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Theorem cvlexchb1 30029
Description: An atomic covering lattice has the exchange property. (Contributed by NM, 16-Nov-2011.)
Hypotheses
Ref Expression
cvlexch.b  |-  B  =  ( Base `  K
)
cvlexch.l  |-  .<_  =  ( le `  K )
cvlexch.j  |-  .\/  =  ( join `  K )
cvlexch.a  |-  A  =  ( Atoms `  K )
Assertion
Ref Expression
cvlexchb1  |-  ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B )  /\  -.  P  .<_  X )  ->  ( P  .<_  ( X  .\/  Q )  <-> 
( X  .\/  P
)  =  ( X 
.\/  Q ) ) )

Proof of Theorem cvlexchb1
StepHypRef Expression
1 cvllat 30025 . . . . . . . . 9  |-  ( K  e.  CvLat  ->  K  e.  Lat )
21adantr 452 . . . . . . . 8  |-  ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B )
)  ->  K  e.  Lat )
3 simpr3 965 . . . . . . . 8  |-  ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B )
)  ->  X  e.  B )
4 simpr2 964 . . . . . . . . 9  |-  ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B )
)  ->  Q  e.  A )
5 cvlexch.b . . . . . . . . . 10  |-  B  =  ( Base `  K
)
6 cvlexch.a . . . . . . . . . 10  |-  A  =  ( Atoms `  K )
75, 6atbase 29988 . . . . . . . . 9  |-  ( Q  e.  A  ->  Q  e.  B )
84, 7syl 16 . . . . . . . 8  |-  ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B )
)  ->  Q  e.  B )
9 cvlexch.l . . . . . . . . 9  |-  .<_  =  ( le `  K )
10 cvlexch.j . . . . . . . . 9  |-  .\/  =  ( join `  K )
115, 9, 10latlej1 14479 . . . . . . . 8  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  Q  e.  B )  ->  X  .<_  ( X  .\/  Q ) )
122, 3, 8, 11syl3anc 1184 . . . . . . 7  |-  ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B )
)  ->  X  .<_  ( X  .\/  Q ) )
13123adant3 977 . . . . . 6  |-  ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B )  /\  -.  P  .<_  X )  ->  X  .<_  ( X 
.\/  Q ) )
1413adantr 452 . . . . 5  |-  ( ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B
)  /\  -.  P  .<_  X )  /\  P  .<_  ( X  .\/  Q
) )  ->  X  .<_  ( X  .\/  Q
) )
15 simpr 448 . . . . 5  |-  ( ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B
)  /\  -.  P  .<_  X )  /\  P  .<_  ( X  .\/  Q
) )  ->  P  .<_  ( X  .\/  Q
) )
16 simpr1 963 . . . . . . . . 9  |-  ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B )
)  ->  P  e.  A )
175, 6atbase 29988 . . . . . . . . 9  |-  ( P  e.  A  ->  P  e.  B )
1816, 17syl 16 . . . . . . . 8  |-  ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B )
)  ->  P  e.  B )
195, 10latjcl 14469 . . . . . . . . 9  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  Q  e.  B )  ->  ( X  .\/  Q
)  e.  B )
202, 3, 8, 19syl3anc 1184 . . . . . . . 8  |-  ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B )
)  ->  ( X  .\/  Q )  e.  B
)
215, 9, 10latjle12 14481 . . . . . . . 8  |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  P  e.  B  /\  ( X  .\/  Q
)  e.  B ) )  ->  ( ( X  .<_  ( X  .\/  Q )  /\  P  .<_  ( X  .\/  Q ) )  <->  ( X  .\/  P )  .<_  ( X  .\/  Q ) ) )
