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Theorem 4atexlemex2 30805
Description: Lemma for 4atexlem7 30809. Show that when  C  =/=  S,  C satisfies the existence condition of the consequent. (Contributed by NM, 25-Nov-2012.)
Hypotheses
Ref Expression
4thatlem.ph  |-  ( ph  <->  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( S  e.  A  /\  ( R  e.  A  /\  -.  R  .<_  W  /\  ( P  .\/  R )  =  ( Q  .\/  R ) )  /\  ( T  e.  A  /\  ( U  .\/  T )  =  ( V  .\/  T ) ) )  /\  ( P  =/=  Q  /\  -.  S  .<_  ( P 
.\/  Q ) ) ) )
4thatlem0.l  |-  .<_  =  ( le `  K )
4thatlem0.j  |-  .\/  =  ( join `  K )
4thatlem0.m  |-  ./\  =  ( meet `  K )
4thatlem0.a  |-  A  =  ( Atoms `  K )
4thatlem0.h  |-  H  =  ( LHyp `  K
)
4thatlem0.u  |-  U  =  ( ( P  .\/  Q )  ./\  W )
4thatlem0.v  |-  V  =  ( ( P  .\/  S )  ./\  W )
4thatlem0.c  |-  C  =  ( ( Q  .\/  T )  ./\  ( P  .\/  S ) )
Assertion
Ref Expression
4atexlemex2  |-  ( (
ph  /\  C  =/=  S )  ->  E. z  e.  A  ( -.  z  .<_  W  /\  ( P  .\/  z )  =  ( S  .\/  z
) ) )
Distinct variable groups:    z, A    z, C    z,  .\/    z,  .<_    z, P    z, S    z, W
Allowed substitution hints:    ph( z)    Q( z)    R( z)    T( z)    U( z)    H( z)    K( z)   
./\ ( z)    V( z)

Proof of Theorem 4atexlemex2
StepHypRef Expression
1 4thatlem.ph . . . 4  |-  ( ph  <->  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( S  e.  A  /\  ( R  e.  A  /\  -.  R  .<_  W  /\  ( P  .\/  R )  =  ( Q  .\/  R ) )  /\  ( T  e.  A  /\  ( U  .\/  T )  =  ( V  .\/  T ) ) )  /\  ( P  =/=  Q  /\  -.  S  .<_  ( P 
.\/  Q ) ) ) )
2 4thatlem0.l . . . 4  |-  .<_  =  ( le `  K )
3 4thatlem0.j . . . 4  |-  .\/  =  ( join `  K )
4 4thatlem0.m . . . 4  |-  ./\  =  ( meet `  K )
5 4thatlem0.a . . . 4  |-  A  =  ( Atoms `  K )
6 4thatlem0.h . . . 4  |-  H  =  ( LHyp `  K
)
7 4thatlem0.u . . . 4  |-  U  =  ( ( P  .\/  Q )  ./\  W )
8 4thatlem0.v . . . 4  |-  V  =  ( ( P  .\/  S )  ./\  W )
9 4thatlem0.c . . . 4  |-  C  =  ( ( Q  .\/  T )  ./\  ( P  .\/  S ) )
101, 2, 3, 4, 5, 6, 7, 8, 94atexlemc 30803 . . 3  |-  ( ph  ->  C  e.  A )
1110adantr 452 . 2  |-  ( (
ph  /\  C  =/=  S )  ->  C  e.  A )
121, 2, 3, 4, 5, 6, 7, 8, 94atexlemnclw 30804 . . 3  |-  ( ph  ->  -.  C  .<_  W )
1312adantr 452 . 2  |-  ( (
ph  /\  C  =/=  S )  ->  -.  C  .<_  W )
141, 2, 3, 4, 5, 6, 7, 84atexlemntlpq 30802 . . . . 5  |-  ( ph  ->  -.  T  .<_  ( P 
.\/  Q ) )
15 id 20 . . . . . . . . . . 11  |-  ( C  =  P  ->  C  =  P )
169, 15syl5eqr 2481 . . . . . . . . . 10  |-  ( C  =  P  ->  (
( Q  .\/  T
)  ./\  ( P  .\/  S ) )  =  P )
1716adantl 453 . . . . . . . . 9  |-  ( (
ph  /\  C  =  P )  ->  (
( Q  .\/  T
)  ./\  ( P  .\/  S ) )  =  P )
1814atexlemkl 30791 . . . . . . . . . . . 12  |-  ( ph  ->  K  e.  Lat )
