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Theorem snlindsntor 42861
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 2938 . . . . 5 ((𝑠 · 𝑋) ≠ 𝑍 ↔ ¬ (𝑠 · 𝑋) = 𝑍)
21ralbii 3127 . . . 4 (∀𝑠 ∈ (𝑆 ∖ { 0 })(𝑠 · 𝑋) ≠ 𝑍 ↔ ∀𝑠 ∈ (𝑆 ∖ { 0 }) ¬ (𝑠 · 𝑋) = 𝑍)
3 raldifsni 4480 . . . 4 (∀𝑠 ∈ (𝑆 ∖ { 0 }) ¬ (𝑠 · 𝑋) = 𝑍 ↔ ∀𝑠𝑆 ((𝑠 · 𝑋) = 𝑍𝑠 = 0 ))
42, 3bitri 266 . . 3 (∀𝑠 ∈ (𝑆 ∖ { 0 })(𝑠 · 𝑋) ≠ 𝑍 ↔ ∀𝑠𝑆 ((𝑠 · 𝑋) = 𝑍𝑠 = 0 ))
5 simpl 474 . . . . . . . . . . . . 13 ((𝑀 ∈ LMod ∧ 𝑋𝐵) → 𝑀 ∈ LMod)
65adantr 472 . . . . . . . . . . . 12 (((𝑀 ∈ LMod ∧ 𝑋𝐵) ∧ ∀𝑠𝑆 ((𝑠 · 𝑋) = 𝑍𝑠 = 0 )) → 𝑀 ∈ LMod)
76adantr 472 . . . . . . . . . . 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 6378 . . . . . . . . . . . . . . . 16 (Base‘𝑅) = (Base‘(Scalar‘𝑀))
118, 10eqtri 2787 . . . . . . . . . . . . . . 15 𝑆 = (Base‘(Scalar‘𝑀))
1211oveq1i 6852 . . . . . . . . . . . . . 14 (𝑆𝑚 {𝑋}) = ((Base‘(Scalar‘𝑀)) ↑𝑚 {𝑋})
1312eleq2i 2836 . . . . . . . . . . . . 13 (𝑓 ∈ (𝑆𝑚 {𝑋}) ↔ 𝑓 ∈ ((Base‘(Scalar‘𝑀)) ↑𝑚 {𝑋}))
1413biimpi 207 . . . . . . . . . . . 12 (𝑓 ∈ (𝑆𝑚 {𝑋}) → 𝑓 ∈ ((Base‘(Scalar‘𝑀)) ↑𝑚 {𝑋}))
1514adantl 473 . . . . . . . . . . 11 ((((𝑀 ∈ LMod ∧ 𝑋𝐵) ∧ ∀𝑠𝑆 ((𝑠 · 𝑋) = 𝑍𝑠 = 0 )) ∧ 𝑓 ∈ (𝑆𝑚 {𝑋})) → 𝑓 ∈ ((Base‘(Scalar‘𝑀)) ↑𝑚 {𝑋}))
16 snelpwi 5068 . . . . . . . . . . . . 13 (𝑋 ∈ (Base‘𝑀) → {𝑋} ∈ 𝒫 (Base‘𝑀))
17 snlindsntor.b . . . . . . . . . . . . 13 𝐵 = (Base‘𝑀)
1816, 17eleq2s 2862 . . . . . . . . . . . 12 (𝑋𝐵 → {𝑋} ∈ 𝒫 (Base‘𝑀))
1918ad3antlr 722 . . . . . . . . . . 11 ((((𝑀 ∈ LMod ∧ 𝑋𝐵) ∧ ∀𝑠𝑆 ((𝑠 · 𝑋) = 𝑍𝑠 = 0 )) ∧ 𝑓 ∈ (𝑆𝑚 {𝑋})) → {𝑋} ∈ 𝒫 (Base‘𝑀))
20 lincval 42799 . . . . . . . . . . 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 1490 . . . . . . . . . 10 ((((𝑀 ∈ LMod ∧ 𝑋𝐵) ∧ ∀𝑠𝑆 ((𝑠 · 𝑋) = 𝑍𝑠 = 0 )) ∧ 𝑓 ∈ (𝑆𝑚 {𝑋})) → (𝑓( linC ‘𝑀){𝑋}) = (𝑀 Σg (𝑥 ∈ {𝑋} ↦ ((𝑓𝑥)( ·𝑠𝑀)𝑥))))
2221eqeq1d 2767 . . . . . . . . 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 622 . . . . . . . 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 19139 . . . . . . . . . . . . . 14 (𝑀 ∈ LMod → 𝑀 ∈ Grp)
