MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  symquadlem Structured version   Unicode version

Theorem symquadlem 23081
Description: Lemma of the symetrial quadrilateral. The diagonals of quadrilaterals with congruent opposing sides intersect at their middle point. In Euclidean geometry, such quadrilaterals are called parallelograms, as opposing sides are parallel. However, this is not true in the case of absolute geometry. Lemma 7.21 of [Schwabhauser] p. 52. (Contributed by Thierry Arnoux, 6-Aug-2019.)
Hypotheses
Ref Expression
mirval.p  |-  P  =  ( Base `  G
)
mirval.d  |-  .-  =  ( dist `  G )
mirval.i  |-  I  =  (Itv `  G )
mirval.l  |-  L  =  (LineG `  G )
mirval.s  |-  S  =  (pInvG `  G )
mirval.g  |-  ( ph  ->  G  e. TarskiG )
symquadlem.m  |-  M  =  ( S `  X
)
symquadlem.a  |-  ( ph  ->  A  e.  P )
symquadlem.b  |-  ( ph  ->  B  e.  P )
symquadlem.c  |-  ( ph  ->  C  e.  P )
symquadlem.d  |-  ( ph  ->  D  e.  P )
symquadlem.x  |-  ( ph  ->  X  e.  P )
symquadlem.1  |-  ( ph  ->  -.  ( A  e.  ( B L C )  \/  B  =  C ) )
symquadlem.2  |-  ( ph  ->  B  =/=  D )
symquadlem.3  |-  ( ph  ->  ( A  .-  B
)  =  ( C 
.-  D ) )
symquadlem.4  |-  ( ph  ->  ( B  .-  C
)  =  ( D 
.-  A ) )
symquadlem.5  |-  ( ph  ->  ( X  e.  ( A L C )  \/  A  =  C ) )
symquadlem.6  |-  ( ph  ->  ( X  e.  ( B L D )  \/  B  =  D ) )
Assertion
Ref Expression
symquadlem  |-  ( ph  ->  A  =  ( M `
 C ) )

Proof of Theorem symquadlem
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 symquadlem.1 . . . . . . . 8  |-  ( ph  ->  -.  ( A  e.  ( B L C )  \/  B  =  C ) )
2 mirval.p . . . . . . . . . . 11  |-  P  =  ( Base `  G
)
3 mirval.l . . . . . . . . . . 11  |-  L  =  (LineG `  G )
4 mirval.i . . . . . . . . . . 11  |-  I  =  (Itv `  G )
5 mirval.g . . . . . . . . . . 11  |-  ( ph  ->  G  e. TarskiG )
6 symquadlem.b . . . . . . . . . . 11  |-  ( ph  ->  B  e.  P )
7 symquadlem.a . . . . . . . . . . 11  |-  ( ph  ->  A  e.  P )
8 mirval.d . . . . . . . . . . . 12  |-  .-  =  ( dist `  G )
92, 8, 4, 5, 6, 7tgbtwntriv2 22939 . . . . . . . . . . 11  |-  ( ph  ->  A  e.  ( B I A ) )
102, 3, 4, 5, 6, 7, 7, 9btwncolg1 22987 . . . . . . . . . 10  |-  ( ph  ->  ( A  e.  ( B L A )  \/  B  =  A ) )
1110adantr 465 . . . . . . . . 9  |-  ( (
ph  /\  A  =  C )  ->  ( A  e.  ( B L A )  \/  B  =  A ) )
12 simpr 461 . . . . . . . . . . . 12  |-  ( (
ph  /\  A  =  C )  ->  A  =  C )
1312oveq2d 6105 . . . . . . . . . . 11  |-  ( (
ph  /\  A  =  C )  ->  ( B L A )  =  ( B L C ) )
1413eleq2d 2508 . . . . . . . . . 10  |-  ( (
ph  /\  A  =  C )  ->  ( A  e.  ( B L A )  <->  A  e.  ( B L C ) ) )
1512eqeq2d 2452 . . . . . . . . . 10  |-  ( (
ph  /\  A  =  C )  ->  ( B  =  A  <->  B  =  C ) )
