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Theorem snlindsntor 47854
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 2931 . . . . 5 ((𝑠 · 𝑋) ≠ 𝑍 ↔ ¬ (𝑠 · 𝑋) = 𝑍)
21ralbii 3083 . . . 4 (∀𝑠 ∈ (𝑆 ∖ { 0 })(𝑠 · 𝑋) ≠ 𝑍 ↔ ∀𝑠 ∈ (𝑆 ∖ { 0 }) ¬ (𝑠 · 𝑋) = 𝑍)
3 raldifsni 4804 . . . 4 (∀𝑠 ∈ (𝑆 ∖ { 0 }) ¬ (𝑠 · 𝑋) = 𝑍 ↔ ∀𝑠𝑆 ((𝑠 · 𝑋) = 𝑍𝑠 = 0 ))
42, 3bitri 274 . . 3 (∀𝑠 ∈ (𝑆 ∖ { 0 })(𝑠 · 𝑋) ≠ 𝑍 ↔ ∀𝑠𝑆 ((𝑠 · 𝑋) = 𝑍𝑠 = 0 ))
5 simpl 481 . . . . . . . . . . . . 13 ((𝑀 ∈ LMod ∧ 𝑋𝐵) → 𝑀 ∈ LMod)
65adantr 479 . . . . . . . . . . . 12 (((𝑀 ∈ LMod ∧ 𝑋𝐵) ∧ ∀𝑠𝑆 ((𝑠 · 𝑋) = 𝑍𝑠 = 0 )) → 𝑀 ∈ LMod)
76adantr 479 . . . . . . . . . . 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 6904 . . . . . . . . . . . . . . . 16 (Base‘𝑅) = (Base‘(Scalar‘𝑀))
118, 10eqtri 2754 . . . . . . . . . . . . . . 15 𝑆 = (Base‘(Scalar‘𝑀))
1211oveq1i 7434 . . . . . . . . . . . . . 14 (𝑆m {𝑋}) = ((Base‘(Scalar‘𝑀)) ↑m {𝑋})
1312eleq2i 2818 . . . . . . . . . . . . 13 (𝑓 ∈ (𝑆m {𝑋}) ↔ 𝑓 ∈ ((Base‘(Scalar‘𝑀)) ↑m {𝑋}))
1413biimpi 215 . . . . . . . . . . . 12 (𝑓 ∈ (𝑆m {𝑋}) → 𝑓 ∈ ((Base‘(Scalar‘𝑀)) ↑m {𝑋}))
1514adantl 480 . . . . . . . . . . 11 ((((𝑀 ∈ LMod ∧ 𝑋𝐵) ∧ ∀𝑠𝑆 ((𝑠 · 𝑋) = 𝑍𝑠 = 0 )) ∧ 𝑓 ∈ (𝑆m {𝑋})) → 𝑓 ∈ ((Base‘(Scalar‘𝑀)) ↑m {𝑋}))
16 snelpwi 5449 . . . . . . . . . . . . 13 (𝑋 ∈ (Base‘𝑀) → {𝑋} ∈ 𝒫 (Base‘𝑀))
17 snlindsntor.b . . . . . . . . . . . . 13 𝐵 = (Base‘𝑀)
1816, 17eleq2s 2844 . . . . . . . . . . . 12 (𝑋𝐵 → {𝑋} ∈ 𝒫 (Base‘𝑀))
1918ad3antlr 729 . . . . . . . . . . 11 ((((𝑀 ∈ LMod ∧ 𝑋𝐵) ∧ ∀𝑠𝑆 ((𝑠 · 𝑋) = 𝑍𝑠 = 0 )) ∧ 𝑓 ∈ (𝑆m {𝑋})) → {𝑋} ∈ 𝒫 (Base‘𝑀))
20 lincval 47792 . . . . . . . . . . 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 2728 . . . . . . . . 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 628 . . . . . . . 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 20843 . . . . . . . . . . . . . 14 (𝑀 ∈ LMod → 𝑀 ∈ Grp)
2524grpmndd 18941 . . . . . . . . . . . . 13 (𝑀 ∈ LMod → 𝑀 ∈ Mnd)
