Theorem snlindsntor 49136

Description: A singleton is linearly independent iff it does not contain a torsion element. According to Wikipedia ("Torsion (algebra)", 15-Apr-2019, https://en.wikipedia.org/wiki/Torsion_(algebra)): "An element m of a module M over a ring R is called a torsion element of the module if there exists a regular element r of the ring (an element that is neither a left nor a right zero divisor) that annihilates m, i.e., (𝑟 · 𝑚) = 0. In an integral domain (a commutative ring without zero divisors), every nonzero element is regular, so a torsion element of a module over an integral domain is one annihilated by a nonzero element of the integral domain." Analogously, the definition in [Lang] p. 147 states that "An element x of [a module] E [over a ring R] is called a torsion element if there exists 𝑎 ∈ 𝑅, 𝑎 ≠ 0, such that 𝑎 · 𝑥 = 0. This definition includes the zero element of the module. Some authors, however, exclude the zero element from the definition of torsion elements. (Contributed by AV, 14-Apr-2019.) (Revised by AV, 27-Apr-2019.)

Hypotheses

Ref	Expression
snlindsntor.b	⊢ 𝐵 = (Base‘𝑀)
snlindsntor.r	⊢ 𝑅 = (Scalar‘𝑀)
snlindsntor.s	⊢ 𝑆 = (Base‘𝑅)
snlindsntor.0	⊢ 0 = (0g‘𝑅)
snlindsntor.z	⊢ 𝑍 = (0g‘𝑀)
snlindsntor.t	⊢ · = ( ·𝑠 ‘𝑀)

Assertion

Ref	Expression
snlindsntor	⊢ ((𝑀 ∈ LMod ∧ 𝑋 ∈ 𝐵) → (∀𝑠 ∈ (𝑆 ∖ { 0 })(𝑠 · 𝑋) ≠ 𝑍 ↔ {𝑋} linIndS 𝑀))

Distinct variable groups:   𝐵,𝑠   𝑀,𝑠   𝑆,𝑠   𝑋,𝑠   𝑍,𝑠   · ,𝑠   0 ,𝑠

Allowed substitution hint:   𝑅(𝑠)

Proof of Theorem snlindsntor

Dummy variables 𝑓 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-ne 2965	. . . . 5 ⊢ ((𝑠 · 𝑋) ≠ 𝑍 ↔ ¬ (𝑠 · 𝑋) = 𝑍)
2	1	ralbii 3117	. . . 4 ⊢ (∀𝑠 ∈ (𝑆 ∖ { 0 })(𝑠 · 𝑋) ≠ 𝑍 ↔ ∀𝑠 ∈ (𝑆 ∖ { 0 }) ¬ (𝑠 · 𝑋) = 𝑍)
3		raldifsni 4767	. . . 4 ⊢ (∀𝑠 ∈ (𝑆 ∖ { 0 }) ¬ (𝑠 · 𝑋) = 𝑍 ↔ ∀𝑠 ∈ 𝑆 ((𝑠 · 𝑋) = 𝑍 → 𝑠 = 0 ))
4	2, 3	bitri 278	. . 3 ⊢ (∀𝑠 ∈ (𝑆 ∖ { 0 })(𝑠 · 𝑋) ≠ 𝑍 ↔ ∀𝑠 ∈ 𝑆 ((𝑠 · 𝑋) = 𝑍 → 𝑠 = 0 ))
5		simpl 487	. . . . . . . . . . . . 13 ⊢ ((𝑀 ∈ LMod ∧ 𝑋 ∈ 𝐵) → 𝑀 ∈ LMod)
