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Theorem cdlemk55 31447
Description: Part of proof of Lemma K of [Crawley] p. 118. Line 11, p. 120.  G,  I stand for g, h.  X represents tau. (Contributed by NM, 26-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 ) )
Assertion
Ref Expression
cdlemk55  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  e.  T  /\  F  =/=  (  _I  |`  B )  /\  N  e.  T
)  /\  G  e.  T  /\  I  e.  T
)  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  [_ ( G  o.  I
)  /  g ]_ X  =  ( [_ G  /  g ]_ X  o.  [_ I  /  g ]_ X ) )
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    I, b, g, z
Allowed substitution hints:    X( z, g, b)    Y( g, b)    Z( z, b)

Proof of Theorem cdlemk55
StepHypRef Expression
1 simpl1 960 . . 3  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  e.  T  /\  F  =/=  (  _I  |`  B )  /\  N  e.  T
)  /\  G  e.  T  /\  I  e.  T
)  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  ( R `  G )  =  ( R `  I ) )  -> 
( ( K  e.  HL  /\  W  e.  H )  /\  ( R `  F )  =  ( R `  N ) ) )
2 simpl21 1035 . . 3  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  e.  T  /\  F  =/=  (  _I  |`  B )  /\  N  e.  T
)  /\  G  e.  T  /\  I  e.  T
)  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  ( R `  G )  =  ( R `  I ) )  -> 
( F  e.  T  /\  F  =/=  (  _I  |`  B )  /\  N  e.  T )
)
3 simpl22 1036 . . 3  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  e.  T  /\  F  =/=  (  _I  |`  B )  /\  N  e.  T
)  /\  G  e.  T  /\  I  e.  T
)  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  ( R `  G )  =  ( R `  I ) )  ->  G  e.  T )
4 simpl3 962 . . 3  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  e.  T  /\  F  =/=  (  _I  |`  B )  /\  N  e.  T
)  /\  G  e.  T  /\  I  e.  T
)  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  ( R `  G )  =  ( R `  I ) )  -> 
( P  e.  A  /\  -.  P  .<_  W ) )
5 simpl23 1037 . . 3  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  e.  T  /\  F  =/=  (  _I  |`  B )  /\  N  e.  T
)  /\  G  e.  T  /\  I  e.  T
)  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  ( R `  G )  =  ( R `  I ) )  ->  I  e.  T )
6 simpr 448 . . 3  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  e.  T  /\  F  =/=  (  _I  |`  B )  /\  N  e.  T
)  /\  G  e.  T  /\  I  e.  T
)  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  ( R `  G )  =  ( R `  I ) )  -> 
( R `  G
)  =  ( R `
 I ) )
7 cdlemk5.b . . . 4  |-  B  =  ( Base `  K
)
8 cdlemk5.l . . . 4  |-  .<_  =  ( le `  K )
9 cdlemk5.j . . . 4  |-  .\/  =  ( join `  K )
10 cdlemk5.m . . . 4  |-  ./\  =  ( meet `  K )
11 cdlemk5.a . . . 4  |-  A  =  ( Atoms `  K )
12 cdlemk5.h . . . 4  |-  H  =  ( LHyp `  K
)
13 cdlemk5.t . . . 4  |-  T  =  ( ( LTrn `  K
) `  W )
14 cdlemk5.r . . . 4  |-  R  =  ( ( trL `  K
) `  W )
15 cdlemk5.z . . . 4  |-  Z  =  ( ( P  .\/  ( R `  b ) )  ./\  ( ( N `  P )  .\/  ( R `  (
b  o.  `' F
) ) ) )
16 cdlemk5.y . . . 4  |-  Y  =  ( ( P  .\/  ( R `  g ) )  ./\  ( Z  .\/  ( R `  (
g  o.  `' b ) ) ) )
17 cdlemk5.x . . . 4  |-  X  =  ( iota_ z  e.  T A. b  e.  T  ( ( b  =/=  (  _I  |`  B )  /\  ( R `  b )  =/=  ( R `  F )  /\  ( R `  b
)  =/=  ( R `
 g ) )  ->  ( z `  P )  =  Y ) )
187, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17cdlemk55b 31446 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  e.  T  /\  F  =/=  (  _I  |`  B )  /\  N  e.  T
)  /\  G  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  ( I  e.  T  /\  ( R `  G
)  =  ( R `
 I ) ) )  ->  [_ ( G  o.  I )  / 
g ]_ X  =  (
[_ G  /  g ]_ X  o.  [_ I  /  g ]_ X
) )
191, 2, 3, 4, 5, 6, 18syl132anc 1202 . 2  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  e.  T  /\  F  =/=  (  _I  |`  B )  /\  N  e.  T
)  /\  G  e.  T  /\  I  e.  T
)  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  ( R `  G )  =  ( R `  I ) )  ->  [_ ( G  o.  I
)  /  g ]_ X  =  ( [_ G  /  g ]_ X  o.  [_ I  /  g ]_ X ) )
20 simpl1 960 . . 3  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  e.  T  /\  F  =/=  (  _I  |`  B )  /\  N  e.  T
)  /\  G  e.  T  /\  I  e.  T
)  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  ( R `  G )  =/=  ( R `  I ) )  -> 
( ( K  e.  HL  /\  W  e.  H )  /\  ( R `  F )  =  ( R `  N ) ) )
