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Theorem snlindsntor 46642
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., (π‘Ÿ Β· π‘š) = 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 π‘Ž ∈ 𝑅, π‘Ž β‰  0, such that π‘Ž Β· π‘₯ = 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 𝐡 = (Baseβ€˜π‘€)
snlindsntor.r 𝑅 = (Scalarβ€˜π‘€)
snlindsntor.s 𝑆 = (Baseβ€˜π‘…)
snlindsntor.0 0 = (0gβ€˜π‘…)
snlindsntor.z 𝑍 = (0gβ€˜π‘€)
snlindsntor.t Β· = ( ·𝑠 β€˜π‘€)
Assertion
Ref Expression
snlindsntor ((𝑀 ∈ LMod ∧ 𝑋 ∈ 𝐡) β†’ (βˆ€π‘  ∈ (𝑆 βˆ– { 0 })(𝑠 Β· 𝑋) β‰  𝑍 ↔ {𝑋} linIndS 𝑀))
Distinct variable groups:   𝐡,𝑠   𝑀,𝑠   𝑆,𝑠   𝑋,𝑠   𝑍,𝑠   Β· ,𝑠   0 ,𝑠
Allowed substitution hint:   𝑅(𝑠)

Proof of Theorem snlindsntor
Dummy variables 𝑓 π‘₯ 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-ne 2941 . . . . 5 ((𝑠 Β· 𝑋) β‰  𝑍 ↔ Β¬ (𝑠 Β· 𝑋) = 𝑍)
21ralbii 3093 . . . 4 (βˆ€π‘  ∈ (𝑆 βˆ– { 0 })(𝑠 Β· 𝑋) β‰  𝑍 ↔ βˆ€π‘  ∈ (𝑆 βˆ– { 0 }) Β¬ (𝑠 Β· 𝑋) = 𝑍)
3 raldifsni 4759 . . . 4 (βˆ€π‘  ∈ (𝑆 βˆ– { 0 }) Β¬ (𝑠 Β· 𝑋) = 𝑍 ↔ βˆ€π‘  ∈ 𝑆 ((𝑠 Β· 𝑋) = 𝑍 β†’ 𝑠 = 0 ))
42, 3bitri 275 . . 3 (βˆ€π‘  ∈ (𝑆 βˆ– { 0 })(𝑠 Β· 𝑋) β‰  𝑍 ↔ βˆ€π‘  ∈ 𝑆 ((𝑠 Β· 𝑋) = 𝑍 β†’ 𝑠 = 0 ))
5 simpl 484 . . . . . . . . . . . . 13 ((𝑀 ∈ LMod ∧ 𝑋 ∈ 𝐡) β†’ 𝑀 ∈ LMod)
65adantr 482 . . . . . . . . . . . 12 (((𝑀 ∈ LMod ∧ 𝑋 ∈ 𝐡) ∧ βˆ€π‘  ∈ 𝑆 ((𝑠 Β· 𝑋) = 𝑍 β†’ 𝑠 = 0 )) β†’ 𝑀 ∈ LMod)
76adantr 482 . . . . . . . . . . 11 ((((𝑀 ∈ LMod ∧ 𝑋 ∈ 𝐡) ∧ βˆ€π‘  ∈ 𝑆 ((𝑠 Β· 𝑋) = 𝑍 β†’ 𝑠 = 0 )) ∧ 𝑓 ∈ (𝑆 ↑m {𝑋})) β†’ 𝑀 ∈ LMod)
8 snlindsntor.s . . . . . . . . . . . . . . . 16 𝑆 = (Baseβ€˜π‘…)
