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Theorem mapdpglem23 31953
Description: Lemma for mapdpg 31965. Baer p. 45, line 10: "and so y' meets all our requirements." Our  h is Baer's y'. (Contributed by NM, 20-Mar-2015.)
Hypotheses
Ref Expression
mapdpglem.h  |-  H  =  ( LHyp `  K
)
mapdpglem.m  |-  M  =  ( (mapd `  K
) `  W )
mapdpglem.u  |-  U  =  ( ( DVecH `  K
) `  W )
mapdpglem.v  |-  V  =  ( Base `  U
)
mapdpglem.s  |-  .-  =  ( -g `  U )
mapdpglem.n  |-  N  =  ( LSpan `  U )
mapdpglem.c  |-  C  =  ( (LCDual `  K
) `  W )
mapdpglem.k  |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )
mapdpglem.x  |-  ( ph  ->  X  e.  V )
mapdpglem.y  |-  ( ph  ->  Y  e.  V )
mapdpglem1.p  |-  .(+)  =  (
LSSum `  C )
mapdpglem2.j  |-  J  =  ( LSpan `  C )
mapdpglem3.f  |-  F  =  ( Base `  C
)
mapdpglem3.te  |-  ( ph  ->  t  e.  ( ( M `  ( N `
 { X }
) )  .(+)  ( M `
 ( N `  { Y } ) ) ) )
mapdpglem3.a  |-  A  =  (Scalar `  U )
mapdpglem3.b  |-  B  =  ( Base `  A
)
mapdpglem3.t  |-  .x.  =  ( .s `  C )
mapdpglem3.r  |-  R  =  ( -g `  C
)
mapdpglem3.g  |-  ( ph  ->  G  e.  F )
mapdpglem3.e  |-  ( ph  ->  ( M `  ( N `  { X } ) )  =  ( J `  { G } ) )
mapdpglem4.q  |-  Q  =  ( 0g `  U
)
mapdpglem.ne  |-  ( ph  ->  ( N `  { X } )  =/=  ( N `  { Y } ) )
mapdpglem4.jt  |-  ( ph  ->  ( M `  ( N `  { ( X  .-  Y ) } ) )  =  ( J `  { t } ) )
mapdpglem4.z  |-  .0.  =  ( 0g `  A )
mapdpglem4.g4  |-  ( ph  ->  g  e.  B )
mapdpglem4.z4  |-  ( ph  ->  z  e.  ( M `
 ( N `  { Y } ) ) )
mapdpglem4.t4  |-  ( ph  ->  t  =  ( ( g  .x.  G ) R z ) )
mapdpglem4.xn  |-  ( ph  ->  X  =/=  Q )
mapdpglem12.yn  |-  ( ph  ->  Y  =/=  Q )
mapdpglem17.ep  |-  E  =  ( ( ( invr `  A ) `  g
)  .x.  z )
Assertion
Ref Expression
mapdpglem23  |-  ( ph  ->  E. h  e.  F  ( ( M `  ( N `  { Y } ) )  =  ( J `  {
h } )  /\  ( M `  ( N `
 { ( X 
.-  Y ) } ) )  =  ( J `  { ( G R h ) } ) ) )
Distinct variable groups:    t,  .-    t, C    t, J    t, M    t, N    t, X    t, Y    B, g    z, g, C    g, F    g, G, z    g, J, z   
g, M, z    g, N, z    R, g, z    .x. , g, z    g, Y, z, t    h, E   
h, F    h, G    h, J    h, M    h, N    R, h    .- , h    h, X    h, Y
Allowed substitution hints:    ph( z, t, g, h)    A( z,
t, g, h)    B( z, t, h)    C( h)    .(+) (
z, t, g, h)    Q( z, t, g, h)    R( t)    .x. ( t, h)    U( z, t, g, h)    E( z, t, g)    F( z, t)    G( t)    H( z, t, g, h)    K( z, t, g, h)    .- ( z,
g)    V( z, t, g, h)    W( z, t, g, h)    X( z, g)    .0. ( z, t, g, h)

Proof of Theorem mapdpglem23
StepHypRef Expression
1 mapdpglem.h . . . 4  |-  H  =  ( LHyp `  K
)
2 mapdpglem.m . . . 4  |-  M  =  ( (mapd `  K
) `  W )
