Theorem inagswap 25730

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.

Step	Hyp	Ref	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 )
10	2, 3, 4, 5, 6, 7, 8, 9	isinag 25729	. . . . . . 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 ) ) ) ) )
11	1, 10	mpbid 222	. . . . . 6 |- ( ph -> ( ( A =/= B /\ C =/= B /\ X =/= B ) /\ E. x e. P ( x e. ( A I C ) /\ ( x = B \/ x ( K ` B ) X ) ) ) )
12	11	simpld 475	. . . . 5 |- ( ph -> ( A =/= B /\ C =/= B /\ X =/= B ) )
13	12	simp2d 1074	. . . 4 |- ( ph -> C =/= B )
14	12	simp1d 1073	. . . 4 |- ( ph -> A =/= B )
15	12	simp3d 1075	. . . 4 |- ( ph -> X =/= B )
16	13, 14, 15	3jca 1242	. . 3 |- ( ph -> ( C =/= B /\ A =/= B /\ X =/= B ) )
17	11	simprd 479	. . . 4 |- ( ph -> E. x e. P ( x e. ( A I C ) /\ ( x = B \/ x ( K ` B ) X ) ) )
18		eqid 2622	. . . . . . . 8 |- ( dist ` G ) = ( dist ` G )
19	9	3ad2ant1 1082	. . . . . . . 8 |- ( ( ph /\ x e. P /\ x e. ( A I C ) ) -> G e. TarskiG )
20	6	3ad2ant1 1082	. . . . . . . 8 |- ( ( ph /\ x e. P /\ x e. ( A I C ) ) -> A e. P )
21		simp2 1062	. . . . . . . 8 |- ( ( ph /\ x e. P /\ x e. ( A I C ) ) -> x e. P )
22	8	3ad2ant1 1082	. . . . . . . 8 |- ( ( ph /\ x e. P /\ x e. ( A I C ) ) -> C e. P )
23		simp3 1063	. . . . . . . 8 |- ( ( ph /\ x e. P /\ x e. ( A I C ) ) -> x e. ( A I C ) )
24	2, 18, 3, 19, 20, 21, 22, 23	tgbtwncom 25383	. . . . . . 7 |- ( ( ph /\ x e. P /\ x e. ( A I C ) ) -> x e. ( C I A ) )
25	24	3expia 1267	. . . . . 6 |- ( ( ph /\ x e. P ) -> ( x e. ( A I C ) -> x e. ( C I A ) ) )
26	25	anim1d 588	. . . . 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 ) ) ) )
27	26	reximdva 3017	. . . 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 ) ) ) )
28	17, 27	mpd 15	. . 3 |- ( ph -> E. x e. P ( x e. ( C I A ) /\ ( x = B \/ x ( K ` B ) X ) ) )
29	16, 28	jca 554	. 2 |- ( ph -> ( ( C =/= B /\ A =/= B /\ X =/= B ) /\ E. x e. P ( x e. ( C I A ) /\ ( x = B \/ x ( K ` B ) X ) ) ) )
30	2, 3, 4, 5, 8, 7, 6, 9	isinag 25729	. 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 ) ) ) ) )
31	29, 30	mpbird 247	1 |- ( ph -> X (inA ` G ) <" C B A "> )

Syntax hints:   -> wi 4   \/ wo 383   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   E.wrex 2913   class class class wbr 4653   ` cfv 5888  (class class class)co 6650   <"cs3 13587   Basecbs 15857   distcds 15950  TarskiGcstrkg 25329  Itvcitv 25335  hlGchlg 25495  inAcinag 25726

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-map 7859  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-card 8765  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-2 11079  df-3 11080  df-n0 11293  df-z 11378  df-uz 11688  df-fz 12327  df-fzo 12466  df-hash 13118  df-word 13299  df-concat 13301  df-s1 13302  df-s2 13593  df-s3 13594  df-trkgc 25347  df-trkgb 25348  df-trkgcb 25349  df-trkg 25352  df-inag 25728

This theorem is referenced by: (None)