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Theorem lmiisolem 24378
Description: Lemma for lmiiso 24379 (Contributed by Thierry Arnoux, 14-Dec-2019.)
Hypotheses
Ref Expression
ismid.p  |-  P  =  ( Base `  G
)
ismid.d  |-  .-  =  ( dist `  G )
ismid.i  |-  I  =  (Itv `  G )
ismid.g  |-  ( ph  ->  G  e. TarskiG )
ismid.1  |-  ( ph  ->  GDimTarskiG 2 )
lmif.m  |-  M  =  ( (lInvG `  G
) `  D )
lmif.l  |-  L  =  (LineG `  G )
lmif.d  |-  ( ph  ->  D  e.  ran  L
)
lmiiso.1  |-  ( ph  ->  A  e.  P )
lmiiso.2  |-  ( ph  ->  B  e.  P )
lmiisolem.s  |-  S  =  ( (pInvG `  G
) `  Z )
lmiisolem.z  |-  Z  =  ( ( A (midG `  G ) ( M `
 A ) ) (midG `  G )
( B (midG `  G ) ( M `
 B ) ) )
Assertion
Ref Expression
lmiisolem  |-  ( ph  ->  ( ( M `  A )  .-  ( M `  B )
)  =  ( A 
.-  B ) )

Proof of Theorem lmiisolem
StepHypRef Expression
1 ismid.p . . . . . . . 8  |-  P  =  ( Base `  G
)
2 ismid.d . . . . . . . 8  |-  .-  =  ( dist `  G )
3 ismid.i . . . . . . . 8  |-  I  =  (Itv `  G )
4 ismid.g . . . . . . . . 9  |-  ( ph  ->  G  e. TarskiG )
54adantr 465 . . . . . . . 8  |-  ( (
ph  /\  ( S `  A )  =  Z )  ->  G  e. TarskiG )
6 lmiisolem.z . . . . . . . . . 10  |-  Z  =  ( ( A (midG `  G ) ( M `
 A ) ) (midG `  G )
( B (midG `  G ) ( M `
 B ) ) )
7 ismid.1 . . . . . . . . . . 11  |-  ( ph  ->  GDimTarskiG 2 )
8 lmiiso.1 . . . . . . . . . . . 12  |-  ( ph  ->  A  e.  P )
9 lmif.m . . . . . . . . . . . . 13  |-  M  =  ( (lInvG `  G
) `  D )
10 lmif.l . . . . . . . . . . . . 13  |-  L  =  (LineG `  G )
11 lmif.d . . . . . . . . . . . . 13  |-  ( ph  ->  D  e.  ran  L
)
121, 2, 3, 4, 7, 9, 10, 11, 8lmicl 24369 . . . . . . . . . . . 12  |-  ( ph  ->  ( M `  A
)  e.  P )
131, 2, 3, 4, 7, 8, 12midcl 24360 . . . . . . . . . . 11  |-  ( ph  ->  ( A (midG `  G ) ( M `
 A ) )  e.  P )
14 lmiiso.2 . . . . . . . . . . . 12  |-  ( ph  ->  B  e.  P )
151, 2, 3, 4, 7, 9, 10, 11, 14lmicl 24369 . . . . . . . . . . . 12  |-  ( ph  ->  ( M `  B
)  e.  P )
161, 2, 3, 4, 7, 14, 15midcl 24360 . . . . . . . . . . 11  |-  ( ph  ->  ( B (midG `  G ) ( M `
 B ) )  e.  P )
171, 2, 3, 4, 7, 13, 16midcl 24360 . . . . . . . . . 10  |-  ( ph  ->  ( ( A (midG `  G ) ( M `
 A ) ) (midG `  G )
( B (midG `  G ) ( M `
 B ) ) )  e.  P )
186, 17syl5eqel 2549 . . . . . . . . 9  |-  ( ph  ->  Z  e.  P )
1918adantr 465 . . . . . . . 8  |-  ( (
ph  /\  ( S `  A )  =  Z )  ->  Z  e.  P )
20 eqid 2457 . . . . . . . . . 10  |-  (pInvG `  G )  =  (pInvG `  G )
21 lmiisolem.s . . . . . . . . . 10  |-  S  =  ( (pInvG `  G
) `  Z )
221, 2, 3, 10, 20, 4, 18, 21, 8mircl 24259 . . . . . . . . 9  |-  ( ph  ->  ( S `  A
)  e.  P )
2322adantr 465 . . . . . . . 8  |-  ( (
ph  /\  ( S `  A )  =  Z )  ->  ( S `  A )  e.  P
)
248adantr 465 . . . . . . . 8  |-  ( (
ph  /\  ( S `  A )  =  Z )  ->  A  e.  P )
251, 2, 3, 10, 20, 5, 19, 21, 24mircgr 24255 . . . . . . . 8  |-  ( (
ph  /\  ( S `  A )  =  Z )  ->  ( Z  .-  ( S `  A
) )  =  ( Z  .-  A ) )
26 simpr 461 . . . . . . . . 9  |-  ( (
ph  /\  ( S `  A )  =  Z )  ->  ( S `  A )  =  Z )
2726eqcomd 2465 . . . . . . . 8  |-  ( (
ph  /\  ( S `  A )  =  Z )  ->  Z  =  ( S `  A ) )
281, 2, 3, 5, 19, 23, 19, 24, 25, 27tgcgreq 24090 . . . . . . 7  |-  ( (
ph  /\  ( S `  A )  =  Z )  ->  Z  =  A )
29 simpr 461 . . . . . . . . . . . . 13  |-  ( (
ph  /\  ( A
(midG `  G )
( M `  A
) )  =  ( B (midG `  G
) ( M `  B ) ) )  ->  ( A (midG `  G ) ( M `
 A ) )  =  ( B (midG `  G ) ( M `
 B ) ) )
3029oveq2d 6312 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( A
(midG `  G )
( M `  A
) )  =  ( B (midG `  G
) ( M `  B ) ) )  ->  ( ( A (midG `  G )
( M `  A
) ) (midG `  G ) ( A (midG `  G )
( M `  A
) ) )  =  ( ( A (midG `  G ) ( M `
 A ) ) (midG `  G )
( B (midG `  G ) ( M `
 B ) ) ) )
3130, 6syl6reqr 2517 . . . . . . . . . . 11  |-  ( (
