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Theorem cvrval2 29537
Description: Binary relation expressing  Y covers  X. Definition of covers in [Kalmbach] p. 15. (cvbr2 22865 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 29532 . 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 413 . . . . . . . 8  |-  ( ( ( X  .<  z  /\  z  .<_  Y )  ->  z  =  Y )  <->  -.  ( ( X  .<  z  /\  z  .<_  Y )  /\  -.  z  =  Y )
)
6 df-ne 2450 . . . . . . . . 9  |-  ( z  =/=  Y  <->  -.  z  =  Y )
76anbi2i 675 . . . . . . . 8  |-  ( ( ( X  .<  z  /\  z  .<_  Y )  /\  z  =/=  Y
)  <->  ( ( X 
.<  z  /\  z  .<_  Y )  /\  -.  z  =  Y )
)
85, 7xchbinxr 302 . . . . . . 7  |-  ( ( ( X  .<  z  /\  z  .<_  Y )  ->  z  =  Y )  <->  -.  ( ( X  .<  z  /\  z  .<_  Y )  /\  z  =/=  Y ) )
9 cvrletr.l . . . . . . . . . . . . 13  |-  .<_  =  ( le `  K )
109, 2pltval 14096 . . . . . . . . . . . 12  |-  ( ( K  e.  A  /\  z  e.  B  /\  Y  e.  B )  ->  ( z  .<  Y  <->  ( z  .<_  Y  /\  z  =/= 
Y ) ) )
11103com23 1157 . . . . . . . . . . 11  |-  ( ( K  e.  A  /\  Y  e.  B  /\  z  e.  B )  ->  ( z  .<  Y  <->  ( z  .<_  Y  /\  z  =/= 
Y ) ) )
12113expa 1151 . . . . . . . . . 10  |-  ( ( ( K  e.  A  /\  Y  e.  B
)  /\  z  e.  B )  ->  (
z  .<  Y  <->  ( z  .<_  Y  /\  z  =/= 
Y ) ) )
1312anbi2d 684 . . . . . . . . 9  |-  ( ( ( K  e.  A  /\  Y  e.  B
)  /\  z  e.  B )  ->  (
( X  .<  z  /\  z  .<  Y )  <-> 
( X  .<  z  /\  ( z  .<_  Y  /\  z  =/=  Y ) ) ) )
14 anass 630 . . . . . . . . 9  |-  ( ( ( X  .<  z  /\  z  .<_  Y )  /\  z  =/=  Y
)  <->  ( X  .<  z  /\  ( z  .<_  Y  /\  z  =/=  Y
) ) )
1513, 14syl6rbbr 255 . . . . . . . 8  |-  ( ( ( K  e.  A  /\  Y  e.  B
)  /\  z  e.  B )  ->  (
( ( X  .<  z  /\  z  .<_  Y )  /\  z  =/=  Y
)  <->  ( X  .<  z  /\  z  .<  Y ) ) )
1615notbid 285 . . . . . . 7  |-  ( ( ( K  e.  A  /\  Y  e.  B
)  /\  z  e.  B )  ->  ( -.  ( ( X  .<  z  /\  z  .<_  Y )  /\  z  =/=  Y
)  <->  -.  ( X  .<  z  /\  z  .<  Y ) ) )
178, 16syl5bb 248 . . . . . 6  |-  ( ( ( K  e.  A  /\  Y  e.  B
)  /\  z  e.  B )  ->  (
( ( X  .<  z  /\  z  .<_  Y )  ->  z  =  Y )  <->  -.  ( X  .<  z  /\  z  .<  Y ) ) )
1817ralbidva 2561 . . . . 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 2555 . . . . 5  |-  ( A. z  e.  B  -.  ( X  .<  z  /\  z  .<  Y )  <->  -.  E. z  e.  B  ( X  .<  z  /\  z  .<  Y ) )
2018, 19syl6bb 252 . . . 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 684 . . 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 974 . 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 247 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 176    /\ wa 358    /\ w3a 934    = wceq 1625    e. wcel 1686    =/= wne 2448   A.wral 2545   E.wrex 2546   class class class wbr 4025   ` cfv 5257   Basecbs 13150   lecple 13217   ltcplt 14077    <o ccvr 29525
This theorem is referenced by:  isat3  29570  cvlcvr1  29602
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1535  ax-5 1546  ax-17 1605  ax-9 1637  ax-8 1645  ax-13 1688  ax-14 1690  ax-6 1705  ax-7 1710  ax-11 1717  ax-12 1868  ax-ext 2266  ax-sep 4143  ax-nul 4151  ax-pow 4190  ax-pr 4216  ax-un 4514
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1531  df-nf 1534  df-sb 1632  df-eu 2149  df-mo 2150  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-ne 2450  df-ral 2550  df-rex 2551  df-rab 2554  df-v 2792  df-sbc 2994  df-dif 3157  df-un 3159  df-in 3161  df-ss 3168  df-nul 3458  df-if 3568  df-pw 3629  df-sn 3648  df-pr 3649  df-op 3651  df-uni 3830  df-br 4026  df-opab 4080  df-mpt 4081  df-id 4311  df-xp 4697  df-rel 4698  df-cnv 4699  df-co 4700  df-dm 4701  df-iota 5221  df-fun 5259  df-fv 5265  df-plt 14094  df-covers 29529
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