Theorem lmod0rng 45314

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) = {(0g‘𝑀)})

Proof of Theorem lmod0rng

Dummy variable 𝑣 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2738	. . . 4 ⊢ (Scalar‘𝑀) = (Scalar‘𝑀)
2	1	lmodring 20046	. . 3 ⊢ (𝑀 ∈ LMod → (Scalar‘𝑀) ∈ Ring)
3		0ringnnzr 20453	. . . . 5 ⊢ ((Scalar‘𝑀) ∈ Ring → ((♯‘(Base‘(Scalar‘𝑀))) = 1 ↔ ¬ (Scalar‘𝑀) ∈ NzRing))
4		eqid 2738	. . . . . . . 8 ⊢ (Base‘(Scalar‘𝑀)) = (Base‘(Scalar‘𝑀))
5		eqid 2738	. . . . . . . 8 ⊢ (0g‘(Scalar‘𝑀)) = (0g‘(Scalar‘𝑀))
6		eqid 2738	. . . . . . . 8 ⊢ (1r‘(Scalar‘𝑀)) = (1r‘(Scalar‘𝑀))
7	4, 5, 6	0ring01eq 20455	. . . . . . 7 ⊢ (((Scalar‘𝑀) ∈ Ring ∧ (♯‘(Base‘(Scalar‘𝑀))) = 1) → (0g‘(Scalar‘𝑀)) = (1r‘(Scalar‘𝑀)))
8		eqid 2738	. . . . . . . . . . . . . 14 ⊢ (Base‘𝑀) = (Base‘𝑀)
9		eqid 2738	. . . . . . . . . . . . . 14 ⊢ ( ·𝑠 ‘𝑀) = ( ·𝑠 ‘𝑀)
10	8, 1, 9, 6	lmodvs1 20066	. . . . . . . . . . . . 13 ⊢ ((𝑀 ∈ LMod ∧ 𝑣 ∈ (Base‘𝑀)) → ((1r‘(Scalar‘𝑀))( ·𝑠 ‘𝑀)𝑣) = 𝑣)
11		eqcom 2745	. . . . . . . . . . . . . . . 16 ⊢ (((1r‘(Scalar‘𝑀))( ·𝑠 ‘𝑀)𝑣) = 𝑣 ↔ 𝑣 = ((1r‘(Scalar‘𝑀))( ·𝑠 ‘𝑀)𝑣))
12	11	biimpi 215	. . . . . . . . . . . . . . 15 ⊢ (((1r‘(Scalar‘𝑀))( ·𝑠 ‘𝑀)𝑣) = 𝑣 → 𝑣 = ((1r‘(Scalar‘𝑀))( ·𝑠 ‘𝑀)𝑣))
13		oveq1 7262	. . . . . . . . . . . . . . . . 17 ⊢ ((1r‘(Scalar‘𝑀)) = (0g‘(Scalar‘𝑀)) → ((1r‘(Scalar‘𝑀))( ·𝑠 ‘𝑀)𝑣) = ((0g‘(Scalar‘𝑀))( ·𝑠 ‘𝑀)𝑣))
14	13	eqcoms 2746	. . . . . . . . . . . . . . . 16 ⊢ ((0g‘(Scalar‘𝑀)) = (1r‘(Scalar‘𝑀)) → ((1r‘(Scalar‘𝑀))( ·𝑠 ‘𝑀)𝑣) = ((0g‘(Scalar‘𝑀))( ·𝑠 ‘𝑀)𝑣))
15		eqid 2738	. . . . . . . . . . . . . . . . 17 ⊢ (0g‘𝑀) = (0g‘𝑀)
16	8, 1, 9, 5, 15	lmod0vs 20071	. . . . . . . . . . . . . . . 16 ⊢ ((𝑀 ∈ LMod ∧ 𝑣 ∈ (Base‘𝑀)) → ((0g‘(Scalar‘𝑀))( ·𝑠 ‘𝑀)𝑣) = (0g‘𝑀))
17	14, 16	sylan9eqr 2801	. . . . . . . . . . . . . . 15 ⊢ (((𝑀 ∈ LMod ∧ 𝑣 ∈ (Base‘𝑀)) ∧ (0g‘(Scalar‘𝑀)) = (1r‘(Scalar‘𝑀))) → ((1r‘(Scalar‘𝑀))( ·𝑠 ‘𝑀)𝑣) = (0g‘𝑀))
18	12, 17	sylan9eq 2799	. . . . . . . . . . . . . 14 ⊢ ((((1r‘(Scalar‘𝑀))( ·𝑠 ‘𝑀)𝑣) = 𝑣 ∧ ((𝑀 ∈ LMod ∧ 𝑣 ∈ (Base‘𝑀)) ∧ (0g‘(Scalar‘𝑀)) = (1r‘(Scalar‘𝑀)))) → 𝑣 = (0g‘𝑀))
