Theorem trgcopyeu 25698

Description: Triangle construction: a copy of a given triangle can always be constructed in such a way that one side is lying on a half-line, and the third vertex is on a given half-plane: uniqueness part. Second part of Theorem 10.16 of [Schwabhauser] p. 92. (Contributed by Thierry Arnoux, 8-Aug-2020.)

Hypotheses

Ref	Expression
trgcopy.p	|- P = ( Base ` G )
trgcopy.m	|- .- = ( dist ` G )
trgcopy.i	|- I = (Itv ` G )
trgcopy.l	|- L = (LineG ` G )
trgcopy.k	|- K = (hlG ` G )
trgcopy.g	|- ( ph -> G e. TarskiG )
trgcopy.a	|- ( ph -> A e. P )
trgcopy.b	|- ( ph -> B e. P )
trgcopy.c	|- ( ph -> C e. P )
trgcopy.d	|- ( ph -> D e. P )
trgcopy.e	|- ( ph -> E e. P )
trgcopy.f	|- ( ph -> F e. P )
trgcopy.1	|- ( ph -> -. ( A e. ( B L C ) \/ B = C ) )
trgcopy.2	|- ( ph -> -. ( D e. ( E L F ) \/ E = F ) )
trgcopy.3	|- ( ph -> ( A .- B ) = ( D .- E ) )

Assertion

Ref	Expression
trgcopyeu	|- ( ph -> E! f e. P ( <" A B C "> (cgrG ` G ) <" D E f "> /\ f ( (hpG ` G ) ` ( D L E ) ) F ) )

Distinct variable groups:   .- , f   A, f   B, f   C, f   D, f   f, E   f, F   f, G   f, I   f, L   P, f   ph, f   f, K

