Theorem lmod0rng 45099

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 2737	. . . 4 ⊢ (Scalar‘𝑀) = (Scalar‘𝑀)
2	1	lmodring 19907	. . 3 ⊢ (𝑀 ∈ LMod → (Scalar‘𝑀) ∈ Ring)
3		0ringnnzr 20307	. . . . 5 ⊢ ((Scalar‘𝑀) ∈ Ring → ((♯‘(Base‘(Scalar‘𝑀))) = 1 ↔ ¬ (Scalar‘𝑀) ∈ NzRing))
4		eqid 2737	. . . . . . . 8 ⊢ (Base‘(Scalar‘𝑀)) = (Base‘(Scalar‘𝑀))
5		eqid 2737	. . . . . . . 8 ⊢ (0g‘(Scalar‘𝑀)) = (0g‘(Scalar‘𝑀))
6		eqid 2737	. . . . . . . 8 ⊢ (1r‘(Scalar‘𝑀)) = (1r‘(Scalar‘𝑀))
7	4, 5, 6	0ring01eq 20309	. . . . . . 7 ⊢ (((Scalar‘𝑀) ∈ Ring ∧ (♯‘(Base‘(Scalar‘𝑀))) = 1) → (0g‘(Scalar‘𝑀)) = (1r‘(Scalar‘𝑀)))
8		eqid 2737	. . . . . . . . . . . . . 14 ⊢ (Base‘𝑀) = (Base‘𝑀)
9		eqid 2737	. . . . . . . . . . . . . 14 ⊢ ( ·𝑠 ‘𝑀) = ( ·𝑠 ‘𝑀)
10	8, 1, 9, 6	lmodvs1 19927	. . . . . . . . . . . . 13 ⊢ ((𝑀 ∈ LMod ∧ 𝑣 ∈ (Base‘𝑀)) → ((1r‘(Scalar‘𝑀))( ·𝑠 ‘𝑀)𝑣) = 𝑣)
11		eqcom 2744	. . . . . . . . . . . . . . . 16 ⊢ (((1r‘(Scalar‘𝑀))( ·𝑠 ‘𝑀)𝑣) = 𝑣 ↔ 𝑣 = ((1r‘(Scalar‘𝑀))( ·𝑠 ‘𝑀)𝑣))
12	11	biimpi 219	. . . . . . . . . . . . . . 15 ⊢ (((1r‘(Scalar‘𝑀))( ·𝑠 ‘𝑀)𝑣) = 𝑣 → 𝑣 = ((1r‘(Scalar‘𝑀))( ·𝑠 ‘𝑀)𝑣))
13		oveq1 7220	. . . . . . . . . . . . . . . . 17 ⊢ ((1r‘(Scalar‘𝑀)) = (0g‘(Scalar‘𝑀)) → ((1r‘(Scalar‘𝑀))( ·𝑠 ‘𝑀)𝑣) = ((0g‘(Scalar‘𝑀))( ·𝑠 ‘𝑀)𝑣))
14	13	eqcoms 2745	. . . . . . . . . . . . . . . 16 ⊢ ((0g‘(Scalar‘𝑀)) = (1r‘(Scalar‘𝑀)) → ((1r‘(Scalar‘𝑀))( ·𝑠 ‘𝑀)𝑣) = ((0g‘(Scalar‘𝑀))( ·𝑠 ‘𝑀)𝑣))
15		eqid 2737	. . . . . . . . . . . . . . . . 17 ⊢ (0g‘𝑀) = (0g‘𝑀)
16	8, 1, 9, 5, 15	lmod0vs 19932	. . . . . . . . . . . . . . . 16 ⊢ ((𝑀 ∈ LMod ∧ 𝑣 ∈ (Base‘𝑀)) → ((0g‘(Scalar‘𝑀))( ·𝑠 ‘𝑀)𝑣) = (0g‘𝑀))
17	14, 16	sylan9eqr 2800	. . . . . . . . . . . . . . 15 ⊢ (((𝑀 ∈ LMod ∧ 𝑣 ∈ (Base‘𝑀)) ∧ (0g‘(Scalar‘𝑀)) = (1r‘(Scalar‘𝑀))) → ((1r‘(Scalar‘𝑀))( ·𝑠 ‘𝑀)𝑣) = (0g‘𝑀))
18	12, 17	sylan9eq 2798	. . . . . . . . . . . . . 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 424	. . . . . . . . . . . . 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 459	. . . . . . . . . 10 ⊢ ((((0g‘(Scalar‘𝑀)) = (1r‘(Scalar‘𝑀)) ∧ 𝑀 ∈ LMod) ∧ 𝑣 ∈ (Base‘𝑀)) → 𝑣 = (0g‘𝑀))
23	22	ralrimiva 3105	. . . . . . . . 9 ⊢ (((0g‘(Scalar‘𝑀)) = (1r‘(Scalar‘𝑀)) ∧ 𝑀 ∈ LMod) → ∀𝑣 ∈ (Base‘𝑀)𝑣 = (0g‘𝑀))
24	8	lmodbn0 19909	. . . . . . . . . . 11 ⊢ (𝑀 ∈ LMod → (Base‘𝑀) ≠ ∅)
25		eqsn 4742	. . . . . . . . . . 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 485	. . . . . . . . 9 ⊢ (((0g‘(Scalar‘𝑀)) = (1r‘(Scalar‘𝑀)) ∧ 𝑀 ∈ LMod) → ((Base‘𝑀) = {(0g‘𝑀)} ↔ ∀𝑣 ∈ (Base‘𝑀)𝑣 = (0g‘𝑀)))
28	23, 27	mpbird 260	. . . . . . . 8 ⊢ (((0g‘(Scalar‘𝑀)) = (1r‘(Scalar‘𝑀)) ∧ 𝑀 ∈ LMod) → (Base‘𝑀) = {(0g‘𝑀)})
29	28	ex 416	. . . . . . 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 416	. . . . 5 ⊢ ((Scalar‘𝑀) ∈ Ring → ((♯‘(Base‘(Scalar‘𝑀))) = 1 → (𝑀 ∈ LMod → (Base‘𝑀) = {(0g‘𝑀)})))
32	3, 31	sylbird 263	. . . 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 410	1 ⊢ ((𝑀 ∈ LMod ∧ ¬ (Scalar‘𝑀) ∈ NzRing) → (Base‘𝑀) = {(0g‘𝑀)})