21 simpl21 1035 . . 3  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  e.  T  /\  F  =/=  (  _I  |`  B )  /\  N  e.  T
)  /\  G  e.  T  /\  I  e.  T
)  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  ( R `  G )  =/=  ( R `  I ) )  -> 
( F  e.  T  /\  F  =/=  (  _I  |`  B )  /\  N  e.  T )
)
22 simpl22 1036 . . 3  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  e.  T  /\  F  =/=  (  _I  |`  B )  /\  N  e.  T
)  /\  G  e.  T  /\  I  e.  T
)  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  ( R `  G )  =/=  ( R `  I ) )  ->  G  e.  T )
23 simpl3 962 . . 3  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  e.  T  /\  F  =/=  (  _I  |`  B )  /\  N  e.  T
)  /\  G  e.  T  /\  I  e.  T
)  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  ( R `  G )  =/=  ( R `  I ) )  -> 
( P  e.  A  /\  -.  P  .<_  W ) )
24 simpl23 1037 . . 3  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  e.  T  /\  F  =/=  (  _I  |`  B )  /\  N  e.  T
)  /\  G  e.  T  /\  I  e.  T
)  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  ( R `  G )  =/=  ( R `  I ) )  ->  I  e.  T )
25 simpr 448 . . 3  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  e.  T  /\  F  =/=  (  _I  |`  B )  /\  N  e.  T
)  /\  G  e.  T  /\  I  e.  T
)  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  ( R `  G )  =/=  ( R `  I ) )  -> 
( R `  G
)  =/=  ( R `
 I ) )
267, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17cdlemk53 31443 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  e.  T  /\  F  =/=  (  _I  |`  B )  /\  N  e.  T
)  /\  G  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  ( I  e.  T  /\  ( R `  G
)  =/=  ( R `
 I ) ) )  ->  [_ ( G  o.  I )  / 
g ]_ X  =  (
[_ G  /  g ]_ X  o.  [_ I  /  g ]_ X
) )
2720, 21, 22, 23, 24, 25, 26syl132anc 1202 . 2  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  e.  T  /\  F  =/=  (  _I  |`  B )  /\  N  e.  T
)  /\  G  e.  T  /\  I  e.  T
)  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  ( R `  G )  =/=  ( R `  I ) )  ->  [_ ( G  o.  I
)  /  g ]_ X  =  ( [_ G  /  g ]_ X  o.  [_ I  /  g ]_ X ) )
2819, 27pm2.61dane 2649 1  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  e.  T  /\  F  =/=  (  _I  |`  B )  /\  N  e.  T
)  /\  G  e.  T  /\  I  e.  T
)  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  [_ ( G  o.  I
)  /  g ]_ X  =  ( [_ G  /  g ]_ X  o.  [_ I  /  g ]_ X ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1721    =/= wne 2571   A.wral 2670   [_csb 3215   class class class wbr 4176    _I cid 4457   `'ccnv 4840    |` cres 4843    o. ccom 4845   ` cfv 5417  (class class class)co 6044   iota_crio 6505   Basecbs 13428   lecple 13495   joincjn 14360   meetcmee 14361   Atomscatm 29750   HLchlt 29837   LHypclh 30470   LTrncltrn 30587   trLctrl 30644
This theorem is referenced by:  cdlemk55u1  31451
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2389  ax-rep 4284  ax-sep 4294  ax-nul 4302  ax-pow 4341  ax-pr 4367  ax-un 4664
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2262  df-mo 2263  df-clab 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-ne 2573  df-nel 2574  df-ral 2675  df-rex 2676  df-reu 2677  df-rmo 2678  df-rab 2679  df-v 2922  df-sbc 3126  df-csb 3216  df-dif 3287  df-un 3289  df-in 3291  df-ss 3298  df-nul 3593  df-if 3704  df-pw 3765  df-sn 3784  df-pr 3785  df-op 3787  df-uni 3980  df-iun 4059  df-iin 4060  df-br 4177  df-opab 4231  df-mpt 4232  df-id 4462  df-xp 4847  df-rel 4848  df-cnv 4849  df-co 4850  df-dm 4851  df-rn 4852  df-res 4853  df-ima 4854  df-iota 5381  df-fun 5419  df-fn 5420  df-f 5421  df-f1 5422  df-fo 5423  df-f1o 5424  df-fv 5425  df-ov 6047  df-oprab 6048  df-mpt2 6049  df-1st 6312  df-2nd 6313  df-undef 6506  df-riota 6512  df-map 6983  df-poset 14362  df-plt 14374  df-lub 14390  df-glb 14391  df-join 14392  df-meet 14393  df-p0 14427  df-p1 14428  df-lat 14434  df-clat 14496  df-oposet 29663  df-ol 29665  df-oml 29666  df-covers 29753  df-ats 29754  df-atl 29785  df-cvlat 29809  df-hlat 29838  df-llines 29984  df-lplanes 29985  df-lvols 29986  df-lines 29987  df-psubsp 29989  df-pmap 29990  df-padd 30282  df-lhyp 30474  df-laut 30475  df-ldil 30590  df-ltrn 30591  df-trl 30645
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