9 snlindsntor.r . . . . . . . . . . . . . . . . 17 𝑅 = (Scalarβ€˜π‘€)
109fveq2i 6849 . . . . . . . . . . . . . . . 16 (Baseβ€˜π‘…) = (Baseβ€˜(Scalarβ€˜π‘€))
118, 10eqtri 2761 . . . . . . . . . . . . . . 15 𝑆 = (Baseβ€˜(Scalarβ€˜π‘€))
1211oveq1i 7371 . . . . . . . . . . . . . 14 (𝑆 ↑m {𝑋}) = ((Baseβ€˜(Scalarβ€˜π‘€)) ↑m {𝑋})
1312eleq2i 2826 . . . . . . . . . . . . 13 (𝑓 ∈ (𝑆 ↑m {𝑋}) ↔ 𝑓 ∈ ((Baseβ€˜(Scalarβ€˜π‘€)) ↑m {𝑋}))
1413biimpi 215 . . . . . . . . . . . 12 (𝑓 ∈ (𝑆 ↑m {𝑋}) β†’ 𝑓 ∈ ((Baseβ€˜(Scalarβ€˜π‘€)) ↑m {𝑋}))
1514adantl 483 . . . . . . . . . . 11 ((((𝑀 ∈ LMod ∧ 𝑋 ∈ 𝐡) ∧ βˆ€π‘  ∈ 𝑆 ((𝑠 Β· 𝑋) = 𝑍 β†’ 𝑠 = 0 )) ∧ 𝑓 ∈ (𝑆 ↑m {𝑋})) β†’ 𝑓 ∈ ((Baseβ€˜(Scalarβ€˜π‘€)) ↑m {𝑋}))
16 snelpwi 5404 . . . . . . . . . . . . 13 (𝑋 ∈ (Baseβ€˜π‘€) β†’ {𝑋} ∈ 𝒫 (Baseβ€˜π‘€))
17 snlindsntor.b . . . . . . . . . . . . 13 𝐡 = (Baseβ€˜π‘€)
1816, 17eleq2s 2852 . . . . . . . . . . . 12 (𝑋 ∈ 𝐡 β†’ {𝑋} ∈ 𝒫 (Baseβ€˜π‘€))
1918ad3antlr 730 . . . . . . . . . . 11 ((((𝑀 ∈ LMod ∧ 𝑋 ∈ 𝐡) ∧ βˆ€π‘  ∈ 𝑆 ((𝑠 Β· 𝑋) = 𝑍 β†’ 𝑠 = 0 )) ∧ 𝑓 ∈ (𝑆 ↑m {𝑋})) β†’ {𝑋} ∈ 𝒫 (Baseβ€˜π‘€))
20 lincval 46580 . . . . . . . . . . 11 ((𝑀 ∈ LMod ∧ 𝑓 ∈ ((Baseβ€˜(Scalarβ€˜π‘€)) ↑m {𝑋}) ∧ {𝑋} ∈ 𝒫 (Baseβ€˜π‘€)) β†’ (𝑓( linC β€˜π‘€){𝑋}) = (𝑀 Ξ£g (π‘₯ ∈ {𝑋} ↦ ((π‘“β€˜π‘₯)( ·𝑠 β€˜π‘€)π‘₯))))
217, 15, 19, 20syl3anc 1372 . . . . . . . . . 10 ((((𝑀 ∈ LMod ∧ 𝑋 ∈ 𝐡) ∧ βˆ€π‘  ∈ 𝑆 ((𝑠 Β· 𝑋) = 𝑍 β†’ 𝑠 = 0 )) ∧ 𝑓 ∈ (𝑆 ↑m {𝑋})) β†’ (𝑓( linC β€˜π‘€){𝑋}) = (𝑀 Ξ£g (π‘₯ ∈ {𝑋} ↦ ((π‘“β€˜π‘₯)( ·𝑠 β€˜π‘€)π‘₯))))
