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Theorem inagswap 24873
Description: Swap the order of the half lines delimiting the angle. Theorem 11.24 of [Schwabhauser] p. 101. (Contributed by Thierry Arnoux, 15-Aug-2020.)
Hypotheses
Ref Expression
isinag.p  |-  P  =  ( Base `  G
)
isinag.i  |-  I  =  (Itv `  G )
isinag.k  |-  K  =  (hlG `  G )
isinag.x  |-  ( ph  ->  X  e.  P )
isinag.a  |-  ( ph  ->  A  e.  P )
isinag.b  |-  ( ph  ->  B  e.  P )
isinag.c  |-  ( ph  ->  C  e.  P )
inagswap.g  |-  ( ph  ->  G  e. TarskiG )
inagswap.1  |-  ( ph  ->  X (inA `  G
) <" A B C "> )
Assertion
Ref Expression
inagswap  |-  ( ph  ->  X (inA `  G
) <" C B A "> )

Proof of Theorem inagswap
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 inagswap.1 . . . . . . 7  |-  ( ph  ->  X (inA `  G
) <" A B C "> )
2 isinag.p . . . . . . . 8  |-  P  =  ( Base `  G
)
3 isinag.i . . . . . . . 8  |-  I  =  (Itv `  G )
4 isinag.k . . . . . . . 8  |-  K  =  (hlG `  G )
5 isinag.x . . . . . . . 8  |-  ( ph  ->  X  e.  P )
6 isinag.a . . . . . . . 8  |-  ( ph  ->  A  e.  P )
7 isinag.b . . . . . . . 8  |-  ( ph  ->  B  e.  P )
8 isinag.c . . . . . . . 8  |-  ( ph  ->  C  e.  P )
9 inagswap.g . . . . . . . 8  |-  ( ph  ->  G  e. TarskiG )
102, 3, 4, 5, 6, 7, 8, 9isinag 24872 . . . . . . 7  |-  ( ph  ->  ( X (inA `  G ) <" A B C ">  <->  ( ( A  =/=  B  /\  C  =/=  B  /\  X  =/= 
B )  /\  E. x  e.  P  (
x  e.  ( A I C )  /\  ( x  =  B  \/  x ( K `  B ) X ) ) ) ) )
111, 10mpbid 214 . . . . . 6  |-  ( ph  ->  ( ( A  =/= 
B  /\  C  =/=  B  /\  X  =/=  B
)  /\  E. x  e.  P  ( x  e.  ( A I C )  /\  ( x  =  B  \/  x
( K `  B
) X ) ) ) )
1211simpld 461 . . . . 5  |-  ( ph  ->  ( A  =/=  B  /\  C  =/=  B  /\  X  =/=  B
) )
1312simp2d 1020 . . . 4  |-  ( ph  ->  C  =/=  B )
1412simp1d 1019 . . . 4  |-  ( ph  ->  A  =/=  B )
1512simp3d 1021 . . . 4  |-  ( ph  ->  X  =/=  B )
1613, 14, 153jca 1187 . . 3  |-  ( ph  ->  ( C  =/=  B  /\  A  =/=  B  /\  X  =/=  B
) )
1711simprd 465 . . . 4  |-  ( ph  ->  E. x  e.  P  ( x  e.  ( A I C )  /\  ( x  =  B  \/  x ( K `  B ) X ) ) )
18 eqid 2450 . . . . . . . 8  |-  ( dist `  G )  =  (
dist `  G )
1993ad2ant1 1028 . . . . . . . 8  |-  ( (
ph  /\  x  e.  P  /\  x  e.  ( A I C ) )  ->  G  e. TarskiG )
2063ad2ant1 1028 . . . . . . . 8  |-  ( (
ph  /\  x  e.  P  /\  x  e.  ( A I C ) )  ->  A  e.  P )
21 simp2 1008 . . . . . . . 8  |-  ( (
ph  /\  x  e.  P  /\  x  e.  ( A I C ) )  ->  x  e.  P )
2283ad2ant1 1028 . . . . . . . 8  |-  ( (
ph  /\  x  e.  P  /\  x  e.  ( A I C ) )  ->  C  e.  P )
23 simp3 1009 . . . . . . . 8  |-  ( (
ph  /\  x  e.  P  /\  x  e.  ( A I C ) )  ->  x  e.  ( A I C ) )
242, 18, 3, 19, 20, 21, 22, 23tgbtwncom 24525 . . . . . . 7  |-  ( (
ph  /\  x  e.  P  /\  x  e.  ( A I C ) )  ->  x  e.  ( C I A ) )
25243expia 1209 . . . . . 6  |-  ( (
ph  /\  x  e.  P )  ->  (
x  e.  ( A I C )  ->  x  e.  ( C I A ) ) )
2625anim1d 567 . . . . 5  |-  ( (
ph  /\  x  e.  P )  ->  (
( x  e.  ( A I C )  /\  ( x  =  B  \/  x ( K `  B ) X ) )  -> 
( x  e.  ( C I A )  /\  ( x  =  B  \/  x ( K `  B ) X ) ) ) )
2726reximdva 2861 . . . 4  |-  ( ph  ->  ( E. x  e.  P  ( x  e.  ( A I C )  /\  ( x  =  B  \/  x
( K `  B
