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Theorem pexmidlem1N 34641
Description: Lemma for pexmidN 34640. 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 3783 . . 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 34604 . . . 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 2532 . . . . . 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 3680 . . . 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 2672 . 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 1374    e. wcel 1762    =/= wne 2655    i^i cin 3468    C_ wss 3469   (/)c0 3778   {csn 4020   ` cfv 5579  (class class class)co 6275   lecple 14551   joincjn 15420   Atomscatm 33935   HLchlt 34022   +Pcpadd 34466   _|_PcpolN 34573
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-rep 4551  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567  ax-riotaBAD 33631
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-nel 2658  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-op 4027  df-uni 4239  df-iun 4320  df-iin 4321  df-br 4441  df-opab 4499  df-mpt 4500  df-id 4788  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-riota 6236  df-ov 6278  df-oprab 6279  df-undef 6992  df-poset 15422  df-plt 15434  df-lub 15450  df-glb 15451  df-join 15452  df-meet 15453  df-p0 15515  df-p1 15516  df-lat 15522  df-clat 15584  df-oposet 33848  df-ol 33850  df-oml 33851  df-covers 33938  df-ats 33939  df-atl 33970  df-cvlat 33994  df-hlat 34023  df-pmap 34175  df-polarityN 34574
This theorem is referenced by:  pexmidlem3N  34643
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