Theorem tendoeq2 36062

Description: Condition determining equality of two trace-preserving endomorphisms, showing it is unnecessary to consider the identity translation. In tendocan 36112, we show that we only need to consider a single non-identity translation. (Contributed by NM, 21-Jun-2013.)

Hypotheses

Ref	Expression
tendoeq2.b	|- B = ( Base ` K )
tendoeq2.h	|- H = ( LHyp ` K )
tendoeq2.t	|- T = ( ( LTrn ` K ) ` W )
tendoeq2.e	|- E = ( ( TEndo ` K ) ` W )

Assertion

Ref	Expression
tendoeq2	|- ( ( ( K e. HL /\ W e. H ) /\ ( U e. E /\ V e. E ) /\ A. f e. T ( f =/= ( _I |` B ) -> ( U ` f ) = ( V ` f ) ) ) -> U = V )

Distinct variable groups:   f, E   f, H   f, K   T, f   f, W   U, f   f, V

Allowed substitution hint:   B( f)

Proof of Theorem tendoeq2

Step	Hyp	Ref	Expression
1		tendoeq2.b	. . . . . . . 8 |- B = ( Base ` K )
2		tendoeq2.h	. . . . . . . 8 |- H = ( LHyp ` K )
3		tendoeq2.e	. . . . . . . 8 |- E = ( ( TEndo ` K ) ` W )
4	1, 2, 3	tendoid 36061	. . . . . . 7 |- ( ( ( K e. HL /\ W e. H ) /\ U e. E ) -> ( U ` ( _I |` B ) ) = ( _I |` B ) )
5	4	adantrr 753	. . . . . 6 |- ( ( ( K e. HL /\ W e. H ) /\ ( U e. E /\ V e. E ) ) -> ( U ` ( _I |` B ) ) = ( _I |` B ) )
6	1, 2, 3	tendoid 36061	. . . . . . 7 |- ( ( ( K e. HL /\ W e. H ) /\ V e. E ) -> ( V ` ( _I |` B ) ) = ( _I |` B ) )
7	6	adantrl 752	. . . . . 6 |- ( ( ( K e. HL /\ W e. H ) /\ ( U e. E /\ V e. E ) ) -> ( V ` ( _I |` B ) ) = ( _I |` B ) )
8	5, 7	eqtr4d 2659	. . . . 5 |- ( ( ( K e. HL /\ W e. H ) /\ ( U e. E /\ V e. E ) ) -> ( U ` ( _I |` B ) ) = ( V ` ( _I |` B ) ) )
9		fveq2 6191	. . . . . 6 |- ( f = ( _I |` B ) -> ( U ` f ) = ( U ` ( _I |` B ) ) )
10		fveq2 6191	. . . . . 6 |- ( f = ( _I |` B ) -> ( V ` f ) = ( V ` ( _I |` B ) ) )
11	9, 10	eqeq12d 2637	. . . . 5 |- ( f = ( _I |` B ) -> ( ( U ` f ) = ( V ` f ) <-> ( U ` ( _I |` B ) ) = ( V ` ( _I |` B ) ) ) )
12	8, 11	syl5ibrcom 237	. . . 4 |- ( ( ( K e. HL /\ W e. H ) /\ ( U e. E /\ V e. E ) ) -> ( f = ( _I |` B ) -> ( U ` f ) = ( V ` f ) ) )
13	12	ralrimivw 2967	. . 3 |- ( ( ( K e. HL /\ W e. H ) /\ ( U e. E /\ V e. E ) ) -> A. f e. T ( f = ( _I |` B ) -> ( U ` f ) = ( V ` f ) ) )
14		r19.26 3064	. . . . 5 |- ( A. f e. T ( ( f = ( _I |` B ) -> ( U ` f ) = ( V ` f ) ) /\ ( f =/= ( _I |` B ) -> ( U ` f ) = ( V ` f ) ) ) <-> ( A. f e. T ( f = ( _I |` B ) -> ( U ` f ) = ( V ` f ) ) /\ A. f e. T ( f =/= ( _I |` B ) -> ( U ` f ) = ( V ` f ) ) ) )
15		jaob 822	. . . . . . 7 |- ( ( ( f = ( _I |` B ) \/ f =/= ( _I |` B ) ) -> ( U ` f ) = ( V ` f ) ) <-> ( ( f = ( _I |` B ) -> ( U ` f ) = ( V ` f ) ) /\ ( f =/= ( _I |` B ) -> ( U ` f ) = ( V ` f ) ) ) )
