ( ( K  e.  A  /\  Y  e.  B
)  /\  z  e.  B )  ->  (
( ( X  .<  z  /\  z  .<_  Y )  ->  z  =  Y )  <->  -.  ( X  .<  z  /\  z  .<  Y ) ) )
1817ralbidva 2682 . . . . 5  |-  ( ( K  e.  A  /\  Y  e.  B )  ->  ( A. z  e.  B  ( ( X 
.<  z  /\  z  .<_  Y )  ->  z  =  Y )  <->  A. z  e.  B  -.  ( X  .<  z  /\  z  .<  Y ) ) )
19 ralnex 2676 . . . . 5  |-  ( A. z  e.  B  -.  ( X  .<  z  /\  z  .<  Y )  <->  -.  E. z  e.  B  ( X  .<  z  /\  z  .<  Y ) )
2018, 19syl6bb 253 . . . 4  |-  ( ( K  e.  A  /\  Y  e.  B )  ->  ( A. z  e.  B  ( ( X 
.<  z  /\  z  .<_  Y )  ->  z  =  Y )  <->  -.  E. z  e.  B  ( X  .<  z  /\  z  .<  Y ) ) )
2120anbi2d 685 . . 3  |-  ( ( K  e.  A  /\  Y  e.  B )  ->  ( ( X  .<  Y  /\  A. z  e.  B  ( ( X 
.<  z  /\  z  .<_  Y )  ->  z  =  Y ) )  <->  ( X  .<  Y  /\  -.  E. z  e.  B  ( X  .<  z  /\  z  .<  Y ) ) ) )
22213adant2 976 . 2  |-  ( ( K  e.  A  /\  X  e.  B  /\  Y  e.  B )  ->  ( ( X  .<  Y  /\  A. z  e.  B  ( ( X 
.<  z  /\  z  .<_  Y )  ->  z  =  Y ) )  <->  ( X  .<  Y  /\  -.  E. z  e.  B  ( X  .<  z  /\  z  .<  Y ) ) ) )
234, 22bitr4d 248 1  |-  ( ( K  e.  A  /\  X  e.  B  /\  Y  e.  B )  ->  ( X C Y  <-> 
( X  .<  Y  /\  A. z  e.  B  ( ( X  .<  z  /\  z  .<_  Y )  ->  z  =  Y ) ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1721    =/= wne 2567   A.wral 2666   E.wrex 2667   class class class wbr 4172   ` cfv 5413   Basecbs 13424   lecple 13491   ltcplt 14353    <o ccvr 29745
This theorem is referenced by:  isat3  29790  cvlcvr1  29822
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-rab 2675  df-v 2918  df-sbc 3122  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-br 4173  df-opab 4227  df-mpt 4228  df-id 4458  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-iota 5377  df-fun 5415  df-fv 5421  df-plt 14370  df-covers 29749
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