Metamath Proof Explorer

Theorem erngdvlem4-rN

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

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

Proof

Step Hyp Ref Expression
1 ernggrp.h-r 
 |-  H = ( LHyp ` K )
2 ernggrp.d-r 
 |-  D = ( ( EDRingR ` K ) ` W )
3 ernggrplem.b-r 
 |-  B = ( Base ` K )
4 ernggrplem.t-r 
 |-  T = ( ( LTrn ` K ) ` W )
5 ernggrplem.e-r 
 |-  E = ( ( TEndo ` K ) ` W )
6 ernggrplem.p-r 
 |-  P = ( a e. E , b e. E |-> ( f e. T |-> ( ( a ` f ) o. ( b ` f ) ) ) )
7 ernggrplem.o-r 
 |-  O = ( f e. T |-> ( _I |` B ) )
8 ernggrplem.i-r 
 |-  I = ( a e. E |-> ( f e. T |-> `' ( a ` f ) ) )
9 erngrnglem.m-r 
 |-  M = ( a e. E , b e. E |-> ( b o. a ) )
10 edlemk6.j-r 
 |-  .\/ = ( join ` K )
11 edlemk6.m-r 
 |-  ./\ = ( meet ` K )
12 edlemk6.r-r 
 |-  R = ( ( trL ` K ) ` W )
13 edlemk6.p-r 
 |-  Q = ( ( oc ` K ) ` W )
14 edlemk6.z-r 
 |-  Z = ( ( Q .\/ ( R ` b ) ) ./\ ( ( h ` Q ) .\/ ( R ` ( b o. `' ( s ` h ) ) ) ) )
15 edlemk6.y-r 
 |-  Y = ( ( Q .\/ ( R ` g ) ) ./\ ( Z .\/ ( R ` ( g o. `' b ) ) ) )
16 edlemk6.x-r 
 |-  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-r 
 |-  U = ( g e. T |-> if ( ( s ` h ) = h , g , X ) )
18 eqid 
 |-  ( Base ` D ) = ( Base ` D )
19 1 4 5 2 18 erngbase-rN 
 |-  ( ( 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-rN 
 |-  ( ( K e. HL /\ W e. H ) -> ( .r ` D ) = ( a e. E , b e. E |-> ( b o. a ) ) )
24 9 23 eqtr4id 
 |-  ( ( K e. HL /\ W e. H ) -> M = ( .r ` D ) )
25 24 adantr 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( h e. T /\ h =/= ( _I |` B ) ) ) -> M = ( .r ` D ) )
26 3 1 4 5 7 tendo0cl 
 |-  ( ( K e. HL /\ W e. H ) -> O e. E )
27 26 19 eleqtrrd 
 |-  ( ( K e. HL /\ W e. H ) -> O e. ( Base ` D ) )
28 eqid 
 |-  ( +g ` D ) = ( +g ` D )
29 1 4 5 2 28 erngfplus-rN 
 |-  ( ( 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 ) -> ( O P O ) = ( O ( +g ` D ) O ) )
32 3 1 4 5 7 6 tendo0pl 
 |-  ( ( ( K e. HL /\ W e. H ) /\ O e. E ) -> ( O P O ) = O )
33 26 32 mpdan 
 |-  ( ( K e. HL /\ W e. H ) -> ( O P O ) = O )
34 31 33 eqtr3d 
 |-  ( ( K e. HL /\ W e. H ) -> ( O ( +g ` D ) O ) = O )
35 1 2 3 4 5 6 7 8 erngdvlem1-rN 
 |-  ( ( K e. HL /\ W e. H ) -> D e. Grp )
36 eqid 
 |-  ( 0g ` D ) = ( 0g ` D )
37 18 28 36 isgrpid2 
 |-  ( D e. Grp -> ( ( O e. ( Base ` D ) /\ ( O ( +g ` D ) O ) = O ) <-> ( 0g ` D ) = O ) )
38 35 37 syl 
 |-  ( ( K e. HL /\ W e. H ) -> ( ( O e. ( Base ` D ) /\ ( O ( +g ` D ) O ) = O ) <-> ( 0g ` D ) = O ) )
39 27 34 38 mpbi2and 
 |-  ( ( K e. HL /\ W e. H ) -> ( 0g ` D ) = O )
40 39 eqcomd 
 |-  ( ( K e. HL /\ W e. H ) -> O = ( 0g ` D ) )
