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Theorem snlindsntor 42025
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 𝑎𝑅, 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 = (0g𝑅)
snlindsntor.z 𝑍 = (0g𝑀)
snlindsntor.t · = ( ·𝑠𝑀)
Assertion
Ref Expression
snlindsntor ((𝑀 ∈ LMod ∧ 𝑋𝐵) → (∀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 𝑥 𝑓 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-ne 2792 . . . . 5 ((𝑠 · 𝑋) ≠ 𝑍 ↔ ¬ (𝑠 · 𝑋) = 𝑍)
21ralbii 2977 . . . 4 (∀𝑠 ∈ (𝑆 ∖ { 0 })(𝑠 · 𝑋) ≠ 𝑍 ↔ ∀𝑠 ∈ (𝑆 ∖ { 0 }) ¬ (𝑠 · 𝑋) = 𝑍)
3 raldifsni 4315 . . . 4 (∀𝑠 ∈ (𝑆 ∖ { 0 }) ¬ (𝑠 · 𝑋) = 𝑍 ↔ ∀𝑠𝑆 ((𝑠 · 𝑋) = 𝑍𝑠 = 0 ))
42, 3bitri 264 . . 3 (∀𝑠 ∈ (𝑆 ∖ { 0 })(𝑠 · 𝑋) ≠ 𝑍 ↔ ∀𝑠𝑆 ((𝑠 · 𝑋) = 𝑍𝑠 = 0 ))
5 simpl 473 . . . . . . . . . . . . 13 ((𝑀 ∈ LMod ∧ 𝑋𝐵) → 𝑀 ∈ LMod)
65adantr 481 . . . . . . . . . . . 12 (((𝑀 ∈ LMod ∧ 𝑋𝐵) ∧ ∀𝑠𝑆 ((𝑠 · 𝑋) = 𝑍𝑠 = 0 )) → 𝑀 ∈ LMod)
76adantr 481 . . . . . . . . . . 11 ((((𝑀 ∈ LMod ∧ 𝑋𝐵) ∧ ∀𝑠𝑆 ((𝑠 · 𝑋) = 𝑍𝑠 = 0 )) ∧ 𝑓 ∈ (𝑆𝑚 {𝑋})) → 𝑀 ∈ LMod)
8 snlindsntor.s . . . . . . . . . . . . . . . 16 𝑆 = (Base‘𝑅)
9 snlindsntor.r . . . . . . . . . . . . . . . . 17 𝑅 = (Scalar‘𝑀)
109fveq2i 6181 . . . . . . . . . . . . . . . 16 (Base‘𝑅) = (Base‘(Scalar‘𝑀))
118, 10eqtri 2642 . . . . . . . . . . . . . . 15 𝑆 = (Base‘(Scalar‘𝑀))
1211oveq1i 6645 . . . . . . . . . . . . . 14 (𝑆𝑚 {𝑋}) = ((Base‘(Scalar‘𝑀)) ↑𝑚 {𝑋})
1312eleq2i 2691 . . . . . . . . . . . . 13 (𝑓 ∈ (𝑆𝑚 {𝑋}) ↔ 𝑓 ∈ ((Base‘(Scalar‘𝑀)) ↑𝑚 {𝑋}))
1413biimpi 206 . . . . . . . . . . . 12 (𝑓 ∈ (𝑆𝑚 {𝑋}) → 𝑓 ∈ ((Base‘(Scalar‘𝑀)) ↑𝑚 {𝑋}))
1514adantl 482 . . . . . . . . . . 11 ((((𝑀 ∈ LMod ∧ 𝑋𝐵) ∧ ∀𝑠𝑆 ((𝑠 · 𝑋) = 𝑍𝑠 = 0 )) ∧ 𝑓 ∈ (𝑆𝑚 {𝑋})) → 𝑓 ∈ ((Base‘(Scalar‘𝑀)) ↑𝑚 {𝑋}))
16 snelpwi 4903 . . . . . . . . . . . . 13 (𝑋 ∈ (Base‘𝑀) → {𝑋} ∈ 𝒫 (Base‘𝑀))