ph  /\  ( A
(midG `  G )
( M `  A
) )  =  ( B (midG `  G
) ( M `  B ) ) )  ->  Z  =  ( ( A (midG `  G ) ( M `
 A ) ) (midG `  G )
( A (midG `  G ) ( M `
 A ) ) ) )
324adantr 465 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( A
(midG `  G )
( M `  A
) )  =  ( B (midG `  G
) ( M `  B ) ) )  ->  G  e. TarskiG )
337adantr 465 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( A
(midG `  G )
( M `  A
) )  =  ( B (midG `  G
) ( M `  B ) ) )  ->  GDimTarskiG 2 )
3413adantr 465 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( A
(midG `  G )
( M `  A
) )  =  ( B (midG `  G
) ( M `  B ) ) )  ->  ( A (midG `  G ) ( M `
 A ) )  e.  P )
351, 2, 3, 32, 33, 34, 34midid 24364 . . . . . . . . . . 11  |-  ( (
ph  /\  ( A
(midG `  G )
( M `  A
) )  =  ( B (midG `  G
) ( M `  B ) ) )  ->  ( ( A (midG `  G )
( M `  A
) ) (midG `  G ) ( A (midG `  G )
( M `  A
) ) )  =  ( A (midG `  G ) ( M `
 A ) ) )
3631, 35eqtrd 2498 . . . . . . . . . 10  |-  ( (
ph  /\  ( A
(midG `  G )
( M `  A
) )  =  ( B (midG `  G
) ( M `  B ) ) )  ->  Z  =  ( A (midG `  G
) ( M `  A ) ) )
37 eqidd 2458 . . . . . . . . . . . . 13  |-  ( ph  ->  ( M `  A
)  =  ( M `
 A ) )
381, 2, 3, 4, 7, 9, 10, 11, 8, 12islmib 24370 . . . . . . . . . . . . 13  |-  ( ph  ->  ( ( M `  A )  =  ( M `  A )  <-> 
( ( A (midG `  G ) ( M `
 A ) )  e.  D  /\  ( D (⟂G `  G )
( A L ( M `  A ) )  \/  A  =  ( M `  A
) ) ) ) )
3937, 38mpbid 210 . . . . . . . . . . . 12  |-  ( ph  ->  ( ( A (midG `  G ) ( M `
 A ) )  e.  D  /\  ( D (⟂G `  G )
( A L ( M `  A ) )  \/  A  =  ( M `  A
) ) ) )
4039simpld 459 . . . . . . . . . . 11  |-  ( ph  ->  ( A (midG `  G ) ( M `
 A ) )  e.  D )
4140adantr 465 . . . . . . . . . 10  |-  ( (
ph  /\  ( A
(midG `  G )
( M `  A
) )  =  ( B (midG `  G
) ( M `  B ) ) )  ->  ( A (midG `  G ) ( M `
 A ) )  e.  D )
4236, 41eqeltrd 2545 . . . . . . . . 9  |-  ( (
ph  /\  ( A
(midG `  G )
( M `  A
) )  =  ( B (midG `  G
) ( M `  B ) ) )  ->  Z  e.  D
)
434adantr 465 . . . . . . . . . . 11  |-  ( (
ph  /\  ( A
(midG `  G )
( M `  A
) )  =/=  ( B (midG `  G )
( M `  B
) ) )  ->  G  e. TarskiG )
4413adantr 465 . . . . . . . . . . 11  |-  ( (
ph  /\  ( A
(midG `  G )
( M `  A
) )  =/=  ( B (midG `  G )
( M `  B
) ) )  -> 
( A (midG `  G ) ( M `
 A ) )  e.  P )
4516adantr 465 . . . . . . . . . . 11  |-  ( (
ph  /\  ( A
(midG `  G )
( M `  A
) )  =/=  ( B (midG `  G )
( M `  B
) ) )  -> 
( B (midG `  G ) ( M `
 B ) )  e.  P )
4618adantr 465 . . . . . . . . . . 11  |-  ( (
ph  /\  ( A
(midG `  G )
( M `  A
) )  =/=  ( B (midG `  G )
( M `  B
) ) )  ->  Z  e.  P )
47 simpr 461 . . . . . . . . . . 11  |-  ( (
ph  /\  ( A
(midG `  G )
( M `  A
) )  =/=  ( B (midG `  G )
( M `  B
) ) )  -> 
( A (midG `  G ) ( M `
 A ) )  =/=  ( B (midG `  G ) ( M `
 B ) ) )
481, 2, 3, 4, 7, 13, 16midbtwn 24362 . . . . . . . . . . . . 13  |-  ( ph  ->  ( ( A (midG `  G ) ( M `
 A ) ) (midG `  G )
( B (midG `  G ) ( M `
 B ) ) )  e.  ( ( A (midG `  G
) ( M `  A ) ) I ( B (midG `  G ) ( M `
 B ) ) ) )
496, 48syl5eqel 2549 . . . . . . . . . . . 12  |-  ( ph  ->  Z  e.  ( ( A (midG `  G
) ( M `  A ) ) I ( B (midG `  G ) ( M `
 B ) ) ) )
5049adantr 465 . . . . . . . . . . 11  |-  ( (
ph  /\  ( A
(midG `  G )
( M `  A
) )  =/=  ( B (midG `  G )
( M `  B
) ) )  ->  Z  e.  ( ( A (midG `  G )
( M `  A
) ) I ( B (midG `  G
) ( M `  B ) ) ) )
511, 3, 10, 43, 44, 45, 46, 47, 50btwnlng1 24216 . . . . . . . . . 10  |-  ( (
ph  /\  ( A
(midG `  G )
( M `  A
) )  =/=  ( B (midG `  G )
( M `  B
) ) )  ->  Z  e.  ( ( A (midG `  G )
( M `  A
) ) L ( B (midG `  G
) ( M `  B ) ) ) )
5211adantr 465 . . . . . . . . . . 11  |-  ( (
ph  /\  ( A
(midG `  G )
( M `  A
) )  =/=  ( B (midG `  G )
( M `  B
) ) )  ->  D  e.  ran  L )
5340adantr 465 . . . . . . . . . . 11  |-  ( (
