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Theorem mideulem 24635
Description: Lemma for mideu 24637. We can assume mideulem.9 "without loss of generality" (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 )
mideulem.1  |-  ( ph  ->  A  =/=  B )
mideulem.2  |-  ( ph  ->  Q  e.  P )
mideulem.3  |-  ( ph  ->  O  e.  P )
mideulem.4  |-  ( ph  ->  T  e.  P )
mideulem.5  |-  ( ph  ->  ( A L B ) (⟂G `  G
) ( Q L B ) )
mideulem.6  |-  ( ph  ->  ( A L B ) (⟂G `  G
) ( A L O ) )
mideulem.7  |-  ( ph  ->  T  e.  ( A L B ) )
mideulem.8  |-  ( ph  ->  T  e.  ( Q I O ) )
mideulem.9  |-  ( ph  ->  ( A  .-  O
) (≤G `  G
) ( B  .-  Q ) )
Assertion
Ref Expression
mideulem  |-  ( ph  ->  E. x  e.  P  B  =  ( ( S `  x ) `  A ) )
Distinct variable groups:    x,  .-    x, A   
x, B    x, I    x, O    x, P    x, Q    x, T    ph, x
Allowed substitution hints:    S( x)    G( x)    L( x)

Proof of Theorem mideulem
Dummy variable  r is distinct from all other variables.
StepHypRef Expression
1 simprrl 772 . . 3  |-  ( ( ( ( ph  /\  r  e.  P )  /\  ( r  e.  ( B I Q )  /\  ( A  .-  O )  =  ( B  .-  r ) ) )  /\  (
x  e.  P  /\  ( B  =  (
( S `  x
) `  A )  /\  O  =  (
( S `  x
) `  r )
) ) )  ->  B  =  ( ( S `  x ) `  A ) )
2 colperpex.p . . . 4  |-  P  =  ( Base `  G
)
3 colperpex.d . . . 4  |-  .-  =  ( dist `  G )
4 colperpex.i . . . 4  |-  I  =  (Itv `  G )
5 colperpex.l . . . 4  |-  L  =  (LineG `  G )
6 colperpex.g . . . . 5  |-  ( ph  ->  G  e. TarskiG )
76ad2antrr 730 . . . 4  |-  ( ( ( ph  /\  r  e.  P )  /\  (
r  e.  ( B I Q )  /\  ( A  .-  O )  =  ( B  .-  r ) ) )  ->  G  e. TarskiG )
8 mideu.s . . . 4  |-  S  =  (pInvG `  G )
9 mideu.1 . . . . 5  |-  ( ph  ->  A  e.  P )
109ad2antrr 730 . . . 4  |-  ( ( ( ph  /\  r  e.  P )  /\  (
r  e.  ( B I Q )  /\  ( A  .-  O )  =  ( B  .-  r ) ) )  ->  A  e.  P
)
11 mideu.2 . . . . 5  |-  ( ph  ->  B  e.  P )
1211ad2antrr 730 . . . 4  |-  ( ( ( ph  /\  r  e.  P )  /\  (
r  e.  ( B I Q )  /\  ( A  .-  O )  =  ( B  .-  r ) ) )  ->  B  e.  P
)
13 mideulem.1 . . . . 5  |-  ( ph  ->  A  =/=  B )
1413ad2antrr 730 . . . 4  |-  ( ( ( ph  /\  r  e.  P )  /\  (
r  e.  ( B I Q )  /\  ( A  .-  O )  =  ( B  .-  r ) ) )  ->  A  =/=  B
)
15 mideulem.2 . . . . 5  |-  ( ph  ->  Q  e.  P )
1615ad2antrr 730 . . . 4  |-  ( ( ( ph  /\  r  e.  P )  /\  (
r  e.  ( B I Q )  /\  ( A  .-  O )  =  ( B  .-  r ) ) )  ->  Q  e.  P
)
17 mideulem.3 . . . . 5  |-  ( ph  ->  O  e.  P )
1817ad2antrr 730 . . . 4  |-  ( ( ( ph  /\  r  e.  P )  /\  (
r  e.  ( B I Q )  /\  ( A  .-  O )  =  ( B  .-  r ) ) )  ->  O  e.  P
)
19 mideulem.4 . . . . 5  |-  ( ph  ->  T  e.  P )
2019ad2antrr 730 . . . 4  |-  ( ( ( ph  /\  r  e.  P )  /\  (
r  e.  ( B I Q )  /\  ( A  .-  O )  =  ( B  .-  r ) ) )  ->  T  e.  P
)
21 mideulem.5 . . . . 5  |-  ( ph  ->  ( A L B ) (⟂G `  G
) ( Q L B ) )
2221ad2antrr 730 . . . 4  |-  ( ( ( ph  /\  r  e.  P )  /\  (
r  e.  ( B I Q )  /\  ( A  .-  O )  =  ( B  .-  r ) ) )  ->  ( A L B ) (⟂G `  G
) ( Q L B ) )
23 mideulem.6 . . . . 5  |-  ( ph  ->  ( A L B ) (⟂G `  G
) ( A L O ) )
2423ad2antrr 730 . . . 4  |-  ( ( ( ph  /\  r  e.  P )  /\  (
r  e.  ( B I Q )  /\  ( A  .-  O )  =  ( B  .-  r ) ) )  ->  ( A L B ) (⟂G `  G
) ( A L O ) )
25 mideulem.7 . . . . 5  |-  ( ph  ->  T  e.  ( A L B ) )
2625ad2antrr 730 . . . 4  |-  ( ( ( ph  /\  r  e.  P )  /\  (
r  e.  ( B I Q )  /\  ( A  .-  O )  =  ( B  .-  r ) ) )  ->  T  e.  ( A L B ) )
27 mideulem.8 . . . . 5  |-  ( ph  ->  T  e.  ( Q I O ) )
2827ad2antrr 730 . . . 4  |-  ( ( ( ph  /\  r  e.  P )  /\  (