17 snlindsntor.b . . . . . . . . . . . . 13 𝐵 = (Base‘𝑀)
1816, 17eleq2s 2717 . . . . . . . . . . . 12 (𝑋𝐵 → {𝑋} ∈ 𝒫 (Base‘𝑀))
1918ad3antlr 766 . . . . . . . . . . 11 ((((𝑀 ∈ LMod ∧ 𝑋𝐵) ∧ ∀𝑠𝑆 ((𝑠 · 𝑋) = 𝑍𝑠 = 0 )) ∧ 𝑓 ∈ (𝑆𝑚 {𝑋})) → {𝑋} ∈ 𝒫 (Base‘𝑀))
20 lincval 41963 . . . . . . . . . . 11 ((𝑀 ∈ LMod ∧ 𝑓 ∈ ((Base‘(Scalar‘𝑀)) ↑𝑚 {𝑋}) ∧ {𝑋} ∈ 𝒫 (Base‘𝑀)) → (𝑓( linC ‘𝑀){𝑋}) = (𝑀 Σg (𝑥 ∈ {𝑋} ↦ ((𝑓𝑥)( ·𝑠𝑀)𝑥))))
217, 15, 19, 20syl3anc 1324 . . . . . . . . . 10 ((((𝑀 ∈ LMod ∧ 𝑋𝐵) ∧ ∀𝑠𝑆 ((𝑠 · 𝑋) = 𝑍𝑠 = 0 )) ∧ 𝑓 ∈ (𝑆𝑚 {𝑋})) → (𝑓( linC ‘𝑀){𝑋}) = (𝑀 Σg (𝑥 ∈ {𝑋} ↦ ((𝑓𝑥)( ·𝑠𝑀)𝑥))))
2221eqeq1d 2622 . . . . . . . . 9 ((((𝑀 ∈ LMod ∧ 𝑋𝐵) ∧ ∀𝑠𝑆 ((𝑠 · 𝑋) = 𝑍𝑠 = 0 )) ∧ 𝑓 ∈ (𝑆𝑚 {𝑋})) → ((𝑓( linC ‘𝑀){𝑋}) = 𝑍 ↔ (𝑀 Σg (𝑥 ∈ {𝑋} ↦ ((𝑓𝑥)( ·𝑠𝑀)𝑥))) = 𝑍))
2322anbi2d 739 . . . . . . . 8 ((((𝑀 ∈ LMod ∧ 𝑋𝐵) ∧ ∀𝑠𝑆 ((𝑠 · 𝑋) = 𝑍𝑠 = 0 )) ∧ 𝑓 ∈ (𝑆𝑚 {𝑋})) → ((𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀){𝑋}) = 𝑍) ↔ (𝑓 finSupp 0 ∧ (𝑀 Σg (𝑥 ∈ {𝑋} ↦ ((𝑓𝑥)( ·𝑠𝑀)𝑥))) = 𝑍)))
24 lmodgrp 18851 . . . . . . . . . . . . . 14 (𝑀 ∈ LMod → 𝑀 ∈ Grp)
25 grpmnd 17410 . . . . . . . . . . . . . 14 (𝑀 ∈ Grp → 𝑀 ∈ Mnd)
2624, 25syl 17 . . . . . . . . . . . . 13 (𝑀 ∈ LMod → 𝑀 ∈ Mnd)
2726ad3antrrr 765 . . . . . . . . . . . 12 ((((𝑀 ∈ LMod ∧ 𝑋𝐵) ∧ ∀𝑠𝑆 ((𝑠 · 𝑋) = 𝑍𝑠 = 0 )) ∧ 𝑓 ∈ (𝑆𝑚 {𝑋})) → 𝑀 ∈ Mnd)
28 simpllr 798 . . . . . . . . . . . 12 ((((𝑀 ∈ LMod ∧ 𝑋𝐵) ∧ ∀𝑠𝑆 ((𝑠 · 𝑋) = 𝑍𝑠 = 0 )) ∧ 𝑓 ∈ (𝑆𝑚 {𝑋})) → 𝑋𝐵)
29 elmapi 7864 . . . . . . . . . . . . . 14 (𝑓 ∈ (𝑆𝑚 {𝑋}) → 𝑓:{𝑋}⟶𝑆)
306adantl 482 . . . . . . . . . . . . . . . 16 ((𝑓:{𝑋}⟶𝑆 ∧ ((𝑀 ∈ LMod ∧ 𝑋𝐵) ∧ ∀𝑠𝑆 ((𝑠 · 𝑋) = 𝑍𝑠 = 0 ))) → 𝑀 ∈ LMod)
31 snidg 4197 . . . . . . . . . . . . . . . . . . 19 (𝑋𝐵𝑋 ∈ {𝑋})
3231adantl 482 . . . . . . . . . . . . . . . . . 18 ((𝑀 ∈ LMod ∧ 𝑋𝐵) → 𝑋 ∈ {𝑋})
3332adantr 481 . . . . . . . . . . . . . . . . 17 (((𝑀 ∈ LMod ∧ 𝑋𝐵) ∧ ∀𝑠𝑆 ((𝑠 · 𝑋) = 𝑍𝑠 = 0 )) → 𝑋 ∈ {𝑋})
