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Theorem cdlemefrs32fva 31211
Description: Part of proof of Lemma E in [Crawley] p. 113. Value of  F at an atom not under  W. TODO: FIX COMMENT TODO: consolidate uses of lhpmat 30841 here and elsewhere, and presence/absence of  s 
.<_  ( P  .\/  Q
) term. Also, why can proof be shortened with cdleme29cl 31188? What is difference from cdlemefs27cl 31224? (Contributed by NM, 29-Mar-2013.)
Hypotheses
Ref Expression
cdlemefrs27.b  |-  B  =  ( Base `  K
)
cdlemefrs27.l  |-  .<_  =  ( le `  K )
cdlemefrs27.j  |-  .\/  =  ( join `  K )
cdlemefrs27.m  |-  ./\  =  ( meet `  K )
cdlemefrs27.a  |-  A  =  ( Atoms `  K )
cdlemefrs27.h  |-  H  =  ( LHyp `  K
)
cdlemefrs27.eq  |-  ( s  =  R  ->  ( ph 
<->  ps ) )
cdlemefrs27.nb  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  P  =/=  Q  /\  (
s  e.  A  /\  ( -.  s  .<_  W  /\  ph ) ) )  ->  N  e.  B )
cdlemefrs27.rnb  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  ps )  ->  [_ R  /  s ]_ N  e.  B
)
cdleme29frs.o  |-  O  =  ( iota_ z  e.  B A. s  e.  A  ( ( -.  s  .<_  W  /\  ( s 
.\/  ( x  ./\  W ) )  =  x )  ->  z  =  ( N  .\/  ( x 
./\  W ) ) ) )
Assertion
Ref Expression
cdlemefrs32fva  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  ps )  ->  [_ R  /  x ]_ O  =  [_ R  /  s ]_ N
)
Distinct variable groups:    z, s, A    H, s    .\/ , s    K, s    .<_ , s    P, s    Q, s    R, s    W, s    ps, s    z, A    z, B    z, H    z, K    z, 
.<_    z, N    z, P    z, Q    z, R    z, W    ps, z    B, s   
z,  .\/    ./\ , s, z    ph, z    x, z, A   
x, B    x,  .\/    x, 
.<_    x,  ./\    x, N    x, s, R    x, W
Allowed substitution hints:    ph( x, s)    ps( x)    P( x)    Q( x)    H( x)    K( x)    N( s)    O( x, z, s)

Proof of Theorem cdlemefrs32fva
StepHypRef Expression
1 simp2rl 1024 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  ps )  ->  R  e.  A )
2 cdlemefrs27.b . . . 4  |-  B  =  ( Base `  K
)
3 cdlemefrs27.a . . . 4  |-  A  =  ( Atoms `  K )
42, 3atbase 30101 . . 3  |-  ( R  e.  A  ->  R  e.  B )
5 cdleme29frs.o . . . 4  |-  O  =  ( iota_ z  e.  B A. s  e.  A  ( ( -.  s  .<_  W  /\  ( s 
.\/  ( x  ./\  W ) )  =  x )  ->  z  =  ( N  .\/  ( x 
./\  W ) ) ) )
6 eqid 2296 . . . 4  |-  ( iota_ z  e.  B A. s  e.  A  ( ( -.  s  .<_  W  /\  ( s  .\/  ( R  ./\  W ) )  =  R )  -> 
z  =  ( N 
.\/  ( R  ./\  W ) ) ) )  =  ( iota_ z  e.  B A. s  e.  A  ( ( -.  s  .<_  W  /\  ( s  .\/  ( R  ./\  W ) )  =  R )  -> 
z  =  ( N 
.\/  ( R  ./\  W ) ) ) )
75, 6cdleme31so 31190 . . 3  |-  ( R  e.  B  ->  [_ R  /  x ]_ O  =  ( iota_ z  e.  B A. s  e.  A  ( ( -.  s  .<_  W  /\  ( s 
.\/  ( R  ./\  W ) )  =  R )  ->  z  =  ( N  .\/  ( R 
./\  W ) ) ) ) )
81, 4, 73syl 18 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  ps )  ->  [_ R  /  x ]_ O  =  ( iota_ z  e.  B A. s  e.  A  (
