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Theorem 4atexlemex2 34078
Description: Lemma for 4atexlem7 34082. 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 34076 . . 3  |-  ( ph  ->  C  e.  A )
1110adantr 465 . 2  |-  ( (
ph  /\  C  =/=  S )  ->  C  e.  A )
121, 2, 3, 4, 5, 6, 7, 8, 94atexlemnclw 34077 . . 3  |-  ( ph  ->  -.  C  .<_  W )
1312adantr 465 . 2  |-  ( (
ph  /\  C  =/=  S )  ->  -.  C  .<_  W )
141, 2, 3, 4, 5, 6, 7, 84atexlemntlpq 34075 . . . . 5  |-  ( ph  ->  -.  T  .<_  ( P 
.\/  Q ) )
15 id 22 . . . . . . . . . . 11  |-  ( C  =  P  ->  C  =  P )
169, 15syl5eqr 2509 . . . . . . . . . 10  |-  ( C  =  P  ->  (
( Q  .\/  T
)  ./\  ( P  .\/  S ) )  =  P )
1716adantl 466 . . . . . . . . 9  |-  ( (
ph  /\  C  =  P )  ->  (
( Q  .\/  T
)  ./\  ( P  .\/  S ) )  =  P )
1814atexlemkl 34064 . . . . . . . . . . . 12  |-  ( ph  ->  K  e.  Lat )
191, 3, 54atexlemqtb 34068 . . . . . . . . . . . 12  |-  ( ph  ->  ( Q  .\/  T
)  e.  ( Base `  K ) )
201, 3, 54atexlempsb 34067 . . . . . . . . . . . 12  |-  ( ph  ->  ( P  .\/  S
)  e.  ( Base `  K ) )
21 eqid 2454 . . . . . . . . . . . . 13  |-  ( Base `  K )  =  (
Base `  K )
2221, 2, 4latmle1 15369 . . . . . . . . . . . 12  |-  ( ( K  e.  Lat  /\  ( Q  .\/  T )  e.  ( Base `  K
)  /\  ( P  .\/  S )  e.  (
Base `  K )
)  ->  ( ( Q  .\/  T )  ./\  ( P  .\/  S ) )  .<_  ( Q  .\/  T ) )
2318, 19, 20, 22syl3anc 1219 . . . . . . . . . . 11  |-  ( ph  ->  ( ( Q  .\/  T )  ./\  ( P  .\/  S ) )  .<_  ( Q  .\/  T ) )
2414atexlemk 34054 . . . . . . . . . . . 12  |-  ( ph  ->  K  e.  HL )
2514atexlemq 34058 . . . . . . . . . . . 12  |-  ( ph  ->  Q  e.  A )
2614atexlemt 34060 . . . . . . . . . . . 12  |-  ( ph  ->  T  e.  A )
273, 5hlatjcom 33375 . . . . . . . . . . . 12  |-  ( ( K  e.  HL  /\  Q  e.  A  /\  T  e.  A )  ->  ( Q  .\/  T
)  =  ( T 
.\/  Q ) )
2824, 25, 26, 27syl3anc 1219 . . . . . . . . . . 11  |-  ( ph  ->  ( Q  .\/  T
)  =  ( T 
.\/  Q ) )
2923, 28breqtrd 4427 . . . . . . . . . 10  |-  ( ph  ->  ( ( Q  .\/  T )  ./\  ( P  .\/  S ) )  .<_  ( T  .\/  Q ) )
3029adantr 465 . . . . . . . . 9  |-  ( (
ph  /\  C  =  P )  ->  (
( Q  .\/  T
)  ./\  ( P  .\/  S ) )  .<_  ( T  .\/  Q ) )
3117, 30eqbrtrrd 4425 . . . . . . . 8  |-  ( (
ph  /\  C  =  P )  ->  P  .<_  ( T  .\/  Q
) )
3214atexlemkc 34065 . . . . . . . . . 10  |-  ( ph  ->  K  e.  CvLat )
3314atexlemp 34057 . . . . . . . . . 10  |-  ( ph  ->  P  e.  A )
3414atexlempnq 34062 . . . . . . . . . 10  |-  ( ph  ->  P  =/=  Q )
352, 3, 5cvlatexch2 33345 . . . . . . . . . 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 1232 . . . . . . . . 9  |-  ( ph  ->  ( P  .<_  ( T 
.\/  Q )  ->  T  .<_  ( P  .\/  Q ) ) )
3736adantr 465 . . . . . . . 8  |-  ( (
ph  /\  C  =  P )  ->  ( P  .<_  ( T  .\/  Q )  ->  T  .<_  ( P  .\/  Q ) ) )
3831, 37mpd 15 . . . . . . 7  |-  ( (
ph  /\  C  =  P )  ->  T  .<_  ( P  .\/  Q
) )
3938ex 434 . . . . . 6  |-  ( ph  ->  ( C  =  P  ->  T  .<_  ( P 
.\/  Q ) ) )
4039necon3bd 2664 . . . . 5  |-  ( ph  ->  ( -.  T  .<_  ( P  .\/  Q )  ->  C  =/=  P
) )
4114, 40mpd 15 . . . 4  |-  ( ph  ->  C  =/=  P )
4241adantr 465 . . 3  |-  ( (
ph  /\  C  =/=  S )  ->  C  =/=  P )
43 simpr 461 . . 3  |-  ( (