2625ad3antrrr 728 . . . . . . . . . . . 12 ((((𝑀 ∈ LMod ∧ 𝑋𝐵) ∧ ∀𝑠𝑆 ((𝑠 · 𝑋) = 𝑍𝑠 = 0 )) ∧ 𝑓 ∈ (𝑆m {𝑋})) → 𝑀 ∈ Mnd)
27 simpllr 774 . . . . . . . . . . . 12 ((((𝑀 ∈ LMod ∧ 𝑋𝐵) ∧ ∀𝑠𝑆 ((𝑠 · 𝑋) = 𝑍𝑠 = 0 )) ∧ 𝑓 ∈ (𝑆m {𝑋})) → 𝑋𝐵)
28 elmapi 8878 . . . . . . . . . . . . . 14 (𝑓 ∈ (𝑆m {𝑋}) → 𝑓:{𝑋}⟶𝑆)
296adantl 480 . . . . . . . . . . . . . . . 16 ((𝑓:{𝑋}⟶𝑆 ∧ ((𝑀 ∈ LMod ∧ 𝑋𝐵) ∧ ∀𝑠𝑆 ((𝑠 · 𝑋) = 𝑍𝑠 = 0 ))) → 𝑀 ∈ LMod)
30 snidg 4667 . . . . . . . . . . . . . . . . . . 19 (𝑋𝐵𝑋 ∈ {𝑋})
3130adantl 480 . . . . . . . . . . . . . . . . . 18 ((𝑀 ∈ LMod ∧ 𝑋𝐵) → 𝑋 ∈ {𝑋})
3231adantr 479 . . . . . . . . . . . . . . . . 17 (((𝑀 ∈ LMod ∧ 𝑋𝐵) ∧ ∀𝑠𝑆 ((𝑠 · 𝑋) = 𝑍𝑠 = 0 )) → 𝑋 ∈ {𝑋})
33 ffvelcdm 7095 . . . . . . . . . . . . . . . . 17 ((𝑓:{𝑋}⟶𝑆𝑋 ∈ {𝑋}) → (𝑓𝑋) ∈ 𝑆)
3432, 33sylan2 591 . . . . . . . . . . . . . . . 16 ((𝑓:{𝑋}⟶𝑆 ∧ ((𝑀 ∈ LMod ∧ 𝑋𝐵) ∧ ∀𝑠𝑆 ((𝑠 · 𝑋) = 𝑍𝑠 = 0 ))) → (𝑓𝑋) ∈ 𝑆)
35 simprlr 778 . . . . . . . . . . . . . . . 16 ((𝑓:{𝑋}⟶𝑆 ∧ ((𝑀 ∈ LMod ∧ 𝑋𝐵) ∧ ∀𝑠𝑆 ((𝑠 · 𝑋) = 𝑍𝑠 = 0 ))) → 𝑋𝐵)
36 eqid 2726 . . . . . . . . . . . . . . . . 17 ( ·𝑠𝑀) = ( ·𝑠𝑀)
3717, 9, 36, 8lmodvscl 20854 . . . . . . . . . . . . . . . 16 ((𝑀 ∈ LMod ∧ (𝑓𝑋) ∈ 𝑆𝑋𝐵) → ((𝑓𝑋)( ·𝑠𝑀)𝑋) ∈ 𝐵)
3829, 34, 35, 37syl3anc 1368 . . . . . . . . . . . . . . 15 ((𝑓:{𝑋}⟶𝑆 ∧ ((𝑀 ∈ LMod ∧ 𝑋𝐵) ∧ ∀𝑠𝑆 ((𝑠 · 𝑋) = 𝑍𝑠 = 0 ))) → ((𝑓𝑋)( ·𝑠𝑀)𝑋) ∈ 𝐵)
3938expcom 412 . . . . . . . . . . . . . 14 (((𝑀 ∈ LMod ∧ 𝑋𝐵) ∧ ∀𝑠𝑆 ((𝑠 · 𝑋) = 𝑍𝑠 = 0 )) → (𝑓:{𝑋}⟶𝑆 → ((𝑓𝑋)( ·𝑠𝑀)𝑋) ∈ 𝐵))
4028, 39syl5com 31 . . . . . . . . . . . . 13 (𝑓 ∈ (𝑆m {𝑋}) → (((𝑀 ∈ LMod ∧ 𝑋𝐵) ∧ ∀𝑠𝑆 ((𝑠 · 𝑋) = 𝑍𝑠 = 0 )) → ((𝑓𝑋)( ·𝑠𝑀)𝑋) ∈ 𝐵))
4140impcom 406 . . . . . . . . . . . 12 ((((𝑀 ∈ LMod ∧ 𝑋𝐵) ∧ ∀𝑠𝑆 ((𝑠 · 𝑋) = 𝑍𝑠 = 0 )) ∧ 𝑓 ∈ (𝑆m {𝑋})) → ((𝑓𝑋)( ·𝑠𝑀)𝑋) ∈ 𝐵)
42 fveq2 6901 . . . . . . . . . . . . . 14 (𝑥 = 𝑋 → (𝑓𝑥) = (𝑓𝑋))
43 id 22 . . . . . . . . . . . . . 14 (𝑥 = 𝑋𝑥 = 𝑋)
4442, 43oveq12d 7442 . . . . . . . . . . . . 13 (𝑥 = 𝑋 → ((𝑓𝑥)( ·𝑠𝑀)𝑥) = ((𝑓𝑋)( ·𝑠𝑀)𝑋))
4517, 44gsumsn 19952 . . . . . . . . . . . 12 ((𝑀 ∈ Mnd ∧ 𝑋𝐵 ∧ ((𝑓𝑋)( ·𝑠𝑀)𝑋) ∈ 𝐵) → (𝑀 Σg (𝑥 ∈ {𝑋} ↦ ((𝑓𝑥)( ·𝑠𝑀)𝑥))) = ((𝑓𝑋)( ·𝑠𝑀)𝑋))