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1543   ∈ wcel 2110   ≠ wne 2940  ∀wral 3061  ∅c0 4237  {csn 4541  ‘cfv 6380  (class class class)co 7213  1c1 10730  ♯chash 13896  Basecbs 16760  Scalarcsca 16805   ·_𝑠 cvsca 16806  0_gc0g 16944  1_rcur 19516  Ringcrg 19562  LModclmod 19899  NzRingcnzr 20295

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-sep 5192  ax-nul 5199  ax-pow 5258  ax-pr 5322  ax-un 7523  ax-cnex 10785  ax-resscn 10786  ax-1cn 10787  ax-icn 10788  ax-addcl 10789  ax-addrcl 10790  ax-mulcl 10791  ax-mulrcl 10792  ax-mulcom 10793  ax-addass 10794  ax-mulass 10795  ax-distr 10796  ax-i2m1 10797  ax-1ne0 10798  ax-1rid 10799  ax-rnegex 10800  ax-rrecex 10801  ax-cnre 10802  ax-pre-lttri 10803  ax-pre-lttrn 10804  ax-pre-ltadd 10805  ax-pre-mulgt0 10806

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3or 1090  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ne 2941  df-nel 3047  df-ral 3066  df-rex 3067  df-reu 3068  df-rmo 3069  df-rab 3070  df-v 3410  df-sbc 3695  df-csb 3812  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-pss 3885  df-nul 4238  df-if 4440  df-pw 4515  df-sn 4542  df-pr 4544  df-tp 4546  df-op 4548  df-uni 4820  df-int 4860  df-iun 4906  df-br 5054  df-opab 5116  df-mpt 5136  df-tr 5162  df-id 5455  df-eprel 5460  df-po 5468  df-so 5469  df-fr 5509  df-we 5511  df-xp 5557  df-rel 5558  df-cnv 5559  df-co 5560  df-dm 5561  df-rn 5562  df-res 5563  df-ima 5564  df-pred 6160  df-ord 6216  df-on 6217  df-lim 6218  df-suc 6219  df-iota 6338  df-fun 6382  df-fn 6383  df-f 6384  df-f1 6385  df-fo 6386  df-f1o 6387  df-fv 6388  df-riota 7170  df-ov 7216  df-oprab 7217  df-mpo 7218  df-om 7645  df-1st 7761  df-2nd 7762  df-wrecs 8047  df-recs 8108  df-rdg 8146  df-1o 8202  df-oadd 8206  df-er 8391  df-en 8627  df-dom 8628  df-sdom 8629  df-fin 8630  df-dju 9517  df-card 9555  df-pnf 10869  df-mnf 10870  df-xr 10871  df-ltxr 10872  df-le 10873  df-sub 11064  df-neg 11065  df-nn 11831  df-2 11893  df-n0 12091  df-xnn0 12163  df-z 12177  df-uz 12439  df-fz 13096  df-hash 13897  df-sets 16717  df-slot 16735  df-ndx 16745  df-base 16761  df-plusg 16815  df-0g 16946  df-mgm 18114  df-sgrp 18163  df-mnd 18174  df-grp 18368  df-minusg 18369  df-mgp 19505  df-ur 19517  df-ring 19564  df-lmod 19901  df-nzr 20296

This theorem is referenced by: (None)