( -.  s  .<_  W  /\  ( s  .\/  ( R  ./\  W ) )  =  R )  ->  z  =  ( N  .\/  ( R 
./\  W ) ) ) ) )
9 ssid 3210 . . . 4  |-  B  C_  B
109a1i 10 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  ps )  ->  B  C_  B )
11 simpll 730 . . . . . . . 8  |-  ( ( ( -.  s  .<_  W  /\  ph )  /\  ( s  .\/  ( R  ./\  W ) )  =  R )  ->  -.  s  .<_  W )
12 simpr 447 . . . . . . . 8  |-  ( ( ( -.  s  .<_  W  /\  ph )  /\  ( s  .\/  ( R  ./\  W ) )  =  R )  -> 
( s  .\/  ( R  ./\  W ) )  =  R )
1311, 12jca 518 . . . . . . 7  |-  ( ( ( -.  s  .<_  W  /\  ph )  /\  ( s  .\/  ( R  ./\  W ) )  =  R )  -> 
( -.  s  .<_  W  /\  ( s  .\/  ( R  ./\  W ) )  =  R ) )
1413imim1i 54 . . . . . 6  |-  ( ( ( -.  s  .<_  W  /\  ( s  .\/  ( R  ./\  W ) )  =  R )  ->  z  =  ( N  .\/  ( R 
./\  W ) ) )  ->  ( (
( -.  s  .<_  W  /\  ph )  /\  ( s  .\/  ( R  ./\  W ) )  =  R )  -> 
z  =  ( N 
.\/  ( R  ./\  W ) ) ) )
1514ralimi 2631 . . . . 5  |-  ( A. s  e.  A  (
( -.  s  .<_  W  /\  ( s  .\/  ( R  ./\  W ) )  =  R )  ->  z  =  ( N  .\/  ( R 
./\  W ) ) )  ->  A. s  e.  A  ( (
( -.  s  .<_  W  /\  ph )  /\  ( s  .\/  ( R  ./\  W ) )  =  R )  -> 
z  =  ( N 
.\/  ( R  ./\  W ) ) ) )
1615rgenw 2623 . . . 4  |-  A. z  e.  B  ( A. s  e.  A  (
( -.  s  .<_  W  /\  ( s  .\/  ( R  ./\  W ) )  =  R )  ->  z  =  ( N  .\/  ( R 
./\  W ) ) )  ->  A. s  e.  A  ( (
( -.  s  .<_  W  /\  ph )  /\  ( s  .\/  ( R  ./\  W ) )  =  R )  -> 
z  =  ( N 
.\/  ( R  ./\  W ) ) ) )
1716a1i 10 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  ps )  ->  A. z  e.  B  ( A. s  e.  A  ( ( -.  s  .<_  W  /\  ( s 
.\/  ( R  ./\  W ) )  =  R )  ->  z  =  ( N  .\/  ( R 
./\  W ) ) )  ->  A. s  e.  A  ( (
( -.  s  .<_  W  /\  ph )  /\  ( s  .\/  ( R  ./\  W ) )  =  R )  -> 
z  =  ( N 
.\/  ( R  ./\  W ) ) ) ) )
18 cdlemefrs27.l . . . . 5  |-  .<_  =  ( le `  K )
19 cdlemefrs27.j . . . . 5  |-  .\/  =  ( join `  K )
20 cdlemefrs27.m . . . . 5  |-  ./\  =  ( meet `  K )
21 cdlemefrs27.h . . . . 5  |-  H  =  ( LHyp `  K
)
22 cdlemefrs27.eq . . . . 5  |-  ( s  =  R  ->  ( ph 
<->  ps ) )
23 cdlemefrs27.nb . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  P  =/=  Q  /\  (
s  e.  A  /\  ( -.  s  .<_  W  /\  ph ) ) )  ->  N  e.  B )
24 cdlemefrs27.rnb . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  ps )  ->  [_ R  /  s ]_ N  e.  B
)
252, 18, 19, 20, 3, 21, 22, 23, 24cdlemefrs29bpre1 31208 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  ps )  ->  E. z  e.  B  A. s  e.  A  ( ( ( -.  s  .<_  W  /\  ph )  /\  ( s 
.\/  ( R  ./\  W ) )  =  R )  ->  z  =  ( N  .\/  ( R 
./\  W ) ) ) )
26 simpl11 1030 . . . . . . . 8  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  ps )  /\  s  e.  A
)  ->  ( K  e.  HL  /\  W  e.  H ) )
27 simpl2r 1009 . . . . . . . 8  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  ps )  /\  s  e.  A
