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Theorem cvlexchb1 32361
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 32357 . . . . . . . . 9  |-  ( K  e.  CvLat  ->  K  e.  Lat )
21adantr 465 . . . . . . . 8  |-  ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B )
)  ->  K  e.  Lat )
3 simpr3 1007 . . . . . . . 8  |-  ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B )
)  ->  X  e.  B )
4 simpr2 1006 . . . . . . . . 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 32320 . . . . . . . . 9  |-  ( Q  e.  A  ->  Q  e.  B )
84, 7syl 17 . . . . . . . 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 16016 . . . . . . . 8  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  Q  e.  B )  ->  X  .<_  ( X  .\/  Q ) )
122, 3, 8, 11syl3anc 1232 . . . . . . 7  |-  ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B )
)  ->  X  .<_  ( X  .\/  Q ) )
13123adant3 1019 . . . . . 6  |-  ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B )  /\  -.  P  .<_  X )  ->  X  .<_  ( X 
.\/  Q ) )
1413adantr 465 . . . . 5  |-  ( ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B
)  /\  -.  P  .<_  X )  /\  P  .<_  ( X  .\/  Q
) )  ->  X  .<_  ( X  .\/  Q
) )
15 simpr 461 . . . . 5  |-  ( ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B
)  /\  -.  P  .<_  X )  /\  P  .<_  ( X  .\/  Q
) )  ->  P  .<_  ( X  .\/  Q
) )
16 simpr1 1005 . . . . . . . . 9  |-  ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B )
)  ->  P  e.  A )
175, 6atbase 32320 . . . . . . . . 9  |-  ( P  e.  A  ->  P  e.  B )
1816, 17syl 17 . . . . . . . 8  |-  ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B )
)  ->  P  e.  B )
195, 10latjcl 16007 . . . . . . . . 9  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  Q  e.  B )  ->  ( X  .\/  Q
)  e.  B )
202, 3, 8, 19syl3anc 1232 . . . . . . . 8  |-  ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B )
)  ->  ( X  .\/  Q )  e.  B
)
215, 9, 10latjle12 16018 . . . . . . . 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 1234 . . . . . . 7  |-  ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B )
)  ->  ( ( X  .<_  ( X  .\/  Q )  /\  P  .<_  ( X  .\/  Q ) )  <->  ( X  .\/  P )  .<_  ( X  .\/  Q ) ) )
23223adant3 1019 . . . . . 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 465 . . . . 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 924 . . . 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 16016 . . . . . . . 8  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  P  e.  B )  ->  X  .<_  ( X  .\/  P ) )
272, 3, 18, 26syl3anc 1232 . . . . . . 7  |-  ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B )
)  ->  X  .<_  ( X  .\/  P ) )
28273adant3 1019 . . . . . 6  |-  ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B )  /\  -.  P  .<_  X )  ->  X  .<_  ( X 
.\/  P ) )
2928adantr 465 . . . . 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 32359 . . . . . 6  |-  ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B )  /\  -.  P  .<_  X )  ->  ( P  .<_  ( X  .\/  Q )  ->  Q  .<_  ( X 
.\/  P ) ) )
3130imp 429 . . . . 5  |-  ( ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B
)  /\  -.  P  .<_  X )  /\  P  .<_  ( X  .\/  Q
) )  ->  Q  .<_  ( X  .\/  P
) )
325, 10latjcl 16007 . . . . . . . . 9  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  P  e.  B )  ->  ( X  .\/  P
)  e.  B )
332, 3, 18, 32syl3anc 1232 . . . . . . . 8  |-  ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B )
)  ->  ( X  .\/  P )  e.  B
)
345, 9, 10latjle12 16018 . . . . . . . 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 1234 . . . . . . 7  |-  ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B )
)  ->  ( ( X  .<_  ( X  .\/  P )  /\  Q  .<_  ( X  .\/  P ) )  <->  ( X  .\/  Q )  .<_  ( X  .\/  P ) ) )
36353adant3 1019 . . . . . 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 465 . . . . 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 924 . . . 4  |-  ( ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B
)  /\  -.  P  .<_  X )  /\  P  .<_  ( X  .\/  Q
) )  ->  ( X  .\/  Q )  .<_  ( X  .\/  P ) )
395, 9latasymb 16010 . . . . . . 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 1232 . . . . . 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 1019 . . . . 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 465 . . . 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 924 . . 3  |-  ( ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B
)  /\  -.  P  .<_  X )  /\  P  .<_  ( X  .\/  Q
) )  ->  ( X  .\/  P )  =  ( X  .\/  Q
) )
4443ex 434 . 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 16017 . . . . 5  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  P  e.  B )  ->  P  .<_  ( X  .\/  P ) )
462, 3, 18, 45syl3anc 1232 . . . 4  |-  ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B )
)  ->  P  .<_  ( X  .\/  P ) )
47 breq2 4401 . . . 4  |-  ( ( X  .\/  P )  =  ( X  .\/  Q )  ->  ( P  .<_  ( X  .\/  P
)  <->  P  .<_  ( X 
.\/  Q ) ) )
4846, 47syl5ibcom 222 . . 3  |-  ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B )
)  ->  ( ( X  .\/  P )  =  ( X  .\/  Q
)  ->  P  .<_  ( X  .\/  Q ) ) )
49483adant3 1019 . 2  |-  ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B )  /\  -.  P  .<_  X )  ->  ( ( X 
.\/  P )  =  ( X  .\/  Q
)  ->  P  .<_  ( X  .\/  Q ) ) )
5044, 49impbid 192 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 setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 186    /\ wa 369    /\ w3a 976    = wceq 1407    e. wcel 1844   class class class wbr 4397   ` cfv 5571  (class class class)co 6280   Basecbs 14843   lecple 14918   joincjn 15899   Latclat 16001   Atomscatm 32294   CvLatclc 32296
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1641  ax-4 1654  ax-5 1727  ax-6 1773  ax-7 1816  ax-8 1846  ax-9 1848  ax-10 1863  ax-11 1868  ax-12 1880  ax-13 2028  ax-ext 2382  ax-rep 4509  ax-sep 4519  ax-nul 4527  ax-pow 4574  ax-pr 4632  ax-un 6576
This theorem depends on definitions:  df-bi 187  df-or 370  df-an 371  df-3an 978  df-tru 1410  df-ex 1636  df-nf 1640  df-sb 1766  df-eu 2244  df-mo 2245  df-clab 2390  df-cleq 2396  df-clel 2399  df-nfc 2554  df-ne 2602  df-ral 2761  df-rex 2762  df-reu 2763  df-rab 2765  df-v 3063  df-sbc 3280  df-csb 3376  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-nul 3741  df-if 3888  df-pw 3959  df-sn 3975  df-pr 3977  df-op 3981  df-uni 4194  df-iun 4275  df-br 4398  df-opab 4456  df-mpt 4457  df-id 4740  df-xp 4831  df-rel 4832  df-cnv 4833  df-co 4834  df-dm 4835  df-rn 4836  df-res 4837  df-ima 4838  df-iota 5535  df-fun 5573  df-fn 5574  df-f 5575  df-f1 5576  df-fo 5577  df-f1o 5578  df-fv 5579  df-riota 6242  df-ov 6283  df-oprab 6284  df-preset 15883  df-poset 15901  df-lub 15930  df-glb 15931  df-join 15932  df-meet 15933  df-lat 16002  df-ats 32298  df-atl 32329  df-cvlat 32353
This theorem is referenced by:  cvlexchb2  32362  cvlexch4N  32364  cvlatexchb1  32365  cvlcvr1  32370  hlexchb1  32414
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