Theorem snlindsntor 48582

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 2929	. . . . 5 ⊢ ((𝑠 · 𝑋) ≠ 𝑍 ↔ ¬ (𝑠 · 𝑋) = 𝑍)
2	1	ralbii 3078	. . . 4 ⊢ (∀𝑠 ∈ (𝑆 ∖ { 0 })(𝑠 · 𝑋) ≠ 𝑍 ↔ ∀𝑠 ∈ (𝑆 ∖ { 0 }) ¬ (𝑠 · 𝑋) = 𝑍)
3		raldifsni 4744	. . . 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 2754	. . . . . . . . . . . . . . 15 ⊢ 𝑆 = (Base‘(Scalar‘𝑀))
12	11	oveq1i 7356	. . . . . . . . . . . . . 14 ⊢ (𝑆 ↑m {𝑋}) = ((Base‘(Scalar‘𝑀)) ↑m {𝑋})
13	12	eleq2i 2823	. . . . . . . . . . . . 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 5383	. . . . . . . . . . . . 13 ⊢ (𝑋 ∈ (Base‘𝑀) → {𝑋} ∈ 𝒫 (Base‘𝑀))
17		snlindsntor.b	. . . . . . . . . . . . 13 ⊢ 𝐵 = (Base‘𝑀)
18	16, 17	eleq2s 2849	. . . . . . . . . . . 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 48520	. . . . . . . . . . 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 2733	. . . . . . . . 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 20800	. . . . . . . . . . . . . 14 ⊢ (𝑀 ∈ LMod → 𝑀 ∈ Grp)
25	24	grpmndd 18859	. . . . . . . . . . . . 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 8773	. . . . . . . . . . . . . 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 4610	. . . . . . . . . . . . . . . . . . 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 7014	. . . . . . . . . . . . . . . . 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 2731	. . . . . . . . . . . . . . . . 17 ⊢ ( ·𝑠 ‘𝑀) = ( ·𝑠 ‘𝑀)
37	17, 9, 36, 8	lmodvscl 20811	. . . . . . . . . . . . . . . 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 7364	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑋 → ((𝑓‘𝑥)( ·𝑠 ‘𝑀)𝑥) = ((𝑓‘𝑋)( ·𝑠 ‘𝑀)𝑋))
45	17, 44	gsumsn 19866	. . . . . . . . . . . 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 2733	. . . . . . . . . 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 7359	. . . . . . . . . . . . . . . 16 ⊢ ((𝑓‘𝑋) · 𝑋) = ((𝑓‘𝑋)( ·𝑠 ‘𝑀)𝑋)
53	52	eqeq1i 2736	. . . . . . . . . . . . . . 15 ⊢ (((𝑓‘𝑋) · 𝑋) = 𝑍 ↔ ((𝑓‘𝑋)( ·𝑠 ‘𝑀)𝑋) = 𝑍)
54		oveq1 7353	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑠 = (𝑓‘𝑋) → (𝑠 · 𝑋) = ((𝑓‘𝑋) · 𝑋))
55	54	eqeq1d 2733	. . . . . . . . . . . . . . . . 17 ⊢ (𝑠 = (𝑓‘𝑋) → ((𝑠 · 𝑋) = 𝑍 ↔ ((𝑓‘𝑋) · 𝑋) = 𝑍))
56		eqeq1 2735	. . . . . . . . . . . . . . . . 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 3570	. . . . . . . . . . . . . . 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 3124	. . . . . 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 8965	. . . . . . . . . . 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 9259	. . . . . . . . 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 3144	. . . . . 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 48581	. . . . . 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 4625	. . . . . . . 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 3155	. . . 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 5383	. . . . . 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 5372	. . 3 ⊢ {𝑋} ∈ V
97	17, 81, 9, 8, 73	islininds 48557	. . 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 1541   ∈ wcel 2111   ≠ wne 2928  ∀wral 3047  Vcvv 3436   ∖ cdif 3894  𝒫 cpw 4547  {csn 4573   class class class wbr 5089   ↦ cmpt 5170  ⟶wf 6477  ‘cfv 6481  (class class class)co 7346   ↑_m cmap 8750  Fincfn 8869   finSupp cfsupp 9245  Basecbs 17120  Scalarcsca 17164   ·_𝑠 cvsca 17165  0_gc0g 17343   Σ_g cgsu 17344  Mndcmnd 18642  LModclmod 20793   linC clinc 48515   linIndS clininds 48551

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-rep 5215  ax-sep 5232  ax-nul 5242  ax-pow 5301  ax-pr 5368  ax-un 7668  ax-cnex 11062  ax-resscn 11063  ax-1cn 11064  ax-icn 11065  ax-addcl 11066  ax-addrcl 11067  ax-mulcl 11068  ax-mulrcl 11069  ax-mulcom 11070  ax-addass 11071  ax-mulass 11072  ax-distr 11073  ax-i2m1 11074  ax-1ne0 11075  ax-1rid 11076  ax-rnegex 11077  ax-rrecex 11078  ax-cnre 11079  ax-pre-lttri 11080  ax-pre-lttrn 11081  ax-pre-ltadd 11082  ax-pre-mulgt0 11083

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-nel 3033  df-ral 3048  df-rex 3057  df-rmo 3346  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3737  df-csb 3846  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-pss 3917  df-nul 4281  df-if 4473  df-pw 4549  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4857  df-int 4896  df-iun 4941  df-br 5090  df-opab 5152  df-mpt 5171  df-tr 5197  df-id 5509  df-eprel 5514  df-po 5522  df-so 5523  df-fr 5567  df-se 5568  df-we 5569  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-res 5626  df-ima 5627  df-pred 6248  df-ord 6309  df-on 6310  df-lim 6311  df-suc 6312  df-iota 6437  df-fun 6483  df-fn 6484  df-f 6485  df-f1 6486  df-fo 6487  df-f1o 6488  df-fv 6489  df-isom 6490  df-riota 7303  df-ov 7349  df-oprab 7350  df-mpo 7351  df-om 7797  df-1st 7921  df-2nd 7922  df-supp 8091  df-frecs 8211  df-wrecs 8242  df-recs 8291  df-rdg 8329  df-1o 8385  df-er 8622  df-map 8752  df-en 8870  df-dom 8871  df-sdom 8872  df-fin 8873  df-fsupp 9246  df-oi 9396  df-card 9832  df-pnf 11148  df-mnf 11149  df-xr 11150  df-ltxr 11151  df-le 11152  df-sub 11346  df-neg 11347  df-nn 12126  df-n0 12382  df-z 12469  df-uz 12733  df-fz 13408  df-fzo 13555  df-seq 13909  df-hash 14238  df-0g 17345  df-gsum 17346  df-mgm 18548  df-sgrp 18627  df-mnd 18643  df-grp 18849  df-mulg 18981  df-cntz 19229  df-lmod 20795  df-linc 48517  df-lininds 48553

This theorem is referenced by:  lindssnlvec  48597