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Theorem cvrval2 28714
Description: Binary relation expressing  Y covers  X. Definition of covers in [Kalmbach] p. 15. (cvbr2 22824 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 28709 . 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 415 . . . . . . . 8  |-  ( ( ( X  .<  z  /\  z  .<_  Y )  ->  z  =  Y )  <->  -.  ( ( X  .<  z  /\  z  .<_  Y )  /\  -.  z  =  Y )
)
6 df-ne 2423 . . . . . . . . 9  |-  ( z  =/=  Y  <->  -.  z  =  Y )
76anbi2i 678 . . . . . . . 8  |-  ( ( ( X  .<  z  /\  z  .<_  Y )  /\  z  =/=  Y
)  <->  ( ( X 
.<  z  /\  z  .<_  Y )  /\  -.  z  =  Y )
)
85, 7xchbinxr 304 . . . . . . 7  |-  ( ( ( X  .<  z  /\  z  .<_  Y )  ->  z  =  Y )  <->  -.  ( ( X  .<  z  /\  z  .<_  Y )  /\  z  =/=  Y ) )
9 cvrletr.l . . . . . . . . . . . . 13  |-  .<_  =  ( le `  K )
109, 2pltval 14057 . . . . . . . . . . . 12  |-  ( ( K  e.  A  /\  z  e.  B  /\  Y  e.  B )  ->  ( z  .<  Y  <->  ( z  .<_  Y  /\  z  =/= 
Y ) ) )
11103com23 1162 . . . . . . . . . . 11  |-  ( ( K  e.  A  /\  Y  e.  B  /\  z  e.  B )  ->  ( z  .<  Y  <->  ( z  .<_  Y  /\  z  =/= 
Y ) ) )
12113expa 1156 . . . . . . . . . 10  |-  ( ( ( K  e.  A  /\  Y  e.  B
)  /\  z  e.  B )  ->  (
z  .<  Y  <->  ( z  .<_  Y  /\  z  =/= 
Y ) ) )
1312anbi2d 687 . . . . . . . . 9  |-  ( ( ( K  e.  A  /\  Y  e.  B
)  /\  z  e.  B )  ->  (
( X  .<  z  /\  z  .<  Y )  <-> 
( X  .<  z  /\  ( z  .<_  Y  /\  z  =/=  Y ) ) ) )
14 anass 633 . . . . . . . . 9  |-  ( ( ( X  .<  z  /\  z  .<_  Y )  /\  z  =/=  Y
)  <->  ( X  .<  z  /\  ( z  .<_  Y  /\  z  =/=  Y
) ) )
1513, 14syl6rbbr 257 . . . . . . . 8  |-  ( ( ( K  e.  A  /\  Y  e.  B
)  /\  z  e.  B )  ->  (
( ( X  .<  z  /\  z  .<_  Y )  /\  z  =/=  Y
)  <->  ( X  .<  z  /\  z  .<  Y ) ) )
1615notbid 287 . . . . . . 7  |-  ( ( ( K  e.  A  /\  Y  e.  B
)  /\  z  e.  B )  ->  ( -.  ( ( X  .<  z  /\  z  .<_  Y )  /\  z  =/=  Y
)  <->  -.  ( X  .<  z  /\  z  .<  Y ) ) )
178, 16syl5bb 250 . . . . . 6  |-  ( ( ( K  e.  A  /\  Y  e.  B
)  /\  z  e.  B )  ->  (
( ( X  .<  z  /\  z  .<_  Y )  ->  z  =  Y )  <->  -.  ( X  .<  z  /\  z  .<  Y ) ) )
1817ralbidva 2534 . . . . 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 2528 . . . . 5  |-  ( A. z  e.  B  -.  ( X  .<  z  /\  z  .<  Y )  <->  -.  E. z  e.  B  ( X  .<  z  /\  z  .<  Y ) )
2018, 19syl6bb 254 . . . 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 687 . . 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 979 . 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 249 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 5    -> wi 6    <-> wb 178    /\ wa 360    /\ w3a 939    = wceq 1619    e. wcel 1621    =/= wne 2421   A.wral 2518   E.wrex 2519   class class class wbr 3997   ` cfv 4673   Basecbs 13111   lecple 13178   ltcplt 14038    <o ccvr 28702
This theorem is referenced by:  isat3  28747  cvlcvr1  28779
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2239  ax-sep 4115  ax-nul 4123  ax-pow 4160  ax-pr 4186  ax-un 4484
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2122  df-mo 2123  df-clab 2245  df-cleq 2251  df-clel 2254  df-nfc 2383  df-ne 2423  df-ral 2523  df-rex 2524  df-rab 2527  df-v 2765  df-sbc 2967  df-dif 3130  df-un 3132  df-in 3134  df-ss 3141  df-nul 3431  df-if 3540  df-pw 3601  df-sn 3620  df-pr 3621  df-op 3623  df-uni 3802  df-br 3998  df-opab 4052  df-mpt 4053  df-id 4281  df-xp 4675  df-rel 4676  df-cnv 4677  df-co 4678  df-dm 4679  df-rn 4680  df-res 4681  df-ima 4682  df-fun 4683  df-fv 4689  df-plt 14055  df-covers 28706
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