)  ->  ( R  e.  A  /\  -.  R  .<_  W ) )
28 simpl3 960 . . . . . . . 8  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  ps )  /\  s  e.  A
)  ->  ps )
29 simpr 447 . . . . . . . 8  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  ps )  /\  s  e.  A
)  ->  s  e.  A )
302, 18, 19, 20, 3, 21, 22cdlemefrs29pre00 31206 . . . . . . . 8  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R  e.  A  /\  -.  R  .<_  W )  /\  ps )  /\  s  e.  A )  ->  ( ( ( -.  s  .<_  W  /\  ph )  /\  ( s 
.\/  ( R  ./\  W ) )  =  R )  <->  ( -.  s  .<_  W  /\  ( s 
.\/  ( R  ./\  W ) )  =  R ) ) )
3126, 27, 28, 29, 30syl31anc 1185 . . . . . . 7  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  ps )  /\  s  e.  A
)  ->  ( (
( -.  s  .<_  W  /\  ph )  /\  ( s  .\/  ( R  ./\  W ) )  =  R )  <->  ( -.  s  .<_  W  /\  (
s  .\/  ( R  ./\ 
W ) )  =  R ) ) )
3231imbi1d 308 . . . . . 6  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  ps )  /\  s  e.  A
)  ->  ( (
( ( -.  s  .<_  W  /\  ph )  /\  ( s  .\/  ( R  ./\  W ) )  =  R )  -> 
z  =  ( N 
.\/  ( R  ./\  W ) ) )  <->  ( ( -.  s  .<_  W  /\  ( s  .\/  ( R  ./\  W ) )  =  R )  -> 
z  =  ( N 
.\/  ( R  ./\  W ) ) ) ) )
3332ralbidva 2572 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  ps )  ->  ( A. s  e.  A  ( ( ( -.  s  .<_  W  /\  ph )  /\  ( s 
.\/  ( R  ./\  W ) )  =  R )  ->  z  =  ( N  .\/  ( R 
./\  W ) ) )  <->  A. s  e.  A  ( ( -.  s  .<_  W  /\  ( s 
.\/  ( R  ./\  W ) )  =  R )  ->  z  =  ( N  .\/  ( R 
./\  W ) ) ) ) )
3433rexbidv 2577 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  ps )  ->  ( E. z  e.  B  A. s  e.  A  ( ( ( -.  s  .<_  W  /\  ph )  /\  ( s 
.\/  ( R  ./\  W ) )  =  R )  ->  z  =  ( N  .\/  ( R 
./\  W ) ) )  <->  E. z  e.  B  A. s  e.  A  ( ( -.  s  .<_  W  /\  ( s 
.\/  ( R  ./\  W ) )  =  R )  ->  z  =  ( N  .\/  ( R 
./\  W ) ) ) ) )
3525, 34mpbid 201 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  ps )  ->  E. z  e.  B  A. s  e.  A  ( ( -.  s  .<_  W  /\  ( s 
.\/  ( R  ./\  W ) )  =  R )  ->  z  =  ( N  .\/  ( R 
./\  W ) ) ) )
362, 18, 19, 20, 3, 21, 22, 23, 24cdlemefrs29cpre1 31209 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  ps )  ->  E! z  e.  B  A. s  e.  A  ( ( ( -.  s  .<_  W  /\  ph )  /\  ( s 
.\/  ( R  ./\  W ) )  =  R )  ->  z  =  ( N  .\/  ( R 
./\  W ) ) ) )
37 riotass2 6348 . . 3  |-  ( ( ( B  C_  B  /\  A. z  e.  B  ( A. s  e.  A  ( ( -.  s  .<_  W  /\  ( s 
.\/  ( R  ./\  W ) )  =  R )  ->  z  =  ( N  .\/  ( R 
./\  W ) ) )  ->  A. s  e.  A  ( (
( -.  s  .<_  W  /\  ph )  /\  ( s  .\/  ( R  ./\  W ) )  =  R )  -> 
z  =  ( N 
.\/  ( R  ./\  W ) ) ) ) )  /\  ( E. z  e.  B  A. s  e.  A  (
( -.  s  .<_  W  /\  ( s  .\/  ( R  ./\  W ) )  =  R )  ->  z  =  ( N  .\/  ( R 
./\  W ) ) )  /\  E! z  e.  B  A. s  e.  A  ( (
