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Theorem pexmidlem1N 33923
Description: Lemma for pexmidN 33922. Holland's proof implicitly requires  q  =/=  r, which we prove here. (Contributed by NM, 2-Feb-2012.) (New usage is discouraged.)
Hypotheses
Ref Expression
pexmidlem.l  |-  .<_  =  ( le `  K )
pexmidlem.j  |-  .\/  =  ( join `  K )
pexmidlem.a  |-  A  =  ( Atoms `  K )
pexmidlem.p  |-  .+  =  ( +P `  K
)
pexmidlem.o  |-  ._|_  =  ( _|_P `  K
)
pexmidlem.m  |-  M  =  ( X  .+  {
p } )
Assertion
Ref Expression
pexmidlem1N  |-  ( ( ( K  e.  HL  /\  X  C_  A )  /\  ( r  e.  X  /\  q  e.  (  ._|_  `  X ) ) )  ->  q  =/=  r )

Proof of Theorem pexmidlem1N
StepHypRef Expression
1 n0i 3743 . . 3  |-  ( r  e.  ( X  i^i  (  ._|_  `  X )
)  ->  -.  ( X  i^i  (  ._|_  `  X
) )  =  (/) )
2 pexmidlem.a . . . . 5  |-  A  =  ( Atoms `  K )
3 pexmidlem.o . . . . 5  |-  ._|_  =  ( _|_P `  K
)
42, 3pnonsingN 33886 . . . 4  |-  ( ( K  e.  HL  /\  X  C_  A )  -> 
( X  i^i  (  ._|_  `  X ) )  =  (/) )
54adantr 465 . . 3  |-  ( ( ( K  e.  HL  /\  X  C_  A )  /\  ( r  e.  X  /\  q  e.  (  ._|_  `  X ) ) )  ->  ( X  i^i  (  ._|_  `  X
) )  =  (/) )
61, 5nsyl3 119 . 2  |-  ( ( ( K  e.  HL  /\  X  C_  A )  /\  ( r  e.  X  /\  q  e.  (  ._|_  `  X ) ) )  ->  -.  r  e.  ( X  i^i  (  ._|_  `  X ) ) )
7 simprr 756 . . . . . 6  |-  ( ( ( K  e.  HL  /\  X  C_  A )  /\  ( r  e.  X  /\  q  e.  (  ._|_  `  X ) ) )  ->  q  e.  (  ._|_  `  X )
)
8 eleq1 2523 . . . . . 6  |-  ( q  =  r  ->  (
q  e.  (  ._|_  `  X )  <->  r  e.  (  ._|_  `  X )
) )
97, 8syl5ibcom 220 . . . . 5  |-  ( ( ( K  e.  HL  /\  X  C_  A )  /\  ( r  e.  X  /\  q  e.  (  ._|_  `  X ) ) )  ->  ( q  =  r  ->  r  e.  (  ._|_  `  X ) ) )
10 simprl 755 . . . . 5  |-  ( ( ( K  e.  HL  /\  X  C_  A )  /\  ( r  e.  X  /\  q  e.  (  ._|_  `  X ) ) )  ->  r  e.  X )
119, 10jctild 543 . . . 4  |-  ( ( ( K  e.  HL  /\  X  C_  A )  /\  ( r  e.  X  /\  q  e.  (  ._|_  `  X ) ) )  ->  ( q  =  r  ->  ( r  e.  X  /\  r  e.  (  ._|_  `  X
) ) ) )
12 elin 3640 . . . 4  |-  ( r  e.  ( X  i^i  (  ._|_  `  X )
)  <->  ( r  e.  X  /\  r  e.  (  ._|_  `  X ) ) )
1311, 12syl6ibr 227 . . 3  |-  ( ( ( K  e.  HL  /\  X  C_  A )  /\  ( r  e.  X  /\  q  e.  (  ._|_  `  X ) ) )  ->  ( q  =  r  ->  r  e.  ( X  i^i  (  ._|_  `  X ) ) ) )
1413necon3bd 2660 . 2  |-  ( ( ( K  e.  HL  /\  X  C_  A )  /\  ( r  e.  X  /\  q  e.  (  ._|_  `  X ) ) )  ->  ( -.  r  e.  ( X  i^i  (  ._|_  `  X
) )  ->  q  =/=  r ) )
156, 14mpd 15 1  |-  ( ( ( K  e.  HL  /\  X  C_  A )  /\  ( r  e.  X  /\  q  e.  (  ._|_  `  X ) ) )  ->  q  =/=  r )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 369    = wceq 1370    e. wcel 1758    =/= wne 2644    i^i cin 3428    C_ wss 3429   (/)c0 3738   {csn 3978   ` cfv 5519  (class class class)co 6193   lecple 14356   joincjn 15225   Atomscatm 33217   HLchlt 33304   +Pcpadd 33748   _|_PcpolN 33855
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4504  ax-sep 4514  ax-nul 4522  ax-pow 4571  ax-pr 4632  ax-un 6475  ax-riotaBAD 32913
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-nel 2647  df-ral 2800  df-rex 2801  df-reu 2802  df-rmo 2803  df-rab 2804  df-v 3073  df-sbc 3288  df-csb 3390  df-dif 3432  df-un 3434  df-in 3436  df-ss 3443  df-nul 3739  df-if 3893  df-pw 3963  df-sn 3979  df-pr 3981  df-op 3985  df-uni 4193  df-iun 4274  df-iin 4275  df-br 4394  df-opab 4452  df-mpt 4453  df-id 4737  df-xp 4947  df-rel 4948  df-cnv 4949  df-co 4950  df-dm 4951  df-rn 4952  df-res 4953  df-ima 4954  df-iota 5482  df-fun 5521  df-fn 5522  df-f 5523  df-f1 5524  df-fo 5525  df-f1o 5526  df-fv 5527  df-riota 6154  df-ov 6196  df-oprab 6197  df-undef 6895  df-poset 15227  df-plt 15239  df-lub 15255  df-glb 15256  df-join 15257  df-meet 15258  df-p0 15320  df-p1 15321  df-lat 15327  df-clat 15389  df-oposet 33130  df-ol 33132  df-oml 33133  df-covers 33220  df-ats 33221  df-atl 33252  df-cvlat 33276  df-hlat 33305  df-pmap 33457  df-polarityN 33856
This theorem is referenced by:  pexmidlem3N  33925
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