19	18	exp32 420	. . . . . . . . . . . . 13 ⊢ (((1r‘(Scalar‘𝑀))( ·𝑠 ‘𝑀)𝑣) = 𝑣 → ((𝑀 ∈ LMod ∧ 𝑣 ∈ (Base‘𝑀)) → ((0g‘(Scalar‘𝑀)) = (1r‘(Scalar‘𝑀)) → 𝑣 = (0g‘𝑀))))
20	10, 19	mpcom 38	. . . . . . . . . . . 12 ⊢ ((𝑀 ∈ LMod ∧ 𝑣 ∈ (Base‘𝑀)) → ((0g‘(Scalar‘𝑀)) = (1r‘(Scalar‘𝑀)) → 𝑣 = (0g‘𝑀)))
21	20	com12 32	. . . . . . . . . . 11 ⊢ ((0g‘(Scalar‘𝑀)) = (1r‘(Scalar‘𝑀)) → ((𝑀 ∈ LMod ∧ 𝑣 ∈ (Base‘𝑀)) → 𝑣 = (0g‘𝑀)))
22	21	impl 455	. . . . . . . . . 10 ⊢ ((((0g‘(Scalar‘𝑀)) = (1r‘(Scalar‘𝑀)) ∧ 𝑀 ∈ LMod) ∧ 𝑣 ∈ (Base‘𝑀)) → 𝑣 = (0g‘𝑀))
23	22	ralrimiva 3107	. . . . . . . . 9 ⊢ (((0g‘(Scalar‘𝑀)) = (1r‘(Scalar‘𝑀)) ∧ 𝑀 ∈ LMod) → ∀𝑣 ∈ (Base‘𝑀)𝑣 = (0g‘𝑀))
24	8	lmodbn0 20048	. . . . . . . . . . 11 ⊢ (𝑀 ∈ LMod → (Base‘𝑀) ≠ ∅)
25		eqsn 4759	. . . . . . . . . . 11 ⊢ ((Base‘𝑀) ≠ ∅ → ((Base‘𝑀) = {(0g‘𝑀)} ↔ ∀𝑣 ∈ (Base‘𝑀)𝑣 = (0g‘𝑀)))
26	24, 25	syl 17	. . . . . . . . . 10 ⊢ (𝑀 ∈ LMod → ((Base‘𝑀) = {(0g‘𝑀)} ↔ ∀𝑣 ∈ (Base‘𝑀)𝑣 = (0g‘𝑀)))
27	26	adantl 481	. . . . . . . . 9 ⊢ (((0g‘(Scalar‘𝑀)) = (1r‘(Scalar‘𝑀)) ∧ 𝑀 ∈ LMod) → ((Base‘𝑀) = {(0g‘𝑀)} ↔ ∀𝑣 ∈ (Base‘𝑀)𝑣 = (0g‘𝑀)))
28	23, 27	mpbird 256	. . . . . . . 8 ⊢ (((0g‘(Scalar‘𝑀)) = (1r‘(Scalar‘𝑀)) ∧ 𝑀 ∈ LMod) → (Base‘𝑀) = {(0g‘𝑀)})
29	28	ex 412	. . . . . . 7 ⊢ ((0g‘(Scalar‘𝑀)) = (1r‘(Scalar‘𝑀)) → (𝑀 ∈ LMod → (Base‘𝑀) = {(0g‘𝑀)}))
30	7, 29	syl 17	. . . . . 6 ⊢ (((Scalar‘𝑀) ∈ Ring ∧ (♯‘(Base‘(Scalar‘𝑀))) = 1) → (𝑀 ∈ LMod → (Base‘𝑀) = {(0g‘𝑀)}))
31	30	ex 412	. . . . 5 ⊢ ((Scalar‘𝑀) ∈ Ring → ((♯‘(Base‘(Scalar‘𝑀))) = 1 → (𝑀 ∈ LMod → (Base‘𝑀) = {(0g‘𝑀)})))
32	3, 31	sylbird 259	. . . 4 ⊢ ((Scalar‘𝑀) ∈ Ring → (¬ (Scalar‘𝑀) ∈ NzRing → (𝑀 ∈ LMod → (Base‘𝑀) = {(0g‘𝑀)})))
33	32	com23 86	. . 3 ⊢ ((Scalar‘𝑀) ∈ Ring → (𝑀 ∈ LMod → (¬ (Scalar‘𝑀) ∈ NzRing → (Base‘𝑀) = {(0g‘𝑀)})))
34	2, 33	mpcom 38	. 2 ⊢ (𝑀 ∈ LMod → (¬ (Scalar‘𝑀) ∈ NzRing → (Base‘𝑀) = {(0g‘𝑀)}))
35	34	imp 406	1 ⊢ ((𝑀 ∈ LMod ∧ ¬ (Scalar‘𝑀) ∈ NzRing) → (Base‘𝑀) = {(0g‘𝑀)})

