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Theorem 4atexlemex2 33288
Description: Lemma for 4atexlem7 33292. 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 33286 . . 3  |-  ( ph  ->  C  e.  A )
1110adantr 462 . 2  |-  ( (
ph  /\  C  =/=  S )  ->  C  e.  A )
121, 2, 3, 4, 5, 6, 7, 8, 94atexlemnclw 33287 . . 3  |-  ( ph  ->  -.  C  .<_  W )
1312adantr 462 . 2  |-  ( (
ph  /\  C  =/=  S )  ->  -.  C  .<_  W )
141, 2, 3, 4, 5, 6, 7, 84atexlemntlpq 33285 . . . . 5  |-  ( ph  ->  -.  T  .<_  ( P 
.\/  Q ) )
15 id 22 . . . . . . . . . . 11  |-  ( C  =  P  ->  C  =  P )
169, 15syl5eqr 2479 . . . . . . . . . 10  |-  ( C  =  P  ->  (
( Q  .\/  T
)  ./\  ( P  .\/  S ) )  =  P )
1716adantl 463 . . . . . . . . 9  |-  ( (
ph  /\  C  =  P )  ->  (
( Q  .\/  T
)  ./\  ( P  .\/  S ) )  =  P )
1814atexlemkl 33274 . . . . . . . . . . . 12  |-  ( ph  ->  K  e.  Lat )
191, 3, 54atexlemqtb 33278 . . . . . . . . . . . 12  |-  ( ph  ->  ( Q  .\/  T
)  e.  ( Base `  K ) )
201, 3, 54atexlempsb 33277 . . . . . . . . . . . 12  |-  ( ph  ->  ( P  .\/  S
)  e.  ( Base `  K ) )
21 eqid 2433 . . . . . . . . . . . . 13  |-  ( Base `  K )  =  (
Base `  K )
2221, 2, 4latmle1 15229 . . . . . . . . . . . 12  |-  ( ( K  e.  Lat  /\  ( Q  .\/  T )  e.  ( Base `  K
)  /\  ( P  .\/  S )  e.  (
Base `  K )
)  ->  ( ( Q  .\/  T )  ./\  ( P  .\/  S ) )  .<_  ( Q  .\/  T ) )
2318, 19, 20, 22syl3anc 1211 . . . . . . . . . . 11  |-  ( ph  ->  ( ( Q  .\/  T )  ./\  ( P  .\/  S ) )  .<_  ( Q  .\/  T ) )
2414atexlemk 33264 . . . . . . . . . . . 12  |-  ( ph  ->  K  e.  HL )
2514atexlemq 33268 . . . . . . . . . . . 12  |-  ( ph  ->  Q  e.  A )
2614atexlemt 33270 . . . . . . . . . . . 12  |-  ( ph  ->  T  e.  A )
273, 5hlatjcom 32585 . . . . . . . . . . . 12  |-  ( ( K  e.  HL  /\  Q  e.  A  /\  T  e.  A )  ->  ( Q  .\/  T
)  =  ( T 
.\/  Q ) )
2824, 25, 26, 27syl3anc 1211 . . . . . . . . . . 11  |-  ( ph  ->  ( Q  .\/  T
)  =  ( T 
.\/  Q ) )
2923, 28breqtrd 4304 . . . . . . . . . 10  |-  ( ph  ->  ( ( Q  .\/  T )  ./\  ( P  .\/  S ) )  .<_  ( T  .\/  Q ) )
3029adantr 462 . . . . . . . . 9  |-  ( (
ph  /\  C  =  P )  ->  (
( Q  .\/  T
)  ./\  ( P  .\/  S ) )  .<_  ( T  .\/  Q ) )
3117, 30eqbrtrrd 4302 . . . . . . . 8  |-  ( (
ph  /\  C  =  P )  ->  P  .<_  ( T  .\/  Q
) )
3214atexlemkc 33275 . . . . . . . . . 10  |-  ( ph  ->  K  e.  CvLat )
3314atexlemp 33267 . . . . . . . . . 10  |-  ( ph  ->  P  e.  A )
3414atexlempnq 33272 . . . . . . . . . 10  |-  ( ph  ->  P  =/=  Q )
352, 3, 5cvlatexch2 32555 . . . . . . . . . 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 1224 . . . . . . . . 9  |-  ( ph  ->  ( P  .<_  ( T 
.\/  Q )  ->  T  .<_  ( P  .\/  Q ) ) )
3736adantr 462 . . . . . . . 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 2635 . . . . 5  |-  ( ph  ->  ( -.  T  .<_  ( P  .\/  Q )  ->  C  =/=  P
) )
4114, 40mpd 15 . . . 4  |-  ( ph  ->  C  =/=  P )
4241adantr 462 . . 3  |-  ( (
ph  /\  C  =/=  S )  ->  C  =/=  P )
43 simpr 458 . . 3  |-  ( (
ph  /\  C  =/=  S )  ->  C  =/=  S )
4421, 2, 4latmle2 15230 . . . . . 6  |-  ( ( K  e.  Lat  /\  ( Q  .\/  T )  e.  ( Base `  K
)  /\  ( P  .\/  S )  e.  (
Base `  K )
)  ->  ( ( Q  .\/  T )  ./\  ( P  .\/  S ) )  .<_  ( P  .\/  S ) )
4518, 19, 20, 44syl3anc 1211 . . . . 5  |-  ( ph  ->  ( ( Q  .\/  T )  ./\  ( P  .\/  S ) )  .<_  ( P  .\/  S ) )
