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Theorem atlelt 33087
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 1017 . . 3  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B
)  /\  ( P  .<_  X  /\  Q  .<  X ) )  ->  Q  .<  X )
2 breq1 4300 . . 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 988 . . . 4  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B
)  /\  ( P  .<_  X  /\  Q  .<  X ) )  ->  K  e.  HL )
5 simp21 1021 . . . 4  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B
)  /\  ( P  .<_  X  /\  Q  .<  X ) )  ->  P  e.  A )
6 simp22 1022 . . . 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 2443 . . . . 5  |-  ( join `  K )  =  (
join `  K )
9 atlelt.a . . . . 5  |-  A  =  ( Atoms `  K )
107, 8, 9atlt 33086 . . . 4  |-  ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  ->  ( P  .<  ( P ( join `  K
) Q )  <->  P  =/=  Q ) )
114, 5, 6, 10syl3anc 1218 . . 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 1016 . . . . 5  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B
)  /\  ( P  .<_  X  /\  Q  .<  X ) )  ->  P  .<_  X )
13 simp23 1023 . . . . . . 7  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B
)  /\  ( P  .<_  X  /\  Q  .<  X ) )  ->  X  e.  B )
144, 6, 133jca 1168 . . . . . 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 15136 . . . . . 6  |-  ( ( K  e.  HL  /\  Q  e.  A  /\  X  e.  B )  ->  ( Q  .<  X  ->  Q  .<_  X ) )
1714, 1, 16sylc 60 . . . . 5  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B
)  /\  ( P  .<_  X  /\  Q  .<  X ) )  ->  Q  .<_  X )
18 hllat 33013 . . . . . . 7  |-  ( K  e.  HL  ->  K  e.  Lat )
19183ad2ant1 1009 . . . . . 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 32939 . . . . . . 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 32939 . . . . . . 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 15237 . . . . . 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 1220 . . . . 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 912 . . . 4  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B
)  /\  ( P  .<_  X  /\  Q  .<  X ) )  ->  ( P ( join `  K
) Q )  .<_  X )
28 hlpos 33015 . . . . . 6  |-  ( K  e.  HL  ->  K  e.  Poset )
29283ad2ant1 1009 . . . . 5  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B
)  /\  ( P  .<_  X  /\  Q  .<  X ) )  ->  K  e.  Poset )
3020, 8latjcl 15226 . . . . . 6  |-  ( ( K  e.  Lat  /\  P  e.  B  /\  Q  e.  B )  ->  ( P ( join `  K ) Q )  e.  B )
3119, 22, 24, 30syl3anc 1218 . . . . 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 15146 . . . . 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 1220 . . . 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 674 . . 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 2693 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 369    /\ w3a 965    = wceq 1369    e. wcel 1756    =/= wne 2611   class class class wbr 4297   ` cfv 5423  (class class class)co 6096   Basecbs 14179   lecple 14250   Posetcpo 15115   ltcplt 15116   joincjn 15119   Latclat 15220   Atomscatm 32913   HLchlt 33000
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4408  ax-sep 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536  ax-un 6377
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-ral 2725  df-rex 2726  df-reu 2727  df-rab 2729  df-v 2979  df-sbc 3192  df-csb 3294  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-nul 3643  df-if 3797  df-pw 3867  df-sn 3883  df-pr 3885  df-op 3889  df-uni 4097  df-iun 4178  df-br 4298  df-opab 4356  df-mpt 4357  df-id 4641  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5386  df-fun 5425  df-fn 5426  df-f 5427  df-f1 5428  df-fo 5429  df-f1o 5430  df-fv 5431  df-riota 6057  df-ov 6099  df-oprab 6100  df-poset 15121  df-plt 15133  df-lub 15149  df-glb 15150  df-join 15151  df-meet 15152  df-p0 15214  df-lat 15221  df-clat 15283  df-oposet 32826  df-ol 32828  df-oml 32829  df-covers 32916  df-ats 32917  df-atl 32948  df-cvlat 32972  df-hlat 33001
This theorem is referenced by:  1cvratlt  33123
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