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Theorem lindslinindimp2lem1 33313
Description: Lemma 1 for lindslinindsimp2 33318. (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 17716 . . . 4  |-  ( M  e.  LMod  ->  R  e. 
Grp )
43adantl 464 . . 3  |-  ( ( S  e.  V  /\  M  e.  LMod )  ->  R  e.  Grp )
5 elmapi 7433 . . . . . 6  |-  ( f  e.  ( B  ^m  S )  ->  f : S --> B )
6 ffvelrn 6005 . . . . . . . 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 432 . . . . . 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 1188 . . 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 2454 . . . 4  |-  ( invg `  R )  =  ( invg `  R )
1412, 13grpinvcl 16294 . . 3  |-  ( ( R  e.  Grp  /\  ( f `  x
)  e.  B )  ->  ( ( invg `  R ) `
 ( f `  x ) )  e.  B )
154, 11, 14syl2an 475 . 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 2546 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 367    /\ w3a 971    = wceq 1398    e. wcel 1823    \ cdif 3458    C_ wss 3461   {csn 4016    |` cres 4990   -->wf 5566   ` cfv 5570  (class class class)co 6270    ^m cmap 7412   Basecbs 14716  Scalarcsca 14787   0gc0g 14929   Grpcgrp 16252   invgcminusg 16253   LModclmod 17707
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-1st 6773  df-2nd 6774  df-map 7414  df-0g 14931  df-mgm 16071  df-sgrp 16110  df-mnd 16120  df-grp 16256  df-minusg 16257  df-ring 17395  df-lmod 17709
This theorem is referenced by: (None)
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