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Theorem cdlemkid4 34572
Description: Lemma for cdlemkid 34574. (Contributed by NM, 25-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
cdlemkid4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  N  e.  T  /\  ( R `
 F )  =  ( R `  N
) )  /\  (
( P  e.  A  /\  -.  P  .<_  W )  /\  G  =  (  _I  |`  B )
) )  ->  [_ G  /  g ]_ X  =  ( iota_ z  e.  T  A. b  e.  T  ( ( b  =/=  (  _I  |`  B )  /\  ( R `  b )  =/=  ( R `  F )  /\  ( R `  b
)  =/=  ( R `
 G ) )  ->  z  =  (  _I  |`  B )
) ) )
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 cdlemkid4
StepHypRef Expression
1 simp3r 1059 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  N  e.  T  /\  ( R `
 F )  =  ( R `  N
) )  /\  (
( P  e.  A  /\  -.  P  .<_  W )  /\  G  =  (  _I  |`  B )
) )  ->  G  =  (  _I  |`  B ) )
2 cdlemk5.b . . . . . 6  |-  B  =  ( Base `  K
)
3 cdlemk5.h . . . . . 6  |-  H  =  ( LHyp `  K
)
4 cdlemk5.t . . . . . 6  |-  T  =  ( ( LTrn `  K
) `  W )
52, 3, 4idltrn 33786 . . . . 5  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  (  _I  |`  B )  e.  T )
653ad2ant1 1051 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  N  e.  T  /\  ( R `
 F )  =  ( R `  N
) )  /\  (
( P  e.  A  /\  -.  P  .<_  W )  /\  G  =  (  _I  |`  B )
) )  ->  (  _I  |`  B )  e.  T )
71, 6eqeltrd 2549 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  N  e.  T  /\  ( R `
 F )  =  ( R `  N
) )  /\  (
( P  e.  A  /\  -.  P  .<_  W )  /\  G  =  (  _I  |`  B )
) )  ->  G  e.  T )
8 cdlemk5.x . . . . . . 7  |-  X  =  ( iota_ z  e.  T  A. b  e.  T  ( ( b  =/=  (  _I  |`  B )  /\  ( R `  b )  =/=  ( R `  F )  /\  ( R `  b
)  =/=  ( R `
 g ) )  ->  ( z `  P )  =  Y ) )
98csbeq2i 3786 . . . . . 6  |-  [_ G  /  g ]_ X  =  [_ G  /  g ]_ ( iota_ z  e.  T  A. b  e.  T  ( ( b  =/=  (  _I  |`  B )  /\  ( R `  b )  =/=  ( R `  F )  /\  ( R `  b
)  =/=  ( R `
 g ) )  ->  ( z `  P )  =  Y ) )
10 csbriota 6282 . . . . . 6  |-  [_ G  /  g ]_ ( iota_ z  e.  T  A. b  e.  T  (
( b  =/=  (  _I  |`  B )  /\  ( R `  b )  =/=  ( R `  F )  /\  ( R `  b )  =/=  ( R `  g
) )  ->  (
z `  P )  =  Y ) )  =  ( iota_ z  e.  T  [. G  /  g ]. A. b  e.  T  ( ( b  =/=  (  _I  |`  B )  /\  ( R `  b )  =/=  ( R `  F )  /\  ( R `  b
)  =/=  ( R `
 g ) )  ->  ( z `  P )  =  Y ) )
119, 10eqtri 2493 . . . . 5  |-  [_ G  /  g ]_ X  =  ( iota_ z  e.  T  [. G  / 
g ]. A. b  e.  T  ( ( b  =/=  (  _I  |`  B )  /\  ( R `  b )  =/=  ( R `  F )  /\  ( R `  b
)  =/=  ( R `
 g ) )  ->  ( z `  P )  =  Y ) )
1211a1i 11 . . . 4  |-  ( G  e.  T  ->  [_ G  /  g ]_ X  =  ( iota_ z  e.  T  [. G  / 
g ]. A. b  e.  T  ( ( b  =/=  (  _I  |`  B )  /\  ( R `  b )  =/=  ( R `  F )  /\  ( R `  b
)  =/=  ( R `
 g ) )  ->  ( z `  P )  =  Y ) ) )
13 sbcralg 3330 . . . . . 6  |-  ( G  e.  T  ->  ( [. G  /  g ]. A. b  e.  T  ( ( b  =/=  (  _I  |`  B )  /\  ( R `  b )  =/=  ( R `  F )  /\  ( R `  b