25 grpmnd 17698 . . . . . . . . . . . . . 14 (𝑀 ∈ Grp → 𝑀 ∈ Mnd)
2624, 25syl 17 . . . . . . . . . . . . 13 (𝑀 ∈ LMod → 𝑀 ∈ Mnd)
2726ad3antrrr 721 . . . . . . . . . . . 12 ((((𝑀 ∈ LMod ∧ 𝑋𝐵) ∧ ∀𝑠𝑆 ((𝑠 · 𝑋) = 𝑍𝑠 = 0 )) ∧ 𝑓 ∈ (𝑆𝑚 {𝑋})) → 𝑀 ∈ Mnd)
28 simpllr 793 . . . . . . . . . . . 12 ((((𝑀 ∈ LMod ∧ 𝑋𝐵) ∧ ∀𝑠𝑆 ((𝑠 · 𝑋) = 𝑍𝑠 = 0 )) ∧ 𝑓 ∈ (𝑆𝑚 {𝑋})) → 𝑋𝐵)
29 elmapi 8082 . . . . . . . . . . . . . 14 (𝑓 ∈ (𝑆𝑚 {𝑋}) → 𝑓:{𝑋}⟶𝑆)
306adantl 473 . . . . . . . . . . . . . . . 16 ((𝑓:{𝑋}⟶𝑆 ∧ ((𝑀 ∈ LMod ∧ 𝑋𝐵) ∧ ∀𝑠𝑆 ((𝑠 · 𝑋) = 𝑍𝑠 = 0 ))) → 𝑀 ∈ LMod)
31 snidg 4364 . . . . . . . . . . . . . . . . . . 19 (𝑋𝐵𝑋 ∈ {𝑋})
3231adantl 473 . . . . . . . . . . . . . . . . . 18 ((𝑀 ∈ LMod ∧ 𝑋𝐵) → 𝑋 ∈ {𝑋})
3332adantr 472 . . . . . . . . . . . . . . . . 17 (((𝑀 ∈ LMod ∧ 𝑋𝐵) ∧ ∀𝑠𝑆 ((𝑠 · 𝑋) = 𝑍𝑠 = 0 )) → 𝑋 ∈ {𝑋})
34 ffvelrn 6547 . . . . . . . . . . . . . . . . 17 ((𝑓:{𝑋}⟶𝑆𝑋 ∈ {𝑋}) → (𝑓𝑋) ∈ 𝑆)
3533, 34sylan2 586 . . . . . . . . . . . . . . . 16 ((𝑓:{𝑋}⟶𝑆 ∧ ((𝑀 ∈ LMod ∧ 𝑋𝐵) ∧ ∀𝑠𝑆 ((𝑠 · 𝑋) = 𝑍𝑠 = 0 ))) → (𝑓𝑋) ∈ 𝑆)
36 simprlr 798 . . . . . . . . . . . . . . . 16 ((𝑓:{𝑋}⟶𝑆 ∧ ((𝑀 ∈ LMod ∧ 𝑋𝐵) ∧ ∀𝑠𝑆 ((𝑠 · 𝑋) = 𝑍𝑠 = 0 ))) → 𝑋𝐵)
37 eqid 2765 . . . . . . . . . . . . . . . . 17 ( ·𝑠𝑀) = ( ·𝑠𝑀)
3817, 9, 37, 8lmodvscl 19149 . . . . . . . . . . . . . . . 16 ((𝑀 ∈ LMod ∧ (𝑓𝑋) ∈ 𝑆𝑋𝐵) → ((𝑓𝑋)( ·𝑠𝑀)𝑋) ∈ 𝐵)
3930, 35, 36, 38syl3anc 1490 . . . . . . . . . . . . . . 15 ((𝑓:{𝑋}⟶𝑆 ∧ ((𝑀 ∈ LMod ∧ 𝑋𝐵) ∧ ∀𝑠𝑆 ((𝑠 · 𝑋) = 𝑍𝑠 = 0 ))) → ((𝑓𝑋)( ·𝑠𝑀)𝑋) ∈ 𝐵)
4039expcom 402 . . . . . . . . . . . . . 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 396 . . . . . . . . . . . 12 ((((𝑀 ∈ LMod ∧ 𝑋𝐵) ∧ ∀𝑠𝑆 ((𝑠 · 𝑋) = 𝑍𝑠 = 0 )) ∧ 𝑓 ∈ (𝑆𝑚 {𝑋})) → ((𝑓𝑋)( ·𝑠𝑀)𝑋) ∈ 𝐵)
43 fveq2 6375 . . . . . . . . . . . . . 14 (𝑥 = 𝑋 → (𝑓𝑥) = (𝑓𝑋))
44 id 22 . . . . . . . . . . . . . 14 (𝑥 = 𝑋𝑥 = 𝑋)
4543, 44oveq12d 6860 . . . . . . . . . . . . 13 (𝑥 = 𝑋 → ((𝑓𝑥)( ·𝑠𝑀)𝑥) = ((𝑓𝑋)( ·𝑠𝑀)𝑋))