4626, 27, 41, 45syl3anc 1368 . . . . . . . . . . 11 ((((𝑀 ∈ LMod ∧ 𝑋𝐵) ∧ ∀𝑠𝑆 ((𝑠 · 𝑋) = 𝑍𝑠 = 0 )) ∧ 𝑓 ∈ (𝑆m {𝑋})) → (𝑀 Σg (𝑥 ∈ {𝑋} ↦ ((𝑓𝑥)( ·𝑠𝑀)𝑥))) = ((𝑓𝑋)( ·𝑠𝑀)𝑋))
4746eqeq1d 2728 . . . . . . . . . 10 ((((𝑀 ∈ LMod ∧ 𝑋𝐵) ∧ ∀𝑠𝑆 ((𝑠 · 𝑋) = 𝑍𝑠 = 0 )) ∧ 𝑓 ∈ (𝑆m {𝑋})) → ((𝑀 Σg (𝑥 ∈ {𝑋} ↦ ((𝑓𝑥)( ·𝑠𝑀)𝑥))) = 𝑍 ↔ ((𝑓𝑋)( ·𝑠𝑀)𝑋) = 𝑍))
4830, 33sylan2 591 . . . . . . . . . . . . . . 15 ((𝑓:{𝑋}⟶𝑆𝑋𝐵) → (𝑓𝑋) ∈ 𝑆)
4948expcom 412 . . . . . . . . . . . . . 14 (𝑋𝐵 → (𝑓:{𝑋}⟶𝑆 → (𝑓𝑋) ∈ 𝑆))
5049adantl 480 . . . . . . . . . . . . 13 ((𝑀 ∈ LMod ∧ 𝑋𝐵) → (𝑓:{𝑋}⟶𝑆 → (𝑓𝑋) ∈ 𝑆))
51 snlindsntor.t . . . . . . . . . . . . . . . . 17 · = ( ·𝑠𝑀)
5251oveqi 7437 . . . . . . . . . . . . . . . 16 ((𝑓𝑋) · 𝑋) = ((𝑓𝑋)( ·𝑠𝑀)𝑋)
5352eqeq1i 2731 . . . . . . . . . . . . . . 15 (((𝑓𝑋) · 𝑋) = 𝑍 ↔ ((𝑓𝑋)( ·𝑠𝑀)𝑋) = 𝑍)
54 oveq1 7431 . . . . . . . . . . . . . . . . . 18 (𝑠 = (𝑓𝑋) → (𝑠 · 𝑋) = ((𝑓𝑋) · 𝑋))
5554eqeq1d 2728 . . . . . . . . . . . . . . . . 17 (𝑠 = (𝑓𝑋) → ((𝑠 · 𝑋) = 𝑍 ↔ ((𝑓𝑋) · 𝑋) = 𝑍))
56 eqeq1 2730 . . . . . . . . . . . . . . . . 17 (𝑠 = (𝑓𝑋) → (𝑠 = 0 ↔ (𝑓𝑋) = 0 ))
5755, 56imbi12d 343 . . . . . . . . . . . . . . . 16 (𝑠 = (𝑓𝑋) → (((𝑠 · 𝑋) = 𝑍𝑠 = 0 ) ↔ (((𝑓𝑋) · 𝑋) = 𝑍 → (𝑓𝑋) = 0 )))
5857rspcva 3606 . . . . . . . . . . . . . . 15 (((𝑓𝑋) ∈ 𝑆 ∧ ∀𝑠𝑆 ((𝑠 · 𝑋) = 𝑍𝑠 = 0 )) → (((𝑓𝑋) · 𝑋) = 𝑍 → (𝑓𝑋) = 0 ))
5953, 58biimtrrid 242 . . . . . . . . . . . . . 14 (((𝑓𝑋) ∈ 𝑆 ∧ ∀𝑠𝑆 ((𝑠 · 𝑋) = 𝑍𝑠 = 0 )) → (((𝑓𝑋)( ·𝑠𝑀)𝑋) = 𝑍 → (𝑓𝑋) = 0 ))
6059ex 411 . . . . . . . . . . . . 13 ((𝑓𝑋) ∈ 𝑆 → (∀𝑠𝑆 ((𝑠 · 𝑋) = 𝑍𝑠 = 0 ) → (((𝑓𝑋)( ·𝑠𝑀)𝑋) = 𝑍 → (𝑓𝑋) = 0 )))
6128, 50, 60syl56 36 . . . . . . . . . . . 12 ((𝑀 ∈ LMod ∧ 𝑋𝐵) → (𝑓 ∈ (𝑆m {𝑋}) → (∀𝑠𝑆 ((𝑠 · 𝑋) = 𝑍𝑠 = 0 ) → (((𝑓𝑋)( ·𝑠𝑀)𝑋) = 𝑍 → (𝑓𝑋) = 0 ))))
6261com23 86 . . . . . . . . . . 11 ((𝑀 ∈ LMod ∧ 𝑋𝐵) → (∀𝑠𝑆 ((𝑠 · 𝑋) = 𝑍𝑠 = 0 ) → (𝑓 ∈ (𝑆m {𝑋}) → (((𝑓𝑋)( ·𝑠𝑀)𝑋) = 𝑍 → (𝑓𝑋) = 0 ))))
6362imp31 416 . . . . . . . . . 10 ((((𝑀 ∈ LMod ∧ 𝑋𝐵) ∧ ∀𝑠𝑆 ((𝑠 · 𝑋) = 𝑍𝑠 = 0 )) ∧ 𝑓 ∈ (𝑆m {𝑋})) → (((𝑓𝑋)( ·𝑠𝑀)𝑋) = 𝑍 → (𝑓𝑋) = 0 ))