34 ffvelrn 6343 . . . . . . . . . . . . . . . . 17 ((𝑓:{𝑋}⟶𝑆𝑋 ∈ {𝑋}) → (𝑓𝑋) ∈ 𝑆)
3533, 34sylan2 491 . . . . . . . . . . . . . . . 16 ((𝑓:{𝑋}⟶𝑆 ∧ ((𝑀 ∈ LMod ∧ 𝑋𝐵) ∧ ∀𝑠𝑆 ((𝑠 · 𝑋) = 𝑍𝑠 = 0 ))) → (𝑓𝑋) ∈ 𝑆)
36 simprlr 802 . . . . . . . . . . . . . . . 16 ((𝑓:{𝑋}⟶𝑆 ∧ ((𝑀 ∈ LMod ∧ 𝑋𝐵) ∧ ∀𝑠𝑆 ((𝑠 · 𝑋) = 𝑍𝑠 = 0 ))) → 𝑋𝐵)
37 eqid 2620 . . . . . . . . . . . . . . . . 17 ( ·𝑠𝑀) = ( ·𝑠𝑀)
3817, 9, 37, 8lmodvscl 18861 . . . . . . . . . . . . . . . 16 ((𝑀 ∈ LMod ∧ (𝑓𝑋) ∈ 𝑆𝑋𝐵) → ((𝑓𝑋)( ·𝑠𝑀)𝑋) ∈ 𝐵)
3930, 35, 36, 38syl3anc 1324 . . . . . . . . . . . . . . 15 ((𝑓:{𝑋}⟶𝑆 ∧ ((𝑀 ∈ LMod ∧ 𝑋𝐵) ∧ ∀𝑠𝑆 ((𝑠 · 𝑋) = 𝑍𝑠 = 0 ))) → ((𝑓𝑋)( ·𝑠𝑀)𝑋) ∈ 𝐵)
4039expcom 451 . . . . . . . . . . . . . 14 (((𝑀 ∈ LMod ∧ 𝑋𝐵) ∧ ∀𝑠𝑆 ((𝑠 · 𝑋) = 𝑍𝑠 = 0 )) → (𝑓:{𝑋}⟶𝑆 → ((𝑓𝑋)( ·𝑠𝑀)𝑋) ∈ 𝐵))
4129, 40syl5com 31 . . . . . . . . . . . . 13 (𝑓 ∈ (𝑆𝑚 {𝑋}) → (((𝑀 ∈ LMod ∧ 𝑋𝐵) ∧ ∀𝑠𝑆 ((𝑠 · 𝑋) = 𝑍𝑠 = 0 )) → ((𝑓𝑋)( ·𝑠𝑀)𝑋) ∈ 𝐵))
4241impcom 446 . . . . . . . . . . . 12 ((((𝑀 ∈ LMod ∧ 𝑋𝐵) ∧ ∀𝑠𝑆 ((𝑠 · 𝑋) = 𝑍𝑠 = 0 )) ∧ 𝑓 ∈ (𝑆𝑚 {𝑋})) → ((𝑓𝑋)( ·𝑠𝑀)𝑋) ∈ 𝐵)
43 fveq2 6178 . . . . . . . . . . . . . 14 (𝑥 = 𝑋 → (𝑓𝑥) = (𝑓𝑋))
44 id 22 . . . . . . . . . . . . . 14 (𝑥 = 𝑋𝑥 = 𝑋)
4543, 44oveq12d 6653 . . . . . . . . . . . . 13 (𝑥 = 𝑋 → ((𝑓𝑥)( ·𝑠𝑀)𝑥) = ((𝑓𝑋)( ·𝑠𝑀)𝑋))
4617, 45gsumsn 18335 . . . . . . . . . . . 12 ((𝑀 ∈ Mnd ∧ 𝑋𝐵 ∧ ((𝑓𝑋)( ·𝑠𝑀)𝑋) ∈ 𝐵) → (𝑀 Σg (𝑥 ∈ {𝑋} ↦ ((𝑓𝑥)( ·𝑠𝑀)𝑥))) = ((𝑓𝑋)( ·𝑠𝑀)𝑋))
4727, 28, 42, 46syl3anc 1324 . . . . . . . . . . 11 ((((𝑀 ∈ LMod ∧ 𝑋𝐵) ∧ ∀𝑠𝑆 ((𝑠 · 𝑋) = 𝑍𝑠 = 0 )) ∧ 𝑓 ∈ (𝑆𝑚 {𝑋})) → (𝑀 Σg (𝑥 ∈ {𝑋} ↦ ((𝑓𝑥)( ·𝑠𝑀)𝑥))) = ((𝑓𝑋)( ·𝑠𝑀)𝑋))
4847eqeq1d 2622 . . . . . . . . . 10 ((((𝑀 ∈ LMod ∧ 𝑋𝐵) ∧ ∀𝑠𝑆 ((𝑠 · 𝑋) = 𝑍𝑠 = 0 )) ∧ 𝑓 ∈ (𝑆𝑚 {𝑋})) → ((𝑀 Σg (𝑥 ∈ {𝑋} ↦ ((𝑓𝑥)( ·𝑠𝑀)𝑥))) = 𝑍 ↔ ((𝑓𝑋)( ·𝑠𝑀)𝑋) = 𝑍))
4931, 34sylan2 491 . . . . . . . . . . . . . . 15 ((𝑓:{𝑋}⟶𝑆𝑋𝐵) → (𝑓𝑋) ∈ 𝑆)
