Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  pexmidlem1N Unicode version

Theorem pexmidlem1N 29977
Description: Lemma for pexmidN 29976. 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 3494 . . 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 29940 . . . 4  |-  ( ( K  e.  HL  /\  X  C_  A )  -> 
( X  i^i  (  ._|_  `  X ) )  =  (/) )
54adantr 451 . . 3  |-  ( ( ( K  e.  HL  /\  X  C_  A )  /\  ( r  e.  X  /\  q  e.  (  ._|_  `  X ) ) )  ->  ( X  i^i  (  ._|_  `  X
) )  =  (/) )
61, 5nsyl3 111 . 2  |-  ( ( ( K  e.  HL  /\  X  C_  A )  /\  ( r  e.  X  /\  q  e.  (  ._|_  `  X ) ) )  ->  -.  r  e.  ( X  i^i  (  ._|_  `  X ) ) )
7 simprr 733 . . . . . 6  |-  ( ( ( K  e.  HL  /\  X  C_  A )  /\  ( r  e.  X  /\  q  e.  (  ._|_  `  X ) ) )  ->  q  e.  (  ._|_  `  X )
)
8 eleq1 2376 . . . . . 6  |-  ( q  =  r  ->  (
q  e.  (  ._|_  `  X )  <->  r  e.  (  ._|_  `  X )
) )
97, 8syl5ibcom 211 . . . . 5  |-  ( ( ( K  e.  HL  /\  X  C_  A )  /\  ( r  e.  X  /\  q  e.  (  ._|_  `  X ) ) )  ->  ( q  =  r  ->  r  e.  (  ._|_  `  X ) ) )
10 simprl 732 . . . . 5  |-  ( ( ( K  e.  HL  /\  X  C_  A )  /\  ( r  e.  X  /\  q  e.  (  ._|_  `  X ) ) )  ->  r  e.  X )
119, 10jctild 527 . . . 4  |-  ( ( ( K  e.  HL  /\  X  C_  A )  /\  ( r  e.  X  /\  q  e.  (  ._|_  `  X ) ) )  ->  ( q  =  r  ->  ( r  e.  X  /\  r  e.  (  ._|_  `  X
) ) ) )
12 elin 3392 . . . 4  |-  ( r  e.  ( X  i^i  (  ._|_  `  X )
)  <->  ( r  e.  X  /\  r  e.  (  ._|_  `  X ) ) )
1311, 12syl6ibr 218 . . 3  |-  ( ( ( K  e.  HL  /\  X  C_  A )  /\  ( r  e.  X  /\  q  e.  (  ._|_  `  X ) ) )  ->  ( q  =  r  ->  r  e.  ( X  i^i  (  ._|_  `  X ) ) ) )
1413necon3bd 2516 . 2  |-  ( ( ( K  e.  HL  /\  X  C_  A )  /\  ( r  e.  X  /\  q  e.  (  ._|_  `  X ) ) )  ->  ( -.  r  e.  ( X  i^i  (  ._|_  `  X
) )  ->  q  =/=  r ) )
156, 14mpd 14 1  |-  ( ( ( K  e.  HL  /\  X  C_  A )  /\  ( r  e.  X  /\  q  e.  (  ._|_  `  X ) ) )  ->  q  =/=  r )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 358    = wceq 1633    e. wcel 1701    =/= wne 2479    i^i cin 3185    C_ wss 3186   (/)c0 3489   {csn 3674   ` cfv 5292  (class class class)co 5900   lecple 13262   joincjn 14127   Atomscatm 29271   HLchlt 29358   + Pcpadd 29802   _|_ PcpolN 29909
This theorem is referenced by:  pexmidlem3N  29979
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-13 1703  ax-14 1705  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297  ax-rep 4168  ax-sep 4178  ax-nul 4186  ax-pow 4225  ax-pr 4251  ax-un 4549
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1533  df-nf 1536  df-sb 1640  df-eu 2180  df-mo 2181  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ne 2481  df-nel 2482  df-ral 2582  df-rex 2583  df-reu 2584  df-rmo 2585  df-rab 2586  df-v 2824  df-sbc 3026  df-csb 3116  df-dif 3189  df-un 3191  df-in 3193  df-ss 3200  df-nul 3490  df-if 3600  df-pw 3661  df-sn 3680  df-pr 3681  df-op 3683  df-uni 3865  df-iun 3944  df-iin 3945  df-br 4061  df-opab 4115  df-mpt 4116  df-id 4346  df-xp 4732  df-rel 4733  df-cnv 4734  df-co 4735  df-dm 4736  df-rn 4737  df-res 4738  df-ima 4739  df-iota 5256  df-fun 5294  df-fn 5295  df-f 5296  df-f1 5297  df-fo 5298  df-f1o 5299  df-fv 5300  df-ov 5903  df-oprab 5904  df-mpt2 5905  df-1st 6164  df-2nd 6165  df-undef 6340  df-riota 6346  df-poset 14129  df-plt 14141  df-lub 14157  df-glb 14158  df-join 14159  df-meet 14160  df-p0 14194  df-p1 14195  df-lat 14201  df-clat 14263  df-oposet 29184  df-ol 29186  df-oml 29187  df-covers 29274  df-ats 29275  df-atl 29306  df-cvlat 29330  df-hlat 29359  df-pmap 29511  df-polarityN 29910
  Copyright terms: Public domain W3C validator