191, 3, 54atexlemqtb 30795 . . . . . . . . . . . 12  |-  ( ph  ->  ( Q  .\/  T
)  e.  ( Base `  K ) )
201, 3, 54atexlempsb 30794 . . . . . . . . . . . 12  |-  ( ph  ->  ( P  .\/  S
)  e.  ( Base `  K ) )
21 eqid 2435 . . . . . . . . . . . . 13  |-  ( Base `  K )  =  (
Base `  K )
2221, 2, 4latmle1 14497 . . . . . . . . . . . 12  |-  ( ( K  e.  Lat  /\  ( Q  .\/  T )  e.  ( Base `  K
)  /\  ( P  .\/  S )  e.  (
Base `  K )
)  ->  ( ( Q  .\/  T )  ./\  ( P  .\/  S ) )  .<_  ( Q  .\/  T ) )
2318, 19, 20, 22syl3anc 1184 . . . . . . . . . . 11  |-  ( ph  ->  ( ( Q  .\/  T )  ./\  ( P  .\/  S ) )  .<_  ( Q  .\/  T ) )
2414atexlemk 30781 . . . . . . . . . . . 12  |-  ( ph  ->  K  e.  HL )
2514atexlemq 30785 . . . . . . . . . . . 12  |-  ( ph  ->  Q  e.  A )
2614atexlemt 30787 . . . . . . . . . . . 12  |-  ( ph  ->  T  e.  A )
273, 5hlatjcom 30102 . . . . . . . . . . . 12  |-  ( ( K  e.  HL  /\  Q  e.  A  /\  T  e.  A )  ->  ( Q  .\/  T
)  =  ( T 
.\/  Q ) )
2824, 25, 26, 27syl3anc 1184 . . . . . . . . . . 11  |-  ( ph  ->  ( Q  .\/  T
)  =  ( T 
.\/  Q ) )
2923, 28breqtrd 4228 . . . . . . . . . 10  |-  ( ph  ->  ( ( Q  .\/  T )  ./\  ( P  .\/  S ) )  .<_  ( T  .\/  Q ) )
3029adantr 452 . . . . . . . . 9  |-  ( (
ph  /\  C  =  P )  ->  (
( Q  .\/  T
)  ./\  ( P  .\/  S ) )  .<_  ( T  .\/  Q ) )
3117, 30eqbrtrrd 4226 . . . . . . . 8  |-  ( (
ph  /\  C  =  P )  ->  P  .<_  ( T  .\/  Q
) )
3214atexlemkc 30792 . . . . . . . . . 10  |-  ( ph  ->  K  e.  CvLat )
3314atexlemp 30784 . . . . . . . . . 10  |-  ( ph  ->  P  e.  A )
3414atexlempnq 30789 . . . . . . . . . 10  |-  ( ph  ->  P  =/=  Q )
352, 3, 5cvlatexch2 30072 . . . . . . . . . 10  |-  ( ( K  e.  CvLat  /\  ( P  e.  A  /\  T  e.  A  /\  Q  e.  A )  /\  P  =/=  Q
)  ->  ( P  .<_  ( T  .\/  Q
)  ->  T  .<_  ( P  .\/  Q ) ) )
3632, 33, 26, 25, 34, 35syl131anc 1197 . . . . . . . . 9  |-  ( ph  ->  ( P  .<_  ( T 
.\/  Q )  ->  T  .<_  ( P  .\/  Q ) ) )
3736adantr 452 . . . . . . . 8  |-  ( (
ph  /\  C  =  P )  ->  ( P  .<_  ( T  .\/  Q )  ->  T  .<_  ( P  .\/  Q ) ) )
3831, 37mpd 15 . . . . . . 7  |-  ( (
ph  /\  C  =  P )  ->  T  .<_  ( P  .\/  Q
) )
3938ex 424 . . . . . 6  |-  ( ph  ->  ( C  =  P  ->  T  .<_  ( P 
.\/  Q ) ) )
4039necon3bd 2635 . . . . 5  |-  ( ph  ->  ( -.  T  .<_  ( P  .\/  Q )  ->  C  =/=  P
) )
4114, 40mpd 15 . . . 4  |-  ( ph  ->  C  =/=  P )
4241adantr 452 . . 3  |-  ( (
ph  /\  C  =/=  S )  ->  C  =/=  P )
43 simpr 448 . . 3  |-  ( (
ph  /\  C  =/=  S )  ->  C  =/=  S )
4421, 2, 4latmle2 14498 . . . . . 6  |-  ( ( K  e.  Lat  /\  ( Q  .\/  T )  e.  ( Base `  K
)  /\  ( P  .\/  S )  e.  (
Base `  K )
)  ->  ( ( Q  .\/  T )  ./\  ( P  .\/  S ) )  .<_  ( P  .\/  S ) )
