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Theorem symquadlem 23767
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 23599 . . . . . . . . . . 11  |-  ( ph  ->  A  e.  ( B I A ) )
102, 3, 4, 5, 6, 7, 7, 9btwncolg1 23663 . . . . . . . . . 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 6291 . . . . . . . . . . 11  |-  ( (
ph  /\  A  =  C )  ->  ( B L A )  =  ( B L C ) )
1413eleq2d 2530 . . . . . . . . . 10  |-  ( (
ph  /\  A  =  C )  ->  ( A  e.  ( B L A )  <->  A  e.  ( B L C ) ) )
1512eqeq2d 2474 . . . . . . . . . 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 )
1918neqned 2663 . . . . . 6  |-  ( ph  ->  A  =/=  C )
2019ad2antrr 725 . . . . 5  |-  ( ( ( ph  /\  x  e.  P )  /\  <" B D X "> (cgrG `  G ) <" D B x "> )  ->  A  =/=  C )
2120necomd 2731 . . . 4  |-  ( ( ( ph  /\  x  e.  P )  /\  <" B D X "> (cgrG `  G ) <" D B x "> )  ->  C  =/=  A )
2221neneqd 2662 . . 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 23666 . . . . . 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 23666 . . . . . . . . . . . . . 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 2460 . . . . . . . . . . . . . . 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 23668 . . . . . . . . . . . . . . . . . . . 20  |-  ( ph  ->  ( D  e.  ( X L B )  \/  X  =  B ) )
562, 3, 4, 5, 29, 6, 35, 55colcom 23666 . . . . . . . . . . . . . . . . . . 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 23592 . . . . . . . . . . . . . . . . . . 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 2468 . . . . . . . . . . . . . . . . . 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 23675 . . . . . . . . . . . . . . . . 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 23592 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  x  e.  P )  /\  <" B D X "> (cgrG `  G ) <" D B x "> )  -> 
( A  .-  X
)  =  ( C 
.-  x ) )
6961eqcomd 2468 . . . . . . . . . . . . . . . . 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 23675 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  x  e.  P )  /\  <" B D X "> (cgrG `  G ) <" D B x "> )  -> 
( X  .-  C
)  =  ( x 
.-  A ) )
712, 8, 4, 24, 27, 28axtgcgrrflx 23580 . . . . . . . . . . . . . . . 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 23628 . . . . . . . . . . . . . . 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 23673 . . . . . . . . . . . . . 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 23666 . . . . . . . . . . . . 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 2662 . . . . . . . . . . . 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 23636 . . . . . . . . . . . . . . 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 23673 . . . . . . . . . . . . . 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 23666 . . . . . . . . . . . . 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 23731 . . . . . . . . 9  |-  ( ( ( ph  /\  x  e.  P )  /\  <" B D X "> (cgrG `  G ) <" D B x "> )  ->  X  =  x )
9291oveq1d 6290 . . . . . . . 8  |-  ( ( ( ph  /\  x  e.  P )  /\  <" B D X "> (cgrG `  G ) <" D B x "> )  -> 
( X  .-  A
)  =  ( x 
.-  A ) )
9392, 70eqtr4d 2504 . . . . . . 7  |-  ( ( ( ph  /\  x  e.  P )  /\  <" B D X "> (cgrG `  G ) <" D B x "> )  -> 
( X  .-  A
)  =  ( X 
.-  C ) )
9493eqcomd 2468 . . . . . 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 23766 . . . . 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 23580 . . 3  |-  ( ph  ->  ( B  .-  D
)  =  ( D 
.-  B ) )
1002, 3, 4, 5, 6, 35, 29, 53, 35, 6, 8, 56, 99lnext 23674 . 2  |-  ( ph  ->  E. x  e.  P  <" B D X "> (cgrG `  G ) <" D B x "> )
10198, 100r19.29a 2996 1  |-  ( ph  ->  A  =  ( M `
 C ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    \/ wo 368    /\ wa 369    = wceq 1374    e. wcel 1762    =/= wne 2655   class class class wbr 4440   ` cfv 5579  (class class class)co 6275   <"cs3 12757   Basecbs 14479   distcds 14553  TarskiGcstrkg 23546  Itvcitv 23553  LineGclng 23554  cgrGccgrg 23623  pInvGcmir 23739
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-rep 4551  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-nel 2658  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-pss 3485  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-tp 4025  df-op 4027  df-uni 4239  df-int 4276  df-iun 4320  df-br 4441  df-opab 4499  df-mpt 4500  df-tr 4534  df-eprel 4784  df-id 4788  df-po 4793  df-so 4794  df-fr 4831  df-we 4833  df-ord 4874  df-on 4875  df-lim 4876  df-suc 4877  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-riota 6236  df-ov 6278  df-oprab 6279  df-mpt2 6280  df-om 6672  df-1st 6774  df-2nd 6775  df-recs 7032  df-rdg 7066  df-1o 7120  df-oadd 7124  df-er 7301  df-pm 7413  df-en 7507  df-dom 7508  df-sdom 7509  df-fin 7510  df-card 8309  df-cda 8537  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9796  df-neg 9797  df-nn 10526  df-2 10583  df-3 10584  df-n0 10785  df-z 10854  df-uz 11072  df-fz 11662  df-fzo 11782  df-hash 12361  df-word 12495  df-concat 12497  df-s1 12498  df-s2 12763  df-s3 12764  df-trkgc 23565  df-trkgb 23566  df-trkgcb 23567  df-trkg 23571  df-cgrg 23624  df-mir 23740
This theorem is referenced by:  mideulem  23806
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