Theorem hlcgreulem 25512

Description: Lemma for hlcgreu 25513. (Contributed by Thierry Arnoux, 9-Aug-2020.)

Hypotheses

Ref	Expression
ishlg.p	|- P = ( Base ` G )
ishlg.i	|- I = (Itv ` G )
ishlg.k	|- K = (hlG ` G )
ishlg.a	|- ( ph -> A e. P )
ishlg.b	|- ( ph -> B e. P )
ishlg.c	|- ( ph -> C e. P )
hlln.1	|- ( ph -> G e. TarskiG )
hltr.d	|- ( ph -> D e. P )
hlcgrex.m	|- .- = ( dist ` G )
hlcgrex.1	|- ( ph -> D =/= A )
hlcgrex.2	|- ( ph -> B =/= C )
hlcgreulem.x	|- ( ph -> X e. P )
hlcgreulem.y	|- ( ph -> Y e. P )
hlcgreulem.1	|- ( ph -> X ( K ` A ) D )
hlcgreulem.2	|- ( ph -> Y ( K ` A ) D )
hlcgreulem.3	|- ( ph -> ( A .- X ) = ( B .- C ) )
hlcgreulem.4	|- ( ph -> ( A .- Y ) = ( B .- C ) )

Assertion

Ref	Expression
hlcgreulem	|- ( ph -> X = Y )