1614, 15orbi12d 709 . . . . . . . . 9  |-  ( (
ph  /\  A  =  C )  ->  (
( A  e.  ( B L A )  \/  B  =  A )  <->  ( A  e.  ( B L C )  \/  B  =  C ) ) )
1711, 16mpbid 210 . . . . . . . 8  |-  ( (
ph  /\  A  =  C )  ->  ( A  e.  ( B L C )  \/  B  =  C ) )
181, 17mtand 659 . . . . . . 7  |-  ( ph  ->  -.  A  =  C )
1918neneqad 2679 . . . . . 6  |-  ( ph  ->  A  =/=  C )
2019ad2antrr 725 . . . . 5  |-  ( ( ( ph  /\  x  e.  P )  /\  <" B D X "> (cgrG `  G ) <" D B x "> )  ->  A  =/=  C )
2120necomd 2693 . . . 4  |-  ( ( ( ph  /\  x  e.  P )  /\  <" B D X "> (cgrG `  G ) <" D B x "> )  ->  C  =/=  A )
2221neneqd 2622 . . 3  |-  ( ( ( ph  /\  x  e.  P )  /\  <" B D X "> (cgrG `  G ) <" D B x "> )  ->  -.  C  =  A
)
23 mirval.s . . . . . 6  |-  S  =  (pInvG `  G )
245ad2antrr 725 . . . . . 6  |-  ( ( ( ph  /\  x  e.  P )  /\  <" B D X "> (cgrG `  G ) <" D B x "> )  ->  G  e. TarskiG )
25 symquadlem.m . . . . . 6  |-  M  =  ( S `  X
)
26 symquadlem.c . . . . . . 7  |-  ( ph  ->  C  e.  P )
2726ad2antrr 725 . . . . . 6  |-  ( ( ( ph  /\  x  e.  P )  /\  <" B D X "> (cgrG `  G ) <" D B x "> )  ->  C  e.  P )
287ad2antrr 725 . . . . . 6  |-  ( ( ( ph  /\  x  e.  P )  /\  <" B D X "> (cgrG `  G ) <" D B x "> )  ->  A  e.  P )
29 symquadlem.x . . . . . . 7  |-  ( ph  ->  X  e.  P )
3029ad2antrr 725 . . . . . 6  |-  ( ( ( ph  /\  x  e.  P )  /\  <" B D X "> (cgrG `  G ) <" D B x "> )  ->  X  e.  P )
31 symquadlem.5 . . . . . . . 8  |-  ( ph  ->  ( X  e.  ( A L C )  \/  A  =  C ) )
3231ad2antrr 725 . . . . . . 7  |-  ( ( ( ph  /\  x  e.  P )  /\  <" B D X "> (cgrG `  G ) <" D B x "> )  -> 
( X  e.  ( A L C )  \/  A  =  C ) )
332, 3, 4, 24, 28, 27, 30, 32colcom 22990 . . . . . 6  |-  ( ( ( ph  /\  x  e.  P )  /\  <" B D X "> (cgrG `  G ) <" D B x "> )  -> 
( X  e.  ( C L A )  \/  C  =  A ) )
346ad2antrr 725 . . . . . . . . . 10  |-  ( ( ( ph  /\  x  e.  P )  /\  <" B D X "> (cgrG `  G ) <" D B x "> )  ->  B  e.  P )
35 symquadlem.d . . . . . . . . . . 11  |-  ( ph  ->  D  e.  P )
3635ad2antrr 725 . . . . . . . . . 10  |-  ( ( ( ph  /\  x  e.  P )  /\  <" B D X "> (cgrG `  G ) <" D B x "> )  ->  D  e.  P )
375adantr 465 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  ( A  e.  ( C L B )  \/  C  =  B ) )  ->  G  e. TarskiG )
3826adantr 465 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  ( A  e.  ( C L B )  \/  C  =  B ) )  ->  C  e.  P )
396adantr 465 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  ( A  e.  ( C L B )  \/  C  =  B ) )  ->  B  e.  P )
407adantr 465 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  ( A  e.  ( C L B )  \/  C  =  B ) )  ->  A  e.  P )
41 simpr 461 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  ( A  e.  ( C L B )  \/  C  =  B ) )  -> 