)  =/=  ( R `
 g ) )  ->  ( z `  P )  =  Y )  <->  A. b  e.  T  [. G  /  g ]. ( ( b  =/=  (  _I  |`  B )  /\  ( R `  b )  =/=  ( R `  F )  /\  ( R `  b
)  =/=  ( R `
 g ) )  ->  ( z `  P )  =  Y ) ) )
14 sbcimg 3297 . . . . . . . 8  |-  ( G  e.  T  ->  ( [. G  /  g ]. ( ( b  =/=  (  _I  |`  B )  /\  ( R `  b )  =/=  ( R `  F )  /\  ( R `  b
)  =/=  ( R `
 g ) )  ->  ( z `  P )  =  Y )  <->  ( [. G  /  g ]. (
b  =/=  (  _I  |`  B )  /\  ( R `  b )  =/=  ( R `  F
)  /\  ( R `  b )  =/=  ( R `  g )
)  ->  [. G  / 
g ]. ( z `  P )  =  Y ) ) )
15 sbc3an 3313 . . . . . . . . . 10  |-  ( [. G  /  g ]. (
b  =/=  (  _I  |`  B )  /\  ( R `  b )  =/=  ( R `  F
)  /\  ( R `  b )  =/=  ( R `  g )
)  <->  ( [. G  /  g ]. b  =/=  (  _I  |`  B )  /\  [. G  / 
g ]. ( R `  b )  =/=  ( R `  F )  /\  [. G  /  g ]. ( R `  b
)  =/=  ( R `
 g ) ) )
16 sbcg 3321 . . . . . . . . . . 11  |-  ( G  e.  T  ->  ( [. G  /  g ]. b  =/=  (  _I  |`  B )  <->  b  =/=  (  _I  |`  B ) ) )
17 sbcg 3321 . . . . . . . . . . 11  |-  ( G  e.  T  ->  ( [. G  /  g ]. ( R `  b
)  =/=  ( R `
 F )  <->  ( R `  b )  =/=  ( R `  F )
) )
18 sbcne12 3779 . . . . . . . . . . . 12  |-  ( [. G  /  g ]. ( R `  b )  =/=  ( R `  g
)  <->  [_ G  /  g ]_ ( R `  b
)  =/=  [_ G  /  g ]_ ( R `  g )
)
19 csbconstg 3362 . . . . . . . . . . . . 13  |-  ( G  e.  T  ->  [_ G  /  g ]_ ( R `  b )  =  ( R `  b ) )
20 csbfv 5916 . . . . . . . . . . . . . 14  |-  [_ G  /  g ]_ ( R `  g )  =  ( R `  G )
2120a1i 11 . . . . . . . . . . . . 13  |-  ( G  e.  T  ->  [_ G  /  g ]_ ( R `  g )  =  ( R `  G ) )
2219, 21neeq12d 2704 . . . . . . . . . . . 12  |-  ( G  e.  T  ->  ( [_ G  /  g ]_ ( R `  b
)  =/=  [_ G  /  g ]_ ( R `  g )  <->  ( R `  b )  =/=  ( R `  G ) ) )
2318, 22syl5bb 265 . . . . . . . . . . 11  |-  ( G  e.  T  ->  ( [. G  /  g ]. ( R `  b
)  =/=  ( R `
 g )  <->  ( R `  b )  =/=  ( R `  G )
) )
2416, 17, 233anbi123d 1365 . . . . . . . . . 10  |-  ( G  e.  T  ->  (
( [. G  /  g ]. b  =/=  (  _I  |`  B )  /\  [. G  /  g ]. ( R `  b )  =/=  ( R `  F )  /\  [. G  /  g ]. ( R `  b )  =/=  ( R `  g
) )  <->  ( b  =/=  (  _I  |`  B )  /\  ( R `  b )  =/=  ( R `  F )  /\  ( R `  b
)  =/=  ( R `
 G ) ) ) )
2515, 24syl5bb 265 . . . . . . . . 9  |-  ( G  e.  T  ->  ( [. G  /  g ]. ( b  =/=  (  _I  |`  B )  /\  ( R `  b )  =/=  ( R `  F )  /\  ( R `  b )  =/=  ( R `  g
) )  <->  ( b  =/=  (  _I  |`  B )  /\  ( R `  b )  =/=  ( R `  F )  /\  ( R `  b
)  =/=  ( R `
 G ) ) ) )
26 sbceq2g 3783 . . . . . . . . 9  |-  ( G  e.  T  ->  ( [. G  /  g ]. ( z `  P
)  =  Y  <->  ( z `  P )  =  [_ G  /  g ]_ Y
) )
2725, 26imbi12d 327 . . . . . . . 8  |-  ( G  e.  T  ->  (
( [. G  /  g ]. ( b  =/=  (  _I  |`  B )  /\  ( R `  b )  =/=  ( R `  F )  /\  ( R `  b )  =/=  ( R `  g
) )  ->  [. G  /  g ]. (
z `  P )  =  Y )  <->  ( (
b  =/=  (  _I  |`  B )  /\  ( R `  b )  =/=  ( R `  F
)  /\  ( R `  b )  =/=  ( R `  G )
