Theorem snlindsntor 48460

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 2926	. . . . 5 ⊢ ((𝑠 · 𝑋) ≠ 𝑍 ↔ ¬ (𝑠 · 𝑋) = 𝑍)
2	1	ralbii 3075	. . . 4 ⊢ (∀𝑠 ∈ (𝑆 ∖ { 0 })(𝑠 · 𝑋) ≠ 𝑍 ↔ ∀𝑠 ∈ (𝑆 ∖ { 0 }) ¬ (𝑠 · 𝑋) = 𝑍)
3		raldifsni 4746	. . . 4 ⊢ (∀𝑠 ∈ (𝑆 ∖ { 0 }) ¬ (𝑠 · 𝑋) = 𝑍 ↔ ∀𝑠 ∈ 𝑆 ((𝑠 · 𝑋) = 𝑍 → 𝑠 = 0 ))
4	2, 3	bitri 275	. . 3 ⊢ (∀𝑠 ∈ (𝑆 ∖ { 0 })(𝑠 · 𝑋) ≠ 𝑍 ↔ ∀𝑠 ∈ 𝑆 ((𝑠 · 𝑋) = 𝑍 → 𝑠 = 0 ))
5		simpl 482	. . . . . . . . . . . . 13 ⊢ ((𝑀 ∈ LMod ∧ 𝑋 ∈ 𝐵) → 𝑀 ∈ LMod)
6	5	adantr 480	. . . . . . . . . . . 12 ⊢ (((𝑀 ∈ LMod ∧ 𝑋 ∈ 𝐵) ∧ ∀𝑠 ∈ 𝑆 ((𝑠 · 𝑋) = 𝑍 → 𝑠 = 0 )) → 𝑀 ∈ LMod)
7	6	adantr 480	. . . . . . . . . . 11 ⊢ ((((𝑀 ∈ LMod ∧ 𝑋 ∈ 𝐵) ∧ ∀𝑠 ∈ 𝑆 ((𝑠 · 𝑋) = 𝑍 → 𝑠 = 0 )) ∧ 𝑓 ∈ (𝑆 ↑m {𝑋})) → 𝑀 ∈ LMod)
8		snlindsntor.s	. . . . . . . . . . . . . . . 16 ⊢ 𝑆 = (Base‘𝑅)
9		snlindsntor.r	. . . . . . . . . . . . . . . . 17 ⊢ 𝑅 = (Scalar‘𝑀)
10	9	fveq2i 6825	. . . . . . . . . . . . . . . 16 ⊢ (Base‘𝑅) = (Base‘(Scalar‘𝑀))
11	8, 10	eqtri 2752	. . . . . . . . . . . . . . 15 ⊢ 𝑆 = (Base‘(Scalar‘𝑀))
12	11	oveq1i 7359	. . . . . . . . . . . . . 14 ⊢ (𝑆 ↑m {𝑋}) = ((Base‘(Scalar‘𝑀)) ↑m {𝑋})
13	12	eleq2i 2820	. . . . . . . . . . . . 13 ⊢ (𝑓 ∈ (𝑆 ↑m {𝑋}) ↔ 𝑓 ∈ ((Base‘(Scalar‘𝑀)) ↑m {𝑋}))
14	13	biimpi 216	. . . . . . . . . . . 12 ⊢ (𝑓 ∈ (𝑆 ↑m {𝑋}) → 𝑓 ∈ ((Base‘(Scalar‘𝑀)) ↑m {𝑋}))
15	14	adantl 481	. . . . . . . . . . 11 ⊢ ((((𝑀 ∈ LMod ∧ 𝑋 ∈ 𝐵) ∧ ∀𝑠 ∈ 𝑆 ((𝑠 · 𝑋) = 𝑍 → 𝑠 = 0 )) ∧ 𝑓 ∈ (𝑆 ↑m {𝑋})) → 𝑓 ∈ ((Base‘(Scalar‘𝑀)) ↑m {𝑋}))
16		snelpwi 5386	. . . . . . . . . . . . 13 ⊢ (𝑋 ∈ (Base‘𝑀) → {𝑋} ∈ 𝒫 (Base‘𝑀))
17		snlindsntor.b	. . . . . . . . . . . . 13 ⊢ 𝐵 = (Base‘𝑀)
18	16, 17	eleq2s 2846	. . . . . . . . . . . 12 ⊢ (𝑋 ∈ 𝐵 → {𝑋} ∈ 𝒫 (Base‘𝑀))
19	18	ad3antlr 731	. . . . . . . . . . 11 ⊢ ((((𝑀 ∈ LMod ∧ 𝑋 ∈ 𝐵) ∧ ∀𝑠 ∈ 𝑆 ((𝑠 · 𝑋) = 𝑍 → 𝑠 = 0 )) ∧ 𝑓 ∈ (𝑆 ↑m {𝑋})) → {𝑋} ∈ 𝒫 (Base‘𝑀))
20		lincval 48398	. . . . . . . . . . 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))))
21	7, 15, 19, 20	syl3anc 1373	. . . . . . . . . 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))))