Proof of Theorem trgcopyeu

Dummy variables a b k t x y z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		trgcopy.p	. . 3 |- P = ( Base ` G )
2		trgcopy.m	. . 3 |- .- = ( dist ` G )
3		trgcopy.i	. . 3 |- I = (Itv ` G )
4		trgcopy.l	. . 3 |- L = (LineG ` G )
5		trgcopy.k	. . 3 |- K = (hlG ` G )
6		trgcopy.g	. . 3 |- ( ph -> G e. TarskiG )
7		trgcopy.a	. . 3 |- ( ph -> A e. P )
8		trgcopy.b	. . 3 |- ( ph -> B e. P )
9		trgcopy.c	. . 3 |- ( ph -> C e. P )
10		trgcopy.d	. . 3 |- ( ph -> D e. P )
11		trgcopy.e	. . 3 |- ( ph -> E e. P )
12		trgcopy.f	. . 3 |- ( ph -> F e. P )
13		trgcopy.1	. . 3 |- ( ph -> -. ( A e. ( B L C ) \/ B = C ) )
14		trgcopy.2	. . 3 |- ( ph -> -. ( D e. ( E L F ) \/ E = F ) )
15		trgcopy.3	. . 3 |- ( ph -> ( A .- B ) = ( D .- E ) )
16	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15	trgcopy 25696	. 2 |- ( ph -> E. f e. P ( <" A B C "> (cgrG ` G ) <" D E f "> /\ f ( (hpG ` G ) ` ( D L E ) ) F ) )
17	6	ad5antr 770	. . . . . . . 8 |- ( ( ( ( ( ( ph /\ f e. P ) /\ k e. P ) /\ ( <" A B C "> (cgrG ` G ) <" D E f "> /\ f ( (hpG ` G ) ` ( D L E ) ) F ) ) /\ <" A B C "> (cgrG ` G ) <" D E k "> ) /\ k ( (hpG ` G ) ` ( D L E ) ) F ) -> G e. TarskiG )
18	7	ad5antr 770	. . . . . . . 8 |- ( ( ( ( ( ( ph /\ f e. P ) /\ k e. P ) /\ ( <" A B C "> (cgrG ` G ) <" D E f "> /\ f ( (hpG ` G ) ` ( D L E ) ) F ) ) /\ <" A B C "> (cgrG ` G ) <" D E k "> ) /\ k ( (hpG ` G ) ` ( D L E ) ) F ) -> A e. P )
19	8	ad5antr 770	. . . . . . . 8 |- ( ( ( ( ( ( ph /\ f e. P ) /\ k e. P ) /\ ( <" A B C "> (cgrG ` G ) <" D E f "> /\ f ( (hpG ` G ) ` ( D L E ) ) F ) ) /\ <" A B C "> (cgrG ` G ) <" D E k "> ) /\ k ( (hpG ` G ) ` ( D L E ) ) F ) -> B e. P )
20	9	ad5antr 770	. . . . . . . 8 |- ( ( ( ( ( ( ph /\ f e. P ) /\ k e. P ) /\ ( <" A B C "> (cgrG ` G ) <" D E f "> /\ f ( (hpG ` G ) ` ( D L E ) ) F ) ) /\ <" A B C "> (cgrG ` G ) <" D E k "> ) /\ k ( (hpG ` G ) ` ( D L E ) ) F ) -> C e. P )
21	10	ad5antr 770	. . . . . . . 8 |- ( ( ( ( ( ( ph /\ f e. P ) /\ k e. P ) /\ ( <" A B C "> (cgrG ` G ) <" D E f "> /\ f ( (hpG ` G ) ` ( D L E ) ) F ) ) /\ <" A B C "> (cgrG ` G ) <" D E k "> ) /\ k ( (hpG ` G ) ` ( D L E ) ) F ) -> D e. P )
22	11	ad5antr 770	. . . . . . . 8 |- ( ( ( ( ( ( ph /\ f e. P ) /\ k e. P ) /\ ( <" A B C "> (cgrG ` G ) <" D E f "> /\ f ( (hpG ` G ) ` ( D L E ) ) F ) ) /\ <" A B C "> (cgrG ` G ) <" D E k "> ) /\ k ( (hpG ` G ) ` ( D L E ) ) F ) -> E e. P )
23	12	ad5antr 770	. . . . . . . 8 |- ( ( ( ( ( ( ph /\ f e. P ) /\ k e. P ) /\ ( <" A B C "> (cgrG ` G ) <" D E f "> /\ f ( (hpG ` G ) ` ( D L E ) ) F ) ) /\ <" A B C "> (cgrG ` G ) <" D E k "> ) /\ k ( (hpG ` G ) ` ( D L E ) ) F ) -> F e. P )