)  ->  ( z `  P )  =  [_ G  /  g ]_ Y
) ) )
2814, 27bitrd 261 . . . . . . 7  |-  ( G  e.  T  ->  ( [. G  /  g ]. ( ( b  =/=  (  _I  |`  B )  /\  ( R `  b )  =/=  ( R `  F )  /\  ( R `  b
)  =/=  ( R `
 g ) )  ->  ( z `  P )  =  Y )  <->  ( ( b  =/=  (  _I  |`  B )  /\  ( R `  b )  =/=  ( R `  F )  /\  ( R `  b
)  =/=  ( R `
 G ) )  ->  ( z `  P )  =  [_ G  /  g ]_ Y
) ) )
2928ralbidv 2829 . . . . . 6  |-  ( G  e.  T  ->  ( A. b  e.  T  [. G  /  g ]. ( ( b  =/=  (  _I  |`  B )  /\  ( R `  b )  =/=  ( R `  F )  /\  ( R `  b
)  =/=  ( R `
 g ) )  ->  ( z `  P )  =  Y )  <->  A. b  e.  T  ( ( b  =/=  (  _I  |`  B )  /\  ( R `  b )  =/=  ( R `  F )  /\  ( R `  b
)  =/=  ( R `
 G ) )  ->  ( z `  P )  =  [_ G  /  g ]_ Y
) ) )
3013, 29bitrd 261 . . . . 5  |-  ( G  e.  T  ->  ( [. G  /  g ]. A. b  e.  T  ( ( b  =/=  (  _I  |`  B )  /\  ( R `  b )  =/=  ( R `  F )  /\  ( R `  b
)  =/=  ( R `
 g ) )  ->  ( z `  P )  =  Y )  <->  A. b  e.  T  ( ( b  =/=  (  _I  |`  B )  /\  ( R `  b )  =/=  ( R `  F )  /\  ( R `  b
)  =/=  ( R `
 G ) )  ->  ( z `  P )  =  [_ G  /  g ]_ Y
) ) )
3130riotabidv 6272 . . . 4  |-  ( G  e.  T  ->  ( iota_ z  e.  T  [. G  /  g ]. A. b  e.  T  (
( b  =/=  (  _I  |`  B )  /\  ( R `  b )  =/=  ( R `  F )  /\  ( R `  b )  =/=  ( R `  g
) )  ->  (
z `  P )  =  Y ) )  =  ( iota_ z  e.  T  A. b  e.  T  ( ( b  =/=  (  _I  |`  B )  /\  ( R `  b )  =/=  ( R `  F )  /\  ( R `  b
)  =/=  ( R `
 G ) )  ->  ( z `  P )  =  [_ G  /  g ]_ Y
) ) )
3212, 31eqtrd 2505 . . 3  |-  ( G  e.  T  ->  [_ G  /  g ]_ X  =  ( iota_ z  e.  T  A. b  e.  T  ( ( b  =/=  (  _I  |`  B )  /\  ( R `  b )  =/=  ( R `  F )  /\  ( R `  b
)  =/=  ( R `
 G ) )  ->  ( z `  P )  =  [_ G  /  g ]_ Y
) ) )
337, 32syl 17 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  N  e.  T  /\  ( R `
 F )  =  ( R `  N
) )  /\  (
( P  e.  A  /\  -.  P  .<_  W )  /\  G  =  (  _I  |`  B )
) )  ->  [_ G  /  g ]_ X  =  ( iota_ z  e.  T  A. b  e.  T  ( ( b  =/=  (  _I  |`  B )  /\  ( R `  b )  =/=  ( R `  F )  /\  ( R `  b
)  =/=  ( R `
 G ) )  ->  ( z `  P )  =  [_ G  /  g ]_ Y
) ) )
34 simpl1 1033 . . . . . . . . . 10  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  N  e.  T  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  G  =  (  _I  |`  B ) ) )  /\  (
( z  e.  T  /\  b  e.  T
)  /\  ( b  =/=  (  _I  |`  B )  /\  ( R `  b )  =/=  ( R `  F )  /\  ( R `  b
)  =/=  ( R `
 G ) ) ) )  ->  ( K  e.  HL  /\  W  e.  H ) )
35 simpl2 1034 . . . . . . . . . 10  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  N  e.  T  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  G  =  (  _I  |`  B ) ) )  /\  (
( z  e.  T  /\  b  e.  T
)  /\  ( b  =/=  (  _I  |`  B )  /\  ( R `  b )  =/=  ( R `  F )  /\  ( R `  b
)  =/=  ( R `
 G ) ) ) )  ->  ( F  e.  T  /\  N  e.  T  /\  ( R `  F )  =  ( R `  N ) ) )