22	21	eqeq1d 2731	. . . . . . . . 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))
23	22	anbi2d 630	. . . . . . . 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)))
24		lmodgrp 20770	. . . . . . . . . . . . . 14 ⊢ (𝑀 ∈ LMod → 𝑀 ∈ Grp)
25	24	grpmndd 18825	. . . . . . . . . . . . 13 ⊢ (𝑀 ∈ LMod → 𝑀 ∈ Mnd)
26	25	ad3antrrr 730	. . . . . . . . . . . 12 ⊢ ((((𝑀 ∈ LMod ∧ 𝑋 ∈ 𝐵) ∧ ∀𝑠 ∈ 𝑆 ((𝑠 · 𝑋) = 𝑍 → 𝑠 = 0 )) ∧ 𝑓 ∈ (𝑆 ↑m {𝑋})) → 𝑀 ∈ Mnd)
27		simpllr 775	. . . . . . . . . . . 12 ⊢ ((((𝑀 ∈ LMod ∧ 𝑋 ∈ 𝐵) ∧ ∀𝑠 ∈ 𝑆 ((𝑠 · 𝑋) = 𝑍 → 𝑠 = 0 )) ∧ 𝑓 ∈ (𝑆 ↑m {𝑋})) → 𝑋 ∈ 𝐵)
28		elmapi 8776	. . . . . . . . . . . . . 14 ⊢ (𝑓 ∈ (𝑆 ↑m {𝑋}) → 𝑓:{𝑋}⟶𝑆)
29	6	adantl 481	. . . . . . . . . . . . . . . 16 ⊢ ((𝑓:{𝑋}⟶𝑆 ∧ ((𝑀 ∈ LMod ∧ 𝑋 ∈ 𝐵) ∧ ∀𝑠 ∈ 𝑆 ((𝑠 · 𝑋) = 𝑍 → 𝑠 = 0 ))) → 𝑀 ∈ LMod)
30		snidg 4612	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑋 ∈ 𝐵 → 𝑋 ∈ {𝑋})
31	30	adantl 481	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑀 ∈ LMod ∧ 𝑋 ∈ 𝐵) → 𝑋 ∈ {𝑋})
32	31	adantr 480	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑀 ∈ LMod ∧ 𝑋 ∈ 𝐵) ∧ ∀𝑠 ∈ 𝑆 ((𝑠 · 𝑋) = 𝑍 → 𝑠 = 0 )) → 𝑋 ∈ {𝑋})
33		ffvelcdm 7015	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑓:{𝑋}⟶𝑆 ∧ 𝑋 ∈ {𝑋}) → (𝑓‘𝑋) ∈ 𝑆)
34	32, 33	sylan2 593	. . . . . . . . . . . . . . . 16 ⊢ ((𝑓:{𝑋}⟶𝑆 ∧ ((𝑀 ∈ LMod ∧ 𝑋 ∈ 𝐵) ∧ ∀𝑠 ∈ 𝑆 ((𝑠 · 𝑋) = 𝑍 → 𝑠 = 0 ))) → (𝑓‘𝑋) ∈ 𝑆)
35		simprlr 779	. . . . . . . . . . . . . . . 16 ⊢ ((𝑓:{𝑋}⟶𝑆 ∧ ((𝑀 ∈ LMod ∧ 𝑋 ∈ 𝐵) ∧ ∀𝑠 ∈ 𝑆 ((𝑠 · 𝑋) = 𝑍 → 𝑠 = 0 ))) → 𝑋 ∈ 𝐵)
36		eqid 2729	. . . . . . . . . . . . . . . . 17 ⊢ ( ·𝑠 ‘𝑀) = ( ·𝑠 ‘𝑀)
37	17, 9, 36, 8	lmodvscl 20781	. . . . . . . . . . . . . . . 16 ⊢ ((𝑀 ∈ LMod ∧ (𝑓‘𝑋) ∈ 𝑆 ∧ 𝑋 ∈ 𝐵) → ((𝑓‘𝑋)( ·𝑠 ‘𝑀)𝑋) ∈ 𝐵)
38	29, 34, 35, 37	syl3anc 1373	. . . . . . . . . . . . . . 15 ⊢ ((𝑓:{𝑋}⟶𝑆 ∧ ((𝑀 ∈ LMod ∧ 𝑋 ∈ 𝐵) ∧ ∀𝑠 ∈ 𝑆 ((𝑠 · 𝑋) = 𝑍 → 𝑠 = 0 ))) → ((𝑓‘𝑋)( ·𝑠 ‘𝑀)𝑋) ∈ 𝐵)
39	38	expcom 413	. . . . . . . . . . . . . 14 ⊢ (((𝑀 ∈ LMod ∧ 𝑋 ∈ 𝐵) ∧ ∀𝑠 ∈ 𝑆 ((𝑠 · 𝑋) = 𝑍 → 𝑠 = 0 )) → (𝑓:{𝑋}⟶𝑆 → ((𝑓‘𝑋)( ·𝑠 ‘𝑀)𝑋) ∈ 𝐵))
