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Theorem cdlemk42 35614
Description: Part of proof of Lemma K of [Crawley] p. 118. TODO: fix comment. (Contributed by NM, 20-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
cdlemk42  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  F  =/=  (  _I  |`  B ) )  /\  ( G  e.  T  /\  G  =/=  (  _I  |`  B ) ) )  /\  ( N  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( b  e.  T  /\  (
b  =/=  (  _I  |`  B )  /\  ( R `  b )  =/=  ( R `  F
)  /\  ( R `  b )  =/=  ( R `  G )
) ) )  -> 
( [_ G  /  g ]_ X `  P )  =  [_ G  / 
g ]_ Y )
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:    X( z, g, b)    Y( g, b)    Z( z, b)

Proof of Theorem cdlemk42
StepHypRef Expression
1 simp13l 1106 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  F  =/=  (  _I  |`  B ) )  /\  ( G  e.  T  /\  G  =/=  (  _I  |`  B ) ) )  /\  ( N  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( b  e.  T  /\  (
b  =/=  (  _I  |`  B )  /\  ( R `  b )  =/=  ( R `  F
)  /\  ( R `  b )  =/=  ( R `  G )
) ) )  ->  G  e.  T )
2 cdlemk5.b . . . . . 6  |-  B  =  ( Base `  K
)
3 cdlemk5.l . . . . . 6  |-  .<_  =  ( le `  K )
4 cdlemk5.j . . . . . 6  |-  .\/  =  ( join `  K )
5 cdlemk5.m . . . . . 6  |-  ./\  =  ( meet `  K )
6 cdlemk5.a . . . . . 6  |-  A  =  ( Atoms `  K )
7 cdlemk5.h . . . . . 6  |-  H  =  ( LHyp `  K
)
8 cdlemk5.t . . . . . 6  |-  T  =  ( ( LTrn `  K
) `  W )
9 cdlemk5.r . . . . . 6  |-  R  =  ( ( trL `  K
) `  W )
10 cdlemk5.z . . . . . 6  |-  Z  =  ( ( P  .\/  ( R `  b ) )  ./\  ( ( N `  P )  .\/  ( R `  (
b  o.  `' F
) ) ) )
11 cdlemk5.y . . . . . 6  |-  Y  =  ( ( P  .\/  ( R `  g ) )  ./\  ( Z  .\/  ( R `  (
g  o.  `' b ) ) ) )
12 cdlemk5.x . . . . . 6  |-  X  =  ( iota_ z  e.  T  A. b  e.  T  ( ( b  =/=  (  _I  |`  B )  /\  ( R `  b )  =/=  ( R `  F )  /\  ( R `  b
)  =/=  ( R `
 g ) )  ->  ( z `  P )  =  Y ) )
132, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12cdlemk36 35586 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  F  =/=  (  _I  |`  B ) )  /\  ( g  e.  T  /\  g  =/=  (  _I  |`  B ) ) )  /\  ( N  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( b  e.  T  /\  (
b  =/=  (  _I  |`  B )  /\  ( R `  b )  =/=  ( R `  F
)  /\  ( R `  b )  =/=  ( R `  g )
) ) )  -> 
( X `  P
)  =  Y )
1413sbcth 3341 . . . 4  |-  ( G  e.  T  ->  [. G  /  g ]. (
( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  F  =/=  (  _I  |`  B ) )  /\  ( g  e.  T  /\  g  =/=  (  _I  |`  B ) ) )  /\  ( N  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( b  e.  T  /\  (
b  =/=  (  _I  |`  B )  /\  ( R `  b )  =/=  ( R `  F
)  /\  ( R `  b )  =/=  ( R `  g )
) ) )  -> 
( X `  P
)  =  Y ) )
15 sbcimg 3368 . . . 4  |-  ( G  e.  T  ->  ( [. G  /  g ]. ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  F  =/=  (  _I  |`  B ) )  /\  ( g  e.  T  /\  g  =/=  (  _I  |`  B ) ) )  /\  ( N  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( b  e.  T  /\  (
b  =/=  (  _I  |`  B )  /\  ( R `  b )  =/=  ( R `  F
)  /\  ( R `  b )  =/=  ( R `  g )
) ) )  -> 
( X `  P
)  =  Y )  <-> 
( [. G  /  g ]. ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  F  =/=  (  _I  |`  B ) )  /\  ( g  e.  T  /\  g  =/=  (  _I  |`  B ) ) )  /\  ( N  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( b  e.  T  /\  (
b  =/=  (  _I  |`  B )  /\  ( R `  b )  =/=  ( R `  F
)  /\  ( R `  b )  =/=  ( R `  g )