36 simpl3l 1085 . . . . . . . . . 10  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  N  e.  T  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  G  =  (  _I  |`  B ) ) )  /\  (
( z  e.  T  /\  b  e.  T
)  /\  ( b  =/=  (  _I  |`  B )  /\  ( R `  b )  =/=  ( R `  F )  /\  ( R `  b
)  =/=  ( R `
 G ) ) ) )  ->  ( P  e.  A  /\  -.  P  .<_  W ) )
37 simpl3r 1086 . . . . . . . . . 10  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  N  e.  T  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  G  =  (  _I  |`  B ) ) )  /\  (
( z  e.  T  /\  b  e.  T
)  /\  ( b  =/=  (  _I  |`  B )  /\  ( R `  b )  =/=  ( R `  F )  /\  ( R `  b
)  =/=  ( R `
 G ) ) ) )  ->  G  =  (  _I  |`  B ) )
38 simprlr 781 . . . . . . . . . . 11  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  N  e.  T  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  G  =  (  _I  |`  B ) ) )  /\  (
( z  e.  T  /\  b  e.  T
)  /\  ( b  =/=  (  _I  |`  B )  /\  ( R `  b )  =/=  ( R `  F )  /\  ( R `  b
)  =/=  ( R `
 G ) ) ) )  ->  b  e.  T )
39 simprr1 1078 . . . . . . . . . . 11  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  N  e.  T  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  G  =  (  _I  |`  B ) ) )  /\  (
( z  e.  T  /\  b  e.  T
)  /\  ( b  =/=  (  _I  |`  B )  /\  ( R `  b )  =/=  ( R `  F )  /\  ( R `  b
)  =/=  ( R `
 G ) ) ) )  ->  b  =/=  (  _I  |`  B ) )
4038, 39jca 541 . . . . . . . . . 10  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  N  e.  T  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  G  =  (  _I  |`  B ) ) )  /\  (
( z  e.  T  /\  b  e.  T
)  /\  ( b  =/=  (  _I  |`  B )  /\  ( R `  b )  =/=  ( R `  F )  /\  ( R `  b
)  =/=  ( R `
 G ) ) ) )  ->  (
b  e.  T  /\  b  =/=  (  _I  |`  B ) ) )
41 cdlemk5.l . . . . . . . . . . 11  |-  .<_  =  ( le `  K )
42 cdlemk5.j . . . . . . . . . . 11  |-  .\/  =  ( join `  K )
43 cdlemk5.m . . . . . . . . . . 11  |-  ./\  =  ( meet `  K )
44 cdlemk5.a . . . . . . . . . . 11  |-  A  =  ( Atoms `  K )
45 cdlemk5.r . . . . . . . . . . 11  |-  R  =  ( ( trL `  K
) `  W )
46 cdlemk5.z . . . . . . . . . . 11  |-  Z  =  ( ( P  .\/  ( R `  b ) )  ./\  ( ( N `  P )  .\/  ( R `  (
b  o.  `' F
) ) ) )
47 cdlemk5.y . . . . . . . . . . 11  |-  Y  =  ( ( P  .\/  ( R `  g ) )  ./\  ( Z  .\/  ( R `  (
g  o.  `' b ) ) ) )
482, 41, 42, 43, 44, 3, 4, 45, 46, 47cdlemkid2 34562 . . . . . . . . . 10  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  N  e.  T  /\  ( R `
 F )  =  ( R `  N
) )  /\  (
( P  e.  A  /\  -.  P  .<_  W )  /\  G  =  (  _I  |`  B )  /\  ( b  e.  T  /\  b  =/=  (  _I  |`  B ) ) ) )  ->  [_ G  /  g ]_ Y  =  P )
4934, 35, 36, 37, 40, 48syl113anc 1304 . . . . . . . . 9  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  N  e.  T  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  G  =  (  _I  |`  B ) ) )  /\  (
( z  e.  T  /\  b  e.  T
)  /\  ( b  =/=  (  _I  |`  B )  /\  ( R `  b )  =/=  ( R `  F )  /\  ( R `  b
)  =/=  ( R `