Proof of Theorem hlcgreulem

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ishlg.p	. . 3 |- P = ( Base ` G )
2		hlcgrex.m	. . 3 |- .- = ( dist ` G )
3		ishlg.i	. . 3 |- I = (Itv ` G )
4		hlln.1	. . . 4 |- ( ph -> G e. TarskiG )
5	4	ad2antrr 762	. . 3 |- ( ( ( ph /\ y e. P ) /\ ( A e. ( D I y ) /\ A =/= y ) ) -> G e. TarskiG )
6		ishlg.a	. . . 4 |- ( ph -> A e. P )
7	6	ad2antrr 762	. . 3 |- ( ( ( ph /\ y e. P ) /\ ( A e. ( D I y ) /\ A =/= y ) ) -> A e. P )
8		ishlg.b	. . . 4 |- ( ph -> B e. P )
9	8	ad2antrr 762	. . 3 |- ( ( ( ph /\ y e. P ) /\ ( A e. ( D I y ) /\ A =/= y ) ) -> B e. P )
10		ishlg.c	. . . 4 |- ( ph -> C e. P )
11	10	ad2antrr 762	. . 3 |- ( ( ( ph /\ y e. P ) /\ ( A e. ( D I y ) /\ A =/= y ) ) -> C e. P )
12		simplr 792	. . 3 |- ( ( ( ph /\ y e. P ) /\ ( A e. ( D I y ) /\ A =/= y ) ) -> y e. P )
13		hlcgreulem.x	. . . 4 |- ( ph -> X e. P )
14	13	ad2antrr 762	. . 3 |- ( ( ( ph /\ y e. P ) /\ ( A e. ( D I y ) /\ A =/= y ) ) -> X e. P )
15		hlcgreulem.y	. . . 4 |- ( ph -> Y e. P )
16	15	ad2antrr 762	. . 3 |- ( ( ( ph /\ y e. P ) /\ ( A e. ( D I y ) /\ A =/= y ) ) -> Y e. P )
17		simprr 796	. . . 4 |- ( ( ( ph /\ y e. P ) /\ ( A e. ( D I y ) /\ A =/= y ) ) -> A =/= y )
18	17	necomd 2849	. . 3 |- ( ( ( ph /\ y e. P ) /\ ( A e. ( D I y ) /\ A =/= y ) ) -> y =/= A )
19		ishlg.k	. . . . 5 |- K = (hlG ` G )
20		hltr.d	. . . . . 6 |- ( ph -> D e. P )
21	20	ad2antrr 762	. . . . 5 |- ( ( ( ph /\ y e. P ) /\ ( A e. ( D I y ) /\ A =/= y ) ) -> D e. P )
22		hlcgreulem.1	. . . . . . 7 |- ( ph -> X ( K ` A ) D )
23	1, 3, 19, 13, 20, 6, 4, 22	hlcomd 25499	. . . . . 6 |- ( ph -> D ( K ` A ) X )
24	23	ad2antrr 762	. . . . 5 |- ( ( ( ph /\ y e. P ) /\ ( A e. ( D I y ) /\ A =/= y ) ) -> D ( K ` A ) X )
25		simprl 794	. . . . 5 |- ( ( ( ph /\ y e. P ) /\ ( A e. ( D I y ) /\ A =/= y ) ) -> A e. ( D I y ) )
26	1, 3, 19, 21, 14, 12, 5, 7, 24, 25	btwnhl 25509	. . . 4 |- ( ( ( ph /\ y e. P ) /\ ( A e. ( D I y ) /\ A =/= y ) ) -> A e. ( X I y ) )
27	1, 2, 3, 5, 14, 7, 12, 26	tgbtwncom 25383	. . 3 |- ( ( ( ph /\ y e. P ) /\ ( A e. ( D I y ) /\ A =/= y ) ) -> A e. ( y I X ) )
28		hlcgreulem.2	. . . . . . 7 |- ( ph -> Y ( K ` A ) D )
29	1, 3, 19, 15, 20, 6, 4, 28	hlcomd 25499	. . . . . 6 |- ( ph -> D ( K ` A ) Y )
30	29	ad2antrr 762	. . . . 5 |- ( ( ( ph /\ y e. P ) /\ ( A e. ( D I y ) /\ A =/= y ) ) -> D ( K ` A ) Y )
31	1, 3, 19, 21, 16, 12, 5, 7, 30, 25	btwnhl 25509	. . . 4 |- ( ( ( ph /\ y e. P ) /\ ( A e. ( D I y ) /\ A =/= y ) ) -> A e. ( Y I y ) )
32	1, 2, 3, 5, 16, 7, 12, 31	tgbtwncom 25383	. . 3 |- ( ( ( ph /\ y e. P ) /\ ( A e. ( D I y ) /\ A =/= y ) ) -> A e. ( y I Y ) )
33		hlcgreulem.3	. . . 4 |- ( ph -> ( A .- X ) = ( B .- C ) )
34	33	ad2antrr 762	. . 3 |- ( ( ( ph /\ y e. P ) /\ ( A e. ( D I y ) /\ A =/= y ) ) -> ( A .- X ) = ( B .- C ) )
35		hlcgreulem.4	. . . 4 |- ( ph -> ( A .- Y ) = ( B .- C ) )
36	35	ad2antrr 762	. . 3 |- ( ( ( ph /\ y e. P ) /\ ( A e. ( D I y ) /\ A =/= y ) ) -> ( A .- Y ) = ( B .- C ) )
37	1, 2, 3, 5, 7, 9, 11, 12, 14, 16, 18, 27, 32, 34, 36	tgsegconeq 25381	. 2 |- ( ( ( ph /\ y e. P ) /\ ( A e. ( D I y ) /\ A =/= y ) ) -> X = Y )
38		fvex 6201	. . . . . 6 |- ( Base ` G ) e. _V
39	1, 38	eqeltri 2697	. . . . 5 |- P e. _V
40	39	a1i 11	. . . 4 |- ( ph -> P e. _V )
41		hlcgrex.2	. . . 4 |- ( ph -> B =/= C )
42	40, 8, 10, 41	nehash2 13256	. . 3 |- ( ph -> 2 <_ ( # ` P ) )
43	1, 2, 3, 4, 20, 6, 42	tgbtwndiff 25401	. 2 |- ( ph -> E. y e. P ( A e. ( D I y ) /\ A =/= y ) )
44	37, 43	r19.29a 3078	1 |- ( ph -> X = Y )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   _Vcvv 3200   class class class wbr 4653   ` cfv 5888  (class class class)co 6650   Basecbs 15857   distcds 15950  TarskiGcstrkg 25329  Itvcitv 25335  hlGchlg 25495

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-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-n0 11293  df-xnn0 11364  df-z 11378  df-uz 11688  df-fz 12327  df-hash 13118  df-trkgc 25347  df-trkgb 25348  df-trkgcb 25349  df-trkg 25352  df-hlg 25496

This theorem is referenced by:  hlcgreu  25513  iscgra1  25702