4518, 19, 20, 44syl3anc 1184 . . . . 5  |-  ( ph  ->  ( ( Q  .\/  T )  ./\  ( P  .\/  S ) )  .<_  ( P  .\/  S ) )
469, 45syl5eqbr 4237 . . . 4  |-  ( ph  ->  C  .<_  ( P  .\/  S ) )
4746adantr 452 . . 3  |-  ( (
ph  /\  C  =/=  S )  ->  C  .<_  ( P  .\/  S ) )
4814atexlems 30786 . . . . 5  |-  ( ph  ->  S  e.  A )
491, 2, 3, 54atexlempns 30796 . . . . 5  |-  ( ph  ->  P  =/=  S )
505, 2, 3cvlsupr2 30078 . . . . 5  |-  ( ( K  e.  CvLat  /\  ( P  e.  A  /\  S  e.  A  /\  C  e.  A )  /\  P  =/=  S
)  ->  ( ( P  .\/  C )  =  ( S  .\/  C
)  <->  ( C  =/= 
P  /\  C  =/=  S  /\  C  .<_  ( P 
.\/  S ) ) ) )
5132, 33, 48, 10, 49, 50syl131anc 1197 . . . 4  |-  ( ph  ->  ( ( P  .\/  C )  =  ( S 
.\/  C )  <->  ( C  =/=  P  /\  C  =/= 
S  /\  C  .<_  ( P  .\/  S ) ) ) )
5251adantr 452 . . 3  |-  ( (
ph  /\  C  =/=  S )  ->  ( ( P  .\/  C )  =  ( S  .\/  C
)  <->  ( C  =/= 
P  /\  C  =/=  S  /\  C  .<_  ( P 
.\/  S ) ) ) )
5342, 43, 47, 52mpbir3and 1137 . 2  |-  ( (
ph  /\  C  =/=  S )  ->  ( P  .\/  C )  =  ( S  .\/  C ) )
54 breq1 4207 . . . . 5  |-  ( z  =  C  ->  (
z  .<_  W  <->  C  .<_  W ) )
5554notbid 286 . . . 4  |-  ( z  =  C  ->  ( -.  z  .<_  W  <->  -.  C  .<_  W ) )
56 oveq2 6081 . . . . 5  |-  ( z  =  C  ->  ( P  .\/  z )  =  ( P  .\/  C
) )
57 oveq2 6081 . . . . 5  |-  ( z  =  C  ->  ( S  .\/  z )  =  ( S  .\/  C
) )
5856, 57eqeq12d 2449 . . . 4  |-  ( z  =  C  ->  (
( P  .\/  z
)  =  ( S 
.\/  z )  <->  ( P  .\/  C )  =  ( S  .\/  C ) ) )
5955, 58anbi12d 692 . . 3  |-  ( z  =  C  ->  (
( -.  z  .<_  W  /\  ( P  .\/  z )  =  ( S  .\/  z ) )  <->  ( -.  C  .<_  W  /\  ( P 
.\/  C )  =  ( S  .\/  C
) ) ) )
6059rspcev 3044 . 2  |-  ( ( C  e.  A  /\  ( -.  C  .<_  W  /\  ( P  .\/  C )  =  ( S 
.\/  C ) ) )  ->  E. z  e.  A  ( -.  z  .<_  W  /\  ( P  .\/  z )  =  ( S  .\/  z
) ) )
6111, 13, 53, 60syl12anc 1182 1  |-  ( (
ph  /\  C  =/=  S )  ->  E. z  e.  A  ( -.  z  .<_  W  /\  ( P  .\/  z )  =  ( S  .\/  z
) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725    =/= wne 2598   E.wrex 2698   class class class wbr 4204   ` cfv 5446  (class class class)co 6073   Basecbs 13461   lecple 13528   joincjn 14393   meetcmee 14394   Latclat 14466   Atomscatm 29998   CvLatclc 30000   HLchlt 30085   LHypclh 30718
This theorem is referenced by:  4atexlemex4  30807  4atexlemex6  30808
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 14395  df-plt 14407  df-lub 14423  df-glb 14424  df-join 14425  df-meet 14426  df-p0 14460  df-p1 14461  df-lat 14467  df-clat 14529  df-oposet 29911  df-ol 29913  df-oml 29914  df-covers 30001  df-ats 30002  df-atl 30033  df-cvlat 30057  df-hlat 30086  df-llines 30232  df-lplanes 30233  df-lhyp 30722
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