5049expcom 451 . . . . . . . . . . . . . 14 (𝑋𝐵 → (𝑓:{𝑋}⟶𝑆 → (𝑓𝑋) ∈ 𝑆))
5150adantl 482 . . . . . . . . . . . . 13 ((𝑀 ∈ LMod ∧ 𝑋𝐵) → (𝑓:{𝑋}⟶𝑆 → (𝑓𝑋) ∈ 𝑆))
52 snlindsntor.t . . . . . . . . . . . . . . . . 17 · = ( ·𝑠𝑀)
5352oveqi 6648 . . . . . . . . . . . . . . . 16 ((𝑓𝑋) · 𝑋) = ((𝑓𝑋)( ·𝑠𝑀)𝑋)
5453eqeq1i 2625 . . . . . . . . . . . . . . 15 (((𝑓𝑋) · 𝑋) = 𝑍 ↔ ((𝑓𝑋)( ·𝑠𝑀)𝑋) = 𝑍)
55 oveq1 6642 . . . . . . . . . . . . . . . . . 18 (𝑠 = (𝑓𝑋) → (𝑠 · 𝑋) = ((𝑓𝑋) · 𝑋))
5655eqeq1d 2622 . . . . . . . . . . . . . . . . 17 (𝑠 = (𝑓𝑋) → ((𝑠 · 𝑋) = 𝑍 ↔ ((𝑓𝑋) · 𝑋) = 𝑍))
57 eqeq1 2624 . . . . . . . . . . . . . . . . 17 (𝑠 = (𝑓𝑋) → (𝑠 = 0 ↔ (𝑓𝑋) = 0 ))
5856, 57imbi12d 334 . . . . . . . . . . . . . . . 16 (𝑠 = (𝑓𝑋) → (((𝑠 · 𝑋) = 𝑍𝑠 = 0 ) ↔ (((𝑓𝑋) · 𝑋) = 𝑍 → (𝑓𝑋) = 0 )))
5958rspcva 3302 . . . . . . . . . . . . . . 15 (((𝑓𝑋) ∈ 𝑆 ∧ ∀𝑠𝑆 ((𝑠 · 𝑋) = 𝑍𝑠 = 0 )) → (((𝑓𝑋) · 𝑋) = 𝑍 → (𝑓𝑋) = 0 ))
6054, 59syl5bir 233 . . . . . . . . . . . . . 14 (((𝑓𝑋) ∈ 𝑆 ∧ ∀𝑠𝑆 ((𝑠 · 𝑋) = 𝑍𝑠 = 0 )) → (((𝑓𝑋)( ·𝑠𝑀)𝑋) = 𝑍 → (𝑓𝑋) = 0 ))
6160ex 450 . . . . . . . . . . . . 13 ((𝑓𝑋) ∈ 𝑆 → (∀𝑠𝑆 ((𝑠 · 𝑋) = 𝑍𝑠 = 0 ) → (((𝑓𝑋)( ·𝑠𝑀)𝑋) = 𝑍 → (𝑓𝑋) = 0 )))
6229, 51, 61syl56 36 . . . . . . . . . . . 12 ((𝑀 ∈ LMod ∧ 𝑋𝐵) → (𝑓 ∈ (𝑆𝑚 {𝑋}) → (∀𝑠𝑆 ((𝑠 · 𝑋) = 𝑍𝑠 = 0 ) → (((𝑓𝑋)( ·𝑠𝑀)𝑋) = 𝑍 → (𝑓𝑋) = 0 ))))
6362com23 86 . . . . . . . . . . 11 ((𝑀 ∈ LMod ∧ 𝑋𝐵) → (∀𝑠𝑆 ((𝑠 · 𝑋) = 𝑍𝑠 = 0 ) → (𝑓 ∈ (𝑆𝑚 {𝑋}) → (((𝑓𝑋)( ·𝑠𝑀)𝑋) = 𝑍 → (𝑓𝑋) = 0 ))))
6463imp31 448 . . . . . . . . . 10 ((((𝑀 ∈ LMod ∧ 𝑋𝐵) ∧ ∀𝑠𝑆 ((𝑠 · 𝑋) = 𝑍𝑠 = 0 )) ∧ 𝑓 ∈ (𝑆𝑚 {𝑋})) → (((𝑓𝑋)( ·𝑠𝑀)𝑋) = 𝑍 → (𝑓𝑋) = 0 ))
6548, 64sylbid 230 . . . . . . . . 9 ((((𝑀 ∈ LMod ∧ 𝑋𝐵) ∧ ∀𝑠𝑆 ((𝑠 · 𝑋) = 𝑍𝑠 = 0 )) ∧ 𝑓 ∈ (𝑆𝑚 {𝑋})) → ((𝑀 Σg (𝑥 ∈ {𝑋} ↦ ((𝑓𝑥)( ·𝑠𝑀)𝑥))) = 𝑍 → (𝑓𝑋) = 0 ))
