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Theorem cvrval2 32834
Description: Binary relation expressing  Y covers  X. Definition of covers in [Kalmbach] p. 15. (cvbr2 27929 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 32829 . 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 426 . . . . . . . 8  |-  ( ( ( X  .<  z  /\  z  .<_  Y )  ->  z  =  Y )  <->  -.  ( ( X  .<  z  /\  z  .<_  Y )  /\  -.  z  =  Y )
)
6 df-ne 2623 . . . . . . . . 9  |-  ( z  =/=  Y  <->  -.  z  =  Y )
76anbi2i 699 . . . . . . . 8  |-  ( ( ( X  .<  z  /\  z  .<_  Y )  /\  z  =/=  Y
)  <->  ( ( X 
.<  z  /\  z  .<_  Y )  /\  -.  z  =  Y )
)
85, 7xchbinxr 313 . . . . . . 7  |-  ( ( ( X  .<  z  /\  z  .<_  Y )  ->  z  =  Y )  <->  -.  ( ( X  .<  z  /\  z  .<_  Y )  /\  z  =/=  Y ) )
9 cvrletr.l . . . . . . . . . . . . 13  |-  .<_  =  ( le `  K )
109, 2pltval 16199 . . . . . . . . . . . 12  |-  ( ( K  e.  A  /\  z  e.  B  /\  Y  e.  B )  ->  ( z  .<  Y  <->  ( z  .<_  Y  /\  z  =/= 
Y ) ) )
11103com23 1213 . . . . . . . . . . 11  |-  ( ( K  e.  A  /\  Y  e.  B  /\  z  e.  B )  ->  ( z  .<  Y  <->  ( z  .<_  Y  /\  z  =/= 
Y ) ) )
12113expa 1207 . . . . . . . . . 10  |-  ( ( ( K  e.  A  /\  Y  e.  B
)  /\  z  e.  B )  ->  (
z  .<  Y  <->  ( z  .<_  Y  /\  z  =/= 
Y ) ) )
1312anbi2d 709 . . . . . . . . 9  |-  ( ( ( K  e.  A  /\  Y  e.  B
)  /\  z  e.  B )  ->  (
( X  .<  z  /\  z  .<  Y )  <-> 
( X  .<  z  /\  ( z  .<_  Y  /\  z  =/=  Y ) ) ) )
14 anass 654 . . . . . . . . 9  |-  ( ( ( X  .<  z  /\  z  .<_  Y )  /\  z  =/=  Y
)  <->  ( X  .<  z  /\  ( z  .<_  Y  /\  z  =/=  Y
) ) )
1513, 14syl6rbbr 268 . . . . . . . 8  |-  ( ( ( K  e.  A  /\  Y  e.  B
)  /\  z  e.  B )  ->  (
( ( X  .<  z  /\  z  .<_  Y )  /\  z  =/=  Y
)  <->  ( X  .<  z  /\  z  .<  Y ) ) )
1615notbid 296 . . . . . . 7  |-  ( ( ( K  e.  A  /\  Y  e.  B
)  /\  z  e.  B )  ->  ( -.  ( ( X  .<  z  /\  z  .<_  Y )  /\  z  =/=  Y
)  <->  -.  ( X  .<  z  /\  z  .<  Y ) ) )
178, 16syl5bb 261 . . . . . 6  |-  ( ( ( K  e.  A  /\  Y  e.  B
)  /\  z  e.  B )  ->  (
( ( X  .<  z  /\  z  .<_  Y )  ->  z  =  Y )  <->  -.  ( X  .<  z  /\  z  .<  Y ) ) )
1817ralbidva 2823 . . . . 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 2833 . . . . 5  |-  ( A. z  e.  B  -.  ( X  .<  z  /\  z  .<  Y )  <->  -.  E. z  e.  B  ( X  .<  z  /\  z  .<  Y ) )
2018, 19syl6bb 265 . . . 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 709 . . 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 1026 . 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 260 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 setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 188    /\ wa 371    /\ w3a 984    = wceq 1443    e. wcel 1886    =/= wne 2621   A.wral 2736   E.wrex 2737   class class class wbr 4401   ` cfv 5581   Basecbs 15114   lecple 15190   ltcplt 16179    <o ccvr 32822
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1668  ax-4 1681  ax-5 1757  ax-6 1804  ax-7 1850  ax-8 1888  ax-9 1895  ax-10 1914  ax-11 1919  ax-12 1932  ax-13 2090  ax-ext 2430  ax-sep 4524  ax-nul 4533  ax-pow 4580  ax-pr 4638  ax-un 6580
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 986  df-tru 1446  df-ex 1663  df-nf 1667  df-sb 1797  df-eu 2302  df-mo 2303  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2580  df-ne 2623  df-ral 2741  df-rex 2742  df-rab 2745  df-v 3046  df-sbc 3267  df-dif 3406  df-un 3408  df-in 3410  df-ss 3417  df-nul 3731  df-if 3881  df-pw 3952  df-sn 3968  df-pr 3970  df-op 3974  df-uni 4198  df-br 4402  df-opab 4461  df-mpt 4462  df-id 4748  df-xp 4839  df-rel 4840  df-cnv 4841  df-co 4842  df-dm 4843  df-iota 5545  df-fun 5583  df-fv 5589  df-plt 16197  df-covers 32826
This theorem is referenced by:  isat3  32867  cvlcvr1  32899
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