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Theorem trgcopyeu 24897
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.
StepHypRef 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 ) )
161, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15trgcopy 24895 . 2  |-  ( ph  ->  E. f  e.  P  ( <" A B C "> (cgrG `  G ) <" D E f ">  /\  f ( (hpG `  G ) `  ( D L E ) ) F ) )
176ad5antr 745 . . . . . . . 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 )
187ad5antr 745 . . . . . . . 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 )
198ad5antr 745 . . . . . . . 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 )
209ad5antr 745 . . . . . . . 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 )
2110ad5antr 745 . . . . . . . 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 )
2211ad5antr 745 . . . . . . . 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 )
2312ad5antr 745 . . . . . . . 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 )
2413ad5antr 745 . . . . . . . 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 ) )
2514ad5antr 745 . . . . . . . 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 ) )
2615ad5antr 745 . . . . . . . 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 463 . . . . . . . . . . . 12  |-  ( ( x  =  a  /\  y  =  b )  ->  x  =  a )
2827eleq1d 2524 . . . . . . . . . . 11  |-  ( ( x  =  a  /\  y  =  b )  ->  ( x  e.  ( P  \  ( D L E ) )  <-> 
a  e.  ( P 
\  ( D L E ) ) ) )
29 simpr 467 . . . . . . . . . . . 12  |-  ( ( x  =  a  /\  y  =  b )  ->  y  =  b )
3029eleq1d 2524 . . . . . . . . . . 11  |-  ( ( x  =  a  /\  y  =  b )  ->  ( y  e.  ( P  \  ( D L E ) )  <-> 
b  e.  ( P 
\  ( D L E ) ) ) )
3128, 30anbi12d 722 . . . . . . . . . 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 467 . . . . . . . . . . . 12  |-  ( ( ( x  =  a  /\  y  =  b )  /\  z  =  t )  ->  z  =  t )
33 simpll 765 . . . . . . . . . . . . 13  |-  ( ( ( x  =  a  /\  y  =  b )  /\  z  =  t )  ->  x  =  a )
34 simplr 767 . . . . . . . . . . . . 13  |-  ( ( ( x  =  a  /\  y  =  b )  /\  z  =  t )  ->  y  =  b )
3533, 34oveq12d 6333 . . . . . . . . . . . 12  |-  ( ( ( x  =  a  /\  y  =  b )  /\  z  =  t )  ->  (
x I y )  =  ( a I b ) )
3632, 35eleq12d 2534 . . . . . . . . . . 11  |-  ( ( ( x  =  a  /\  y  =  b )  /\  z  =  t )  ->  (
z  e.  ( x I y )  <->  t  e.  ( a I b ) ) )
3736cbvrexdva 3038 . . . . . . . . . 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 ) ) )
3831, 37anbi12d 722 . . . . . . . . 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 ) ) ) )
3938cbvopabv 4486 . . . . . . . 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 784 . . . . . . . 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 782 . . . . . . . 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 774 . . . . . . . . 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 ) )
4342simpld 465 . . . . . . . 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 767 . . . . . . . 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 "> )
4542simprd 469 . . . . . . . 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 467 . . . . . . . 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 )
471, 2, 3, 4, 5, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 39, 40, 41, 43, 44, 45, 46trgcopyeulem 24896 . . . . . . 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 )
4847anasss 657 . . . . . 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 )
4948anasss 657 . . . . 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 )
5049ex 440 . . . 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 ) )