ph  /\  ( A
(midG `  G )
( M `  A
) )  =/=  ( B (midG `  G )
( M `  B
) ) )  -> 
( A (midG `  G ) ( M `
 A ) )  e.  D )
54 eqidd 2458 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( M `  B
)  =  ( M `
 B ) )
551, 2, 3, 4, 7, 9, 10, 11, 14, 15islmib 24370 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( ( M `  B )  =  ( M `  B )  <-> 
( ( B (midG `  G ) ( M `
 B ) )  e.  D  /\  ( D (⟂G `  G )
( B L ( M `  B ) )  \/  B  =  ( M `  B
) ) ) ) )
5654, 55mpbid 210 . . . . . . . . . . . . 13  |-  ( ph  ->  ( ( B (midG `  G ) ( M `
 B ) )  e.  D  /\  ( D (⟂G `  G )
( B L ( M `  B ) )  \/  B  =  ( M `  B
) ) ) )
5756simpld 459 . . . . . . . . . . . 12  |-  ( ph  ->  ( B (midG `  G ) ( M `
 B ) )  e.  D )
5857adantr 465 . . . . . . . . . . 11  |-  ( (
ph  /\  ( A
(midG `  G )
( M `  A
) )  =/=  ( B (midG `  G )
( M `  B
) ) )  -> 
( B (midG `  G ) ( M `
 B ) )  e.  D )
591, 3, 10, 43, 44, 45, 47, 47, 52, 53, 58tglinethru 24233 . . . . . . . . . 10  |-  ( (
ph  /\  ( A
(midG `  G )
( M `  A
) )  =/=  ( B (midG `  G )
( M `  B
) ) )  ->  D  =  ( ( A (midG `  G )
( M `  A
) ) L ( B (midG `  G
) ( M `  B ) ) ) )
6051, 59eleqtrrd 2548 . . . . . . . . 9  |-  ( (
ph  /\  ( A
(midG `  G )
( M `  A
) )  =/=  ( B (midG `  G )
( M `  B
) ) )  ->  Z  e.  D )
6142, 60pm2.61dane 2775 . . . . . . . 8  |-  ( ph  ->  Z  e.  D )
6261adantr 465 . . . . . . 7  |-  ( (
ph  /\  ( S `  A )  =  Z )  ->  Z  e.  D )
6328, 62eqeltrrd 2546 . . . . . 6  |-  ( (
ph  /\  ( S `  A )  =  Z )  ->  A  e.  D )
641, 2, 3, 4, 7, 9, 10, 11, 8lmiinv 24375 . . . . . . 7  |-  ( ph  ->  ( ( M `  A )  =  A  <-> 
A  e.  D ) )
6564biimpar 485 . . . . . 6  |-  ( (
ph  /\  A  e.  D )  ->  ( M `  A )  =  A )
6663, 65syldan 470 . . . . 5  |-  ( (
ph  /\  ( S `  A )  =  Z )  ->  ( M `  A )  =  A )
6766, 28eqtr4d 2501 . . . 4  |-  ( (
ph  /\  ( S `  A )  =  Z )  ->  ( M `  A )  =  Z )
6867oveq1d 6311 . . 3  |-  ( (
ph  /\  ( S `  A )  =  Z )  ->  ( ( M `  A )  .-  ( M `  B
) )  =  ( Z  .-  ( M `
 B ) ) )
69 eqidd 2458 . . . . . . . . 9  |-  ( (
ph  /\  B  =  ( M `  B ) )  ->  Z  =  Z )
704adantr 465 . . . . . . . . . 10  |-  ( (
ph  /\  B  =  ( M `  B ) )  ->  G  e. TarskiG )
7114adantr 465 . . . . . . . . . 10  |-  ( (
ph  /\  B  =  ( M `  B ) )  ->  B  e.  P )
7216adantr 465 . . . . . . . . . 10  |-  ( (
ph  /\  B  =  ( M `  B ) )  ->  ( B
(midG `  G )
( M `  B
) )  e.  P
)
731, 2, 3, 4, 7, 14, 15midbtwn 24362 . . . . . . . . . . . 12  |-  ( ph  ->  ( B (midG `  G ) ( M `
 B ) )  e.  ( B I ( M `  B
) ) )
7473adantr 465 . . . . . . . . . . 11  |-  ( (
ph  /\  B  =  ( M `  B ) )  ->  ( B
(midG `  G )
( M `  B
) )  e.  ( B I ( M `
 B ) ) )
75 simpr 461 . . . . . . . . . . . 12  |-  ( (
ph  /\  B  =  ( M `  B ) )  ->  B  =  ( M `  B ) )
7675oveq2d 6312 . . . . . . . . . . 11  |-  ( (
ph  /\  B  =  ( M `  B ) )  ->  ( B I B )  =  ( B I ( M `
 B ) ) )
7774, 76eleqtrrd 2548 . . . . . . . . . 10  |-  ( (
ph  /\  B  =  ( M `  B ) )  ->  ( B
(midG `  G )
( M `  B
) )  e.  ( B I B ) )
781, 2, 3, 70, 71, 72, 77axtgbtwnid 24080 . . . . . . . . 9  |-  ( (
ph  /\  B  =  ( M `  B ) )  ->  B  =  ( B (midG `  G
) ( M `  B ) ) )
79 eqidd 2458 . . . . . . . . 9  |-  ( (
ph  /\  B  =  ( M `  B ) )  ->  B  =  B )
8069, 78, 79s3eqd 12840 . . . . . . . 8  |-  ( (
ph  /\  B  =  ( M `  B ) )  ->  <" Z B B ">  =  <" Z ( B (midG `  G )
( M `  B
) ) B "> )
811, 2, 3, 10, 20, 4, 18, 14, 14ragtrivb 24296 . . . . . . . . 9  |-  ( ph  ->  <" Z B B ">  e.  (∟G `  G ) )
8281adantr 465 . . . . . . . 8  |-  ( (
ph  /\  B  =  ( M `  B ) )  ->  <" Z B B ">  e.  (∟G `  G ) )
8380, 82eqeltrrd 2546 . . . . . . 7  |-  ( (
ph  /\  B  =  ( M `  B ) )  ->  <" Z
( B (midG `  G ) ( M `
 B ) ) B ">  e.  (∟G `  G ) )
844adantr 465 . . . . . . . 8  |-  ( (
