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Theorem opphllem4 24242
Description: Lemma for opphl 24245. (Contributed by Thierry Arnoux, 22-Feb-2020.)
Hypotheses
Ref Expression
hpg.p  |-  P  =  ( Base `  G
)
hpg.d  |-  .-  =  ( dist `  G )
hpg.i  |-  I  =  (Itv `  G )
hpg.o  |-  O  =  { <. a ,  b
>.  |  ( (
a  e.  ( P 
\  D )  /\  b  e.  ( P  \  D ) )  /\  E. t  e.  D  t  e.  ( a I b ) ) }
opphl.l  |-  L  =  (LineG `  G )
opphl.d  |-  ( ph  ->  D  e.  ran  L
)
opphl.g  |-  ( ph  ->  G  e. TarskiG )
opphl.k  |-  K  =  ( c  e.  P  |->  { <. a ,  b
>.  |  ( (
a  e.  P  /\  b  e.  P )  /\  ( a  =/=  c  /\  b  =/=  c  /\  ( a  e.  ( c I b )  \/  b  e.  ( c I a ) ) ) ) } )
opphllem5.n  |-  N  =  ( (pInvG `  G
) `  M )
opphllem5.a  |-  ( ph  ->  A  e.  P )
opphllem5.c  |-  ( ph  ->  C  e.  P )
opphllem5.r  |-  ( ph  ->  R  e.  D )
opphllem5.s  |-  ( ph  ->  S  e.  D )
opphllem5.m  |-  ( ph  ->  M  e.  P )
opphllem5.o  |-  ( ph  ->  A O C )
opphllem5.p  |-  ( ph  ->  D (⟂G `  G
) ( A L R ) )
opphllem5.q  |-  ( ph  ->  D (⟂G `  G
) ( C L S ) )
opphllem3.t  |-  ( ph  ->  R  =/=  S )
opphllem3.l  |-  ( ph  ->  ( S  .-  C
) (≤G `  G
) ( R  .-  A ) )
opphllem3.u  |-  ( ph  ->  U  e.  P )
opphllem3.v  |-  ( ph  ->  ( N `  R
)  =  S )
opphllem4.u  |-  ( ph  ->  V  e.  P )
opphllem4.1  |-  ( ph  ->  U ( K `  R ) A )
opphllem4.2  |-  ( ph  ->  V ( K `  S ) C )
Assertion
Ref Expression
opphllem4  |-  ( ph  ->  U O V )
Distinct variable groups:    A, a,
b, c, t    D, a, b, t    C, a, b, c, t    G, a, b, c, t    L, a, b, t    I, a, b, c, t    t, K    M, a, b, c, t    t, O    N, a, b, c, t    P, a, b, c, t    R, a, b, c, t    S, a, b, c, t    U, a, b, c, t    V, a, b, c, t    ph, a,
b, t    .- , a, b, t
Allowed substitution hints:    ph( c)    D( c)    K( a, b, c)    L( c)    .- ( c)    O( a, b, c)

Proof of Theorem opphllem4
StepHypRef Expression
1 hpg.p . 2  |-  P  =  ( Base `  G
)
2 hpg.d . 2  |-  .-  =  ( dist `  G )
3 hpg.i . 2  |-  I  =  (Itv `  G )
4 hpg.o . 2  |-  O  =  { <. a ,  b
>.  |  ( (
a  e.  ( P 
\  D )  /\  b  e.  ( P  \  D ) )  /\  E. t  e.  D  t  e.  ( a I b ) ) }
5 opphl.l . 2  |-  L  =  (LineG `  G )
6 opphl.d . 2  |-  ( ph  ->  D  e.  ran  L
)
7 opphl.g . 2  |-  ( ph  ->  G  e. TarskiG )
8 opphllem4.u . 2  |-  ( ph  ->  V  e.  P )
9 opphllem3.u . 2  |-  ( ph  ->  U  e.  P )
10 opphllem5.n . . 3  |-  N  =  ( (pInvG `  G
) `  M )
11 eqid 2382 . . . 4  |-  (pInvG `  G )  =  (pInvG `  G )