40	28, 39	syl5com 31	. . . . . . . . . . . . 13 ⊢ (𝑓 ∈ (𝑆 ↑m {𝑋}) → (((𝑀 ∈ LMod ∧ 𝑋 ∈ 𝐵) ∧ ∀𝑠 ∈ 𝑆 ((𝑠 · 𝑋) = 𝑍 → 𝑠 = 0 )) → ((𝑓‘𝑋)( ·𝑠 ‘𝑀)𝑋) ∈ 𝐵))
41	40	impcom 407	. . . . . . . . . . . 12 ⊢ ((((𝑀 ∈ LMod ∧ 𝑋 ∈ 𝐵) ∧ ∀𝑠 ∈ 𝑆 ((𝑠 · 𝑋) = 𝑍 → 𝑠 = 0 )) ∧ 𝑓 ∈ (𝑆 ↑m {𝑋})) → ((𝑓‘𝑋)( ·𝑠 ‘𝑀)𝑋) ∈ 𝐵)
42		fveq2 6822	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑋 → (𝑓‘𝑥) = (𝑓‘𝑋))
43		id 22	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑋 → 𝑥 = 𝑋)
44	42, 43	oveq12d 7367	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑋 → ((𝑓‘𝑥)( ·𝑠 ‘𝑀)𝑥) = ((𝑓‘𝑋)( ·𝑠 ‘𝑀)𝑋))
45	17, 44	gsumsn 19833	. . . . . . . . . . . 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))
46	26, 27, 41, 45	syl3anc 1373	. . . . . . . . . . 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))
47	46	eqeq1d 2731	. . . . . . . . . 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))
48	30, 33	sylan2 593	. . . . . . . . . . . . . . 15 ⊢ ((𝑓:{𝑋}⟶𝑆 ∧ 𝑋 ∈ 𝐵) → (𝑓‘𝑋) ∈ 𝑆)
49	48	expcom 413	. . . . . . . . . . . . . 14 ⊢ (𝑋 ∈ 𝐵 → (𝑓:{𝑋}⟶𝑆 → (𝑓‘𝑋) ∈ 𝑆))
50	49	adantl 481	. . . . . . . . . . . . 13 ⊢ ((𝑀 ∈ LMod ∧ 𝑋 ∈ 𝐵) → (𝑓:{𝑋}⟶𝑆 → (𝑓‘𝑋) ∈ 𝑆))
51		snlindsntor.t	. . . . . . . . . . . . . . . . 17 ⊢ · = ( ·𝑠 ‘𝑀)
52	51	oveqi 7362	. . . . . . . . . . . . . . . 16 ⊢ ((𝑓‘𝑋) · 𝑋) = ((𝑓‘𝑋)( ·𝑠 ‘𝑀)𝑋)
53	52	eqeq1i 2734	. . . . . . . . . . . . . . 15 ⊢ (((𝑓‘𝑋) · 𝑋) = 𝑍 ↔ ((𝑓‘𝑋)( ·𝑠 ‘𝑀)𝑋) = 𝑍)
54		oveq1 7356	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑠 = (𝑓‘𝑋) → (𝑠 · 𝑋) = ((𝑓‘𝑋) · 𝑋))
55	54	eqeq1d 2731	. . . . . . . . . . . . . . . . 17 ⊢ (𝑠 = (𝑓‘𝑋) → ((𝑠 · 𝑋) = 𝑍 ↔ ((𝑓‘𝑋) · 𝑋) = 𝑍))
56		eqeq1 2733	. . . . . . . . . . . . . . . . 17 ⊢ (𝑠 = (𝑓‘𝑋) → (𝑠 = 0 ↔ (𝑓‘𝑋) = 0 ))
57	55, 56	imbi12d 344	. . . . . . . . . . . . . . . 16 ⊢ (𝑠 = (𝑓‘𝑋) → (((𝑠 · 𝑋) = 𝑍 → 𝑠 = 0 ) ↔ (((𝑓‘𝑋) · 𝑋) = 𝑍 → (𝑓‘𝑋) = 0 )))
58	57	rspcva 3575	. . . . . . . . . . . . . . 15 ⊢ (((𝑓‘𝑋) ∈ 𝑆 ∧ ∀𝑠 ∈ 𝑆 ((𝑠 · 𝑋) = 𝑍 → 𝑠 = 0 )) → (((𝑓‘𝑋) · 𝑋) = 𝑍 → (𝑓‘𝑋) = 0 ))
