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Theorem lmod0rng 32694
Description: If the scalar ring of a module is the zero ring, the module is the zero module, i.e. the base set of the module is the singleton consisting of the identity element only. (Contributed by AV, 17-Apr-2019.)
Assertion
Ref Expression
lmod0rng  |-  ( ( M  e.  LMod  /\  -.  (Scalar `  M )  e. NzRing
)  ->  ( Base `  M )  =  {
( 0g `  M
) } )

Proof of Theorem lmod0rng
Dummy variable  v is distinct from all other variables.
StepHypRef Expression
1 eqid 2443 . . . 4  |-  (Scalar `  M )  =  (Scalar `  M )
21lmodring 17394 . . 3  |-  ( M  e.  LMod  ->  (Scalar `  M )  e.  Ring )
3 0ringnnzr 17791 . . . . 5  |-  ( (Scalar `  M )  e.  Ring  -> 
( ( # `  ( Base `  (Scalar `  M
) ) )  =  1  <->  -.  (Scalar `  M
)  e. NzRing ) )
4 eqid 2443 . . . . . . . 8  |-  ( Base `  (Scalar `  M )
)  =  ( Base `  (Scalar `  M )
)
5 eqid 2443 . . . . . . . 8  |-  ( 0g
`  (Scalar `  M )
)  =  ( 0g
`  (Scalar `  M )
)
6 eqid 2443 . . . . . . . 8  |-  ( 1r
`  (Scalar `  M )
)  =  ( 1r
`  (Scalar `  M )
)
74, 5, 60ring01eq 17793 . . . . . . 7  |-  ( ( (Scalar `  M )  e.  Ring  /\  ( # `  ( Base `  (Scalar `  M
) ) )  =  1 )  ->  ( 0g `  (Scalar `  M
) )  =  ( 1r `  (Scalar `  M ) ) )
8 eqid 2443 . . . . . . . . . . . . . 14  |-  ( Base `  M )  =  (
Base `  M )
9 eqid 2443 . . . . . . . . . . . . . 14  |-  ( .s
`  M )  =  ( .s `  M
)
108, 1, 9, 6lmodvs1 17414 . . . . . . . . . . . . 13  |-  ( ( M  e.  LMod  /\  v  e.  ( Base `  M
) )  ->  (
( 1r `  (Scalar `  M ) ) ( .s `  M ) v )  =  v )
11 eqcom 2452 . . . . . . . . . . . . . . . 16  |-  ( ( ( 1r `  (Scalar `  M ) ) ( .s `  M ) v )  =  v  <-> 
v  =  ( ( 1r `  (Scalar `  M ) ) ( .s `  M ) v ) )
1211biimpi 194 . . . . . . . . . . . . . . 15  |-  ( ( ( 1r `  (Scalar `  M ) ) ( .s `  M ) v )  =  v  ->  v  =  ( ( 1r `  (Scalar `  M ) ) ( .s `  M ) v ) )
13 oveq1 6288 . . . . . . . . . . . . . . . . 17  |-  ( ( 1r `  (Scalar `  M ) )  =  ( 0g `  (Scalar `  M ) )  -> 
( ( 1r `  (Scalar `  M ) ) ( .s `  M
) v )  =  ( ( 0g `  (Scalar `  M ) ) ( .s `  M
) v ) )
1413eqcoms 2455 . . . . . . . . . . . . . . . 16  |-  ( ( 0g `  (Scalar `  M ) )  =  ( 1r `  (Scalar `  M ) )  -> 
( ( 1r `  (Scalar `  M ) ) ( .s `  M
) v )  =  ( ( 0g `  (Scalar `  M ) ) ( .s `  M
) v ) )
15 eqid 2443 . . . . . . . . . . . . . . . . 17  |-  ( 0g
`  M )  =  ( 0g `  M
)
168, 1, 9, 5, 15lmod0vs 17419 . . . . . . . . . . . . . . . 16  |-  ( ( M  e.  LMod  /\  v  e.  ( Base `  M
) )  ->  (
( 0g `  (Scalar `  M ) ) ( .s `  M ) v )  =  ( 0g `  M ) )