2221eqeq1d 2735 . . . . . . . . 9 ((((𝑀 ∈ LMod ∧ 𝑋 ∈ 𝐡) ∧ βˆ€π‘  ∈ 𝑆 ((𝑠 Β· 𝑋) = 𝑍 β†’ 𝑠 = 0 )) ∧ 𝑓 ∈ (𝑆 ↑m {𝑋})) β†’ ((𝑓( linC β€˜π‘€){𝑋}) = 𝑍 ↔ (𝑀 Ξ£g (π‘₯ ∈ {𝑋} ↦ ((π‘“β€˜π‘₯)( ·𝑠 β€˜π‘€)π‘₯))) = 𝑍))
2322anbi2d 630 . . . . . . . 8 ((((𝑀 ∈ LMod ∧ 𝑋 ∈ 𝐡) ∧ βˆ€π‘  ∈ 𝑆 ((𝑠 Β· 𝑋) = 𝑍 β†’ 𝑠 = 0 )) ∧ 𝑓 ∈ (𝑆 ↑m {𝑋})) β†’ ((𝑓 finSupp 0 ∧ (𝑓( linC β€˜π‘€){𝑋}) = 𝑍) ↔ (𝑓 finSupp 0 ∧ (𝑀 Ξ£g (π‘₯ ∈ {𝑋} ↦ ((π‘“β€˜π‘₯)( ·𝑠 β€˜π‘€)π‘₯))) = 𝑍)))
24 lmodgrp 20372 . . . . . . . . . . . . . 14 (𝑀 ∈ LMod β†’ 𝑀 ∈ Grp)
2524grpmndd 18768 . . . . . . . . . . . . 13 (𝑀 ∈ LMod β†’ 𝑀 ∈ Mnd)
2625ad3antrrr 729 . . . . . . . . . . . 12 ((((𝑀 ∈ LMod ∧ 𝑋 ∈ 𝐡) ∧ βˆ€π‘  ∈ 𝑆 ((𝑠 Β· 𝑋) = 𝑍 β†’ 𝑠 = 0 )) ∧ 𝑓 ∈ (𝑆 ↑m {𝑋})) β†’ 𝑀 ∈ Mnd)
27 simpllr 775 . . . . . . . . . . . 12 ((((𝑀 ∈ LMod ∧ 𝑋 ∈ 𝐡) ∧ βˆ€π‘  ∈ 𝑆 ((𝑠 Β· 𝑋) = 𝑍 β†’ 𝑠 = 0 )) ∧ 𝑓 ∈ (𝑆 ↑m {𝑋})) β†’ 𝑋 ∈ 𝐡)
28 elmapi 8793 . . . . . . . . . . . . . 14 (𝑓 ∈ (𝑆 ↑m {𝑋}) β†’ 𝑓:{𝑋}βŸΆπ‘†)
296adantl 483 . . . . . . . . . . . . . . . 16 ((𝑓:{𝑋}βŸΆπ‘† ∧ ((𝑀 ∈ LMod ∧ 𝑋 ∈ 𝐡) ∧ βˆ€π‘  ∈ 𝑆 ((𝑠 Β· 𝑋) = 𝑍 β†’ 𝑠 = 0 ))) β†’ 𝑀 ∈ LMod)
30 snidg 4624 . . . . . . . . . . . . . . . . . . 19 (𝑋 ∈ 𝐡 β†’ 𝑋 ∈ {𝑋})
3130adantl 483 . . . . . . . . . . . . . . . . . 18 ((𝑀 ∈ LMod ∧ 𝑋 ∈ 𝐡) β†’ 𝑋 ∈ {𝑋})
3231adantr 482 . . . . . . . . . . . . . . . . 17 (((𝑀 ∈ LMod ∧ 𝑋 ∈ 𝐡) ∧ βˆ€π‘  ∈ 𝑆 ((𝑠 Β· 𝑋) = 𝑍 β†’ 𝑠 = 0 )) β†’ 𝑋 ∈ {𝑋})
33 ffvelcdm 7036 . . . . . . . . . . . . . . . . 17 ((𝑓:{𝑋}βŸΆπ‘† ∧ 𝑋 ∈ {𝑋}) β†’ (π‘“β€˜π‘‹) ∈ 𝑆)
3432, 33sylan2 594 . . . . . . . . . . . . . . . 16 ((𝑓:{𝑋}βŸΆπ‘† ∧ ((𝑀 ∈ LMod ∧ 𝑋 ∈ 𝐡) ∧ βˆ€π‘  ∈ 𝑆 ((𝑠 Β· 𝑋) = 𝑍 β†’ 𝑠 = 0 ))) β†’ (π‘“β€˜π‘‹) ∈ 𝑆)