24	13	ad5antr 770	. . . . . . . 8 |- ( ( ( ( ( ( ph /\ f e. P ) /\ k e. P ) /\ ( <" A B C "> (cgrG ` G ) <" D E f "> /\ f ( (hpG ` G ) ` ( D L E ) ) F ) ) /\ <" A B C "> (cgrG ` G ) <" D E k "> ) /\ k ( (hpG ` G ) ` ( D L E ) ) F ) -> -. ( A e. ( B L C ) \/ B = C ) )
25	14	ad5antr 770	. . . . . . . 8 |- ( ( ( ( ( ( ph /\ f e. P ) /\ k e. P ) /\ ( <" A B C "> (cgrG ` G ) <" D E f "> /\ f ( (hpG ` G ) ` ( D L E ) ) F ) ) /\ <" A B C "> (cgrG ` G ) <" D E k "> ) /\ k ( (hpG ` G ) ` ( D L E ) ) F ) -> -. ( D e. ( E L F ) \/ E = F ) )
26	15	ad5antr 770	. . . . . . . 8 |- ( ( ( ( ( ( ph /\ f e. P ) /\ k e. P ) /\ ( <" A B C "> (cgrG ` G ) <" D E f "> /\ f ( (hpG ` G ) ` ( D L E ) ) F ) ) /\ <" A B C "> (cgrG ` G ) <" D E k "> ) /\ k ( (hpG ` G ) ` ( D L E ) ) F ) -> ( A .- B ) = ( D .- E ) )
27		simpl 473	. . . . . . . . . . . 12 |- ( ( x = a /\ y = b ) -> x = a )
28	27	eleq1d 2686	. . . . . . . . . . 11 |- ( ( x = a /\ y = b ) -> ( x e. ( P \ ( D L E ) ) <-> a e. ( P \ ( D L E ) ) ) )
29		simpr 477	. . . . . . . . . . . 12 |- ( ( x = a /\ y = b ) -> y = b )
30	29	eleq1d 2686	. . . . . . . . . . 11 |- ( ( x = a /\ y = b ) -> ( y e. ( P \ ( D L E ) ) <-> b e. ( P \ ( D L E ) ) ) )
31	28, 30	anbi12d 747	. . . . . . . . . 10 |- ( ( x = a /\ y = b ) -> ( ( x e. ( P \ ( D L E ) ) /\ y e. ( P \ ( D L E ) ) ) <-> ( a e. ( P \ ( D L E ) ) /\ b e. ( P \ ( D L E ) ) ) ) )
32		simpr 477	. . . . . . . . . . . 12 |- ( ( ( x = a /\ y = b ) /\ z = t ) -> z = t )
33		simpll 790	. . . . . . . . . . . . 13 |- ( ( ( x = a /\ y = b ) /\ z = t ) -> x = a )
34		simplr 792	. . . . . . . . . . . . 13 |- ( ( ( x = a /\ y = b ) /\ z = t ) -> y = b )
35	33, 34	oveq12d 6668	. . . . . . . . . . . 12 |- ( ( ( x = a /\ y = b ) /\ z = t ) -> ( x I y ) = ( a I b ) )
36	32, 35	eleq12d 2695	. . . . . . . . . . 11 |- ( ( ( x = a /\ y = b ) /\ z = t ) -> ( z e. ( x I y ) <-> t e. ( a I b ) ) )
37	36	cbvrexdva 3178	. . . . . . . . . 10 |- ( ( x = a /\ y = b ) -> ( E. z e. ( D L E ) z e. ( x I y ) <-> E. t e. ( D L E ) t e. ( a I b ) ) )
38	31, 37	anbi12d 747	. . . . . . . . 9 |- ( ( x = a /\ y = b ) -> ( ( ( x e. ( P \ ( D L E ) ) /\ y e. ( P \ ( D L E ) ) ) /\ E. z e. ( D L E ) z e. ( x I y ) ) <-> ( ( a e. ( P \ ( D L E ) ) /\ b e. ( P \ ( D L E ) ) ) /\ E. t e. ( D L E ) t e. ( a I b ) ) ) )
39	38	cbvopabv 4722	. . . . . . . 8 |- { <. x , y >. | ( ( x e. ( P \ ( D L E ) ) /\ y e. ( P \ ( D L E ) ) ) /\ E. z e. ( D L E ) z e. ( x I y ) ) } = { <. a , b >. | ( ( a e. ( P \ ( D L E ) ) /\ b e. ( P \ ( D L E ) ) ) /\ E. t e. ( D L E ) t e. ( a I b ) ) }
