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Theorem cdlemkid4 31123
Description: Lemma for cdlemkid 31125. (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 984 . . . 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 30339 . . . . 5  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  (  _I  |`  B )  e.  T )
653ad2ant1 976 . . . 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 2357 . . 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 . . . . . 6  |-  X  =  ( iota_ z  e.  T A. b  e.  T  ( ( b  =/=  (  _I  |`  B )  /\  ( R `  b )  =/=  ( R `  F )  /\  ( R `  b
)  =/=  ( R `
 g ) )  ->  ( z `  P )  =  Y ) )
98csbeq2i 3107 . . . . 5  |-  [_ 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 csbriotag 6317 . . . . 5  |-  ( G  e.  T  ->  [_ 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, 10syl5eq 2327 . . . 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 ) ) )
12 sbcralg 3065 . . . . . 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 ) ) )
13 sbcimg 3032 . . . . . . . 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 ) ) )
14 sbc3ang 3049 . . . . . . . . . 10  |-  ( G  e.  T  ->  ( [. 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 ) ) ) )
15 sbcg 3056 . . . . . . . . . . 11  |-  ( G  e.  T  ->  ( [. G  /  g ]. b  =/=  (  _I  |`  B )  <->  b  =/=  (  _I  |`  B ) ) )
16 sbcg 3056 . . . . . . . . . . 11  |-  ( G  e.  T  ->  ( [. G  /  g ]. ( R `  b
)  =/=  ( R `
 F )  <->  ( R `  b )  =/=  ( R `  F )
) )
17 sbcne12g 3099 . . . . . . . . . . . 12  |-  ( G  e.  T  ->  ( [. G  /  g ]. ( R `  b
)  =/=  ( R `
 g )  <->  [_ G  / 
g ]_ ( R `  b )  =/=  [_ G  /  g ]_ ( R `  g )
) )
18 csbconstg 3095 . . . . . . . . . . . . 13  |-  ( G  e.  T  ->  [_ G  /  g ]_ ( R `  b )  =  ( R `  b ) )
19 csbfvg 5538 . . . . . . . . . . . . 13  |-  ( G  e.  T  ->  [_ G  /  g ]_ ( R `  g )  =  ( R `  G ) )
2018, 19neeq12d 2461 . . . . . . . . . . . 12  |-  ( G  e.  T  ->  ( [_ G  /  g ]_ ( R `  b
)  =/=  [_ G  /  g ]_ ( R `  g )  <->  ( R `  b )  =/=  ( R `  G ) ) )
2117, 20bitrd 244 . . . . . . . . . . 11  |-  ( G  e.  T  ->  ( [. G  /  g ]. ( R `  b
)  =/=  ( R `
 g )  <->  ( R `  b )  =/=  ( R `  G )
) )
2215, 16, 213anbi123d 1252 . . . . . . . . . 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 ) ) ) )
2314, 22bitrd 244 . . . . . . . . 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 ) ) ) )
24 sbceq2g 3103 . . . . . . . . 9  |-  ( G  e.  T  ->  ( [. G  /  g ]. ( z `  P
)  =  Y  <->  ( z `  P )  =  [_ G  /  g ]_ Y
) )
2523, 24imbi12d 311 . . . . . . . 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
) ) )
2613, 25bitrd 244 . . . . . . 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
) ) )
2726ralbidv 2563 . . . . . 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
) ) )
2812, 27bitrd 244 . . . . 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
) ) )
2928riotabidv 6306 . . . 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
) ) )
3011, 29eqtrd 2315 . . 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
) ) )
317, 30syl 15 . 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
) ) )
32 simpl1 958 . . . . . . . . . 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 ) )
33 simpl2 959 . . . . . . . . . 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 ) ) )
34 simpl3l 1010 . . . . . . . . . 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 ) )
35 simpl3r 1011 . . . . . . . . . 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 ) )
36 simprlr 739 . . . . . . . . . . 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 )
37 simprr1 1003 . . . . . . . . . . 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 ) )
3836, 37jca 518 . . . . . . . . . 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 ) ) )
39 cdlemk5.l . . . . . . . . . . 11  |-  .<_  =  ( le `  K )
40 cdlemk5.j . . . . . . . . . . 11  |-  .\/  =  ( join `  K )
41 cdlemk5.m . . . . . . . . . . 11  |-  ./\  =  ( meet `  K )
42 cdlemk5.a . . . . . . . . . . 11  |-  A  =  ( Atoms `  K )
43 cdlemk5.r . . . . . . . . . . 11  |-  R  =  ( ( trL `  K
) `  W )
44 cdlemk5.z . . . . . . . . . . 11  |-  Z  =  ( ( P  .\/  ( R `  b ) )  ./\  ( ( N `  P )  .\/  ( R `  (
b  o.  `' F
) ) ) )
45 cdlemk5.y . . . . . . . . . . 11  |-  Y  =  ( ( P  .\/  ( R `  g ) )  ./\  ( Z  .\/  ( R `  (
g  o.  `' b ) ) ) )
462, 39, 40, 41, 42, 3, 4, 43, 44, 45cdlemkid2 31113 . . . . . . . . . 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 )
4732, 33, 34, 35, 38, 46syl113anc 1194 . . . . . . . . 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 )
4847eqeq2d 2294 . . . . . . . 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 ) )
49 simprll 738 . . . . . . . . 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 )
502, 39, 42, 3, 4ltrnideq 30364 . . . . . . . . 9  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  z  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  ( z  =  (  _I  |`  B )  <-> 
( z `  P
)  =  P ) )
5132, 49, 34, 50syl3anc 1182 . . . . . . . 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 ) )
5248, 51bitr4d 247 . . . . . . 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 ) ) )
5352exp44 596 . . . . . 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 )
) ) ) ) )
5453imp41 576 . . . . 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 ) ) )
5554pm5.74da 668 . . . 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 )
) ) )
5655ralbidva 2559 . . 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 )
) ) )
5756riotabidva 6321 . 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 )
) ) )
5831, 57eqtrd 2315 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 set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684    =/= wne 2446   A.wral 2543   [.wsbc 2991   [_csb 3081   class class class wbr 4023    _I cid 4304   `'ccnv 4688    |` cres 4691    o. ccom 4693   ` cfv 5255  (class class class)co 5858   iota_crio 6297   Basecbs 13148   lecple 13215   joincjn 14078   meetcmee 14079   Atomscatm 29453   HLchlt 29540   LHypclh 30173   LTrncltrn 30290   trLctrl 30347
This theorem is referenced by:  cdlemkid5  31124  cdlemkid  31125
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-iin 3908  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-undef 6298  df-riota 6304  df-map 6774  df-poset 14080  df-plt 14092  df-lub 14108  df-glb 14109  df-join 14110  df-meet 14111  df-p0 14145  df-p1 14146  df-lat 14152  df-clat 14214  df-oposet 29366  df-ol 29368  df-oml 29369  df-covers 29456  df-ats 29457  df-atl 29488  df-cvlat 29512  df-hlat 29541  df-llines 29687  df-lplanes 29688  df-lvols 29689  df-lines 29690  df-psubsp 29692  df-pmap 29693  df-padd 29985  df-lhyp 30177  df-laut 30178  df-ldil 30293  df-ltrn 30294  df-trl 30348
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