6447, 63sylbid 239 . . . . . . . . 9 ((((𝑀 ∈ LMod ∧ 𝑋𝐵) ∧ ∀𝑠𝑆 ((𝑠 · 𝑋) = 𝑍𝑠 = 0 )) ∧ 𝑓 ∈ (𝑆m {𝑋})) → ((𝑀 Σg (𝑥 ∈ {𝑋} ↦ ((𝑓𝑥)( ·𝑠𝑀)𝑥))) = 𝑍 → (𝑓𝑋) = 0 ))
6564adantld 489 . . . . . . . 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 ))
6623, 65sylbid 239 . . . . . . 7 ((((𝑀 ∈ LMod ∧ 𝑋𝐵) ∧ ∀𝑠𝑆 ((𝑠 · 𝑋) = 𝑍𝑠 = 0 )) ∧ 𝑓 ∈ (𝑆m {𝑋})) → ((𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀){𝑋}) = 𝑍) → (𝑓𝑋) = 0 ))
6766ralrimiva 3136 . . . . . 6 (((𝑀 ∈ LMod ∧ 𝑋𝐵) ∧ ∀𝑠𝑆 ((𝑠 · 𝑋) = 𝑍𝑠 = 0 )) → ∀𝑓 ∈ (𝑆m {𝑋})((𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀){𝑋}) = 𝑍) → (𝑓𝑋) = 0 ))
6867ex 411 . . . . 5 ((𝑀 ∈ LMod ∧ 𝑋𝐵) → (∀𝑠𝑆 ((𝑠 · 𝑋) = 𝑍𝑠 = 0 ) → ∀𝑓 ∈ (𝑆m {𝑋})((𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀){𝑋}) = 𝑍) → (𝑓𝑋) = 0 )))
69 impexp 449 . . . . . . . 8 (((𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀){𝑋}) = 𝑍) → (𝑓𝑋) = 0 ) ↔ (𝑓 finSupp 0 → ((𝑓( linC ‘𝑀){𝑋}) = 𝑍 → (𝑓𝑋) = 0 )))
7028adantl 480 . . . . . . . . . 10 (((𝑀 ∈ LMod ∧ 𝑋𝐵) ∧ 𝑓 ∈ (𝑆m {𝑋})) → 𝑓:{𝑋}⟶𝑆)
71 snfi 9081 . . . . . . . . . . 11 {𝑋} ∈ Fin
7271a1i 11 . . . . . . . . . 10 (((𝑀 ∈ LMod ∧ 𝑋𝐵) ∧ 𝑓 ∈ (𝑆m {𝑋})) → {𝑋} ∈ Fin)
73 snlindsntor.0 . . . . . . . . . . . 12 0 = (0g𝑅)
7473fvexi 6915 . . . . . . . . . . 11 0 ∈ V
7574a1i 11 . . . . . . . . . 10 (((𝑀 ∈ LMod ∧ 𝑋𝐵) ∧ 𝑓 ∈ (𝑆m {𝑋})) → 0 ∈ V)
7670, 72, 75fdmfifsupp 9418 . . . . . . . . 9 (((𝑀 ∈ LMod ∧ 𝑋𝐵) ∧ 𝑓 ∈ (𝑆m {𝑋})) → 𝑓 finSupp 0 )
77 pm2.27 42 . . . . . . . . 9 (𝑓 finSupp 0 → ((𝑓 finSupp 0 → ((𝑓( linC ‘𝑀){𝑋}) = 𝑍 → (𝑓𝑋) = 0 )) → ((𝑓( linC ‘𝑀){𝑋}) = 𝑍 → (𝑓𝑋) = 0 )))
7876, 77syl 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 )))
7969, 78biimtrid 241 . . . . . . 7 (((𝑀 ∈ LMod ∧ 𝑋𝐵) ∧ 𝑓 ∈ (𝑆m {𝑋})) → (((𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀){𝑋}) = 𝑍) → (𝑓𝑋) = 0 ) → ((𝑓( linC ‘𝑀){𝑋}) = 𝑍 → (𝑓𝑋) = 0 )))
8079ralimdva 3157 . . . . . 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 )))
81 snlindsntor.z . . . . . . 7 𝑍 = (0g𝑀)
