Theorem snlindsntor 45700

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 2943	. . . . 5 ⊢ ((𝑠 · 𝑋) ≠ 𝑍 ↔ ¬ (𝑠 · 𝑋) = 𝑍)
2	1	ralbii 3090	. . . 4 ⊢ (∀𝑠 ∈ (𝑆 ∖ { 0 })(𝑠 · 𝑋) ≠ 𝑍 ↔ ∀𝑠 ∈ (𝑆 ∖ { 0 }) ¬ (𝑠 · 𝑋) = 𝑍)
3		raldifsni 4725	. . . 4 ⊢ (∀𝑠 ∈ (𝑆 ∖ { 0 }) ¬ (𝑠 · 𝑋) = 𝑍 ↔ ∀𝑠 ∈ 𝑆 ((𝑠 · 𝑋) = 𝑍 → 𝑠 = 0 ))
4	2, 3	bitri 274	. . 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 6759	. . . . . . . . . . . . . . . 16 ⊢ (Base‘𝑅) = (Base‘(Scalar‘𝑀))
11	8, 10	eqtri 2766	. . . . . . . . . . . . . . 15 ⊢ 𝑆 = (Base‘(Scalar‘𝑀))
12	11	oveq1i 7265	. . . . . . . . . . . . . 14 ⊢ (𝑆 ↑m {𝑋}) = ((Base‘(Scalar‘𝑀)) ↑m {𝑋})
13	12	eleq2i 2830	. . . . . . . . . . . . 13 ⊢ (𝑓 ∈ (𝑆 ↑m {𝑋}) ↔ 𝑓 ∈ ((Base‘(Scalar‘𝑀)) ↑m {𝑋}))
14	13	biimpi 215	. . . . . . . . . . . 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 5354	. . . . . . . . . . . . 13 ⊢ (𝑋 ∈ (Base‘𝑀) → {𝑋} ∈ 𝒫 (Base‘𝑀))
17		snlindsntor.b	. . . . . . . . . . . . 13 ⊢ 𝐵 = (Base‘𝑀)
18	16, 17	eleq2s 2857	. . . . . . . . . . . 12 ⊢ (𝑋 ∈ 𝐵 → {𝑋} ∈ 𝒫 (Base‘𝑀))
19	18	ad3antlr 727	. . . . . . . . . . 11 ⊢ ((((𝑀 ∈ LMod ∧ 𝑋 ∈ 𝐵) ∧ ∀𝑠 ∈ 𝑆 ((𝑠 · 𝑋) = 𝑍 → 𝑠 = 0 )) ∧ 𝑓 ∈ (𝑆 ↑m {𝑋})) → {𝑋} ∈ 𝒫 (Base‘𝑀))
20		lincval 45638	. . . . . . . . . . 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 1369	. . . . . . . . . 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 2740	. . . . . . . . 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 628	. . . . . . . 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 20045	. . . . . . . . . . . . . 14 ⊢ (𝑀 ∈ LMod → 𝑀 ∈ Grp)
25	24	grpmndd 18504	. . . . . . . . . . . . 13 ⊢ (𝑀 ∈ LMod → 𝑀 ∈ Mnd)
26	25	ad3antrrr 726	. . . . . . . . . . . 12 ⊢ ((((𝑀 ∈ LMod ∧ 𝑋 ∈ 𝐵) ∧ ∀𝑠 ∈ 𝑆 ((𝑠 · 𝑋) = 𝑍 → 𝑠 = 0 )) ∧ 𝑓 ∈ (𝑆 ↑m {𝑋})) → 𝑀 ∈ Mnd)
27		simpllr 772	. . . . . . . . . . . 12 ⊢ ((((𝑀 ∈ LMod ∧ 𝑋 ∈ 𝐵) ∧ ∀𝑠 ∈ 𝑆 ((𝑠 · 𝑋) = 𝑍 → 𝑠 = 0 )) ∧ 𝑓 ∈ (𝑆 ↑m {𝑋})) → 𝑋 ∈ 𝐵)
28		elmapi 8595	. . . . . . . . . . . . . 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 4592	. . . . . . . . . . . . . . . . . . 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		ffvelrn 6941	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑓:{𝑋}⟶𝑆 ∧ 𝑋 ∈ {𝑋}) → (𝑓‘𝑋) ∈ 𝑆)
34	32, 33	sylan2 592	. . . . . . . . . . . . . . . 16 ⊢ ((𝑓:{𝑋}⟶𝑆 ∧ ((𝑀 ∈ LMod ∧ 𝑋 ∈ 𝐵) ∧ ∀𝑠 ∈ 𝑆 ((𝑠 · 𝑋) = 𝑍 → 𝑠 = 0 ))) → (𝑓‘𝑋) ∈ 𝑆)
35		simprlr 776	. . . . . . . . . . . . . . . 16 ⊢ ((𝑓:{𝑋}⟶𝑆 ∧ ((𝑀 ∈ LMod ∧ 𝑋 ∈ 𝐵) ∧ ∀𝑠 ∈ 𝑆 ((𝑠 · 𝑋) = 𝑍 → 𝑠 = 0 ))) → 𝑋 ∈ 𝐵)
36		eqid 2738	. . . . . . . . . . . . . . . . 17 ⊢ ( ·𝑠 ‘𝑀) = ( ·𝑠 ‘𝑀)
37	17, 9, 36, 8	lmodvscl 20055	. . . . . . . . . . . . . . . 16 ⊢ ((𝑀 ∈ LMod ∧ (𝑓‘𝑋) ∈ 𝑆 ∧ 𝑋 ∈ 𝐵) → ((𝑓‘𝑋)( ·𝑠 ‘𝑀)𝑋) ∈ 𝐵)
38	29, 34, 35, 37	syl3anc 1369	. . . . . . . . . . . . . . 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 6756	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑋 → (𝑓‘𝑥) = (𝑓‘𝑋))
43		id 22	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑋 → 𝑥 = 𝑋)
44	42, 43	oveq12d 7273	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑋 → ((𝑓‘𝑥)( ·𝑠 ‘𝑀)𝑥) = ((𝑓‘𝑋)( ·𝑠 ‘𝑀)𝑋))
45	17, 44	gsumsn 19470	. . . . . . . . . . . 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 1369	. . . . . . . . . . 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 2740	. . . . . . . . . 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 592	. . . . . . . . . . . . . . 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 7268	. . . . . . . . . . . . . . . 16 ⊢ ((𝑓‘𝑋) · 𝑋) = ((𝑓‘𝑋)( ·𝑠 ‘𝑀)𝑋)