12 opphllem5.m . . . 4  |-  ( ph  ->  M  e.  P )
131, 2, 3, 5, 11, 7, 12, 10, 9mircl 24162 . . 3  |-  ( ph  ->  ( N `  U
)  e.  P )
14 opphllem5.s . . 3  |-  ( ph  ->  S  e.  D )
15 opphllem5.o . . . . . . . . . 10  |-  ( ph  ->  A O C )
16 opphllem5.a . . . . . . . . . . 11  |-  ( ph  ->  A  e.  P )
17 opphllem5.c . . . . . . . . . . 11  |-  ( ph  ->  C  e.  P )
181, 2, 3, 4, 16, 17islnopp 24233 . . . . . . . . . 10  |-  ( ph  ->  ( A O C  <-> 
( ( -.  A  e.  D  /\  -.  C  e.  D )  /\  E. t  e.  D  t  e.  ( A I C ) ) ) )
1915, 18mpbid 210 . . . . . . . . 9  |-  ( ph  ->  ( ( -.  A  e.  D  /\  -.  C  e.  D )  /\  E. t  e.  D  t  e.  ( A I C ) ) )
2019simplld 752 . . . . . . . 8  |-  ( ph  ->  -.  A  e.  D
)
21 opphllem5.r . . . . . . . . . . . 12  |-  ( ph  ->  R  e.  D )
221, 5, 3, 7, 6, 21tglnpt 24056 . . . . . . . . . . 11  |-  ( ph  ->  R  e.  P )
23 opphl.k . . . . . . . . . . . . 13  |-  K  =  ( c  e.  P  |->  { <. a ,  b
>.  |  ( (
a  e.  P  /\  b  e.  P )  /\  ( a  =/=  c  /\  b  =/=  c  /\  ( a  e.  ( c I b )  \/  b  e.  ( c I a ) ) ) ) } )
24 opphllem4.1 . . . . . . . . . . . . 13  |-  ( ph  ->  U ( K `  R ) A )
251, 3, 23, 9, 16, 22, 24hlne1 24109 . . . . . . . . . . . 12  |-  ( ph  ->  U  =/=  R )
2625necomd 2653 . . . . . . . . . . 11  |-  ( ph  ->  R  =/=  U )
271, 3, 23, 9, 16, 22, 24, 5, 7hlln 24111 . . . . . . . . . . 11  |-  ( ph  ->  U  e.  ( A L R ) )
281, 3, 23, 9, 16, 22ishlg 24106 . . . . . . . . . . . . 13  |-  ( ph  ->  ( U ( K `
 R ) A  <-> 
( U  =/=  R  /\  A  =/=  R  /\  ( U  e.  ( R I A )  \/  A  e.  ( R I U ) ) ) ) )
2924, 28mpbid 210 . . . . . . . . . . . 12  |-  ( ph  ->  ( U  =/=  R  /\  A  =/=  R  /\  ( U  e.  ( R I A )  \/  A  e.  ( R I U ) ) ) )
3029simp2d 1007 . . . . . . . . . . 11  |-  ( ph  ->  A  =/=  R )
311, 3, 5, 7, 22, 9, 16, 26, 27, 30lnrot1 24123 . . . . . . . . . 10  |-  ( ph  ->  A  e.  ( R L U ) )
3231adantr 463 . . . . . . . . 9  |-  ( (
ph  /\  U  e.  D )  ->  A  e.  ( R L U ) )
337adantr 463 . . . . . . . . . 10  |-  ( (
ph  /\  U  e.  D )  ->  G  e. TarskiG )
3422adantr 463 . . . . . . . . . 10  |-  ( (
ph  /\  U  e.  D )  ->  R  e.  P )
359adantr 463 . . . . . . . . . 10  |-  ( (
ph  /\  U  e.  D )  ->  U  e.  P )
3626adantr 463 . . . . . . . . . 10  |-  ( (
ph  /\  U  e.  D )  ->  R  =/=  U )
376adantr 463 . . . . . . . . . 10  |-  ( (
ph  /\  U  e.  D )  ->  D  e.  ran  L )
