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Theorem frlmphllem 18606
Description: Lemma for frlmphl 18607. (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 1017 . . . . . . . 8  |-  ( (
ph  /\  g  e.  V  /\  h  e.  V
)  ->  I  e.  W )
3 simp2 997 . . . . . . . 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 18588 . . . . . . . 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 7440 . . . . . . 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 5731 . . . . . 6  |-  ( g : I --> B  -> 
g  Fn  I )
1210, 11syl 16 . . . . 5  |-  ( (
ph  /\  g  e.  V  /\  h  e.  V
)  ->  g  Fn  I )
13 simp3 998 . . . . . . . 8  |-  ( (
ph  /\  g  e.  V  /\  h  e.  V
)  ->  h  e.  V )
144, 5, 6frlmbasmap 18588 . . . . . . . 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 7440 . . . . . . 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 5731 . . . . . 6  |-  ( h : I --> B  ->  h  Fn  I )
1917, 18syl 16 . . . . 5  |-  ( (
ph  /\  g  e.  V  /\  h  e.  V
)  ->  h  Fn  I )
20 inidm 3707 . . . . 5  |-  ( I  i^i  I )  =  I
21 eqidd 2468 . . . . 5  |-  ( ( ( ph  /\  g  e.  V  /\  h  e.  V )  /\  x  e.  I )  ->  (
g `  x )  =  ( g `  x ) )
22 eqidd 2468 . . . . 5  |-  ( ( ( ph  /\  g  e.  V  /\  h  e.  V )  /\  x  e.  I )  ->  (
h `  x )  =  ( h `  x ) )
2312, 19, 2, 2, 20, 21, 22offval 6531 . . . 4  |-  ( (
ph  /\  g  e.  V  /\  h  e.  V
)  ->  ( g  oF  .x.  h )  =  ( x  e.  I  |->  ( ( g `
 x )  .x.  ( h `  x
) ) ) )
2423oveq1d 6299 . . 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 6309 . . . . 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 5624 . . . . . . 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 5607 . . . . . 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 18586 . . . . 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 17205 . . . . . . . . . 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 17203 . . . . . . . 8  |-  ( R  e.  DivRing  ->  R  e.  Ring )
4038, 39syl 16 . . . . . . 7  |-  ( ph  ->  R  e.  Ring )
41403ad2ant1 1017 . . . . . 6  |-  ( (
ph  /\  g  e.  V  /\  h  e.  V
)  ->  R  e.  Ring )
425, 32rng0cl 17021 . . . . . 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 17036 . . . . . 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 6937 . . . 4  |-  ( (
ph  /\  g  e.  V  /\  h  e.  V
)  ->  ( (
g  oF  .x.  h ) supp  .0.  )  C_  ( g supp  .0.  )
)
48 fsuppsssupp 7845 . . . . 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 7836 . . . 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 1229 . . 3  |-  ( (
ph  /\  g  e.  V  /\  h  e.  V
)  ->  ( (
g  oF  .x.  h ) supp  .0.  )  e.  Fin )
5124, 50eqeltrrd 2556 . 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 6130 . . . 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 5876 . . . . 5  |-  ( 0g
`  R )  e. 
_V
5632, 55eqeltri 2551 . . . 4  |-  .0.  e.  _V
5756a1i 11 . . 3  |-  ( (
ph  /\  g  e.  V  /\  h  e.  V
)  ->  .0.  e.  _V )
58 funisfsupp 7834 . . 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 1228 . 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 973    = wceq 1379    e. wcel 1767   _Vcvv 3113    C_ wss 3476   class class class wbr 4447    |-> cmpt 4505   Fun wfun 5582    Fn wfn 5583   -->wf 5584   ` cfv 5588  (class class class)co 6284    oFcof 6522   supp csupp 6901    ^m cmap 7420   Fincfn 7516   finSupp cfsupp 7829   Basecbs 14490   .rcmulr 14556   *rcstv 14557   .icip 14560   0gc0g 14695   Ringcrg 17000   CRingccrg 17001   DivRingcdr 17196  Fieldcfield 17197   freeLMod cfrlm 18572
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6576  ax-cnex 9548  ax-resscn 9549  ax-1cn 9550  ax-icn 9551  ax-addcl 9552  ax-addrcl 9553  ax-mulcl 9554  ax-mulrcl 9555  ax-mulcom 9556  ax-addass 9557  ax-mulass 9558  ax-distr 9559  ax-i2m1 9560  ax-1ne0 9561  ax-1rid 9562  ax-rnegex 9563  ax-rrecex 9564  ax-cnre 9565  ax-pre-lttri 9566  ax-pre-lttrn 9567  ax-pre-ltadd 9568  ax-pre-mulgt0 9569
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-riota 6245  df-ov 6287  df-oprab 6288  df-mpt2 6289  df-of 6524  df-om 6685  df-1st 6784  df-2nd 6785  df-supp 6902  df-recs 7042  df-rdg 7076  df-1o 7130  df-oadd 7134  df-er 7311  df-map 7422  df-ixp 7470  df-en 7517  df-dom 7518  df-sdom 7519  df-fin 7520  df-fsupp 7830  df-sup 7901  df-pnf 9630  df-mnf 9631  df-xr 9632  df-ltxr 9633  df-le 9634  df-sub 9807  df-neg 9808  df-nn 10537  df-2 10594  df-3 10595  df-4 10596  df-5 10597  df-6 10598  df-7 10599  df-8 10600  df-9 10601  df-10 10602  df-n0 10796  df-z 10865  df-dec 10977  df-uz 11083  df-fz 11673  df-struct 14492  df-ndx 14493  df-slot 14494  df-base 14495  df-sets 14496  df-ress 14497  df-plusg 14568  df-mulr 14569  df-sca 14571  df-vsca 14572  df-ip 14573  df-tset 14574  df-ple 14575  df-ds 14577  df-hom 14579  df-cco 14580  df-0g 14697  df-prds 14703  df-pws 14705  df-mnd 15732  df-grp 15867  df-minusg 15868  df-mgp 16944  df-rng 17002  df-drng 17198  df-field 17199  df-sra 17618  df-rgmod 17619  df-dsmm 18558  df-frlm 18573
This theorem is referenced by:  frlmphl  18607
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