( A  e.  ( C L B )  \/  C  =  B ) )
422, 3, 4, 37, 38, 39, 40, 41colcom 22990 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  ( A  e.  ( C L B )  \/  C  =  B ) )  -> 
( A  e.  ( B L C )  \/  B  =  C ) )
4342ex 434 . . . . . . . . . . . . 13  |-  ( ph  ->  ( ( A  e.  ( C L B )  \/  C  =  B )  ->  ( A  e.  ( B L C )  \/  B  =  C ) ) )
4443con3d 133 . . . . . . . . . . . 12  |-  ( ph  ->  ( -.  ( A  e.  ( B L C )  \/  B  =  C )  ->  -.  ( A  e.  ( C L B )  \/  C  =  B ) ) )
451, 44mpd 15 . . . . . . . . . . 11  |-  ( ph  ->  -.  ( A  e.  ( C L B )  \/  C  =  B ) )
4645ad2antrr 725 . . . . . . . . . 10  |-  ( ( ( ph  /\  x  e.  P )  /\  <" B D X "> (cgrG `  G ) <" D B x "> )  ->  -.  ( A  e.  ( C L B )  \/  C  =  B ) )
4731orcomd 388 . . . . . . . . . . . . 13  |-  ( ph  ->  ( A  =  C  \/  X  e.  ( A L C ) ) )
4847ord 377 . . . . . . . . . . . 12  |-  ( ph  ->  ( -.  A  =  C  ->  X  e.  ( A L C ) ) )
4918, 48mpd 15 . . . . . . . . . . 11  |-  ( ph  ->  X  e.  ( A L C ) )
5049ad2antrr 725 . . . . . . . . . 10  |-  ( ( ( ph  /\  x  e.  P )  /\  <" B D X "> (cgrG `  G ) <" D B x "> )  ->  X  e.  ( A L C ) )
5118ad2antrr 725 . . . . . . . . . . 11  |-  ( ( ( ph  /\  x  e.  P )  /\  <" B D X "> (cgrG `  G ) <" D B x "> )  ->  -.  A  =  C
)
52 simplr 754 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  x  e.  P )  /\  <" B D X "> (cgrG `  G ) <" D B x "> )  ->  x  e.  P )
53 eqid 2441 . . . . . . . . . . . . . . 15  |-  (cgrG `  G )  =  (cgrG `  G )
54 symquadlem.6 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ph  ->  ( X  e.  ( B L D )  \/  B  =  D ) )
552, 3, 4, 5, 6, 35, 29, 54colrot2 22992 . . . . . . . . . . . . . . . . . . . 20  |-  ( ph  ->  ( D  e.  ( X L B )  \/  X  =  B ) )
562, 3, 4, 5, 29, 6, 35, 55colcom 22990 . . . . . . . . . . . . . . . . . . 19  |-  ( ph  ->  ( D  e.  ( B L X )  \/  B  =  X ) )
5756ad2antrr 725 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ph  /\  x  e.  P )  /\  <" B D X "> (cgrG `  G ) <" D B x "> )  -> 
( D  e.  ( B L X )  \/  B  =  X ) )
58 simpr 461 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ph  /\  x  e.  P )  /\  <" B D X "> (cgrG `  G ) <" D B x "> )  ->  <" B D X "> (cgrG `  G ) <" D B x "> )
59 symquadlem.3 . . . . . . . . . . . . . . . . . . . 20  |-  ( ph  ->  ( A  .-  B
)  =  ( C 
.-  D ) )
602, 8, 4, 5, 7, 6, 26, 35, 59tgcgrcomlr 22932 . . . . . . . . . . . . . . . . . . 19  |-  ( ph  ->  ( B  .-  A
)  =  ( D 
.-  C ) )
6160ad2antrr 725 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ph  /\  x  e.  P )  /\  <" B D X "> (cgrG `  G ) <" D B x "> )  -> 
( B  .-  A
)  =  ( D 
.-  C ) )
62 symquadlem.4 . . . . . . . . . . . . . . . . . . . 20  |-  ( ph  ->  ( B  .-  C
)  =  ( D 