3821adantr 463 . . . . . . . . . 10  |-  ( (
ph  /\  U  e.  D )  ->  R  e.  D )
39 simpr 459 . . . . . . . . . 10  |-  ( (
ph  /\  U  e.  D )  ->  U  e.  D )
401, 3, 5, 33, 34, 35, 36, 36, 37, 38, 39tglinethru 24136 . . . . . . . . 9  |-  ( (
ph  /\  U  e.  D )  ->  D  =  ( R L U ) )
4132, 40eleqtrrd 2473 . . . . . . . 8  |-  ( (
ph  /\  U  e.  D )  ->  A  e.  D )
4220, 41mtand 657 . . . . . . 7  |-  ( ph  ->  -.  U  e.  D
)
437adantr 463 . . . . . . . . 9  |-  ( (
ph  /\  ( N `  U )  e.  D
)  ->  G  e. TarskiG )
4412adantr 463 . . . . . . . . 9  |-  ( (
ph  /\  ( N `  U )  e.  D
)  ->  M  e.  P )
459adantr 463 . . . . . . . . 9  |-  ( (
ph  /\  ( N `  U )  e.  D
)  ->  U  e.  P )
461, 2, 3, 5, 11, 43, 44, 10, 45mirmir 24163 . . . . . . . 8  |-  ( (
ph  /\  ( N `  U )  e.  D
)  ->  ( N `  ( N `  U
) )  =  U )
476adantr 463 . . . . . . . . 9  |-  ( (
ph  /\  ( N `  U )  e.  D
)  ->  D  e.  ran  L )
481, 5, 3, 7, 6, 14tglnpt 24056 . . . . . . . . . . . 12  |-  ( ph  ->  S  e.  P )
49 opphllem3.t . . . . . . . . . . . . 13  |-  ( ph  ->  R  =/=  S )
5049necomd 2653 . . . . . . . . . . . 12  |-  ( ph  ->  S  =/=  R )
511, 2, 3, 5, 11, 7, 12, 10, 22mirbtwn 24159 . . . . . . . . . . . . 13  |-  ( ph  ->  M  e.  ( ( N `  R ) I R ) )
52 opphllem3.v . . . . . . . . . . . . . 14  |-  ( ph  ->  ( N `  R
)  =  S )
5352oveq1d 6211 . . . . . . . . . . . . 13  |-  ( ph  ->  ( ( N `  R ) I R )  =  ( S I R ) )
5451, 53eleqtrd 2472 . . . . . . . . . . . 12  |-  ( ph  ->  M  e.  ( S I R ) )
551, 3, 5, 7, 48, 22, 12, 50, 54btwnlng1 24119 . . . . . . . . . . 11  |-  ( ph  ->  M  e.  ( S L R ) )
561, 3, 5, 7, 48, 22, 50, 50, 6, 14, 21tglinethru 24136 . . . . . . . . . . 11  |-  ( ph  ->  D  =  ( S L R ) )
5755, 56eleqtrrd 2473 . . . . . . . . . 10  |-  ( ph  ->  M  e.  D )
5857adantr 463 . . . . . . . . 9  |-  ( (
ph  /\  ( N `  U )  e.  D
)  ->  M  e.  D )
59 simpr 459 . . . . . . . . 9  |-  ( (
ph  /\  ( N `  U )  e.  D
)  ->  ( N `  U )  e.  D
)
601, 2, 3, 5, 11, 43, 10, 47, 58, 59mirln 24176 . . . . . . . 8  |-  ( (
ph  /\  ( N `  U )  e.  D
)  ->  ( N `  ( N `  U
) )  e.  D
)
6146, 60eqeltrrd 2471 . . . . . . 7  |-  ( (
ph  /\  ( N `  U )  e.  D
)  ->  U  e.  D )
6242, 61mtand 657 . . . . . 6  |-  ( ph  ->  -.  ( N `  U )  e.  D
)
6362, 42jca 530 . . . . 5  |-  ( ph  ->  ( -.  ( N `
 U )  e.  D  /\  -.  U  e.  D ) )
641, 2, 3, 5, 11, 7, 12, 10, 9mirbtwn 24159 . . . . . 6  |-  ( ph  ->  M  e.  ( ( N `  U ) I U ) )
