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Theorem lmod0rng 41639
 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 ∈ LMod ∧ ¬ (Scalar‘𝑀) ∈ NzRing) → (Base‘𝑀) = {(0g𝑀)})

Proof of Theorem lmod0rng
Dummy variable 𝑣 is distinct from all other variables.
StepHypRef Expression
1 eqid 2621 . . . 4 (Scalar‘𝑀) = (Scalar‘𝑀)
21lmodring 18865 . . 3 (𝑀 ∈ LMod → (Scalar‘𝑀) ∈ Ring)
3 0ringnnzr 19263 . . . . 5 ((Scalar‘𝑀) ∈ Ring → ((#‘(Base‘(Scalar‘𝑀))) = 1 ↔ ¬ (Scalar‘𝑀) ∈ NzRing))
4 eqid 2621 . . . . . . . 8 (Base‘(Scalar‘𝑀)) = (Base‘(Scalar‘𝑀))
5 eqid 2621 . . . . . . . 8 (0g‘(Scalar‘𝑀)) = (0g‘(Scalar‘𝑀))
6 eqid 2621 . . . . . . . 8 (1r‘(Scalar‘𝑀)) = (1r‘(Scalar‘𝑀))
74, 5, 60ring01eq 19265 . . . . . . 7 (((Scalar‘𝑀) ∈ Ring ∧ (#‘(Base‘(Scalar‘𝑀))) = 1) → (0g‘(Scalar‘𝑀)) = (1r‘(Scalar‘𝑀)))
8 eqid 2621 . . . . . . . . . . . . . 14 (Base‘𝑀) = (Base‘𝑀)
9 eqid 2621 . . . . . . . . . . . . . 14 ( ·𝑠𝑀) = ( ·𝑠𝑀)
108, 1, 9, 6lmodvs1 18885 . . . . . . . . . . . . 13 ((𝑀 ∈ LMod ∧ 𝑣 ∈ (Base‘𝑀)) → ((1r‘(Scalar‘𝑀))( ·𝑠𝑀)𝑣) = 𝑣)
11 eqcom 2628 . . . . . . . . . . . . . . . 16 (((1r‘(Scalar‘𝑀))( ·𝑠𝑀)𝑣) = 𝑣𝑣 = ((1r‘(Scalar‘𝑀))( ·𝑠𝑀)𝑣))
1211biimpi 206 . . . . . . . . . . . . . . 15 (((1r‘(Scalar‘𝑀))( ·𝑠𝑀)𝑣) = 𝑣𝑣 = ((1r‘(Scalar‘𝑀))( ·𝑠𝑀)𝑣))
13 oveq1 6654 . . . . . . . . . . . . . . . . 17 ((1r‘(Scalar‘𝑀)) = (0g‘(Scalar‘𝑀)) → ((1r‘(Scalar‘𝑀))( ·𝑠𝑀)𝑣) = ((0g‘(Scalar‘𝑀))( ·𝑠𝑀)𝑣))
1413eqcoms 2629 . . . . . . . . . . . . . . . 16 ((0g‘(Scalar‘𝑀)) = (1r‘(Scalar‘𝑀)) → ((1r‘(Scalar‘𝑀))( ·𝑠𝑀)𝑣) = ((0g‘(Scalar‘𝑀))( ·𝑠𝑀)𝑣))
15 eqid 2621 . . . . . . . . . . . . . . . . 17 (0g𝑀) = (0g𝑀)
168, 1, 9, 5, 15lmod0vs 18890 . . . . . . . . . . . . . . . 16 ((𝑀 ∈ LMod ∧ 𝑣 ∈ (Base‘𝑀)) → ((0g‘(Scalar‘𝑀))( ·𝑠𝑀)𝑣) = (0g𝑀))
1714, 16sylan9eqr 2677 . . . . . . . . . . . . . . 15 (((𝑀 ∈ LMod ∧ 𝑣 ∈ (Base‘𝑀)) ∧ (0g‘(Scalar‘𝑀)) = (1r‘(Scalar‘𝑀))) → ((1r‘(Scalar‘𝑀))( ·𝑠𝑀)𝑣) = (0g𝑀))
1812, 17sylan9eq 2675 . . . . . . . . . . . . . 14 ((((1r‘(Scalar‘𝑀))( ·𝑠𝑀)𝑣) = 𝑣 ∧ ((𝑀 ∈ LMod ∧ 𝑣 ∈ (Base‘𝑀)) ∧ (0g‘(Scalar‘𝑀)) = (1r‘(Scalar‘𝑀)))) → 𝑣 = (0g𝑀))
1918exp32 631 . . . . . . . . . . . . 13 (((1r‘(Scalar‘𝑀))( ·𝑠𝑀)𝑣) = 𝑣 → ((𝑀 ∈ LMod ∧ 𝑣 ∈ (Base‘𝑀)) → ((0g‘(Scalar‘𝑀)) = (1r‘(Scalar‘𝑀)) → 𝑣 = (0g𝑀))))
