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Theorem atlelt 29603
Description: Transfer less-than relation from one atom to another. (Contributed by NM, 7-May-2012.)
Hypotheses
Ref Expression
atlelt.b  |-  B  =  ( Base `  K
)
atlelt.l  |-  .<_  =  ( le `  K )
atlelt.s  |-  .<  =  ( lt `  K )
atlelt.a  |-  A  =  ( Atoms `  K )
Assertion
Ref Expression
atlelt  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B
)  /\  ( P  .<_  X  /\  Q  .<  X ) )  ->  P  .<  X )

Proof of Theorem atlelt
StepHypRef Expression
1 simp3r 986 . . 3  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B
)  /\  ( P  .<_  X  /\  Q  .<  X ) )  ->  Q  .<  X )
2 breq1 4149 . . 3  |-  ( P  =  Q  ->  ( P  .<  X  <->  Q  .<  X ) )
31, 2syl5ibrcom 214 . 2  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B
)  /\  ( P  .<_  X  /\  Q  .<  X ) )  ->  ( P  =  Q  ->  P 
.<  X ) )
4 simp1 957 . . . 4  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B
)  /\  ( P  .<_  X  /\  Q  .<  X ) )  ->  K  e.  HL )
5 simp21 990 . . . 4  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B
)  /\  ( P  .<_  X  /\  Q  .<  X ) )  ->  P  e.  A )
6 simp22 991 . . . 4  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B
)  /\  ( P  .<_  X  /\  Q  .<  X ) )  ->  Q  e.  A )
7 atlelt.s . . . . 5  |-  .<  =  ( lt `  K )
8 eqid 2380 . . . . 5  |-  ( join `  K )  =  (
join `  K )
9 atlelt.a . . . . 5  |-  A  =  ( Atoms `  K )
107, 8, 9atlt 29602 . . . 4  |-  ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  ->  ( P  .<  ( P ( join `  K
) Q )  <->  P  =/=  Q ) )
114, 5, 6, 10syl3anc 1184 . . 3  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B
)  /\  ( P  .<_  X  /\  Q  .<  X ) )  ->  ( P  .<  ( P (
join `  K ) Q )  <->  P  =/=  Q ) )
12 simp3l 985 . . . . 5  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B
)  /\  ( P  .<_  X  /\  Q  .<  X ) )  ->  P  .<_  X )
13 simp23 992 . . . . . . 7  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B
)  /\  ( P  .<_  X  /\  Q  .<  X ) )  ->  X  e.  B )
144, 6, 133jca 1134 . . . . . 6  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B
)  /\  ( P  .<_  X  /\  Q  .<  X ) )  ->  ( K  e.  HL  /\  Q  e.  A  /\  X  e.  B ) )
15 atlelt.l . . . . . . 7  |-  .<_  =  ( le `  K )
1615, 7pltle 14338 . . . . . 6  |-  ( ( K  e.  HL  /\  Q  e.  A  /\  X  e.  B )  ->  ( Q  .<  X  ->  Q  .<_  X ) )
1714, 1, 16sylc 58 . . . . 5  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B
)  /\  ( P  .<_  X  /\  Q  .<  X ) )  ->  Q  .<_  X )
18 hllat 29529 . . . . . . 7  |-  ( K  e.  HL  ->  K  e.  Lat )
19183ad2ant1 978 . . . . . 6  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B
)  /\  ( P  .<_  X  /\  Q  .<  X ) )  ->  K  e.  Lat )
20 atlelt.b . . . . . . . 8  |-  B  =  ( Base `  K
)
2120, 9atbase 29455 . . . . . . 7  |-  ( P  e.  A  ->  P  e.  B )
225, 21syl 16 . . . . . 6  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B
)  /\  ( P  .<_  X  /\  Q  .<  X ) )  ->  P  e.  B )
2320, 9atbase 29455 . . . . . . 7  |-  ( Q  e.  A  ->  Q  e.  B )
246, 23syl 16 . . . . . 6  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B
)  /\  ( P  .<_  X  /\  Q  .<  X ) )  ->  Q  e.  B )
2520, 15, 8latjle12 14411 . . . . . 6  |-  ( ( K  e.  Lat  /\  ( P  e.  B  /\  Q  e.  B  /\  X  e.  B