6	5	adantr 485	. . . . . . . . . . . 12 ⊢ (((𝑀 ∈ LMod ∧ 𝑋 ∈ 𝐵) ∧ ∀𝑠 ∈ 𝑆 ((𝑠 · 𝑋) = 𝑍 → 𝑠 = 0 )) → 𝑀 ∈ LMod)
7	6	adantr 485	. . . . . . . . . . 11 ⊢ ((((𝑀 ∈ LMod ∧ 𝑋 ∈ 𝐵) ∧ ∀𝑠 ∈ 𝑆 ((𝑠 · 𝑋) = 𝑍 → 𝑠 = 0 )) ∧ 𝑓 ∈ (𝑆 ↑m {𝑋})) → 𝑀 ∈ LMod)
8		snlindsntor.s	. . . . . . . . . . . . . . 15 ⊢ 𝑆 = (Base‘𝑅)
9		snlindsntor.r	. . . . . . . . . . . . . . . 16 ⊢ 𝑅 = (Scalar‘𝑀)
10	9	fveq2i 6885	. . . . . . . . . . . . . . 15 ⊢ (Base‘𝑅) = (Base‘(Scalar‘𝑀))
11	8, 10	eqtri 2792	. . . . . . . . . . . . . 14 ⊢ 𝑆 = (Base‘(Scalar‘𝑀))
12	11	oveq1i 7421	. . . . . . . . . . . . 13 ⊢ (𝑆 ↑m {𝑋}) = ((Base‘(Scalar‘𝑀)) ↑m {𝑋})
13	12	eleq2i 2861	. . . . . . . . . . . 12 ⊢ (𝑓 ∈ (𝑆 ↑m {𝑋}) ↔ 𝑓 ∈ ((Base‘(Scalar‘𝑀)) ↑m {𝑋}))
14	13	bilani 509	. . . . . . . . . . 11 ⊢ ((((𝑀 ∈ LMod ∧ 𝑋 ∈ 𝐵) ∧ ∀𝑠 ∈ 𝑆 ((𝑠 · 𝑋) = 𝑍 → 𝑠 = 0 )) ∧ 𝑓 ∈ (𝑆 ↑m {𝑋})) → 𝑓 ∈ ((Base‘(Scalar‘𝑀)) ↑m {𝑋}))
15		snelpwi 5426	. . . . . . . . . . . . 13 ⊢ (𝑋 ∈ (Base‘𝑀) → {𝑋} ∈ 𝒫 (Base‘𝑀))
16		snlindsntor.b	. . . . . . . . . . . . 13 ⊢ 𝐵 = (Base‘𝑀)
17	15, 16	eleq2s 2887	. . . . . . . . . . . 12 ⊢ (𝑋 ∈ 𝐵 → {𝑋} ∈ 𝒫 (Base‘𝑀))
18	17	ad3antlr 743	. . . . . . . . . . 11 ⊢ ((((𝑀 ∈ LMod ∧ 𝑋 ∈ 𝐵) ∧ ∀𝑠 ∈ 𝑆 ((𝑠 · 𝑋) = 𝑍 → 𝑠 = 0 )) ∧ 𝑓 ∈ (𝑆 ↑m {𝑋})) → {𝑋} ∈ 𝒫 (Base‘𝑀))
19		lincval 49074	. . . . . . . . . . 11 ⊢ ((𝑀 ∈ LMod ∧ 𝑓 ∈ ((Base‘(Scalar‘𝑀)) ↑m {𝑋}) ∧ {𝑋} ∈ 𝒫 (Base‘𝑀)) → (𝑓( linC ‘𝑀){𝑋}) = (𝑀 Σg (𝑥 ∈ {𝑋} ↦ ((𝑓‘𝑥)( ·𝑠 ‘𝑀)𝑥))))
20	7, 14, 18, 19	syl3anc 1396	. . . . . . . . . 10 ⊢ ((((𝑀 ∈ LMod ∧ 𝑋 ∈ 𝐵) ∧ ∀𝑠 ∈ 𝑆 ((𝑠 · 𝑋) = 𝑍 → 𝑠 = 0 )) ∧ 𝑓 ∈ (𝑆 ↑m {𝑋})) → (𝑓( linC ‘𝑀){𝑋}) = (𝑀 Σg (𝑥 ∈ {𝑋} ↦ ((𝑓‘𝑥)( ·𝑠 ‘𝑀)𝑥))))
21	20	eqeq1d 2771	. . . . . . . . 9 ⊢ ((((𝑀 ∈ LMod ∧ 𝑋 ∈ 𝐵) ∧ ∀𝑠 ∈ 𝑆 ((𝑠 · 𝑋) = 𝑍 → 𝑠 = 0 )) ∧ 𝑓 ∈ (𝑆 ↑m {𝑋})) → ((𝑓( linC ‘𝑀){𝑋}) = 𝑍 ↔ (𝑀 Σg (𝑥 ∈ {𝑋} ↦ ((𝑓‘𝑥)( ·𝑠 ‘𝑀)𝑥))) = 𝑍))
22	21	anbi2d 641	. . . . . . . 8 ⊢ ((((𝑀 ∈ LMod ∧ 𝑋 ∈ 𝐵) ∧ ∀𝑠 ∈ 𝑆 ((𝑠 · 𝑋) = 𝑍 → 𝑠 = 0 )) ∧ 𝑓 ∈ (𝑆 ↑m {𝑋})) → ((𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀){𝑋}) = 𝑍) ↔ (𝑓 finSupp 0 ∧ (𝑀 Σg (𝑥 ∈ {𝑋} ↦ ((𝑓‘𝑥)( ·𝑠 ‘𝑀)𝑥))) = 𝑍)))