35 simprlr 779 . . . . . . . . . . . . . . . 16 ((𝑓:{𝑋}βŸΆπ‘† ∧ ((𝑀 ∈ LMod ∧ 𝑋 ∈ 𝐡) ∧ βˆ€π‘  ∈ 𝑆 ((𝑠 Β· 𝑋) = 𝑍 β†’ 𝑠 = 0 ))) β†’ 𝑋 ∈ 𝐡)
36 eqid 2733 . . . . . . . . . . . . . . . . 17 ( ·𝑠 β€˜π‘€) = ( ·𝑠 β€˜π‘€)
3717, 9, 36, 8lmodvscl 20383 . . . . . . . . . . . . . . . 16 ((𝑀 ∈ LMod ∧ (π‘“β€˜π‘‹) ∈ 𝑆 ∧ 𝑋 ∈ 𝐡) β†’ ((π‘“β€˜π‘‹)( ·𝑠 β€˜π‘€)𝑋) ∈ 𝐡)
3829, 34, 35, 37syl3anc 1372 . . . . . . . . . . . . . . 15 ((𝑓:{𝑋}βŸΆπ‘† ∧ ((𝑀 ∈ LMod ∧ 𝑋 ∈ 𝐡) ∧ βˆ€π‘  ∈ 𝑆 ((𝑠 Β· 𝑋) = 𝑍 β†’ 𝑠 = 0 ))) β†’ ((π‘“β€˜π‘‹)( ·𝑠 β€˜π‘€)𝑋) ∈ 𝐡)
3938expcom 415 . . . . . . . . . . . . . 14 (((𝑀 ∈ LMod ∧ 𝑋 ∈ 𝐡) ∧ βˆ€π‘  ∈ 𝑆 ((𝑠 Β· 𝑋) = 𝑍 β†’ 𝑠 = 0 )) β†’ (𝑓:{𝑋}βŸΆπ‘† β†’ ((π‘“β€˜π‘‹)( ·𝑠 β€˜π‘€)𝑋) ∈ 𝐡))
4028, 39syl5com 31 . . . . . . . . . . . . 13 (𝑓 ∈ (𝑆 ↑m {𝑋}) β†’ (((𝑀 ∈ LMod ∧ 𝑋 ∈ 𝐡) ∧ βˆ€π‘  ∈ 𝑆 ((𝑠 Β· 𝑋) = 𝑍 β†’ 𝑠 = 0 )) β†’ ((π‘“β€˜π‘‹)( ·𝑠 β€˜π‘€)𝑋) ∈ 𝐡))
4140impcom 409 . . . . . . . . . . . 12 ((((𝑀 ∈ LMod ∧ 𝑋 ∈ 𝐡) ∧ βˆ€π‘  ∈ 𝑆 ((𝑠 Β· 𝑋) = 𝑍 β†’ 𝑠 = 0 )) ∧ 𝑓 ∈ (𝑆 ↑m {𝑋})) β†’ ((π‘“β€˜π‘‹)( ·𝑠 β€˜π‘€)𝑋) ∈ 𝐡)
42 fveq2 6846 . . . . . . . . . . . . . 14 (π‘₯ = 𝑋 β†’ (π‘“β€˜π‘₯) = (π‘“β€˜π‘‹))
43 id 22 . . . . . . . . . . . . . 14 (π‘₯ = 𝑋 β†’ π‘₯ = 𝑋)
4442, 43oveq12d 7379 . . . . . . . . . . . . 13 (π‘₯ = 𝑋 β†’ ((π‘“β€˜π‘₯)( ·𝑠 β€˜π‘€)π‘₯) = ((π‘“β€˜π‘‹)( ·𝑠 β€˜π‘€)𝑋))
4517, 44gsumsn 19739 . . . . . . . . . . . 12 ((𝑀 ∈ Mnd ∧ 𝑋 ∈ 𝐡 ∧ ((π‘“β€˜π‘‹)( ·𝑠 β€˜π‘€)𝑋) ∈ 𝐡) β†’ (𝑀 Ξ£g (π‘₯ ∈ {𝑋} ↦ ((π‘“β€˜π‘₯)( ·𝑠 β€˜π‘€)π‘₯))) = ((π‘“β€˜π‘‹)( ·𝑠 β€˜π‘€)𝑋))