1714, 16sylan9eqr 2506 . . . . . . . . . . . . . . 15  |-  ( ( ( M  e.  LMod  /\  v  e.  ( Base `  M ) )  /\  ( 0g `  (Scalar `  M ) )  =  ( 1r `  (Scalar `  M ) ) )  ->  ( ( 1r
`  (Scalar `  M )
) ( .s `  M ) v )  =  ( 0g `  M ) )
1812, 17sylan9eq 2504 . . . . . . . . . . . . . 14  |-  ( ( ( ( 1r `  (Scalar `  M ) ) ( .s `  M
) v )  =  v  /\  ( ( M  e.  LMod  /\  v  e.  ( Base `  M
) )  /\  ( 0g `  (Scalar `  M
) )  =  ( 1r `  (Scalar `  M ) ) ) )  ->  v  =  ( 0g `  M ) )
1918exp32 605 . . . . . . . . . . . . 13  |-  ( ( ( 1r `  (Scalar `  M ) ) ( .s `  M ) v )  =  v  ->  ( ( M  e.  LMod  /\  v  e.  ( Base `  M
) )  ->  (
( 0g `  (Scalar `  M ) )  =  ( 1r `  (Scalar `  M ) )  -> 
v  =  ( 0g
`  M ) ) ) )
2010, 19mpcom 36 . . . . . . . . . . . 12  |-  ( ( M  e.  LMod  /\  v  e.  ( Base `  M
) )  ->  (
( 0g `  (Scalar `  M ) )  =  ( 1r `  (Scalar `  M ) )  -> 
v  =  ( 0g
`  M ) ) )
2120com12 31 . . . . . . . . . . 11  |-  ( ( 0g `  (Scalar `  M ) )  =  ( 1r `  (Scalar `  M ) )  -> 
( ( M  e. 
LMod  /\  v  e.  (
Base `  M )
)  ->  v  =  ( 0g `  M ) ) )
2221impl 620 . . . . . . . . . 10  |-  ( ( ( ( 0g `  (Scalar `  M ) )  =  ( 1r `  (Scalar `  M ) )  /\  M  e.  LMod )  /\  v  e.  (
Base `  M )
)  ->  v  =  ( 0g `  M ) )
2322ralrimiva 2857 . . . . . . . . 9  |-  ( ( ( 0g `  (Scalar `  M ) )  =  ( 1r `  (Scalar `  M ) )  /\  M  e.  LMod )  ->  A. v  e.  ( Base `  M ) v  =  ( 0g `  M ) )
248lmodbn0 17396 . . . . . . . . . . 11  |-  ( M  e.  LMod  ->  ( Base `  M )  =/=  (/) )
25 eqsn 4176 . . . . . . . . . . 11  |-  ( (
Base `  M )  =/=  (/)  ->  ( ( Base `  M )  =  { ( 0g `  M ) }  <->  A. v  e.  ( Base `  M
) v  =  ( 0g `  M ) ) )
2624, 25syl 16 . . . . . . . . . 10  |-  ( M  e.  LMod  ->  ( (
Base `  M )  =  { ( 0g `  M ) }  <->  A. v  e.  ( Base `  M
) v  =  ( 0g `  M ) ) )
2726adantl 466 . . . . . . . . 9  |-  ( ( ( 0g `  (Scalar `  M ) )  =  ( 1r `  (Scalar `  M ) )  /\  M  e.  LMod )  -> 
( ( Base `  M
)  =  { ( 0g `  M ) }  <->  A. v  e.  (
Base `  M )
v  =  ( 0g
`  M ) ) )
2823, 27mpbird 232 . . . . . . . 8  |-  ( ( ( 0g `  (Scalar `  M ) )  =  ( 1r `  (Scalar `  M ) )  /\  M  e.  LMod )  -> 
( Base `  M )  =  { ( 0g `  M ) } )
2928ex 434 . . . . . . 7  |-  ( ( 0g `  (Scalar `  M ) )  =  ( 1r `  (Scalar `  M ) )  -> 
( M  e.  LMod  -> 
( Base `  M )  =  { ( 0g `  M ) } ) )
307, 29syl 16 . . . . . 6  |-  ( ( (Scalar `  M )  e.  Ring  /\  ( # `  ( Base `  (Scalar `  M
) ) )  =  1 )  ->  ( M  e.  LMod  ->  ( Base `  M )  =  { ( 0g `  M ) } ) )
3130ex 434 . . . . 5  |-  ( (Scalar `  M )  e.  Ring  -> 
( ( # `  ( Base `  (Scalar `  M
) ) )  =  1  ->  ( M  e.  LMod  ->  ( Base `  M )  =  {
( 0g `  M
) } ) ) )