4617, 45gsumsn 18620 . . . . . . . . . . . 12 ((𝑀 ∈ Mnd ∧ 𝑋𝐵 ∧ ((𝑓𝑋)( ·𝑠𝑀)𝑋) ∈ 𝐵) → (𝑀 Σg (𝑥 ∈ {𝑋} ↦ ((𝑓𝑥)( ·𝑠𝑀)𝑥))) = ((𝑓𝑋)( ·𝑠𝑀)𝑋))
4727, 28, 42, 46syl3anc 1490 . . . . . . . . . . 11 ((((𝑀 ∈ LMod ∧ 𝑋𝐵) ∧ ∀𝑠𝑆 ((𝑠 · 𝑋) = 𝑍𝑠 = 0 )) ∧ 𝑓 ∈ (𝑆𝑚 {𝑋})) → (𝑀 Σg (𝑥 ∈ {𝑋} ↦ ((𝑓𝑥)( ·𝑠𝑀)𝑥))) = ((𝑓𝑋)( ·𝑠𝑀)𝑋))
4847eqeq1d 2767 . . . . . . . . . 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 586 . . . . . . . . . . . . . . 15 ((𝑓:{𝑋}⟶𝑆𝑋𝐵) → (𝑓𝑋) ∈ 𝑆)
5049expcom 402 . . . . . . . . . . . . . 14 (𝑋𝐵 → (𝑓:{𝑋}⟶𝑆 → (𝑓𝑋) ∈ 𝑆))
5150adantl 473 . . . . . . . . . . . . 13 ((𝑀 ∈ LMod ∧ 𝑋𝐵) → (𝑓:{𝑋}⟶𝑆 → (𝑓𝑋) ∈ 𝑆))
52 snlindsntor.t . . . . . . . . . . . . . . . . 17 · = ( ·𝑠𝑀)
5352oveqi 6855 . . . . . . . . . . . . . . . 16 ((𝑓𝑋) · 𝑋) = ((𝑓𝑋)( ·𝑠𝑀)𝑋)
5453eqeq1i 2770 . . . . . . . . . . . . . . 15 (((𝑓𝑋) · 𝑋) = 𝑍 ↔ ((𝑓𝑋)( ·𝑠𝑀)𝑋) = 𝑍)
55 oveq1 6849 . . . . . . . . . . . . . . . . . 18 (𝑠 = (𝑓𝑋) → (𝑠 · 𝑋) = ((𝑓𝑋) · 𝑋))
5655eqeq1d 2767 . . . . . . . . . . . . . . . . 17 (𝑠 = (𝑓𝑋) → ((𝑠 · 𝑋) = 𝑍 ↔ ((𝑓𝑋) · 𝑋) = 𝑍))
57 eqeq1 2769 . . . . . . . . . . . . . . . . 17 (𝑠 = (𝑓𝑋) → (𝑠 = 0 ↔ (𝑓𝑋) = 0 ))
5856, 57imbi12d 335 . . . . . . . . . . . . . . . 16 (𝑠 = (𝑓𝑋) → (((𝑠 · 𝑋) = 𝑍𝑠 = 0 ) ↔ (((𝑓𝑋) · 𝑋) = 𝑍 → (𝑓𝑋) = 0 )))
5958rspcva 3459 . . . . . . . . . . . . . . 15 (((𝑓𝑋) ∈ 𝑆 ∧ ∀𝑠𝑆 ((𝑠 · 𝑋) = 𝑍𝑠 = 0 )) → (((𝑓𝑋) · 𝑋) = 𝑍 → (𝑓𝑋) = 0 ))
6054, 59syl5bir 234 . . . . . . . . . . . . . 14 (((𝑓𝑋) ∈ 𝑆 ∧ ∀𝑠𝑆 ((𝑠 · 𝑋) = 𝑍𝑠 = 0 )) → (((𝑓𝑋)( ·𝑠𝑀)𝑋) = 𝑍 → (𝑓𝑋) = 0 ))
6160ex 401 . . . . . . . . . . . . 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 408 . . . . . . . . . 10 ((((𝑀 ∈ LMod ∧ 𝑋𝐵) ∧ ∀𝑠𝑆 ((𝑠 · 𝑋) = 𝑍𝑠 = 0 )) ∧ 𝑓 ∈ (𝑆𝑚 {𝑋})) → (((𝑓𝑋)( ·𝑠𝑀)𝑋) = 𝑍 → (𝑓𝑋) = 0 ))
6548, 64sylbid 231 . . . . . . . . 9 ((((𝑀 ∈ LMod ∧ 𝑋𝐵) ∧ ∀𝑠𝑆 ((𝑠 · 𝑋) = 𝑍𝑠 = 0 )) ∧ 𝑓 ∈ (𝑆𝑚 {𝑋})) → ((𝑀 Σg (𝑥 ∈ {𝑋} ↦ ((𝑓𝑥)( ·𝑠𝑀)𝑥))) = 𝑍 → (𝑓𝑋) = 0 ))
6665adantld 484 . . . . . . . 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 231 . . . . . . 7 ((((𝑀 ∈ LMod ∧ 𝑋𝐵) ∧ ∀𝑠𝑆 ((𝑠 · 𝑋) = 𝑍𝑠 = 0 )) ∧ 𝑓 ∈ (𝑆𝑚 {𝑋})) → ((𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀){𝑋}) = 𝑍) → (𝑓𝑋) = 0 ))