ph  /\  B  =/=  ( M `  B ) )  ->  G  e. TarskiG )
8561adantr 465 . . . . . . . 8  |-  ( (
ph  /\  B  =/=  ( M `  B ) )  ->  Z  e.  D )
8657adantr 465 . . . . . . . 8  |-  ( (
ph  /\  B  =/=  ( M `  B ) )  ->  ( B
(midG `  G )
( M `  B
) )  e.  D
)
8714adantr 465 . . . . . . . 8  |-  ( (
ph  /\  B  =/=  ( M `  B ) )  ->  B  e.  P )
88 df-ne 2654 . . . . . . . . . 10  |-  ( B  =/=  ( M `  B )  <->  -.  B  =  ( M `  B ) )
8956simprd 463 . . . . . . . . . . . 12  |-  ( ph  ->  ( D (⟂G `  G
) ( B L ( M `  B
) )  \/  B  =  ( M `  B ) ) )
9089orcomd 388 . . . . . . . . . . 11  |-  ( ph  ->  ( B  =  ( M `  B )  \/  D (⟂G `  G
) ( B L ( M `  B
) ) ) )
9190orcanai 913 . . . . . . . . . 10  |-  ( (
ph  /\  -.  B  =  ( M `  B ) )  ->  D (⟂G `  G )
( B L ( M `  B ) ) )
9288, 91sylan2b 475 . . . . . . . . 9  |-  ( (
ph  /\  B  =/=  ( M `  B ) )  ->  D (⟂G `  G ) ( B L ( M `  B ) ) )
9315adantr 465 . . . . . . . . . . 11  |-  ( (
ph  /\  B  =/=  ( M `  B ) )  ->  ( M `  B )  e.  P
)
94 simpr 461 . . . . . . . . . . 11  |-  ( (
ph  /\  B  =/=  ( M `  B ) )  ->  B  =/=  ( M `  B ) )
9516adantr 465 . . . . . . . . . . 11  |-  ( (
ph  /\  B  =/=  ( M `  B ) )  ->  ( B
(midG `  G )
( M `  B
) )  e.  P
)
964adantr 465 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  ( B
(midG `  G )
( M `  B
) )  =  B )  ->  G  e. TarskiG )
9714adantr 465 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  ( B
(midG `  G )
( M `  B
) )  =  B )  ->  B  e.  P )
9815adantr 465 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  ( B
(midG `  G )
( M `  B
) )  =  B )  ->  ( M `  B )  e.  P
)
997adantr 465 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  ( B
(midG `  G )
( M `  B
) )  =  B )  ->  GDimTarskiG 2 )
100 simpr 461 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  ( B
(midG `  G )
( M `  B
) )  =  B )  ->  ( B
(midG `  G )
( M `  B
) )  =  B )
1011, 2, 3, 96, 99, 97, 98, 100midcgr 24363 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  ( B
(midG `  G )
( M `  B
) )  =  B )  ->  ( B  .-  B )  =  ( B  .-  ( M `
 B ) ) )
102101eqcomd 2465 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  ( B
(midG `  G )
( M `  B
) )  =  B )  ->  ( B  .-  ( M `  B
) )  =  ( B  .-  B ) )
1031, 2, 3, 96, 97, 98, 97, 102axtgcgrid 24077 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  ( B
(midG `  G )
( M `  B
) )  =  B )  ->  B  =  ( M `  B ) )
104103ex 434 . . . . . . . . . . . . 13  |-  ( ph  ->  ( ( B (midG `  G ) ( M `
 B ) )  =  B  ->  B  =  ( M `  B ) ) )
105104necon3d 2681 . . . . . . . . . . . 12  |-  ( ph  ->  ( B  =/=  ( M `  B )  ->  ( B (midG `  G ) ( M `
 B ) )  =/=  B ) )
106105imp 429 . . . . . . . . . . 11  |-  ( (
ph  /\  B  =/=  ( M `  B ) )  ->  ( B
(midG `  G )
( M `  B
) )  =/=  B
)
10773adantr 465 . . . . . . . . . . . 12  |-  ( (
ph  /\  B  =/=  ( M `  B ) )  ->  ( B
(midG `  G )
( M `  B
) )  e.  ( B I ( M `
 B ) ) )
1081, 3, 10, 84, 87, 93, 95, 94, 107btwnlng1 24216 . . . . . . . . . . 11  |-  ( (
ph  /\  B  =/=  ( M `  B ) )  ->  ( B
(midG `  G )
( M `  B
) )  e.  ( B L ( M `
 B ) ) )
1091, 3, 10, 84, 87, 93, 94, 95, 106, 108tglineelsb2 24229 . . . . . . . . . 10  |-  ( (
ph  /\  B  =/=  ( M `  B ) )  ->  ( B L ( M `  B ) )  =  ( B L ( B (midG `  G
) ( M `  B ) ) ) )
1101, 3, 10, 84, 95, 87, 106tglinecom 24232 . . . . . . . . . 10  |-  ( (
ph  /\  B  =/=  ( M `  B ) )  ->  ( ( B (midG `  G )
( M `  B
) ) L B )  =  ( B L ( B (midG `  G ) ( M `
 B ) ) ) )
111109, 110eqtr4d 2501 . . . . . . . . 9  |-  ( (
ph  /\  B  =/=  ( M `  B ) )  ->  ( B L ( M `  B ) )  =  ( ( B (midG `  G ) ( M `
 B ) ) L B ) )
11292, 111breqtrd 4480 . . . . . . . 8  |-  ( (
ph  /\  B  =/=  ( M `  B ) )  ->  D (⟂G `  G ) ( ( B (midG `  G
) ( M `  B ) ) L B ) )
1131, 2, 3, 10, 84, 85, 86, 87, 112perpdrag 24319 . . . . . . 7  |-  ( (