323, 31sylbird 235 . . . 4  |-  ( (Scalar `  M )  e.  Ring  -> 
( -.  (Scalar `  M )  e. NzRing  ->  ( M  e.  LMod  ->  (
Base `  M )  =  { ( 0g `  M ) } ) ) )
3332com23 78 . . 3  |-  ( (Scalar `  M )  e.  Ring  -> 
( M  e.  LMod  -> 
( -.  (Scalar `  M )  e. NzRing  ->  (
Base `  M )  =  { ( 0g `  M ) } ) ) )
342, 33mpcom 36 . 2  |-  ( M  e.  LMod  ->  ( -.  (Scalar `  M )  e. NzRing  ->  ( Base `  M
)  =  { ( 0g `  M ) } ) )
3534imp 429 1  |-  ( ( M  e.  LMod  /\  -.  (Scalar `  M )  e. NzRing
)  ->  ( Base `  M )  =  {
( 0g `  M
) } )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1383    e. wcel 1804    =/= wne 2638   A.wral 2793   (/)c0 3770   {csn 4014   ` cfv 5578  (class class class)co 6281   1c1 9496   #chash 12384   Basecbs 14509  Scalarcsca 14577   .scvsca 14578   0gc0g 14714   1rcur 17027   Ringcrg 17072   LModclmod 17386  NzRingcnzr 17779
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-rep 4548  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577  ax-cnex 9551  ax-resscn 9552  ax-1cn 9553  ax-icn 9554  ax-addcl 9555  ax-addrcl 9556  ax-mulcl 9557  ax-mulrcl 9558  ax-mulcom 9559  ax-addass 9560  ax-mulass 9561  ax-distr 9562  ax-i2m1 9563  ax-1ne0 9564  ax-1rid 9565  ax-rnegex 9566  ax-rrecex 9567  ax-cnre 9568  ax-pre-lttri 9569  ax-pre-lttrn 9570  ax-pre-ltadd 9571  ax-pre-mulgt0 9572
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 975  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-nel 2641  df-ral 2798  df-rex 2799  df-reu 2800  df-rmo 2801  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-tp 4019  df-op 4021  df-uni 4235  df-int 4272  df-iun 4317  df-br 4438  df-opab 4496  df-mpt 4497  df-tr 4531  df-eprel 4781  df-id 4785  df-po 4790  df-so 4791  df-fr 4828  df-we 4830  df-ord 4871  df-on 4872  df-lim 4873  df-suc 4874  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586  df-riota 6242  df-ov 6284  df-oprab 6285  df-mpt2 6286  df-om 6686  df-1st 6785  df-2nd 6786  df-recs 7044  df-rdg 7078  df-1o 7132  df-oadd 7136  df-er 7313  df-en 7519  df-dom 7520  df-sdom 7521  df-fin 7522  df-card 8323  df-cda 8551  df-pnf 9633  df-mnf 9634  df-xr 9635  df-ltxr 9636  df-le 9637  df-sub 9812  df-neg 9813  df-nn 10543  df-2 10600  df-n0 10802  df-z 10871  df-uz 11091  df-fz 11682  df-hash 12385  df-ndx 14512  df-slot 14513  df-base 14514  df-sets 14515  df-plusg 14587  df-0g 14716  df-mgm 15746  df-sgrp 15785  df-mnd 15795  df-grp 15931  df-minusg 15932  df-mgp 17016  df-ur 17028  df-ring 17074  df-lmod 17388  df-nzr 17780
This theorem is referenced by: (None)
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