23		lmodgrp 20966	. . . . . . . . . . . . . 14 ⊢ (𝑀 ∈ LMod → 𝑀 ∈ Grp)
24	23	grpmndd 19013	. . . . . . . . . . . . 13 ⊢ (𝑀 ∈ LMod → 𝑀 ∈ Mnd)
25	24	ad3antrrr 742	. . . . . . . . . . . 12 ⊢ ((((𝑀 ∈ LMod ∧ 𝑋 ∈ 𝐵) ∧ ∀𝑠 ∈ 𝑆 ((𝑠 · 𝑋) = 𝑍 → 𝑠 = 0 )) ∧ 𝑓 ∈ (𝑆 ↑m {𝑋})) → 𝑀 ∈ Mnd)
26		simpllr 787	. . . . . . . . . . . 12 ⊢ ((((𝑀 ∈ LMod ∧ 𝑋 ∈ 𝐵) ∧ ∀𝑠 ∈ 𝑆 ((𝑠 · 𝑋) = 𝑍 → 𝑠 = 0 )) ∧ 𝑓 ∈ (𝑆 ↑m {𝑋})) → 𝑋 ∈ 𝐵)
27		elmapi 8846	. . . . . . . . . . . . . 14 ⊢ (𝑓 ∈ (𝑆 ↑m {𝑋}) → 𝑓:{𝑋}⟶𝑆)
28	6	adantl 486	. . . . . . . . . . . . . . . 16 ⊢ ((𝑓:{𝑋}⟶𝑆 ∧ ((𝑀 ∈ LMod ∧ 𝑋 ∈ 𝐵) ∧ ∀𝑠 ∈ 𝑆 ((𝑠 · 𝑋) = 𝑍 → 𝑠 = 0 ))) → 𝑀 ∈ LMod)
29		snidg 4631	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑋 ∈ 𝐵 → 𝑋 ∈ {𝑋})
30	29	adantl 486	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑀 ∈ LMod ∧ 𝑋 ∈ 𝐵) → 𝑋 ∈ {𝑋})
31	30	adantr 485	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑀 ∈ LMod ∧ 𝑋 ∈ 𝐵) ∧ ∀𝑠 ∈ 𝑆 ((𝑠 · 𝑋) = 𝑍 → 𝑠 = 0 )) → 𝑋 ∈ {𝑋})
32		ffvelcdm 7077	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑓:{𝑋}⟶𝑆 ∧ 𝑋 ∈ {𝑋}) → (𝑓‘𝑋) ∈ 𝑆)
33	31, 32	sylan2 604	. . . . . . . . . . . . . . . 16 ⊢ ((𝑓:{𝑋}⟶𝑆 ∧ ((𝑀 ∈ LMod ∧ 𝑋 ∈ 𝐵) ∧ ∀𝑠 ∈ 𝑆 ((𝑠 · 𝑋) = 𝑍 → 𝑠 = 0 ))) → (𝑓‘𝑋) ∈ 𝑆)
34		simprlr 791	. . . . . . . . . . . . . . . 16 ⊢ ((𝑓:{𝑋}⟶𝑆 ∧ ((𝑀 ∈ LMod ∧ 𝑋 ∈ 𝐵) ∧ ∀𝑠 ∈ 𝑆 ((𝑠 · 𝑋) = 𝑍 → 𝑠 = 0 ))) → 𝑋 ∈ 𝐵)
35		eqid 2769	. . . . . . . . . . . . . . . . 17 ⊢ ( ·𝑠 ‘𝑀) = ( ·𝑠 ‘𝑀)
36	16, 9, 35, 8	lmodvscl 20977	. . . . . . . . . . . . . . . 16 ⊢ ((𝑀 ∈ LMod ∧ (𝑓‘𝑋) ∈ 𝑆 ∧ 𝑋 ∈ 𝐵) → ((𝑓‘𝑋)( ·𝑠 ‘𝑀)𝑋) ∈ 𝐵)
37	28, 33, 34, 36	syl3anc 1396	. . . . . . . . . . . . . . 15 ⊢ ((𝑓:{𝑋}⟶𝑆 ∧ ((𝑀 ∈ LMod ∧ 𝑋 ∈ 𝐵) ∧ ∀𝑠 ∈ 𝑆 ((𝑠 · 𝑋) = 𝑍 → 𝑠 = 0 ))) → ((𝑓‘𝑋)( ·𝑠 ‘𝑀)𝑋) ∈ 𝐵)
38	37	expcom 418	. . . . . . . . . . . . . 14 ⊢ (((𝑀 ∈ LMod ∧ 𝑋 ∈ 𝐵) ∧ ∀𝑠 ∈ 𝑆 ((𝑠 · 𝑋) = 𝑍 → 𝑠 = 0 )) → (𝑓:{𝑋}⟶𝑆 → ((𝑓‘𝑋)( ·𝑠 ‘𝑀)𝑋) ∈ 𝐵))
39	27, 38	syl5com 32	. . . . . . . . . . . . 13 ⊢ (𝑓 ∈ (𝑆 ↑m {𝑋}) → (((𝑀 ∈ LMod ∧ 𝑋 ∈ 𝐵) ∧ ∀𝑠 ∈ 𝑆 ((𝑠 · 𝑋) = 𝑍 → 𝑠 = 0 )) → ((𝑓‘𝑋)( ·𝑠 ‘𝑀)𝑋) ∈ 𝐵))
40	39	impcom 412	. . . . . . . . . . . 12 ⊢ ((((𝑀 ∈ LMod ∧ 𝑋 ∈ 𝐵) ∧ ∀𝑠 ∈ 𝑆 ((𝑠 · 𝑋) = 𝑍 → 𝑠 = 0 )) ∧ 𝑓 ∈ (𝑆 ↑m {𝑋})) → ((𝑓‘𝑋)( ·𝑠 ‘𝑀)𝑋) ∈ 𝐵)