3 mapdpglem.u . . . 4  |-  U  =  ( ( DVecH `  K
) `  W )
4 eqid 2358 . . . 4  |-  ( LSubSp `  U )  =  (
LSubSp `  U )
5 mapdpglem.c . . . 4  |-  C  =  ( (LCDual `  K
) `  W )
6 eqid 2358 . . . 4  |-  ( LSubSp `  C )  =  (
LSubSp `  C )
7 mapdpglem.k . . . 4  |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )
81, 3, 7dvhlmod 31369 . . . . 5  |-  ( ph  ->  U  e.  LMod )
9 mapdpglem.y . . . . 5  |-  ( ph  ->  Y  e.  V )
10 mapdpglem.v . . . . . 6  |-  V  =  ( Base `  U
)
11 mapdpglem.n . . . . . 6  |-  N  =  ( LSpan `  U )
1210, 4, 11lspsncl 15833 . . . . 5  |-  ( ( U  e.  LMod  /\  Y  e.  V )  ->  ( N `  { Y } )  e.  (
LSubSp `  U ) )
138, 9, 12syl2anc 642 . . . 4  |-  ( ph  ->  ( N `  { Y } )  e.  (
LSubSp `  U ) )
141, 2, 3, 4, 5, 6, 7, 13mapdcl2 31915 . . 3  |-  ( ph  ->  ( M `  ( N `  { Y } ) )  e.  ( LSubSp `  C )
)
15 mapdpglem.s . . . 4  |-  .-  =  ( -g `  U )
16 mapdpglem.x . . . 4  |-  ( ph  ->  X  e.  V )
17 mapdpglem1.p . . . 4  |-  .(+)  =  (
LSSum `  C )
18 mapdpglem2.j . . . 4  |-  J  =  ( LSpan `  C )
19 mapdpglem3.f . . . 4  |-  F  =  ( Base `  C
)
20 mapdpglem3.te . . . 4  |-  ( ph  ->  t  e.  ( ( M `  ( N `
 { X }
) )  .(+)  ( M `
 ( N `  { Y } ) ) ) )
21 mapdpglem3.a . . . 4  |-  A  =  (Scalar `  U )
22 mapdpglem3.b . . . 4  |-  B  =  ( Base `  A
)
23 mapdpglem3.t . . . 4  |-  .x.  =  ( .s `  C )
24 mapdpglem3.r . . . 4  |-  R  =  ( -g `  C
)
25 mapdpglem3.g . . . 4  |-  ( ph  ->  G  e.  F )
26 mapdpglem3.e . . . 4  |-  ( ph  ->  ( M `  ( N `  { X } ) )  =  ( J `  { G } ) )
27 mapdpglem4.q . . . 4  |-  Q  =  ( 0g `  U
)
28 mapdpglem.ne . . . 4  |-  ( ph  ->  ( N `  { X } )  =/=  ( N `  { Y } ) )
29 mapdpglem4.jt . . . 4  |-  ( ph  ->  ( M `  ( N `  { ( X  .-  Y ) } ) )  =  ( J `  { t } ) )
30 mapdpglem4.z . . . 4  |-  .0.  =  ( 0g `  A )
31 mapdpglem4.g4 . . . 4  |-  ( ph  ->  g  e.  B )
32 mapdpglem4.z4 . . . 4  |-  ( ph  ->  z  e.  ( M `
 ( N `  { Y } ) ) )
33 mapdpglem4.t4 . . . 4  |-  ( ph  ->  t  =  ( ( g  .x.  G ) R z ) )
34 mapdpglem4.xn . . . 4  |-  ( ph  ->  X  =/=  Q )
35 mapdpglem12.yn . . . 4  |-  ( ph  ->  Y  =/=  Q )
36 mapdpglem17.ep . . . 4  |-  E  =  ( ( ( invr `  A ) `  g
)  .x.  z )
371, 2, 3, 10, 15, 11, 5, 7, 16, 9, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36mapdpglem19 31949 . . 3  |-  ( ph  ->  E  e.  ( M `
 ( N `  { Y } ) ) )
3819, 6lssel 15794 . . 3  |-  ( ( ( M `  ( N `  { Y } ) )  e.  ( LSubSp `  C )  /\  E  e.  ( M `  ( N `  { Y } ) ) )  ->  E  e.  F )
3914, 37, 38syl2anc 642 . 2  |-  ( ph  ->  E  e.  F )
401, 2, 3, 10, 15, 11, 5, 7, 16, 9, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36mapdpglem20 31950 . 2  |-  ( ph  ->  ( M `  ( N `  { Y } ) )  =  ( J `  { E } ) )
