Mathbox for Alexander van der Vekens < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  snlindsntor Structured version   Visualization version   GIF version

Theorem snlindsntor 45267
 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 = (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 2952 . . . . 5 ((𝑠 · 𝑋) ≠ 𝑍 ↔ ¬ (𝑠 · 𝑋) = 𝑍)
21ralbii 3097 . . . 4 (∀𝑠 ∈ (𝑆 ∖ { 0 })(𝑠 · 𝑋) ≠ 𝑍 ↔ ∀𝑠 ∈ (𝑆 ∖ { 0 }) ¬ (𝑠 · 𝑋) = 𝑍)
3 raldifsni 4685 . . . 4 (∀𝑠 ∈ (𝑆 ∖ { 0 }) ¬ (𝑠 · 𝑋) = 𝑍 ↔ ∀𝑠𝑆 ((𝑠 · 𝑋) = 𝑍𝑠 = 0 ))
42, 3bitri 278 . . 3 (∀𝑠 ∈ (𝑆 ∖ { 0 })(𝑠 · 𝑋) ≠ 𝑍 ↔ ∀𝑠𝑆 ((𝑠 · 𝑋) = 𝑍𝑠 = 0 ))
5 simpl 486 . . . . . . . . . . . . 13 ((𝑀 ∈ LMod ∧ 𝑋𝐵) → 𝑀 ∈ LMod)
65adantr 484 . . . . . . . . . . . 12 (((𝑀 ∈ LMod ∧ 𝑋𝐵) ∧ ∀𝑠𝑆 ((𝑠 · 𝑋) = 𝑍𝑠 = 0 )) → 𝑀 ∈ LMod)
76adantr 484 . . . . . . . . . . 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 6661 . . . . . . . . . . . . . . . 16 (Base‘𝑅) = (Base‘(Scalar‘𝑀))
118, 10eqtri 2781 . . . . . . . . . . . . . . 15 𝑆 = (Base‘(Scalar‘𝑀))
1211oveq1i 7160 . . . . . . . . . . . . . 14 (𝑆m {𝑋}) = ((Base‘(Scalar‘𝑀)) ↑m {𝑋})
1312eleq2i 2843 . . . . . . . . . . . . 13 (𝑓 ∈ (𝑆m {𝑋}) ↔ 𝑓 ∈ ((Base‘(Scalar‘𝑀)) ↑m {𝑋}))
1413biimpi 219 . . . . . . . . . . . 12 (𝑓 ∈ (𝑆m {𝑋}) → 𝑓 ∈ ((Base‘(Scalar‘𝑀)) ↑m {𝑋}))
1514adantl 485 . . . . . . . . . . 11 ((((𝑀 ∈ LMod ∧ 𝑋𝐵) ∧ ∀𝑠𝑆 ((𝑠 · 𝑋) = 𝑍𝑠 = 0 )) ∧ 𝑓 ∈ (𝑆m {𝑋})) → 𝑓 ∈ ((Base‘(Scalar‘𝑀)) ↑m {𝑋}))
16 snelpwi 5305 . . . . . . . . . . . . 13 (𝑋 ∈ (Base‘𝑀) → {𝑋} ∈ 𝒫 (Base‘𝑀))
17 snlindsntor.b . . . . . . . . . . . . 13 𝐵 = (Base‘𝑀)
1816, 17eleq2s 2870 . . . . . . . . . . . 12 (𝑋𝐵 → {𝑋} ∈ 𝒫 (Base‘𝑀))
1918ad3antlr 730 . . . . . . . . . . 11 ((((𝑀 ∈ LMod ∧ 𝑋𝐵) ∧ ∀𝑠𝑆 ((𝑠 · 𝑋) = 𝑍𝑠 = 0 )) ∧ 𝑓 ∈ (𝑆m {𝑋})) → {𝑋} ∈ 𝒫 (Base‘𝑀))
20 lincval 45205 . . . . . . . . . . 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 1368 . . . . . . . . . 10 ((((𝑀 ∈ LMod ∧ 𝑋𝐵) ∧ ∀𝑠𝑆 ((𝑠 · 𝑋) = 𝑍𝑠 = 0 )) ∧ 𝑓 ∈ (𝑆m {𝑋})) → (𝑓( linC ‘𝑀){𝑋}) = (𝑀 Σg (𝑥 ∈ {𝑋} ↦ ((𝑓𝑥)( ·𝑠𝑀)𝑥))))
2221eqeq1d 2760 . . . . . . . . 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 631 . . . . . . . 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 19709 . . . . . . . . . . . . . 14 (𝑀 ∈ LMod → 𝑀 ∈ Grp)
