Metamath Proof Explorer

Theorem erngdvlem4

Description: Lemma for erngdv . (Contributed by NM, 11-Aug-2013)

Ref Expression
Hypotheses ernggrp.h 
|- H = ( LHyp ` K )
ernggrp.d 
|- D = ( ( EDRing ` K ) ` W )
erngdv.b 
|- B = ( Base ` K )
erngdv.t 
|- T = ( ( LTrn ` K ) ` W )
erngdv.e 
|- E = ( ( TEndo ` K ) ` W )
erngdv.p 
|- P = ( a e. E , b e. E |-> ( f e. T |-> ( ( a ` f ) o. ( b ` f ) ) ) )
erngdv.o 
|- .0. = ( f e. T |-> ( _I |` B ) )
erngdv.i 
|- I = ( a e. E |-> ( f e. T |-> `' ( a ` f ) ) )
erngrnglem.m 
|- .+ = ( a e. E , b e. E |-> ( a o. b ) )
edlemk6.j 
|- .\/ = ( join ` K )
edlemk6.m 
|- ./\ = ( meet ` K )
edlemk6.r 
|- R = ( ( trL ` K ) ` W )
edlemk6.p 
|- Q = ( ( oc ` K ) ` W )
edlemk6.z 
|- Z = ( ( Q .\/ ( R ` b ) ) ./\ ( ( h ` Q ) .\/ ( R ` ( b o. `' ( s ` h ) ) ) ) )
edlemk6.y 
|- Y = ( ( Q .\/ ( R ` g ) ) ./\ ( Z .\/ ( R ` ( g o. `' b ) ) ) )
edlemk6.x 
|- X = ( iota_ z e. T A. b e. T ( ( b =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` ( s ` h ) ) /\ ( R ` b ) =/= ( R ` g ) ) -> ( z ` Q ) = Y ) )
edlemk6.u 
|- U = ( g e. T |-> if ( ( s ` h ) = h , g , X ) )
Assertion erngdvlem4 
|- ( ( ( K e. HL /\ W e. H ) /\ ( h e. T /\ h =/= ( _I |` B ) ) ) -> D e. DivRing )