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 395   = wceq 1539   ∈ wcel 2108   ≠ wne 2942  ∀wral 3063  ∅c0 4253  {csn 4558  ‘cfv 6418  (class class class)co 7255  1c1 10803  ♯chash 13972  Basecbs 16840  Scalarcsca 16891   ·_𝑠 cvsca 16892  0_gc0g 17067  1_rcur 19652  Ringcrg 19698  LModclmod 20038  NzRingcnzr 20441

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566  ax-cnex 10858  ax-resscn 10859  ax-1cn 10860  ax-icn 10861  ax-addcl 10862  ax-addrcl 10863  ax-mulcl 10864  ax-mulrcl 10865  ax-mulcom 10866  ax-addass 10867  ax-mulass 10868  ax-distr 10869  ax-i2m1 10870  ax-1ne0 10871  ax-1rid 10872  ax-rnegex 10873  ax-rrecex 10874  ax-cnre 10875  ax-pre-lttri 10876  ax-pre-lttrn 10877  ax-pre-ltadd 10878  ax-pre-mulgt0 10879

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-nel 3049  df-ral 3068  df-rex 3069  df-reu 3070  df-rmo 3071  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-tp 4563  df-op 4565  df-uni 4837  df-int 4877  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-tr 5188  df-id 5480  df-eprel 5486  df-po 5494  df-so 5495  df-fr 5535  df-we 5537  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-pred 6191  df-ord 6254  df-on 6255  df-lim 6256  df-suc 6257  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-riota 7212  df-ov 7258  df-oprab 7259  df-mpo 7260  df-om 7688  df-1st 7804  df-2nd 7805  df-frecs 8068  df-wrecs 8099  df-recs 8173  df-rdg 8212  df-1o 8267  df-oadd 8271  df-er 8456  df-en 8692  df-dom 8693  df-sdom 8694  df-fin 8695  df-dju 9590  df-card 9628  df-pnf 10942  df-mnf 10943  df-xr 10944  df-ltxr 10945  df-le 10946  df-sub 11137  df-neg 11138  df-nn 11904  df-2 11966  df-n0 12164  df-xnn0 12236  df-z 12250  df-uz 12512  df-fz 13169  df-hash 13973  df-sets 16793  df-slot 16811  df-ndx 16823  df-base 16841  df-plusg 16901  df-0g 17069  df-mgm 18241  df-sgrp 18290  df-mnd 18301  df-grp 18495  df-minusg 18496  df-mgp 19636  df-ur 19653  df-ring 19700  df-lmod 20040  df-nzr 20442

This theorem is referenced by: (None)