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Theorem iscgra1 24931
Description: A special version of iscgra 24930 where one distance is known to be equal. In this case, angle congruence can be written with only one quantifier. (Contributed by Thierry Arnoux, 9-Aug-2020.)
Hypotheses
Ref Expression
iscgra.p  |-  P  =  ( Base `  G
)
iscgra.i  |-  I  =  (Itv `  G )
iscgra.k  |-  K  =  (hlG `  G )
iscgra.g  |-  ( ph  ->  G  e. TarskiG )
iscgra.a  |-  ( ph  ->  A  e.  P )
iscgra.b  |-  ( ph  ->  B  e.  P )
iscgra.c  |-  ( ph  ->  C  e.  P )
iscgra.d  |-  ( ph  ->  D  e.  P )
iscgra.e  |-  ( ph  ->  E  e.  P )
iscgra.f  |-  ( ph  ->  F  e.  P )
iscgra1.m  |-  .-  =  ( dist `  G )
iscgra1.1  |-  ( ph  ->  A  =/=  B )
iscgra1.2  |-  ( ph  ->  ( A  .-  B
)  =  ( D 
.-  E ) )
Assertion
Ref Expression
iscgra1  |-  ( ph  ->  ( <" A B C "> (cgrA `  G ) <" D E F ">  <->  E. x  e.  P  ( <" A B C "> (cgrG `  G ) <" D E x ">  /\  x
( K `  E
) F ) ) )
Distinct variable groups:    x, A    x, B    x, C    x, D    x, E    x, F    x, K    ph, x    x, G    x, I    x, P
Allowed substitution hint:    .- ( x)

Proof of Theorem iscgra1
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 iscgra.p . . 3  |-  P  =  ( Base `  G
)
2 iscgra.i . . 3  |-  I  =  (Itv `  G )
3 iscgra.k . . 3  |-  K  =  (hlG `  G )
4 iscgra.g . . 3  |-  ( ph  ->  G  e. TarskiG )
5 iscgra.a . . 3  |-  ( ph  ->  A  e.  P )
6 iscgra.b . . 3  |-  ( ph  ->  B  e.  P )
7 iscgra.c . . 3  |-  ( ph  ->  C  e.  P )
8 iscgra.d . . 3  |-  ( ph  ->  D  e.  P )
9 iscgra.e . . 3  |-  ( ph  ->  E  e.  P )
10 iscgra.f . . 3  |-  ( ph  ->  F  e.  P )
111, 2, 3, 4, 5, 6, 7, 8, 9, 10iscgra 24930 . 2  |-  ( ph  ->  ( <" A B C "> (cgrA `  G ) <" D E F ">  <->  E. y  e.  P  E. x  e.  P  ( <" A B C "> (cgrG `  G ) <" y E x ">  /\  y
( K `  E
) D  /\  x
( K `  E
) F ) ) )
129ad3antrrr 744 . . . . . . . 8  |-  ( ( ( ( ph  /\  y  e.  P )  /\  x  e.  P
)  /\  ( <" A B C "> (cgrG `  G ) <" y E x ">  /\  y
( K `  E
) D  /\  x
( K `  E
) F ) )  ->  E  e.  P
)
136ad3antrrr 744 . . . . . . . 8  |-  ( ( ( ( ph  /\  y  e.  P )  /\  x  e.  P
)  /\  ( <" A B C "> (cgrG `  G ) <" y E x ">  /\  y
( K `  E
) D  /\  x
( K `  E
) F ) )  ->  B  e.  P
)
145ad3antrrr 744 . . . . . . . 8  |-  ( ( ( ( ph  /\  y  e.  P )  /\  x  e.  P
)  /\  ( <" A B C "> (cgrG `  G ) <" y E x ">  /\  y
( K `  E
) D  /\  x
( K `  E
) F ) )  ->  A  e.  P
)
154ad3antrrr 744 . . . . . . . 8  |-  ( ( ( ( ph  /\  y  e.  P )  /\  x  e.  P
)  /\  ( <" A B C "> (cgrG `  G ) <" y E x ">  /\  y
( K `  E
) D  /\  x
( K `  E
) F ) )  ->  G  e. TarskiG )
168ad3antrrr 744 . . . . . . . 8  |-  ( ( ( ( ph  /\  y  e.  P )  /\  x  e.  P