Proof

Step Hyp Ref Expression
1 ernggrp.h 
 |-  H = ( LHyp ` K )
2 ernggrp.d 
 |-  D = ( ( EDRing ` K ) ` W )
3 erngdv.b 
 |-  B = ( Base ` K )
4 erngdv.t 
 |-  T = ( ( LTrn ` K ) ` W )
5 erngdv.e 
 |-  E = ( ( TEndo ` K ) ` W )
6 erngdv.p 
 |-  P = ( a e. E , b e. E |-> ( f e. T |-> ( ( a ` f ) o. ( b ` f ) ) ) )
7 erngdv.o 
 |-  .0. = ( f e. T |-> ( _I |` B ) )
8 erngdv.i 
 |-  I = ( a e. E |-> ( f e. T |-> `' ( a ` f ) ) )
9 erngrnglem.m 
 |-  .+ = ( a e. E , b e. E |-> ( a o. b ) )
10 edlemk6.j 
 |-  .\/ = ( join ` K )
11 edlemk6.m 
 |-  ./\ = ( meet ` K )
12 edlemk6.r 
 |-  R = ( ( trL ` K ) ` W )
13 edlemk6.p 
 |-  Q = ( ( oc ` K ) ` W )
14 edlemk6.z 
 |-  Z = ( ( Q .\/ ( R ` b ) ) ./\ ( ( h ` Q ) .\/ ( R ` ( b o. `' ( s ` h ) ) ) ) )
15 edlemk6.y 
 |-  Y = ( ( Q .\/ ( R ` g ) ) ./\ ( Z .\/ ( R ` ( g o. `' b ) ) ) )
16 edlemk6.x 
 |-  X = ( iota_ z e. T A. b e. T ( ( b =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` ( s ` h ) ) /\ ( R ` b ) =/= ( R ` g ) ) -> ( z ` Q ) = Y ) )
17 edlemk6.u 
 |-  U = ( g e. T |-> if ( ( s ` h ) = h , g , X ) )
18 eqid 
 |-  ( Base ` D ) = ( Base ` D )
19 1 4 5 2 18 erngbase 
 |-  ( ( K e. HL /\ W e. H ) -> ( Base ` D ) = E )
20 19 eqcomd 
 |-  ( ( K e. HL /\ W e. H ) -> E = ( Base ` D ) )
21 20 adantr 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( h e. T /\ h =/= ( _I |` B ) ) ) -> E = ( Base ` D ) )
22 eqid 
 |-  ( .r ` D ) = ( .r ` D )
23 1 4 5 2 22 erngfmul 
 |-  ( ( K e. HL /\ W e. H ) -> ( .r ` D ) = ( a e. E , b e. E |-> ( a o. b ) ) )
24 9 23 eqtr4id 
 |-  ( ( K e. HL /\ W e. H ) -> .+ = ( .r ` D ) )
25 24 adantr 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( h e. T /\ h =/= ( _I |` B ) ) ) -> .+ = ( .r ` D ) )
26 3 1 4 5 7 tendo0cl 
 |-  ( ( K e. HL /\ W e. H ) -> .0. e. E )
27 26 19 eleqtrrd 
 |-  ( ( K e. HL /\ W e. H ) -> .0. e. ( Base ` D ) )
28 eqid 
 |-  ( +g ` D ) = ( +g ` D )
29 1 4 5 2 28 erngfplus 
 |-  ( ( K e. HL /\ W e. H ) -> ( +g ` D ) = ( a e. E , b e. E |-> ( f e. T |-> ( ( a ` f ) o. ( b ` f ) ) ) ) )
30 6 29 eqtr4id 
 |-  ( ( K e. HL /\ W e. H ) -> P = ( +g ` D ) )
31 30 oveqd 
 |-  ( ( K e. HL /\ W e. H ) -> ( .0. P .0. ) = ( .0. ( +g ` D ) .0. ) )
32 3 1 4 5 7 6 tendo0pl 
 |-  ( ( ( K e. HL /\ W e. H ) /\ .0. e. E ) -> ( .0. P .0. ) = .0. )
33 26 32 mpdan 
 |-  ( ( K e. HL /\ W e. H ) -> ( .0. P .0. ) = .0. )
34 31 33 eqtr3d 
 |-  ( ( K e. HL /\ W e. H ) -> ( .0. ( +g ` D ) .0. ) = .0. )
35 1 2 3 4 5 6 7 8 erngdvlem1 
 |-  ( ( K e. HL /\ W e. H ) -> D e. Grp )
36 eqid 
 |-  ( 0g ` D ) = ( 0g ` D )
37 18 28 36 isgrpid2 
 |-  ( D e. Grp -> ( ( .0. e. ( Base ` D ) /\ ( .0. ( +g ` D ) .0. ) = .0. ) <-> ( 0g ` D ) = .0. ) )
38 35 37 syl 
 |-  ( ( K e. HL /\ W e. H ) -> ( ( .0. e. ( Base ` D ) /\ ( .0. ( +g ` D ) .0. ) = .0. ) <-> ( 0g ` D ) = .0. ) )
39 27 34 38 mpbi2and 
 |-  ( ( K e. HL /\ W e. H ) -> ( 0g ` D ) = .0. )
40 39 eqcomd 
 |-  ( ( K e. HL /\ W e. H ) -> .0. = ( 0g ` D ) )
41 40 adantr 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( h e. T /\ h =/= ( _I |` B ) ) ) -> .0. = ( 0g ` D ) )
42 1 2 3 4 5 6 7 8 9 erngdvlem3 
 |-  ( ( K e. HL /\ W e. H ) -> D e. Ring )
43 1 4 5 2 42 erng1lem 
 |-  ( ( K e. HL /\ W e. H ) -> ( 1r ` D ) = ( _I |` T ) )
44 43 eqcomd 
 |-  ( ( K e. HL /\ W e. H ) -> ( _I |` T ) = ( 1r ` D ) )
45 44 adantr 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( h e. T /\ h =/= ( _I |` B ) ) ) -> ( _I |` T ) = ( 1r ` D ) )
46 42 adantr 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( h e. T /\ h =/= ( _I |` B ) ) ) -> D e. Ring )
47 simp1l 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( h e. T /\ h =/= ( _I |` B ) ) ) /\ ( s e. E /\ s =/= .0. ) /\ ( t e. E /\ t =/= .0. ) ) -> ( K e. HL /\ W e. H ) )
48 24 oveqd 
 |-  ( ( K e. HL /\ W e. H ) -> ( s .+ t ) = ( s ( .r ` D ) t ) )