) ) )  ->  [. G  /  g ]. ( X `  P
)  =  Y ) ) )
1614, 15mpbid 210 . . 3  |-  ( G  e.  T  ->  ( [. G  /  g ]. ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  F  =/=  (  _I  |`  B ) )  /\  ( g  e.  T  /\  g  =/=  (  _I  |`  B ) ) )  /\  ( N  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( b  e.  T  /\  (
b  =/=  (  _I  |`  B )  /\  ( R `  b )  =/=  ( R `  F
)  /\  ( R `  b )  =/=  ( R `  g )
) ) )  ->  [. G  /  g ]. ( X `  P
)  =  Y ) )
17 eleq1 2534 . . . . . . 7  |-  ( g  =  G  ->  (
g  e.  T  <->  G  e.  T ) )
18 neeq1 2743 . . . . . . 7  |-  ( g  =  G  ->  (
g  =/=  (  _I  |`  B )  <->  G  =/=  (  _I  |`  B ) ) )
1917, 18anbi12d 710 . . . . . 6  |-  ( g  =  G  ->  (
( g  e.  T  /\  g  =/=  (  _I  |`  B ) )  <-> 
( G  e.  T  /\  G  =/=  (  _I  |`  B ) ) ) )
20193anbi3d 1300 . . . . 5  |-  ( g  =  G  ->  (
( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  F  =/=  (  _I  |`  B ) )  /\  ( g  e.  T  /\  g  =/=  (  _I  |`  B ) ) )  <->  ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  F  =/=  (  _I  |`  B ) )  /\  ( G  e.  T  /\  G  =/=  (  _I  |`  B ) ) ) ) )
21 fveq2 5859 . . . . . . . 8  |-  ( g  =  G  ->  ( R `  g )  =  ( R `  G ) )
2221neeq2d 2740 . . . . . . 7  |-  ( g  =  G  ->  (
( R `  b
)  =/=  ( R `
 g )  <->  ( R `  b )  =/=  ( R `  G )
) )
23223anbi3d 1300 . . . . . 6  |-  ( g  =  G  ->  (
( b  =/=  (  _I  |`  B )  /\  ( R `  b )  =/=  ( R `  F )  /\  ( R `  b )  =/=  ( R `  g
) )  <->  ( b  =/=  (  _I  |`  B )  /\  ( R `  b )  =/=  ( R `  F )  /\  ( R `  b
)  =/=  ( R `
 G ) ) ) )
2423anbi2d 703 . . . . 5  |-  ( g  =  G  ->  (
( b  e.  T  /\  ( b  =/=  (  _I  |`  B )  /\  ( R `  b )  =/=  ( R `  F )  /\  ( R `  b )  =/=  ( R `  g
) ) )  <->  ( b  e.  T  /\  (
b  =/=  (  _I  |`  B )  /\  ( R `  b )  =/=  ( R `  F
)  /\  ( R `  b )  =/=  ( R `  G )
) ) ) )
2520, 243anbi13d 1296 . . . 4  |-  ( g  =  G  ->  (
( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  F  =/=  (  _I  |`  B ) )  /\  ( g  e.  T  /\  g  =/=  (  _I  |`  B ) ) )  /\  ( N  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( b  e.  T  /\  (
b  =/=  (  _I  |`  B )  /\  ( R `  b )  =/=  ( R `  F
)  /\  ( R `  b )  =/=  ( R `  g )
) ) )  <->  ( (
( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  F  =/=  (  _I  |`  B ) )  /\  ( G  e.  T  /\  G  =/=  (  _I  |`  B ) ) )  /\  ( N  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( b  e.  T  /\  (
b  =/=  (  _I  |`  B )  /\  ( R `  b )  =/=  ( R `  F
)  /\  ( R `  b )  =/=  ( R `  G )
) ) ) ) )
2625sbcieg 3359 . . 3  |-  ( G  e.  T  ->  ( [. G  /  g ]. ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  F  =/=  (  _I  |`  B ) )  /\  ( g  e.  T  /\  g  =/=  (  _I  |`  B ) ) )  /\  ( N  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( b  e.  T  /\  (
b  =/=  (  _I  |`  B )  /\  ( R `  b )  =/=  ( R `  F
)  /\  ( R `  b )  =/=  ( R `  g )
) ) )  <->  ( (
( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  F  =/=  (  _I  |`  B ) )  /\  ( G  e.  T  /\  G  =/=  (  _I  |`  B ) ) )  /\  ( N  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( b  e.  T  /\  (
b  =/=  (  _I  |`  B )  /\  ( R `  b )  =/=  ( R `  F
)  /\  ( R `  b )  =/=  ( R `  G )
) ) ) ) )
27 sbceqg 3820 . . . 4  |-  ( G  e.  T  ->  ( [. G  /  g ]. ( X `  P
)  =  Y  <->  [_ G  / 
g ]_ ( X `  P )  =  [_ G  /  g ]_ Y
) )
28 csbfv12 5894 . . . . . 6  |-  [_ G  /  g ]_ ( X `  P )  =  ( [_ G  /  g ]_ X `  [_ G  /  g ]_ P )
