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Theorem cdlemkid3N 31193
Description: Lemma for cdlemkid 31196. (Contributed by NM, 25-Jul-2013.) (New usage is discouraged.)
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
cdlemkid3N  |-  ( ( ( 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 )  =  P ) ) )
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 cdlemkid3N
StepHypRef Expression
1 simp3r 985 . . . 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 30410 . . . . 5  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  (  _I  |`  B )  e.  T )
653ad2ant1 977 . . . 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 2440 . . 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 3193 . . . . 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 6459 . . . . 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 2410 . . . 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 3151 . . . . . 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 3118 . . . . . . . 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 3135 . . . . . . . . . 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 3142 . . . . . . . . . . 11  |-  ( G  e.  T  ->  ( [. G  /  g ]. b  =/=  (  _I  |`  B )  <->  b  =/=  (  _I  |`  B ) ) )
16 sbcg 3142 . . . . . . . . . . 11  |-  ( G  e.  T  ->  ( [. G  /  g ]. ( R `  b
)  =/=  ( R `
 F )  <->  ( R `  b )  =/=  ( R `  F )
) )
17 sbcne12g 3185 . . . . . . . . . . . 12  |-  ( G  e.  T  ->  ( [. G  /  g ]. ( R `  b
)  =/=  ( R `
 g )  <->  [_ G  / 
g ]_ ( R `  b )  =/=  [_ G  /  g ]_ ( R `  g )
) )
18 csbconstg 3181 . . . . . . . . . . . . 13  |-  ( G  e.  T  ->  [_ G  /  g ]_ ( R `  b )  =  ( R `  b ) )
19 csbfvg 5645 . . . . . . . . . . . . 13  |-  ( G  e.  T  ->  [_ G  /  g ]_ ( R `  g )  =  ( R `  G ) )
2018, 19neeq12d 2544 . . . . . . . . . . . 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 1253 . . . . . . . . . 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 3189 . . . . . . . . 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 2648 . . . . . 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 6448 . . . 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 2398 . . 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 simp11 986 . . . . . . . 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 ) ) )  /\  b  e.  T  /\  (
b  =/=  (  _I  |`  B )  /\  ( R `  b )  =/=  ( R `  F
)  /\  ( R `  b )  =/=  ( R `  G )
) )  ->  ( K  e.  HL  /\  W  e.  H ) )
33 simp12 987 . . . . . . . 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 ) ) )  /\  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 simp13l 1071 . . . . . . . 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 ) ) )  /\  b  e.  T  /\  (
b  =/=  (  _I  |`  B )  /\  ( R `  b )  =/=  ( R `  F
)  /\  ( R `  b )  =/=  ( R `  G )
) )  ->  ( P  e.  A  /\  -.  P  .<_  W ) )
35 simp13r 1072 . . . . . . . 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 ) ) )  /\  b  e.  T  /\  (
b  =/=  (  _I  |`  B )  /\  ( R `  b )  =/=  ( R `  F
)  /\  ( R `  b )  =/=  ( R `  G )
) )  ->  G  =  (  _I  |`  B ) )
36 simp2 957 . . . . . . . . 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 ) ) )  /\  b  e.  T  /\  (
b  =/=  (  _I  |`  B )  /\  ( R `  b )  =/=  ( R `  F
)  /\  ( R `  b )  =/=  ( R `  G )
) )  ->  b  e.  T )
37 simp31 992 . . . . . . . . 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 ) ) )  /\  b  e.  T  /\  (
b  =/=  (  _I  |`  B )  /\  ( R `  b )  =/=  ( R `  F
)  /\  ( R `  b )  =/=  ( R `  G )
) )  ->  b  =/=  (  _I  |`  B ) )
3836, 37jca 518 . . . . . . . 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 ) ) )  /\  b  e.  T  /\  (
b  =/=  (  _I  |`  B )  /\  ( R `  b )  =/=  ( R `  F
)  /\  ( R `  b )  =/=  ( R `  G )
) )  ->  (
b  e.  T  /\  b  =/=  (  _I  |`  B ) ) )
39 cdlemk5.l . . . . . . . . 9  |-  .<_  =  ( le `  K )
40 cdlemk5.j . . . . . . . . 9  |-  .\/  =  ( join `  K )
