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.

Step	Hyp	Ref	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 )
11	1, 2, 3, 4, 5, 6, 7, 8, 9, 10	iscgra 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 ) ) )
12	9	ad3antrrr 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 )
13	6	ad3antrrr 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 )
14	5	ad3antrrr 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 )
15	4	ad3antrrr 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 )
16	8	ad3antrrr 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 )
20	1, 2, 3, 18, 16, 12, 15, 19	hlne2 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 ) )
22	21	ad3antrrr 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 ) )
23	22	eqcomd 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 ) )
24	1, 17, 2, 15, 16, 12, 14, 13, 23, 20	tgcgrneq 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 )
25	24	necomd 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 )
26	1, 2, 3, 16, 12, 12, 15, 20	hlid 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 )
28	7	ad3antrrr 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 "> )
31	1, 17, 2, 27, 15, 14, 13, 28, 18, 12, 29, 30	cgr3simp1 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 ) )
32	31	eqcomd 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 ) )
33	1, 17, 2, 15, 18, 12, 14, 13, 32	tgcgrcomlr 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 ) )
34	1, 17, 2, 15, 16, 12, 14, 13, 23	tgcgrcomlr 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 ) )
35	1, 2, 3, 12, 13, 14, 15, 16, 17, 20, 25, 18, 16, 19, 26, 33, 34	hlcgreulem 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 )
37	35, 30, 36	jca32 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 )
40	39	ad2antrl 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 )
41	8	ad3antrrr 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 )
42	9	ad3antrrr 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 )
43	4	ad3antrrr 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 )
45	1, 17, 2, 4, 5, 6, 8, 9, 21, 44	tgcgrneq 24606	. . . . . . . . . 10 |- ( ph -> D =/= E )
46	45	ad3antrrr 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 )
47	1, 2, 3, 41, 41, 42, 43, 46	hlid 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 )
48	40, 47	eqbrtrd 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 )
50	38, 48, 49	3jca 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 ) )
51	37, 50	impbida 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 ) ) ) )
52	51	rexbidva 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 ) ) )
54	52, 53	syl6bb 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 ) ) ) )
55	54	rexbidva 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 )
58	39, 56, 57	s3eqd 13019	. . . . . . 7 |- ( y = D -> <" y E x "> = <" D E x "> )
59	58	breq2d 4407	. . . . . 6 |- ( y = D -> ( <" A B C "> (cgrG ` G ) <" y E x "> <-> <" A B C "> (cgrG ` G ) <" D E x "> ) )
60	59	anbi1d 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 ) ) )
61	60	rexbidv 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 ) ) )
62	61	ceqsrexv 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 ) ) )
63	8, 62	syl 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 ) ) )
64	11, 55, 63	3bitrd 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 ) ) )

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