40		simp-5r 809	. . . . . . . 8 |- ( ( ( ( ( ( ph /\ f e. P ) /\ k e. P ) /\ ( <" A B C "> (cgrG ` G ) <" D E f "> /\ f ( (hpG ` G ) ` ( D L E ) ) F ) ) /\ <" A B C "> (cgrG ` G ) <" D E k "> ) /\ k ( (hpG ` G ) ` ( D L E ) ) F ) -> f e. P )
41		simp-4r 807	. . . . . . . 8 |- ( ( ( ( ( ( ph /\ f e. P ) /\ k e. P ) /\ ( <" A B C "> (cgrG ` G ) <" D E f "> /\ f ( (hpG ` G ) ` ( D L E ) ) F ) ) /\ <" A B C "> (cgrG ` G ) <" D E k "> ) /\ k ( (hpG ` G ) ` ( D L E ) ) F ) -> k e. P )
42		simpllr 799	. . . . . . . . 9 |- ( ( ( ( ( ( ph /\ f e. P ) /\ k e. P ) /\ ( <" A B C "> (cgrG ` G ) <" D E f "> /\ f ( (hpG ` G ) ` ( D L E ) ) F ) ) /\ <" A B C "> (cgrG ` G ) <" D E k "> ) /\ k ( (hpG ` G ) ` ( D L E ) ) F ) -> ( <" A B C "> (cgrG ` G ) <" D E f "> /\ f ( (hpG ` G ) ` ( D L E ) ) F ) )
43	42	simpld 475	. . . . . . . 8 |- ( ( ( ( ( ( ph /\ f e. P ) /\ k e. P ) /\ ( <" A B C "> (cgrG ` G ) <" D E f "> /\ f ( (hpG ` G ) ` ( D L E ) ) F ) ) /\ <" A B C "> (cgrG ` G ) <" D E k "> ) /\ k ( (hpG ` G ) ` ( D L E ) ) F ) -> <" A B C "> (cgrG ` G ) <" D E f "> )
44		simplr 792	. . . . . . . 8 |- ( ( ( ( ( ( ph /\ f e. P ) /\ k e. P ) /\ ( <" A B C "> (cgrG ` G ) <" D E f "> /\ f ( (hpG ` G ) ` ( D L E ) ) F ) ) /\ <" A B C "> (cgrG ` G ) <" D E k "> ) /\ k ( (hpG ` G ) ` ( D L E ) ) F ) -> <" A B C "> (cgrG ` G ) <" D E k "> )
45	42	simprd 479	. . . . . . . 8 |- ( ( ( ( ( ( ph /\ f e. P ) /\ k e. P ) /\ ( <" A B C "> (cgrG ` G ) <" D E f "> /\ f ( (hpG ` G ) ` ( D L E ) ) F ) ) /\ <" A B C "> (cgrG ` G ) <" D E k "> ) /\ k ( (hpG ` G ) ` ( D L E ) ) F ) -> f ( (hpG ` G ) ` ( D L E ) ) F )
46		simpr 477	. . . . . . . 8 |- ( ( ( ( ( ( ph /\ f e. P ) /\ k e. P ) /\ ( <" A B C "> (cgrG ` G ) <" D E f "> /\ f ( (hpG ` G ) ` ( D L E ) ) F ) ) /\ <" A B C "> (cgrG ` G ) <" D E k "> ) /\ k ( (hpG ` G ) ` ( D L E ) ) F ) -> k ( (hpG ` G ) ` ( D L E ) ) F )
47	1, 2, 3, 4, 5, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 39, 40, 41, 43, 44, 45, 46	trgcopyeulem 25697	. . . . . . 7 |- ( ( ( ( ( ( ph /\ f e. P ) /\ k e. P ) /\ ( <" A B C "> (cgrG ` G ) <" D E f "> /\ f ( (hpG ` G ) ` ( D L E ) ) F ) ) /\ <" A B C "> (cgrG ` G ) <" D E k "> ) /\ k ( (hpG ` G ) ` ( D L E ) ) F ) -> f = k )
48	47	anasss 679	. . . . . 6 |- ( ( ( ( ( ph /\ f e. P ) /\ k e. P ) /\ ( <" A B C "> (cgrG ` G ) <" D E f "> /\ f ( (hpG ` G ) ` ( D L E ) ) F ) ) /\ ( <" A B C "> (cgrG ` G ) <" D E k "> /\ k ( (hpG ` G ) ` ( D L E ) ) F ) ) -> f = k )
49	48	anasss 679	. . . . 5 |- ( ( ( ( ph /\ f e. P ) /\ k e. P ) /\ ( ( <" A B C "> (cgrG ` G ) <" D E f "> /\ f ( (hpG ` G ) ` ( D L E ) ) F ) /\ ( <" A B C "> (cgrG ` G ) <" D E k "> /\ k ( (hpG ` G ) ` ( D L E ) ) F ) ) ) -> f = k )
