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Theorem mideu 24779
Description: Existence and uniqueness of the midpoint, Theorem 8.22 of [Schwabhauser] p. 64. (Contributed by Thierry Arnoux, 25-Nov-2019.)
Hypotheses
Ref Expression
colperpex.p  |-  P  =  ( Base `  G
)
colperpex.d  |-  .-  =  ( dist `  G )
colperpex.i  |-  I  =  (Itv `  G )
colperpex.l  |-  L  =  (LineG `  G )
colperpex.g  |-  ( ph  ->  G  e. TarskiG )
mideu.s  |-  S  =  (pInvG `  G )
mideu.1  |-  ( ph  ->  A  e.  P )
mideu.2  |-  ( ph  ->  B  e.  P )
mideu.3  |-  ( ph  ->  GDimTarskiG 2 )
Assertion
Ref Expression
mideu  |-  ( ph  ->  E! x  e.  P  B  =  ( ( S `  x ) `  A ) )
Distinct variable groups:    x,  .-    x, A   
x, B    x, G    x, I    x, L    x, P    x, S    ph, x

Proof of Theorem mideu
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 colperpex.p . . 3  |-  P  =  ( Base `  G
)
2 colperpex.d . . 3  |-  .-  =  ( dist `  G )
3 colperpex.i . . 3  |-  I  =  (Itv `  G )
4 colperpex.l . . 3  |-  L  =  (LineG `  G )
5 colperpex.g . . 3  |-  ( ph  ->  G  e. TarskiG )
6 mideu.s . . 3  |-  S  =  (pInvG `  G )
7 mideu.1 . . 3  |-  ( ph  ->  A  e.  P )
8 mideu.2 . . 3  |-  ( ph  ->  B  e.  P )
9 mideu.3 . . 3  |-  ( ph  ->  GDimTarskiG 2 )
101, 2, 3, 4, 5, 6, 7, 8, 9midex 24778 . 2  |-  ( ph  ->  E. x  e.  P  B  =  ( ( S `  x ) `  A ) )
115ad2antrr 730 . . . . . 6  |-  ( ( ( ph  /\  (
x  e.  P  /\  y  e.  P )
)  /\  ( B  =  ( ( S `
 x ) `  A )  /\  B  =  ( ( S `
 y ) `  A ) ) )  ->  G  e. TarskiG )
12 simplrl 768 . . . . . 6  |-  ( ( ( ph  /\  (
x  e.  P  /\  y  e.  P )
)  /\  ( B  =  ( ( S `
 x ) `  A )  /\  B  =  ( ( S `
 y ) `  A ) ) )  ->  x  e.  P
)
13 simplrr 769 . . . . . 6  |-  ( ( ( ph  /\  (
x  e.  P  /\  y  e.  P )
)  /\  ( B  =  ( ( S `
 x ) `  A )  /\  B  =  ( ( S `
 y ) `  A ) ) )  ->  y  e.  P
)
147ad2antrr 730 . . . . . 6  |-  ( ( ( ph  /\  (
x  e.  P  /\  y  e.  P )
)  /\  ( B  =  ( ( S `
 x ) `  A )  /\  B  =  ( ( S `
 y ) `  A ) ) )  ->  A  e.  P
)
158ad2antrr 730 . . . . . 6  |-  ( ( ( ph  /\  (
x  e.  P  /\  y  e.  P )
)  /\  ( B  =  ( ( S `
 x ) `  A )  /\  B  =  ( ( S `
 y ) `  A ) ) )  ->  B  e.  P
)
16 simprl 762 . . . . . . 7  |-  ( ( ( ph  /\  (
x  e.  P  /\  y  e.  P )
)  /\  ( B  =  ( ( S `
 x ) `  A )  /\  B  =  ( ( S `
 y ) `  A ) ) )  ->  B  =  ( ( S `  x
) `  A )
)
1716eqcomd 2430 . . . . . 6  |-  ( ( ( ph  /\  (
x  e.  P  /\  y  e.  P )
)  /\  ( B  =  ( ( S `
 x ) `  A )  /\  B  =  ( ( S `
 y ) `  A ) ) )  ->  ( ( S `
 x ) `  A )  =  B )
18 simprr 764 . . . . . . 7  |-  ( ( ( ph  /\  (
x  e.  P  /\  y  e.  P )
)  /\  ( B  =  ( ( S `
 x ) `  A )  /\  B  =  ( ( S `
 y ) `  A ) ) )  ->  B  =  ( ( S `  y
) `  A )
)
1918eqcomd 2430 . . . . . 6  |-  ( ( ( ph  /\  (
x  e.  P  /\  y  e.  P )
)  /\  ( B  =  ( ( S `
 x ) `  A )  /\  B  =  ( ( S `
 y ) `  A ) ) )  ->  ( ( S `
 y ) `  A )  =  B )
201, 2, 3, 4, 6, 11, 12, 13, 14, 15, 17, 19miduniq 24729 . . . . 5  |-  ( ( ( ph  /\  (
x  e.  P  /\  y  e.  P )
)  /\  ( B  =  ( ( S `
 x ) `  A )  /\  B  =  ( ( S `
 y ) `  A ) ) )  ->  x  =  y )
2120ex 435 . . . 4  |-  ( (
ph  /\  ( x  e.  P  /\  y  e.  P ) )  -> 
( ( B  =  ( ( S `  x ) `  A
