Theorem tendoeq1 36052

Description: Condition determining equality of two trace-preserving endomorphisms. (Contributed by NM, 11-Jun-2013.)

Hypotheses

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

Assertion

Ref	Expression
tendoeq1	|- ( ( ( K e. HL /\ W e. H ) /\ ( U e. E /\ V e. E ) /\ A. f e. T ( U ` f ) = ( V ` f ) ) -> U = V )

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

Allowed substitution hints:   E( f)   H( f)

Proof of Theorem tendoeq1

Step	Hyp	Ref	Expression
1		simp3 1063	. 2 |- ( ( ( K e. HL /\ W e. H ) /\ ( U e. E /\ V e. E ) /\ A. f e. T ( U ` f ) = ( V ` f ) ) -> A. f e. T ( U ` f ) = ( V ` f ) )
2		simp1 1061	. . . . 5 |- ( ( ( K e. HL /\ W e. H ) /\ ( U e. E /\ V e. E ) /\ A. f e. T ( U ` f ) = ( V ` f ) ) -> ( K e. HL /\ W e. H ) )
3		simp2l 1087	. . . . 5 |- ( ( ( K e. HL /\ W e. H ) /\ ( U e. E /\ V e. E ) /\ A. f e. T ( U ` f ) = ( V ` f ) ) -> U e. E )
4		tendof.h	. . . . . 6 |- H = ( LHyp ` K )
5		tendof.t	. . . . . 6 |- T = ( ( LTrn ` K ) ` W )
6		tendof.e	. . . . . 6 |- E = ( ( TEndo ` K ) ` W )
7	4, 5, 6	tendof 36051	. . . . 5 |- ( ( ( K e. HL /\ W e. H ) /\ U e. E ) -> U : T --> T )
8	2, 3, 7	syl2anc 693	. . . 4 |- ( ( ( K e. HL /\ W e. H ) /\ ( U e. E /\ V e. E ) /\ A. f e. T ( U ` f ) = ( V ` f ) ) -> U : T --> T )
9		ffn 6045	. . . 4 |- ( U : T --> T -> U Fn T )
10	8, 9	syl 17	. . 3 |- ( ( ( K e. HL /\ W e. H ) /\ ( U e. E /\ V e. E ) /\ A. f e. T ( U ` f ) = ( V ` f ) ) -> U Fn T )
11		simp2r 1088	. . . . 5 |- ( ( ( K e. HL /\ W e. H ) /\ ( U e. E /\ V e. E ) /\ A. f e. T ( U ` f ) = ( V ` f ) ) -> V e. E )
12	4, 5, 6	tendof 36051	. . . . 5 |- ( ( ( K e. HL /\ W e. H ) /\ V e. E ) -> V : T --> T )
13	2, 11, 12	syl2anc 693	. . . 4 |- ( ( ( K e. HL /\ W e. H ) /\ ( U e. E /\ V e. E ) /\ A. f e. T ( U ` f ) = ( V ` f ) ) -> V : T --> T )
14		ffn 6045	. . . 4 |- ( V : T --> T -> V Fn T )
15	13, 14	syl 17	. . 3 |- ( ( ( K e. HL /\ W e. H ) /\ ( U e. E /\ V e. E ) /\ A. f e. T ( U ` f ) = ( V ` f ) ) -> V Fn T )
16		eqfnfv 6311	. . 3 |- ( ( U Fn T /\ V Fn T ) -> ( U = V <-> A. f e. T ( U ` f ) = ( V ` f ) ) )
17	10, 15, 16	syl2anc 693	. 2 |- ( ( ( K e. HL /\ W e. H ) /\ ( U e. E /\ V e. E ) /\ A. f e. T ( U ` f ) = ( V ` f ) ) -> ( U = V <-> A. f e. T ( U ` f ) = ( V ` f ) ) )
18	1, 17	mpbird 247	1 |- ( ( ( K e. HL /\ W e. H ) /\ ( U e. E /\ V e. E ) /\ A. f e. T ( U ` f ) = ( V ` f ) ) -> U = V )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   A.wral 2912   Fn wfn 5883   -->wf 5884   ` cfv 5888   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-ov 6653  df-oprab 6654  df-mpt2 6655  df-map 7859  df-tendo 36043

This theorem is referenced by:  tendoeq2  36062  tendoplcom  36070  tendoplass  36071  tendodi1  36072  tendodi2  36073  tendo0pl  36079  tendoipl  36085