25 grpmnd 18176 . . . . . . . . . . . . . 14 (𝑀 ∈ Grp → 𝑀 ∈ Mnd)
2624, 25syl 17 . . . . . . . . . . . . 13 (𝑀 ∈ LMod → 𝑀 ∈ Mnd)
2726ad3antrrr 729 . . . . . . . . . . . 12 ((((𝑀 ∈ LMod ∧ 𝑋𝐵) ∧ ∀𝑠𝑆 ((𝑠 · 𝑋) = 𝑍𝑠 = 0 )) ∧ 𝑓 ∈ (𝑆m {𝑋})) → 𝑀 ∈ Mnd)
28 simpllr 775 . . . . . . . . . . . 12 ((((𝑀 ∈ LMod ∧ 𝑋𝐵) ∧ ∀𝑠𝑆 ((𝑠 · 𝑋) = 𝑍𝑠 = 0 )) ∧ 𝑓 ∈ (𝑆m {𝑋})) → 𝑋𝐵)
29 elmapi 8438 . . . . . . . . . . . . . 14 (𝑓 ∈ (𝑆m {𝑋}) → 𝑓:{𝑋}⟶𝑆)
306adantl 485 . . . . . . . . . . . . . . . 16 ((𝑓:{𝑋}⟶𝑆 ∧ ((𝑀 ∈ LMod ∧ 𝑋𝐵) ∧ ∀𝑠𝑆 ((𝑠 · 𝑋) = 𝑍𝑠 = 0 ))) → 𝑀 ∈ LMod)
31 snidg 4556 . . . . . . . . . . . . . . . . . . 19 (𝑋𝐵𝑋 ∈ {𝑋})
3231adantl 485 . . . . . . . . . . . . . . . . . 18 ((𝑀 ∈ LMod ∧ 𝑋𝐵) → 𝑋 ∈ {𝑋})
3332adantr 484 . . . . . . . . . . . . . . . . 17 (((𝑀 ∈ LMod ∧ 𝑋𝐵) ∧ ∀𝑠𝑆 ((𝑠 · 𝑋) = 𝑍𝑠 = 0 )) → 𝑋 ∈ {𝑋})
34 ffvelrn 6840 . . . . . . . . . . . . . . . . 17 ((𝑓:{𝑋}⟶𝑆𝑋 ∈ {𝑋}) → (𝑓𝑋) ∈ 𝑆)
3533, 34sylan2 595 . . . . . . . . . . . . . . . 16 ((𝑓:{𝑋}⟶𝑆 ∧ ((𝑀 ∈ LMod ∧ 𝑋𝐵) ∧ ∀𝑠𝑆 ((𝑠 · 𝑋) = 𝑍𝑠 = 0 ))) → (𝑓𝑋) ∈ 𝑆)
36 simprlr 779 . . . . . . . . . . . . . . . 16 ((𝑓:{𝑋}⟶𝑆 ∧ ((𝑀 ∈ LMod ∧ 𝑋𝐵) ∧ ∀𝑠𝑆 ((𝑠 · 𝑋) = 𝑍𝑠 = 0 ))) → 𝑋𝐵)
37 eqid 2758 . . . . . . . . . . . . . . . . 17 ( ·𝑠𝑀) = ( ·𝑠𝑀)
3817, 9, 37, 8lmodvscl 19719 . . . . . . . . . . . . . . . 16 ((𝑀 ∈ LMod ∧ (𝑓𝑋) ∈ 𝑆𝑋𝐵) → ((𝑓𝑋)( ·𝑠𝑀)𝑋) ∈ 𝐵)
3930, 35, 36, 38syl3anc 1368 . . . . . . . . . . . . . . 15 ((𝑓:{𝑋}⟶𝑆 ∧ ((𝑀 ∈ LMod ∧ 𝑋𝐵) ∧ ∀𝑠𝑆 ((𝑠 · 𝑋) = 𝑍𝑠 = 0 ))) → ((𝑓𝑋)( ·𝑠𝑀)𝑋) ∈ 𝐵)
4039expcom 417 . . . . . . . . . . . . . 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 411 . . . . . . . . . . . 12 ((((𝑀 ∈ LMod ∧ 𝑋𝐵) ∧ ∀𝑠𝑆 ((𝑠 · 𝑋) = 𝑍𝑠 = 0 )) ∧ 𝑓 ∈ (𝑆m {𝑋})) → ((𝑓𝑋)( ·𝑠𝑀)𝑋) ∈ 𝐵)
43 fveq2 6658 . . . . . . . . . . . . . 14 (𝑥 = 𝑋 → (𝑓𝑥) = (𝑓𝑋))
44 id 22 . . . . . . . . . . . . . 14 (𝑥 = 𝑋𝑥 = 𝑋)
4543, 44oveq12d 7168 . . . . . . . . . . . . 13 (𝑥 = 𝑋 → ((𝑓𝑥)( ·𝑠𝑀)𝑥) = ((𝑓𝑋)( ·𝑠𝑀)𝑋))
4617, 45gsumsn 19142 . . . . . . . . . . . 12 ((𝑀 ∈ Mnd ∧ 𝑋𝐵 ∧ ((𝑓𝑋)( ·𝑠𝑀)𝑋) ∈ 𝐵) → (𝑀 Σg (𝑥 ∈ {𝑋} ↦ ((𝑓𝑥)( ·𝑠𝑀)𝑥))) = ((𝑓𝑋)( ·𝑠𝑀)𝑋))
4727, 28, 42, 46syl3anc 1368 . . . . . . . . . . 11 ((((𝑀 ∈ LMod ∧ 𝑋𝐵) ∧ ∀𝑠𝑆 ((𝑠 · 𝑋) = 𝑍𝑠 = 0 )) ∧ 𝑓 ∈ (𝑆m {𝑋})) → (𝑀 Σg (𝑥 ∈ {𝑋} ↦ ((𝑓𝑥)( ·𝑠𝑀)𝑥))) = ((𝑓𝑋)( ·𝑠𝑀)𝑋))