8217, 9, 8, 73, 81, 51snlindsntorlem 47853 . . . . . 6 ((𝑀 ∈ LMod ∧ 𝑋𝐵) → (∀𝑓 ∈ (𝑆m {𝑋})((𝑓( linC ‘𝑀){𝑋}) = 𝑍 → (𝑓𝑋) = 0 ) → ∀𝑠𝑆 ((𝑠 · 𝑋) = 𝑍𝑠 = 0 )))
8380, 82syld 47 . . . . 5 ((𝑀 ∈ LMod ∧ 𝑋𝐵) → (∀𝑓 ∈ (𝑆m {𝑋})((𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀){𝑋}) = 𝑍) → (𝑓𝑋) = 0 ) → ∀𝑠𝑆 ((𝑠 · 𝑋) = 𝑍𝑠 = 0 )))
8468, 83impbid 211 . . . 4 ((𝑀 ∈ LMod ∧ 𝑋𝐵) → (∀𝑠𝑆 ((𝑠 · 𝑋) = 𝑍𝑠 = 0 ) ↔ ∀𝑓 ∈ (𝑆m {𝑋})((𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀){𝑋}) = 𝑍) → (𝑓𝑋) = 0 )))
85 fveqeq2 6910 . . . . . . . . 9 (𝑦 = 𝑋 → ((𝑓𝑦) = 0 ↔ (𝑓𝑋) = 0 ))
8685ralsng 4682 . . . . . . . 8 (𝑋𝐵 → (∀𝑦 ∈ {𝑋} (𝑓𝑦) = 0 ↔ (𝑓𝑋) = 0 ))
8786adantl 480 . . . . . . 7 ((𝑀 ∈ LMod ∧ 𝑋𝐵) → (∀𝑦 ∈ {𝑋} (𝑓𝑦) = 0 ↔ (𝑓𝑋) = 0 ))
8887bicomd 222 . . . . . 6 ((𝑀 ∈ LMod ∧ 𝑋𝐵) → ((𝑓𝑋) = 0 ↔ ∀𝑦 ∈ {𝑋} (𝑓𝑦) = 0 ))
8988imbi2d 339 . . . . 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 )))
9089ralbidv 3168 . . . 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 )))
91 snelpwi 5449 . . . . . 6 (𝑋𝐵 → {𝑋} ∈ 𝒫 𝐵)
9291adantl 480 . . . . 5 ((𝑀 ∈ LMod ∧ 𝑋𝐵) → {𝑋} ∈ 𝒫 𝐵)
9392biantrurd 531 . . . 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 ))))
9484, 90, 933bitrd 304 . . 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 ))))
954, 94bitrid 282 . 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 ))))
96 snex 5437 . . 3 {𝑋} ∈ V
9717, 81, 9, 8, 73islininds 47829 . . 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 ))))
9896, 5, 97sylancr 585 . 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 ))))
9995, 98bitr4d 281 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 205  wa 394   = wceq 1534  wcel 2099  wne 2930  wral 3051  Vcvv 3462  cdif 3944  𝒫 cpw 4607  {csn 4633   class class class wbr 5153  cmpt 5236  wf 6550  cfv 6554  (class class class)co 7424  m cmap 8855  Fincfn 8974   finSupp cfsupp 9405  Basecbs 17213  Scalarcsca 17269   ·𝑠 cvsca 17270  0gc0g 17454   Σg cgsu 17455  Mndcmnd 18727  LModclmod 20836   linC clinc 47787   linIndS clininds 47823
