Theorem ltrncoidN 35414

Description: Two translations are equal if the composition of one with the converse of the other is the zero translation. This is an analogue of vector subtraction. (Contributed by NM, 7-Apr-2014.) (New usage is discouraged.)

Hypotheses

Ref	Expression
ltrn1o.b	|- B = ( Base ` K )
ltrn1o.h	|- H = ( LHyp ` K )
ltrn1o.t	|- T = ( ( LTrn ` K ) ` W )

Assertion

Ref	Expression
ltrncoidN	|- ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) -> ( ( F o. `' G ) = ( _I |` B ) <-> F = G ) )

Proof of Theorem ltrncoidN

Step	Hyp	Ref	Expression
1		simpl1 1064	. . . . . . . 8 |- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) /\ ( F o. `' G ) = ( _I |` B ) ) -> ( K e. HL /\ W e. H ) )
2		simpl3 1066	. . . . . . . 8 |- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) /\ ( F o. `' G ) = ( _I |` B ) ) -> G e. T )
3		ltrn1o.b	. . . . . . . . 9 |- B = ( Base ` K )
4		ltrn1o.h	. . . . . . . . 9 |- H = ( LHyp ` K )
5		ltrn1o.t	. . . . . . . . 9 |- T = ( ( LTrn ` K ) ` W )
6	3, 4, 5	ltrn1o 35410	. . . . . . . 8 |- ( ( ( K e. HL /\ W e. H ) /\ G e. T ) -> G : B -1-1-onto-> B )
7	1, 2, 6	syl2anc 693	. . . . . . 7 |- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) /\ ( F o. `' G ) = ( _I |` B ) ) -> G : B -1-1-onto-> B )
8		f1ococnv1 6165	. . . . . . 7 |- ( G : B -1-1-onto-> B -> ( `' G o. G ) = ( _I |` B ) )
9	7, 8	syl 17	. . . . . 6 |- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) /\ ( F o. `' G ) = ( _I |` B ) ) -> ( `' G o. G ) = ( _I |` B ) )
10	9	coeq2d 5284	. . . . 5 |- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) /\ ( F o. `' G ) = ( _I |` B ) ) -> ( F o. ( `' G o. G ) ) = ( F o. ( _I |` B ) ) )
11		simpl2 1065	. . . . . . 7 |- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) /\ ( F o. `' G ) = ( _I |` B ) ) -> F e. T )
12	3, 4, 5	ltrn1o 35410	. . . . . . 7 |- ( ( ( K e. HL /\ W e. H ) /\ F e. T ) -> F : B -1-1-onto-> B )
13	1, 11, 12	syl2anc 693	. . . . . 6 |- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) /\ ( F o. `' G ) = ( _I |` B ) ) -> F : B -1-1-onto-> B )
14		f1of 6137	. . . . . 6 |- ( F : B -1-1-onto-> B -> F : B --> B )
15		fcoi1 6078	. . . . . 6 |- ( F : B --> B -> ( F o. ( _I |` B ) ) = F )
16	13, 14, 15	3syl 18	. . . . 5 |- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) /\ ( F o. `' G ) = ( _I |` B ) ) -> ( F o. ( _I |` B ) ) = F )
17	10, 16	eqtr2d 2657	. . . 4 |- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) /\ ( F o. `' G ) = ( _I |` B ) ) -> F = ( F o. ( `' G o. G ) ) )
18		coass 5654	. . . 4 |- ( ( F o. `' G ) o. G ) = ( F o. ( `' G o. G ) )
19	17, 18	syl6eqr 2674	. . 3 |- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) /\ ( F o. `' G ) = ( _I |` B ) ) -> F = ( ( F o. `' G ) o. G ) )
20		simpr 477	. . . . 5 |- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) /\ ( F o. `' G ) = ( _I |` B ) ) -> ( F o. `' G ) = ( _I |` B ) )
21	20	coeq1d 5283	. . . 4 |- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) /\ ( F o. `' G ) = ( _I |` B ) ) -> ( ( F o. `' G ) o. G ) = ( ( _I |` B ) o. G ) )
22		f1of 6137	. . . . 5 |- ( G : B -1-1-onto-> B -> G : B --> B )
23		fcoi2 6079	. . . . 5 |- ( G : B --> B -> ( ( _I |` B ) o. G ) = G )
24	7, 22, 23	3syl 18	. . . 4 |- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) /\ ( F o. `' G ) = ( _I |` B ) ) -> ( ( _I |` B ) o. G ) = G )
25	21, 24	eqtrd 2656	. . 3 |- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) /\ ( F o. `' G ) = ( _I |` B ) ) -> ( ( F o. `' G ) o. G ) = G )
26	19, 25	eqtrd 2656	. 2 |- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) /\ ( F o. `' G ) = ( _I |` B ) ) -> F = G )
27		simpr 477	. . . 4 |- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) /\ F = G ) -> F = G )
28	27	coeq1d 5283	. . 3 |- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) /\ F = G ) -> ( F o. `' G ) = ( G o. `' G ) )
29		simpl1 1064	. . . . 5 |- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) /\ F = G ) -> ( K e. HL /\ W e. H ) )
30		simpl3 1066	. . . . 5 |- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) /\ F = G ) -> G e. T )
31	29, 30, 6	syl2anc 693	. . . 4 |- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) /\ F = G ) -> G : B -1-1-onto-> B )
32		f1ococnv2 6163	. . . 4 |- ( G : B -1-1-onto-> B -> ( G o. `' G ) = ( _I |` B ) )
33	31, 32	syl 17	. . 3 |- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) /\ F = G ) -> ( G o. `' G ) = ( _I |` B ) )
34	28, 33	eqtrd 2656	. 2 |- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) /\ F = G ) -> ( F o. `' G ) = ( _I |` B ) )
35	26, 34	impbida 877	1 |- ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) -> ( ( F o. `' G ) = ( _I |` B ) <-> F = G ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   _I cid 5023   `'ccnv 5113   |` cres 5116   o. ccom 5118   -->wf 5884   -1-1-onto->wf1o 5887   ` cfv 5888   Basecbs 15857   HLchlt 34637   LHypclh 35270   LTrncltrn 35387

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

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  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-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-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  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-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-map 7859  df-laut 35275  df-ldil 35390  df-ltrn 35391

This theorem is referenced by:  tendospcanN  36312