6867ralrimiva 3113 . . . . . 6 (((𝑀 ∈ LMod ∧ 𝑋𝐵) ∧ ∀𝑠𝑆 ((𝑠 · 𝑋) = 𝑍𝑠 = 0 )) → ∀𝑓 ∈ (𝑆𝑚 {𝑋})((𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀){𝑋}) = 𝑍) → (𝑓𝑋) = 0 ))
6968ex 401 . . . . 5 ((𝑀 ∈ LMod ∧ 𝑋𝐵) → (∀𝑠𝑆 ((𝑠 · 𝑋) = 𝑍𝑠 = 0 ) → ∀𝑓 ∈ (𝑆𝑚 {𝑋})((𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀){𝑋}) = 𝑍) → (𝑓𝑋) = 0 )))
70 impexp 441 . . . . . . . 8 (((𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀){𝑋}) = 𝑍) → (𝑓𝑋) = 0 ) ↔ (𝑓 finSupp 0 → ((𝑓( linC ‘𝑀){𝑋}) = 𝑍 → (𝑓𝑋) = 0 )))
7129adantl 473 . . . . . . . . . 10 (((𝑀 ∈ LMod ∧ 𝑋𝐵) ∧ 𝑓 ∈ (𝑆𝑚 {𝑋})) → 𝑓:{𝑋}⟶𝑆)
72 snfi 8245 . . . . . . . . . . 11 {𝑋} ∈ Fin
7372a1i 11 . . . . . . . . . 10 (((𝑀 ∈ LMod ∧ 𝑋𝐵) ∧ 𝑓 ∈ (𝑆𝑚 {𝑋})) → {𝑋} ∈ Fin)
74 snlindsntor.0 . . . . . . . . . . . 12 0 = (0g𝑅)
7574fvexi 6389 . . . . . . . . . . 11 0 ∈ V
7675a1i 11 . . . . . . . . . 10 (((𝑀 ∈ LMod ∧ 𝑋𝐵) ∧ 𝑓 ∈ (𝑆𝑚 {𝑋})) → 0 ∈ V)
7771, 73, 76fdmfifsupp 8492 . . . . . . . . 9 (((𝑀 ∈ LMod ∧ 𝑋𝐵) ∧ 𝑓 ∈ (𝑆𝑚 {𝑋})) → 𝑓 finSupp 0 )
78 pm2.27 42 . . . . . . . . 9 (𝑓 finSupp 0 → ((𝑓 finSupp 0 → ((𝑓( linC ‘𝑀){𝑋}) = 𝑍 → (𝑓𝑋) = 0 )) → ((𝑓( linC ‘𝑀){𝑋}) = 𝑍 → (𝑓𝑋) = 0 )))
7977, 78syl 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 )))
8070, 79syl5bi 233 . . . . . . 7 (((𝑀 ∈ LMod ∧ 𝑋𝐵) ∧ 𝑓 ∈ (𝑆𝑚 {𝑋})) → (((𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀){𝑋}) = 𝑍) → (𝑓𝑋) = 0 ) → ((𝑓( linC ‘𝑀){𝑋}) = 𝑍 → (𝑓𝑋) = 0 )))
8180ralimdva 3109 . . . . . 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 )))
82 snlindsntor.z . . . . . . 7 𝑍 = (0g𝑀)
8317, 9, 8, 74, 82, 52snlindsntorlem 42860 . . . . . 6 ((𝑀 ∈ LMod ∧ 𝑋𝐵) → (∀𝑓 ∈ (𝑆𝑚 {𝑋})((𝑓( linC ‘𝑀){𝑋}) = 𝑍 → (𝑓𝑋) = 0 ) → ∀𝑠𝑆 ((𝑠 · 𝑋) = 𝑍𝑠 = 0 )))
8481, 83syld 47 . . . . 5 ((𝑀 ∈ LMod ∧ 𝑋𝐵) → (∀𝑓 ∈ (𝑆𝑚 {𝑋})((𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀){𝑋}) = 𝑍) → (𝑓𝑋) = 0 ) → ∀𝑠𝑆 ((𝑠 · 𝑋) = 𝑍𝑠 = 0 )))