222, 3, 18, 20, 21syl13anc 1186 . . . . . . 7  |-  ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B )
)  ->  ( ( X  .<_  ( X  .\/  Q )  /\  P  .<_  ( X  .\/  Q ) )  <->  ( X  .\/  P )  .<_  ( X  .\/  Q ) ) )
23223adant3 977 . . . . . 6  |-  ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B )  /\  -.  P  .<_  X )  ->  ( ( X 
.<_  ( X  .\/  Q
)  /\  P  .<_  ( X  .\/  Q ) )  <->  ( X  .\/  P )  .<_  ( X  .\/  Q ) ) )
2423adantr 452 . . . . 5  |-  ( ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B
)  /\  -.  P  .<_  X )  /\  P  .<_  ( X  .\/  Q
) )  ->  (
( X  .<_  ( X 
.\/  Q )  /\  P  .<_  ( X  .\/  Q ) )  <->  ( X  .\/  P )  .<_  ( X 
.\/  Q ) ) )
2514, 15, 24mpbi2and 888 . . . 4  |-  ( ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B
)  /\  -.  P  .<_  X )  /\  P  .<_  ( X  .\/  Q
) )  ->  ( X  .\/  P )  .<_  ( X  .\/  Q ) )
265, 9, 10latlej1 14479 . . . . . . . 8  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  P  e.  B )  ->  X  .<_  ( X  .\/  P ) )
272, 3, 18, 26syl3anc 1184 . . . . . . 7  |-  ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B )
)  ->  X  .<_  ( X  .\/  P ) )
28273adant3 977 . . . . . 6  |-  ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B )  /\  -.  P  .<_  X )  ->  X  .<_  ( X 
.\/  P ) )
2928adantr 452 . . . . 5  |-  ( ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B
)  /\  -.  P  .<_  X )  /\  P  .<_  ( X  .\/  Q
) )  ->  X  .<_  ( X  .\/  P
) )
305, 9, 10, 6cvlexch1 30027 . . . . . 6  |-  ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B )  /\  -.  P  .<_  X )  ->  ( P  .<_  ( X  .\/  Q )  ->  Q  .<_  ( X 
.\/  P ) ) )
3130imp 419 . . . . 5  |-  ( ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B
)  /\  -.  P  .<_  X )  /\  P  .<_  ( X  .\/  Q
) )  ->  Q  .<_  ( X  .\/  P
) )
325, 10latjcl 14469 . . . . . . . . 9  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  P  e.  B )  ->  ( X  .\/  P
)  e.  B )
332, 3, 18, 32syl3anc 1184 . . . . . . . 8  |-  ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B )
)  ->  ( X  .\/  P )  e.  B
)
345, 9, 10latjle12 14481 . . . . . . . 8  |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Q  e.  B  /\  ( X  .\/  P
)  e.  B ) )  ->  ( ( X  .<_  ( X  .\/  P )  /\  Q  .<_  ( X  .\/  P ) )  <->  ( X  .\/  Q )  .<_  ( X  .\/  P ) ) )
352, 3, 8, 33, 34syl13anc 1186 . . . . . . 7  |-  ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B )
)  ->  ( ( X  .<_  ( X  .\/  P )  /\  Q  .<_  ( X  .\/  P ) )  <->  ( X  .\/  Q )  .<_  ( X  .\/  P ) ) )
36353adant3 977 . . . . . 6  |-  ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B )  /\  -.  P  .<_  X )  ->  ( ( X 
.<_  ( X  .\/  P
)  /\  Q  .<_  ( X  .\/  P ) )  <->  ( X  .\/  Q )  .<_  ( X  .\/  P ) ) )
3736adantr 452 . . . . 5  |-  ( ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B
)  /\  -.  P  .<_  X )  /\  P  .<_  ( X  .\/  Q
) )  ->  (
( X  .<_  ( X 
.\/  P )  /\  Q  .<_  ( X  .\/  P ) )  <->  ( X  .\/  Q )  .<_  ( X 
.\/  P ) ) )
