This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

Theorem cdleml4N

Description: Part of proof of Lemma L of Crawley p. 120. TODO: fix comment. (Contributed by NM, 1-Aug-2013) (New usage is discouraged.)

Ref Expression
Hypotheses cdleml1.b 
|- B = ( Base ` K )
cdleml1.h 
|- H = ( LHyp ` K )
cdleml1.t 
|- T = ( ( LTrn ` K ) ` W )
cdleml1.r 
|- R = ( ( trL ` K ) ` W )
cdleml1.e 
|- E = ( ( TEndo ` K ) ` W )
cdleml3.o 
|- .0. = ( g e. T |-> ( _I |` B ) )
Assertion cdleml4N 
|- ( ( ( K e. HL /\ W e. H ) /\ ( U e. E /\ V e. E ) /\ ( U =/= .0. /\ V =/= .0. ) ) -> E. s e. E ( s o. U ) = V )

Proof

Step Hyp Ref Expression
1 cdleml1.b 
 |-  B = ( Base ` K )
2 cdleml1.h 
 |-  H = ( LHyp ` K )
3 cdleml1.t 
 |-  T = ( ( LTrn ` K ) ` W )
4 cdleml1.r 
 |-  R = ( ( trL ` K ) ` W )
5 cdleml1.e 
 |-  E = ( ( TEndo ` K ) ` W )
6 cdleml3.o 
 |-  .0. = ( g e. T |-> ( _I |` B ) )
7 1 2 3 cdlemftr0 
 |-  ( ( K e. HL /\ W e. H ) -> E. f e. T f =/= ( _I |` B ) )
8 7 3ad2ant1 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( U e. E /\ V e. E ) /\ ( U =/= .0. /\ V =/= .0. ) ) -> E. f e. T f =/= ( _I |` B ) )
9 simp11 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( U e. E /\ V e. E ) /\ ( U =/= .0. /\ V =/= .0. ) ) /\ f e. T /\ f =/= ( _I |` B ) ) -> ( K e. HL /\ W e. H ) )
10 simp12l 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( U e. E /\ V e. E ) /\ ( U =/= .0. /\ V =/= .0. ) ) /\ f e. T /\ f =/= ( _I |` B ) ) -> U e. E )
11 simp12r 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( U e. E /\ V e. E ) /\ ( U =/= .0. /\ V =/= .0. ) ) /\ f e. T /\ f =/= ( _I |` B ) ) -> V e. E )
12 simp2 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( U e. E /\ V e. E ) /\ ( U =/= .0. /\ V =/= .0. ) ) /\ f e. T /\ f =/= ( _I |` B ) ) -> f e. T )
13 simp3 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( U e. E /\ V e. E ) /\ ( U =/= .0. /\ V =/= .0. ) ) /\ f e. T /\ f =/= ( _I |` B ) ) -> f =/= ( _I |` B ) )
14 simp13l 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( U e. E /\ V e. E ) /\ ( U =/= .0. /\ V =/= .0. ) ) /\ f e. T /\ f =/= ( _I |` B ) ) -> U =/= .0. )
15 simp13r 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( U e. E /\ V e. E ) /\ ( U =/= .0. /\ V =/= .0. ) ) /\ f e. T /\ f =/= ( _I |` B ) ) -> V =/= .0. )
16 1 2 3 4 5 6 cdleml3N 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( U e. E /\ V e. E /\ f e. T ) /\ ( f =/= ( _I |` B ) /\ U =/= .0. /\ V =/= .0. ) ) -> E. s e. E ( s o. U ) = V )
17 9 10 11 12 13 14 15 16 syl133anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( U e. E /\ V e. E ) /\ ( U =/= .0. /\ V =/= .0. ) ) /\ f e. T /\ f =/= ( _I |` B ) ) -> E. s e. E ( s o. U ) = V )
18 17 rexlimdv3a 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( U e. E /\ V e. E ) /\ ( U =/= .0. /\ V =/= .0. ) ) -> ( E. f e. T f =/= ( _I |` B ) -> E. s e. E ( s o. U ) = V ) )
19 8 18 mpd 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( U e. E /\ V e. E ) /\ ( U =/= .0. /\ V =/= .0. ) ) -> E. s e. E ( s o. U ) = V )