4847eqeq1d 2760 . . . . . . . . . 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 595 . . . . . . . . . . . . . . 15 ((𝑓:{𝑋}⟶𝑆𝑋𝐵) → (𝑓𝑋) ∈ 𝑆)
5049expcom 417 . . . . . . . . . . . . . 14 (𝑋𝐵 → (𝑓:{𝑋}⟶𝑆 → (𝑓𝑋) ∈ 𝑆))
5150adantl 485 . . . . . . . . . . . . 13 ((𝑀 ∈ LMod ∧ 𝑋𝐵) → (𝑓:{𝑋}⟶𝑆 → (𝑓𝑋) ∈ 𝑆))
52 snlindsntor.t . . . . . . . . . . . . . . . . 17 · = ( ·𝑠𝑀)
5352oveqi 7163 . . . . . . . . . . . . . . . 16 ((𝑓𝑋) · 𝑋) = ((𝑓𝑋)( ·𝑠𝑀)𝑋)
5453eqeq1i 2763 . . . . . . . . . . . . . . 15 (((𝑓𝑋) · 𝑋) = 𝑍 ↔ ((𝑓𝑋)( ·𝑠𝑀)𝑋) = 𝑍)
55 oveq1 7157 . . . . . . . . . . . . . . . . . 18 (𝑠 = (𝑓𝑋) → (𝑠 · 𝑋) = ((𝑓𝑋) · 𝑋))
5655eqeq1d 2760 . . . . . . . . . . . . . . . . 17 (𝑠 = (𝑓𝑋) → ((𝑠 · 𝑋) = 𝑍 ↔ ((𝑓𝑋) · 𝑋) = 𝑍))
57 eqeq1 2762 . . . . . . . . . . . . . . . . 17 (𝑠 = (𝑓𝑋) → (𝑠 = 0 ↔ (𝑓𝑋) = 0 ))
5856, 57imbi12d 348 . . . . . . . . . . . . . . . 16 (𝑠 = (𝑓𝑋) → (((𝑠 · 𝑋) = 𝑍𝑠 = 0 ) ↔ (((𝑓𝑋) · 𝑋) = 𝑍 → (𝑓𝑋) = 0 )))
5958rspcva 3539 . . . . . . . . . . . . . . 15 (((𝑓𝑋) ∈ 𝑆 ∧ ∀𝑠𝑆 ((𝑠 · 𝑋) = 𝑍𝑠 = 0 )) → (((𝑓𝑋) · 𝑋) = 𝑍 → (𝑓𝑋) = 0 ))
6054, 59syl5bir 246 . . . . . . . . . . . . . 14 (((𝑓𝑋) ∈ 𝑆 ∧ ∀𝑠𝑆 ((𝑠 · 𝑋) = 𝑍𝑠 = 0 )) → (((𝑓𝑋)( ·𝑠𝑀)𝑋) = 𝑍 → (𝑓𝑋) = 0 ))
6160ex 416 . . . . . . . . . . . . 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 421 . . . . . . . . . 10 ((((𝑀 ∈ LMod ∧ 𝑋𝐵) ∧ ∀𝑠𝑆 ((𝑠 · 𝑋) = 𝑍𝑠 = 0 )) ∧ 𝑓 ∈ (𝑆m {𝑋})) → (((𝑓𝑋)( ·𝑠𝑀)𝑋) = 𝑍 → (𝑓𝑋) = 0 ))
6548, 64sylbid 243 . . . . . . . . 9 ((((𝑀 ∈ LMod ∧ 𝑋𝐵) ∧ ∀𝑠𝑆 ((𝑠 · 𝑋) = 𝑍𝑠 = 0 )) ∧ 𝑓 ∈ (𝑆m {𝑋})) → ((𝑀 Σg (𝑥 ∈ {𝑋} ↦ ((𝑓𝑥)( ·𝑠𝑀)𝑥))) = 𝑍 → (𝑓𝑋) = 0 ))
6665adantld 494 . . . . . . . 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 243 . . . . . . 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 416 . . . . 5 ((𝑀 ∈ LMod ∧ 𝑋𝐵) → (∀𝑠𝑆 ((𝑠 · 𝑋) = 𝑍𝑠 = 0 ) → ∀𝑓 ∈ (𝑆m {𝑋})((𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀){𝑋}) = 𝑍) → (𝑓𝑋) = 0 )))
70 impexp 454 . . . . . . . 8 (((𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀){𝑋}) = 𝑍) → (𝑓𝑋) = 0 ) ↔ (𝑓 finSupp 0 → ((𝑓( linC ‘𝑀){𝑋}) = 𝑍 → (𝑓𝑋) = 0 )))
