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Theorem cdlemn11c 30666
Description: Part of proof of Lemma N of [Crawley] p. 121 line 37. (Contributed by NM, 27-Feb-2014.)
Hypotheses
Ref Expression
cdlemn11a.b  |-  B  =  ( Base `  K
)
cdlemn11a.l  |-  .<_  =  ( le `  K )
cdlemn11a.j  |-  .\/  =  ( join `  K )
cdlemn11a.a  |-  A  =  ( Atoms `  K )
cdlemn11a.h  |-  H  =  ( LHyp `  K
)
cdlemn11a.p  |-  P  =  ( ( oc `  K ) `  W
)
cdlemn11a.o  |-  O  =  ( h  e.  T  |->  (  _I  |`  B ) )
cdlemn11a.t  |-  T  =  ( ( LTrn `  K
) `  W )
cdlemn11a.r  |-  R  =  ( ( trL `  K
) `  W )
cdlemn11a.e  |-  E  =  ( ( TEndo `  K
) `  W )
cdlemn11a.i  |-  I  =  ( ( DIsoB `  K
) `  W )
cdlemn11a.J  |-  J  =  ( ( DIsoC `  K
) `  W )
cdlemn11a.u  |-  U  =  ( ( DVecH `  K
) `  W )
cdlemn11a.d  |-  .+  =  ( +g  `  U )
cdlemn11a.s  |-  .(+)  =  (
LSSum `  U )
cdlemn11a.f  |-  F  =  ( iota_ h  e.  T
( h `  P
)  =  Q )
cdlemn11a.g  |-  G  =  ( iota_ h  e.  T
( h `  P
)  =  N )
Assertion
Ref Expression
cdlemn11c  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( N  e.  A  /\  -.  N  .<_  W )  /\  ( X  e.  B  /\  X  .<_  W ) )  /\  ( J `  N )  C_  ( ( J `  Q )  .(+)  ( I `
 X ) ) )  ->  E. y  e.  ( J `  Q
) E. z  e.  ( I `  X
) <. G ,  (  _I  |`  T ) >.  =  ( y  .+  z ) )
Distinct variable groups:    y, z,  .+    y,  .\/ , z    y, h, z,  .<_    A, h, y, z    B, h, y, z    y, G, z    y, I, z   
y,  .(+) , z    h, H, y, z    h, K, y, z    h, N, y, z    y, J, z    P, h    Q, h, y, z    T, h, y, z   
y, U, z    h, W, y, z    y, X, z
Allowed substitution hints:    P( y, z)    .+ ( h)    .(+) ( h)    R( y,
z, h)    U( h)    E( y, z, h)    F( y, z, h)    G( h)    I( h)    J( h)    .\/ ( h)    O( y, z, h)    X( h)

Proof of Theorem cdlemn11c
StepHypRef Expression
1 cdlemn11a.b . . 3  |-  B  =  ( Base `  K
)
2 cdlemn11a.l . . 3  |-  .<_  =  ( le `  K )
3 cdlemn11a.j . . 3  |-  .\/  =  ( join `  K )
4 cdlemn11a.a . . 3  |-  A  =  ( Atoms `  K )
5 cdlemn11a.h . . 3  |-  H  =  ( LHyp `  K
)
6 cdlemn11a.p . . 3  |-  P  =  ( ( oc `  K ) `  W
)
7 cdlemn11a.o . . 3  |-  O  =  ( h  e.  T  |->  (  _I  |`  B ) )
8 cdlemn11a.t . . 3  |-  T  =  ( ( LTrn `  K
) `  W )
9 cdlemn11a.r . . 3  |-  R  =  ( ( trL `  K
) `  W )
10 cdlemn11a.e . . 3  |-  E  =  ( ( TEndo `  K
) `  W )
11 cdlemn11a.i . . 3  |-  I  =  ( ( DIsoB `  K
) `  W )
12 cdlemn11a.J . . 3  |-  J  =  ( ( DIsoC `  K
) `  W )
13 cdlemn11a.u . . 3  |-  U  =  ( ( DVecH `  K
) `  W )
14 cdlemn11a.d . . 3  |-  .+  =  ( +g  `  U )
15 cdlemn11a.s . . 3  |-  .(+)  =  (
LSSum `  U )
16 cdlemn11a.f . . 3  |-  F  =  ( iota_ h  e.  T
( h `  P
)  =  Q )
17 cdlemn11a.g . . 3  |-  G  =  ( iota_ h  e.  T
( h `  P
)  =  N )
181, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17cdlemn11b 30665 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( N  e.  A  /\  -.  N  .<_  W )  /\  ( X  e.  B  /\  X  .<_  W ) )  /\  ( J `  N )  C_  ( ( J `  Q )  .(+)  ( I `
 X ) ) )  ->  <. G , 
(  _I  |`  T )
>.  e.  ( ( J `
 Q )  .(+)  ( I `  X ) ) )
19 simp1 960 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( N  e.  A  /\  -.  N  .<_  W )  /\  ( X  e.  B  /\  X  .<_  W ) )  /\  ( J `  N )  C_  ( ( J `  Q )  .(+)  ( I `
 X ) ) )  ->  ( K  e.  HL  /\  W  e.  H ) )
205, 13, 19dvhlmod 30567 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( N  e.  A  /\  -.  N  .<_  W )  /\  ( X  e.  B  /\  X  .<_  W ) )  /\  ( J `  N )  C_  ( ( J `  Q )  .(+)  ( I `