469, 45syl5eqbr 4313 . . . 4  |-  ( ph  ->  C  .<_  ( P  .\/  S ) )
4746adantr 462 . . 3  |-  ( (
ph  /\  C  =/=  S )  ->  C  .<_  ( P  .\/  S ) )
4814atexlems 33269 . . . . 5  |-  ( ph  ->  S  e.  A )
491, 2, 3, 54atexlempns 33279 . . . . 5  |-  ( ph  ->  P  =/=  S )
505, 2, 3cvlsupr2 32561 . . . . 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 1224 . . . 4  |-  ( ph  ->  ( ( P  .\/  C )  =  ( S 
.\/  C )  <->  ( C  =/=  P  /\  C  =/= 
S  /\  C  .<_  ( P  .\/  S ) ) ) )
5251adantr 462 . . 3  |-  ( (
ph  /\  C  =/=  S )  ->  ( ( P  .\/  C )  =  ( S  .\/  C
)  <->  ( C  =/= 
P  /\  C  =/=  S  /\  C  .<_  ( P 
.\/  S ) ) ) )
5342, 43, 47, 52mpbir3and 1164 . 2  |-  ( (
ph  /\  C  =/=  S )  ->  ( P  .\/  C )  =  ( S  .\/  C ) )
54 breq1 4283 . . . . 5  |-  ( z  =  C  ->  (
z  .<_  W  <->  C  .<_  W ) )
5554notbid 294 . . . 4  |-  ( z  =  C  ->  ( -.  z  .<_  W  <->  -.  C  .<_  W ) )
56 oveq2 6088 . . . . 5  |-  ( z  =  C  ->  ( P  .\/  z )  =  ( P  .\/  C
) )
57 oveq2 6088 . . . . 5  |-  ( z  =  C  ->  ( S  .\/  z )  =  ( S  .\/  C
) )
5856, 57eqeq12d 2447 . . . 4  |-  ( z  =  C  ->  (
( P  .\/  z
)  =  ( S 
.\/  z )  <->  ( P  .\/  C )  =  ( S  .\/  C ) ) )
5955, 58anbi12d 703 . . 3  |-  ( z  =  C  ->  (
( -.  z  .<_  W  /\  ( P  .\/  z )  =  ( S  .\/  z ) )  <->  ( -.  C  .<_  W  /\  ( P 
.\/  C )  =  ( S  .\/  C
) ) ) )
6059rspcev 3062 . 2  |-  ( ( C  e.  A  /\  ( -.  C  .<_  W  /\  ( P  .\/  C )  =  ( S 
.\/  C ) ) )  ->  E. z  e.  A  ( -.  z  .<_  W  /\  ( P  .\/  z )  =  ( S  .\/  z
) ) )
6111, 13, 53, 60syl12anc 1209 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 958    = wceq 1362    e. wcel 1755    =/= wne 2596   E.wrex 2706   class class class wbr 4280   ` cfv 5406  (class class class)co 6080   Basecbs 14157   lecple 14228   joincjn 15097   meetcmee 15098   Latclat 15198   Atomscatm 32481   CvLatclc 32483   HLchlt 32568   LHypclh 33201
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1594  ax-4 1605  ax-5 1669  ax-6 1707  ax-7 1727  ax-8 1757  ax-9 1759  ax-10 1774  ax-11 1779  ax-12 1791  ax-13 1942  ax-ext 2414  ax-rep 4391  ax-sep 4401  ax-nul 4409  ax-pow 4458  ax-pr 4519  ax-un 6361
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 960  df-tru 1365  df-ex 1590  df-nf 1593  df-sb 1700  df-eu 2258  df-mo 2259  df-clab 2420  df-cleq 2426  df-clel 2429  df-nfc 2558  df-ne 2598  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2964  df-sbc 3176  df-csb 3277  df-dif 3319  df-un 3321  df-in 3323  df-ss 3330  df-nul 3626  df-if 3780  df-pw 3850  df-sn 3866  df-pr 3868  df-op 3872  df-uni 4080  df-iun 4161  df-br 4281  df-opab 4339  df-mpt 4340  df-id 4623  df-xp 4833  df-rel 4834  df-cnv 4835  df-co 4836  df-dm 4837  df-rn 4838  df-res 4839  df-ima 4840  df-iota 5369  df-fun 5408  df-fn 5409  df-f 5410  df-f1 5411  df-fo 5412  df-f1o 5413  df-fv 5414  df-riota 6039  df-ov 6083  df-oprab 6084  df-poset 15099  df-plt 15111  df-lub 15127  df-glb 15128  df-join 15129  df-meet 15130  df-p0 15192  df-p1 15193  df-lat 15199  df-clat 15261  df-oposet 32394  df-ol 32396  df-oml 32397  df-covers 32484  df-ats 32485  df-atl 32516  df-cvlat 32540  df-hlat 32569  df-llines 32715  df-lplanes 32716  df-lhyp 33205
This theorem is referenced by:  4atexlemex4  33290  4atexlemex6  33291
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