5150anasss 657 . . 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 ) )
5251ralrimivva 2821 . 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 2463 . . . . . 6  |-  ( f  =  k  ->  D  =  D )
54 eqidd 2463 . . . . . 6  |-  ( f  =  k  ->  E  =  E )
55 id 22 . . . . . 6  |-  ( f  =  k  ->  f  =  k )
5653, 54, 55s3eqd 12996 . . . . 5  |-  ( f  =  k  ->  <" D E f ">  =  <" D E k "> )
5756breq2d 4428 . . . 4  |-  ( f  =  k  ->  ( <" A B C "> (cgrG `  G ) <" D E f ">  <->  <" A B C "> (cgrG `  G ) <" D E k "> ) )
58 breq1 4419 . . . 4  |-  ( f  =  k  ->  (
f ( (hpG `  G ) `  ( D L E ) ) F  <->  k ( (hpG
`  G ) `  ( D L E ) ) F ) )
5957, 58anbi12d 722 . . 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 ) ) )
6059reu4 3244 . 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 ) ) )
6116, 52, 60sylanbrc 675 1  |-  ( ph  ->  E! f  e.  P  ( <" A B C "> (cgrG `  G ) <" D E f ">  /\  f ( (hpG `  G ) `  ( D L E ) ) F ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    \/ wo 374    /\ wa 375    = wceq 1455    e. wcel 1898   A.wral 2749   E.wrex 2750   E!wreu 2751    \ cdif 3413   class class class wbr 4416   {copab 4474   ` cfv 5601  (class class class)co 6315   <"cs3 12975   Basecbs 15170   distcds 15248  TarskiGcstrkg 24527  Itvcitv 24533  LineGclng 24534  cgrGccgrg 24604  hlGchlg 24694  hpGchpg 24848
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1680  ax-4 1693  ax-5 1769  ax-6 1816  ax-7 1862  ax-8 1900  ax-9 1907  ax-10 1926  ax-11 1931  ax-12 1944  ax-13 2102  ax-ext 2442  ax-rep 4529  ax-sep 4539  ax-nul 4548  ax-pow 4595  ax-pr 4653  ax-un 6610  ax-cnex 9621  ax-resscn 9622  ax-1cn 9623  ax-icn 9624  ax-addcl 9625  ax-addrcl 9626  ax-mulcl 9627  ax-mulrcl 9628  ax-mulcom 9629  ax-addass 9630  ax-mulass 9631  ax-distr 9632  ax-i2m1 9633  ax-1ne0 9634  ax-1rid 9635  ax-rnegex 9636  ax-rrecex 9637  ax-cnre 9638  ax-pre-lttri 9639  ax-pre-lttrn 9640  ax-pre-ltadd 9641  ax-pre-mulgt0 9642
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3or 992  df-3an 993  df-tru 1458  df-fal 1461  df-ex 1675  df-nf 1679  df-sb 1809  df-eu 2314  df-mo 2315  df-clab 2449  df-cleq 2455  df-clel 2458  df-nfc 2592  df-ne 2635  df-nel 2636  df-ral 2754  df-rex 2755  df-reu 2756  df-rmo 2757  df-rab 2758  df-v 3059  df-sbc 3280  df-csb 3376  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-pss 3432  df-nul 3744  df-if 3894  df-pw 3965  df-sn 3981  df-pr 3983  df-tp 3985  df-op 3987  df-uni 4213  df-int 4249  df-iun 4294  df-br 4417  df-opab 4476  df-mpt 4477  df-tr 4512  df-eprel 4764  df-id 4768  df-po 4774  df-so 4775  df-fr 4812  df-we 4814  df-xp 4859  df-rel 4860  df-cnv 4861  df-co 4862  df-dm 4863  df-rn 4864  df-res 4865  df-ima 4866  df-pred 5399  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-riota 6277  df-ov 6318  df-oprab 6319  df-mpt2 6320  df-om 6720  df-1st 6820  df-2nd 6821  df-wrecs 7054  df-recs 7116  df-rdg 7154  df-1o 7208  df-oadd 7212  df-er 7389  df-map 7500  df-pm 7501  df-en 7596  df-dom 7597  df-sdom 7598  df-fin 7599  df-card 8399  df-cda 8624  df-pnf 9703  df-mnf 9704  df-xr 9705  df-ltxr 9706  df-le 9707  df-sub 9888  df-neg 9889  df-nn 10638  df-2 10696  df-3 10697  df-n0 10899  df-z 10967  df-uz 11189  df-fz 11814  df-fzo 11947  df-hash 12548  df-word 12697  df-concat 12699  df-s1 12700  df-s2 12981  df-s3 12982  df-trkgc 24545  df-trkgb 24546  df-trkgcb 24547  df-trkgld 24549  df-trkg 24550  df-cgrg 24605  df-ismt 24627  df-leg 24677  df-hlg 24695  df-mir 24747  df-rag 24788  df-perpg 24790  df-hpg 24849  df-mid 24865  df-lmi 24866
This theorem is referenced by: (None)
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