7129adantl 485 . . . . . . . . . 10 (((𝑀 ∈ LMod ∧ 𝑋𝐵) ∧ 𝑓 ∈ (𝑆m {𝑋})) → 𝑓:{𝑋}⟶𝑆)
72 snfi 8614 . . . . . . . . . . 11 {𝑋} ∈ Fin
7372a1i 11 . . . . . . . . . 10 (((𝑀 ∈ LMod ∧ 𝑋𝐵) ∧ 𝑓 ∈ (𝑆m {𝑋})) → {𝑋} ∈ Fin)
74 snlindsntor.0 . . . . . . . . . . . 12 0 = (0g𝑅)
7574fvexi 6672 . . . . . . . . . . 11 0 ∈ V
7675a1i 11 . . . . . . . . . 10 (((𝑀 ∈ LMod ∧ 𝑋𝐵) ∧ 𝑓 ∈ (𝑆m {𝑋})) → 0 ∈ V)
7771, 73, 76fdmfifsupp 8876 . . . . . . . . 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 245 . . . . . . 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 3108 . . . . . 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 45266 . . . . . 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 215 . . . 4 ((𝑀 ∈ LMod ∧ 𝑋𝐵) → (∀𝑠𝑆 ((𝑠 · 𝑋) = 𝑍𝑠 = 0 ) ↔ ∀𝑓 ∈ (𝑆m {𝑋})((𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀){𝑋}) = 𝑍) → (𝑓𝑋) = 0 )))
86 fveqeq2 6667 . . . . . . . . 9 (𝑦 = 𝑋 → ((𝑓𝑦) = 0 ↔ (𝑓𝑋) = 0 ))
8786ralsng 4570 . . . . . . . 8 (𝑋𝐵 → (∀𝑦 ∈ {𝑋} (𝑓𝑦) = 0 ↔ (𝑓𝑋) = 0 ))
8887adantl 485 . . . . . . 7 ((𝑀 ∈ LMod ∧ 𝑋𝐵) → (∀𝑦 ∈ {𝑋} (𝑓𝑦) = 0 ↔ (𝑓𝑋) = 0 ))
8988bicomd 226 . . . . . 6 ((𝑀 ∈ LMod ∧ 𝑋𝐵) → ((𝑓𝑋) = 0 ↔ ∀𝑦 ∈ {𝑋} (𝑓𝑦) = 0 ))
9089imbi2d 344 . . . . 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 3126 . . . 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 5305 . . . . . 6 (𝑋𝐵 → {𝑋} ∈ 𝒫 𝐵)
9392adantl 485 . . . . 5 ((𝑀 ∈ LMod ∧ 𝑋𝐵) → {𝑋} ∈ 𝒫 𝐵)
9493biantrurd 536 . . . 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 308 . . 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 286 . 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 5300 . . 3 {𝑋} ∈ V
9817, 82, 9, 8, 74islininds 45242 . . 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 590 . 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 285 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 209   ∧ wa 399   = wceq 1538   ∈ wcel 2111   ≠ wne 2951  ∀wral 3070  Vcvv 3409   ∖ cdif 3855  𝒫 cpw 4494  {csn 4522   class class class wbr 5032   ↦ cmpt 5112  ⟶wf 6331  ‘cfv 6335  (class class class)co 7150   ↑m cmap 8416  Fincfn 8527   finSupp cfsupp 8866  Basecbs 16541  Scalarcsca 16626   ·𝑠 cvsca 16627  0gc0g 16771   Σg cgsu 16772  