50	49	ex 450	. . . 4 |- ( ( ( ph /\ f e. P ) /\ k e. P ) -> ( ( ( <" A B C "> (cgrG ` G ) <" D E f "> /\ f ( (hpG ` G ) ` ( D L E ) ) F ) /\ ( <" A B C "> (cgrG ` G ) <" D E k "> /\ k ( (hpG ` G ) ` ( D L E ) ) F ) ) -> f = k ) )
51	50	anasss 679	. . 3 |- ( ( ph /\ ( f e. P /\ k e. P ) ) -> ( ( ( <" A B C "> (cgrG ` G ) <" D E f "> /\ f ( (hpG ` G ) ` ( D L E ) ) F ) /\ ( <" A B C "> (cgrG ` G ) <" D E k "> /\ k ( (hpG ` G ) ` ( D L E ) ) F ) ) -> f = k ) )
52	51	ralrimivva 2971	. 2 |- ( ph -> A. f e. P A. k e. P ( ( ( <" A B C "> (cgrG ` G ) <" D E f "> /\ f ( (hpG ` G ) ` ( D L E ) ) F ) /\ ( <" A B C "> (cgrG ` G ) <" D E k "> /\ k ( (hpG ` G ) ` ( D L E ) ) F ) ) -> f = k ) )
53		eqidd 2623	. . . . . 6 |- ( f = k -> D = D )
54		eqidd 2623	. . . . . 6 |- ( f = k -> E = E )
55		id 22	. . . . . 6 |- ( f = k -> f = k )
56	53, 54, 55	s3eqd 13609	. . . . 5 |- ( f = k -> <" D E f "> = <" D E k "> )
57	56	breq2d 4665	. . . 4 |- ( f = k -> ( <" A B C "> (cgrG ` G ) <" D E f "> <-> <" A B C "> (cgrG ` G ) <" D E k "> ) )
58		breq1 4656	. . . 4 |- ( f = k -> ( f ( (hpG ` G ) ` ( D L E ) ) F <-> k ( (hpG ` G ) ` ( D L E ) ) F ) )
59	57, 58	anbi12d 747	. . 3 |- ( f = k -> ( ( <" A B C "> (cgrG ` G ) <" D E f "> /\ f ( (hpG ` G ) ` ( D L E ) ) F ) <-> ( <" A B C "> (cgrG ` G ) <" D E k "> /\ k ( (hpG ` G ) ` ( D L E ) ) F ) ) )
60	59	reu4 3400	. 2 |- ( E! f e. P ( <" A B C "> (cgrG ` G ) <" D E f "> /\ f ( (hpG ` G ) ` ( D L E ) ) F ) <-> ( E. f e. P ( <" A B C "> (cgrG ` G ) <" D E f "> /\ f ( (hpG ` G ) ` ( D L E ) ) F ) /\ A. f e. P A. k e. P ( ( ( <" A B C "> (cgrG ` G ) <" D E f "> /\ f ( (hpG ` G ) ` ( D L E ) ) F ) /\ ( <" A B C "> (cgrG ` G ) <" D E k "> /\ k ( (hpG ` G ) ` ( D L E ) ) F ) ) -> f = k ) ) )
61	16, 52, 60	sylanbrc 698	1 |- ( ph -> E! f e. P ( <" A B C "> (cgrG ` G ) <" D E f "> /\ f ( (hpG ` G ) ` ( D L E ) ) F ) )

Syntax hints:   -. wn 3   -> wi 4   \/ wo 383   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   E.wrex 2913   E!wreu 2914   \ cdif 3571   class class class wbr 4653   {copab 4712   ` cfv 5888  (class class class)co 6650   <"cs3 13587   Basecbs 15857   distcds 15950  TarskiGcstrkg 25329  Itvcitv 25335  LineGclng 25336  cgrGccgrg 25405  hlGchlg 25495  hpGchpg 25649

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-fal 1489  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-trkgld 25351  df-trkg 25352  df-cgrg 25406  df-ismt 25428  df-leg 25478  df-hlg 25496  df-mir 25548  df-rag 25589  df-perpg 25591  df-hpg 25650  df-mid 25666  df-lmi 25667

This theorem is referenced by: (None)