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Theorem cdlemk39s 31910
Description: Substitution version of cdlemk39 31887. TODO: Can any commonality with cdlemk35s 31908 be exploited? (Contributed by NM, 23-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
cdlemk39s  |-  ( ( ( 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 ) ) )  ->  ( R `  [_ G  / 
g ]_ X )  .<_  ( 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:    X( z, g, b)    Y( g, b)    Z( z, b)

Proof of Theorem cdlemk39s
StepHypRef Expression
1 simp22l 1077 . 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 ) ) )  ->  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, 12cdlemk39 31887 . . . . 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 ) ) )  ->  ( R `  X )  .<_  ( R `  g
) )
1413sbcth 3184 . . . 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 ) ) )  ->  ( R `  X )  .<_  ( R `  g
) ) )
15 sbcimg 3211 . . . 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 ) ) )  ->  ( R `  X )  .<_  ( R `  g
) )  <->  ( [. 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 ) ) )  ->  [. G  /  g ]. ( R `  X )  .<_  ( R `  g
) ) ) )
1614, 15mpbid 203 . . 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 ) ) )  ->  [. G  /  g ]. ( R `  X )  .<_  ( R `  g
) ) )
17 eleq1 2503 . . . . . . 7  |-  ( g  =  G  ->  (
g  e.  T  <->  G  e.  T ) )
18 neeq1 2616 . . . . . . 7  |-  ( g  =  G  ->  (
g  =/=  (  _I  |`  B )  <->  G  =/=  (  _I  |`  B ) ) )
1917, 18anbi12d 693 . . . . . 6  |-  ( g  =  G  ->  (
( g  e.  T  /\  g  =/=  (  _I  |`  B ) )  <-> 
( G  e.  T  /\  G  =/=  (  _I  |`  B ) ) ) )
20193anbi2d 1260 . . . . 5  |-  ( g  =  G  ->  (
( ( F  e.  T  /\  F  =/=  (  _I  |`  B ) )  /\  ( g  e.  T  /\  g  =/=  (  _I  |`  B ) )  /\  N  e.  T )  <->  ( ( F  e.  T  /\  F  =/=  (  _I  |`  B ) )  /\  ( G  e.  T  /\  G  =/=  (  _I  |`  B ) )  /\  N  e.  T ) ) )
21203anbi2d 1260 . . . 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 ) ) )  <->  ( ( 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 ) ) ) ) )
2221sbcieg 3202 . . 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 ) ) )  <->  ( ( 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 ) ) ) ) )
23 sbcbr12g 4293 . . . 4  |-  ( G  e.  T  ->  ( [. G  /  g ]. ( R `  X
)  .<_  ( R `  g )  <->  [_ G  / 
g ]_ ( R `  X )  .<_  [_ G  /  g ]_ ( R `  g )
) )
24 csbfv2g 5798 . . . . 5  |-  ( G  e.  T  ->  [_ G  /  g ]_ ( R `  X )  =  ( R `  [_ G  /  g ]_ X ) )
25 csbfvgOLD 5800 . . . . 5  |-  ( G  e.  T  ->  [_ G  /  g ]_ ( R `  g )  =  ( R `  G ) )
2624, 25breq12d 4256 . . . 4  |-  ( G  e.  T  ->  ( [_ G  /  g ]_ ( R `  X
)  .<_  [_ G  /  g ]_ ( R `  g
)  <->  ( R `  [_ G  /  g ]_ X )  .<_  ( R `
 G ) ) )
2723, 26bitrd 246 . . 3  |-  ( G  e.  T  ->  ( [. G  /  g ]. ( R `  X
)  .<_  ( R `  g )  <->  ( R `  [_ G  /  g ]_ X )  .<_  ( R `
 G ) ) )
2816, 22, 273imtr3d 260 . 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 ) ) )  ->  ( R `  [_ G  / 
g ]_ X )  .<_  ( R `  G ) ) )
291, 28mpcom 35 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 ) ) )  ->  ( R `  [_ G  / 
g ]_ X )  .<_  ( R `  G ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 360    /\ w3a 937    = wceq 1654    e. wcel 1728    =/= wne 2606   A.wral 2712   [.wsbc 3170   [_csb 3270   class class class wbr 4243    _I cid 4528   `'ccnv 4912    |` cres 4915    o. ccom 4917   ` cfv 5489  (class class class)co 6117   iota_crio 6578   Basecbs 13507   lecple 13574   joincjn 14439   meetcmee 14440   Atomscatm 30235   HLchlt 30322   LHypclh 30955   LTrncltrn 31072   trLctrl 31129
This theorem is referenced by:  cdlemk39s-id  31911  cdlemk51  31924
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1628  ax-9 1669  ax-8 1690  ax-13 1730  ax-14 1732  ax-6 1747  ax-7 1752  ax-11 1764  ax-12 1954  ax-ext 2424  ax-rep 4351  ax-sep 4361  ax-nul 4369  ax-pow 4412  ax-pr 4438  ax-un 4736
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-fal 1330  df-ex 1552  df-nf 1555  df-sb 1661  df-eu 2292  df-mo 2293  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-nel 2609  df-ral 2717  df-rex 2718  df-reu 2719  df-rmo 2720  df-rab 2721  df-v 2967  df-sbc 3171  df-csb 3271  df-dif 3312  df-un 3314  df-in 3316  df-ss 3323  df-nul 3617  df-if 3768  df-pw 3830  df-sn 3849  df-pr 3850  df-op 3852  df-uni 4045  df-iun 4124  df-iin 4125  df-br 4244  df-opab 4298  df-mpt 4299  df-id 4533  df-xp 4919  df-rel 4920  df-cnv 4921  df-co 4922  df-dm 4923  df-rn 4924  df-res 4925  df-ima 4926  df-iota 5453  df-fun 5491  df-fn 5492  df-f 5493  df-f1 5494  df-fo 5495  df-f1o 5496  df-fv 5497  df-ov 6120  df-oprab 6121  df-mpt2 6122  df-1st 6385  df-2nd 6386  df-undef 6579  df-riota 6585  df-map 7056  df-poset 14441  df-plt 14453  df-lub 14469  df-glb 14470  df-join 14471  df-meet 14472  df-p0 14506  df-p1 14507  df-lat 14513  df-clat 14575  df-oposet 30148  df-ol 30150  df-oml 30151  df-covers 30238  df-ats 30239  df-atl 30270  df-cvlat 30294  df-hlat 30323  df-llines 30469  df-lplanes 30470  df-lvols 30471  df-lines 30472  df-psubsp 30474  df-pmap 30475  df-padd 30767  df-lhyp 30959  df-laut 30960  df-ldil 31075  df-ltrn 31076  df-trl 31130
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