 X ) ) )  ->  U  e.  LMod )
21 eqid 2284 . . . . . 6  |-  ( LSubSp `  U )  =  (
LSubSp `  U )
2221lsssssubg 15709 . . . . 5  |-  ( U  e.  LMod  ->  ( LSubSp `  U )  C_  (SubGrp `  U ) )
2320, 22syl 17 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( N  e.  A  /\  -.  N  .<_  W )  /\  ( X  e.  B  /\  X  .<_  W ) )  /\  ( J `  N )  C_  ( ( J `  Q )  .(+)  ( I `
 X ) ) )  ->  ( LSubSp `  U )  C_  (SubGrp `  U ) )
24 simp21 993 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( N  e.  A  /\  -.  N  .<_  W )  /\  ( X  e.  B  /\  X  .<_  W ) )  /\  ( J `  N )  C_  ( ( J `  Q )  .(+)  ( I `
 X ) ) )  ->  ( Q  e.  A  /\  -.  Q  .<_  W ) )
252, 4, 5, 13, 12, 21diclss 30650 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  -> 
( J `  Q
)  e.  ( LSubSp `  U ) )
2619, 24, 25syl2anc 645 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( N  e.  A  /\  -.  N  .<_  W )  /\  ( X  e.  B  /\  X  .<_  W ) )  /\  ( J `  N )  C_  ( ( J `  Q )  .(+)  ( I `
 X ) ) )  ->  ( J `  Q )  e.  (
LSubSp `  U ) )
2723, 26sseldd 3182 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( N  e.  A  /\  -.  N  .<_  W )  /\  ( X  e.  B  /\  X  .<_  W ) )  /\  ( J `  N )  C_  ( ( J `  Q )  .(+)  ( I `
 X ) ) )  ->  ( J `  Q )  e.  (SubGrp `  U ) )
28 simp23l 1081 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( N  e.  A  /\  -.  N  .<_  W )  /\  ( X  e.  B  /\  X  .<_  W ) )  /\  ( J `  N )  C_  ( ( J `  Q )  .(+)  ( I `
 X ) ) )  ->  X  e.  B )
29 simp23r 1082 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( N  e.  A  /\  -.  N  .<_  W )  /\  ( X  e.  B  /\  X  .<_  W ) )  /\  ( J `  N )  C_  ( ( J `  Q )  .(+)  ( I `
 X ) ) )  ->  X  .<_  W )
301, 2, 5, 13, 11, 21diblss 30627 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  X  .<_  W ) )  ->  (
I `  X )  e.  ( LSubSp `  U )
)
3119, 28, 29, 30syl12anc 1185 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( N  e.  A  /\  -.  N  .<_  W )  /\  ( X  e.  B  /\  X  .<_  W ) )  /\  ( J `  N )  C_  ( ( J `  Q )  .(+)  ( I `
 X ) ) )  ->  ( I `  X )  e.  (
LSubSp `  U ) )
3223, 31sseldd 3182 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( N  e.  A  /\  -.  N  .<_  W )  /\  ( X  e.  B  /\  X  .<_  W ) )  /\  ( J `  N )  C_  ( ( J `  Q )  .(+)  ( I `
 X ) ) )  ->  ( I `  X )  e.  (SubGrp `  U ) )
3314, 15lsmelval 14954 . . 3  |-  ( ( ( J `  Q
)  e.  (SubGrp `  U )  /\  (
I `  X )  e.  (SubGrp `  U )
)  ->  ( <. G ,  (  _I  |`  T )
>.  e.  ( ( J `
 Q )  .(+)  ( I `  X ) )  <->  E. y  e.  ( J `  Q ) E. z  e.  ( I `  X )
<. G ,  (  _I  |`  T ) >.  =  ( y  .+  z ) ) )
3427, 32, 33syl2anc 645 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( N  e.  A  /\  -.  N  .<_  W )  /\  ( X  e.  B  /\  X  .<_  W ) )  /\  ( J `  N )  C_  ( ( J `  Q )  .(+)  ( I `
 X ) ) )  ->  ( <. G ,  (  _I  |`  T )
>.  e.  ( ( J `
 Q )  .(+)  ( I `  X ) )  <->  E. y  e.  ( J `  Q ) E. z  e.  ( I `  X )
<. G ,  (  _I  |`  T ) >.  =  ( y  .+  z ) ) )
3518, 34mpbid 203 1  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( N  e.  A  /\  -.  N  .<_  W )  /\  ( X  e.  B  /\  X  .<_  W ) )  /\  ( J `  N )  C_  ( ( J `  Q )  .(+)  ( I `
 X ) ) )  ->  E. y  e.  ( J `  Q
) E. z  e.  ( I `  X
) <. G ,  (  _I  |`  T ) >.  =  ( y  .+  z ) )
Colors of variables: wff set class