16		exmidne 2804	. . . . . . . 8 |- ( f = ( _I |` B ) \/ f =/= ( _I |` B ) )
17		pm5.5 351	. . . . . . . 8 |- ( ( f = ( _I |` B ) \/ f =/= ( _I |` B ) ) -> ( ( ( f = ( _I |` B ) \/ f =/= ( _I |` B ) ) -> ( U ` f ) = ( V ` f ) ) <-> ( U ` f ) = ( V ` f ) ) )
18	16, 17	ax-mp 5	. . . . . . 7 |- ( ( ( f = ( _I |` B ) \/ f =/= ( _I |` B ) ) -> ( U ` f ) = ( V ` f ) ) <-> ( U ` f ) = ( V ` f ) )
19	15, 18	bitr3i 266	. . . . . 6 |- ( ( ( f = ( _I |` B ) -> ( U ` f ) = ( V ` f ) ) /\ ( f =/= ( _I |` B ) -> ( U ` f ) = ( V ` f ) ) ) <-> ( U ` f ) = ( V ` f ) )
20	19	ralbii 2980	. . . . 5 |- ( A. f e. T ( ( f = ( _I |` B ) -> ( U ` f ) = ( V ` f ) ) /\ ( f =/= ( _I |` B ) -> ( U ` f ) = ( V ` f ) ) ) <-> A. f e. T ( U ` f ) = ( V ` f ) )
21	14, 20	bitr3i 266	. . . 4 |- ( ( A. f e. T ( f = ( _I |` B ) -> ( U ` f ) = ( V ` f ) ) /\ A. f e. T ( f =/= ( _I |` B ) -> ( U ` f ) = ( V ` f ) ) ) <-> A. f e. T ( U ` f ) = ( V ` f ) )
22		tendoeq2.t	. . . . . 6 |- T = ( ( LTrn ` K ) ` W )
23	2, 22, 3	tendoeq1 36052	. . . . 5 |- ( ( ( K e. HL /\ W e. H ) /\ ( U e. E /\ V e. E ) /\ A. f e. T ( U ` f ) = ( V ` f ) ) -> U = V )
24	23	3expia 1267	. . . 4 |- ( ( ( K e. HL /\ W e. H ) /\ ( U e. E /\ V e. E ) ) -> ( A. f e. T ( U ` f ) = ( V ` f ) -> U = V ) )
25	21, 24	syl5bi 232	. . 3 |- ( ( ( K e. HL /\ W e. H ) /\ ( U e. E /\ V e. E ) ) -> ( ( A. f e. T ( f = ( _I |` B ) -> ( U ` f ) = ( V ` f ) ) /\ A. f e. T ( f =/= ( _I |` B ) -> ( U ` f ) = ( V ` f ) ) ) -> U = V ) )
26	13, 25	mpand 711	. 2 |- ( ( ( K e. HL /\ W e. H ) /\ ( U e. E /\ V e. E ) ) -> ( A. f e. T ( f =/= ( _I |` B ) -> ( U ` f ) = ( V ` f ) ) -> U = V ) )
27	26	3impia 1261	1 |- ( ( ( K e. HL /\ W e. H ) /\ ( U e. E /\ V e. E ) /\ A. f e. T ( f =/= ( _I |` B ) -> ( U ` f ) = ( V ` f ) ) ) -> U = V )

Syntax hints:   -> wi 4   <-> wb 196   \/ wo 383   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   A.wral 2912   _I cid 5023   |` cres 5116   ` cfv 5888   Basecbs 15857   HLchlt 34637   LHypclh 35270   LTrncltrn 35387   TEndoctendo 36040

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-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-map 7859  df-preset 16928  df-poset 16946  df-plt 16958  df-lub 16974  df-glb 16975  df-join 16976  df-meet 16977  df-p0 17039  df-p1 17040  df-lat 17046  df-clat 17108  df-oposet 34463  df-ol 34465  df-oml 34466  df-covers 34553  df-ats 34554  df-atl 34585  df-cvlat 34609  df-hlat 34638  df-lhyp 35274  df-laut 35275  df-ldil 35390  df-ltrn 35391  df-trl 35446  df-tendo 36043

This theorem is referenced by:  tendocan  36112