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Theorem cvlexchb1 30142
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 30138 . . . . . . . . 9  |-  ( K  e.  CvLat  ->  K  e.  Lat )
21adantr 451 . . . . . . . 8  |-  ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B )
)  ->  K  e.  Lat )
3 simpr3 963 . . . . . . . 8  |-  ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B )
)  ->  X  e.  B )
4 simpr2 962 . . . . . . . . 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 30101 . . . . . . . . 9  |-  ( Q  e.  A  ->  Q  e.  B )
84, 7syl 15 . . . . . . . 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 14182 . . . . . . . 8  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  Q  e.  B )  ->  X  .<_  ( X  .\/  Q ) )
122, 3, 8, 11syl3anc 1182 . . . . . . 7  |-  ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B )
)  ->  X  .<_  ( X  .\/  Q ) )
13123adant3 975 . . . . . 6  |-  ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B )  /\  -.  P  .<_  X )  ->  X  .<_  ( X 
.\/  Q ) )
1413adantr 451 . . . . 5  |-  ( ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B
)  /\  -.  P  .<_  X )  /\  P  .<_  ( X  .\/  Q
) )  ->  X  .<_  ( X  .\/  Q
) )
15 simpr 447 . . . . 5  |-  ( ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B
)  /\  -.  P  .<_  X )  /\  P  .<_  ( X  .\/  Q
) )  ->  P  .<_  ( X  .\/  Q
) )
16 simpr1 961 . . . . . . . . 9  |-  ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B )
)  ->  P  e.  A )
175, 6atbase 30101 . . . . . . . . 9  |-  ( P  e.  A  ->  P  e.  B )
1816, 17syl 15 . . . . . . . 8  |-  ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B )
)  ->  P  e.  B )
195, 10latjcl 14172 . . . . . . . . 9  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  Q  e.  B )  ->  ( X  .\/  Q
)  e.  B )
202, 3, 8, 19syl3anc 1182 . . . . . . . 8  |-  ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B )
)  ->  ( X  .\/  Q )  e.  B
)
215, 9, 10latjle12 14184 . . . . . . . 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 1184 . . . . . . 7  |-  ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B )
)  ->  ( ( X  .<_  ( X  .\/  Q )  /\  P  .<_  ( X  .\/  Q ) )  <->  ( X  .\/  P )  .<_  ( X  .\/  Q ) ) )
23223adant3 975 . . . . . 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 451 . . . . 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 887 . . . 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 14182 . . . . . . . 8  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  P  e.  B )  ->  X  .<_  ( X  .\/  P ) )
272, 3, 18, 26syl3anc 1182 . . . . . . 7  |-  ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B )
)  ->  X  .<_  ( X  .\/  P ) )
28273adant3 975 . . . . . 6  |-  ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B )  /\  -.  P  .<_  X )  ->  X  .<_  ( X 
.\/  P ) )
2928adantr 451 . . . . 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 30140 . . . . . 6  |-  ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B )  /\  -.  P  .<_  X )  ->  ( P  .<_  ( X  .\/  Q )  ->  Q  .<_  ( X 
.\/  P ) ) )
3130imp 418 . . . . 5  |-  ( ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B
)  /\  -.  P  .<_  X )  /\  P  .<_  ( X  .\/  Q
) )  ->  Q  .<_  ( X  .\/  P
) )
325, 10latjcl 14172 . . . . . . . . 9  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  P  e.  B )  ->  ( X  .\/  P
)  e.  B )
332, 3, 18, 32syl3anc 1182 . . . . . . . 8  |-  ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B )
)  ->  ( X  .\/  P )  e.  B
)
345, 9, 10latjle12 14184 . . . . . . . 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 1184 . . . . . . 7  |-  ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B )
)  ->  ( ( X  .<_  ( X  .\/  P )  /\  Q  .<_  ( X  .\/  P ) )  <->  ( X  .\/  Q )  .<_  ( X  .\/  P ) ) )
36353adant3 975 . . . . . 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 451 . . . . 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 887 . . . 4  |-  ( ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B
)  /\  -.  P  .<_  X )  /\  P  .<_  ( X  .\/  Q
) )  ->  ( X  .\/  Q )  .<_  ( X  .\/  P ) )
395, 9latasymb 14176 . . . . . . 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 1182 . . . . . 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 975 . . . . 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 451 . . . 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 887 . . 3  |-  ( ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B
)  /\  -.  P  .<_  X )  /\  P  .<_  ( X  .\/  Q
) )  ->  ( X  .\/  P )  =  ( X  .\/  Q
) )
4443ex 423 . 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 14183 . . . . 5  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  P  e.  B )  ->  P  .<_  ( X  .\/  P ) )
462, 3, 18, 45syl3anc 1182 . . . 4  |-  ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B )
)  ->  P  .<_  ( X  .\/  P ) )
47 breq2 4043 . . . 4  |-  ( ( X  .\/  P )  =  ( X  .\/  Q )  ->  ( P  .<_  ( X  .\/  P
)  <->  P  .<_  ( X 
.\/  Q ) ) )
4846, 47syl5ibcom 211 . . 3  |-  ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B )
)  ->  ( ( X  .\/  P )  =  ( X  .\/  Q
)  ->  P  .<_  ( X  .\/  Q ) ) )
49483adant3 975 . 2  |-  ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B )  /\  -.  P  .<_  X )  ->  ( ( X 
.\/  P )  =  ( X  .\/  Q
)  ->  P  .<_  ( X  .\/  Q ) ) )
5044, 49impbid 183 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 176    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696   class class class wbr 4039   ` cfv 5271  (class class class)co 5874   Basecbs 13164   lecple 13231   joincjn 14094   Latclat 14167   Atomscatm 30075   CvLatclc 30077
This theorem is referenced by:  cvlexchb2  30143  cvlexch4N  30145  cvlatexchb1  30146  cvlcvr1  30151  hlexchb1  30195
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-undef 6314  df-riota 6320  df-poset 14096  df-lub 14124  df-join 14126  df-lat 14168  df-ats 30079  df-atl 30110  df-cvlat 30134
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