65 eleq1 2454 . . . . . . 7  |-  ( t  =  M  ->  (
t  e.  ( ( N `  U ) I U )  <->  M  e.  ( ( N `  U ) I U ) ) )
6665rspcev 3135 . . . . . 6  |-  ( ( M  e.  D  /\  M  e.  ( ( N `  U )
I U ) )  ->  E. t  e.  D  t  e.  ( ( N `  U )
I U ) )
6757, 64, 66syl2anc 659 . . . . 5  |-  ( ph  ->  E. t  e.  D  t  e.  ( ( N `  U )
I U ) )
6863, 67jca 530 . . . 4  |-  ( ph  ->  ( ( -.  ( N `  U )  e.  D  /\  -.  U  e.  D )  /\  E. t  e.  D  t  e.  ( ( N `  U ) I U ) ) )
691, 2, 3, 4, 13, 9islnopp 24233 . . . 4  |-  ( ph  ->  ( ( N `  U ) O U  <-> 
( ( -.  ( N `  U )  e.  D  /\  -.  U  e.  D )  /\  E. t  e.  D  t  e.  ( ( N `  U ) I U ) ) ) )
7068, 69mpbird 232 . . 3  |-  ( ph  ->  ( N `  U
) O U )
71 eqidd 2383 . . 3  |-  ( ph  ->  ( N `  U
)  =  ( N `
 U ) )
72 opphllem5.p . . . . . . . 8  |-  ( ph  ->  D (⟂G `  G
) ( A L R ) )
73 opphllem5.q . . . . . . . 8  |-  ( ph  ->  D (⟂G `  G
) ( C L S ) )
74 opphllem3.l . . . . . . . 8  |-  ( ph  ->  ( S  .-  C
) (≤G `  G
) ( R  .-  A ) )
751, 2, 3, 4, 5, 6, 7, 23, 10, 16, 17, 21, 14, 12, 15, 72, 73, 49, 74, 9, 52opphllem3 24241 . . . . . . 7  |-  ( ph  ->  ( U ( K `
 R ) A  <-> 
( N `  U
) ( K `  S ) C ) )
7624, 75mpbid 210 . . . . . 6  |-  ( ph  ->  ( N `  U
) ( K `  S ) C )
77 opphllem4.2 . . . . . . 7  |-  ( ph  ->  V ( K `  S ) C )
781, 3, 23, 8, 17, 48, 77hlcomd 24108 . . . . . 6  |-  ( ph  ->  C ( K `  S ) V )
791, 3, 23, 13, 17, 8, 7, 48, 76, 78hltr 24114 . . . . 5  |-  ( ph  ->  ( N `  U
) ( K `  S ) V )
801, 3, 23, 13, 8, 48ishlg 24106 . . . . 5  |-  ( ph  ->  ( ( N `  U ) ( K `
 S ) V  <-> 
( ( N `  U )  =/=  S  /\  V  =/=  S  /\  ( ( N `  U )  e.  ( S I V )  \/  V  e.  ( S I ( N `
 U ) ) ) ) ) )
8179, 80mpbid 210 . . . 4  |-  ( ph  ->  ( ( N `  U )  =/=  S  /\  V  =/=  S  /\  ( ( N `  U )  e.  ( S I V )  \/  V  e.  ( S I ( N `
 U ) ) ) ) )
8281simp1d 1006 . . 3  |-  ( ph  ->  ( N `  U
)  =/=  S )
8381simp2d 1007 . . 3  |-  ( ph  ->  V  =/=  S )
8481simp3d 1008 . . 3  |-  ( ph  ->  ( ( N `  U )  e.  ( S I V )  \/  V  e.  ( S I ( N `
 U ) ) ) )
851, 2, 3, 4, 5, 6, 7, 10, 13, 8, 9, 14, 70, 57, 71, 82, 83, 84opphllem2 24240 . 2  |-  ( ph  ->  V O U )
861, 2, 3, 4, 5, 6, 7, 8, 9, 85oppcom 24236 1  |-  ( ph  ->  U O V )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    \/ wo 366    /\ wa 367    /\ w3a 971    = wceq 1399    e. wcel 1826    =/= wne 2577   E.wrex 2733    \ cdif 3386   class class class wbr 4367   {copab 4424    |-> cmpt 4425   ran crn 4914   ` cfv 5496  (class class class)co 6196   Basecbs 14634   distcds 14711  TarskiGcstrkg 23942  Itvcitv 23949  LineGclng 23950  ≤Gcleg 24089  pInvGcmir 24153  ⟂Gcperpg 24192