49 47 48 syl 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( h e. T /\ h =/= ( _I |` B ) ) ) /\ ( s e. E /\ s =/= .0. ) /\ ( t e. E /\ t =/= .0. ) ) -> ( s .+ t ) = ( s ( .r ` D ) t ) )
50 simp2l 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( h e. T /\ h =/= ( _I |` B ) ) ) /\ ( s e. E /\ s =/= .0. ) /\ ( t e. E /\ t =/= .0. ) ) -> s e. E )
51 simp3l 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( h e. T /\ h =/= ( _I |` B ) ) ) /\ ( s e. E /\ s =/= .0. ) /\ ( t e. E /\ t =/= .0. ) ) -> t e. E )
52 1 4 5 2 22 erngmul 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( s e. E /\ t e. E ) ) -> ( s ( .r ` D ) t ) = ( s o. t ) )
53 47 50 51 52 syl12anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( h e. T /\ h =/= ( _I |` B ) ) ) /\ ( s e. E /\ s =/= .0. ) /\ ( t e. E /\ t =/= .0. ) ) -> ( s ( .r ` D ) t ) = ( s o. t ) )
54 49 53 eqtrd 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( h e. T /\ h =/= ( _I |` B ) ) ) /\ ( s e. E /\ s =/= .0. ) /\ ( t e. E /\ t =/= .0. ) ) -> ( s .+ t ) = ( s o. t ) )
55 3 1 4 5 7 tendoconid 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( s e. E /\ s =/= .0. ) /\ ( t e. E /\ t =/= .0. ) ) -> ( s o. t ) =/= .0. )
56 55 3adant1r 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( h e. T /\ h =/= ( _I |` B ) ) ) /\ ( s e. E /\ s =/= .0. ) /\ ( t e. E /\ t =/= .0. ) ) -> ( s o. t ) =/= .0. )
57 54 56 eqnetrd 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( h e. T /\ h =/= ( _I |` B ) ) ) /\ ( s e. E /\ s =/= .0. ) /\ ( t e. E /\ t =/= .0. ) ) -> ( s .+ t ) =/= .0. )
58 3 1 4 5 7 tendo1ne0 
 |-  ( ( K e. HL /\ W e. H ) -> ( _I |` T ) =/= .0. )
59 58 adantr 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( h e. T /\ h =/= ( _I |` B ) ) ) -> ( _I |` T ) =/= .0. )
60 simpll 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( h e. T /\ h =/= ( _I |` B ) ) ) /\ ( s e. E /\ s =/= .0. ) ) -> ( K e. HL /\ W e. H ) )
61 simplrl 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( h e. T /\ h =/= ( _I |` B ) ) ) /\ ( s e. E /\ s =/= .0. ) ) -> h e. T )
62 simpr 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( h e. T /\ h =/= ( _I |` B ) ) ) /\ ( s e. E /\ s =/= .0. ) ) -> ( s e. E /\ s =/= .0. ) )
63 3 10 11 1 4 12 13 14 15 16 17 5 7 cdleml6 
 |-  ( ( ( K e. HL /\ W e. H ) /\ h e. T /\ ( s e. E /\ s =/= .0. ) ) -> ( U e. E /\ ( U ` ( s ` h ) ) = h ) )
64 63 simpld 
 |-  ( ( ( K e. HL /\ W e. H ) /\ h e. T /\ ( s e. E /\ s =/= .0. ) ) -> U e. E )
65 60 61 62 64 syl3anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( h e. T /\ h =/= ( _I |` B ) ) ) /\ ( s e. E /\ s =/= .0. ) ) -> U e. E )
66 24 oveqd 
 |-  ( ( K e. HL /\ W e. H ) -> ( U .+ s ) = ( U ( .r ` D ) s ) )
67 66 ad2antrr 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( h e. T /\ h =/= ( _I |` B ) ) ) /\ ( s e. E /\ s =/= .0. ) ) -> ( U .+ s ) = ( U ( .r ` D ) s ) )
68 simprl 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( h e. T /\ h =/= ( _I |` B ) ) ) /\ ( s e. E /\ s =/= .0. ) ) -> s e. E )
69 1 4 5 2 22 erngmul 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( U e. E /\ s e. E ) ) -> ( U ( .r ` D ) s ) = ( U o. s ) )
70 60 65 68 69 syl12anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( h e. T /\ h =/= ( _I |` B ) ) ) /\ ( s e. E /\ s =/= .0. ) ) -> ( U ( .r ` D ) s ) = ( U o. s ) )
71 3 10 11 1 4 12 13 14 15 16 17 5 7 cdleml8 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( h e. T /\ h =/= ( _I |` B ) ) /\ ( s e. E /\ s =/= .0. ) ) -> ( U o. s ) = ( _I |` T ) )
72 71 3expa 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( h e. T /\ h =/= ( _I |` B ) ) ) /\ ( s e. E /\ s =/= .0. ) ) -> ( U o. s ) = ( _I |` T ) )
73 67 70 72 3eqtrd 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( h e. T /\ h =/= ( _I |` B ) ) ) /\ ( s e. E /\ s =/= .0. ) ) -> ( U .+ s ) = ( _I |` T ) )
74 21 25 41 45 46 57 59 65 73 isdrngd 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( h e. T /\ h =/= ( _I |` B ) ) ) -> D e. DivRing )