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Theorem cdlemk39u1 34919
Description: Lemma for cdlemk39u 34920. (Contributed by NM, 31-Jul-2013.)
Hypotheses
Ref Expression
cdlemk5.b  |-  B  =  ( Base `  K
)
cdlemk5.l  |-  .<_  =  ( le `  K )
cdlemk5.j  |-  .\/  =  ( join `  K )
cdlemk5.m  |-  ./\  =  ( meet `  K )
cdlemk5.a  |-  A  =  ( Atoms `  K )
cdlemk5.h  |-  H  =  ( LHyp `  K
)
cdlemk5.t  |-  T  =  ( ( LTrn `  K
) `  W )
cdlemk5.r  |-  R  =  ( ( trL `  K
) `  W )
cdlemk5.z  |-  Z  =  ( ( P  .\/  ( R `  b ) )  ./\  ( ( N `  P )  .\/  ( R `  (
b  o.  `' F
) ) ) )
cdlemk5.y  |-  Y  =  ( ( P  .\/  ( R `  g ) )  ./\  ( Z  .\/  ( R `  (
g  o.  `' b ) ) ) )
cdlemk5.x  |-  X  =  ( iota_ z  e.  T  A. b  e.  T  ( ( b  =/=  (  _I  |`  B )  /\  ( R `  b )  =/=  ( R `  F )  /\  ( R `  b
)  =/=  ( R `
 g ) )  ->  ( z `  P )  =  Y ) )
cdlemk5.u  |-  U  =  ( g  e.  T  |->  if ( F  =  N ,  g ,  X ) )
Assertion
Ref Expression
cdlemk39u1  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  N  e.  T )  /\  (
( R `  F
)  =  ( R `
 N )  /\  F  =/=  N  /\  G  e.  T )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  ( R `  ( U `  G
) )  .<_  ( R `
 G ) )
Distinct variable groups:    ./\ , g    .\/ , g    B, g    P, g    R, g    T, g    g, Z    g, b, G, z    ./\ , b, z    .<_ , b    z,
g,  .<_    .\/ , b, z    A, b, g, z    B, b, z    F, b, g, z   
z, G    H, b,
g, z    K, b,
g, z    N, b,
g, z    P, b,
z    R, b, z    T, b, z    W, b, g, z    z, Y    G, b
Allowed substitution hints:    U( z, g, b)    X( z, g, b)    Y( g, b)    Z( z, b)

Proof of Theorem cdlemk39u1
StepHypRef Expression
1 simp22 1022 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  N  e.  T )  /\  (
( R `  F
)  =  ( R `
 N )  /\  F  =/=  N  /\  G  e.  T )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  F  =/=  N )
2 simp23 1023 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  N  e.  T )  /\  (
( R `  F
)  =  ( R `
 N )  /\  F  =/=  N  /\  G  e.  T )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  G  e.  T )
3 cdlemk5.x . . . . 5  |-  X  =  ( iota_ z  e.  T  A. b  e.  T  ( ( b  =/=  (  _I  |`  B )  /\  ( R `  b )  =/=  ( R `  F )  /\  ( R `  b
)  =/=  ( R `
 g ) )  ->  ( z `  P )  =  Y ) )
4 cdlemk5.u . . . . 5  |-  U  =  ( g  e.  T  |->  if ( F  =  N ,  g ,  X ) )
53, 4cdlemk40f 34871 . . . 4  |-  ( ( F  =/=  N  /\  G  e.  T )  ->  ( U `  G
)  =  [_ G  /  g ]_ X
)
61, 2, 5syl2anc 661 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  N  e.  T )  /\  (
( R `  F
)  =  ( R `
 N )  /\  F  =/=  N  /\  G  e.  T )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  ( U `  G )  =  [_ G  /  g ]_ X
)
76fveq2d 5795 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  N  e.  T )  /\  (
( R `  F
)  =  ( R `
 N )  /\  F  =/=  N  /\  G  e.  T )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  ( R `  ( U `  G
) )  =  ( R `  [_ G  /  g ]_ X
) )
8 simp11 1018 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  N  e.  T )  /\  (
( R `  F
)  =  ( R `
 N )  /\  F  =/=  N  /\  G  e.  T )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  ( K  e.  HL  /\  W  e.  H ) )
9 simp12 1019 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  N  e.  T )  /\  (
( R `  F
)  =  ( R `
 N )  /\  F  =/=  N  /\  G  e.  T )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  F  e.  T )
10 simp13 1020 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  N  e.  T )  /\  (
( R `  F
)  =  ( R `
 N )  /\  F  =/=  N  /\  G  e.  T )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  N  e.  T )
11 simp21 1021 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  N  e.  T )  /\  (
( R `  F
)  =  ( R `
 N )  /\  F  =/=  N  /\  G  e.  T )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  ( R `  F )  =  ( R `  N ) )
12 cdlemk5.b . . . . . 6  |-  B  =  ( Base `  K
)
13 cdlemk5.h . . . . . 6  |-  H  =  ( LHyp `  K
)
14 cdlemk5.t . . . . . 6  |-  T  =  ( ( LTrn `  K
) `  W )
15 cdlemk5.r . . . . . 6  |-  R  =  ( ( trL `  K
) `  W )
