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Theorem cdlemefrs29clN 34406
Description: TODO: NOT USED? Show closure of the unique element in cdlemefrs29cpre1 34405. (Contributed by NM, 29-Mar-2013.) (New usage is discouraged.)
Hypotheses
Ref Expression
cdlemefrs27.b  |-  B  =  ( Base `  K
)
cdlemefrs27.l  |-  .<_  =  ( le `  K )
cdlemefrs27.j  |-  .\/  =  ( join `  K )
cdlemefrs27.m  |-  ./\  =  ( meet `  K )
cdlemefrs27.a  |-  A  =  ( Atoms `  K )
cdlemefrs27.h  |-  H  =  ( LHyp `  K
)
cdlemefrs27.eq  |-  ( s  =  R  ->  ( ph 
<->  ps ) )
cdlemefrs27.nb  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  P  =/=  Q  /\  (
s  e.  A  /\  ( -.  s  .<_  W  /\  ph ) ) )  ->  N  e.  B )
cdlemefrs27.rnb  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  ps )  ->  [_ R  /  s ]_ N  e.  B
)
cdlemefrs29cl.o  |-  O  =  ( iota_ z  e.  B  A. s  e.  A  ( ( -.  s  .<_  W  /\  ( s 
.\/  ( R  ./\  W ) )  =  R )  ->  z  =  ( N  .\/  ( R 
./\  W ) ) ) )
Assertion
Ref Expression
cdlemefrs29clN  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  ps )  ->  O  e.  B )
Distinct variable groups:    z, s, A    H, s    .\/ , s    K, s    .<_ , s    P, s    Q, s    R, s    W, s    ps, s    z, A    z, B    z, H    z, K    z, 
.<_    z, N    z, P    z, Q    z, R    z, W    ps, z    B, s   
z,  .\/    ./\ , s, z    ph, z
Allowed substitution hints:    ph( s)    N( s)    O( z, s)

Proof of Theorem cdlemefrs29clN
StepHypRef Expression
1 simpl11 1063 . . . . . . 7  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  ps )  /\  s  e.  A
)  ->  ( K  e.  HL  /\  W  e.  H ) )
2 simpl2r 1042 . . . . . . 7  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  ps )  /\  s  e.  A
)  ->  ( R  e.  A  /\  -.  R  .<_  W ) )
3 simpl3 993 . . . . . . 7  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  ps )  /\  s  e.  A
)  ->  ps )
4 simpr 461 . . . . . . 7  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  ps )  /\  s  e.  A
)  ->  s  e.  A )
5 cdlemefrs27.b . . . . . . . 8  |-  B  =  ( Base `  K
)
6 cdlemefrs27.l . . . . . . . 8  |-  .<_  =  ( le `  K )
7 cdlemefrs27.j . . . . . . . 8  |-  .\/  =  ( join `  K )
8 cdlemefrs27.m . . . . . . . 8  |-  ./\  =  ( meet `  K )
9 cdlemefrs27.a . . . . . . . 8  |-  A  =  ( Atoms `  K )
10 cdlemefrs27.h . . . . . . . 8  |-  H  =  ( LHyp `  K
)
11 cdlemefrs27.eq . . . . . . . 8  |-  ( s  =  R  ->  ( ph 
<->  ps ) )
125, 6, 7, 8, 9, 10, 11cdlemefrs29pre00 34402 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R  e.  A  /\  -.  R  .<_  W )  /\  ps )  /\  s  e.  A )  ->  ( ( ( -.  s  .<_  W  /\  ph )  /\  ( s 
.\/  ( R  ./\  W ) )  =  R )  <->  ( -.  s  .<_  W  /\  ( s 
.\/  ( R  ./\  W ) )  =  R ) ) )
131, 2, 3, 4, 12syl31anc 1222 . . . . . 6  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  ps )  /\  s  e.  A
)  ->  ( (
( -.  s  .<_  W  /\  ph )  /\  ( s  .\/  ( R  ./\  W ) )  =  R )  <->  ( -.  s  .<_  W  /\  (
s  .\/  ( R  ./\ 
W ) )  =  R ) ) )
1413imbi1d 317 . . . . 5  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  ps )  /\  s  e.  A