ph  /\  B  =/=  ( M `  B ) )  ->  <" Z
( B (midG `  G ) ( M `
 B ) ) B ">  e.  (∟G `  G ) )
11483, 113pm2.61dane 2775 . . . . . 6  |-  ( ph  ->  <" Z ( B (midG `  G
) ( M `  B ) ) B ">  e.  (∟G `  G ) )
1151, 2, 3, 10, 20, 4, 18, 16, 14israg 24291 . . . . . 6  |-  ( ph  ->  ( <" Z
( B (midG `  G ) ( M `
 B ) ) B ">  e.  (∟G `  G )  <->  ( Z  .-  B )  =  ( Z  .-  ( ( (pInvG `  G ) `  ( B (midG `  G ) ( M `
 B ) ) ) `  B ) ) ) )
116114, 115mpbid 210 . . . . 5  |-  ( ph  ->  ( Z  .-  B
)  =  ( Z 
.-  ( ( (pInvG `  G ) `  ( B (midG `  G )
( M `  B
) ) ) `  B ) ) )
117 eqidd 2458 . . . . . . 7  |-  ( ph  ->  ( B (midG `  G ) ( M `
 B ) )  =  ( B (midG `  G ) ( M `
 B ) ) )
1181, 2, 3, 4, 7, 14, 15, 20, 16ismidb 24361 . . . . . . 7  |-  ( ph  ->  ( ( M `  B )  =  ( ( (pInvG `  G
) `  ( B
(midG `  G )
( M `  B
) ) ) `  B )  <->  ( B
(midG `  G )
( M `  B
) )  =  ( B (midG `  G
) ( M `  B ) ) ) )
119117, 118mpbird 232 . . . . . 6  |-  ( ph  ->  ( M `  B
)  =  ( ( (pInvG `  G ) `  ( B (midG `  G ) ( M `
 B ) ) ) `  B ) )
120119oveq2d 6312 . . . . 5  |-  ( ph  ->  ( Z  .-  ( M `  B )
)  =  ( Z 
.-  ( ( (pInvG `  G ) `  ( B (midG `  G )
( M `  B
) ) ) `  B ) ) )
121116, 120eqtr4d 2501 . . . 4  |-  ( ph  ->  ( Z  .-  B
)  =  ( Z 
.-  ( M `  B ) ) )
122121adantr 465 . . 3  |-  ( (
ph  /\  ( S `  A )  =  Z )  ->  ( Z  .-  B )  =  ( Z  .-  ( M `
 B ) ) )
12328oveq1d 6311 . . 3  |-  ( (
ph  /\  ( S `  A )  =  Z )  ->  ( Z  .-  B )  =  ( A  .-  B ) )
12468, 122, 1233eqtr2d 2504 . 2  |-  ( (
ph  /\  ( S `  A )  =  Z )  ->  ( ( M `  A )  .-  ( M `  B
) )  =  ( A  .-  B ) )
1254adantr 465 . . . 4  |-  ( (
ph  /\  ( S `  A )  =/=  Z
)  ->  G  e. TarskiG )
12622adantr 465 . . . 4  |-  ( (
ph  /\  ( S `  A )  =/=  Z
)  ->  ( S `  A )  e.  P
)
12718adantr 465 . . . 4  |-  ( (
ph  /\  ( S `  A )  =/=  Z
)  ->  Z  e.  P )
1288adantr 465 . . . 4  |-  ( (
ph  /\  ( S `  A )  =/=  Z
)  ->  A  e.  P )
1291, 2, 3, 10, 20, 4, 18, 21, 12mircl 24259 . . . . 5  |-  ( ph  ->  ( S `  ( M `  A )
)  e.  P )
130129adantr 465 . . . 4  |-  ( (
ph  /\  ( S `  A )  =/=  Z
)  ->  ( S `  ( M `  A
) )  e.  P
)
13112adantr 465 . . . 4  |-  ( (
ph  /\  ( S `  A )  =/=  Z
)  ->  ( M `  A )  e.  P
)
13214adantr 465 . . . 4  |-  ( (
ph  /\  ( S `  A )  =/=  Z
)  ->  B  e.  P )
13315adantr 465 . . . 4  |-  ( (
ph  /\  ( S `  A )  =/=  Z
)  ->  ( M `  B )  e.  P
)
134 simpr 461 . . . 4  |-  ( (
ph  /\  ( S `  A )  =/=  Z
)  ->  ( S `  A )  =/=  Z
)
1351, 2, 3, 10, 20, 125, 127, 21, 128mirbtwn 24256 . . . 4  |-  ( (
ph  /\  ( S `  A )  =/=  Z
)  ->  Z  e.  ( ( S `  A ) I A ) )
1361, 2, 3, 10, 20, 125, 127, 21, 131mirbtwn 24256 . . . 4  |-  ( (
ph  /\  ( S `  A )  =/=  Z
)  ->  Z  e.  ( ( S `  ( M `  A ) ) I ( M `
 A ) ) )
137 eqidd 2458 . . . . . . . . . . . 12  |-  ( (
ph  /\  A  =  ( M `  A ) )  ->  Z  =  Z )
1384adantr 465 . . . . . . . . . . . . 13  |-  ( (
ph  /\  A  =  ( M `  A ) )  ->  G  e. TarskiG )
1398adantr 465 . . . . . . . . . . . . 13  |-  ( (
ph  /\  A  =  ( M `  A ) )  ->  A  e.  P )
14013adantr 465 . . . . . . . . . . . . 13  |-  ( (
ph  /\  A  =  ( M `  A ) )  ->  ( A
(midG `  G )
( M `  A
) )  e.  P
)
1411, 2, 3, 4, 7, 8, 12midbtwn 24362 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( A (midG `  G ) ( M `
 A ) )  e.  ( A I ( M `  A
) ) )
142141adantr 465 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  A  =  ( M `  A ) )  ->  ( A
(midG `  G )
( M `  A
) )  e.  ( A I ( M `
 A ) ) )
143 simpr 461 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  A  =  ( M `  A ) )  ->  A  =  ( M `  A ) )
144143oveq2d 6312 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  A  =  ( M `  A ) )  ->  ( A I A )  =  ( A I ( M `
 A ) ) )
145142, 144eleqtrrd 2548 . . . . . . . . . . . . 13  |-  ( (