59	53, 58	biimtrrid 243	. . . . . . . . . . . . . 14 ⊢ (((𝑓‘𝑋) ∈ 𝑆 ∧ ∀𝑠 ∈ 𝑆 ((𝑠 · 𝑋) = 𝑍 → 𝑠 = 0 )) → (((𝑓‘𝑋)( ·𝑠 ‘𝑀)𝑋) = 𝑍 → (𝑓‘𝑋) = 0 ))
60	59	ex 412	. . . . . . . . . . . . 13 ⊢ ((𝑓‘𝑋) ∈ 𝑆 → (∀𝑠 ∈ 𝑆 ((𝑠 · 𝑋) = 𝑍 → 𝑠 = 0 ) → (((𝑓‘𝑋)( ·𝑠 ‘𝑀)𝑋) = 𝑍 → (𝑓‘𝑋) = 0 )))
61	28, 50, 60	syl56 36	. . . . . . . . . . . 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 ))))
62	61	com23 86	. . . . . . . . . . 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 ))))
63	62	imp31 417	. . . . . . . . . 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 ))
64	47, 63	sylbid 240	. . . . . . . . 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 ))
65	64	adantld 490	. . . . . . . 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 ))
66	23, 65	sylbid 240	. . . . . . 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 ))
67	66	ralrimiva 3121	. . . . . 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 ))
68	67	ex 412	. . . . 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 )))
69		impexp 450	. . . . . . . 8 ⊢ (((𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀){𝑋}) = 𝑍) → (𝑓‘𝑋) = 0 ) ↔ (𝑓 finSupp 0 → ((𝑓( linC ‘𝑀){𝑋}) = 𝑍 → (𝑓‘𝑋) = 0 )))
70	28	adantl 481	. . . . . . . . . 10 ⊢ (((𝑀 ∈ LMod ∧ 𝑋 ∈ 𝐵) ∧ 𝑓 ∈ (𝑆 ↑m {𝑋})) → 𝑓:{𝑋}⟶𝑆)
71		snfi 8968	. . . . . . . . . . 11 ⊢ {𝑋} ∈ Fin
72	71	a1i 11	. . . . . . . . . 10 ⊢ (((𝑀 ∈ LMod ∧ 𝑋 ∈ 𝐵) ∧ 𝑓 ∈ (𝑆 ↑m {𝑋})) → {𝑋} ∈ Fin)
73		snlindsntor.0	. . . . . . . . . . . 12 ⊢ 0 = (0g‘𝑅)
74	73	fvexi 6836	. . . . . . . . . . 11 ⊢ 0 ∈ V
75	74	a1i 11	. . . . . . . . . 10 ⊢ (((𝑀 ∈ LMod ∧ 𝑋 ∈ 𝐵) ∧ 𝑓 ∈ (𝑆 ↑m {𝑋})) → 0 ∈ V)
76	70, 72, 75	fdmfifsupp 9265	. . . . . . . . 9 ⊢ (((𝑀 ∈ LMod ∧ 𝑋 ∈ 𝐵) ∧ 𝑓 ∈ (𝑆 ↑m {𝑋})) → 𝑓 finSupp 0 )
77		pm2.27 42	. . . . . . . . 9 ⊢ (𝑓 finSupp 0 → ((𝑓 finSupp 0 → ((𝑓( linC ‘𝑀){𝑋}) = 𝑍 → (𝑓‘𝑋) = 0 )) → ((𝑓( linC ‘𝑀){𝑋}) = 𝑍 → (𝑓‘𝑋) = 0 )))
78	76, 77	syl 17	. . . . . . . 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 )))
79	69, 78	biimtrid 242	. . . . . . 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 )))