41 40 adantr 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( h e. T /\ h =/= ( _I |` B ) ) ) -> O = ( 0g ` D ) )
42 1 4 5 tendoidcl 
 |-  ( ( K e. HL /\ W e. H ) -> ( _I |` T ) e. E )
43 42 19 eleqtrrd 
 |-  ( ( K e. HL /\ W e. H ) -> ( _I |` T ) e. ( Base ` D ) )
44 19 eleq2d 
 |-  ( ( K e. HL /\ W e. H ) -> ( u e. ( Base ` D ) <-> u e. E ) )
45 simpl 
 |-  ( ( ( K e. HL /\ W e. H ) /\ u e. E ) -> ( K e. HL /\ W e. H ) )
46 42 adantr 
 |-  ( ( ( K e. HL /\ W e. H ) /\ u e. E ) -> ( _I |` T ) e. E )
47 simpr 
 |-  ( ( ( K e. HL /\ W e. H ) /\ u e. E ) -> u e. E )
48 1 4 5 2 22 erngmul-rN 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( _I |` T ) e. E /\ u e. E ) ) -> ( ( _I |` T ) ( .r ` D ) u ) = ( u o. ( _I |` T ) ) )
49 45 46 47 48 syl12anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ u e. E ) -> ( ( _I |` T ) ( .r ` D ) u ) = ( u o. ( _I |` T ) ) )
50 1 4 5 tendo1mulr 
 |-  ( ( ( K e. HL /\ W e. H ) /\ u e. E ) -> ( u o. ( _I |` T ) ) = u )
51 49 50 eqtrd 
 |-  ( ( ( K e. HL /\ W e. H ) /\ u e. E ) -> ( ( _I |` T ) ( .r ` D ) u ) = u )
52 1 4 5 2 22 erngmul-rN 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( u e. E /\ ( _I |` T ) e. E ) ) -> ( u ( .r ` D ) ( _I |` T ) ) = ( ( _I |` T ) o. u ) )
53 45 47 46 52 syl12anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ u e. E ) -> ( u ( .r ` D ) ( _I |` T ) ) = ( ( _I |` T ) o. u ) )
54 1 4 5 tendo1mul 
 |-  ( ( ( K e. HL /\ W e. H ) /\ u e. E ) -> ( ( _I |` T ) o. u ) = u )
55 53 54 eqtrd 
 |-  ( ( ( K e. HL /\ W e. H ) /\ u e. E ) -> ( u ( .r ` D ) ( _I |` T ) ) = u )
56 51 55 jca 
 |-  ( ( ( K e. HL /\ W e. H ) /\ u e. E ) -> ( ( ( _I |` T ) ( .r ` D ) u ) = u /\ ( u ( .r ` D ) ( _I |` T ) ) = u ) )
57 56 ex 
 |-  ( ( K e. HL /\ W e. H ) -> ( u e. E -> ( ( ( _I |` T ) ( .r ` D ) u ) = u /\ ( u ( .r ` D ) ( _I |` T ) ) = u ) ) )
58 44 57 sylbid 
 |-  ( ( K e. HL /\ W e. H ) -> ( u e. ( Base ` D ) -> ( ( ( _I |` T ) ( .r ` D ) u ) = u /\ ( u ( .r ` D ) ( _I |` T ) ) = u ) ) )
59 58 ralrimiv 
 |-  ( ( K e. HL /\ W e. H ) -> A. u e. ( Base ` D ) ( ( ( _I |` T ) ( .r ` D ) u ) = u /\ ( u ( .r ` D ) ( _I |` T ) ) = u ) )
60 1 2 3 4 5 6 7 8 9 erngdvlem3-rN 
 |-  ( ( K e. HL /\ W e. H ) -> D e. Ring )
61 eqid 
 |-  ( 1r ` D ) = ( 1r ` D )
62 18 22 61 isringid 
 |-  ( D e. Ring -> ( ( ( _I |` T ) e. ( Base ` D ) /\ A. u e. ( Base ` D ) ( ( ( _I |` T ) ( .r ` D ) u ) = u /\ ( u ( .r ` D ) ( _I |` T ) ) = u ) ) <-> ( 1r ` D ) = ( _I |` T ) ) )
63 60 62 syl 
 |-  ( ( K e. HL /\ W e. H ) -> ( ( ( _I |` T ) e. ( Base ` D ) /\ A. u e. ( Base ` D ) ( ( ( _I |` T ) ( .r ` D ) u ) = u /\ ( u ( .r ` D ) ( _I |` T ) ) = u ) ) <-> ( 1r ` D ) = ( _I |` T ) ) )