)  ->  ( (
( ( -.  s  .<_  W  /\  ph )  /\  ( s  .\/  ( R  ./\  W ) )  =  R )  -> 
z  =  ( N 
.\/  ( R  ./\  W ) ) )  <->  ( ( -.  s  .<_  W  /\  ( s  .\/  ( R  ./\  W ) )  =  R )  -> 
z  =  ( N 
.\/  ( R  ./\  W ) ) ) ) )
1514ralbidva 2844 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  ps )  ->  ( A. s  e.  A  ( ( ( -.  s  .<_  W  /\  ph )  /\  ( s 
.\/  ( R  ./\  W ) )  =  R )  ->  z  =  ( N  .\/  ( R 
./\  W ) ) )  <->  A. s  e.  A  ( ( -.  s  .<_  W  /\  ( s 
.\/  ( R  ./\  W ) )  =  R )  ->  z  =  ( N  .\/  ( R 
./\  W ) ) ) ) )
1615riotabidv 6166 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  ps )  ->  ( iota_ z  e.  B  A. s  e.  A  ( ( ( -.  s  .<_  W  /\  ph )  /\  ( s 
.\/  ( R  ./\  W ) )  =  R )  ->  z  =  ( N  .\/  ( R 
./\  W ) ) ) )  =  (
iota_ z  e.  B  A. s  e.  A  ( ( -.  s  .<_  W  /\  ( s 
.\/  ( R  ./\  W ) )  =  R )  ->  z  =  ( N  .\/  ( R 
./\  W ) ) ) ) )
17 cdlemefrs29cl.o . . 3  |-  O  =  ( iota_ z  e.  B  A. s  e.  A  ( ( -.  s  .<_  W  /\  ( s 
.\/  ( R  ./\  W ) )  =  R )  ->  z  =  ( N  .\/  ( R 
./\  W ) ) ) )
1816, 17syl6reqr 2514 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  ps )  ->  O  =  ( iota_ z  e.  B  A. s  e.  A  ( (
( -.  s  .<_  W  /\  ph )  /\  ( s  .\/  ( R  ./\  W ) )  =  R )  -> 
z  =  ( N 
.\/  ( R  ./\  W ) ) ) ) )
19 cdlemefrs27.nb . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  P  =/=  Q  /\  (
s  e.  A  /\  ( -.  s  .<_  W  /\  ph ) ) )  ->  N  e.  B )
20 cdlemefrs27.rnb . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  ps )  ->  [_ R  /  s ]_ N  e.  B
)
215, 6, 7, 8, 9, 10, 11, 19, 20cdlemefrs29cpre1 34405 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  ps )  ->  E! z  e.  B  A. s  e.  A  ( ( ( -.  s  .<_  W  /\  ph )  /\  ( s 
.\/  ( R  ./\  W ) )  =  R )  ->  z  =  ( N  .\/  ( R 
./\  W ) ) ) )
22 riotacl 6179 . . 3  |-  ( E! z  e.  B  A. s  e.  A  (
( ( -.  s  .<_  W  /\  ph )  /\  ( s  .\/  ( R  ./\  W ) )  =  R )  -> 
z  =  ( N 
.\/  ( R  ./\  W ) ) )  -> 
( iota_ z  e.  B  A. s  e.  A  ( ( ( -.  s  .<_  W  /\  ph )  /\  ( s 
.\/  ( R  ./\  W ) )  =  R )  ->  z  =  ( N  .\/  ( R 
./\  W ) ) ) )  e.  B
)
2321, 22syl 16 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  ps )  ->  ( iota_ z  e.  B  A. s  e.  A  ( ( ( -.  s  .<_  W  /\  ph )  /\  ( s 
.\/  ( R  ./\  W ) )  =  R )  ->  z  =  ( N  .\/  ( R 
./\  W ) ) ) )  e.  B
)
2418, 23eqeltrd 2542 1  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  ps )  ->  O  e.  B )
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   A.wral 2799   E!wreu 2801   [_csb 3398   class class class wbr 4403   ` cfv 5529   iota_crio 6163  (class class class)co 6203   Basecbs 14296   lecple 14368   joincjn 15237   meetcmee 15238   Atomscatm 33271   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-rmo 2807  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-lhyp 33995
This theorem is referenced by:  cdlemefr29clN  34414  cdlemefs29clN  34426
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