411, 2, 3, 10, 15, 11, 5, 7, 16, 9, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36mapdpglem22 31952 . 2  |-  ( ph  ->  ( M `  ( N `  { ( X  .-  Y ) } ) )  =  ( J `  { ( G R E ) } ) )
42 sneq 3727 . . . . . 6  |-  ( h  =  E  ->  { h }  =  { E } )
4342fveq2d 5612 . . . . 5  |-  ( h  =  E  ->  ( J `  { h } )  =  ( J `  { E } ) )
4443eqeq2d 2369 . . . 4  |-  ( h  =  E  ->  (
( M `  ( N `  { Y } ) )  =  ( J `  {
h } )  <->  ( M `  ( N `  { Y } ) )  =  ( J `  { E } ) ) )
45 oveq2 5953 . . . . . . 7  |-  ( h  =  E  ->  ( G R h )  =  ( G R E ) )
4645sneqd 3729 . . . . . 6  |-  ( h  =  E  ->  { ( G R h ) }  =  { ( G R E ) } )
4746fveq2d 5612 . . . . 5  |-  ( h  =  E  ->  ( J `  { ( G R h ) } )  =  ( J `
 { ( G R E ) } ) )
4847eqeq2d 2369 . . . 4  |-  ( h  =  E  ->  (
( M `  ( N `  { ( X  .-  Y ) } ) )  =  ( J `  { ( G R h ) } )  <->  ( M `  ( N `  {
( X  .-  Y
) } ) )  =  ( J `  { ( G R E ) } ) ) )
4944, 48anbi12d 691 . . 3  |-  ( h  =  E  ->  (
( ( M `  ( N `  { Y } ) )  =  ( J `  {
h } )  /\  ( M `  ( N `
 { ( X 
.-  Y ) } ) )  =  ( J `  { ( G R h ) } ) )  <->  ( ( M `  ( N `  { Y } ) )  =  ( J `
 { E }
)  /\  ( M `  ( N `  {
( X  .-  Y
) } ) )  =  ( J `  { ( G R E ) } ) ) ) )
5049rspcev 2960 . 2  |-  ( ( E  e.  F  /\  ( ( M `  ( N `  { Y } ) )  =  ( J `  { E } )  /\  ( M `  ( N `  { ( X  .-  Y ) } ) )  =  ( J `
 { ( G R E ) } ) ) )  ->  E. h  e.  F  ( ( M `  ( N `  { Y } ) )  =  ( J `  {
h } )  /\  ( M `  ( N `
 { ( X 
.-  Y ) } ) )  =  ( J `  { ( G R h ) } ) ) )
5139, 40, 41, 50syl12anc 1180 1  |-  ( ph  ->  E. h  e.  F  ( ( M `  ( N `  { Y } ) )  =  ( J `  {
h } )  /\  ( M `  ( N `
 { ( X 
.-  Y ) } ) )  =  ( J `  { ( G R h ) } ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1642    e. wcel 1710    =/= wne 2521   E.wrex 2620   {csn 3716   ` cfv 5337  (class class class)co 5945   Basecbs 13245  Scalarcsca 13308   .scvsca 13309   0gc0g 13499   -gcsg 14464   LSSumclsm 15044   invrcinvr 15552   LModclmod 15726   LSubSpclss 15788   LSpanclspn 15827   HLchlt 29609   LHypclh 30242   DVecHcdvh 31337  LCDualclcd 31845  mapdcmpd 31883
This theorem is referenced by:  mapdpglem24  31963
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-rep 4212  ax-sep 4222  ax-nul 4230  ax-pow 4269  ax-pr 4295  ax-un 4594  ax-cnex 8883  ax-resscn 8884  ax-1cn 8885  ax-icn 8886  ax-addcl 8887  ax-addrcl 8888  ax-mulcl 8889  ax-mulrcl 8890  ax-mulcom 8891  ax-addass 8892  ax-mulass 8893  ax-distr 8894  ax-i2m1 8895  ax-1ne0 8896  ax-1rid 8897  ax-rnegex 8898  ax-rrecex 8899  ax-cnre 8900  ax-pre-lttri 8901  ax-pre-lttrn 8902  ax-pre-ltadd 8903  ax-pre-mulgt0 8904