41 cdlemk5.m . . . . . . . . 9  |-  ./\  =  ( meet `  K )
42 cdlemk5.a . . . . . . . . 9  |-  A  =  ( Atoms `  K )
43 cdlemk5.r . . . . . . . . 9  |-  R  =  ( ( trL `  K
) `  W )
44 cdlemk5.z . . . . . . . . 9  |-  Z  =  ( ( P  .\/  ( R `  b ) )  ./\  ( ( N `  P )  .\/  ( R `  (
b  o.  `' F
) ) ) )
45 cdlemk5.y . . . . . . . . 9  |-  Y  =  ( ( P  .\/  ( R `  g ) )  ./\  ( Z  .\/  ( R `  (
g  o.  `' b ) ) ) )
462, 39, 40, 41, 42, 3, 4, 43, 44, 45cdlemkid2 31184 . . . . . . . 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 )  /\  ( b  e.  T  /\  b  =/=  (  _I  |`  B ) ) ) )  ->  [_ G  /  g ]_ Y  =  P )
4732, 33, 34, 35, 38, 46syl113anc 1195 . . . . . . 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 ) ) )  /\  b  e.  T  /\  (
b  =/=  (  _I  |`  B )  /\  ( R `  b )  =/=  ( R `  F
)  /\  ( R `  b )  =/=  ( R `  G )
) )  ->  [_ G  /  g ]_ Y  =  P )
48473expa 1152 . . . . . 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 ) ) )  /\  b  e.  T )  /\  (
b  =/=  (  _I  |`  B )  /\  ( R `  b )  =/=  ( R `  F
)  /\  ( R `  b )  =/=  ( R `  G )
) )  ->  [_ G  /  g ]_ Y  =  P )
4948eqeq2d 2377 . . . . 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 ) ) )  /\  b  e.  T )  /\  (
b  =/=  (  _I  |`  B )  /\  ( R `  b )  =/=  ( R `  F
)  /\  ( R `  b )  =/=  ( R `  G )
) )  ->  (
( z `  P
)  =  [_ G  /  g ]_ Y  <->  ( z `  P )  =  P ) )
5049pm5.74da 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 ) ) )  /\  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 `  P )  =  P ) ) )
5150ralbidva 2644 . . 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 )
) )  ->  ( 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 `  P )  =  P ) ) )
5251riotabidv 6448 . 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 `  P )  =  P ) ) )
5331, 52eqtrd 2398 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 `  P )  =  P ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 358    /\ w3a 935    = wceq 1647    e. wcel 1715    =/= wne 2529   A.wral 2628   [.wsbc 3077   [_csb 3167   class class class wbr 4125    _I cid 4407   `'ccnv 4791    |` cres 4794    o. ccom 4796   ` cfv 5358  (class class class)co 5981   iota_crio 6439   Basecbs 13356   lecple 13423   joincjn 14288   meetcmee 14289   Atomscatm 29524   HLchlt 29611   LHypclh 30244   LTrncltrn 30361   trLctrl 30418
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-13 1717  ax-14 1719  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347  ax-rep 4233  ax-sep 4243  ax-nul 4251  ax-pow 4290  ax-pr 4316  ax-un 4615
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 936  df-3an 937  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-eu 2221  df-mo 2222  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-ne 2531  df-nel 2532  df-ral 2633  df-rex 2634  df-reu 2635  df-rmo 2636  df-rab 2637  df-v 2875  df-sbc 3078  df-csb 3168  df-dif 3241  df-un 3243  df-in 3245  df-ss 3252  df-nul 3544  df-if 3655  df-pw 3716  df-sn 3735  df-pr 3736  df-op 3738  df-uni 3930  df-iun 4009  df-iin 4010  df-br 4126  df-opab 4180  df-mpt 4181  df-id 4412  df-xp 4798  df-rel 4799  df-cnv 4800  df-co 4801  df-dm 4802  df-rn 4803  df-res 4804  df-ima 4805  df-iota 5322  df-fun 5360  df-fn 5361  df-f 5362  df-f1 5363  df-fo 5364  df-f1o 5365  df-fv 5366  df-ov 5984  df-oprab 5985  df-mpt2 5986  df-1st 6249  df-2nd 6250  df-undef 6440  df-riota 6446  df-map 6917  df-poset 14290  df-plt 14302  df-lub 14318  df-glb 14319  df-join 14320  df-meet 14321  df-p0 14355  df-p1 14356  df-lat 14362  df-clat 14424  df-oposet 29437  df-ol 29439  df-oml 29440  df-covers 29527  df-ats 29528  df-atl 29559  df-cvlat 29583  df-hlat 29612  df-llines 29758  df-lplanes 29759  df-lvols 29760  df-lines 29761  df-psubsp 29763  df-pmap 29764  df-padd 30056  df-lhyp 30248  df-laut 30249  df-ldil 30364  df-ltrn 30365  df-trl 30419
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