.-  A ) )
6362ad2antrr 725 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( ph  /\  x  e.  P )  /\  <" B D X "> (cgrG `  G ) <" D B x "> )  -> 
( B  .-  C
)  =  ( D 
.-  A ) )
6463eqcomd 2446 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ph  /\  x  e.  P )  /\  <" B D X "> (cgrG `  G ) <" D B x "> )  -> 
( D  .-  A
)  =  ( B 
.-  C ) )
65 symquadlem.2 . . . . . . . . . . . . . . . . . . 19  |-  ( ph  ->  B  =/=  D )
6665ad2antrr 725 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ph  /\  x  e.  P )  /\  <" B D X "> (cgrG `  G ) <" D B x "> )  ->  B  =/=  D )
672, 3, 4, 24, 34, 36, 30, 53, 36, 34, 8, 28, 52, 27, 57, 58, 61, 64, 66tgfscgr 22998 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ph  /\  x  e.  P )  /\  <" B D X "> (cgrG `  G ) <" D B x "> )  -> 
( X  .-  A
)  =  ( x 
.-  C ) )
682, 8, 4, 24, 30, 28, 52, 27, 67tgcgrcomlr 22932 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  x  e.  P )  /\  <" B D X "> (cgrG `  G ) <" D B x "> )  -> 
( A  .-  X
)  =  ( C 
.-  x ) )
6961eqcomd 2446 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ph  /\  x  e.  P )  /\  <" B D X "> (cgrG `  G ) <" D B x "> )  -> 
( D  .-  C
)  =  ( B 
.-  A ) )
702, 3, 4, 24, 34, 36, 30, 53, 36, 34, 8, 27, 52, 28, 57, 58, 63, 69, 66tgfscgr 22998 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  x  e.  P )  /\  <" B D X "> (cgrG `  G ) <" D B x "> )  -> 
( X  .-  C
)  =  ( x 
.-  A ) )
712, 8, 4, 24, 27, 28axtgcgrrflx 22921 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  x  e.  P )  /\  <" B D X "> (cgrG `  G ) <" D B x "> )  -> 
( C  .-  A
)  =  ( A 
.-  C ) )
722, 8, 53, 24, 28, 30, 27, 27, 52, 28, 68, 70, 71trgcgr 22966 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  x  e.  P )  /\  <" B D X "> (cgrG `  G ) <" D B x "> )  ->  <" A X C "> (cgrG `  G ) <" C x A "> )
732, 3, 4, 24, 28, 30, 27, 53, 27, 52, 28, 32, 72lnxfr 22996 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  x  e.  P )  /\  <" B D X "> (cgrG `  G ) <" D B x "> )  -> 
( x  e.  ( C L A )  \/  C  =  A ) )
742, 3, 4, 24, 27, 28, 52, 73colcom 22990 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  x  e.  P )  /\  <" B D X "> (cgrG `  G ) <" D B x "> )  -> 
( x  e.  ( A L C )  \/  A  =  C ) )
7574orcomd 388 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  x  e.  P )  /\  <" B D X "> (cgrG `  G ) <" D B x "> )  -> 
( A  =  C  \/  x  e.  ( A L C ) ) )
7675ord 377 . . . . . . . . . . 11  |-  ( ( ( ph  /\  x  e.  P )  /\  <" B D X "> (cgrG `  G ) <" D B x "> )  -> 
( -.  A  =  C  ->  x  e.  ( A L C ) ) )
7751, 76mpd 15 . . . . . . . . . 10  |-  ( ( ( ph  /\  x  e.  P )  /\  <" B D X "> (cgrG `  G ) <" D B x "> )  ->  x  e.  ( A L C ) )