)  /\  ( <" A B C "> (cgrG `  G ) <" y E x ">  /\  y
( K `  E
) D  /\  x
( K `  E
) F ) )  ->  D  e.  P
)
17 iscgra1.m . . . . . . . 8  |-  .-  =  ( dist `  G )
18 simpllr 777 . . . . . . . . 9  |-  ( ( ( ( ph  /\  y  e.  P )  /\  x  e.  P
)  /\  ( <" A B C "> (cgrG `  G ) <" y E x ">  /\  y
( K `  E
) D  /\  x
( K `  E
) F ) )  ->  y  e.  P
)
19 simpr2 1037 . . . . . . . . 9  |-  ( ( ( ( ph  /\  y  e.  P )  /\  x  e.  P
)  /\  ( <" A B C "> (cgrG `  G ) <" y E x ">  /\  y
( K `  E
) D  /\  x
( K `  E
) F ) )  ->  y ( K `
 E ) D )
201, 2, 3, 18, 16, 12, 15, 19hlne2 24730 . . . . . . . 8  |-  ( ( ( ( ph  /\  y  e.  P )  /\  x  e.  P
)  /\  ( <" A B C "> (cgrG `  G ) <" y E x ">  /\  y
( K `  E
) D  /\  x
( K `  E
) F ) )  ->  D  =/=  E
)
21 iscgra1.2 . . . . . . . . . . . 12  |-  ( ph  ->  ( A  .-  B
)  =  ( D 
.-  E ) )
2221ad3antrrr 744 . . . . . . . . . . 11  |-  ( ( ( ( ph  /\  y  e.  P )  /\  x  e.  P
)  /\  ( <" A B C "> (cgrG `  G ) <" y E x ">  /\  y
( K `  E
) D  /\  x
( K `  E
) F ) )  ->  ( A  .-  B )  =  ( D  .-  E ) )
2322eqcomd 2477 . . . . . . . . . 10  |-  ( ( ( ( ph  /\  y  e.  P )  /\  x  e.  P
)  /\  ( <" A B C "> (cgrG `  G ) <" y E x ">  /\  y
( K `  E
) D  /\  x
( K `  E
) F ) )  ->  ( D  .-  E )  =  ( A  .-  B ) )
241, 17, 2, 15, 16, 12, 14, 13, 23, 20tgcgrneq 24606 . . . . . . . . 9  |-  ( ( ( ( ph  /\  y  e.  P )  /\  x  e.  P
)  /\  ( <" A B C "> (cgrG `  G ) <" y E x ">  /\  y
( K `  E
) D  /\  x
( K `  E
) F ) )  ->  A  =/=  B
)
2524necomd 2698 . . . . . . . 8  |-  ( ( ( ( ph  /\  y  e.  P )  /\  x  e.  P
)  /\  ( <" A B C "> (cgrG `  G ) <" y E x ">  /\  y
( K `  E
) D  /\  x
( K `  E
) F ) )  ->  B  =/=  A
)
261, 2, 3, 16, 12, 12, 15, 20hlid 24733 . . . . . . . 8  |-  ( ( ( ( ph  /\  y  e.  P )  /\  x  e.  P
)  /\  ( <" A B C "> (cgrG `  G ) <" y E x ">  /\  y
( K `  E
) D  /\  x
( K `  E
) F ) )  ->  D ( K `
 E ) D )
27 eqid 2471 . . . . . . . . . . 11  |-  (cgrG `  G )  =  (cgrG `  G )
287ad3antrrr 744 . . . . . . . . . . 11  |-  ( ( ( ( ph  /\  y  e.  P )  /\  x  e.  P
)  /\  ( <" A B C "> (cgrG `  G ) <" y E x ">  /\  y
( K `  E
) D  /\  x
( K `  E
) F ) )  ->  C  e.  P
)
29 simplr 770 . . . . . . . . . . 11  |-  ( ( ( ( ph  /\  y  e.  P )  /\  x  e.  P
)  /\  ( <" A B C "> (cgrG `  G ) <" y E x ">  /\  y
( K `  E
) D  /\  x
( K `  E
) F ) )  ->  x  e.  P
)
30 simpr1 1036 . . . . . . . . . . 11  |-  ( ( ( ( ph  /\  y  e.  P )  /\  x  e.  P
)  /\  ( <" A B C "> (cgrG `  G ) <" y E x ">  /\  y
( K `  E
) D  /\  x
( K `  E
) F ) )  ->  <" A B C "> (cgrG `  G ) <" y E x "> )
311, 17, 2, 27, 15, 14, 13, 28, 18, 12, 29, 30cgr3simp1 24644 . . . . . . . . . 10  |-  ( ( ( ( ph  /\  y  e.  P )  /\  x  e.  P