64 43 59 63 mpbi2and 
 |-  ( ( K e. HL /\ W e. H ) -> ( 1r ` D ) = ( _I |` T ) )
65 64 eqcomd 
 |-  ( ( K e. HL /\ W e. H ) -> ( _I |` T ) = ( 1r ` D ) )
66 65 adantr 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( h e. T /\ h =/= ( _I |` B ) ) ) -> ( _I |` T ) = ( 1r ` D ) )
67 60 adantr 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( h e. T /\ h =/= ( _I |` B ) ) ) -> D e. Ring )
68 simp1l 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( h e. T /\ h =/= ( _I |` B ) ) ) /\ ( s e. E /\ s =/= O ) /\ ( t e. E /\ t =/= O ) ) -> ( K e. HL /\ W e. H ) )
69 24 oveqd 
 |-  ( ( K e. HL /\ W e. H ) -> ( s M t ) = ( s ( .r ` D ) t ) )
70 68 69 syl 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( h e. T /\ h =/= ( _I |` B ) ) ) /\ ( s e. E /\ s =/= O ) /\ ( t e. E /\ t =/= O ) ) -> ( s M t ) = ( s ( .r ` D ) t ) )
71 simp2l 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( h e. T /\ h =/= ( _I |` B ) ) ) /\ ( s e. E /\ s =/= O ) /\ ( t e. E /\ t =/= O ) ) -> s e. E )
72 simp3l 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( h e. T /\ h =/= ( _I |` B ) ) ) /\ ( s e. E /\ s =/= O ) /\ ( t e. E /\ t =/= O ) ) -> t e. E )
73 1 4 5 2 22 erngmul-rN 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( s e. E /\ t e. E ) ) -> ( s ( .r ` D ) t ) = ( t o. s ) )
74 68 71 72 73 syl12anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( h e. T /\ h =/= ( _I |` B ) ) ) /\ ( s e. E /\ s =/= O ) /\ ( t e. E /\ t =/= O ) ) -> ( s ( .r ` D ) t ) = ( t o. s ) )
75 70 74 eqtrd 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( h e. T /\ h =/= ( _I |` B ) ) ) /\ ( s e. E /\ s =/= O ) /\ ( t e. E /\ t =/= O ) ) -> ( s M t ) = ( t o. s ) )
76 simp3 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( h e. T /\ h =/= ( _I |` B ) ) ) /\ ( s e. E /\ s =/= O ) /\ ( t e. E /\ t =/= O ) ) -> ( t e. E /\ t =/= O ) )
77 simp2 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( h e. T /\ h =/= ( _I |` B ) ) ) /\ ( s e. E /\ s =/= O ) /\ ( t e. E /\ t =/= O ) ) -> ( s e. E /\ s =/= O ) )
78 3 1 4 5 7 tendoconid 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( t e. E /\ t =/= O ) /\ ( s e. E /\ s =/= O ) ) -> ( t o. s ) =/= O )
79 68 76 77 78 syl3anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( h e. T /\ h =/= ( _I |` B ) ) ) /\ ( s e. E /\ s =/= O ) /\ ( t e. E /\ t =/= O ) ) -> ( t o. s ) =/= O )
80 75 79 eqnetrd 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( h e. T /\ h =/= ( _I |` B ) ) ) /\ ( s e. E /\ s =/= O ) /\ ( t e. E /\ t =/= O ) ) -> ( s M t ) =/= O )
81 3 1 4 5 7 tendo1ne0 
 |-  ( ( K e. HL /\ W e. H ) -> ( _I |` T ) =/= O )