 G ) ) ) )  ->  [_ G  /  g ]_ Y  =  P )
5049eqeq2d 2481 . . . . . . . 8  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  N  e.  T  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  G  =  (  _I  |`  B ) ) )  /\  (
( z  e.  T  /\  b  e.  T
)  /\  ( b  =/=  (  _I  |`  B )  /\  ( R `  b )  =/=  ( R `  F )  /\  ( R `  b
)  =/=  ( R `
 G ) ) ) )  ->  (
( z `  P
)  =  [_ G  /  g ]_ Y  <->  ( z `  P )  =  P ) )
51 simprll 780 . . . . . . . . 9  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  N  e.  T  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  G  =  (  _I  |`  B ) ) )  /\  (
( z  e.  T  /\  b  e.  T
)  /\  ( b  =/=  (  _I  |`  B )  /\  ( R `  b )  =/=  ( R `  F )  /\  ( R `  b
)  =/=  ( R `
 G ) ) ) )  ->  z  e.  T )
522, 41, 44, 3, 4ltrnideq 33812 . . . . . . . . 9  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  z  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  ( z  =  (  _I  |`  B )  <-> 
( z `  P
)  =  P ) )
5334, 51, 36, 52syl3anc 1292 . . . . . . . 8  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  N  e.  T  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  G  =  (  _I  |`  B ) ) )  /\  (
( z  e.  T  /\  b  e.  T
)  /\  ( b  =/=  (  _I  |`  B )  /\  ( R `  b )  =/=  ( R `  F )  /\  ( R `  b
)  =/=  ( R `
 G ) ) ) )  ->  (
z  =  (  _I  |`  B )  <->  ( z `  P )  =  P ) )
5450, 53bitr4d 264 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  N  e.  T  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  G  =  (  _I  |`  B ) ) )  /\  (
( z  e.  T  /\  b  e.  T
)  /\  ( b  =/=  (  _I  |`  B )  /\  ( R `  b )  =/=  ( R `  F )  /\  ( R `  b
)  =/=  ( R `
 G ) ) ) )  ->  (
( z `  P
)  =  [_ G  /  g ]_ Y  <->  z  =  (  _I  |`  B ) ) )
5554exp44 624 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  N  e.  T  /\  ( R `
 F )  =  ( R `  N
) )  /\  (
( P  e.  A  /\  -.  P  .<_  W )  /\  G  =  (  _I  |`  B )
) )  ->  (
z  e.  T  -> 
( b  e.  T  ->  ( ( b  =/=  (  _I  |`  B )  /\  ( R `  b )  =/=  ( R `  F )  /\  ( R `  b
)  =/=  ( R `
 G ) )  ->  ( ( z `
 P )  = 
[_ G  /  g ]_ Y  <->  z  =  (  _I  |`  B )
) ) ) ) )
5655imp41 604 . . . . 5  |-  ( ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  N  e.  T  /\  ( R `  F
)  =  ( R `
 N ) )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  G  =  (  _I  |`  B ) ) )  /\  z  e.  T
)  /\  b  e.  T )  /\  (
b  =/=  (  _I  |`  B )  /\  ( R `  b )  =/=  ( R `  F
)  /\  ( R `  b )  =/=  ( R `  G )
) )  ->  (
( z `  P
)  =  [_ G  /  g ]_ Y  <->  z  =  (  _I  |`  B ) ) )
5756pm5.74da 701 . . . 4  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  N  e.  T  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  G  =  (  _I  |`  B ) ) )  /\  z  e.  T )  /\  b  e.  T )  ->  (
( ( b  =/=  (  _I  |`  B )  /\  ( R `  b )  =/=  ( R `  F )  /\  ( R `  b
)  =/=  ( R `
 G ) )  ->  ( z `  P )  =  [_ G  /  g ]_ Y
)  <->  ( ( b  =/=  (  _I  |`  B )  /\  ( R `  b )  =/=  ( R `  F )  /\  ( R `  b