3829, 31, 37mpbi2and 888 . . . 4  |-  ( ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B
)  /\  -.  P  .<_  X )  /\  P  .<_  ( X  .\/  Q
) )  ->  ( X  .\/  Q )  .<_  ( X  .\/  P ) )
395, 9latasymb 14473 . . . . . . 7  |-  ( ( K  e.  Lat  /\  ( X  .\/  P )  e.  B  /\  ( X  .\/  Q )  e.  B )  ->  (
( ( X  .\/  P )  .<_  ( X  .\/  Q )  /\  ( X  .\/  Q )  .<_  ( X  .\/  P ) )  <->  ( X  .\/  P )  =  ( X 
.\/  Q ) ) )
402, 33, 20, 39syl3anc 1184 . . . . . 6  |-  ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B )
)  ->  ( (
( X  .\/  P
)  .<_  ( X  .\/  Q )  /\  ( X 
.\/  Q )  .<_  ( X  .\/  P ) )  <->  ( X  .\/  P )  =  ( X 
.\/  Q ) ) )
41403adant3 977 . . . . 5  |-  ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B )  /\  -.  P  .<_  X )  ->  ( ( ( X  .\/  P ) 
.<_  ( X  .\/  Q
)  /\  ( X  .\/  Q )  .<_  ( X 
.\/  P ) )  <-> 
( X  .\/  P
)  =  ( X 
.\/  Q ) ) )
4241adantr 452 . . . 4  |-  ( ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B
)  /\  -.  P  .<_  X )  /\  P  .<_  ( X  .\/  Q
) )  ->  (
( ( X  .\/  P )  .<_  ( X  .\/  Q )  /\  ( X  .\/  Q )  .<_  ( X  .\/  P ) )  <->  ( X  .\/  P )  =  ( X 
.\/  Q ) ) )
4325, 38, 42mpbi2and 888 . . 3  |-  ( ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B
)  /\  -.  P  .<_  X )  /\  P  .<_  ( X  .\/  Q
) )  ->  ( X  .\/  P )  =  ( X  .\/  Q
) )
4443ex 424 . 2  |-  ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B )  /\  -.  P  .<_  X )  ->  ( P  .<_  ( X  .\/  Q )  ->  ( X  .\/  P )  =  ( X 
.\/  Q ) ) )
455, 9, 10latlej2 14480 . . . . 5  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  P  e.  B )  ->  P  .<_  ( X  .\/  P ) )
462, 3, 18, 45syl3anc 1184 . . . 4  |-  ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B )
)  ->  P  .<_  ( X  .\/  P ) )
47 breq2 4208 . . . 4  |-  ( ( X  .\/  P )  =  ( X  .\/  Q )  ->  ( P  .<_  ( X  .\/  P
)  <->  P  .<_  ( X 
.\/  Q ) ) )
4846, 47syl5ibcom 212 . . 3  |-  ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B )
)  ->  ( ( X  .\/  P )  =  ( X  .\/  Q
)  ->  P  .<_  ( X  .\/  Q ) ) )
49483adant3 977 . 2  |-  ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B )  /\  -.  P  .<_  X )  ->  ( ( X 
.\/  P )  =  ( X  .\/  Q
)  ->  P  .<_  ( X  .\/  Q ) ) )
5044, 49impbid 184 1  |-  ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B )  /\  -.  P  .<_  X )  ->  ( P  .<_  ( X  .\/  Q )  <-> 
( X  .\/  P
)  =  ( X 
.\/  Q ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725   class class class wbr 4204   ` cfv 5446  (class class class)co 6073   Basecbs 13459   lecple 13526   joincjn 14391   Latclat 14464   Atomscatm 29962   CvLatclc 29964
This theorem is referenced by:  cvlexchb2  30030  cvlexch4N  30032  cvlatexchb1  30033  cvlcvr1  30038  hlexchb1  30082
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-undef 6535  df-riota 6541  df-poset 14393  df-lub 14421  df-join 14423  df-lat 14465  df-ats 29966  df-atl 29997  df-cvlat 30021
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