Syntax hints:   -. wn 5    -> wi 6    <-> wb 178    /\ wa 360    /\ w3a 939    = wceq 1628    e. wcel 1688   E.wrex 2545    C_ wss 3153   <.cop 3644   class class class wbr 4024    e. cmpt 4078    _I cid 4303    |` cres 4690   ` cfv 5221  (class class class)co 5819   iota_crio 6290   Basecbs 13142   +g cplusg 13202   lecple 13209   occoc 13210   joincjn 14072  SubGrpcsubg 14609   LSSumclsm 14939   LModclmod 15621   LSubSpclss 15683   Atomscatm 28720   HLchlt 28807   LHypclh 29440   LTrncltrn 29557   trLctrl 29614   TEndoctendo 30208   DVecHcdvh 30535   DIsoBcdib 30595   DIsoCcdic 30629
This theorem is referenced by:  cdlemn11pre  30667
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1538  ax-5 1549  ax-17 1608  ax-9 1641  ax-8 1648  ax-13 1690  ax-14 1692  ax-6 1707  ax-7 1712  ax-11 1719  ax-12 1869  ax-ext 2265  ax-rep 4132  ax-sep 4142  ax-nul 4150  ax-pow 4187  ax-pr 4213  ax-un 4511  ax-cnex 8788  ax-resscn 8789  ax-1cn 8790  ax-icn 8791  ax-addcl 8792  ax-addrcl 8793  ax-mulcl 8794  ax-mulrcl 8795  ax-mulcom 8796  ax-addass 8797  ax-mulass 8798  ax-distr 8799  ax-i2m1 8800  ax-1ne0 8801  ax-1rid 8802  ax-rnegex 8803  ax-rrecex 8804  ax-cnre 8805  ax-pre-lttri 8806  ax-pre-lttrn 8807  ax-pre-ltadd 8808  ax-pre-mulgt0 8809
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-fal 1316  df-ex 1534  df-nf 1537  df-sb 1636  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-nel 2450  df-ral 2549  df-rex 2550  df-reu 2551  df-rmo 2552  df-rab 2553  df-v 2791  df-sbc 2993  df-csb 3083  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-pss 3169  df-nul 3457  df-if 3567  df-pw 3628  df-sn 3647  df-pr 3648  df-tp 3649  df-op 3650  df-uni 3829  df-int 3864  df-iun 3908  df-iin 3909  df-br 4025  df-opab 4079  df-mpt 4080  df-tr 4115  df-eprel 4304  df-id 4308  df-po 4313  df-so 4314  df-fr 4351  df-we 4353  df-ord 4394  df-on 4395  df-lim 4396  df-suc 4397  df-om 4656  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-fun 5223  df-fn 5224  df-f 5225  df-f1 5226  df-fo 5227  df-f1o 5228  df-fv 5229  df-ov 5822  df-oprab 5823  df-mpt2 5824  df-1st 6083  df-2nd 6084  df-tpos 6195  df-iota 6252  df-undef 6291  df-riota 6299  df-recs 6383  df-rdg 6418  df-1o 6474  df-oadd 6478  df-er 6655  df-map 6769  df-en 6859  df-dom 6860  df-sdom 6861  df-fin 6862  df-pnf 8864  df-mnf 8865  df-xr 8866  df-ltxr 8867  df-le 8868  df-sub 9034  df-neg 9035  df-nn 9742  df-2 9799  df-3 9800  df-4 9801  df-5 9802  df-6 9803  df-n0 9961  df-z 10020  df-uz 10226  df-fz 10777  df-struct 13144  df-ndx 13145  df-slot 13146  df-base 13147  df-sets 13148  df-ress 13149  df-plusg 13215  df-mulr 13216  df-sca 13218  df-vsca 13219  df-0g 13398  df-poset 14074  df-plt 14086  df-lub 14102  df-glb 14103  df-join 14104  df-meet 14105  df-p0 14139  df-p1 14140  df-lat 14146  df-clat 14208  df-mnd 14361  df-grp 14483  df-minusg 14484  df-sbg 14485  df-subg 14612  df-lsm 14941  df-mgp 15320  df-rng 15334  df-ur 15336  df-oppr 15399  df-dvdsr 15417  df-unit 15418  df-invr 15448  df-dvr 15459  df-drng 15508  df-lmod 15623  df-lss 15684  df-lvec 15850  df-oposet 28633  df-ol 28635  df-oml 28636  df-covers 28723  df-ats 28724  df-atl 28755  df-cvlat 28779  df-hlat 28808  df-llines 28954  df-lplanes 28955  df-lvols 28956  df-lines 28957  df-psubsp 28959  df-pmap 28960  df-padd 29252  df-lhyp 29444  df-laut 29445  df-ldil 29560  df-ltrn 29561  df-trl 29615  df-tendo 30211  df-edring 30213  df-disoa 30486  df-dvech 30536  df-dib 30596  df-dic 30630
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