41		fveq2 6882	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑋 → (𝑓‘𝑥) = (𝑓‘𝑋))
42		id 23	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑋 → 𝑥 = 𝑋)
43	41, 42	oveq12d 7429	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑋 → ((𝑓‘𝑥)( ·𝑠 ‘𝑀)𝑥) = ((𝑓‘𝑋)( ·𝑠 ‘𝑀)𝑋))
44	16, 43	gsumsn 20024	. . . . . . . . . . . 12 ⊢ ((𝑀 ∈ Mnd ∧ 𝑋 ∈ 𝐵 ∧ ((𝑓‘𝑋)( ·𝑠 ‘𝑀)𝑋) ∈ 𝐵) → (𝑀 Σg (𝑥 ∈ {𝑋} ↦ ((𝑓‘𝑥)( ·𝑠 ‘𝑀)𝑥))) = ((𝑓‘𝑋)( ·𝑠 ‘𝑀)𝑋))
45	25, 26, 40, 44	syl3anc 1396	. . . . . . . . . . 11 ⊢ ((((𝑀 ∈ LMod ∧ 𝑋 ∈ 𝐵) ∧ ∀𝑠 ∈ 𝑆 ((𝑠 · 𝑋) = 𝑍 → 𝑠 = 0 )) ∧ 𝑓 ∈ (𝑆 ↑m {𝑋})) → (𝑀 Σg (𝑥 ∈ {𝑋} ↦ ((𝑓‘𝑥)( ·𝑠 ‘𝑀)𝑥))) = ((𝑓‘𝑋)( ·𝑠 ‘𝑀)𝑋))
46	45	eqeq1d 2771	. . . . . . . . . 10 ⊢ ((((𝑀 ∈ LMod ∧ 𝑋 ∈ 𝐵) ∧ ∀𝑠 ∈ 𝑆 ((𝑠 · 𝑋) = 𝑍 → 𝑠 = 0 )) ∧ 𝑓 ∈ (𝑆 ↑m {𝑋})) → ((𝑀 Σg (𝑥 ∈ {𝑋} ↦ ((𝑓‘𝑥)( ·𝑠 ‘𝑀)𝑥))) = 𝑍 ↔ ((𝑓‘𝑋)( ·𝑠 ‘𝑀)𝑋) = 𝑍))
47	29, 32	sylan2 604	. . . . . . . . . . . . . . 15 ⊢ ((𝑓:{𝑋}⟶𝑆 ∧ 𝑋 ∈ 𝐵) → (𝑓‘𝑋) ∈ 𝑆)
48	47	expcom 418	. . . . . . . . . . . . . 14 ⊢ (𝑋 ∈ 𝐵 → (𝑓:{𝑋}⟶𝑆 → (𝑓‘𝑋) ∈ 𝑆))
49	48	adantl 486	. . . . . . . . . . . . 13 ⊢ ((𝑀 ∈ LMod ∧ 𝑋 ∈ 𝐵) → (𝑓:{𝑋}⟶𝑆 → (𝑓‘𝑋) ∈ 𝑆))
50		snlindsntor.t	. . . . . . . . . . . . . . . . 17 ⊢ · = ( ·𝑠 ‘𝑀)
51	50	oveqi 7424	. . . . . . . . . . . . . . . 16 ⊢ ((𝑓‘𝑋) · 𝑋) = ((𝑓‘𝑋)( ·𝑠 ‘𝑀)𝑋)
52	51	eqeq1i 2774	. . . . . . . . . . . . . . 15 ⊢ (((𝑓‘𝑋) · 𝑋) = 𝑍 ↔ ((𝑓‘𝑋)( ·𝑠 ‘𝑀)𝑋) = 𝑍)
53		oveq1 7418	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑠 = (𝑓‘𝑋) → (𝑠 · 𝑋) = ((𝑓‘𝑋) · 𝑋))
54	53	eqeq1d 2771	. . . . . . . . . . . . . . . . 17 ⊢ (𝑠 = (𝑓‘𝑋) → ((𝑠 · 𝑋) = 𝑍 ↔ ((𝑓‘𝑋) · 𝑋) = 𝑍))
55		eqeq1 2773	. . . . . . . . . . . . . . . . 17 ⊢ (𝑠 = (𝑓‘𝑋) → (𝑠 = 0 ↔ (𝑓‘𝑋) = 0 ))
56	54, 55	imbi12d 347	. . . . . . . . . . . . . . . 16 ⊢ (𝑠 = (𝑓‘𝑋) → (((𝑠 · 𝑋) = 𝑍 → 𝑠 = 0 ) ↔ (((𝑓‘𝑋) · 𝑋) = 𝑍 → (𝑓‘𝑋) = 0 )))
57	56	rspcva 3588	. . . . . . . . . . . . . . 15 ⊢ (((𝑓‘𝑋) ∈ 𝑆 ∧ ∀𝑠 ∈ 𝑆 ((𝑠 · 𝑋) = 𝑍 → 𝑠 = 0 )) → (((𝑓‘𝑋) · 𝑋) = 𝑍 → (𝑓‘𝑋) = 0 ))
58	52, 57	biimtrrid 246	. . . . . . . . . . . . . 14 ⊢ (((𝑓‘𝑋) ∈ 𝑆 ∧ ∀𝑠 ∈ 𝑆 ((𝑠 · 𝑋) = 𝑍 → 𝑠 = 0 )) → (((𝑓‘𝑋)( ·𝑠 ‘𝑀)𝑋) = 𝑍 → (𝑓‘𝑋) = 0 ))