ph  /\  A  =  ( M `  A ) )  ->  ( A
(midG `  G )
( M `  A
) )  e.  ( A I A ) )
1461, 2, 3, 138, 139, 140, 145axtgbtwnid 24080 . . . . . . . . . . . 12  |-  ( (
ph  /\  A  =  ( M `  A ) )  ->  A  =  ( A (midG `  G
) ( M `  A ) ) )
147 eqidd 2458 . . . . . . . . . . . 12  |-  ( (
ph  /\  A  =  ( M `  A ) )  ->  A  =  A )
148137, 146, 147s3eqd 12840 . . . . . . . . . . 11  |-  ( (
ph  /\  A  =  ( M `  A ) )  ->  <" Z A A ">  =  <" Z ( A (midG `  G )
( M `  A
) ) A "> )
1491, 2, 3, 10, 20, 4, 18, 8, 8ragtrivb 24296 . . . . . . . . . . . 12  |-  ( ph  ->  <" Z A A ">  e.  (∟G `  G ) )
150149adantr 465 . . . . . . . . . . 11  |-  ( (
ph  /\  A  =  ( M `  A ) )  ->  <" Z A A ">  e.  (∟G `  G ) )
151148, 150eqeltrrd 2546 . . . . . . . . . 10  |-  ( (
ph  /\  A  =  ( M `  A ) )  ->  <" Z
( A (midG `  G ) ( M `
 A ) ) A ">  e.  (∟G `  G ) )
1524adantr 465 . . . . . . . . . . 11  |-  ( (
ph  /\  A  =/=  ( M `  A ) )  ->  G  e. TarskiG )
15361adantr 465 . . . . . . . . . . 11  |-  ( (
ph  /\  A  =/=  ( M `  A ) )  ->  Z  e.  D )
15440adantr 465 . . . . . . . . . . 11  |-  ( (
ph  /\  A  =/=  ( M `  A ) )  ->  ( A
(midG `  G )
( M `  A
) )  e.  D
)
1558adantr 465 . . . . . . . . . . 11  |-  ( (
ph  /\  A  =/=  ( M `  A ) )  ->  A  e.  P )
156 df-ne 2654 . . . . . . . . . . . . 13  |-  ( A  =/=  ( M `  A )  <->  -.  A  =  ( M `  A ) )
15739simprd 463 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( D (⟂G `  G
) ( A L ( M `  A
) )  \/  A  =  ( M `  A ) ) )
158157orcomd 388 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( A  =  ( M `  A )  \/  D (⟂G `  G
) ( A L ( M `  A
) ) ) )
159158orcanai 913 . . . . . . . . . . . . 13  |-  ( (
ph  /\  -.  A  =  ( M `  A ) )  ->  D (⟂G `  G )
( A L ( M `  A ) ) )
160156, 159sylan2b 475 . . . . . . . . . . . 12  |-  ( (
ph  /\  A  =/=  ( M `  A ) )  ->  D (⟂G `  G ) ( A L ( M `  A ) ) )
16112adantr 465 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  A  =/=  ( M `  A ) )  ->  ( M `  A )  e.  P
)
162 simpr 461 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  A  =/=  ( M `  A ) )  ->  A  =/=  ( M `  A ) )
16313adantr 465 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  A  =/=  ( M `  A ) )  ->  ( A
(midG `  G )
( M `  A
) )  e.  P
)
1644adantr 465 . . . . . . . . . . . . . . . . . 18  |-  ( (
ph  /\  ( A
(midG `  G )
( M `  A
) )  =  A )  ->  G  e. TarskiG )
1658adantr 465 . . . . . . . . . . . . . . . . . 18  |-  ( (
ph  /\  ( A
(midG `  G )
( M `  A
) )  =  A )  ->  A  e.  P )
16612adantr 465 . . . . . . . . . . . . . . . . . 18  |-  ( (
ph  /\  ( A
(midG `  G )
( M `  A
) )  =  A )  ->  ( M `  A )  e.  P
)
1677adantr 465 . . . . . . . . . . . . . . . . . . . 20  |-  ( (
ph  /\  ( A
(midG `  G )
( M `  A
) )  =  A )  ->  GDimTarskiG 2 )
168 simpr 461 . . . . . . . . . . . . . . . . . . . 20  |-  ( (
ph  /\  ( A
(midG `  G )
( M `  A
) )  =  A )  ->  ( A
(midG `  G )
( M `  A
) )  =  A )
1691, 2, 3, 164, 167, 165, 166, 168midcgr 24363 . . . . . . . . . . . . . . . . . . 19  |-  ( (
ph  /\  ( A
(midG `  G )
( M `  A
) )  =  A )  ->  ( A  .-  A )  =  ( A  .-  ( M `
 A ) ) )
170169eqcomd 2465 . . . . . . . . . . . . . . . . . 18  |-  ( (
ph  /\  ( A
(midG `  G )
( M `  A
) )  =  A )  ->  ( A  .-  ( M `  A
) )  =  ( A  .-  A ) )
1711, 2, 3, 164, 165, 166, 165, 170axtgcgrid 24077 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  ( A
(midG `  G )
( M `  A
) )  =  A )  ->  A  =  ( M `  A ) )
172171ex 434 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  ( ( A (midG `  G ) ( M `
 A ) )  =  A  ->  A  =  ( M `  A ) ) )
173172necon3d 2681 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( A  =/=  ( M `  A )  ->  ( A (midG `  G ) ( M `
 A ) )  =/=  A ) )
174173imp 429 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  A  =/=  ( M `  A ) )  ->  ( A