)  /\  ( <" A B C "> (cgrG `  G ) <" y E x ">  /\  y
( K `  E
) D  /\  x
( K `  E
) F ) )  ->  ( A  .-  B )  =  ( y  .-  E ) )
3231eqcomd 2477 . . . . . . . . 9  |-  ( ( ( ( ph  /\  y  e.  P )  /\  x  e.  P
)  /\  ( <" A B C "> (cgrG `  G ) <" y E x ">  /\  y
( K `  E
) D  /\  x
( K `  E
) F ) )  ->  ( y  .-  E )  =  ( A  .-  B ) )
331, 17, 2, 15, 18, 12, 14, 13, 32tgcgrcomlr 24603 . . . . . . . 8  |-  ( ( ( ( ph  /\  y  e.  P )  /\  x  e.  P
)  /\  ( <" A B C "> (cgrG `  G ) <" y E x ">  /\  y
( K `  E
) D  /\  x
( K `  E
) F ) )  ->  ( E  .-  y )  =  ( B  .-  A ) )
341, 17, 2, 15, 16, 12, 14, 13, 23tgcgrcomlr 24603 . . . . . . . 8  |-  ( ( ( ( ph  /\  y  e.  P )  /\  x  e.  P
)  /\  ( <" A B C "> (cgrG `  G ) <" y E x ">  /\  y
( K `  E
) D  /\  x
( K `  E
) F ) )  ->  ( E  .-  D )  =  ( B  .-  A ) )
351, 2, 3, 12, 13, 14, 15, 16, 17, 20, 25, 18, 16, 19, 26, 33, 34hlcgreulem 24741 . . . . . . 7  |-  ( ( ( ( ph  /\  y  e.  P )  /\  x  e.  P
)  /\  ( <" A B C "> (cgrG `  G ) <" y E x ">  /\  y
( K `  E
) D  /\  x
( K `  E
) F ) )  ->  y  =  D )
36 simpr3 1038 . . . . . . 7  |-  ( ( ( ( ph  /\  y  e.  P )  /\  x  e.  P
)  /\  ( <" A B C "> (cgrG `  G ) <" y E x ">  /\  y
( K `  E
) D  /\  x
( K `  E
) F ) )  ->  x ( K `
 E ) F )
3735, 30, 36jca32 544 . . . . . 6  |-  ( ( ( ( ph  /\  y  e.  P )  /\  x  e.  P
)  /\  ( <" A B C "> (cgrG `  G ) <" y E x ">  /\  y
( K `  E
) D  /\  x
( K `  E
) F ) )  ->  ( y  =  D  /\  ( <" A B C "> (cgrG `  G ) <" y E x ">  /\  x ( K `  E ) F ) ) )
38 simprrl 782 . . . . . . 7  |-  ( ( ( ( ph  /\  y  e.  P )  /\  x  e.  P
)  /\  ( y  =  D  /\  ( <" A B C "> (cgrG `  G ) <" y E x ">  /\  x ( K `  E ) F ) ) )  ->  <" A B C "> (cgrG `  G ) <" y E x "> )
39 id 22 . . . . . . . . 9  |-  ( y  =  D  ->  y  =  D )
4039ad2antrl 742 . . . . . . . 8  |-  ( ( ( ( ph  /\  y  e.  P )  /\  x  e.  P
)  /\  ( y  =  D  /\  ( <" A B C "> (cgrG `  G ) <" y E x ">  /\  x ( K `  E ) F ) ) )  ->  y  =  D )
418ad3antrrr 744 . . . . . . . . 9  |-  ( ( ( ( ph  /\  y  e.  P )  /\  x  e.  P
)  /\  ( y  =  D  /\  ( <" A B C "> (cgrG `  G ) <" y E x ">  /\  x ( K `  E ) F ) ) )  ->  D  e.  P )
429ad3antrrr 744 . . . . . . . . 9  |-  ( ( ( ( ph  /\  y  e.  P )  /\  x  e.  P
)  /\  ( y  =  D  /\  ( <" A B C "> (cgrG `  G ) <" y E x ">  /\  x ( K `  E ) F ) ) )  ->  E  e.  P )
434ad3antrrr 744 . . . . . . . . 9  |-  ( ( ( ( ph  /\  y  e.  P )  /\  x  e.  P
)  /\  ( y  =  D  /\  ( <" A B C "> (cgrG `  G ) <" y E x ">  /\  x ( K `  E ) F ) ) )  ->  G  e. TarskiG )
44 iscgra1.1 . . . . . . . . . . 11  |-  ( ph  ->  A  =/=  B )
