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Theorem footeq 24815
Description: Uniqueness of the foot point. (Contributed by Thierry Arnoux, 1-Mar-2020.)
Hypotheses
Ref Expression
isperp.p  |-  P  =  ( Base `  G
)
isperp.d  |-  .-  =  ( dist `  G )
isperp.i  |-  I  =  (Itv `  G )
isperp.l  |-  L  =  (LineG `  G )
isperp.g  |-  ( ph  ->  G  e. TarskiG )
isperp.a  |-  ( ph  ->  A  e.  ran  L
)
footeq.x  |-  ( ph  ->  X  e.  A )
footeq.y  |-  ( ph  ->  Y  e.  A )
footeq.z  |-  ( ph  ->  Z  e.  P )
footeq.1  |-  ( ph  ->  ( X L Z ) (⟂G `  G
) A )
footeq.2  |-  ( ph  ->  ( Y L Z ) (⟂G `  G
) A )
Assertion
Ref Expression
footeq  |-  ( ph  ->  X  =  Y )

Proof of Theorem footeq
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 oveq2 6323 . . 3  |-  ( x  =  X  ->  ( Z L x )  =  ( Z L X ) )
21breq1d 4426 . 2  |-  ( x  =  X  ->  (
( Z L x ) (⟂G `  G
) A  <->  ( Z L X ) (⟂G `  G
) A ) )
3 oveq2 6323 . . 3  |-  ( x  =  Y  ->  ( Z L x )  =  ( Z L Y ) )
43breq1d 4426 . 2  |-  ( x  =  Y  ->  (
( Z L x ) (⟂G `  G
) A  <->  ( Z L Y ) (⟂G `  G
) A ) )
5 isperp.p . . 3  |-  P  =  ( Base `  G
)
6 isperp.d . . 3  |-  .-  =  ( dist `  G )
7 isperp.i . . 3  |-  I  =  (Itv `  G )
8 isperp.l . . 3  |-  L  =  (LineG `  G )
9 isperp.g . . 3  |-  ( ph  ->  G  e. TarskiG )
10 isperp.a . . 3  |-  ( ph  ->  A  e.  ran  L
)
11 footeq.z . . 3  |-  ( ph  ->  Z  e.  P )
12 footeq.x . . . 4  |-  ( ph  ->  X  e.  A )
13 footeq.1 . . . 4  |-  ( ph  ->  ( X L Z ) (⟂G `  G
) A )
145, 6, 7, 8, 9, 10, 12, 11, 13footne 24814 . . 3  |-  ( ph  ->  -.  Z  e.  A
)
155, 6, 7, 8, 9, 10, 11, 14foot 24813 . 2  |-  ( ph  ->  E! x  e.  A  ( Z L x ) (⟂G `  G ) A )
16 footeq.y . 2  |-  ( ph  ->  Y  e.  A )
175, 8, 7, 9, 10, 12tglnpt 24643 . . . 4  |-  ( ph  ->  X  e.  P )
188, 9, 13perpln1 24804 . . . . 5  |-  ( ph  ->  ( X L Z )  e.  ran  L
)
195, 7, 8, 9, 17, 11, 18tglnne 24722 . . . 4  |-  ( ph  ->  X  =/=  Z )
205, 7, 8, 9, 17, 11, 19tglinecom 24729 . . 3  |-  ( ph  ->  ( X L Z )  =  ( Z L X ) )
2120, 13eqbrtrrd 4439 . 2  |-  ( ph  ->  ( Z L X ) (⟂G `  G
) A )
225, 8, 7, 9, 10, 16tglnpt 24643 . . . 4  |-  ( ph  ->  Y  e.  P )
23 footeq.2 . . . . . 6  |-  ( ph  ->  ( Y L Z ) (⟂G `  G
) A )
248, 9, 23perpln1 24804 . . . . 5  |-  ( ph  ->  ( Y L Z )  e.  ran  L
)
255, 7, 8, 9, 22, 11, 24tglnne 24722 . . . 4  |-  ( ph  ->  Y  =/=  Z )
265, 7, 8, 9, 22, 11, 25tglinecom 24729 . . 3  |-  ( ph  ->  ( Y L Z )  =  ( Z L Y ) )
2726, 23eqbrtrrd 4439 . 2  |-  ( ph  ->  ( Z L Y ) (⟂G `  G
) A )
282, 4, 15, 12, 16, 21, 27reu2eqd 3247 1  |-  ( ph  ->  X  =  Y )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1455    e. wcel 1898   class class class wbr 4416   ran crn 4854   ` cfv 5601  (class class class)co 6315   Basecbs 15170   distcds 15248  TarskiGcstrkg 24527  Itvcitv 24533  LineGclng 24534  ⟂Gcperpg 24789
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-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-trkg 24550  df-cgrg 24605  df-leg 24677  df-mir 24747  df-rag 24788  df-perpg 24790
This theorem is referenced by: (None)
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