r  e.  ( B I Q )  /\  ( A  .-  O )  =  ( B  .-  r ) ) )  ->  T  e.  ( Q I O ) )
29 simplr 760 . . . 4  |-  ( ( ( ph  /\  r  e.  P )  /\  (
r  e.  ( B I Q )  /\  ( A  .-  O )  =  ( B  .-  r ) ) )  ->  r  e.  P
)
30 simprl 762 . . . 4  |-  ( ( ( ph  /\  r  e.  P )  /\  (
r  e.  ( B I Q )  /\  ( A  .-  O )  =  ( B  .-  r ) ) )  ->  r  e.  ( B I Q ) )
31 simprr 764 . . . 4  |-  ( ( ( ph  /\  r  e.  P )  /\  (
r  e.  ( B I Q )  /\  ( A  .-  O )  =  ( B  .-  r ) ) )  ->  ( A  .-  O )  =  ( B  .-  r ) )
322, 3, 4, 5, 7, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 29, 30, 31opphllem 24634 . . 3  |-  ( ( ( ph  /\  r  e.  P )  /\  (
r  e.  ( B I Q )  /\  ( A  .-  O )  =  ( B  .-  r ) ) )  ->  E. x  e.  P  ( B  =  (
( S `  x
) `  A )  /\  O  =  (
( S `  x
) `  r )
) )
331, 32reximddv 2908 . 2  |-  ( ( ( ph  /\  r  e.  P )  /\  (
r  e.  ( B I Q )  /\  ( A  .-  O )  =  ( B  .-  r ) ) )  ->  E. x  e.  P  B  =  ( ( S `  x ) `  A ) )
34 mideulem.9 . . 3  |-  ( ph  ->  ( A  .-  O
) (≤G `  G
) ( B  .-  Q ) )
35 eqid 2429 . . . 4  |-  (≤G `  G )  =  (≤G `  G )
362, 3, 4, 35, 6, 9, 17, 11, 15legov 24490 . . 3  |-  ( ph  ->  ( ( A  .-  O ) (≤G `  G ) ( B 
.-  Q )  <->  E. r  e.  P  ( r  e.  ( B I Q )  /\  ( A 
.-  O )  =  ( B  .-  r
) ) ) )
3734, 36mpbid 213 . 2  |-  ( ph  ->  E. r  e.  P  ( r  e.  ( B I Q )  /\  ( A  .-  O )  =  ( B  .-  r ) ) )
3833, 37r19.29a 2977 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 1870    =/= wne 2625   E.wrex 2783   class class class wbr 4426   ` cfv 5601  (class class class)co 6305   Basecbs 15084   distcds 15161  TarskiGcstrkg 24341  Itvcitv 24347  LineGclng 24348  ≤Gcleg 24487  pInvGcmir 24557  ⟂Gcperpg 24597
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-rep 4538  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597  ax-cnex 9594  ax-resscn 9595  ax-1cn 9596  ax-icn 9597  ax-addcl 9598  ax-addrcl 9599  ax-mulcl 9600  ax-mulrcl 9601  ax-mulcom 9602  ax-addass 9603  ax-mulass 9604  ax-distr 9605  ax-i2m1 9606  ax-1ne0 9607  ax-1rid 9608  ax-rnegex 9609  ax-rrecex 9610  ax-cnre 9611  ax-pre-lttri 9612  ax-pre-lttrn 9613  ax-pre-ltadd 9614  ax-pre-mulgt0 9615
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 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-nel 2628  df-ral 2787  df-rex 2788  df-reu 2789  df-rmo 2790  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-pss 3458  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-tp 4007  df-op 4009  df-uni 4223  df-int 4259  df-iun 4304  df-br 4427  df-opab 4485  df-mpt 4486  df-tr 4521  df-eprel 4765  df-id 4769  df-po 4775  df-so 4776  df-fr 4813  df-we 4815  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  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 6267  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-om 6707  df-1st 6807  df-2nd 6808  df-wrecs 7036  df-recs 7098  df-rdg 7136  df-1o 7190  df-oadd 7194  df-er 7371  df-map 7482  df-pm 7483  df-en 7578  df-dom 7579  df-sdom 7580  df-fin 7581  df-card 8372  df-cda 8596  df-pnf 9676  df-mnf 9677  df-xr 9678  df-ltxr 9679  df-le 9680  df-sub 9861  df-neg 9862  df-nn 10610  df-2 10668  df-3 10669  df-n0 10870  df-z 10938  df-uz 11160  df-fz 11783  df-fzo 11914  df-hash 12513  df-word 12651  df-concat 12653  df-s1 12654  df-s2 12929  df-s3 12930  df-trkgc 24359  df-trkgb 24360  df-trkgcb 24361  df-trkg 24364  df-cgrg 24419  df-leg 24488  df-mir 24558  df-rag 24596  df-perpg 24598
This theorem is referenced by:  midex  24636
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