7865neneqd 2622 . . . . . . . . . . . 12  |-  ( ph  ->  -.  B  =  D )
7954orcomd 388 . . . . . . . . . . . . 13  |-  ( ph  ->  ( B  =  D  \/  X  e.  ( B L D ) ) )
8079ord 377 . . . . . . . . . . . 12  |-  ( ph  ->  ( -.  B  =  D  ->  X  e.  ( B L D ) ) )
8178, 80mpd 15 . . . . . . . . . . 11  |-  ( ph  ->  X  e.  ( B L D ) )
8281ad2antrr 725 . . . . . . . . . 10  |-  ( ( ( ph  /\  x  e.  P )  /\  <" B D X "> (cgrG `  G ) <" D B x "> )  ->  X  e.  ( B L D ) )
8378ad2antrr 725 . . . . . . . . . . 11  |-  ( ( ( ph  /\  x  e.  P )  /\  <" B D X "> (cgrG `  G ) <" D B x "> )  ->  -.  B  =  D
)
8454ad2antrr 725 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  x  e.  P )  /\  <" B D X "> (cgrG `  G ) <" D B x "> )  -> 
( X  e.  ( B L D )  \/  B  =  D ) )
852, 8, 4, 53, 24, 34, 36, 30, 36, 34, 52, 58cgr3swap23 22974 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  x  e.  P )  /\  <" B D X "> (cgrG `  G ) <" D B x "> )  ->  <" B X D "> (cgrG `  G ) <" D x B "> )
862, 3, 4, 24, 34, 30, 36, 53, 36, 52, 34, 84, 85lnxfr 22996 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  x  e.  P )  /\  <" B D X "> (cgrG `  G ) <" D B x "> )  -> 
( x  e.  ( D L B )  \/  D  =  B ) )
872, 3, 4, 24, 36, 34, 52, 86colcom 22990 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  x  e.  P )  /\  <" B D X "> (cgrG `  G ) <" D B x "> )  -> 
( x  e.  ( B L D )  \/  B  =  D ) )
8887orcomd 388 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  x  e.  P )  /\  <" B D X "> (cgrG `  G ) <" D B x "> )  -> 
( B  =  D  \/  x  e.  ( B L D ) ) )
8988ord 377 . . . . . . . . . . 11  |-  ( ( ( ph  /\  x  e.  P )  /\  <" B D X "> (cgrG `  G ) <" D B x "> )  -> 
( -.  B  =  D  ->  x  e.  ( B L D ) ) )
9083, 89mpd 15 . . . . . . . . . 10  |-  ( ( ( ph  /\  x  e.  P )  /\  <" B D X "> (cgrG `  G ) <" D B x "> )  ->  x  e.  ( B L D ) )
912, 4, 3, 24, 28, 27, 34, 36, 46, 50, 77, 82, 90tglineinteq 23048 . . . . . . . . 9  |-  ( ( ( ph  /\  x  e.  P )  /\  <" B D X "> (cgrG `  G ) <" D B x "> )  ->  X  =  x )
9291oveq1d 6104 . . . . . . . 8  |-  ( ( ( ph  /\  x  e.  P )  /\  <" B D X "> (cgrG `  G ) <" D B x "> )  -> 
( X  .-  A
)  =  ( x 
.-  A ) )
9392, 70eqtr4d 2476 . . . . . . 7  |-  ( ( ( ph  /\  x  e.  P )  /\  <" B D X "> (cgrG `  G ) <" D B x "> )  -> 
( X  .-  A
)  =  ( X 
.-  C ) )
9493eqcomd 2446 . . . . . 6  |-  ( ( ( ph  /\  x  e.  P )  /\  <" B D X "> (cgrG `  G ) <" D B x "> )  -> 
( X  .-  C
)  =  ( X 
.-  A ) )
952, 8, 4, 3, 23, 24, 25, 27, 28, 30, 33, 94colmid 23080 . . . . 5  |-  ( ( ( ph  /\  x  e.  P )  /\  <" B D X "> (cgrG `  G ) <" D B x "> )  -> 
( A  =  ( M `  C )  \/  C  =  A ) )
9695orcomd 388 . . . 4  |-  ( ( ( ph  /\  x  e.  P )  /\  <" B D X "> (cgrG `  G ) <" D B x "> )  -> 