451, 17, 2, 4, 5, 6, 8, 9, 21, 44tgcgrneq 24606 . . . . . . . . . 10  |-  ( ph  ->  D  =/=  E )
4645ad3antrrr 744 . . . . . . . . 9  |-  ( ( ( ( ph  /\  y  e.  P )  /\  x  e.  P
)  /\  ( y  =  D  /\  ( <" A B C "> (cgrG `  G ) <" y E x ">  /\  x ( K `  E ) F ) ) )  ->  D  =/=  E )
471, 2, 3, 41, 41, 42, 43, 46hlid 24733 . . . . . . . 8  |-  ( ( ( ( ph  /\  y  e.  P )  /\  x  e.  P
)  /\  ( y  =  D  /\  ( <" A B C "> (cgrG `  G ) <" y E x ">  /\  x ( K `  E ) F ) ) )  ->  D
( K `  E
) D )
4840, 47eqbrtrd 4416 . . . . . . 7  |-  ( ( ( ( ph  /\  y  e.  P )  /\  x  e.  P
)  /\  ( y  =  D  /\  ( <" A B C "> (cgrG `  G ) <" y E x ">  /\  x ( K `  E ) F ) ) )  ->  y
( K `  E
) D )
49 simprrr 783 . . . . . . 7  |-  ( ( ( ( ph  /\  y  e.  P )  /\  x  e.  P
)  /\  ( y  =  D  /\  ( <" A B C "> (cgrG `  G ) <" y E x ">  /\  x ( K `  E ) F ) ) )  ->  x
( K `  E
) F )
5038, 48, 493jca 1210 . . . . . 6  |-  ( ( ( ( ph  /\  y  e.  P )  /\  x  e.  P
)  /\  ( y  =  D  /\  ( <" A B C "> (cgrG `  G ) <" y E x ">  /\  x ( K `  E ) F ) ) )  ->  ( <" A B C "> (cgrG `  G ) <" y E x ">  /\  y ( K `  E ) D  /\  x ( K `  E ) F ) )
5137, 50impbida 850 . . . . 5  |-  ( ( ( ph  /\  y  e.  P )  /\  x  e.  P )  ->  (
( <" A B C "> (cgrG `  G ) <" y E x ">  /\  y ( K `  E ) D  /\  x ( K `  E ) F )  <-> 
( y  =  D  /\  ( <" A B C "> (cgrG `  G ) <" y E x ">  /\  x ( K `  E ) F ) ) ) )
5251rexbidva 2889 . . . 4  |-  ( (
ph  /\  y  e.  P )  ->  ( E. x  e.  P  ( <" A B C "> (cgrG `  G ) <" y E x ">  /\  y ( K `  E ) D  /\  x ( K `  E ) F )  <->  E. x  e.  P  ( y  =  D  /\  ( <" A B C "> (cgrG `  G ) <" y E x ">  /\  x ( K `  E ) F ) ) ) )
53 r19.42v 2931 . . . 4  |-  ( E. x  e.  P  ( y  =  D  /\  ( <" A B C "> (cgrG `  G ) <" y E x ">  /\  x ( K `  E ) F ) )  <->  ( y  =  D  /\  E. x  e.  P  ( <" A B C "> (cgrG `  G ) <" y E x ">  /\  x
( K `  E
) F ) ) )
5452, 53syl6bb 269 . . 3  |-  ( (
ph  /\  y  e.  P )  ->  ( E. x  e.  P  ( <" A B C "> (cgrG `  G ) <" y E x ">  /\  y ( K `  E ) D  /\  x ( K `  E ) F )  <-> 
( y  =  D  /\  E. x  e.  P  ( <" A B C "> (cgrG `  G ) <" y E x ">  /\  x ( K `  E ) F ) ) ) )
5554rexbidva 2889 . 2  |-  ( ph  ->  ( E. y  e.  P  E. x  e.  P  ( <" A B C "> (cgrG `  G ) <" y E x ">  /\  y ( K `  E ) D  /\  x ( K `  E ) F )  <->  E. y  e.  P  ( y  =  D  /\  E. x  e.  P  ( <" A B C "> (cgrG `  G ) <" y E x ">  /\  x ( K `  E ) F ) ) ) )
56 eqidd 2472 . . . . . . . 8  |-  ( y  =  D  ->  E  =  E )
57 eqidd 2472 . . . . . . . 8  |-  ( y  =  D  ->  x  =  x )
5839, 56, 57s3eqd 13019 . . . . . . 7  |-  ( y  =  D  ->  <" y E x ">  =  <" D E x "> )
5958breq2d 4407 . . . . . 6  |-  ( y  =  D  ->  ( <" A B C "> (cgrG `  G ) <" y E x ">  <->  <" A B C "> (cgrG `  G ) <" D E x "> ) )
