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Theorem cvlexchb1 34127
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 34123 . . . . . . . . 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 1004 . . . . . . . 8  |-  ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B )
)  ->  X  e.  B )
4 simpr2 1003 . . . . . . . . 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 34086 . . . . . . . . 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 15543 . . . . . . . 8  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  Q  e.  B )  ->  X  .<_  ( X  .\/  Q ) )
122, 3, 8, 11syl3anc 1228 . . . . . . 7  |-  ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B )
)  ->  X  .<_  ( X  .\/  Q ) )
13123adant3 1016 . . . . . 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 1002 . . . . . . . . 9  |-  ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B )
)  ->  P  e.  A )
175, 6atbase 34086 . . . . . . . . 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 15534 . . . . . . . . 9  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  Q  e.  B )  ->  ( X  .\/  Q
)  e.  B )
202, 3, 8, 19syl3anc 1228 . . . . . . . 8  |-  ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B )
)  ->  ( X  .\/  Q )  e.  B
)
215, 9, 10latjle12 15545 . . . . . . . 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 1230 . . . . . . 7  |-  ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B )
)  ->  ( ( X  .<_  ( X  .\/  Q )  /\  P  .<_  ( X  .\/  Q ) )  <->  ( X  .\/  P )  .<_  ( X  .\/  Q ) ) )
23223adant3 1016 . . . . . 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 919 . . . 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 15543 . . . . . . . 8  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  P  e.  B )  ->  X  .<_  ( X  .\/  P ) )
272, 3, 18, 26syl3anc 1228 . . . . . . 7  |-  ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B )
)  ->  X  .<_  ( X  .\/  P ) )
28273adant3 1016 . . . . . 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 34125 . . . . . 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 15534 . . . . . . . . 9  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  P  e.  B )  ->  ( X  .\/  P
)  e.  B )
332, 3, 18, 32syl3anc 1228 . . . . . . . 8  |-  ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B )
)  ->  ( X  .\/  P )  e.  B
)
345, 9, 10latjle12 15545 . . . . . . . 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 1230 . . . . . . 7  |-  ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B )
)  ->  ( ( X  .<_  ( X  .\/  P )  /\  Q  .<_  ( X  .\/  P ) )  <->  ( X  .\/  Q )  .<_  ( X  .\/  P ) ) )
36353adant3 1016 . . . . . 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 919 . . . 4  |-  ( ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B
)  /\  -.  P  .<_  X )  /\  P  .<_  ( X  .\/  Q
) )  ->  ( X  .\/  Q )  .<_  ( X  .\/  P ) )
395, 9latasymb 15537 . . . . . . 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 1228 . . . . . 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 1016 . . . . 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 919 . . 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 15544 . . . . 5  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  P  e.  B )  ->  P  .<_  ( X  .\/  P ) )
462, 3, 18, 45syl3anc 1228 . . . 4  |-  ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B )
)  ->  P  .<_  ( X  .\/  P ) )
47 breq2 4451 . . . 4  |-  ( ( X  .\/  P )  =  ( X  .\/  Q )  ->  ( P  .<_  ( X  .\/  P
)  <->  P  .<_  ( X 
.\/  Q ) ) )
4846, 47syl5ibcom 220 . . 3  |-  ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B )
)  ->  ( ( X  .\/  P )  =  ( X  .\/  Q
)  ->  P  .<_  ( X  .\/  Q ) ) )
49483adant3 1016 . 2  |-  ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B )  /\  -.  P  .<_  X )  ->  ( ( X 
.\/  P )  =  ( X  .\/  Q
)  ->  P  .<_  ( X  .\/  Q ) ) )
5044, 49impbid 191 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 184    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767   class class class wbr 4447   ` cfv 5586  (class class class)co 6282   Basecbs 14486   lecple 14558   joincjn 15427   Latclat 15528   Atomscatm 34060   CvLatclc 34062
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-riota 6243  df-ov 6285  df-oprab 6286  df-poset 15429  df-lub 15457  df-glb 15458  df-join 15459  df-meet 15460  df-lat 15529  df-ats 34064  df-atl 34095  df-cvlat 34119
This theorem is referenced by:  cvlexchb2  34128  cvlexch4N  34130  cvlatexchb1  34131  cvlcvr1  34136  hlexchb1  34180
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