) )  ->  (
( P  .<_  X  /\  Q  .<_  X )  <->  ( P
( join `  K ) Q )  .<_  X ) )
2619, 22, 24, 13, 25syl13anc 1186 . . . . 5  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B
)  /\  ( P  .<_  X  /\  Q  .<  X ) )  ->  (
( P  .<_  X  /\  Q  .<_  X )  <->  ( P
( join `  K ) Q )  .<_  X ) )
2712, 17, 26mpbi2and 888 . . . 4  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B
)  /\  ( P  .<_  X  /\  Q  .<  X ) )  ->  ( P ( join `  K
) Q )  .<_  X )
28 hlpos 29531 . . . . . 6  |-  ( K  e.  HL  ->  K  e.  Poset )
29283ad2ant1 978 . . . . 5  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B
)  /\  ( P  .<_  X  /\  Q  .<  X ) )  ->  K  e.  Poset )
3020, 8latjcl 14399 . . . . . 6  |-  ( ( K  e.  Lat  /\  P  e.  B  /\  Q  e.  B )  ->  ( P ( join `  K ) Q )  e.  B )
3119, 22, 24, 30syl3anc 1184 . . . . 5  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B
)  /\  ( P  .<_  X  /\  Q  .<  X ) )  ->  ( P ( join `  K
) Q )  e.  B )
3220, 15, 7pltletr 14348 . . . . 5  |-  ( ( K  e.  Poset  /\  ( P  e.  B  /\  ( P ( join `  K
) Q )  e.  B  /\  X  e.  B ) )  -> 
( ( P  .<  ( P ( join `  K
) Q )  /\  ( P ( join `  K
) Q )  .<_  X )  ->  P  .<  X ) )
3329, 22, 31, 13, 32syl13anc 1186 . . . 4  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B
)  /\  ( P  .<_  X  /\  Q  .<  X ) )  ->  (
( P  .<  ( P ( join `  K
) Q )  /\  ( P ( join `  K
) Q )  .<_  X )  ->  P  .<  X ) )
3427, 33mpan2d 656 . . 3  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B
)  /\  ( P  .<_  X  /\  Q  .<  X ) )  ->  ( P  .<  ( P (
join `  K ) Q )  ->  P  .<  X ) )
3511, 34sylbird 227 . 2  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B
)  /\  ( P  .<_  X  /\  Q  .<  X ) )  ->  ( P  =/=  Q  ->  P  .<  X ) )
363, 35pm2.61dne 2620 1  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B
)  /\  ( P  .<_  X  /\  Q  .<  X ) )  ->  P  .<  X )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1717    =/= wne 2543   class class class wbr 4146   ` cfv 5387  (class class class)co 6013   Basecbs 13389   lecple 13456   Posetcpo 14317   ltcplt 14318   joincjn 14321   Latclat 14394   Atomscatm 29429   HLchlt 29516
This theorem is referenced by:  1cvratlt  29639
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361  ax-rep 4254  ax-sep 4264  ax-nul 4272  ax-pow 4311  ax-pr 4337  ax-un 4634
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-nel 2546  df-ral 2647  df-rex 2648  df-reu 2649  df-rab 2651  df-v 2894  df-sbc 3098  df-csb 3188  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-nul 3565  df-if 3676  df-pw 3737  df-sn 3756  df-pr 3757  df-op 3759  df-uni 3951  df-iun 4030  df-br 4147  df-opab 4201  df-mpt 4202  df-id 4432  df-xp 4817  df-rel 4818  df-cnv 4819  df-co 4820  df-dm 4821  df-rn 4822  df-res 4823  df-ima 4824  df-iota 5351  df-fun 5389  df-fn 5390  df-f 5391  df-f1 5392  df-fo 5393  df-f1o 5394  df-fv 5395  df-ov 6016  df-oprab 6017  df-mpt2 6018  df-1st 6281  df-2nd 6282  df-undef 6472  df-riota 6478  df-poset 14323  df-plt 14335  df-lub 14351  df-glb 14352  df-join 14353  df-meet 14354  df-p0 14388  df-lat 14395  df-clat 14457  df-oposet 29342  df-ol 29344  df-oml 29345  df-covers 29432  df-ats 29433  df-atl 29464  df-cvlat 29488  df-hlat 29517
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