6059anbi1d 719 . . . . 5  |-  ( y  =  D  ->  (
( <" A B C "> (cgrG `  G ) <" y E x ">  /\  x ( K `  E ) F )  <-> 
( <" A B C "> (cgrG `  G ) <" D E x ">  /\  x ( K `  E ) F ) ) )
6160rexbidv 2892 . . . 4  |-  ( y  =  D  ->  ( E. x  e.  P  ( <" A B C "> (cgrG `  G ) <" y E x ">  /\  x ( K `  E ) F )  <->  E. x  e.  P  ( <" A B C "> (cgrG `  G ) <" D E x ">  /\  x ( K `  E ) F ) ) )
6261ceqsrexv 3160 . . 3  |-  ( D  e.  P  ->  ( E. y  e.  P  ( y  =  D  /\  E. x  e.  P  ( <" A B C "> (cgrG `  G ) <" y E x ">  /\  x ( K `  E ) F ) )  <->  E. x  e.  P  ( <" A B C "> (cgrG `  G ) <" D E x ">  /\  x ( K `  E ) F ) ) )
638, 62syl 17 . 2  |-  ( ph  ->  ( E. y  e.  P  ( y  =  D  /\  E. x  e.  P  ( <" A B C "> (cgrG `  G ) <" y E x ">  /\  x
( K `  E
) F ) )  <->  E. x  e.  P  ( <" A B C "> (cgrG `  G ) <" D E x ">  /\  x ( K `  E ) F ) ) )
6411, 55, 633bitrd 287 1  |-  ( ph  ->  ( <" A B C "> (cgrA `  G ) <" D E F ">  <->  E. x  e.  P  ( <" A B C "> (cgrG `  G ) <" D E x ">  /\  x
( K `  E
) F ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 189    /\ wa 376    /\ w3a 1007    = wceq 1452    e. wcel 1904    =/= wne 2641   E.wrex 2757   class class class wbr 4395   ` cfv 5589  (class class class)co 6308   <"cs3 12997   Basecbs 15199   distcds 15277  TarskiGcstrkg 24557  Itvcitv 24563  cgrGccgrg 24634  hlGchlg 24724  cgrAccgra 24928
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-cnex 9613  ax-resscn 9614  ax-1cn 9615  ax-icn 9616  ax-addcl 9617  ax-addrcl 9618  ax-mulcl 9619  ax-mulrcl 9620  ax-mulcom 9621  ax-addass 9622  ax-mulass 9623  ax-distr 9624  ax-i2m1 9625  ax-1ne0 9626  ax-1rid 9627  ax-rnegex 9628  ax-rrecex 9629  ax-cnre 9630  ax-pre-lttri 9631  ax-pre-lttrn 9632  ax-pre-ltadd 9633  ax-pre-mulgt0 9634
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-int 4227  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-om 6712  df-1st 6812  df-2nd 6813  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-1o 7200  df-oadd 7204  df-er 7381  df-map 7492  df-pm 7493  df-en 7588  df-dom 7589  df-sdom 7590  df-fin 7591  df-card 8391  df-cda 8616  df-pnf 9695  df-mnf 9696  df-xr 9697  df-ltxr 9698  df-le 9699  df-sub 9882  df-neg 9883  df-nn 10632  df-2 10690  df-3 10691  df-n0 10894  df-z 10962  df-uz 11183  df-fz 11811  df-fzo 11943  df-hash 12554  df-word 12711  df-concat 12713  df-s1 12714  df-s2 13003  df-s3 13004  df-trkgc 24575  df-trkgb 24576  df-trkgcb 24577  df-trkg 24580  df-cgrg 24635  df-hlg 24725  df-cgra 24929
This theorem is referenced by:  acopyeu  24954
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