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Theorem atlelt 32804
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 1012 . . 3  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B
)  /\  ( P  .<_  X  /\  Q  .<  X ) )  ->  Q  .<  X )
2 breq1 4292 . . 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 983 . . . 4  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B
)  /\  ( P  .<_  X  /\  Q  .<  X ) )  ->  K  e.  HL )
5 simp21 1016 . . . 4  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B
)  /\  ( P  .<_  X  /\  Q  .<  X ) )  ->  P  e.  A )
6 simp22 1017 . . . 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 2441 . . . . 5  |-  ( join `  K )  =  (
join `  K )
9 atlelt.a . . . . 5  |-  A  =  ( Atoms `  K )
107, 8, 9atlt 32803 . . . 4  |-  ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  ->  ( P  .<  ( P ( join `  K
) Q )  <->  P  =/=  Q ) )
114, 5, 6, 10syl3anc 1213 . . 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 1011 . . . . 5  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B
)  /\  ( P  .<_  X  /\  Q  .<  X ) )  ->  P  .<_  X )
13 simp23 1018 . . . . . . 7  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B
)  /\  ( P  .<_  X  /\  Q  .<  X ) )  ->  X  e.  B )
144, 6, 133jca 1163 . . . . . 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 15127 . . . . . 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 32730 . . . . . . 7  |-  ( K  e.  HL  ->  K  e.  Lat )
19183ad2ant1 1004 . . . . . 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 32656 . . . . . . 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 32656 . . . . . . 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 15228 . . . . . 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 1215 . . . . 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 907 . . . 4  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B
)  /\  ( P  .<_  X  /\  Q  .<  X ) )  ->  ( P ( join `  K
) Q )  .<_  X )
28 hlpos 32732 . . . . . 6  |-  ( K  e.  HL  ->  K  e.  Poset )
29283ad2ant1 1004 . . . . 5  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B
)  /\  ( P  .<_  X  /\  Q  .<  X ) )  ->  K  e.  Poset )
3020, 8latjcl 15217 . . . . . 6  |-  ( ( K  e.  Lat  /\  P  e.  B  /\  Q  e.  B )  ->  ( P ( join `  K ) Q )  e.  B )
3119, 22, 24, 30syl3anc 1213 . . . . 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 15137 . . . . 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 1215 . . . 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 669 . . 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 2686 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 960    = wceq 1364    e. wcel 1761    =/= wne 2604   class class class wbr 4289   ` cfv 5415  (class class class)co 6090   Basecbs 14170   lecple 14241   Posetcpo 15106   ltcplt 15107   joincjn 15110   Latclat 15211   Atomscatm 32630   HLchlt 32717
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-rep 4400  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-ral 2718  df-rex 2719  df-reu 2720  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-op 3881  df-uni 4089  df-iun 4170  df-br 4290  df-opab 4348  df-mpt 4349  df-id 4632  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-riota 6049  df-ov 6093  df-oprab 6094  df-poset 15112  df-plt 15124  df-lub 15140  df-glb 15141  df-join 15142  df-meet 15143  df-p0 15205  df-lat 15212  df-clat 15274  df-oposet 32543  df-ol 32545  df-oml 32546  df-covers 32633  df-ats 32634  df-atl 32665  df-cvlat 32689  df-hlat 32718
This theorem is referenced by:  1cvratlt  32840
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