4626, 27, 41, 45syl3anc 1372 . . . . . . . . . . 11 ((((𝑀 ∈ LMod ∧ 𝑋 ∈ 𝐡) ∧ βˆ€π‘  ∈ 𝑆 ((𝑠 Β· 𝑋) = 𝑍 β†’ 𝑠 = 0 )) ∧ 𝑓 ∈ (𝑆 ↑m {𝑋})) β†’ (𝑀 Ξ£g (π‘₯ ∈ {𝑋} ↦ ((π‘“β€˜π‘₯)( ·𝑠 β€˜π‘€)π‘₯))) = ((π‘“β€˜π‘‹)( ·𝑠 β€˜π‘€)𝑋))
4746eqeq1d 2735 . . . . . . . . . 10 ((((𝑀 ∈ LMod ∧ 𝑋 ∈ 𝐡) ∧ βˆ€π‘  ∈ 𝑆 ((𝑠 Β· 𝑋) = 𝑍 β†’ 𝑠 = 0 )) ∧ 𝑓 ∈ (𝑆 ↑m {𝑋})) β†’ ((𝑀 Ξ£g (π‘₯ ∈ {𝑋} ↦ ((π‘“β€˜π‘₯)( ·𝑠 β€˜π‘€)π‘₯))) = 𝑍 ↔ ((π‘“β€˜π‘‹)( ·𝑠 β€˜π‘€)𝑋) = 𝑍))
4830, 33sylan2 594 . . . . . . . . . . . . . . 15 ((𝑓:{𝑋}βŸΆπ‘† ∧ 𝑋 ∈ 𝐡) β†’ (π‘“β€˜π‘‹) ∈ 𝑆)
4948expcom 415 . . . . . . . . . . . . . 14 (𝑋 ∈ 𝐡 β†’ (𝑓:{𝑋}βŸΆπ‘† β†’ (π‘“β€˜π‘‹) ∈ 𝑆))
5049adantl 483 . . . . . . . . . . . . 13 ((𝑀 ∈ LMod ∧ 𝑋 ∈ 𝐡) β†’ (𝑓:{𝑋}βŸΆπ‘† β†’ (π‘“β€˜π‘‹) ∈ 𝑆))
51 snlindsntor.t . . . . . . . . . . . . . . . . 17 Β· = ( ·𝑠 β€˜π‘€)
5251oveqi 7374 . . . . . . . . . . . . . . . 16 ((π‘“β€˜π‘‹) Β· 𝑋) = ((π‘“β€˜π‘‹)( ·𝑠 β€˜π‘€)𝑋)
5352eqeq1i 2738 . . . . . . . . . . . . . . 15 (((π‘“β€˜π‘‹) Β· 𝑋) = 𝑍 ↔ ((π‘“β€˜π‘‹)( ·𝑠 β€˜π‘€)𝑋) = 𝑍)
54 oveq1 7368 . . . . . . . . . . . . . . . . . 18 (𝑠 = (π‘“β€˜π‘‹) β†’ (𝑠 Β· 𝑋) = ((π‘“β€˜π‘‹) Β· 𝑋))
5554eqeq1d 2735 . . . . . . . . . . . . . . . . 17 (𝑠 = (π‘“β€˜π‘‹) β†’ ((𝑠 Β· 𝑋) = 𝑍 ↔ ((π‘“β€˜π‘‹) Β· 𝑋) = 𝑍))
56 eqeq1 2737 . . . . . . . . . . . . . . . . 17 (𝑠 = (π‘“β€˜π‘‹) β†’ (𝑠 = 0 ↔ (π‘“β€˜π‘‹) = 0 ))
5755, 56imbi12d 345 . . . . . . . . . . . . . . . 16 (𝑠 = (π‘“β€˜π‘‹) β†’ (((𝑠 Β· 𝑋) = 𝑍 β†’ 𝑠 = 0 ) ↔ (((π‘“β€˜π‘‹) Β· 𝑋) = 𝑍 β†’ (π‘“β€˜π‘‹) = 0 )))
