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Theorem snlindsntor 41574
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 𝑓 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-ne 2791 . . . . 5 ((𝑠 · 𝑋) ≠ 𝑍 ↔ ¬ (𝑠 · 𝑋) = 𝑍)
21ralbii 2975 . . . 4 (∀𝑠 ∈ (𝑆 ∖ { 0 })(𝑠 · 𝑋) ≠ 𝑍 ↔ ∀𝑠 ∈ (𝑆 ∖ { 0 }) ¬ (𝑠 · 𝑋) = 𝑍)
3 raldifsni 4298 . . . 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 6156 . . . . . . . . . . . . . . . 16 (Base‘𝑅) = (Base‘(Scalar‘𝑀))
118, 10eqtri 2643 . . . . . . . . . . . . . . 15 𝑆 = (Base‘(Scalar‘𝑀))
1211oveq1i 6620 . . . . . . . . . . . . . 14 (𝑆𝑚 {𝑋}) = ((Base‘(Scalar‘𝑀)) ↑𝑚 {𝑋})
1312eleq2i 2690 . . . . . . . . . . . . 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 4878 . . . . . . . . . . . . 13 (𝑋 ∈ (Base‘𝑀) → {𝑋} ∈ 𝒫 (Base‘𝑀))
17 snlindsntor.b . . . . . . . . . . . . 13 𝐵 = (Base‘𝑀)
1816, 17eleq2s 2716 . . . . . . . . . . . 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 41512 . . . . . . . . . . 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 1323 . . . . . . . . . 10 ((((𝑀 ∈ LMod ∧ 𝑋𝐵) ∧ ∀𝑠𝑆 ((𝑠 · 𝑋) = 𝑍𝑠 = 0 )) ∧ 𝑓 ∈ (𝑆𝑚 {𝑋})) → (𝑓( linC ‘𝑀){𝑋}) = (𝑀 Σg (𝑥 ∈ {𝑋} ↦ ((𝑓𝑥)( ·𝑠𝑀)𝑥))))
2221eqeq1d 2623 . . . . . . . . 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 18802 . . . . . . . . . . . . . 14 (𝑀 ∈ LMod → 𝑀 ∈ Grp)
25 grpmnd 17361 . . . . . . . . . . . . . 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 7831 . . . . . . . . . . . . . 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 4182 . . . . . . . . . . . . . . . . . . 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 6318 . . . . . . . . . . . . . . . . 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 2621 . . . . . . . . . . . . . . . . 17 ( ·𝑠𝑀) = ( ·𝑠𝑀)
3817, 9, 37, 8lmodvscl 18812 . . . . . . . . . . . . . . . 16 ((𝑀 ∈ LMod ∧ (𝑓𝑋) ∈ 𝑆𝑋𝐵) → ((𝑓𝑋)( ·𝑠𝑀)𝑋) ∈ 𝐵)
3930, 35, 36, 38syl3anc 1323 . . . . . . . . . . . . . . 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 6153 . . . . . . . . . . . . . 14 (𝑥 = 𝑋 → (𝑓𝑥) = (𝑓𝑋))
44 id 22 . . . . . . . . . . . . . 14 (𝑥 = 𝑋𝑥 = 𝑋)
4543, 44oveq12d 6628 . . . . . . . . . . . . 13 (𝑥 = 𝑋 → ((𝑓𝑥)( ·𝑠𝑀)𝑥) = ((𝑓𝑋)( ·𝑠𝑀)𝑋))
4617, 45gsumsn 18286 . . . . . . . . . . . 12 ((𝑀 ∈ Mnd ∧ 𝑋𝐵 ∧ ((𝑓𝑋)( ·𝑠𝑀)𝑋) ∈ 𝐵) → (𝑀 Σg (𝑥 ∈ {𝑋} ↦ ((𝑓𝑥)( ·𝑠𝑀)𝑥))) = ((𝑓𝑋)( ·𝑠𝑀)𝑋))
4727, 28, 42, 46syl3anc 1323 . . . . . . . . . . 11 ((((𝑀 ∈ LMod ∧ 𝑋𝐵) ∧ ∀𝑠𝑆 ((𝑠 · 𝑋) = 𝑍𝑠 = 0 )) ∧ 𝑓 ∈ (𝑆𝑚 {𝑋})) → (𝑀 Σg (𝑥 ∈ {𝑋} ↦ ((𝑓𝑥)( ·𝑠𝑀)𝑥))) = ((𝑓𝑋)( ·𝑠𝑀)𝑋))
4847eqeq1d 2623 . . . . . . . . . 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 6623 . . . . . . . . . . . . . . . 16 ((𝑓𝑋) · 𝑋) = ((𝑓𝑋)( ·𝑠𝑀)𝑋)
5453eqeq1i 2626 . . . . . . . . . . . . . . 15 (((𝑓𝑋) · 𝑋) = 𝑍 ↔ ((𝑓𝑋)( ·𝑠𝑀)𝑋) = 𝑍)
55 oveq1 6617 . . . . . . . . . . . . . . . . . 18 (𝑠 = (𝑓𝑋) → (𝑠 · 𝑋) = ((𝑓𝑋) · 𝑋))
5655eqeq1d 2623 . . . . . . . . . . . . . . . . 17 (𝑠 = (𝑓𝑋) → ((𝑠 · 𝑋) = 𝑍 ↔ ((𝑓𝑋) · 𝑋) = 𝑍))
57 eqeq1 2625 . . . . . . . . . . . . . . . . 17 (𝑠 = (𝑓𝑋) → (𝑠 = 0 ↔ (𝑓𝑋) = 0 ))
5856, 57imbi12d 334 . . . . . . . . . . . . . . . 16 (𝑠 = (𝑓𝑋) → (((𝑠 · 𝑋) = 𝑍𝑠 = 0 ) ↔ (((𝑓𝑋) · 𝑋) = 𝑍 → (𝑓𝑋) = 0 )))
5958rspcva 3296 . . . . . . . . . . . . . . 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 2961 . . . . . 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 7990 . . . . . . . . . . 11 {𝑋} ∈ Fin