80	79	ralimdva 3141	. . . . . 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 )))
81		snlindsntor.z	. . . . . . 7 ⊢ 𝑍 = (0g‘𝑀)
82	17, 9, 8, 73, 81, 51	snlindsntorlem 48459	. . . . . 6 ⊢ ((𝑀 ∈ LMod ∧ 𝑋 ∈ 𝐵) → (∀𝑓 ∈ (𝑆 ↑m {𝑋})((𝑓( linC ‘𝑀){𝑋}) = 𝑍 → (𝑓‘𝑋) = 0 ) → ∀𝑠 ∈ 𝑆 ((𝑠 · 𝑋) = 𝑍 → 𝑠 = 0 )))
83	80, 82	syld 47	. . . . 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 )))
84	68, 83	impbid 212	. . . 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 )))
85		fveqeq2 6831	. . . . . . . . 9 ⊢ (𝑦 = 𝑋 → ((𝑓‘𝑦) = 0 ↔ (𝑓‘𝑋) = 0 ))
86	85	ralsng 4627	. . . . . . . 8 ⊢ (𝑋 ∈ 𝐵 → (∀𝑦 ∈ {𝑋} (𝑓‘𝑦) = 0 ↔ (𝑓‘𝑋) = 0 ))
87	86	adantl 481	. . . . . . 7 ⊢ ((𝑀 ∈ LMod ∧ 𝑋 ∈ 𝐵) → (∀𝑦 ∈ {𝑋} (𝑓‘𝑦) = 0 ↔ (𝑓‘𝑋) = 0 ))
88	87	bicomd 223	. . . . . 6 ⊢ ((𝑀 ∈ LMod ∧ 𝑋 ∈ 𝐵) → ((𝑓‘𝑋) = 0 ↔ ∀𝑦 ∈ {𝑋} (𝑓‘𝑦) = 0 ))
89	88	imbi2d 340	. . . . 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 )))
90	89	ralbidv 3152	. . . 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 )))
91		snelpwi 5386	. . . . . 6 ⊢ (𝑋 ∈ 𝐵 → {𝑋} ∈ 𝒫 𝐵)
92	91	adantl 481	. . . . 5 ⊢ ((𝑀 ∈ LMod ∧ 𝑋 ∈ 𝐵) → {𝑋} ∈ 𝒫 𝐵)
93	92	biantrurd 532	. . . 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 ))))
94	84, 90, 93	3bitrd 305	. . 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 ))))
95	4, 94	bitrid 283	. 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 ))))
96		snex 5375	. . 3 ⊢ {𝑋} ∈ V
97	17, 81, 9, 8, 73	islininds 48435	. . 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 ))))
98	96, 5, 97	sylancr 587	. 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 ))))
99	95, 98	bitr4d 282	1 ⊢ ((𝑀 ∈ LMod ∧ 𝑋 ∈ 𝐵) → (∀𝑠 ∈ (𝑆 ∖ { 0 })(𝑠 · 𝑋) ≠ 𝑍 ↔ {𝑋} linIndS 𝑀))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109   ≠ wne 2925  ∀wral 3044  Vcvv 3436   ∖ cdif 3900  𝒫 cpw 4551  {csn 4577   class class class wbr 5092   ↦ cmpt 5173  ⟶wf 6478  ‘cfv 6482  (class class class)co 7349   ↑_m cmap 8753  Fincfn 8872   finSupp cfsupp 9251  Basecbs 17120  Scalarcsca 17164   ·_𝑠 cvsca 17165  0_gc0g 17343   Σ_g cgsu 17344  Mndcmnd 18608  LModclmod 20763   linC clinc 48393   linIndS clininds 48429