)  =/=  ( R `
 G ) )  ->  z  =  (  _I  |`  B )
) ) )
5857ralbidva 2828 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  N  e.  T  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  G  =  (  _I  |`  B ) ) )  /\  z  e.  T )  ->  ( A. b  e.  T  ( ( b  =/=  (  _I  |`  B )  /\  ( R `  b )  =/=  ( R `  F )  /\  ( R `  b
)  =/=  ( R `
 G ) )  ->  ( z `  P )  =  [_ G  /  g ]_ Y
)  <->  A. b  e.  T  ( ( b  =/=  (  _I  |`  B )  /\  ( R `  b )  =/=  ( R `  F )  /\  ( R `  b
)  =/=  ( R `
 G ) )  ->  z  =  (  _I  |`  B )
) ) )
5958riotabidva 6286 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  N  e.  T  /\  ( R `
 F )  =  ( R `  N
) )  /\  (
( P  e.  A  /\  -.  P  .<_  W )  /\  G  =  (  _I  |`  B )
) )  ->  ( iota_ z  e.  T  A. b  e.  T  (
( b  =/=  (  _I  |`  B )  /\  ( R `  b )  =/=  ( R `  F )  /\  ( R `  b )  =/=  ( R `  G
) )  ->  (
z `  P )  =  [_ G  /  g ]_ Y ) )  =  ( iota_ z  e.  T  A. b  e.  T  ( ( b  =/=  (  _I  |`  B )  /\  ( R `  b )  =/=  ( R `  F )  /\  ( R `  b
)  =/=  ( R `
 G ) )  ->  z  =  (  _I  |`  B )
) ) )
6033, 59eqtrd 2505 1  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  N  e.  T  /\  ( R `
 F )  =  ( R `  N
) )  /\  (
( P  e.  A  /\  -.  P  .<_  W )  /\  G  =  (  _I  |`  B )
) )  ->  [_ G  /  g ]_ X  =  ( iota_ z  e.  T  A. b  e.  T  ( ( b  =/=  (  _I  |`  B )  /\  ( R `  b )  =/=  ( R `  F )  /\  ( R `  b
)  =/=  ( R `
 G ) )  ->  z  =  (  _I  |`  B )
) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 189    /\ wa 376    /\ w3a 1007    = wceq 1452    e. wcel 1904    =/= wne 2641   A.wral 2756   [.wsbc 3255   [_csb 3349   class class class wbr 4395    _I cid 4749   `'ccnv 4838    |` cres 4841    o. ccom 4843   ` cfv 5589   iota_crio 6269  (class class class)co 6308   Basecbs 15199   lecple 15275   joincjn 16267   meetcmee 16268   Atomscatm 32900   HLchlt 32987   LHypclh 33620   LTrncltrn 33737   trLctrl 33795
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-riotaBAD 32589
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-fal 1458  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-op 3966  df-uni 4191  df-iun 4271  df-iin 4272  df-br 4396  df-opab 4455  df-mpt 4456  df-id 4754  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-1st 6812  df-2nd 6813  df-undef 7038  df-map 7492  df-preset 16251  df-poset 16269  df-plt 16282  df-lub 16298  df-glb 16299  df-join 16300  df-meet 16301  df-p0 16363  df-p1 16364  df-lat 16370  df-clat 16432  df-oposet 32813  df-ol 32815  df-oml 32816  df-covers 32903  df-ats 32904  df-atl 32935  df-cvlat 32959  df-hlat 32988  df-llines 33134  df-lplanes 33135  df-lvols 33136  df-lines 33137  df-psubsp 33139  df-pmap 33140  df-padd 33432  df-lhyp 33624  df-laut 33625  df-ldil 33740  df-ltrn 33741  df-trl 33796
This theorem is referenced by:  cdlemkid5  34573  cdlemkid  34574
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