( -.  s  .<_  W  /\  ph )  /\  ( s  .\/  ( R  ./\  W ) )  =  R )  -> 
z  =  ( N 
.\/  ( R  ./\  W ) ) ) ) )  ->  ( iota_ z  e.  B A. s  e.  A  ( ( -.  s  .<_  W  /\  ( s  .\/  ( R  ./\  W ) )  =  R )  -> 
z  =  ( N 
.\/  ( R  ./\  W ) ) ) )  =  ( iota_ z  e.  B A. s  e.  A  ( ( ( -.  s  .<_  W  /\  ph )  /\  ( s 
.\/  ( R  ./\  W ) )  =  R )  ->  z  =  ( N  .\/  ( R 
./\  W ) ) ) ) )
3810, 17, 35, 36, 37syl22anc 1183 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  ps )  ->  ( iota_ z  e.  B A. s  e.  A  ( ( -.  s  .<_  W  /\  ( s 
.\/  ( R  ./\  W ) )  =  R )  ->  z  =  ( N  .\/  ( R 
./\  W ) ) ) )  =  (
iota_ z  e.  B A. s  e.  A  ( ( ( -.  s  .<_  W  /\  ph )  /\  ( s 
.\/  ( R  ./\  W ) )  =  R )  ->  z  =  ( N  .\/  ( R 
./\  W ) ) ) ) )
392, 18, 19, 20, 3, 21, 22, 23cdlemefrs29bpre0 31207 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  ps )  ->  ( A. s  e.  A  ( ( ( -.  s  .<_  W  /\  ph )  /\  ( s 
.\/  ( R  ./\  W ) )  =  R )  ->  z  =  ( N  .\/  ( R 
./\  W ) ) )  <->  z  =  [_ R  /  s ]_ N
) )
40393ad2ant1 976 . . . 4  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  ps )  /\  [_ R  /  s ]_ N  e.  B  /\  z  e.  B
)  ->  ( A. s  e.  A  (
( ( -.  s  .<_  W  /\  ph )  /\  ( s  .\/  ( R  ./\  W ) )  =  R )  -> 
z  =  ( N 
.\/  ( R  ./\  W ) ) )  <->  z  =  [_ R  /  s ]_ N ) )
4140riota5OLD 6347 . . 3  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  ps )  /\  [_ R  /  s ]_ N  e.  B
)  ->  ( iota_ z  e.  B A. s  e.  A  ( (
( -.  s  .<_  W  /\  ph )  /\  ( s  .\/  ( R  ./\  W ) )  =  R )  -> 
z  =  ( N 
.\/  ( R  ./\  W ) ) ) )  =  [_ R  / 
s ]_ N )
4224, 41mpdan 649 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  ps )  ->  ( iota_ z  e.  B A. s  e.  A  ( ( ( -.  s  .<_  W  /\  ph )  /\  ( s 
.\/  ( R  ./\  W ) )  =  R )  ->  z  =  ( N  .\/  ( R 
./\  W ) ) ) )  =  [_ R  /  s ]_ N
)
438, 38, 423eqtrd 2332 1  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  ps )  ->  [_ R  /  x ]_ O  =  [_ R  /  s ]_ N
)
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696    =/= wne 2459   A.wral 2556   E.wrex 2557   E!wreu 2558   [_csb 3094    C_ wss 3165   class class class wbr 4039   ` cfv 5271  (class class class)co 5874   iota_crio 6313   Basecbs 13164   lecple 13231   joincjn 14094   meetcmee 14095   Atomscatm 30075   HLchlt 30162   LHypclh 30795
This theorem is referenced by:  cdlemefrs32fva1  31212  cdlemefr32fvaN  31220  cdlemefs32fvaN  31233
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-undef 6314  df-riota 6320  df-poset 14096  df-plt 14108  df-lub 14124  df-glb 14125  df-join 14126  df-meet 14127  df-p0 14161  df-p1 14162  df-lat 14168  df-clat 14230  df-oposet 29988  df-ol 29990  df-oml 29991  df-covers 30078  df-ats 30079  df-atl 30110  df-cvlat 30134  df-hlat 30163  df-lhyp 30799
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