5857rspcva 3581 . . . . . . . . . . . . . . 15 (((π‘“β€˜π‘‹) ∈ 𝑆 ∧ βˆ€π‘  ∈ 𝑆 ((𝑠 Β· 𝑋) = 𝑍 β†’ 𝑠 = 0 )) β†’ (((π‘“β€˜π‘‹) Β· 𝑋) = 𝑍 β†’ (π‘“β€˜π‘‹) = 0 ))
5953, 58biimtrrid 242 . . . . . . . . . . . . . 14 (((π‘“β€˜π‘‹) ∈ 𝑆 ∧ βˆ€π‘  ∈ 𝑆 ((𝑠 Β· 𝑋) = 𝑍 β†’ 𝑠 = 0 )) β†’ (((π‘“β€˜π‘‹)( ·𝑠 β€˜π‘€)𝑋) = 𝑍 β†’ (π‘“β€˜π‘‹) = 0 ))
6059ex 414 . . . . . . . . . . . . 13 ((π‘“β€˜π‘‹) ∈ 𝑆 β†’ (βˆ€π‘  ∈ 𝑆 ((𝑠 Β· 𝑋) = 𝑍 β†’ 𝑠 = 0 ) β†’ (((π‘“β€˜π‘‹)( ·𝑠 β€˜π‘€)𝑋) = 𝑍 β†’ (π‘“β€˜π‘‹) = 0 )))
6128, 50, 60syl56 36 . . . . . . . . . . . 12 ((𝑀 ∈ LMod ∧ 𝑋 ∈ 𝐡) β†’ (𝑓 ∈ (𝑆 ↑m {𝑋}) β†’ (βˆ€π‘  ∈ 𝑆 ((𝑠 Β· 𝑋) = 𝑍 β†’ 𝑠 = 0 ) β†’ (((π‘“β€˜π‘‹)( ·𝑠 β€˜π‘€)𝑋) = 𝑍 β†’ (π‘“β€˜π‘‹) = 0 ))))
6261com23 86 . . . . . . . . . . 11 ((𝑀 ∈ LMod ∧ 𝑋 ∈ 𝐡) β†’ (βˆ€π‘  ∈ 𝑆 ((𝑠 Β· 𝑋) = 𝑍 β†’ 𝑠 = 0 ) β†’ (𝑓 ∈ (𝑆 ↑m {𝑋}) β†’ (((π‘“β€˜π‘‹)( ·𝑠 β€˜π‘€)𝑋) = 𝑍 β†’ (π‘“β€˜π‘‹) = 0 ))))
6362imp31 419 . . . . . . . . . 10 ((((𝑀 ∈ LMod ∧ 𝑋 ∈ 𝐡) ∧ βˆ€π‘  ∈ 𝑆 ((𝑠 Β· 𝑋) = 𝑍 β†’ 𝑠 = 0 )) ∧ 𝑓 ∈ (𝑆 ↑m {𝑋})) β†’ (((π‘“β€˜π‘‹)( ·𝑠 β€˜π‘€)𝑋) = 𝑍 β†’ (π‘“β€˜π‘‹) = 0 ))
6447, 63sylbid 239 . . . . . . . . 9 ((((𝑀 ∈ LMod ∧ 𝑋 ∈ 𝐡) ∧ βˆ€π‘  ∈ 𝑆 ((𝑠 Β· 𝑋) = 𝑍 β†’ 𝑠 = 0 )) ∧ 𝑓 ∈ (𝑆 ↑m {𝑋})) β†’ ((𝑀 Ξ£g (π‘₯ ∈ {𝑋} ↦ ((π‘“β€˜π‘₯)( ·𝑠 β€˜π‘€)π‘₯))) = 𝑍 β†’ (π‘“β€˜π‘‹) = 0 ))
6564adantld 492 . . . . . . . 8 ((((𝑀 ∈ LMod ∧ 𝑋 ∈ 𝐡) ∧ βˆ€π‘  ∈ 𝑆 ((𝑠 Β· 𝑋) = 𝑍 β†’ 𝑠 = 0 )) ∧ 𝑓 ∈ (𝑆 ↑m {𝑋})) β†’ ((𝑓 finSupp 0 ∧ (𝑀 Ξ£g (π‘₯ ∈ {𝑋} ↦ ((π‘“β€˜π‘₯)( ·𝑠 β€˜π‘€)π‘₯))) = 𝑍) β†’ (π‘“β€˜π‘‹) = 0 ))
6623, 65sylbid 239 . . . . . . 7 ((((𝑀 ∈ LMod ∧ 𝑋 ∈ 𝐡) ∧ βˆ€π‘  ∈ 𝑆 ((𝑠 Β· 𝑋) = 𝑍 β†’ 𝑠 = 0 )) ∧ 𝑓 ∈ (𝑆 ↑m {𝑋})) β†’ ((𝑓 finSupp 0 ∧ (𝑓( linC β€˜π‘€){𝑋}) = 𝑍) β†’ (π‘“β€˜π‘‹) = 0 ))