2010, 19mpcom 38 . . . . . . . . . . . 12 ((𝑀 ∈ LMod ∧ 𝑣 ∈ (Base‘𝑀)) → ((0g‘(Scalar‘𝑀)) = (1r‘(Scalar‘𝑀)) → 𝑣 = (0g𝑀)))
2120com12 32 . . . . . . . . . . 11 ((0g‘(Scalar‘𝑀)) = (1r‘(Scalar‘𝑀)) → ((𝑀 ∈ LMod ∧ 𝑣 ∈ (Base‘𝑀)) → 𝑣 = (0g𝑀)))
2221impl 650 . . . . . . . . . 10 ((((0g‘(Scalar‘𝑀)) = (1r‘(Scalar‘𝑀)) ∧ 𝑀 ∈ LMod) ∧ 𝑣 ∈ (Base‘𝑀)) → 𝑣 = (0g𝑀))
2322ralrimiva 2965 . . . . . . . . 9 (((0g‘(Scalar‘𝑀)) = (1r‘(Scalar‘𝑀)) ∧ 𝑀 ∈ LMod) → ∀𝑣 ∈ (Base‘𝑀)𝑣 = (0g𝑀))
248lmodbn0 18867 . . . . . . . . . . 11 (𝑀 ∈ LMod → (Base‘𝑀) ≠ ∅)
25 eqsn 4359 . . . . . . . . . . 11 ((Base‘𝑀) ≠ ∅ → ((Base‘𝑀) = {(0g𝑀)} ↔ ∀𝑣 ∈ (Base‘𝑀)𝑣 = (0g𝑀)))
2624, 25syl 17 . . . . . . . . . 10 (𝑀 ∈ LMod → ((Base‘𝑀) = {(0g𝑀)} ↔ ∀𝑣 ∈ (Base‘𝑀)𝑣 = (0g𝑀)))
2726adantl 482 . . . . . . . . 9 (((0g‘(Scalar‘𝑀)) = (1r‘(Scalar‘𝑀)) ∧ 𝑀 ∈ LMod) → ((Base‘𝑀) = {(0g𝑀)} ↔ ∀𝑣 ∈ (Base‘𝑀)𝑣 = (0g𝑀)))
2823, 27mpbird 247 . . . . . . . 8 (((0g‘(Scalar‘𝑀)) = (1r‘(Scalar‘𝑀)) ∧ 𝑀 ∈ LMod) → (Base‘𝑀) = {(0g𝑀)})
2928ex 450 . . . . . . 7 ((0g‘(Scalar‘𝑀)) = (1r‘(Scalar‘𝑀)) → (𝑀 ∈ LMod → (Base‘𝑀) = {(0g𝑀)}))
307, 29syl 17 . . . . . 6 (((Scalar‘𝑀) ∈ Ring ∧ (#‘(Base‘(Scalar‘𝑀))) = 1) → (𝑀 ∈ LMod → (Base‘𝑀) = {(0g𝑀)}))
3130ex 450 . . . . 5 ((Scalar‘𝑀) ∈ Ring → ((#‘(Base‘(Scalar‘𝑀))) = 1 → (𝑀 ∈ LMod → (Base‘𝑀) = {(0g𝑀)})))
323, 31sylbird 250 . . . 4 ((Scalar‘𝑀) ∈ Ring → (¬ (Scalar‘𝑀) ∈ NzRing → (𝑀 ∈ LMod → (Base‘𝑀) = {(0g𝑀)})))
3332com23 86 . . 3 ((Scalar‘𝑀) ∈ Ring → (𝑀 ∈ LMod → (¬ (Scalar‘𝑀) ∈ NzRing → (Base‘𝑀) = {(0g𝑀)})))
342, 33mpcom 38 . 2 (𝑀 ∈ LMod → (¬ (Scalar‘𝑀) ∈ NzRing → (Base‘𝑀) = {(0g𝑀)}))
3534imp 445 1 ((𝑀 ∈ LMod ∧ ¬ (Scalar‘𝑀) ∈ NzRing) → (Base‘𝑀) = {(0g𝑀)})
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∧ wa 384   = wceq 1482   ∈ wcel 1989   ≠ wne 2793  ∀wral 2911  ∅c0 3913  {csn 4175  ‘cfv 5886  (class class class)co 6647  1c1 9934  #chash 13112  Basecbs 15851  Scalarcsca 15938   ·𝑠 cvsca 15939  0gc0g 16094  1rcur 18495  Ringcrg 18541  LModclmod 18857  NzRingcnzr 19251 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-8 1991  ax-9 1998  ax-10 2018  ax-11 