29 csbconstg 3443 . . . . . . 7  |-  ( G  e.  T  ->  [_ G  /  g ]_ P  =  P )
3029fveq2d 5863 . . . . . 6  |-  ( G  e.  T  ->  ( [_ G  /  g ]_ X `  [_ G  /  g ]_ P
)  =  ( [_ G  /  g ]_ X `  P ) )
3128, 30syl5eq 2515 . . . . 5  |-  ( G  e.  T  ->  [_ G  /  g ]_ ( X `  P )  =  ( [_ G  /  g ]_ X `  P ) )
3231eqeq1d 2464 . . . 4  |-  ( G  e.  T  ->  ( [_ G  /  g ]_ ( X `  P
)  =  [_ G  /  g ]_ Y  <->  (
[_ G  /  g ]_ X `  P )  =  [_ G  / 
g ]_ Y ) )
3327, 32bitrd 253 . . 3  |-  ( G  e.  T  ->  ( [. G  /  g ]. ( X `  P
)  =  Y  <->  ( [_ G  /  g ]_ X `  P )  =  [_ G  /  g ]_ Y
) )
3416, 26, 333imtr3d 267 . 2  |-  ( G  e.  T  ->  (
( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  F  =/=  (  _I  |`  B ) )  /\  ( G  e.  T  /\  G  =/=  (  _I  |`  B ) ) )  /\  ( N  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( b  e.  T  /\  (
b  =/=  (  _I  |`  B )  /\  ( R `  b )  =/=  ( R `  F
)  /\  ( R `  b )  =/=  ( R `  G )
) ) )  -> 
( [_ G  /  g ]_ X `  P )  =  [_ G  / 
g ]_ Y ) )
351, 34mpcom 36 1  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  F  =/=  (  _I  |`  B ) )  /\  ( G  e.  T  /\  G  =/=  (  _I  |`  B ) ) )  /\  ( N  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( b  e.  T  /\  (
b  =/=  (  _I  |`  B )  /\  ( R `  b )  =/=  ( R `  F
)  /\  ( R `  b )  =/=  ( R `  G )
) ) )  -> 
( [_ G  /  g ]_ X `  P )  =  [_ G  / 
g ]_ Y )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 369    /\ w3a 968    = wceq 1374    e. wcel 1762    =/= wne 2657   A.wral 2809   [.wsbc 3326   [_csb 3430   class class class wbr 4442    _I cid 4785   `'ccnv 4993    |` cres 4996    o. ccom 4998   ` cfv 5581   iota_crio 6237  (class class class)co 6277   Basecbs 14481   lecple 14553   joincjn 15422   meetcmee 15423   Atomscatm 33937   HLchlt 34024   LHypclh 34657   LTrncltrn 34774   trLctrl 34831
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1963  ax-ext 2440  ax-rep 4553  ax-sep 4563  ax-nul 4571  ax-pow 4620  ax-pr 4681  ax-un 6569  ax-riotaBAD 33633
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-fal 1380  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2274  df-mo 2275  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-ne 2659  df-nel 2660  df-ral 2814  df-rex 2815  df-reu 2816  df-rmo 2817  df-rab 2818  df-v 3110  df-sbc 3327  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3781  df-if 3935  df-pw 4007  df-sn 4023  df-pr 4025  df-op 4029  df-uni 4241  df-iun 4322  df-iin 4323  df-br 4443  df-opab 4501  df-mpt 4502  df-id 4790  df-xp 5000  df-rel 5001  df-cnv 5002  df-co 5003  df-dm 5004  df-rn 5005  df-res 5006  df-ima 5007  df-iota 5544  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-riota 6238  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-1st 6776  df-2nd 6777  df-undef 6994  df-map 7414  df-poset 15424  df-plt 15436  df-lub 15452  df-glb 15453  df-join 15454  df-meet 15455  df-p0 15517  df-p1 15518  df-lat 15524  df-clat 15586  df-oposet 33850  df-ol 33852  df-oml 33853  df-covers 33940  df-ats 33941  df-atl 33972  df-cvlat 33996  df-hlat 34025  df-llines 34171  df-lplanes 34172  df-lvols 34173  df-lines 34174  df-psubsp 34176  df-pmap 34177  df-padd 34469  df-lhyp 34661  df-laut 34662  df-ldil 34777  df-ltrn 34778  df-trl 34832
This theorem is referenced by:  cdlemk19xlem  35615  cdlemk42yN  35617  cdlemk11tc  35618  cdlemk43N  35636
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