(midG `  G )
( M `  A
) )  =/=  A
)
175141adantr 465 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  A  =/=  ( M `  A ) )  ->  ( A
(midG `  G )
( M `  A
) )  e.  ( A I ( M `
 A ) ) )
1761, 3, 10, 152, 155, 161, 163, 162, 175btwnlng1 24216 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  A  =/=  ( M `  A ) )  ->  ( A
(midG `  G )
( M `  A
) )  e.  ( A L ( M `
 A ) ) )
1771, 3, 10, 152, 155, 161, 162, 163, 174, 176tglineelsb2 24229 . . . . . . . . . . . . 13  |-  ( (
ph  /\  A  =/=  ( M `  A ) )  ->  ( A L ( M `  A ) )  =  ( A L ( A (midG `  G
) ( M `  A ) ) ) )
1781, 3, 10, 152, 163, 155, 174tglinecom 24232 . . . . . . . . . . . . 13  |-  ( (
ph  /\  A  =/=  ( M `  A ) )  ->  ( ( A (midG `  G )
( M `  A
) ) L A )  =  ( A L ( A (midG `  G ) ( M `
 A ) ) ) )
179177, 178eqtr4d 2501 . . . . . . . . . . . 12  |-  ( (
ph  /\  A  =/=  ( M `  A ) )  ->  ( A L ( M `  A ) )  =  ( ( A (midG `  G ) ( M `
 A ) ) L A ) )
180160, 179breqtrd 4480 . . . . . . . . . . 11  |-  ( (
ph  /\  A  =/=  ( M `  A ) )  ->  D (⟂G `  G ) ( ( A (midG `  G
) ( M `  A ) ) L A ) )
1811, 2, 3, 10, 152, 153, 154, 155, 180perpdrag 24319 . . . . . . . . . 10  |-  ( (
ph  /\  A  =/=  ( M `  A ) )  ->  <" Z
( A (midG `  G ) ( M `
 A ) ) A ">  e.  (∟G `  G ) )
182151, 181pm2.61dane 2775 . . . . . . . . 9  |-  ( ph  ->  <" Z ( A (midG `  G
) ( M `  A ) ) A ">  e.  (∟G `  G ) )
1831, 2, 3, 10, 20, 4, 18, 13, 8israg 24291 . . . . . . . . 9  |-  ( ph  ->  ( <" Z
( A (midG `  G ) ( M `
 A ) ) A ">  e.  (∟G `  G )  <->  ( Z  .-  A )  =  ( Z  .-  ( ( (pInvG `  G ) `  ( A (midG `  G ) ( M `
 A ) ) ) `  A ) ) ) )
184182, 183mpbid 210 . . . . . . . 8  |-  ( ph  ->  ( Z  .-  A
)  =  ( Z 
.-  ( ( (pInvG `  G ) `  ( A (midG `  G )
( M `  A
) ) ) `  A ) ) )
185 eqidd 2458 . . . . . . . . . 10  |-  ( ph  ->  ( A (midG `  G ) ( M `
 A ) )  =  ( A (midG `  G ) ( M `
 A ) ) )
1861, 2, 3, 4, 7, 8, 12, 20, 13ismidb 24361 . . . . . . . . . 10  |-  ( ph  ->  ( ( M `  A )  =  ( ( (pInvG `  G
) `  ( A
(midG `  G )
( M `  A
) ) ) `  A )  <->  ( A
(midG `  G )
( M `  A
) )  =  ( A (midG `  G
) ( M `  A ) ) ) )
187185, 186mpbird 232 . . . . . . . . 9  |-  ( ph  ->  ( M `  A
)  =  ( ( (pInvG `  G ) `  ( A (midG `  G ) ( M `
 A ) ) ) `  A ) )
188187oveq2d 6312 . . . . . . . 8  |-  ( ph  ->  ( Z  .-  ( M `  A )
)  =  ( Z 
.-  ( ( (pInvG `  G ) `  ( A (midG `  G )
( M `  A
) ) ) `  A ) ) )
189184, 188eqtr4d 2501 . . . . . . 7  |-  ( ph  ->  ( Z  .-  A
)  =  ( Z 
.-  ( M `  A ) ) )
1901, 2, 3, 10, 20, 4, 18, 21, 8mircgr 24255 . . . . . . 7  |-  ( ph  ->  ( Z  .-  ( S `  A )
)  =  ( Z 
.-  A ) )
1911, 2, 3, 10, 20, 4, 18, 21, 12mircgr 24255 . . . . . . 7  |-  ( ph  ->  ( Z  .-  ( S `  ( M `  A ) ) )  =  ( Z  .-  ( M `  A ) ) )
192189, 190, 1913eqtr4d 2508 . . . . . 6  |-  ( ph  ->  ( Z  .-  ( S `  A )
)  =  ( Z 
.-  ( S `  ( M `  A ) ) ) )
193192adantr 465 . . . . 5  |-  ( (
ph  /\  ( S `  A )  =/=  Z
)  ->  ( Z  .-  ( S `  A
) )  =  ( Z  .-  ( S `
 ( M `  A ) ) ) )
1941, 2, 3, 125, 127, 126, 127, 130, 193tgcgrcomlr 24088 . . . 4  |-  ( (
ph  /\  ( S `  A )  =/=  Z
)  ->  ( ( S `  A )  .-  Z )  =  ( ( S `  ( M `  A )
)  .-  Z )
)
195189adantr 465 . . . 4  |-  ( (
ph  /\  ( S `  A )  =/=  Z
)  ->  ( Z  .-  A )  =  ( Z  .-  ( M `
 A ) ) )
19621fveq1i 5873 . . . . . . . . . 10  |-  ( S `
 ( A (midG `  G ) ( M `
 A ) ) )  =  ( ( (pInvG `  G ) `  Z ) `  ( A (midG `  G )
( M `  A
) ) )
1971, 2, 3, 4, 7, 8, 12, 21, 18mirmid 24366 . . . . . . . . . 10  |-  ( ph  ->  ( ( S `  A ) (midG `  G ) ( S `
 ( M `  A ) ) )  =  ( S `  ( A (midG `  G
) ( M `  A ) ) ) )
1986eqcomi 2470 . . . . . . . . . . 11  |-  ( ( A (midG `  G
) ( M `  A ) ) (midG `  G ) ( B (midG `  G )
( M `  B
) ) )  =  Z
1991, 2, 3, 4, 7, 13, 16, 20, 18ismidb 24361 . . . . . . . . . . 11  |-  ( ph  ->  ( ( B (midG `  G ) ( M `