( C  =  A  \/  A  =  ( M `  C ) ) )
9796ord 377 . . 3  |-  ( ( ( ph  /\  x  e.  P )  /\  <" B D X "> (cgrG `  G ) <" D B x "> )  -> 
( -.  C  =  A  ->  A  =  ( M `  C ) ) )
9822, 97mpd 15 . 2  |-  ( ( ( ph  /\  x  e.  P )  /\  <" B D X "> (cgrG `  G ) <" D B x "> )  ->  A  =  ( M `  C ) )
992, 8, 4, 5, 6, 35axtgcgrrflx 22921 . . 3  |-  ( ph  ->  ( B  .-  D
)  =  ( D 
.-  B ) )
1002, 3, 4, 5, 6, 35, 29, 53, 35, 6, 8, 56, 99lnext 22997 . 2  |-  ( ph  ->  E. x  e.  P  <" B D X "> (cgrG `  G ) <" D B x "> )
10198, 100r19.29a 2860 1  |-  ( ph  ->  A  =  ( M `
 C ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    \/ wo 368    /\ wa 369    = wceq 1369    e. wcel 1756    =/= wne 2604   class class class wbr 4290   ` cfv 5416  (class class class)co 6089   <"cs3 12467   Basecbs 14172   distcds 14245  TarskiGcstrkg 22887  Itvcitv 22895  LineGclng 22896  cgrGccgrg 22961  pInvGcmir 23053
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2422  ax-rep 4401  ax-sep 4411  ax-nul 4419  ax-pow 4468  ax-pr 4529  ax-un 6370  ax-cnex 9336  ax-resscn 9337  ax-1cn 9338  ax-icn 9339  ax-addcl 9340  ax-addrcl 9341  ax-mulcl 9342  ax-mulrcl 9343  ax-mulcom 9344  ax-addass 9345  ax-mulass 9346  ax-distr 9347  ax-i2m1 9348  ax-1ne0 9349  ax-1rid 9350  ax-rnegex 9351  ax-rrecex 9352  ax-cnre 9353  ax-pre-lttri 9354  ax-pre-lttrn 9355  ax-pre-ltadd 9356  ax-pre-mulgt0 9357
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rmo 2721  df-rab 2722  df-v 2972  df-sbc 3185  df-csb 3287  df-dif 3329  df-un 3331  df-in 3333  df-ss 3340  df-pss 3342  df-nul 3636  df-if 3790  df-pw 3860  df-sn 3876  df-pr 3878  df-tp 3880  df-op 3882  df-uni 4090  df-int 4127  df-iun 4171  df-br 4291  df-opab 4349  df-mpt 4350  df-tr 4384  df-eprel 4630  df-id 4634  df-po 4639  df-so 4640  df-fr 4677  df-we 4679  df-ord 4720  df-on 4721  df-lim 4722  df-suc 4723  df-xp 4844  df-rel 4845  df-cnv 4846  df-co 4847  df-dm 4848  df-rn 4849  df-res 4850  df-ima 4851  df-iota 5379  df-fun 5418  df-fn 5419  df-f 5420  df-f1 5421  df-fo 5422  df-f1o 5423  df-fv 5424  df-riota 6050  df-ov 6092  df-oprab 6093  df-mpt2 6094  df-om 6475  df-1st 6575  df-2nd 6576  df-recs 6830  df-rdg 6864  df-1o 6918  df-oadd 6922  df-er 7099  df-pm 7215  df-en 7309  df-dom 7310  df-sdom 7311  df-fin 7312  df-card 8107  df-cda 8335  df-pnf 9418  df-mnf 9419  df-xr 9420  df-ltxr 9421  df-le 9422  df-sub 9595  df-neg 9596  df-nn 10321  df-2 10378  df-3 10379  df-n0 10578  df-z 10645  df-uz 10860  df-fz 11436  df-fzo 11547  df-hash 12102  df-word 12227  df-concat 12229  df-s1 12230  df-s2 12473  df-s3 12474  df-trkgc 22907  df-trkgb 22908  df-trkgcb 22909  df-trkg 22914  df-cgrg 22962  df-mir 23054
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator