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Theorem frlmphllem 18203
Description: Lemma for frlmphl 18204. (Contributed by AV, 21-Jul-2019.)
Hypotheses
Ref Expression
frlmphl.y  |-  Y  =  ( R freeLMod  I )
frlmphl.b  |-  B  =  ( Base `  R
)
frlmphl.t  |-  .x.  =  ( .r `  R )
frlmphl.v  |-  V  =  ( Base `  Y
)
frlmphl.j  |-  .,  =  ( .i `  Y )
frlmphl.o  |-  O  =  ( 0g `  Y
)
frlmphl.0  |-  .0.  =  ( 0g `  R )
frlmphl.s  |-  .*  =  ( *r `  R )
frlmphl.f  |-  ( ph  ->  R  e. Field )
frlmphl.m  |-  ( (
ph  /\  g  e.  V  /\  ( g  .,  g )  =  .0.  )  ->  g  =  O )
frlmphl.u  |-  ( (
ph  /\  x  e.  B )  ->  (  .*  `  x )  =  x )
frlmphl.i  |-  ( ph  ->  I  e.  W )
Assertion
Ref Expression
frlmphllem  |-  ( (
ph  /\  g  e.  V  /\  h  e.  V
)  ->  ( x  e.  I  |->  ( ( g `  x ) 
.x.  ( h `  x ) ) ) finSupp  .0.  )
Distinct variable groups:    B, g, x    g, I, x    R, g, x    g, V, x   
g, W, x    .x. , g, x    B, h, g, x   
h, I    R, h    h, V    h, W    g, Y, h, x    .0. , g, h, x    ph, g, h, x    ., , g, h, x    .x. , h    g, O, h   
x,  .*
Allowed substitution hints:    .* ( g, h)    O( x)

Proof of Theorem frlmphllem
StepHypRef Expression
1 frlmphl.i . . . . . . . . 9  |-  ( ph  ->  I  e.  W )
213ad2ant1 1009 . . . . . . . 8  |-  ( (
ph  /\  g  e.  V  /\  h  e.  V
)  ->  I  e.  W )
3 simp2 989 . . . . . . . 8  |-  ( (
ph  /\  g  e.  V  /\  h  e.  V
)  ->  g  e.  V )
4 frlmphl.y . . . . . . . . 9  |-  Y  =  ( R freeLMod  I )
5 frlmphl.b . . . . . . . . 9  |-  B  =  ( Base `  R
)
6 frlmphl.v . . . . . . . . 9  |-  V  =  ( Base `  Y
)
74, 5, 6frlmbasmap 18185 . . . . . . . 8  |-  ( ( I  e.  W  /\  g  e.  V )  ->  g  e.  ( B  ^m  I ) )
82, 3, 7syl2anc 661 . . . . . . 7  |-  ( (
ph  /\  g  e.  V  /\  h  e.  V
)  ->  g  e.  ( B  ^m  I ) )
9 elmapi 7232 . . . . . . 7  |-  ( g  e.  ( B  ^m  I )  ->  g : I --> B )
108, 9syl 16 . . . . . 6  |-  ( (
ph  /\  g  e.  V  /\  h  e.  V
)  ->  g :
I --> B )
11 ffn 5557 . . . . . 6  |-  ( g : I --> B  -> 
g  Fn  I )
1210, 11syl 16 . . . . 5  |-  ( (
ph  /\  g  e.  V  /\  h  e.  V
)  ->  g  Fn  I )
13 simp3 990 . . . . . . . 8  |-  ( (
ph  /\  g  e.  V  /\  h  e.  V
)  ->  h  e.  V )
144, 5, 6frlmbasmap 18185 . . . . . . . 8  |-  ( ( I  e.  W  /\  h  e.  V )  ->  h  e.  ( B  ^m  I ) )
152, 13, 14syl2anc 661 . . . . . . 7  |-  ( (
ph  /\  g  e.  V  /\  h  e.  V
)  ->  h  e.  ( B  ^m  I ) )
16 elmapi 7232 . . . . . . 7  |-  ( h  e.  ( B  ^m  I )  ->  h : I --> B )
1715, 16syl 16 . . . . . 6  |-  ( (
ph  /\  g  e.  V  /\  h  e.  V
)  ->  h :
I --> B )
18 ffn 5557 . . . . . 6  |-  ( h : I --> B  ->  h  Fn  I )
1917, 18syl 16 . . . . 5  |-  ( (
ph  /\  g  e.  V  /\  h  e.  V
)  ->  h  Fn  I )
20 inidm 3557 . . . . 5  |-  ( I  i^i  I )  =  I
21 eqidd 2442 . . . . 5  |-  ( ( ( ph  /\  g  e.  V  /\  h  e.  V )  /\  x  e.  I )  ->  (
g `  x )  =  ( g `  x ) )
22 eqidd 2442 . . . . 5  |-  ( ( ( ph  /\  g  e.  V  /\  h  e.  V )  /\  x  e.  I )  ->  (
h `  x )  =  ( h `  x ) )
2312, 19, 2, 2, 20, 21, 22offval 6325 . . . 4  |-  ( (
ph  /\  g  e.  V  /\  h  e.  V
)  ->  ( g  oF  .x.  h )  =  ( x  e.  I  |->  ( ( g `
 x )  .x.  ( h `  x
) ) ) )
2423oveq1d 6104 . . 3  |-  ( (
ph  /\  g  e.  V  /\  h  e.  V
)  ->  ( (
g  oF  .x.  h ) supp  .0.  )  =  ( ( x  e.  I  |->  ( ( g `  x ) 
.x.  ( h `  x ) ) ) supp 
.0.  ) )
25 ovex 6114 . . . . 5  |-  ( g  oF  .x.  h
)  e.  _V
2625a1i 11 . . . 4  |-  ( (
ph  /\  g  e.  V  /\  h  e.  V
)  ->  ( g  oF  .x.  h )  e.  _V )
27 funmpt 5452 . . . . . . 7  |-  Fun  (
x  e.  I  |->  ( ( g `  x
)  .x.  ( h `  x ) ) )
2827a1i 11 . . . . . 6  |-  ( ( g  oF  .x.  h )  =  ( x  e.  I  |->  ( ( g `  x
)  .x.  ( h `  x ) ) )  ->  Fun  ( x  e.  I  |->  ( ( g `  x ) 
.x.  ( h `  x ) ) ) )
29 funeq 5435 . . . . . 6  |-  ( ( g  oF  .x.  h )  =  ( x  e.  I  |->  ( ( g `  x
)  .x.  ( h `  x ) ) )  ->  ( Fun  (
g  oF  .x.  h )  <->  Fun  ( x  e.  I  |->  ( ( g `  x ) 
.x.  ( h `  x ) ) ) ) )
3028, 29mpbird 232 . . . . 5  |-  ( ( g  oF  .x.  h )  =  ( x  e.  I  |->  ( ( g `  x
)  .x.  ( h `  x ) ) )  ->  Fun  ( g  oF  .x.  h ) )
3123, 30syl 16 . . . 4  |-  ( (
ph  /\  g  e.  V  /\  h  e.  V
)  ->  Fun  ( g  oF  .x.  h
) )
32 frlmphl.0 . . . . . 6  |-  .0.  =  ( 0g `  R )
334, 32, 6frlmbasfsupp 18183 . . . . 5  |-  ( ( I  e.  W  /\  g  e.  V )  ->  g finSupp  .0.  )
342, 3, 33syl2anc 661 . . . 4  |-  ( (
ph  /\  g  e.  V  /\  h  e.  V
)  ->  g finSupp  .0.  )
35 frlmphl.f . . . . . . . . . 10  |-  ( ph  ->  R  e. Field )
36 isfld 16839 . . . . . . . . . 10  |-  ( R  e. Field 
<->  ( R  e.  DivRing  /\  R  e.  CRing ) )
3735, 36sylib 196 . . . . . . . . 9  |-  ( ph  ->  ( R  e.  DivRing  /\  R  e.  CRing ) )
3837simpld 459 . . . . . . . 8  |-  ( ph  ->  R  e.  DivRing )
39 drngrng 16837 . . . . . . . 8  |-  ( R  e.  DivRing  ->  R  e.  Ring )
4038, 39syl 16 . . . . . . 7  |-  ( ph  ->  R  e.  Ring )
41403ad2ant1 1009 . . . . . 6  |-  ( (
ph  /\  g  e.  V  /\  h  e.  V
)  ->  R  e.  Ring )
425, 32rng0cl 16664 . . . . . 6  |-  ( R  e.  Ring  ->  .0.  e.  B )
4341, 42syl 16 . . . . 5  |-  ( (
ph  /\  g  e.  V  /\  h  e.  V
)  ->  .0.  e.  B )
44 frlmphl.t . . . . . . 7  |-  .x.  =  ( .r `  R )
455, 44, 32rnglz 16679 . . . . . 6  |-  ( ( R  e.  Ring  /\  x  e.  B )  ->  (  .0.  .x.  x )  =  .0.  )
4641, 45sylan 471 . . . . 5  |-  ( ( ( ph  /\  g  e.  V  /\  h  e.  V )  /\  x  e.  B )  ->  (  .0.  .x.  x )  =  .0.  )
472, 43, 10, 17, 46suppofss1d 6724 . . . 4  |-  ( (
ph  /\  g  e.  V  /\  h  e.  V
)  ->  ( (
g  oF  .x.  h ) supp  .0.  )  C_  ( g supp  .0.  )
)
48 fsuppsssupp 7634 . . . . 5  |-  ( ( ( ( g  oF  .x.  h )  e.  _V  /\  Fun  ( g  oF  .x.  h ) )  /\  ( g finSupp  .0.  /\  ( ( g  oF  .x.  h ) supp 
.0.  )  C_  (
g supp  .0.  ) )
)  ->  ( g  oF  .x.  h ) finSupp  .0.  )
4948fsuppimpd 7625 . . . 4  |-  ( ( ( ( g  oF  .x.  h )  e.  _V  /\  Fun  ( g  oF  .x.  h ) )  /\  ( g finSupp  .0.  /\  ( ( g  oF  .x.  h ) supp 
.0.  )  C_  (
g supp  .0.  ) )
)  ->  ( (
g  oF  .x.  h ) supp  .0.  )  e.  Fin )
5026, 31, 34, 47, 49syl22anc 1219 . . 3  |-  ( (
ph  /\  g  e.  V  /\  h  e.  V
)  ->  ( (
g  oF  .x.  h ) supp  .0.  )  e.  Fin )
5124, 50eqeltrrd 2516 . 2  |-  ( (
ph  /\  g  e.  V  /\  h  e.  V
)  ->  ( (
x  e.  I  |->  ( ( g `  x
)  .x.  ( h `  x ) ) ) supp 
.0.  )  e.  Fin )
5227a1i 11 . . 3  |-  ( (
ph  /\  g  e.  V  /\  h  e.  V
)  ->  Fun  ( x  e.  I  |->  ( ( g `  x ) 
.x.  ( h `  x ) ) ) )
53 mptexg 5945 . . . 4  |-  ( I  e.  W  ->  (
x  e.  I  |->  ( ( g `  x
)  .x.  ( h `  x ) ) )  e.  _V )
542, 53syl 16 . . 3  |-  ( (
ph  /\  g  e.  V  /\  h  e.  V
)  ->  ( x  e.  I  |->  ( ( g `  x ) 
.x.  ( h `  x ) ) )  e.  _V )
55 fvex 5699 . . . . 5  |-  ( 0g
`  R )  e. 
_V
5632, 55eqeltri 2511 . . . 4  |-  .0.  e.  _V
5756a1i 11 . . 3  |-  ( (
ph  /\  g  e.  V  /\  h  e.  V
)  ->  .0.  e.  _V )
58 funisfsupp 7623 . . 3  |-  ( ( Fun  ( x  e.  I  |->  ( ( g `
 x )  .x.  ( h `  x
) ) )  /\  ( x  e.  I  |->  ( ( g `  x )  .x.  (
h `  x )
) )  e.  _V  /\  .0.  e.  _V )  ->  ( ( x  e.  I  |->  ( ( g `
 x )  .x.  ( h `  x
) ) ) finSupp  .0.  <->  (
( x  e.  I  |->  ( ( g `  x )  .x.  (
h `  x )
) ) supp  .0.  )  e.  Fin ) )
5952, 54, 57, 58syl3anc 1218 . 2  |-  ( (
ph  /\  g  e.  V  /\  h  e.  V
)  ->  ( (
x  e.  I  |->  ( ( g `  x
)  .x.  ( h `  x ) ) ) finSupp  .0. 
<->  ( ( x  e.  I  |->  ( ( g `
 x )  .x.  ( h `  x
) ) ) supp  .0.  )  e.  Fin )
)
6051, 59mpbird 232 1  |-  ( (
ph  /\  g  e.  V  /\  h  e.  V
)  ->  ( x  e.  I  |->  ( ( g `  x ) 
.x.  ( h `  x ) ) ) finSupp  .0.  )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756   _Vcvv 2970    C_ wss 3326   class class class wbr 4290    e. cmpt 4348   Fun wfun 5410    Fn wfn 5411   -->wf 5412   ` cfv 5416  (class class class)co 6089    oFcof 6316   supp csupp 6688    ^m cmap 7212   Fincfn 7308   finSupp cfsupp 7618   Basecbs 14172   .rcmulr 14237   *rcstv 14238   .icip 14241   0gc0g 14376   Ringcrg 16643   CRingccrg 16644   DivRingcdr 16830  Fieldcfield 16831   freeLMod cfrlm 18169
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2422  ax-rep 4401  ax-sep 4411  ax-nul 4419  ax-pow 4468  ax-pr 4529  ax-un 6370  ax-cnex 9336  ax-resscn 9337  ax-1cn 9338  ax-icn 9339  ax-addcl 9340  ax-addrcl 9341  ax-mulcl 9342  ax-mulrcl 9343  ax-mulcom 9344  ax-addass 9345  ax-mulass 9346  ax-distr 9347  ax-i2m1 9348  ax-1ne0 9349  ax-1rid 9350  ax-rnegex 9351  ax-rrecex 9352  ax-cnre 9353  ax-pre-lttri 9354  ax-pre-lttrn 9355  ax-pre-ltadd 9356  ax-pre-mulgt0 9357
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rmo 2721  df-rab 2722  df-v 2972  df-sbc 3185  df-csb 3287  df-dif 3329  df-un 3331  df-in 3333  df-ss 3340  df-pss 3342  df-nul 3636  df-if 3790  df-pw 3860  df-sn 3876  df-pr 3878  df-tp 3880  df-op 3882  df-uni 4090  df-int 4127  df-iun 4171  df-br 4291  df-opab 4349  df-mpt 4350  df-tr 4384  df-eprel 4630  df-id 4634  df-po 4639  df-so 4640  df-fr 4677  df-we 4679  df-ord 4720  df-on 4721  df-lim 4722  df-suc 4723  df-xp 4844  df-rel 4845  df-cnv 4846  df-co 4847  df-dm 4848  df-rn 4849  df-res 4850  df-ima 4851  df-iota 5379  df-fun 5418  df-fn 5419  df-f 5420  df-f1 5421  df-fo 5422  df-f1o 5423  df-fv 5424  df-riota 6050  df-ov 6092  df-oprab 6093  df-mpt2 6094  df-of 6318  df-om 6475  df-1st 6575  df-2nd 6576  df-supp 6689  df-recs 6830  df-rdg 6864  df-1o 6918  df-oadd 6922  df-er 7099  df-map 7214  df-ixp 7262  df-en 7309  df-dom 7310  df-sdom 7311  df-fin 7312  df-fsupp 7619  df-sup 7689  df-pnf 9418  df-mnf 9419  df-xr 9420  df-ltxr 9421  df-le 9422  df-sub 9595  df-neg 9596  df-nn 10321  df-2 10378  df-3 10379  df-4 10380  df-5 10381  df-6 10382  df-7 10383  df-8 10384  df-9 10385  df-10 10386  df-n0 10578  df-z 10645  df-dec 10754  df-uz 10860  df-fz 11436  df-struct 14174  df-ndx 14175  df-slot 14176  df-base 14177  df-sets 14178  df-ress 14179  df-plusg 14249  df-mulr 14250  df-sca 14252  df-vsca 14253  df-ip 14254  df-tset 14255  df-ple 14256  df-ds 14258  df-hom 14260  df-cco 14261  df-0g 14378  df-prds 14384  df-pws 14386  df-mnd 15413  df-grp 15543  df-minusg 15544  df-mgp 16590  df-rng 16645  df-drng 16832  df-field 16833  df-sra 17251  df-rgmod 17252  df-dsmm 18155  df-frlm 18170
This theorem is referenced by:  frlmphl  18204
  Copyright terms: Public domain W3C validator