6665adantld 483 . . . . . . . 8 ((((𝑀 ∈ LMod ∧ 𝑋𝐵) ∧ ∀𝑠𝑆 ((𝑠 · 𝑋) = 𝑍𝑠 = 0 )) ∧ 𝑓 ∈ (𝑆𝑚 {𝑋})) → ((𝑓 finSupp 0 ∧ (𝑀 Σg (𝑥 ∈ {𝑋} ↦ ((𝑓𝑥)( ·𝑠𝑀)𝑥))) = 𝑍) → (𝑓𝑋) = 0 ))
6723, 66sylbid 230 . . . . . . 7 ((((𝑀 ∈ LMod ∧ 𝑋𝐵) ∧ ∀𝑠𝑆 ((𝑠 · 𝑋) = 𝑍𝑠 = 0 )) ∧ 𝑓 ∈ (𝑆𝑚 {𝑋})) → ((𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀){𝑋}) = 𝑍) → (𝑓𝑋) = 0 ))
6867ralrimiva 2963 . . . . . 6 (((𝑀 ∈ LMod ∧ 𝑋𝐵) ∧ ∀𝑠𝑆 ((𝑠 · 𝑋) = 𝑍𝑠 = 0 )) → ∀𝑓 ∈ (𝑆𝑚 {𝑋})((𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀){𝑋}) = 𝑍) → (𝑓𝑋) = 0 ))
6968ex 450 . . . . 5 ((𝑀 ∈ LMod ∧ 𝑋𝐵) → (∀𝑠𝑆 ((𝑠 · 𝑋) = 𝑍𝑠 = 0 ) → ∀𝑓 ∈ (𝑆𝑚 {𝑋})((𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀){𝑋}) = 𝑍) → (𝑓𝑋) = 0 )))
70 impexp 462 . . . . . . . 8 (((𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀){𝑋}) = 𝑍) → (𝑓𝑋) = 0 ) ↔ (𝑓 finSupp 0 → ((𝑓( linC ‘𝑀){𝑋}) = 𝑍 → (𝑓𝑋) = 0 )))
7129adantl 482 . . . . . . . . . 10 (((𝑀 ∈ LMod ∧ 𝑋𝐵) ∧ 𝑓 ∈ (𝑆𝑚 {𝑋})) → 𝑓:{𝑋}⟶𝑆)
72 snfi 8023 . . . . . . . . . . 11 {𝑋} ∈ Fin
7372a1i 11 . . . . . . . . . 10 (((𝑀 ∈ LMod ∧ 𝑋𝐵) ∧ 𝑓 ∈ (𝑆𝑚 {𝑋})) → {𝑋} ∈ Fin)
74 snlindsntor.0 . . . . . . . . . . . 12 0 = (0g𝑅)
75 fvex 6188 . . . . . . . . . . . 12 (0g𝑅) ∈ V
7674, 75eqeltri 2695 . . . . . . . . . . 11 0 ∈ V
7776a1i 11 . . . . . . . . . 10 (((𝑀 ∈ LMod ∧ 𝑋𝐵) ∧ 𝑓 ∈ (𝑆𝑚 {𝑋})) → 0 ∈ V)
7871, 73, 77fdmfifsupp 8270 . . . . . . . . 9 (((𝑀 ∈ LMod ∧ 𝑋𝐵) ∧ 𝑓 ∈ (𝑆𝑚 {𝑋})) → 𝑓 finSupp 0 )
79 pm2.27 42 . . . . . . . . 9 (𝑓 finSupp 0 → ((𝑓 finSupp 0 → ((𝑓( linC ‘𝑀){𝑋}) = 𝑍 → (𝑓𝑋) = 0 )) → ((𝑓( linC ‘𝑀){𝑋}) = 𝑍 → (𝑓𝑋) = 0 )))
8078, 79syl 17 . . . . . . . 8 (((𝑀 ∈ LMod ∧ 𝑋𝐵) ∧ 𝑓 ∈ (𝑆𝑚 {𝑋})) → ((𝑓 finSupp 0 → ((𝑓( linC ‘𝑀){𝑋}) = 𝑍 → (𝑓𝑋) = 0 )) → ((𝑓( linC ‘𝑀){𝑋}) = 𝑍 → (𝑓𝑋) = 0 )))
8170, 80syl5bi 232 . . . . . . 7 (((𝑀 ∈ LMod ∧ 𝑋𝐵) ∧ 𝑓 ∈ (𝑆𝑚 {𝑋})) → (((𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀){𝑋}) = 𝑍) → (𝑓𝑋) = 0 ) → ((𝑓( linC ‘𝑀){𝑋}) = 𝑍 → (𝑓𝑋) = 0 )))