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-8 1828  ax-9 1830  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360  ax-rep 4478  ax-sep 4488  ax-nul 4496  ax-pow 4543  ax-pr 4601  ax-un 6491  ax-cnex 9459  ax-resscn 9460  ax-1cn 9461  ax-icn 9462  ax-addcl 9463  ax-addrcl 9464  ax-mulcl 9465  ax-mulrcl 9466  ax-mulcom 9467  ax-addass 9468  ax-mulass 9469  ax-distr 9470  ax-i2m1 9471  ax-1ne0 9472  ax-1rid 9473  ax-rnegex 9474  ax-rrecex 9475  ax-cnre 9476  ax-pre-lttri 9477  ax-pre-lttrn 9478  ax-pre-ltadd 9479  ax-pre-mulgt0 9480
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1402  df-ex 1621  df-nf 1625  df-sb 1748  df-eu 2222  df-mo 2223  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ne 2579  df-nel 2580  df-ral 2737  df-rex 2738  df-reu 2739  df-rmo 2740  df-rab 2741  df-v 3036  df-sbc 3253  df-csb 3349  df-dif 3392  df-un 3394  df-in 3396  df-ss 3403  df-pss 3405  df-nul 3712  df-if 3858  df-pw 3929  df-sn 3945  df-pr 3947  df-tp 3949  df-op 3951  df-uni 4164  df-int 4200  df-iun 4245  df-br 4368  df-opab 4426  df-mpt 4427  df-tr 4461  df-eprel 4705  df-id 4709  df-po 4714  df-so 4715  df-fr 4752  df-we 4754  df-ord 4795  df-on 4796  df-lim 4797  df-suc 4798  df-xp 4919  df-rel 4920  df-cnv 4921  df-co 4922  df-dm 4923  df-rn 4924  df-res 4925  df-ima 4926  df-iota 5460  df-fun 5498  df-fn 5499  df-f 5500  df-f1 5501  df-fo 5502  df-f1o 5503  df-fv 5504  df-riota 6158  df-ov 6199  df-oprab 6200  df-mpt2 6201  df-om 6600  df-1st 6699  df-2nd 6700  df-recs 6960  df-rdg 6994  df-1o 7048  df-oadd 7052  df-er 7229  df-map 7340  df-pm 7341  df-en 7436  df-dom 7437  df-sdom 7438  df-fin 7439  df-card 8233  df-cda 8461  df-pnf 9541  df-mnf 9542  df-xr 9543  df-ltxr 9544  df-le 9545  df-sub 9720  df-neg 9721  df-nn 10453  df-2 10511  df-3 10512  df-n0 10713  df-z 10782  df-uz 11002  df-fz 11594  df-fzo 11718  df-hash 12308  df-word 12446  df-concat 12448  df-s1 12449  df-s2 12724  df-s3 12725  df-trkgc 23961  df-trkgb 23962  df-trkgcb 23963  df-trkg 23967  df-cgrg 24023  df-leg 24090  df-mir 24154  df-rag 24191  df-perpg 24193
This theorem is referenced by:  opphllem5  24243
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