8569, 84impbid 203 . . . 4 ((𝑀 ∈ LMod ∧ 𝑋𝐵) → (∀𝑠𝑆 ((𝑠 · 𝑋) = 𝑍𝑠 = 0 ) ↔ ∀𝑓 ∈ (𝑆𝑚 {𝑋})((𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀){𝑋}) = 𝑍) → (𝑓𝑋) = 0 )))
86 fveqeq2 6384 . . . . . . . . 9 (𝑦 = 𝑋 → ((𝑓𝑦) = 0 ↔ (𝑓𝑋) = 0 ))
8786ralsng 4375 . . . . . . . 8 (𝑋𝐵 → (∀𝑦 ∈ {𝑋} (𝑓𝑦) = 0 ↔ (𝑓𝑋) = 0 ))
8887adantl 473 . . . . . . 7 ((𝑀 ∈ LMod ∧ 𝑋𝐵) → (∀𝑦 ∈ {𝑋} (𝑓𝑦) = 0 ↔ (𝑓𝑋) = 0 ))
8988bicomd 214 . . . . . 6 ((𝑀 ∈ LMod ∧ 𝑋𝐵) → ((𝑓𝑋) = 0 ↔ ∀𝑦 ∈ {𝑋} (𝑓𝑦) = 0 ))
9089imbi2d 331 . . . . 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 )))
9190ralbidv 3133 . . . 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 )))
92 snelpwi 5068 . . . . . 6 (𝑋𝐵 → {𝑋} ∈ 𝒫 𝐵)
9392adantl 473 . . . . 5 ((𝑀 ∈ LMod ∧ 𝑋𝐵) → {𝑋} ∈ 𝒫 𝐵)
9493biantrurd 528 . . . 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 ))))
9585, 91, 943bitrd 296 . . 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 ))))
964, 95syl5bb 274 . 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 ))))
97 snex 5064 . . 3 {𝑋} ∈ V
9817, 82, 9, 8, 74islininds 42836 . . 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 ))))
9997, 5, 98sylancr 581 . 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 ))))
10096, 99bitr4d 273 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 197  wa 384   = wceq 1652  wcel 2155  wne 2937  wral 3055  Vcvv 3350  cdif 3729  𝒫 cpw 4315  {csn 4334   class class class wbr 4809  cmpt 4888  wf 6064  cfv 6068  (class class class)co 6842  𝑚 cmap 8060  Fincfn 8160   finSupp cfsupp 8482  Basecbs 16132  Scalarcsca 16219   ·𝑠 cvsca 16220  0gc0g 16368   Σg cgsu 16369  Mndcmnd 17562  Grpcgrp 17691  LModclmod 19132   linC clinc 42794   linIndS clininds 42830
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2069  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-rep 4930  ax-sep 4941  ax-nul 4949  ax-pow 5001  ax-pr 5062  ax-un 7147  ax-inf2 8753  ax-cnex 10245  ax-resscn 10246  ax-1cn 10247  ax-icn 10248  ax-addcl 10249  ax-addrcl 10250  ax-mulcl 10251  ax-mulrcl 10252  ax-mulcom 10253  ax-addass 10254  ax-mulass 10255  ax-distr 10256  ax-i2m1 10257  ax-1ne0 10258  ax-1rid 10259  ax-rnegex 10260  ax-rrecex 10261  ax-cnre 10262  ax-pre-lttri 10263  ax-pre-lttrn 10264  ax-pre-ltadd 10265  ax-pre-mulgt0 10266