82 81 adantr 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( h e. T /\ h =/= ( _I |` B ) ) ) -> ( _I |` T ) =/= O )
83 simpll 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( h e. T /\ h =/= ( _I |` B ) ) ) /\ ( s e. E /\ s =/= O ) ) -> ( K e. HL /\ W e. H ) )
84 simplrl 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( h e. T /\ h =/= ( _I |` B ) ) ) /\ ( s e. E /\ s =/= O ) ) -> h e. T )
85 simpr 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( h e. T /\ h =/= ( _I |` B ) ) ) /\ ( s e. E /\ s =/= O ) ) -> ( s e. E /\ s =/= O ) )
86 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 =/= O ) ) -> ( U e. E /\ ( U ` ( s ` h ) ) = h ) )
87 86 simpld 
 |-  ( ( ( K e. HL /\ W e. H ) /\ h e. T /\ ( s e. E /\ s =/= O ) ) -> U e. E )
88 83 84 85 87 syl3anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( h e. T /\ h =/= ( _I |` B ) ) ) /\ ( s e. E /\ s =/= O ) ) -> U e. E )
89 3 10 11 1 4 12 13 14 15 16 17 5 7 cdleml9 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( h e. T /\ h =/= ( _I |` B ) ) /\ ( s e. E /\ s =/= O ) ) -> U =/= O )
90 89 3expa 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( h e. T /\ h =/= ( _I |` B ) ) ) /\ ( s e. E /\ s =/= O ) ) -> U =/= O )
91 24 oveqd 
 |-  ( ( K e. HL /\ W e. H ) -> ( s M U ) = ( s ( .r ` D ) U ) )
92 91 ad2antrr 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( h e. T /\ h =/= ( _I |` B ) ) ) /\ ( s e. E /\ s =/= O ) ) -> ( s M U ) = ( s ( .r ` D ) U ) )
93 simprl 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( h e. T /\ h =/= ( _I |` B ) ) ) /\ ( s e. E /\ s =/= O ) ) -> s e. E )
94 1 4 5 2 22 erngmul-rN 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( s e. E /\ U e. E ) ) -> ( s ( .r ` D ) U ) = ( U o. s ) )
95 83 93 88 94 syl12anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( h e. T /\ h =/= ( _I |` B ) ) ) /\ ( s e. E /\ s =/= O ) ) -> ( s ( .r ` D ) U ) = ( U o. s ) )
96 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 =/= O ) ) -> ( U o. s ) = ( _I |` T ) )
97 96 3expa 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( h e. T /\ h =/= ( _I |` B ) ) ) /\ ( s e. E /\ s =/= O ) ) -> ( U o. s ) = ( _I |` T ) )
98 95 97 eqtrd 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( h e. T /\ h =/= ( _I |` B ) ) ) /\ ( s e. E /\ s =/= O ) ) -> ( s ( .r ` D ) U ) = ( _I |` T ) )
99 92 98 eqtrd 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( h e. T /\ h =/= ( _I |` B ) ) ) /\ ( s e. E /\ s =/= O ) ) -> ( s M U ) = ( _I |` T ) )
100 21 25 41 66 67 80 82 88 90 99 isdrngrd 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( h e. T /\ h =/= ( _I |` B ) ) ) -> D e. DivRing )