)  /\  B  =  ( ( S `  y ) `  A
) )  ->  x  =  y ) )
2221ralrimivva 2843 . . 3  |-  ( ph  ->  A. x  e.  P  A. y  e.  P  ( ( B  =  ( ( S `  x ) `  A
)  /\  B  =  ( ( S `  y ) `  A
) )  ->  x  =  y ) )
23 fveq2 5882 . . . . . 6  |-  ( x  =  y  ->  ( S `  x )  =  ( S `  y ) )
2423fveq1d 5884 . . . . 5  |-  ( x  =  y  ->  (
( S `  x
) `  A )  =  ( ( S `
 y ) `  A ) )
2524eqeq2d 2436 . . . 4  |-  ( x  =  y  ->  ( B  =  ( ( S `  x ) `  A )  <->  B  =  ( ( S `  y ) `  A
) ) )
2625rmo4 3263 . . 3  |-  ( E* x  e.  P  B  =  ( ( S `
 x ) `  A )  <->  A. x  e.  P  A. y  e.  P  ( ( B  =  ( ( S `  x ) `  A )  /\  B  =  ( ( S `
 y ) `  A ) )  ->  x  =  y )
)
2722, 26sylibr 215 . 2  |-  ( ph  ->  E* x  e.  P  B  =  ( ( S `  x ) `  A ) )
28 reu5 3043 . 2  |-  ( E! x  e.  P  B  =  ( ( S `
 x ) `  A )  <->  ( E. x  e.  P  B  =  ( ( S `
 x ) `  A )  /\  E* x  e.  P  B  =  ( ( S `
 x ) `  A ) ) )
2910, 27, 28sylanbrc 668 1  |-  ( ph  ->  E! x  e.  P  B  =  ( ( S `  x ) `  A ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 370    = wceq 1437    e. wcel 1872   A.wral 2771   E.wrex 2772   E!wreu 2773   E*wrmo 2774   class class class wbr 4423   ` cfv 5601   2c2 10667   Basecbs 15121   distcds 15199  TarskiGcstrkg 24477  DimTarskiGcstrkgld 24481  Itvcitv 24483  LineGclng 24484  pInvGcmir 24696
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2057  ax-ext 2401  ax-rep 4536  ax-sep 4546  ax-nul 4555  ax-pow 4602  ax-pr 4660  ax-un 6598  ax-cnex 9603  ax-resscn 9604  ax-1cn 9605  ax-icn 9606  ax-addcl 9607  ax-addrcl 9608  ax-mulcl 9609  ax-mulrcl 9610  ax-mulcom 9611  ax-addass 9612  ax-mulass 9613  ax-distr 9614  ax-i2m1 9615  ax-1ne0 9616  ax-1rid 9617  ax-rnegex 9618  ax-rrecex 9619  ax-cnre 9620  ax-pre-lttri 9621  ax-pre-lttrn 9622  ax-pre-ltadd 9623  ax-pre-mulgt0 9624
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2273  df-mo 2274  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2568  df-ne 2616  df-nel 2617  df-ral 2776  df-rex 2777  df-reu 2778  df-rmo 2779  df-rab 2780  df-v 3082  df-sbc 3300  df-csb 3396  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-pss 3452  df-nul 3762  df-if 3912  df-pw 3983  df-sn 3999  df-pr 4001  df-tp 4003  df-op 4005  df-uni 4220  df-int 4256  df-iun 4301  df-br 4424  df-opab 4483  df-mpt 4484  df-tr 4519  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 6268  df-ov 6309  df-oprab 6310  df-mpt2 6311  df-om 6708  df-1st 6808  df-2nd 6809  df-wrecs 7040  df-recs 7102  df-rdg 7140  df-1o 7194  df-oadd 7198  df-er 7375  df-map 7486  df-pm 7487  df-en 7582  df-dom 7583  df-sdom 7584  df-fin 7585  df-card 8382  df-cda 8606  df-pnf 9685  df-mnf 9686  df-xr 9687  df-ltxr 9688  df-le 9689  df-sub 9870  df-neg 9871  df-nn 10618  df-2 10676  df-3 10677  df-n0 10878  df-z 10946  df-uz 11168  df-fz 11793  df-fzo 11924  df-hash 12523  df-word 12669  df-concat 12671  df-s1 12672  df-s2 12947  df-s3 12948  df-trkgc 24495  df-trkgb 24496  df-trkgcb 24497  df-trkgld 24499  df-trkg 24500  df-cgrg 24555  df-leg 24627  df-mir 24697  df-rag 24738  df-perpg 24740
This theorem is referenced by:  midf  24817  ismidb  24819
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