8281ralimdva 2959 . . . . . 6 ((𝑀 ∈ LMod ∧ 𝑋𝐵) → (∀𝑓 ∈ (𝑆𝑚 {𝑋})((𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀){𝑋}) = 𝑍) → (𝑓𝑋) = 0 ) → ∀𝑓 ∈ (𝑆𝑚 {𝑋})((𝑓( linC ‘𝑀){𝑋}) = 𝑍 → (𝑓𝑋) = 0 )))
83 snlindsntor.z . . . . . . 7 𝑍 = (0g𝑀)
8417, 9, 8, 74, 83, 52snlindsntorlem 42024 . . . . . 6 ((𝑀 ∈ LMod ∧ 𝑋𝐵) → (∀𝑓 ∈ (𝑆𝑚 {𝑋})((𝑓( linC ‘𝑀){𝑋}) = 𝑍 → (𝑓𝑋) = 0 ) → ∀𝑠𝑆 ((𝑠 · 𝑋) = 𝑍𝑠 = 0 )))
8582, 84syld 47 . . . . 5 ((𝑀 ∈ LMod ∧ 𝑋𝐵) → (∀𝑓 ∈ (𝑆𝑚 {𝑋})((𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀){𝑋}) = 𝑍) → (𝑓𝑋) = 0 ) → ∀𝑠𝑆 ((𝑠 · 𝑋) = 𝑍𝑠 = 0 )))
8669, 85impbid 202 . . . 4 ((𝑀 ∈ LMod ∧ 𝑋𝐵) → (∀𝑠𝑆 ((𝑠 · 𝑋) = 𝑍𝑠 = 0 ) ↔ ∀𝑓 ∈ (𝑆𝑚 {𝑋})((𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀){𝑋}) = 𝑍) → (𝑓𝑋) = 0 )))
87 fveq2 6178 . . . . . . . . . 10 (𝑦 = 𝑋 → (𝑓𝑦) = (𝑓𝑋))
8887eqeq1d 2622 . . . . . . . . 9 (𝑦 = 𝑋 → ((𝑓𝑦) = 0 ↔ (𝑓𝑋) = 0 ))
8988ralsng 4209 . . . . . . . 8 (𝑋𝐵 → (∀𝑦 ∈ {𝑋} (𝑓𝑦) = 0 ↔ (𝑓𝑋) = 0 ))
9089adantl 482 . . . . . . 7 ((𝑀 ∈ LMod ∧ 𝑋𝐵) → (∀𝑦 ∈ {𝑋} (𝑓𝑦) = 0 ↔ (𝑓𝑋) = 0 ))
9190bicomd 213 . . . . . 6 ((𝑀 ∈ LMod ∧ 𝑋𝐵) → ((𝑓𝑋) = 0 ↔ ∀𝑦 ∈ {𝑋} (𝑓𝑦) = 0 ))
9291imbi2d 330 . . . . 5 ((𝑀 ∈ LMod ∧ 𝑋𝐵) → (((𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀){𝑋}) = 𝑍) → (𝑓𝑋) = 0 ) ↔ ((𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀){𝑋}) = 𝑍) → ∀𝑦 ∈ {𝑋} (𝑓𝑦) = 0 )))
9392ralbidv 2983 . . . 4 ((𝑀 ∈ LMod ∧ 𝑋𝐵) → (∀𝑓 ∈ (𝑆𝑚 {𝑋})((𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀){𝑋}) = 𝑍) → (𝑓𝑋) = 0 ) ↔ ∀𝑓 ∈ (𝑆𝑚 {𝑋})((𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀){𝑋}) = 𝑍) → ∀𝑦 ∈ {𝑋} (𝑓𝑦) = 0 )))
94 snelpwi 4903 . . . . . 6 (𝑋𝐵 → {𝑋} ∈ 𝒫 𝐵)
9594adantl 482 . . . . 5 ((𝑀 ∈ LMod ∧ 𝑋𝐵) → {𝑋} ∈ 𝒫 𝐵)