Mndcmnd 17977  Grpcgrp 18169  LModclmod 19702   linC clinc 45200   linIndS clininds 45236 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-rep 5156  ax-sep 5169  ax-nul 5176  ax-pow 5234  ax-pr 5298  ax-un 7459  ax-cnex 10631  ax-resscn 10632  ax-1cn 10633  ax-icn 10634  ax-addcl 10635  ax-addrcl 10636  ax-mulcl 10637  ax-mulrcl 10638  ax-mulcom 10639  ax-addass 10640  ax-mulass 10641  ax-distr 10642  ax-i2m1 10643  ax-1ne0 10644  ax-1rid 10645  ax-rnegex 10646  ax-rrecex 10647  ax-cnre 10648  ax-pre-lttri 10649  ax-pre-lttrn 10650  ax-pre-ltadd 10651  ax-pre-mulgt0 10652 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-nel 3056  df-ral 3075  df-rex 3076  df-reu 3077  df-rmo 3078  df-rab 3079  df-v 3411  df-sbc 3697  df-csb 3806  df-dif 3861  df-un 3863  df-in 3865  df-ss 3875  df-pss 3877  df-nul 4226  df-if 4421  df-pw 4496  df-sn 4523  df-pr 4525  df-tp 4527  df-op 4529  df-uni 4799  df-int 4839  df-iun 4885  df-br 5033  df-opab 5095  df-mpt 5113  df-tr 5139  df-id 5430  df-eprel 5435  df-po 5443  df-so 5444  df-fr 5483  df-se 5484  df-we 5485  df-xp 5530  df-rel 5531  df-cnv 5532  df-co 5533  df-dm 5534  df-rn 5535  df-res 5536  df-ima 5537  df-pred 6126  df-ord 6172  df-on 6173  df-lim 6174  df-suc 6175  df-iota 6294  df-fun 6337  df-fn 6338  df-f 6339  df-f1 6340  df-fo 6341  df-f1o 6342  df-fv 6343  df-isom 6344  df-riota 7108  df-ov 7153  df-oprab 7154  df-mpo 7155  df-om 7580  df-1st 7693  df-2nd 7694  df-supp 7836  df-wrecs 7957  df-recs 8018  df-rdg 8056  df-1o 8112  df-er 8299  df-map 8418  df-en 8528  df-dom 8529  df-sdom 8530  df-fin 8531  df-fsupp 8867  df-oi 9007  df-card 9401  df-pnf 10715  df-mnf 10716  df-xr 10717  df-ltxr 10718  df-le 10719  df-sub 10910  df-neg 10911  df-nn 11675  df-n0 11935  df-z 12021  df-uz 12283  df-fz 12940  df-fzo 13083  df-seq 13419  df-hash 13741  df-0g 16773  df-gsum 16774  df-mgm 17918  df-sgrp 17967  df-mnd 17978  df-grp 18172  df-mulg 18292  df-cntz 18514  df-lmod 19704  df-linc 45202  df-lininds 45238 This theorem is referenced by:  lindssnlvec  45282
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