1612, 13, 14, 15trlnid 34131 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  N  e.  T )  /\  ( F  =/=  N  /\  ( R `  F )  =  ( R `  N ) ) )  ->  F  =/=  (  _I  |`  B ) )
178, 9, 10, 1, 11, 16syl122anc 1228 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  N  e.  T )  /\  (
( R `  F
)  =  ( R `
 N )  /\  F  =/=  N  /\  G  e.  T )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  F  =/=  (  _I  |`  B ) )
189, 17jca 532 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  N  e.  T )  /\  (
( R `  F
)  =  ( R `
 N )  /\  F  =/=  N  /\  G  e.  T )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  ( F  e.  T  /\  F  =/=  (  _I  |`  B ) ) )
19 simp3 990 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  N  e.  T )  /\  (
( R `  F
)  =  ( R `
 N )  /\  F  =/=  N  /\  G  e.  T )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  ( P  e.  A  /\  -.  P  .<_  W ) )
20 cdlemk5.l . . . 4  |-  .<_  =  ( le `  K )
21 cdlemk5.j . . . 4  |-  .\/  =  ( join `  K )
22 cdlemk5.m . . . 4  |-  ./\  =  ( meet `  K )
23 cdlemk5.a . . . 4  |-  A  =  ( Atoms `  K )
24 cdlemk5.z . . . 4  |-  Z  =  ( ( P  .\/  ( R `  b ) )  ./\  ( ( N `  P )  .\/  ( R `  (
b  o.  `' F
) ) ) )
25 cdlemk5.y . . . 4  |-  Y  =  ( ( P  .\/  ( R `  g ) )  ./\  ( Z  .\/  ( R `  (
g  o.  `' b ) ) ) )
2612, 20, 21, 22, 23, 13, 14, 15, 24, 25, 3cdlemk39s-id 34892 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( F  e.  T  /\  F  =/=  (  _I  |`  B ) )  /\  G  e.  T  /\  N  e.  T )  /\  (
( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) ) )  ->  ( R `  [_ G  / 
g ]_ X )  .<_  ( R `  G ) )
278, 18, 2, 10, 19, 11, 26syl132anc 1237 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  N  e.  T )  /\  (
( R `  F
)  =  ( R `
 N )  /\  F  =/=  N  /\  G  e.  T )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  ( R `  [_ G  /  g ]_ X )  .<_  ( R `
 G ) )
287, 27eqbrtrd 4412 1  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  N  e.  T )  /\  (
( R `  F
)  =  ( R `
 N )  /\  F  =/=  N  /\  G  e.  T )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  ( R `  ( U `  G
) )  .<_  ( R `
 G ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 369    /\ w3a 965    = wceq 1370    e. wcel 1758    =/= wne 2644   A.wral 2795   [_csb 3388   ifcif 3891   class class class wbr 4392    |-> cmpt 4450    _I cid 4731   `'ccnv 4939    |` cres 4942    o. ccom 4944   ` cfv 5518   iota_crio 6152  (class class class)co 6192   Basecbs 14278   lecple 14349   joincjn 15218   meetcmee 15219   Atomscatm 33216   HLchlt 33303   LHypclh 33936   LTrncltrn 34053   trLctrl 34110
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 1952  ax-ext 2430  ax-rep 4503  ax-sep 4513  ax-nul 4521  ax-pow 4570  ax-pr 4631  ax-un 6474  ax-riotaBAD 32912
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-fal 1376  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-nel 2647  df-ral 2800  df-rex 2801  df-reu 2802  df-rmo 2803  df-rab 2804  df-v 3072  df-sbc 3287  df-csb 3389  df-dif 3431  df-un 3433  df-in 3435  df-ss 3442  df-nul 3738  df-if 3892  df-pw 3962  df-sn 3978  df-pr 3980  df-op 3984  df-uni 4192  df-iun 4273  df-iin 4274  df-br 4393  df-opab 4451  df-mpt 4452  df-id 4736  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-rn 4951  df-res 4952  df-ima 4953  df-iota 5481  df-fun 5520  df-fn 5521  df-f 5522  df-f1 5523  df-fo 5524  df-f1o 5525  df-fv 5526  df-riota 6153  df-ov 6195  df-oprab 6196  df-mpt2 6197  df-1st 6679  df-2nd 6680  df-undef 6894  df-map 7318  df-poset 15220  df-plt 15232  df-lub 15248  df-glb 15249  df-join 15250  df-meet 15251  df-p0 15313  df-p1 15314  df-lat 15320  df-clat 15382  df-oposet 33129  df-ol 33131  df-oml 33132  df-covers 33219  df-ats 33220  df-atl 33251  df-cvlat 33275  df-hlat 33304  df-llines 33450  df-lplanes 33451  df-lvols 33452  df-lines 33453  df-psubsp 33455  df-pmap 33456  df-padd 33748  df-lhyp 33940  df-laut 33941  df-ldil 34056  df-ltrn 34057  df-trl 34111
This theorem is referenced by:  cdlemk39u  34920
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