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3or 1108  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2062  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-nel 3041  df-ral 3060  df-rex 3061  df-reu 3062  df-rmo 3063  df-rab 3064  df-v 3352  df-sbc 3597  df-csb 3692  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-pss 3748  df-nul 4080  df-if 4244  df-pw 4317  df-sn 4335  df-pr 4337  df-tp 4339  df-op 4341  df-uni 4595  df-int 4634  df-iun 4678  df-br 4810  df-opab 4872  df-mpt 4889  df-tr 4912  df-id 5185  df-eprel 5190  df-po 5198  df-so 5199  df-fr 5236  df-se 5237  df-we 5238  df-xp 5283  df-rel 5284  df-cnv 5285  df-co 5286  df-dm 5287  df-rn 5288  df-res 5289  df-ima 5290  df-pred 5865  df-ord 5911  df-on 5912  df-lim 5913  df-suc 5914  df-iota 6031  df-fun 6070  df-fn 6071  df-f 6072  df-f1 6073  df-fo 6074  df-f1o 6075  df-fv 6076  df-isom 6077  df-riota 6803  df-ov 6845  df-oprab 6846  df-mpt2 6847  df-om 7264  df-1st 7366  df-2nd 7367  df-supp 7498  df-wrecs 7610  df-recs 7672  df-rdg 7710  df-1o 7764  df-oadd 7768  df-er 7947  df-map 8062  df-en 8161  df-dom 8162  df-sdom 8163  df-fin 8164  df-fsupp 8483  df-oi 8622  df-card 9016  df-pnf 10330  df-mnf 10331  df-xr 10332  df-ltxr 10333  df-le 10334  df-sub 10522  df-neg 10523  df-nn 11275  df-n0 11539  df-z 11625  df-uz 11887  df-fz 12534  df-fzo 12674  df-seq 13009  df-hash 13322  df-0g 16370  df-gsum 16371  df-mgm 17510  df-sgrp 17552  df-mnd 17563  df-grp 17694  df-mulg 17810  df-cntz 18015  df-lmod 19134  df-linc 42796  df-lininds 42832
This theorem is referenced by:  lindssnlvec  42876
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