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Theorem lmod0rng 41763
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 2514 . . . 4 (Scalar‘𝑀) = (Scalar‘𝑀)
21lmodring 18601 . . 3 (𝑀 ∈ LMod → (Scalar‘𝑀) ∈ Ring)
3 0ringnnzr 18994 . . . . 5 ((Scalar‘𝑀) ∈ Ring → ((#‘(Base‘(Scalar‘𝑀))) = 1 ↔ ¬ (Scalar‘𝑀) ∈ NzRing))
4 eqid 2514 . . . . . . . 8 (Base‘(Scalar‘𝑀)) = (Base‘(Scalar‘𝑀))
5 eqid 2514 . . . . . . . 8 (0g‘(Scalar‘𝑀)) = (0g‘(Scalar‘𝑀))
6 eqid 2514 . . . . . . . 8 (1r‘(Scalar‘𝑀)) = (1r‘(Scalar‘𝑀))
74, 5, 60ring01eq 18996 . . . . . . 7 (((Scalar‘𝑀) ∈ Ring ∧ (#‘(Base‘(Scalar‘𝑀))) = 1) → (0g‘(Scalar‘𝑀)) = (1r‘(Scalar‘𝑀)))
8 eqid 2514 . . . . . . . . . . . . . 14 (Base‘𝑀) = (Base‘𝑀)
9 eqid 2514 . . . . . . . . . . . . . 14 ( ·𝑠𝑀) = ( ·𝑠𝑀)
108, 1, 9, 6lmodvs1 18621 . . . . . . . . . . . . 13 ((𝑀 ∈ LMod ∧ 𝑣 ∈ (Base‘𝑀)) → ((1r‘(Scalar‘𝑀))( ·𝑠𝑀)𝑣) = 𝑣)
11 eqcom 2521 . . . . . . . . . . . . . . . 16 (((1r‘(Scalar‘𝑀))( ·𝑠𝑀)𝑣) = 𝑣𝑣 = ((1r‘(Scalar‘𝑀))( ·𝑠𝑀)𝑣))
1211biimpi 204 . . . . . . . . . . . . . . 15 (((1r‘(Scalar‘𝑀))( ·𝑠𝑀)𝑣) = 𝑣𝑣 = ((1r‘(Scalar‘𝑀))( ·𝑠𝑀)𝑣))
13 oveq1 6433 . . . . . . . . . . . . . . . . 17 ((1r‘(Scalar‘𝑀)) = (0g‘(Scalar‘𝑀)) → ((1r‘(Scalar‘𝑀))( ·𝑠𝑀)𝑣) = ((0g‘(Scalar‘𝑀))( ·𝑠𝑀)𝑣))
1413eqcoms 2522 . . . . . . . . . . . . . . . 16 ((0g‘(Scalar‘𝑀)) = (1r‘(Scalar‘𝑀)) → ((1r‘(Scalar‘𝑀))( ·𝑠𝑀)𝑣) = ((0g‘(Scalar‘𝑀))( ·𝑠𝑀)𝑣))
15 eqid 2514 . . . . . . . . . . . . . . . . 17 (0g𝑀) = (0g𝑀)
168, 1, 9, 5, 15lmod0vs 18626 . . . . . . . . . . . . . . . 16 ((𝑀 ∈ LMod ∧ 𝑣 ∈ (Base‘𝑀)) → ((0g‘(Scalar‘𝑀))( ·𝑠𝑀)𝑣) = (0g𝑀))
1714, 16sylan9eqr 2570 . . . . . . . . . . . . . . 15 (((𝑀 ∈ LMod ∧ 𝑣 ∈ (Base‘𝑀)) ∧ (0g‘(Scalar‘𝑀)) = (1r‘(Scalar‘𝑀))) → ((1r‘(Scalar‘𝑀))( ·𝑠𝑀)𝑣) = (0g𝑀))
1812, 17sylan9eq 2568 . . . . . . . . . . . . . 14 ((((1r‘(Scalar‘𝑀))( ·𝑠𝑀)𝑣) = 𝑣 ∧ ((𝑀 ∈ LMod ∧ 𝑣 ∈ (Base‘𝑀)) ∧ (0g‘(Scalar‘𝑀)) = (1r‘(Scalar‘𝑀)))) → 𝑣 = (0g𝑀))
1918exp32 628 . . . . . . . . . . . . 13 (((1r‘(Scalar‘𝑀))( ·𝑠𝑀)𝑣) = 𝑣 → ((𝑀 ∈ LMod ∧ 𝑣 ∈ (Base‘𝑀)) → ((0g‘(Scalar‘𝑀)) = (1r‘(Scalar‘𝑀)) → 𝑣 = (0g𝑀))))
2010, 19mpcom 37 . . . . . . . . . . . 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 647 . . . . . . . . . 10 ((((0g‘(Scalar‘𝑀)) = (1r‘(Scalar‘𝑀)) ∧ 𝑀 ∈ LMod) ∧ 𝑣 ∈ (Base‘𝑀)) → 𝑣 = (0g𝑀))
2322ralrimiva 2853 . . . . . . . . 9 (((0g‘(Scalar‘𝑀)) = (1r‘(Scalar‘𝑀)) ∧ 𝑀 ∈ LMod) → ∀𝑣 ∈ (Base‘𝑀)𝑣 = (0g𝑀))
248lmodbn0 18603 . . . . . . . . . . 11 (𝑀 ∈ LMod → (Base‘𝑀) ≠ ∅)
25 eqsn 4202 . . . . . . . . . . 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 480 . . . . . . . . 9 (((0g‘(Scalar‘𝑀)) = (1r‘(Scalar‘𝑀)) ∧ 𝑀 ∈ LMod) → ((Base‘𝑀) = {(0g𝑀)} ↔ ∀𝑣 ∈ (Base‘𝑀)𝑣 = (0g𝑀)))
2823, 27mpbird 245 . . . . . . . 8 (((0g‘(Scalar‘𝑀)) = (1r‘(Scalar‘𝑀)) ∧ 𝑀 ∈ LMod) → (Base‘𝑀) = {(0g𝑀)})
2928ex 448 . . . . . . 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 448 . . . . 5 ((Scalar‘𝑀) ∈ Ring → ((#‘(Base‘(Scalar‘𝑀))) = 1 → (𝑀 ∈ LMod → (Base‘𝑀) = {(0g𝑀)})))
