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Theorem atlelt 30309
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 987 . . 3  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B
)  /\  ( P  .<_  X  /\  Q  .<  X ) )  ->  Q  .<  X )
2 breq1 4218 . . 3  |-  ( P  =  Q  ->  ( P  .<  X  <->  Q  .<  X ) )
31, 2syl5ibrcom 215 . 2  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B
)  /\  ( P  .<_  X  /\  Q  .<  X ) )  ->  ( P  =  Q  ->  P 
.<  X ) )
4 simp1 958 . . . 4  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B
)  /\  ( P  .<_  X  /\  Q  .<  X ) )  ->  K  e.  HL )
5 simp21 991 . . . 4  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B
)  /\  ( P  .<_  X  /\  Q  .<  X ) )  ->  P  e.  A )
6 simp22 992 . . . 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 2438 . . . . 5  |-  ( join `  K )  =  (
join `  K )
9 atlelt.a . . . . 5  |-  A  =  ( Atoms `  K )
107, 8, 9atlt 30308 . . . 4  |-  ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  ->  ( P  .<  ( P ( join `  K
) Q )  <->  P  =/=  Q ) )
114, 5, 6, 10syl3anc 1185 . . 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 986 . . . . 5  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B
)  /\  ( P  .<_  X  /\  Q  .<  X ) )  ->  P  .<_  X )
13 simp23 993 . . . . . . 7  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B
)  /\  ( P  .<_  X  /\  Q  .<  X ) )  ->  X  e.  B )
144, 6, 133jca 1135 . . . . . 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 14423 . . . . . 6  |-  ( ( K  e.  HL  /\  Q  e.  A  /\  X  e.  B )  ->  ( Q  .<  X  ->  Q  .<_  X ) )
1714, 1, 16sylc 59 . . . . 5  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B
)  /\  ( P  .<_  X  /\  Q  .<  X ) )  ->  Q  .<_  X )
18 hllat 30235 . . . . . . 7  |-  ( K  e.  HL  ->  K  e.  Lat )
19183ad2ant1 979 . . . . . 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 30161 . . . . . . 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 30161 . . . . . . 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 14496 . . . . . 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 1187 . . . . 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 889 . . . 4  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B
)  /\  ( P  .<_  X  /\  Q  .<  X ) )  ->  ( P ( join `  K
) Q )  .<_  X )
28 hlpos 30237 . . . . . 6  |-  ( K  e.  HL  ->  K  e.  Poset )
29283ad2ant1 979 . . . . 5  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B
)  /\  ( P  .<_  X  /\  Q  .<  X ) )  ->  K  e.  Poset )
3020, 8latjcl 14484 . . . . . 6  |-  ( ( K  e.  Lat  /\  P  e.  B  /\  Q  e.  B )  ->  ( P ( join `  K ) Q )  e.  B )
3119, 22, 24, 30syl3anc 1185 . . . . 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 14433 . . . . 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 1187 . . . 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 657 . . 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 228 . 2  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B
)  /\  ( P  .<_  X  /\  Q  .<  X ) )  ->  ( P  =/=  Q  ->  P  .<  X ) )
363, 35pm2.61dne 2683 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 178    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726    =/= wne 2601   class class class wbr 4215   ` cfv 5457  (class class class)co 6084   Basecbs 13474   lecple 13541   Posetcpo 14402   ltcplt 14403   joincjn 14406   Latclat 14479   Atomscatm 30135   HLchlt 30222
This theorem is referenced by:  1cvratlt  30345
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4323  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-iun 4097  df-br 4216  df-opab 4270  df-mpt 4271  df-id 4501  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-ov 6087  df-oprab 6088  df-mpt2 6089  df-1st 6352  df-2nd 6353  df-undef 6546  df-riota 6552  df-poset 14408  df-plt 14420  df-lub 14436  df-glb 14437  df-join 14438  df-meet 14439  df-p0 14473  df-lat 14480  df-clat 14542  df-oposet 30048  df-ol 30050  df-oml 30051  df-covers 30138  df-ats 30139  df-atl 30170  df-cvlat 30194  df-hlat 30223
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