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-rep 5290  ax-sep 5304  ax-nul 5311  ax-pow 5369  ax-pr 5433  ax-un 7746  ax-cnex 11214  ax-resscn 11215  ax-1cn 11216  ax-icn 11217  ax-addcl 11218  ax-addrcl 11219  ax-mulcl 11220  ax-mulrcl 11221  ax-mulcom 11222  ax-addass 11223  ax-mulass 11224  ax-distr 11225  ax-i2m1 11226  ax-1ne0 11227  ax-1rid 11228  ax-rnegex 11229  ax-rrecex 11230  ax-cnre 11231  ax-pre-lttri 11232  ax-pre-lttrn 11233  ax-pre-ltadd 11234  ax-pre-mulgt0 11235
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-nel 3037  df-ral 3052  df-rex 3061  df-rmo 3364  df-reu 3365  df-rab 3420  df-v 3464  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3967  df-nul 4326  df-if 4534  df-pw 4609  df-sn 4634  df-pr 4636  df-op 4640  df-uni 4914  df-int 4955  df-iun 5003  df-br 5154  df-opab 5216  df-mpt 5237  df-tr 5271  df-id 5580  df-eprel 5586  df-po 5594  df-so 5595  df-fr 5637  df-se 5638  df-we 5639  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-pred 6312  df-ord 6379  df-on 6380  df-lim 6381  df-suc 6382  df-iota 6506  df-fun 6556  df-fn 6557  df-f 6558  df-f1 6559  df-fo 6560  df-f1o 6561  df-fv 6562  df-isom 6563  df-riota 7380  df-ov 7427  df-oprab 7428  df-mpo 7429  df-om 7877  df-1st 8003  df-2nd 8004  df-supp 8175  df-frecs 8296  df-wrecs 8327  df-recs 8401  df-rdg 8440  df-1o 8496  df-er 8734  df-map 8857  df-en 8975  df-dom 8976  df-sdom 8977  df-fin 8978  df-fsupp 9406  df-oi 9553  df-card 9982  df-pnf 11300  df-mnf 11301  df-xr 11302  df-ltxr 11303  df-le 11304  df-sub 11496  df-neg 11497  df-nn 12265  df-n0 12525  df-z 12611  df-uz 12875  df-fz 13539  df-fzo 13682  df-seq 14022  df-hash 14348  df-0g 17456  df-gsum 17457  df-mgm 18633  df-sgrp 18712  df-mnd 18728  df-grp 18931  df-mulg 19062  df-cntz 19311  df-lmod 20838  df-linc 47789  df-lininds 47825
This theorem is referenced by:  lindssnlvec  47869
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