Theorem miduniq2 25582

Description: If two point inversions commute, they are identical. Theorem 7.19 of [Schwabhauser] p. 52. (Contributed by Thierry Arnoux, 30-Jul-2019.)

Hypotheses

Ref	Expression
mirval.p	|- P = ( Base ` G )
mirval.d	|- .- = ( dist ` G )
mirval.i	|- I = (Itv ` G )
mirval.l	|- L = (LineG ` G )
mirval.s	|- S = (pInvG ` G )
mirval.g	|- ( ph -> G e. TarskiG )
miduniq2.a	|- ( ph -> A e. P )
miduniq2.b	|- ( ph -> B e. P )
miduniq2.x	|- ( ph -> X e. P )
miduniq2.e	|- ( ph -> ( ( S ` A ) ` ( ( S ` B ) ` X ) ) = ( ( S ` B ) ` ( ( S ` A ) ` X ) ) )

Assertion

Ref	Expression
miduniq2	|- ( ph -> A = B )

Proof of Theorem miduniq2

Step	Hyp	Ref	Expression
1		mirval.p	. . . 4 |- P = ( Base ` G )
2		mirval.d	. . . 4 |- .- = ( dist ` G )
3		mirval.i	. . . 4 |- I = (Itv ` G )
4		mirval.l	. . . 4 |- L = (LineG ` G )
5		mirval.s	. . . 4 |- S = (pInvG ` G )
6		mirval.g	. . . 4 |- ( ph -> G e. TarskiG )
7		miduniq2.b	. . . . . 6 |- ( ph -> B e. P )
8		eqid 2622	. . . . . 6 |- ( S ` B ) = ( S ` B )
9	1, 2, 3, 4, 5, 6, 7, 8	mirf 25555	. . . . 5 |- ( ph -> ( S ` B ) : P --> P )
10		miduniq2.a	. . . . 5 |- ( ph -> A e. P )
11	9, 10	ffvelrnd 6360	. . . 4 |- ( ph -> ( ( S ` B ) ` A ) e. P )
12		miduniq2.x	. . . 4 |- ( ph -> X e. P )
13		eqid 2622	. . . . . 6 |- ( ( S ` B ) ` A ) = ( ( S ` B ) ` A )
14		eqid 2622	. . . . . 6 |- ( ( S ` B ) ` ( ( S ` B ) ` X ) ) = ( ( S ` B ) ` ( ( S ` B ) ` X ) )
15		eqid 2622	. . . . . 6 |- ( ( S ` B ) ` ( ( S ` B ) ` ( ( S ` A ) ` X ) ) ) = ( ( S ` B ) ` ( ( S ` B ) ` ( ( S ` A ) ` X ) ) )
16	9, 12	ffvelrnd 6360	. . . . . 6 |- ( ph -> ( ( S ` B ) ` X ) e. P )
17		eqid 2622	. . . . . . . 8 |- ( S ` A ) = ( S ` A )
18	1, 2, 3, 4, 5, 6, 10, 17, 12	mircl 25556	. . . . . . 7 |- ( ph -> ( ( S ` A ) ` X ) e. P )
19	9, 18	ffvelrnd 6360	. . . . . 6 |- ( ph -> ( ( S ` B ) ` ( ( S ` A ) ` X ) ) e. P )
20		miduniq2.e	. . . . . 6 |- ( ph -> ( ( S ` A ) ` ( ( S ` B ) ` X ) ) = ( ( S ` B ) ` ( ( S ` A ) ` X ) ) )
21	1, 2, 3, 4, 5, 6, 8, 13, 14, 15, 7, 10, 16, 19, 20	mirauto 25579	. . . . 5 |- ( ph -> ( ( S ` ( ( S ` B ) ` A ) ) ` ( ( S ` B ) ` ( ( S ` B ) ` X ) ) ) = ( ( S ` B ) ` ( ( S ` B ) ` ( ( S ` A ) ` X ) ) ) )
22	1, 2, 3, 4, 5, 6, 7, 8, 12	mirmir 25557	. . . . . 6 |- ( ph -> ( ( S ` B ) ` ( ( S ` B ) ` X ) ) = X )
23	22	fveq2d 6195	. . . . 5 |- ( ph -> ( ( S ` ( ( S ` B ) ` A ) ) ` ( ( S ` B ) ` ( ( S ` B ) ` X ) ) ) = ( ( S ` ( ( S ` B ) ` A ) ) ` X ) )
24	1, 2, 3, 4, 5, 6, 7, 8, 18	mirmir 25557	. . . . 5 |- ( ph -> ( ( S ` B ) ` ( ( S ` B ) ` ( ( S ` A ) ` X ) ) ) = ( ( S ` A ) ` X ) )
25	21, 23, 24	3eqtr3d 2664	. . . 4 |- ( ph -> ( ( S ` ( ( S ` B ) ` A ) ) ` X ) = ( ( S ` A ) ` X ) )
26	1, 2, 3, 4, 5, 6, 11, 10, 12, 25	miduniq1 25581	. . 3 |- ( ph -> ( ( S ` B ) ` A ) = A )
27	1, 2, 3, 4, 5, 6, 7, 8, 10	mirinv 25561	. . 3 |- ( ph -> ( ( ( S ` B ) ` A ) = A <-> B = A ) )
28	26, 27	mpbid 222	. 2 |- ( ph -> B = A )
29	28	eqcomd 2628	1 |- ( ph -> A = B )

Syntax hints:   -> wi 4   = wceq 1483   e. wcel 1990   ` cfv 5888   Basecbs 15857   distcds 15950  TarskiGcstrkg 25329  Itvcitv 25335  LineGclng 25336  pInvGcmir 25547

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-rmo 2920  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-pm 7860  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-card 8765  df-cda 8990  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-xnn0 11364  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-cgrg 25406  df-mir 25548

This theorem is referenced by: (None)