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Theorem atlelt 32435
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 1026 . . 3  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B
)  /\  ( P  .<_  X  /\  Q  .<  X ) )  ->  Q  .<  X )
2 breq1 4397 . . 3  |-  ( P  =  Q  ->  ( P  .<  X  <->  Q  .<  X ) )
31, 2syl5ibrcom 222 . 2  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B
)  /\  ( P  .<_  X  /\  Q  .<  X ) )  ->  ( P  =  Q  ->  P 
.<  X ) )
4 simp1 997 . . . 4  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B
)  /\  ( P  .<_  X  /\  Q  .<  X ) )  ->  K  e.  HL )
5 simp21 1030 . . . 4  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B
)  /\  ( P  .<_  X  /\  Q  .<  X ) )  ->  P  e.  A )
6 simp22 1031 . . . 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 2402 . . . . 5  |-  ( join `  K )  =  (
join `  K )
9 atlelt.a . . . . 5  |-  A  =  ( Atoms `  K )
107, 8, 9atlt 32434 . . . 4  |-  ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  ->  ( P  .<  ( P ( join `  K
) Q )  <->  P  =/=  Q ) )
114, 5, 6, 10syl3anc 1230 . . 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 1025 . . . . 5  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B
)  /\  ( P  .<_  X  /\  Q  .<  X ) )  ->  P  .<_  X )
13 simp23 1032 . . . . . . 7  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B
)  /\  ( P  .<_  X  /\  Q  .<  X ) )  ->  X  e.  B )
144, 6, 133jca 1177 . . . . . 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 15913 . . . . . 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 32361 . . . . . . 7  |-  ( K  e.  HL  ->  K  e.  Lat )
19183ad2ant1 1018 . . . . . 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 32287 . . . . . . 7  |-  ( P  e.  A  ->  P  e.  B )
225, 21syl 17 . . . . . 6  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B
)  /\  ( P  .<_  X  /\  Q  .<  X ) )  ->  P  e.  B )
2320, 9atbase 32287 . . . . . . 7  |-  ( Q  e.  A  ->  Q  e.  B )
246, 23syl 17 . . . . . 6  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B
)  /\  ( P  .<_  X  /\  Q  .<  X ) )  ->  Q  e.  B )
2520, 15, 8latjle12 16014 . . . . . 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 1232 . . . . 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 922 . . . 4  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B
)  /\  ( P  .<_  X  /\  Q  .<  X ) )  ->  ( P ( join `  K
) Q )  .<_  X )
28 hlpos 32363 . . . . . 6  |-  ( K  e.  HL  ->  K  e.  Poset )
29283ad2ant1 1018 . . . . 5  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B
)  /\  ( P  .<_  X  /\  Q  .<  X ) )  ->  K  e.  Poset )
3020, 8latjcl 16003 . . . . . 6  |-  ( ( K  e.  Lat  /\  P  e.  B  /\  Q  e.  B )  ->  ( P ( join `  K ) Q )  e.  B )
3119, 22, 24, 30syl3anc 1230 . . . . 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 15923 . . . . 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 1232 . . . 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 672 . . 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 235 . 2  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B
)  /\  ( P  .<_  X  /\  Q  .<  X ) )  ->  ( P  =/=  Q  ->  P  .<  X ) )
363, 35pm2.61dne 2720 1  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B
)  /\  ( P  .<_  X  /\  Q  .<  X ) )  ->  P  .<  X )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    /\ w3a 974    = wceq 1405    e. wcel 1842    =/= wne 2598   class class class wbr 4394   ` cfv 5568  (class class class)co 6277   Basecbs 14839   lecple 14914   Posetcpo 15891   ltcplt 15892   joincjn 15895   Latclat 15997   Atomscatm 32261   HLchlt 32348
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4506  ax-sep 4516  ax-nul 4524  ax-pow 4571  ax-pr 4629  ax-un 6573
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2758  df-rex 2759  df-reu 2760  df-rab 2762  df-v 3060  df-sbc 3277  df-csb 3373  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-nul 3738  df-if 3885  df-pw 3956  df-sn 3972  df-pr 3974  df-op 3978  df-uni 4191  df-iun 4272  df-br 4395  df-opab 4453  df-mpt 4454  df-id 4737  df-xp 4828  df-rel 4829  df-cnv 4830  df-co 4831  df-dm 4832  df-rn 4833  df-res 4834  df-ima 4835  df-iota 5532  df-fun 5570  df-fn 5571  df-f 5572  df-f1 5573  df-fo 5574  df-f1o 5575  df-fv 5576  df-riota 6239  df-ov 6280  df-oprab 6281  df-preset 15879  df-poset 15897  df-plt 15910  df-lub 15926  df-glb 15927  df-join 15928  df-meet 15929  df-p0 15991  df-lat 15998  df-clat 16060  df-oposet 32174  df-ol 32176  df-oml 32177  df-covers 32264  df-ats 32265  df-atl 32296  df-cvlat 32320  df-hlat 32349
This theorem is referenced by:  1cvratlt  32471
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