) X ) )  ->  E. x  e.  P  ( x  e.  ( C I A )  /\  ( x  =  B  \/  x ( K `  B ) X ) ) ) )
2817, 27mpd 15 . . 3  |-  ( ph  ->  E. x  e.  P  ( x  e.  ( C I A )  /\  ( x  =  B  \/  x ( K `  B ) X ) ) )
2916, 28jca 535 . 2  |-  ( ph  ->  ( ( C  =/= 
B  /\  A  =/=  B  /\  X  =/=  B
)  /\  E. x  e.  P  ( x  e.  ( C I A )  /\  ( x  =  B  \/  x
( K `  B
) X ) ) ) )
302, 3, 4, 5, 8, 7, 6, 9isinag 24872 . 2  |-  ( ph  ->  ( X (inA `  G ) <" C B A ">  <->  ( ( C  =/=  B  /\  A  =/=  B  /\  X  =/= 
B )  /\  E. x  e.  P  (
x  e.  ( C I A )  /\  ( x  =  B  \/  x ( K `  B ) X ) ) ) ) )
3129, 30mpbird 236 1  |-  ( ph  ->  X (inA `  G
) <" C B A "> )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    \/ wo 370    /\ wa 371    /\ w3a 984    = wceq 1443    e. wcel 1886    =/= wne 2621   E.wrex 2737   class class class wbr 4401   ` cfv 5581  (class class class)co 6288   <"cs3 12933   Basecbs 15114   distcds 15192  TarskiGcstrkg 24471  Itvcitv 24477  hlGchlg 24638  inAcinag 24869
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1668  ax-4 1681  ax-5 1757  ax-6 1804  ax-7 1850  ax-8 1888  ax-9 1895  ax-10 1914  ax-11 1919  ax-12 1932  ax-13 2090  ax-ext 2430  ax-rep 4514  ax-sep 4524  ax-nul 4533  ax-pow 4580  ax-pr 4638  ax-un 6580  ax-cnex 9592  ax-resscn 9593  ax-1cn 9594  ax-icn 9595  ax-addcl 9596  ax-addrcl 9597  ax-mulcl 9598  ax-mulrcl 9599  ax-mulcom 9600  ax-addass 9601  ax-mulass 9602  ax-distr 9603  ax-i2m1 9604  ax-1ne0 9605  ax-1rid 9606  ax-rnegex 9607  ax-rrecex 9608  ax-cnre 9609  ax-pre-lttri 9610  ax-pre-lttrn 9611  ax-pre-ltadd 9612  ax-pre-mulgt0 9613
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 985  df-3an 986  df-tru 1446  df-ex 1663  df-nf 1667  df-sb 1797  df-eu 2302  df-mo 2303  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2580  df-ne 2623  df-nel 2624  df-ral 2741  df-rex 2742  df-reu 2743  df-rmo 2744  df-rab 2745  df-v 3046  df-sbc 3267  df-csb 3363  df-dif 3406  df-un 3408  df-in 3410  df-ss 3417  df-pss 3419  df-nul 3731  df-if 3881  df-pw 3952  df-sn 3968  df-pr 3970  df-tp 3972  df-op 3974  df-uni 4198  df-int 4234  df-iun 4279  df-br 4402  df-opab 4461  df-mpt 4462  df-tr 4497  df-eprel 4744  df-id 4748  df-po 4754  df-so 4755  df-fr 4792  df-we 4794  df-xp 4839  df-rel 4840  df-cnv 4841  df-co 4842  df-dm 4843  df-rn 4844  df-res 4845  df-ima 4846  df-pred 5379  df-ord 5425  df-on 5426  df-lim 5427  df-suc 5428  df-iota 5545  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-riota 6250  df-ov 6291  df-oprab 6292  df-mpt2 6293  df-om 6690  df-1st 6790  df-2nd 6791  df-wrecs 7025  df-recs 7087  df-rdg 7125  df-1o 7179  df-oadd 7183  df-er 7360  df-map 7471  df-en 7567  df-dom 7568  df-sdom 7569  df-fin 7570  df-card 8370  df-cda 8595  df-pnf 9674  df-mnf 9675  df-xr 9676  df-ltxr 9677  df-le 9678  df-sub 9859  df-neg 9860  df-nn 10607  df-2 10665  df-3 10666  df-n0 10867  df-z 10935  df-uz 11157  df-fz 11782  df-fzo 11913  df-hash 12513  df-word 12661  df-concat 12663  df-s1 12664  df-s2 12939  df-s3 12940  df-trkgc 24489  df-trkgb 24490  df-trkgcb 24491  df-trkg 24494  df-inag 24871
This theorem is referenced by: (None)
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