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1319  df-fal 1320  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-nel 2524  df-ral 2624  df-rex 2625  df-reu 2626  df-rmo 2627  df-rab 2628  df-v 2866  df-sbc 3068  df-csb 3158  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-pss 3244  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-tp 3724  df-op 3725  df-uni 3909  df-int 3944  df-iun 3988  df-iin 3989  df-br 4105  df-opab 4159  df-mpt 4160  df-tr 4195  df-eprel 4387  df-id 4391  df-po 4396  df-so 4397  df-fr 4434  df-we 4436  df-ord 4477  df-on 4478  df-lim 4479  df-suc 4480  df-om 4739  df-xp 4777  df-rel 4778  df-cnv 4779  df-co 4780  df-dm 4781  df-rn 4782  df-res 4783  df-ima 4784  df-iota 5301  df-fun 5339  df-fn 5340  df-f 5341  df-f1 5342  df-fo 5343  df-f1o 5344  df-fv 5345  df-ov 5948  df-oprab 5949  df-mpt2 5950  df-of 6165  df-1st 6209  df-2nd 6210  df-tpos 6321  df-undef 6385  df-riota 6391  df-recs 6475  df-rdg 6510  df-1o 6566  df-oadd 6570  df-er 6747  df-map 6862  df-en 6952  df-dom 6953  df-sdom 6954  df-fin 6955  df-pnf 8959  df-mnf 8960  df-xr 8961  df-ltxr 8962  df-le 8963  df-sub 9129  df-neg 9130  df-nn 9837  df-2 9894  df-3 9895  df-4 9896  df-5 9897  df-6 9898  df-n0 10058  df-z 10117  df-uz 10323  df-fz 10875  df-struct 13247  df-ndx 13248  df-slot 13249  df-base 13250  df-sets 13251  df-ress 13252  df-plusg 13318  df-mulr 13319  df-sca 13321  df-vsca 13322  df-0g 13503  df-mre 13587  df-mrc 13588  df-acs 13590  df-poset 14179  df-plt 14191  df-lub 14207  df-glb 14208  df-join 14209  df-meet 14210  df-p0 14244  df-p1 14245  df-lat 14251  df-clat 14313  df-mnd 14466  df-submnd 14515  df-grp 14588  df-minusg 14589  df-sbg 14590  df-subg 14717  df-cntz 14892  df-oppg 14918  df-lsm 15046  df-cmn 15190  df-abl 15191  df-mgp 15425  df-rng 15439  df-ur 15441  df-oppr 15504  df-dvdsr 15522  df-unit 15523  df-invr 15553  df-dvr 15564  df-drng 15613  df-lmod 15728  df-lss 15789  df-lsp 15828  df-lvec 15955  df-lsatoms 29235  df-lshyp 29236  df-lcv 29278  df-lfl 29317  df-lkr 29345  df-ldual 29383  df-oposet 29435  df-ol 29437  df-oml 29438  df-covers 29525  df-ats 29526  df-atl 29557  df-cvlat 29581  df-hlat 29610  df-llines 29756  df-lplanes 29757  df-lvols 29758  df-lines 29759  df-psubsp 29761  df-pmap 29762  df-padd 30054  df-lhyp 30246  df-laut 30247  df-ldil 30362  df-ltrn 30363  df-trl 30417  df-tgrp 31001  df-tendo 31013  df-edring 31015  df-dveca 31261  df-disoa 31288  df-dvech 31338  df-dib 31398  df-dic 31432  df-dih 31488  df-doch 31607  df-djh 31654  df-lcdual 31846  df-mapd 31884
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