53	52	eqeq1i 2743	. . . . . . . . . . . . . . 15 ⊢ (((𝑓‘𝑋) · 𝑋) = 𝑍 ↔ ((𝑓‘𝑋)( ·𝑠 ‘𝑀)𝑋) = 𝑍)
54		oveq1 7262	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑠 = (𝑓‘𝑋) → (𝑠 · 𝑋) = ((𝑓‘𝑋) · 𝑋))
55	54	eqeq1d 2740	. . . . . . . . . . . . . . . . 17 ⊢ (𝑠 = (𝑓‘𝑋) → ((𝑠 · 𝑋) = 𝑍 ↔ ((𝑓‘𝑋) · 𝑋) = 𝑍))
56		eqeq1 2742	. . . . . . . . . . . . . . . . 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 3550	. . . . . . . . . . . . . . 15 ⊢ (((𝑓‘𝑋) ∈ 𝑆 ∧ ∀𝑠 ∈ 𝑆 ((𝑠 · 𝑋) = 𝑍 → 𝑠 = 0 )) → (((𝑓‘𝑋) · 𝑋) = 𝑍 → (𝑓‘𝑋) = 0 ))
59	53, 58	syl5bir 242	. . . . . . . . . . . . . 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 239	. . . . . . . . 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 239	. . . . . . 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 3107	. . . . . 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 8788	. . . . . . . . . . 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 6770	. . . . . . . . . . 11 ⊢ 0 ∈ V
75	74	a1i 11	. . . . . . . . . 10 ⊢ (((𝑀 ∈ LMod ∧ 𝑋 ∈ 𝐵) ∧ 𝑓 ∈ (𝑆 ↑m {𝑋})) → 0 ∈ V)
76	70, 72, 75	fdmfifsupp 9068	. . . . . . . . 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	syl5bi 241	. . . . . . 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 3102	. . . . . 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 45699	. . . . . 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 211	. . . 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 6765	. . . . . . . . 9 ⊢ (𝑦 = 𝑋 → ((𝑓‘𝑦) = 0 ↔ (𝑓‘𝑋) = 0 ))
86	85	ralsng 4606	. . . . . . . 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 222	. . . . . 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 3120	. . . 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 5354	. . . . . 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 304	. . 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	syl5bb 282	. 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 5349	. . 3 ⊢ {𝑋} ∈ V
97	17, 81, 9, 8, 73	islininds 45675	. . 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 586	. 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 281	1 ⊢ ((𝑀 ∈ LMod ∧ 𝑋 ∈ 𝐵) → (∀𝑠 ∈ (𝑆 ∖ { 0 })(𝑠 · 𝑋) ≠ 𝑍 ↔ {𝑋} linIndS 𝑀))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 395   = wceq 1539   ∈ wcel 2108   ≠ wne 2942  ∀wral 3063  Vcvv 3422   ∖ cdif 3880  𝒫 cpw 4530  {csn 4558   class class class wbr 5070   ↦ cmpt 5153  ⟶wf 6414  ‘cfv 6418  (class class class)co 7255   ↑_m cmap 8573  Fincfn 8691   finSupp cfsupp 9058  Basecbs 16840  Scalarcsca 16891   ·_𝑠 cvsca 16892  0_gc0g 17067   Σ_g cgsu 17068  Mndcmnd 18300  LModclmod 20038   linC clinc 45633   linIndS clininds 45669

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-rep 5205  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-se 5536  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-isom 6427  df-riota 7212  df-ov 7258  df-oprab 7259  df-mpo 7260  df-om 7688  df-1st 7804  df-2nd 7805  df-supp 7949  df-frecs 8068  df-wrecs 8099  df-recs 8173  df-rdg 8212  df-1o 8267  df-er 8456  df-map 8575  df-en 8692  df-dom 8693  df-sdom 8694  df-fin 8695  df-fsupp 9059  df-oi 9199  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-n0 12164  df-z 12250  df-uz 12512  df-fz 13169  df-fzo 13312  df-seq 13650  df-hash 13973  df-0g 17069  df-gsum 17070  df-mgm 18241  df-sgrp 18290  df-mnd 18301  df-grp 18495  df-mulg 18616  df-cntz 18838  df-lmod 20040  df-linc 45635  df-lininds 45671

This theorem is referenced by:  lindssnlvec  45715