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Theorem lindslinindimp2lem1 31102
Description: Lemma 1 for lindslinindsimp2 31107. (Contributed by AV, 25-Apr-2019.)
Hypotheses
Ref Expression
lindslinind.r  |-  R  =  (Scalar `  M )
lindslinind.b  |-  B  =  ( Base `  R
)
lindslinind.0  |-  .0.  =  ( 0g `  R )
lindslinind.z  |-  Z  =  ( 0g `  M
)
lindslinind.y  |-  Y  =  ( ( invg `  R ) `  (
f `  x )
)
lindslinind.g  |-  G  =  ( f  |`  ( S  \  { x }
) )
Assertion
Ref Expression
lindslinindimp2lem1  |-  ( ( ( S  e.  V  /\  M  e.  LMod )  /\  ( S  C_  ( Base `  M )  /\  x  e.  S  /\  f  e.  ( B  ^m  S ) ) )  ->  Y  e.  B )
Distinct variable groups:    B, f    f, M    R, f, x    S, f, x    f, Z    .0. , f, x
Allowed substitution hints:    B( x)    G( x, f)    M( x)    V( x, f)    Y( x, f)    Z( x)

Proof of Theorem lindslinindimp2lem1
StepHypRef Expression
1 lindslinind.y . 2  |-  Y  =  ( ( invg `  R ) `  (
f `  x )
)
2 lindslinind.r . . . . 5  |-  R  =  (Scalar `  M )
32lmodfgrp 17072 . . . 4  |-  ( M  e.  LMod  ->  R  e. 
Grp )
43adantl 466 . . 3  |-  ( ( S  e.  V  /\  M  e.  LMod )  ->  R  e.  Grp )
5 elmapi 7337 . . . . . 6  |-  ( f  e.  ( B  ^m  S )  ->  f : S --> B )
6 ffvelrn 5943 . . . . . . . 8  |-  ( ( f : S --> B  /\  x  e.  S )  ->  ( f `  x
)  e.  B )
76a1d 25 . . . . . . 7  |-  ( ( f : S --> B  /\  x  e.  S )  ->  ( S  C_  ( Base `  M )  -> 
( f `  x
)  e.  B ) )
87ex 434 . . . . . 6  |-  ( f : S --> B  -> 
( x  e.  S  ->  ( S  C_  ( Base `  M )  -> 
( f `  x
)  e.  B ) ) )
95, 8syl 16 . . . . 5  |-  ( f  e.  ( B  ^m  S )  ->  (
x  e.  S  -> 
( S  C_  ( Base `  M )  -> 
( f `  x
)  e.  B ) ) )
109com13 80 . . . 4  |-  ( S 
C_  ( Base `  M
)  ->  ( x  e.  S  ->  ( f  e.  ( B  ^m  S )  ->  (
f `  x )  e.  B ) ) )
11103imp 1182 . . 3  |-  ( ( S  C_  ( Base `  M )  /\  x  e.  S  /\  f  e.  ( B  ^m  S
) )  ->  (
f `  x )  e.  B )
12 lindslinind.b . . . 4  |-  B  =  ( Base `  R
)
13 eqid 2451 . . . 4  |-  ( invg `  R )  =  ( invg `  R )
1412, 13grpinvcl 15694 . . 3  |-  ( ( R  e.  Grp  /\  ( f `  x
)  e.  B )  ->  ( ( invg `  R ) `
 ( f `  x ) )  e.  B )
154, 11, 14syl2an 477 . 2  |-  ( ( ( S  e.  V  /\  M  e.  LMod )  /\  ( S  C_  ( Base `  M )  /\  x  e.  S  /\  f  e.  ( B  ^m  S ) ) )  ->  ( ( invg `  R ) `
 ( f `  x ) )  e.  B )
161, 15syl5eqel 2543 1  |-  ( ( ( S  e.  V  /\  M  e.  LMod )  /\  ( S  C_  ( Base `  M )  /\  x  e.  S  /\  f  e.  ( B  ^m  S ) ) )  ->  Y  e.  B )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 965    = wceq 1370    e. wcel 1758    \ cdif 3426    C_ wss 3429   {csn 3978    |` cres 4943   -->wf 5515   ` cfv 5519  (class class class)co 6193    ^m cmap 7317   Basecbs 14285  Scalarcsca 14352   0gc0g 14489   Grpcgrp 15521   invgcminusg 15522   LModclmod 17063
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4504  ax-sep 4514  ax-nul 4522  ax-pow 4571  ax-pr 4632  ax-un 6475
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-reu 2802  df-rmo 2803  df-rab 2804  df-v 3073  df-sbc 3288  df-csb 3390  df-dif 3432  df-un 3434  df-in 3436  df-ss 3443  df-nul 3739  df-if 3893  df-pw 3963  df-sn 3979  df-pr 3981  df-op 3985  df-uni 4193  df-iun 4274  df-br 4394  df-opab 4452  df-mpt 4453  df-id 4737  df-xp 4947  df-rel 4948  df-cnv 4949  df-co 4950  df-dm 4951  df-rn 4952  df-res 4953  df-ima 4954  df-iota 5482  df-fun 5521  df-fn 5522  df-f 5523  df-f1 5524  df-fo 5525  df-f1o 5526  df-fv 5527  df-riota 6154  df-ov 6196  df-oprab 6197  df-mpt2 6198  df-1st 6680  df-2nd 6681  df-map 7319  df-0g 14491  df-mnd 15526  df-grp 15656  df-minusg 15657  df-rng 16762  df-lmod 17065
This theorem is referenced by: (None)
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