59	58	ex 417	. . . . . . . . . . . . 13 ⊢ ((𝑓‘𝑋) ∈ 𝑆 → (∀𝑠 ∈ 𝑆 ((𝑠 · 𝑋) = 𝑍 → 𝑠 = 0 ) → (((𝑓‘𝑋)( ·𝑠 ‘𝑀)𝑋) = 𝑍 → (𝑓‘𝑋) = 0 )))
60	27, 49, 59	syl56 37	. . . . . . . . . . . 12 ⊢ ((𝑀 ∈ LMod ∧ 𝑋 ∈ 𝐵) → (𝑓 ∈ (𝑆 ↑m {𝑋}) → (∀𝑠 ∈ 𝑆 ((𝑠 · 𝑋) = 𝑍 → 𝑠 = 0 ) → (((𝑓‘𝑋)( ·𝑠 ‘𝑀)𝑋) = 𝑍 → (𝑓‘𝑋) = 0 ))))
61	60	com23 87	. . . . . . . . . . 11 ⊢ ((𝑀 ∈ LMod ∧ 𝑋 ∈ 𝐵) → (∀𝑠 ∈ 𝑆 ((𝑠 · 𝑋) = 𝑍 → 𝑠 = 0 ) → (𝑓 ∈ (𝑆 ↑m {𝑋}) → (((𝑓‘𝑋)( ·𝑠 ‘𝑀)𝑋) = 𝑍 → (𝑓‘𝑋) = 0 ))))
62	61	imp31 422	. . . . . . . . . 10 ⊢ ((((𝑀 ∈ LMod ∧ 𝑋 ∈ 𝐵) ∧ ∀𝑠 ∈ 𝑆 ((𝑠 · 𝑋) = 𝑍 → 𝑠 = 0 )) ∧ 𝑓 ∈ (𝑆 ↑m {𝑋})) → (((𝑓‘𝑋)( ·𝑠 ‘𝑀)𝑋) = 𝑍 → (𝑓‘𝑋) = 0 ))
63	46, 62	sylbid 243	. . . . . . . . 9 ⊢ ((((𝑀 ∈ LMod ∧ 𝑋 ∈ 𝐵) ∧ ∀𝑠 ∈ 𝑆 ((𝑠 · 𝑋) = 𝑍 → 𝑠 = 0 )) ∧ 𝑓 ∈ (𝑆 ↑m {𝑋})) → ((𝑀 Σg (𝑥 ∈ {𝑋} ↦ ((𝑓‘𝑥)( ·𝑠 ‘𝑀)𝑥))) = 𝑍 → (𝑓‘𝑋) = 0 ))
64	63	adantld 495	. . . . . . . 8 ⊢ ((((𝑀 ∈ LMod ∧ 𝑋 ∈ 𝐵) ∧ ∀𝑠 ∈ 𝑆 ((𝑠 · 𝑋) = 𝑍 → 𝑠 = 0 )) ∧ 𝑓 ∈ (𝑆 ↑m {𝑋})) → ((𝑓 finSupp 0 ∧ (𝑀 Σg (𝑥 ∈ {𝑋} ↦ ((𝑓‘𝑥)( ·𝑠 ‘𝑀)𝑥))) = 𝑍) → (𝑓‘𝑋) = 0 ))
65	22, 64	sylbid 243	. . . . . . 7 ⊢ ((((𝑀 ∈ LMod ∧ 𝑋 ∈ 𝐵) ∧ ∀𝑠 ∈ 𝑆 ((𝑠 · 𝑋) = 𝑍 → 𝑠 = 0 )) ∧ 𝑓 ∈ (𝑆 ↑m {𝑋})) → ((𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀){𝑋}) = 𝑍) → (𝑓‘𝑋) = 0 ))
66	65	ralrimiva 3163	. . . . . 6 ⊢ (((𝑀 ∈ LMod ∧ 𝑋 ∈ 𝐵) ∧ ∀𝑠 ∈ 𝑆 ((𝑠 · 𝑋) = 𝑍 → 𝑠 = 0 )) → ∀𝑓 ∈ (𝑆 ↑m {𝑋})((𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀){𝑋}) = 𝑍) → (𝑓‘𝑋) = 0 ))
67	66	ex 417	. . . . 5 ⊢ ((𝑀 ∈ LMod ∧ 𝑋 ∈ 𝐵) → (∀𝑠 ∈ 𝑆 ((𝑠 · 𝑋) = 𝑍 → 𝑠 = 0 ) → ∀𝑓 ∈ (𝑆 ↑m {𝑋})((𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀){𝑋}) = 𝑍) → (𝑓‘𝑋) = 0 )))
68		impexp 455	. . . . . . . 8 ⊢ (((𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀){𝑋}) = 𝑍) → (𝑓‘𝑋) = 0 ) ↔ (𝑓 finSupp 0 → ((𝑓( linC ‘𝑀){𝑋}) = 𝑍 → (𝑓‘𝑋) = 0 )))
69	27	adantl 486	. . . . . . . . . 10 ⊢ (((𝑀 ∈ LMod ∧ 𝑋 ∈ 𝐵) ∧ 𝑓 ∈ (𝑆 ↑m {𝑋})) → 𝑓:{𝑋}⟶𝑆)
70		snfi 9040	. . . . . . . . . . 11 ⊢ {𝑋} ∈ Fin
71	70	a1i 11	. . . . . . . . . 10 ⊢ (((𝑀 ∈ LMod ∧ 𝑋 ∈ 𝐵) ∧ 𝑓 ∈ (𝑆 ↑m {𝑋})) → {𝑋} ∈ Fin)
