Theorem iscgra1 25702

Description: A special version of iscgra 25701 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 25701	. 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 766	. . . . . . . 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 766	. . . . . . . 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 766	. . . . . . . 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 766	. . . . . . . 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 766	. . . . . . . 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 799	. . . . . . . . 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 1068	. . . . . . . . 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 25501	. . . . . . . 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 766	. . . . . . . . . . 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 2628	. . . . . . . . . 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 25378	. . . . . . . . 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 2849	. . . . . . . 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 25504	. . . . . . . 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 2622	. . . . . . . . . . 11 |- (cgrG ` G ) = (cgrG ` G )
28	7	ad3antrrr 766	. . . . . . . . . . 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 792	. . . . . . . . . . 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 1067	. . . . . . . . . . 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 25415	. . . . . . . . . 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 2628	. . . . . . . . 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 25375	. . . . . . . 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 25375	. . . . . . . 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 25512	. . . . . . 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 1069	. . . . . . 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 558	. . . . . 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 804	. . . . . . 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 764	. . . . . . . 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 766	. . . . . . . . 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 766	. . . . . . . . 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 766	. . . . . . . . 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 25378	. . . . . . . . . 10 |- ( ph -> D =/= E )
46	45	ad3antrrr 766	. . . . . . . . 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 25504	. . . . . . . 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 4675	. . . . . . 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 805	. . . . . . 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 1242	. . . . . 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 877	. . . . 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 3049	. . . 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 3092	. . . 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 276	. . 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 3049	. 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 2623	. . . . . . . 8 |- ( y = D -> E = E )
57		eqidd 2623	. . . . . . . 8 |- ( y = D -> x = x )
58	39, 56, 57	s3eqd 13609	. . . . . . 7 |- ( y = D -> <" y E x "> = <" D E x "> )
59	58	breq2d 4665	. . . . . 6 |- ( y = D -> ( <" A B C "> (cgrG ` G ) <" y E x "> <-> <" A B C "> (cgrG ` G ) <" D E x "> ) )
60	59	anbi1d 741	. . . . 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 3052	. . . 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 3336	. . 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 294	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 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   E.wrex 2913   class class class wbr 4653   ` cfv 5888  (class class class)co 6650   <"cs3 13587   Basecbs 15857   distcds 15950  TarskiGcstrkg 25329  Itvcitv 25335  cgrGccgrg 25405  hlGchlg 25495  cgrAccgra 25699

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-map 7859  df-pm 7860  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-card 8765  df-cda 8990  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-2 11079  df-3 11080  df-n0 11293  df-xnn0 11364  df-z 11378  df-uz 11688  df-fz 12327  df-fzo 12466  df-hash 13118  df-word 13299  df-concat 13301  df-s1 13302  df-s2 13593  df-s3 13594  df-trkgc 25347  df-trkgb 25348  df-trkgcb 25349  df-trkg 25352  df-cgrg 25406  df-hlg 25496  df-cgra 25700

This theorem is referenced by:  acopyeu  25725