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5218  ax-sep 5235  ax-nul 5245  ax-pow 5304  ax-pr 5371  ax-un 7671  ax-cnex 11065  ax-resscn 11066  ax-1cn 11067  ax-icn 11068  ax-addcl 11069  ax-addrcl 11070  ax-mulcl 11071  ax-mulrcl 11072  ax-mulcom 11073  ax-addass 11074  ax-mulass 11075  ax-distr 11076  ax-i2m1 11077  ax-1ne0 11078  ax-1rid 11079  ax-rnegex 11080  ax-rrecex 11081  ax-cnre 11082  ax-pre-lttri 11083  ax-pre-lttrn 11084  ax-pre-ltadd 11085  ax-pre-mulgt0 11086

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-nel 3030  df-ral 3045  df-rex 3054  df-rmo 3343  df-reu 3344  df-rab 3395  df-v 3438  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4285  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4859  df-int 4897  df-iun 4943  df-br 5093  df-opab 5155  df-mpt 5174  df-tr 5200  df-id 5514  df-eprel 5519  df-po 5527  df-so 5528  df-fr 5572  df-se 5573  df-we 5574  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-pred 6249  df-ord 6310  df-on 6311  df-lim 6312  df-suc 6313  df-iota 6438  df-fun 6484  df-fn 6485  df-f 6486  df-f1 6487  df-fo 6488  df-f1o 6489  df-fv 6490  df-isom 6491  df-riota 7306  df-ov 7352  df-oprab 7353  df-mpo 7354  df-om 7800  df-1st 7924  df-2nd 7925  df-supp 8094  df-frecs 8214  df-wrecs 8245  df-recs 8294  df-rdg 8332  df-1o 8388  df-er 8625  df-map 8755  df-en 8873  df-dom 8874  df-sdom 8875  df-fin 8876  df-fsupp 9252  df-oi 9402  df-card 9835  df-pnf 11151  df-mnf 11152  df-xr 11153  df-ltxr 11154  df-le 11155  df-sub 11349  df-neg 11350  df-nn 12129  df-n0 12385  df-z 12472  df-uz 12736  df-fz 13411  df-fzo 13558  df-seq 13909  df-hash 14238  df-0g 17345  df-gsum 17346  df-mgm 18514  df-sgrp 18593  df-mnd 18609  df-grp 18815  df-mulg 18947  df-cntz 19196  df-lmod 20765  df-linc 48395  df-lininds 48431

This theorem is referenced by:  lindssnlvec  48475