72		snlindsntor.0	. . . . . . . . . . . 12 ⊢ 0 = (0g‘𝑅)
73	72	fvexi 6896	. . . . . . . . . . 11 ⊢ 0 ∈ V
74	73	a1i 11	. . . . . . . . . 10 ⊢ (((𝑀 ∈ LMod ∧ 𝑋 ∈ 𝐵) ∧ 𝑓 ∈ (𝑆 ↑m {𝑋})) → 0 ∈ V)
75	69, 71, 74	fdmfifsupp 9335	. . . . . . . . 9 ⊢ (((𝑀 ∈ LMod ∧ 𝑋 ∈ 𝐵) ∧ 𝑓 ∈ (𝑆 ↑m {𝑋})) → 𝑓 finSupp 0 )
76		pm2.27 43	. . . . . . . . 9 ⊢ (𝑓 finSupp 0 → ((𝑓 finSupp 0 → ((𝑓( linC ‘𝑀){𝑋}) = 𝑍 → (𝑓‘𝑋) = 0 )) → ((𝑓( linC ‘𝑀){𝑋}) = 𝑍 → (𝑓‘𝑋) = 0 )))
77	75, 76	syl 18	. . . . . . . 8 ⊢ (((𝑀 ∈ LMod ∧ 𝑋 ∈ 𝐵) ∧ 𝑓 ∈ (𝑆 ↑m {𝑋})) → ((𝑓 finSupp 0 → ((𝑓( linC ‘𝑀){𝑋}) = 𝑍 → (𝑓‘𝑋) = 0 )) → ((𝑓( linC ‘𝑀){𝑋}) = 𝑍 → (𝑓‘𝑋) = 0 )))
78	68, 77	biimtrid 245	. . . . . . 7 ⊢ (((𝑀 ∈ LMod ∧ 𝑋 ∈ 𝐵) ∧ 𝑓 ∈ (𝑆 ↑m {𝑋})) → (((𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀){𝑋}) = 𝑍) → (𝑓‘𝑋) = 0 ) → ((𝑓( linC ‘𝑀){𝑋}) = 𝑍 → (𝑓‘𝑋) = 0 )))
79	78	ralimdva 3183	. . . . . 6 ⊢ ((𝑀 ∈ LMod ∧ 𝑋 ∈ 𝐵) → (∀𝑓 ∈ (𝑆 ↑m {𝑋})((𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀){𝑋}) = 𝑍) → (𝑓‘𝑋) = 0 ) → ∀𝑓 ∈ (𝑆 ↑m {𝑋})((𝑓( linC ‘𝑀){𝑋}) = 𝑍 → (𝑓‘𝑋) = 0 )))
80		snlindsntor.z	. . . . . . 7 ⊢ 𝑍 = (0g‘𝑀)
81	16, 9, 8, 72, 80, 50	snlindsntorlem 49135	. . . . . 6 ⊢ ((𝑀 ∈ LMod ∧ 𝑋 ∈ 𝐵) → (∀𝑓 ∈ (𝑆 ↑m {𝑋})((𝑓( linC ‘𝑀){𝑋}) = 𝑍 → (𝑓‘𝑋) = 0 ) → ∀𝑠 ∈ 𝑆 ((𝑠 · 𝑋) = 𝑍 → 𝑠 = 0 )))
82	79, 81	syld 48	. . . . 5 ⊢ ((𝑀 ∈ LMod ∧ 𝑋 ∈ 𝐵) → (∀𝑓 ∈ (𝑆 ↑m {𝑋})((𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀){𝑋}) = 𝑍) → (𝑓‘𝑋) = 0 ) → ∀𝑠 ∈ 𝑆 ((𝑠 · 𝑋) = 𝑍 → 𝑠 = 0 )))
83	67, 82	impbid 215	. . . 4 ⊢ ((𝑀 ∈ LMod ∧ 𝑋 ∈ 𝐵) → (∀𝑠 ∈ 𝑆 ((𝑠 · 𝑋) = 𝑍 → 𝑠 = 0 ) ↔ ∀𝑓 ∈ (𝑆 ↑m {𝑋})((𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀){𝑋}) = 𝑍) → (𝑓‘𝑋) = 0 )))
84		fveqeq2 6891	. . . . . . . . 9 ⊢ (𝑦 = 𝑋 → ((𝑓‘𝑦) = 0 ↔ (𝑓‘𝑋) = 0 ))
85	84	ralsng 4646	. . . . . . . 8 ⊢ (𝑋 ∈ 𝐵 → (∀𝑦 ∈ {𝑋} (𝑓‘𝑦) = 0 ↔ (𝑓‘𝑋) = 0 ))
86	85	adantl 486	. . . . . . 7 ⊢ ((𝑀 ∈ LMod ∧ 𝑋 ∈ 𝐵) → (∀𝑦 ∈ {𝑋} (𝑓‘𝑦) = 0 ↔ (𝑓‘𝑋) = 0 ))
87	86	bicomd 226	. . . . . 6 ⊢ ((𝑀 ∈ LMod ∧ 𝑋 ∈ 𝐵) → ((𝑓‘𝑋) = 0 ↔ ∀𝑦 ∈ {𝑋} (𝑓‘𝑦) = 0 ))