 B ) )  =  ( ( (pInvG `  G ) `  Z
) `  ( A
(midG `  G )
( M `  A
) ) )  <->  ( ( A (midG `  G )
( M `  A
) ) (midG `  G ) ( B (midG `  G )
( M `  B
) ) )  =  Z ) )
200198, 199mpbiri 233 . . . . . . . . . 10  |-  ( ph  ->  ( B (midG `  G ) ( M `
 B ) )  =  ( ( (pInvG `  G ) `  Z
) `  ( A
(midG `  G )
( M `  A
) ) ) )
201196, 197, 2003eqtr4a 2524 . . . . . . . . 9  |-  ( ph  ->  ( ( S `  A ) (midG `  G ) ( S `
 ( M `  A ) ) )  =  ( B (midG `  G ) ( M `
 B ) ) )
2021, 2, 3, 4, 7, 22, 129, 20, 16ismidb 24361 . . . . . . . . 9  |-  ( ph  ->  ( ( S `  ( M `  A ) )  =  ( ( (pInvG `  G ) `  ( B (midG `  G ) ( M `
 B ) ) ) `  ( S `
 A ) )  <-> 
( ( S `  A ) (midG `  G ) ( S `
 ( M `  A ) ) )  =  ( B (midG `  G ) ( M `
 B ) ) ) )
203201, 202mpbird 232 . . . . . . . 8  |-  ( ph  ->  ( S `  ( M `  A )
)  =  ( ( (pInvG `  G ) `  ( B (midG `  G ) ( M `
 B ) ) ) `  ( S `
 A ) ) )
204119, 203oveq12d 6314 . . . . . . 7  |-  ( ph  ->  ( ( M `  B )  .-  ( S `  ( M `  A ) ) )  =  ( ( ( (pInvG `  G ) `  ( B (midG `  G ) ( M `
 B ) ) ) `  B ) 
.-  ( ( (pInvG `  G ) `  ( B (midG `  G )
( M `  B
) ) ) `  ( S `  A ) ) ) )
205 eqid 2457 . . . . . . . 8  |-  ( (pInvG `  G ) `  ( B (midG `  G )
( M `  B
) ) )  =  ( (pInvG `  G
) `  ( B
(midG `  G )
( M `  B
) ) )
2061, 2, 3, 10, 20, 4, 16, 205, 14, 22miriso 24267 . . . . . . 7  |-  ( ph  ->  ( ( ( (pInvG `  G ) `  ( B (midG `  G )
( M `  B
) ) ) `  B )  .-  (
( (pInvG `  G
) `  ( B
(midG `  G )
( M `  B
) ) ) `  ( S `  A ) ) )  =  ( B  .-  ( S `
 A ) ) )
207204, 206eqtr2d 2499 . . . . . 6  |-  ( ph  ->  ( B  .-  ( S `  A )
)  =  ( ( M `  B ) 
.-  ( S `  ( M `  A ) ) ) )
208207adantr 465 . . . . 5  |-  ( (
ph  /\  ( S `  A )  =/=  Z
)  ->  ( B  .-  ( S `  A
) )  =  ( ( M `  B
)  .-  ( S `  ( M `  A
) ) ) )
2091, 2, 3, 125, 132, 126, 133, 130, 208tgcgrcomlr 24088 . . . 4  |-  ( (
ph  /\  ( S `  A )  =/=  Z
)  ->  ( ( S `  A )  .-  B )  =  ( ( S `  ( M `  A )
)  .-  ( M `  B ) ) )
210121adantr 465 . . . 4  |-  ( (
ph  /\  ( S `  A )  =/=  Z
)  ->  ( Z  .-  B )  =  ( Z  .-  ( M `
 B ) ) )
2111, 2, 3, 125, 126, 127, 128, 130, 127, 131, 132, 133, 134, 135, 136, 194, 195, 209, 210axtg5seg 24079 . . 3  |-  ( (
ph  /\  ( S `  A )  =/=  Z
)  ->  ( A  .-  B )  =  ( ( M `  A
)  .-  ( M `  B ) ) )
212211eqcomd 2465 . 2  |-  ( (
ph  /\  ( S `  A )  =/=  Z
)  ->  ( ( M `  A )  .-  ( M `  B
) )  =  ( A  .-  B ) )
213124, 212pm2.61dane 2775 1  |-  ( ph  ->  ( ( M `  A )  .-  ( M `  B )
)  =  ( A 
.-  B ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    \/ wo 368    /\ wa 369    = wceq 1395    e. wcel 1819    =/= wne 2652   class class class wbr 4456   ran crn 5009   ` cfv 5594  (class class class)co 6296   2c2 10606   <"cs3 12819   Basecbs 14735   distcds 14812  TarskiGcstrkg 24042  DimTarskiGcstrkgld 24046  Itvcitv 24049  LineGclng 24050  pInvGcmir 24250  ∟Gcrag 24287  ⟂Gcperpg 24289  midGcmid 24355  lInvGclmi 24356
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-int 4289  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6700  df-1st 6799  df-2nd 6800  df-recs 7060  df-rdg 7094  df-1o 7148  df-oadd 7152  df-er 7329  df-map 7440  df-pm 7441  df-en 7536  df-dom 7537  df-sdom 7538  df-fin 7539  df-card 8337  df-cda 8565  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-nn 10557  df-2 10615  df-3 10616  df-n0 10817  df-z 10886  df-uz 11107  df-fz 11698  df-fzo 11822  df-hash 12409  df-word 12546  df-concat 12548  df-s1 12549  df-s2 12825  df-s3 12826  df-trkgc 24061  df-trkgb 24062  df-trkgcb 24063  df-trkgld 24065  df-trkg 24067  df-cgrg 24120  df-leg 24187  df-mir 24251  df-rag 24288  df-perpg 24290  df-mid 24357  df-lmi 24358
This theorem is referenced by:  lmiiso  24379
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