ph  /\  C  =/=  S )  ->  C  =/=  S )
4421, 2, 4latmle2 15370 . . . . . 6  |-  ( ( K  e.  Lat  /\  ( Q  .\/  T )  e.  ( Base `  K
)  /\  ( P  .\/  S )  e.  (
Base `  K )
)  ->  ( ( Q  .\/  T )  ./\  ( P  .\/  S ) )  .<_  ( P  .\/  S ) )
4518, 19, 20, 44syl3anc 1219 . . . . 5  |-  ( ph  ->  ( ( Q  .\/  T )  ./\  ( P  .\/  S ) )  .<_  ( P  .\/  S ) )
469, 45syl5eqbr 4436 . . . 4  |-  ( ph  ->  C  .<_  ( P  .\/  S ) )
4746adantr 465 . . 3  |-  ( (
ph  /\  C  =/=  S )  ->  C  .<_  ( P  .\/  S ) )
4814atexlems 34059 . . . . 5  |-  ( ph  ->  S  e.  A )
491, 2, 3, 54atexlempns 34069 . . . . 5  |-  ( ph  ->  P  =/=  S )
505, 2, 3cvlsupr2 33351 . . . . 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 1232 . . . 4  |-  ( ph  ->  ( ( P  .\/  C )  =  ( S 
.\/  C )  <->  ( C  =/=  P  /\  C  =/= 
S  /\  C  .<_  ( P  .\/  S ) ) ) )
5251adantr 465 . . 3  |-  ( (
ph  /\  C  =/=  S )  ->  ( ( P  .\/  C )  =  ( S  .\/  C
)  <->  ( C  =/= 
P  /\  C  =/=  S  /\  C  .<_  ( P 
.\/  S ) ) ) )
5342, 43, 47, 52mpbir3and 1171 . 2  |-  ( (
ph  /\  C  =/=  S )  ->  ( P  .\/  C )  =  ( S  .\/  C ) )
54 breq1 4406 . . . . 5  |-  ( z  =  C  ->  (
z  .<_  W  <->  C  .<_  W ) )
5554notbid 294 . . . 4  |-  ( z  =  C  ->  ( -.  z  .<_  W  <->  -.  C  .<_  W ) )
56 oveq2 6211 . . . . 5  |-  ( z  =  C  ->  ( P  .\/  z )  =  ( P  .\/  C
) )
57 oveq2 6211 . . . . 5  |-  ( z  =  C  ->  ( S  .\/  z )  =  ( S  .\/  C
) )
5856, 57eqeq12d 2476 . . . 4  |-  ( z  =  C  ->  (
( P  .\/  z
)  =  ( S 
.\/  z )  <->  ( P  .\/  C )  =  ( S  .\/  C ) ) )
5955, 58anbi12d 710 . . 3  |-  ( z  =  C  ->  (
( -.  z  .<_  W  /\  ( P  .\/  z )  =  ( S  .\/  z ) )  <->  ( -.  C  .<_  W  /\  ( P 
.\/  C )  =  ( S  .\/  C
) ) ) )
6059rspcev 3179 . 2  |-  ( ( C  e.  A  /\  ( -.  C  .<_  W  /\  ( P  .\/  C )  =  ( S 
.\/  C ) ) )  ->  E. z  e.  A  ( -.  z  .<_  W  /\  ( P  .\/  z )  =  ( S  .\/  z
) ) )
6111, 13, 53, 60syl12anc 1217 1  |-  ( (
ph  /\  C  =/=  S )  ->  E. z  e.  A  ( -.  z  .<_  W  /\  ( P  .\/  z )  =  ( S  .\/  z
) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1370    e. wcel 1758    =/= wne 2648   E.wrex 2800   class class class wbr 4403   ` cfv 5529  (class class class)co 6203   Basecbs 14296   lecple 14368   joincjn 15237   meetcmee 15238   Latclat 15338   Atomscatm 33271   CvLatclc 33273   HLchlt 33358   LHypclh 33991
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4514  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-reu 2806  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-op 3995  df-uni 4203  df-iun 4284  df-br 4404  df-opab 4462  df-mpt 4463  df-id 4747  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-f1 5534  df-fo 5535  df-f1o 5536  df-fv 5537  df-riota 6164  df-ov 6206  df-oprab 6207  df-poset 15239  df-plt 15251  df-lub 15267  df-glb 15268  df-join 15269  df-meet 15270  df-p0 15332  df-p1 15333  df-lat 15339  df-clat 15401  df-oposet 33184  df-ol 33186  df-oml 33187  df-covers 33274  df-ats 33275  df-atl 33306  df-cvlat 33330  df-hlat 33359  df-llines 33505  df-lplanes 33506  df-lhyp 33995
This theorem is referenced by:  4atexlemex4  34080  4atexlemex6  34081
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