323, 31sylbird 248 . . . 4 ((Scalar‘𝑀) ∈ Ring → (¬ (Scalar‘𝑀) ∈ NzRing → (𝑀 ∈ LMod → (Base‘𝑀) = {(0g𝑀)})))
3332com23 83 . . 3 ((Scalar‘𝑀) ∈ Ring → (𝑀 ∈ LMod → (¬ (Scalar‘𝑀) ∈ NzRing → (Base‘𝑀) = {(0g𝑀)})))
342, 33mpcom 37 . 2 (𝑀 ∈ LMod → (¬ (Scalar‘𝑀) ∈ NzRing → (Base‘𝑀) = {(0g𝑀)}))
3534imp 443 1 ((𝑀 ∈ LMod ∧ ¬ (Scalar‘𝑀) ∈ NzRing) → (Base‘𝑀) = {(0g𝑀)})
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 194  wa 382   = wceq 1474  wcel 1938  wne 2684  wral 2800  c0 3777  {csn 4028  cfv 5689  (class class class)co 6426  1c1 9692  #chash 12847  Basecbs 15579  Scalarcsca 15655   ·𝑠 cvsca 15656  0gc0g 15807  1rcur 18231  Ringcrg 18277  LModclmod 18593  NzRingcnzr 18982
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1700  ax-4 1713  ax-5 1793  ax-6 1838  ax-7 1885  ax-8 1940  ax-9 1947  ax-10 1966  ax-11 1971  ax-12 1983  ax-13 2137  ax-ext 2494  ax-rep 4597  ax-sep 4607  ax-nul 4616  ax-pow 4668  ax-pr 4732  ax-un 6723  ax-cnex 9747  ax-resscn 9748  ax-1cn 9749  ax-icn 9750  ax-addcl 9751  ax-addrcl 9752  ax-mulcl 9753  ax-mulrcl 9754  ax-mulcom 9755  ax-addass 9756  ax-mulass 9757  ax-distr 9758  ax-i2m1 9759  ax-1ne0 9760  ax-1rid 9761  ax-rnegex 9762  ax-rrecex 9763  ax-cnre 9764  ax-pre-lttri 9765  ax-pre-lttrn 9766  ax-pre-ltadd 9767  ax-pre-mulgt0 9768
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1699  df-sb 1831  df-eu 2366  df-mo 2367  df-clab 2501  df-cleq 2507  df-clel 2510  df-nfc 2644  df-ne 2686  df-nel 2687  df-ral 2805  df-rex 2806  df-reu 2807  df-rmo 2808  df-rab 2809  df-v 3079  df-sbc 3307  df-csb 3404  df-dif 3447  df-un 3449  df-in 3451  df-ss 3458  df-pss 3460  df-nul 3778  df-if 3940  df-pw 4013  df-sn 4029  df-pr 4031  df-tp 4033  df-op 4035  df-uni 4271  df-int 4309  df-iun 4355  df-br 4482  df-opab 4542  df-mpt 4543  df-tr 4579  df-eprel 4843  df-id 4847  df-po 4853  df-so 4854  df-fr 4891  df-we 4893  df-xp 4938  df-rel 4939  df-cnv 4940  df-co 4941  df-dm 4942  df-rn 4943  df-res 4944  df-ima 4945  df-pred 5487  df-ord 5533  df-on 5534  df-lim 5535  df-suc 5536  df-iota 5653  df-fun 5691  df-fn 5692  df-f 5693  df-f1 5694  df-fo 5695  df-f1o 5696  df-fv 5697  df-riota 6388  df-ov 6429  df-oprab 6430  df-mpt2 6431  df-om 6834  df-1st 6934  df-2nd 6935  df-wrecs 7169  df-recs 7231  df-rdg 7269  df-1o 7323  df-oadd 7327  df-er 7505  df-en 7718  df-dom 7719  df-sdom 7720  df-fin 7721  df-card 8524  df-cda 8749  df-pnf 9831  df-mnf 9832  df-xr 9833  df-ltxr 9834  df-le 9835  df-sub 10019  df-neg 10020  df-nn 10776  df-2 10834  df-n0 11048  df-z 11119  df-uz 11428  df-fz 12066  df-hash 12848  df-ndx 15582  df-slot 15583  df-base 15584  df-sets 15585  df-plusg 15665  df-0g 15809  df-mgm 16957  df-sgrp 16999  df-mnd 17010  df-grp 17140  df-minusg 17141  df-mgp 18220  df-ur 18232  df-ring 18279  df-lmod 18595  df-nzr 18983
This theorem is referenced by: (None)
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