7372a1i 11 . . . . . . . . . 10 (((𝑀 ∈ LMod ∧ 𝑋𝐵) ∧ 𝑓 ∈ (𝑆𝑚 {𝑋})) → {𝑋} ∈ Fin)
74 snlindsntor.0 . . . . . . . . . . . 12 0 = (0g𝑅)
75 fvex 6163 . . . . . . . . . . . 12 (0g𝑅) ∈ V
7674, 75eqeltri 2694 . . . . . . . . . . 11 0 ∈ V
7776a1i 11 . . . . . . . . . 10 (((𝑀 ∈ LMod ∧ 𝑋𝐵) ∧ 𝑓 ∈ (𝑆𝑚 {𝑋})) → 0 ∈ V)
7871, 73, 77fdmfifsupp 8237 . . . . . . . . 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 2957 . . . . . 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 41573 . . . . . 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 6153 . . . . . . . . . 10 (𝑦 = 𝑋 → (𝑓𝑦) = (𝑓𝑋))
8887eqeq1d 2623 . . . . . . . . 9 (𝑦 = 𝑋 → ((𝑓𝑦) = 0 ↔ (𝑓𝑋) = 0 ))
8988ralsng 4194 . . . . . . . 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 2981 . . . 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 4878 . . . . . 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 4874 . . 3 {𝑋} ∈ V
10017, 83, 9, 8, 74islininds 41549 . . 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 1480  wcel 1987  wne 2790  wral 2907  Vcvv 3189  cdif 3556  𝒫 cpw 4135  {csn 4153   class class class wbr 4618  cmpt 4678  wf 5848  cfv 5852  (class class class)co 6610  𝑚 cmap 7809  Fincfn 7907   finSupp cfsupp 8227  Basecbs 15792  Scalarcsca 15876   ·𝑠 cvsca 15877  0gc0g 16032   Σg cgsu 16033  Mndcmnd 17226  Grpcgrp 17354  LModclmod 18795   linC clinc 41507   linIndS clininds 41543
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6909  ax-inf2 8490  ax-cnex 9944  ax-resscn 9945  ax-1cn 9946  ax-icn 9947  ax-addcl 9948  ax-addrcl 9949  ax-mulcl 9950  ax-mulrcl 9951  ax-mulcom 9952  ax-addass 9953  ax-mulass 9954  ax-distr 9955  ax-i2m1 9956  ax-1ne0 9957  ax-1rid 9958  ax-rnegex 9959  ax-rrecex 9960  ax-cnre 9961  ax-pre-lttri 9962  ax-pre-lttrn 9963  ax-pre-ltadd 9964  ax-pre-mulgt0 9965
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3191  df-sbc 3422  df-csb 3519  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-pss 3575  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-uni 4408  df-int 4446  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-tr 4718  df-eprel 4990  df-id 4994  df-po 5000  df-so 5001  df-fr 5038  df-se 5039  df-we 5040  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-pred 5644  df-ord 5690  df-on 5691  df-lim 5692  df-suc 5693  df-iota 5815  df-fun 5854  df-fn 5855  df-f 5856  df-f1 5857  df-fo 5858  df-f1o 5859  df-fv 5860  df-isom 5861  df-riota 6571  df-ov 6613  df-oprab 6614  df-mpt2 6615  df-om 7020  df-1st 7120  df-2nd 7121  df-supp 7248  df-wrecs 7359  df-recs 7420  df-rdg 7458  df-1o 7512  df-oadd 7516  df-er 7694  df-map 7811  df-en 7908  df-dom 7909  df-sdom 7910  df-fin 7911  df-fsupp 8228  df-oi 8367  df-card 8717  df-pnf 10028  df-mnf 10029  df-xr 10030  df-ltxr 10031  df-le 10032  df-sub 10220  df-neg 10221  df-nn 10973  df-n0 11245  df-z 11330  df-uz 11640  df-fz 12277  df-fzo 12415  df-seq 12750  df-hash 13066  df-0g 16034  df-gsum 16035  df-mgm 17174  df-sgrp 17216  df-mnd 17227  df-grp 17357  df-mulg 17473  df-cntz 17682  df-lmod 18797  df-linc 41509  df-lininds 41545
This theorem is referenced by:  lindssnlvec  41589
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