2033  ax-12 2046  ax-13 2245  ax-ext 2601  ax-rep 4769  ax-sep 4779  ax-nul 4787  ax-pow 4841  ax-pr 4904  ax-un 6946  ax-cnex 9989  ax-resscn 9990  ax-1cn 9991  ax-icn 9992  ax-addcl 9993  ax-addrcl 9994  ax-mulcl 9995  ax-mulrcl 9996  ax-mulcom 9997  ax-addass 9998  ax-mulass 9999  ax-distr 10000  ax-i2m1 10001  ax-1ne0 10002  ax-1rid 10003  ax-rnegex 10004  ax-rrecex 10005  ax-cnre 10006  ax-pre-lttri 10007  ax-pre-lttrn 10008  ax-pre-ltadd 10009  ax-pre-mulgt0 10010 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1485  df-ex 1704  df-nf 1709  df-sb 1880  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2752  df-ne 2794  df-nel 2897  df-ral 2916  df-rex 2917  df-reu 2918  df-rmo 2919  df-rab 2920  df-v 3200  df-sbc 3434  df-csb 3532  df-dif 3575  df-un 3577  df-in 3579  df-ss 3586  df-pss 3588  df-nul 3914  df-if 4085  df-pw 4158  df-sn 4176  df-pr 4178  df-tp 4180  df-op 4182  df-uni 4435  df-int 4474  df-iun 4520  df-br 4652  df-opab 4711  df-mpt 4728  df-tr 4751  df-id 5022  df-eprel 5027  df-po 5033  df-so 5034  df-fr 5071  df-we 5073  df-xp 5118  df-rel 5119  df-cnv 5120  df-co 5121  df-dm 5122  df-rn 5123  df-res 5124  df-ima 5125  df-pred 5678  df-ord 5724  df-on 5725  df-lim 5726  df-suc 5727  df-iota 5849  df-fun 5888  df-fn 5889  df-f 5890  df-f1 5891  df-fo 5892  df-f1o 5893  df-fv 5894  df-riota 6608  df-ov 6650  df-oprab 6651  df-mpt2 6652  df-om 7063  df-1st 7165  df-2nd 7166  df-wrecs 7404  df-recs 7465  df-rdg 7503  df-1o 7557  df-oadd 7561  df-er 7739  df-en 7953  df-dom 7954  df-sdom 7955  df-fin 7956  df-card 8762  df-cda 8987  df-pnf 10073  df-mnf 10074  df-xr 10075  df-ltxr 10076  df-le 10077  df-sub 10265  df-neg 10266  df-nn 11018  df-2 11076  df-n0 11290  df-xnn0 11361  df-z 11375  df-uz 11685  df-fz 12324  df-hash 13113  df-ndx 15854  df-slot 15855  df-base 15857  df-sets 15858  df-plusg 15948  df-0g 16096  df-mgm 17236  df-sgrp 17278  df-mnd 17289  df-grp 17419  df-minusg 17420  df-mgp 18484  df-ur 18496  df-ring 18543  df-lmod 18859  df-nzr 19252 This theorem is referenced by: (None)
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