9695biantrurd 529 . . . 4 ((𝑀 ∈ LMod ∧ 𝑋𝐵) → (∀𝑓 ∈ (𝑆𝑚 {𝑋})((𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀){𝑋}) = 𝑍) → ∀𝑦 ∈ {𝑋} (𝑓𝑦) = 0 ) ↔ ({𝑋} ∈ 𝒫 𝐵 ∧ ∀𝑓 ∈ (𝑆𝑚 {𝑋})((𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀){𝑋}) = 𝑍) → ∀𝑦 ∈ {𝑋} (𝑓𝑦) = 0 ))))
9786, 93, 963bitrd 294 . . 3 ((𝑀 ∈ LMod ∧ 𝑋𝐵) → (∀𝑠𝑆 ((𝑠 · 𝑋) = 𝑍𝑠 = 0 ) ↔ ({𝑋} ∈ 𝒫 𝐵 ∧ ∀𝑓 ∈ (𝑆𝑚 {𝑋})((𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀){𝑋}) = 𝑍) → ∀𝑦 ∈ {𝑋} (𝑓𝑦) = 0 ))))
984, 97syl5bb 272 . 2 ((𝑀 ∈ LMod ∧ 𝑋𝐵) → (∀𝑠 ∈ (𝑆 ∖ { 0 })(𝑠 · 𝑋) ≠ 𝑍 ↔ ({𝑋} ∈ 𝒫 𝐵 ∧ ∀𝑓 ∈ (𝑆𝑚 {𝑋})((𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀){𝑋}) = 𝑍) → ∀𝑦 ∈ {𝑋} (𝑓𝑦) = 0 ))))
99 snex 4899 . . 3 {𝑋} ∈ V
10017, 83, 9, 8, 74islininds 42000 . . 3 (({𝑋} ∈ V ∧ 𝑀 ∈ LMod) → ({𝑋} linIndS 𝑀 ↔ ({𝑋} ∈ 𝒫 𝐵 ∧ ∀𝑓 ∈ (𝑆𝑚 {𝑋})((𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀){𝑋}) = 𝑍) → ∀𝑦 ∈ {𝑋} (𝑓𝑦) = 0 ))))
10199, 5, 100sylancr 694 . 2 ((𝑀 ∈ LMod ∧ 𝑋𝐵) → ({𝑋} linIndS 𝑀 ↔ ({𝑋} ∈ 𝒫 𝐵 ∧ ∀𝑓 ∈ (𝑆𝑚 {𝑋})((𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀){𝑋}) = 𝑍) → ∀𝑦 ∈ {𝑋} (𝑓𝑦) = 0 ))))
10298, 101bitr4d 271 1 ((𝑀 ∈ LMod ∧ 𝑋𝐵) → (∀𝑠 ∈ (𝑆 ∖ { 0 })(𝑠 · 𝑋) ≠ 𝑍 ↔ {𝑋} linIndS 𝑀))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 384   = wceq 1481  wcel 1988  wne 2791  wral 2909  Vcvv 3195  cdif 3564  𝒫 cpw 4149  {csn 4168   class class class wbr 4644  cmpt 4720  wf 5872  cfv 5876  (class class class)co 6635  𝑚 cmap 7842  Fincfn 7940   finSupp cfsupp 8260  Basecbs 15838  Scalarcsca 15925   ·𝑠 cvsca 15926  0gc0g 16081   Σg cgsu 16082  Mndcmnd 17275  Grpcgrp 17403  LModclmod 18844   linC clinc 41958   linIndS clininds 41994
