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Theorem cdlemk39s 34865
Description: Substitution version of cdlemk39 34842. TODO: Can any commonality with cdlemk35s 34863 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 1107 . 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 34842 . . . . 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 3285 . . . 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 3312 . . . 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 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 ) ) )  ->  [. G  /  g ]. ( R `  X )  .<_  ( R `  g
) ) )
17 eleq1 2520 . . . . . . 7  |-  ( g  =  G  ->  (
g  e.  T  <->  G  e.  T ) )
18 neeq1 2726 . . . . . . 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 ) ) ) )
20193anbi2d 1295 . . . . 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 1295 . . . 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 3303 . . 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 4430 . . . 4  |-  ( G  e.  T  ->  ( [. G  /  g ]. ( R `  X
)  .<_  ( R `  g )  <->  [_ G  / 
g ]_ ( R `  X )  .<_  [_ G  /  g ]_ ( R `  g )
) )
24 csbfv2g 5812 . . . . 5  |-  ( G  e.  T  ->  [_ G  /  g ]_ ( R `  X )  =  ( R `  [_ G  /  g ]_ X ) )
25 csbfv 5813 . . . . . 6  |-  [_ G  /  g ]_ ( R `  g )  =  ( R `  G )
2625a1i 11 . . . . 5  |-  ( G  e.  T  ->  [_ G  /  g ]_ ( R `  g )  =  ( R `  G ) )
2724, 26breq12d 4389 . . . 4  |-  ( G  e.  T  ->  ( [_ G  /  g ]_ ( R `  X
)  .<_  [_ G  /  g ]_ ( R `  g
)  <->  ( R `  [_ G  /  g ]_ X )  .<_  ( R `
 G ) ) )
2823, 27bitrd 253 . . 3  |-  ( G  e.  T  ->  ( [. G  /  g ]. ( R `  X
)  .<_  ( R `  g )  <->  ( R `  [_ G  /  g ]_ X )  .<_  ( R `
 G ) ) )
2916, 22, 283imtr3d 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 ) ) )  ->  ( R `  [_ G  / 
g ]_ X )  .<_  ( R `  G ) ) )
301, 29mpcom 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 ) ) )  ->  ( R `  [_ G  / 
g ]_ X )  .<_  ( R `  G ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 369    /\ w3a 965    = wceq 1370    e. wcel 1757    =/= wne 2641   A.wral 2792   [.wsbc 3270   [_csb 3372   class class class wbr 4376    _I cid 4715   `'ccnv 4923    |` cres 4926    o. ccom 4928   ` cfv 5502   iota_crio 6136  (class class class)co 6176   Basecbs 14262   lecple 14333   joincjn 15202   meetcmee 15203   Atomscatm 33190   HLchlt 33277   LHypclh 33910   LTrncltrn 34027   trLctrl 34084
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 1709  ax-7 1729  ax-8 1759  ax-9 1761  ax-10 1776  ax-11 1781  ax-12 1793  ax-13 1944  ax-ext 2429  ax-rep 4487  ax-sep 4497  ax-nul 4505  ax-pow 4554  ax-pr 4615  ax-un 6458  ax-riotaBAD 32886
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 1702  df-eu 2263  df-mo 2264  df-clab 2436  df-cleq 2442  df-clel 2445  df-nfc 2598  df-ne 2643  df-nel 2644  df-ral 2797  df-rex 2798  df-reu 2799  df-rmo 2800  df-rab 2801  df-v 3056  df-sbc 3271  df-csb 3373  df-dif 3415  df-un 3417  df-in 3419  df-ss 3426  df-nul 3722  df-if 3876  df-pw 3946  df-sn 3962  df-pr 3964  df-op 3968  df-uni 4176  df-iun 4257  df-iin 4258  df-br 4377  df-opab 4435  df-mpt 4436  df-id 4720  df-xp 4930  df-rel 4931  df-cnv 4932  df-co 4933  df-dm 4934  df-rn 4935  df-res 4936  df-ima 4937  df-iota 5465  df-fun 5504  df-fn 5505  df-f 5506  df-f1 5507  df-fo 5508  df-f1o 5509  df-fv 5510  df-riota 6137  df-ov 6179  df-oprab 6180  df-mpt2 6181  df-1st 6663  df-2nd 6664  df-undef 6878  df-map 7302  df-poset 15204  df-plt 15216  df-lub 15232  df-glb 15233  df-join 15234  df-meet 15235  df-p0 15297  df-p1 15298  df-lat 15304  df-clat 15366  df-oposet 33103  df-ol 33105  df-oml 33106  df-covers 33193  df-ats 33194  df-atl 33225  df-cvlat 33249  df-hlat 33278  df-llines 33424  df-lplanes 33425  df-lvols 33426  df-lines 33427  df-psubsp 33429  df-pmap 33430  df-padd 33722  df-lhyp 33914  df-laut 33915  df-ldil 34030  df-ltrn 34031  df-trl 34085
This theorem is referenced by:  cdlemk39s-id  34866  cdlemk51  34879
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