88	87	imbi2d 343	. . . . 5 ⊢ ((𝑀 ∈ LMod ∧ 𝑋 ∈ 𝐵) → (((𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀){𝑋}) = 𝑍) → (𝑓‘𝑋) = 0 ) ↔ ((𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀){𝑋}) = 𝑍) → ∀𝑦 ∈ {𝑋} (𝑓‘𝑦) = 0 )))
89	88	ralbidv 3194	. . . 4 ⊢ ((𝑀 ∈ LMod ∧ 𝑋 ∈ 𝐵) → (∀𝑓 ∈ (𝑆 ↑m {𝑋})((𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀){𝑋}) = 𝑍) → (𝑓‘𝑋) = 0 ) ↔ ∀𝑓 ∈ (𝑆 ↑m {𝑋})((𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀){𝑋}) = 𝑍) → ∀𝑦 ∈ {𝑋} (𝑓‘𝑦) = 0 )))
90		snelpwi 5426	. . . . . 6 ⊢ (𝑋 ∈ 𝐵 → {𝑋} ∈ 𝒫 𝐵)
91	90	adantl 486	. . . . 5 ⊢ ((𝑀 ∈ LMod ∧ 𝑋 ∈ 𝐵) → {𝑋} ∈ 𝒫 𝐵)
92	91	biantrurd 541	. . . 4 ⊢ ((𝑀 ∈ LMod ∧ 𝑋 ∈ 𝐵) → (∀𝑓 ∈ (𝑆 ↑m {𝑋})((𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀){𝑋}) = 𝑍) → ∀𝑦 ∈ {𝑋} (𝑓‘𝑦) = 0 ) ↔ ({𝑋} ∈ 𝒫 𝐵 ∧ ∀𝑓 ∈ (𝑆 ↑m {𝑋})((𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀){𝑋}) = 𝑍) → ∀𝑦 ∈ {𝑋} (𝑓‘𝑦) = 0 ))))
93	83, 89, 92	3bitrd 308	. . 3 ⊢ ((𝑀 ∈ LMod ∧ 𝑋 ∈ 𝐵) → (∀𝑠 ∈ 𝑆 ((𝑠 · 𝑋) = 𝑍 → 𝑠 = 0 ) ↔ ({𝑋} ∈ 𝒫 𝐵 ∧ ∀𝑓 ∈ (𝑆 ↑m {𝑋})((𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀){𝑋}) = 𝑍) → ∀𝑦 ∈ {𝑋} (𝑓‘𝑦) = 0 ))))
94	4, 93	bitrid 286	. 2 ⊢ ((𝑀 ∈ LMod ∧ 𝑋 ∈ 𝐵) → (∀𝑠 ∈ (𝑆 ∖ { 0 })(𝑠 · 𝑋) ≠ 𝑍 ↔ ({𝑋} ∈ 𝒫 𝐵 ∧ ∀𝑓 ∈ (𝑆 ↑m {𝑋})((𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀){𝑋}) = 𝑍) → ∀𝑦 ∈ {𝑋} (𝑓‘𝑦) = 0 ))))
95		snex 5411	. . 3 ⊢ {𝑋} ∈ V
96	16, 80, 9, 8, 72	islininds 49111	. . 3 ⊢ (({𝑋} ∈ V ∧ 𝑀 ∈ LMod) → ({𝑋} linIndS 𝑀 ↔ ({𝑋} ∈ 𝒫 𝐵 ∧ ∀𝑓 ∈ (𝑆 ↑m {𝑋})((𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀){𝑋}) = 𝑍) → ∀𝑦 ∈ {𝑋} (𝑓‘𝑦) = 0 ))))
97	95, 5, 96	sylancr 598	. 2 ⊢ ((𝑀 ∈ LMod ∧ 𝑋 ∈ 𝐵) → ({𝑋} linIndS 𝑀 ↔ ({𝑋} ∈ 𝒫 𝐵 ∧ ∀𝑓 ∈ (𝑆 ↑m {𝑋})((𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀){𝑋}) = 𝑍) → ∀𝑦 ∈ {𝑋} (𝑓‘𝑦) = 0 ))))
98	94, 97	bitr4d 285	1 ⊢ ((𝑀 ∈ LMod ∧ 𝑋 ∈ 𝐵) → (∀𝑠 ∈ (𝑆 ∖ { 0 })(𝑠 · 𝑋) ≠ 𝑍 ↔ {𝑋} linIndS 𝑀))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1567   ∈ wcel 2149   ≠ wne 2964  ∀wral 3085  Vcvv 3463   ∖ cdif 3910  𝒫 cpw 4567  {csn 4594   class class class wbr 5113   ↦ cmpt 5196  ⟶wf 6533  ‘cfv 6537  (class class class)co 7411   ↑_m cmap 8824  Fincfn 8943   finSupp cfsupp 9321  Basecbs 17269  Scalarcsca 17313   ·_𝑠 cvsca 17314  0_gc0g 17492   Σ_g cgsu 17493  Mndcmnd 18792  LModclmod 20959   linC clinc 49069   linIndS clininds 49105