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-rep 4762  ax-sep 4772  ax-nul 4780  ax-pow 4834  ax-pr 4897  ax-un 6934  ax-inf2 8523  ax-cnex 9977  ax-resscn 9978  ax-1cn 9979  ax-icn 9980  ax-addcl 9981  ax-addrcl 9982  ax-mulcl 9983  ax-mulrcl 9984  ax-mulcom 9985  ax-addass 9986  ax-mulass 9987  ax-distr 9988  ax-i2m1 9989  ax-1ne0 9990  ax-1rid 9991  ax-rnegex 9992  ax-rrecex 9993  ax-cnre 9994  ax-pre-lttri 9995  ax-pre-lttrn 9996  ax-pre-ltadd 9997  ax-pre-mulgt0 9998
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-nel 2895  df-ral 2914  df-rex 2915  df-reu 2916  df-rmo 2917  df-rab 2918  df-v 3197  df-sbc 3430  df-csb 3527  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-pss 3583  df-nul 3908  df-if 4078  df-pw 4151  df-sn 4169  df-pr 4171  df-tp 4173  df-op 4175  df-uni 4428  df-int 4467  df-iun 4513  df-br 4645  df-opab 4704  df-mpt 4721  df-tr 4744  df-id 5014  df-eprel 5019  df-po 5025  df-so 5026  df-fr 5063  df-se 5064  df-we 5065  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-res 5116  df-ima 5117  df-pred 5668  df-ord 5714  df-on 5715  df-lim 5716  df-suc 5717  df-iota 5839  df-fun 5878  df-fn 5879  df-f 5880  df-f1 5881  df-fo 5882  df-f1o 5883  df-fv 5884  df-isom 5885  df-riota 6596  df-ov 6638  df-oprab 6639  df-mpt2 6640  df-om 7051  df-1st 7153  df-2nd 7154  df-supp 7281  df-wrecs 7392  df-recs 7453  df-rdg 7491  df-1o 7545  df-oadd 7549  df-er 7727  df-map 7844  df-en 7941  df-dom 7942  df-sdom 7943  df-fin 7944  df-fsupp 8261  df-oi 8400  df-card 8750  df-pnf 10061  df-mnf 10062  df-xr 10063  df-ltxr 10064  df-le 10065  df-sub 10253  df-neg 10254  df-nn 11006  df-n0 11278  df-z 11363  df-uz 11673  df-fz 12312  df-fzo 12450  df-seq 12785  df-hash 13101  df-0g 16083  df-gsum 16084  df-mgm 17223  df-sgrp 17265  df-mnd 17276  df-grp 17406  df-mulg 17522  df-cntz 17731  df-lmod 18846  df-linc 41960  df-lininds 41996
This theorem is referenced by:  lindssnlvec  42040
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