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-rep 5242  ax-sep 5261  ax-nul 5271  ax-pow 5337  ax-pr 5405  ax-un 7733  ax-cnex 11156  ax-resscn 11157  ax-1cn 11158  ax-icn 11159  ax-addcl 11160  ax-addrcl 11161  ax-mulcl 11162  ax-mulrcl 11163  ax-mulcom 11164  ax-addass 11165  ax-mulass 11166  ax-distr 11167  ax-i2m1 11168  ax-1ne0 11169  ax-1rid 11170  ax-rnegex 11171  ax-rrecex 11172  ax-cnre 11173  ax-pre-lttri 11174  ax-pre-lttrn 11175  ax-pre-ltadd 11176  ax-pre-mulgt0 11177

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-nel 3071  df-ral 3086  df-rex 3096  df-rmo 3376  df-reu 3377  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-pss 3933  df-nul 4295  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-int 4917  df-iun 4962  df-br 5114  df-opab 5178  df-mpt 5197  df-tr 5223  df-id 5557  df-eprel 5562  df-po 5570  df-so 5571  df-fr 5615  df-se 5616  df-we 5617  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-rn 5673  df-res 5674  df-ima 5675  df-pred 6303  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-isom 6546  df-riota 7368  df-ov 7414  df-oprab 7415  df-mpo 7416  df-om 7863  df-1st 7986  df-2nd 7987  df-supp 8157  df-frecs 8278  df-wrecs 8309  df-recs 8358  df-rdg 8397  df-1o 8453  df-er 8694  df-map 8826  df-en 8944  df-dom 8945  df-sdom 8946  df-fin 8947  df-fsupp 9322  df-oi 9472  df-card 9925  df-pnf 11245  df-mnf 11246  df-xr 11247  df-ltxr 11248  df-le 11249  df-sub 11443  df-neg 11444  df-nn 12234  df-n0 12505  df-z 12592  df-uz 12863  df-fz 13536  df-fzo 13683  df-seq 14038  df-hash 14367  df-0g 17494  df-gsum 17495  df-mgm 18698  df-sgrp 18777  df-mnd 18793  df-grp 19003  df-mulg 19134  df-cntz 19387  df-lmod 20961  df-linc 49071  df-lininds 49107

This theorem is referenced by:  lindssnlvec  49151