6766ralrimiva 3140 . . . . . 6 (((𝑀 ∈ LMod ∧ 𝑋 ∈ 𝐡) ∧ βˆ€π‘  ∈ 𝑆 ((𝑠 Β· 𝑋) = 𝑍 β†’ 𝑠 = 0 )) β†’ βˆ€π‘“ ∈ (𝑆 ↑m {𝑋})((𝑓 finSupp 0 ∧ (𝑓( linC β€˜π‘€){𝑋}) = 𝑍) β†’ (π‘“β€˜π‘‹) = 0 ))
6867ex 414 . . . . 5 ((𝑀 ∈ LMod ∧ 𝑋 ∈ 𝐡) β†’ (βˆ€π‘  ∈ 𝑆 ((𝑠 Β· 𝑋) = 𝑍 β†’ 𝑠 = 0 ) β†’ βˆ€π‘“ ∈ (𝑆 ↑m {𝑋})((𝑓 finSupp 0 ∧ (𝑓( linC β€˜π‘€){𝑋}) = 𝑍) β†’ (π‘“β€˜π‘‹) = 0 )))
69 impexp 452 . . . . . . . 8 (((𝑓 finSupp 0 ∧ (𝑓( linC β€˜π‘€){𝑋}) = 𝑍) β†’ (π‘“β€˜π‘‹) = 0 ) ↔ (𝑓 finSupp 0 β†’ ((𝑓( linC β€˜π‘€){𝑋}) = 𝑍 β†’ (π‘“β€˜π‘‹) = 0 )))
7028adantl 483 . . . . . . . . . 10 (((𝑀 ∈ LMod ∧ 𝑋 ∈ 𝐡) ∧ 𝑓 ∈ (𝑆 ↑m {𝑋})) β†’ 𝑓:{𝑋}βŸΆπ‘†)
71 snfi 8994 . . . . . . . . . . 11 {𝑋} ∈ Fin
7271a1i 11 . . . . . . . . . 10 (((𝑀 ∈ LMod ∧ 𝑋 ∈ 𝐡) ∧ 𝑓 ∈ (𝑆 ↑m {𝑋})) β†’ {𝑋} ∈ Fin)
73 snlindsntor.0 . . . . . . . . . . . 12 0 = (0gβ€˜π‘…)
7473fvexi 6860 . . . . . . . . . . 11 0 ∈ V
7574a1i 11 . . . . . . . . . 10 (((𝑀 ∈ LMod ∧ 𝑋 ∈ 𝐡) ∧ 𝑓 ∈ (𝑆 ↑m {𝑋})) β†’ 0 ∈ V)
7670, 72, 75fdmfifsupp 9323 . . . . . . . . 9 (((𝑀 ∈ LMod ∧ 𝑋 ∈ 𝐡) ∧ 𝑓 ∈ (𝑆 ↑m {𝑋})) β†’ 𝑓 finSupp 0 )
77 pm2.27 42 . . . . . . . . 9 (𝑓 finSupp 0 β†’ ((𝑓 finSupp 0 β†’ ((𝑓( linC β€˜π‘€){𝑋}) = 𝑍 β†’ (π‘“β€˜π‘‹) = 0 )) β†’ ((𝑓( linC β€˜π‘€){𝑋}) = 𝑍 β†’ (π‘“β€˜π‘‹) = 0 )))
7876, 77syl 17 . . . . . . . 8 (((𝑀 ∈ LMod ∧ 𝑋 ∈ 𝐡) ∧ 𝑓 ∈ (𝑆 ↑m {𝑋})) β†’ ((𝑓 finSupp 0 β†’ ((𝑓( linC β€˜π‘€){𝑋}) = 𝑍 β†’ (π‘“β€˜π‘‹) = 0 )) β†’ ((𝑓( linC β€˜π‘€){𝑋}) = 𝑍 β†’ (π‘“β€˜π‘‹) = 0 )))
7969, 78biimtrid 241 . . . . . . 7 (((𝑀 ∈ LMod ∧ 𝑋 ∈ 𝐡) ∧ 𝑓 ∈ (𝑆 ↑m {𝑋})) β†’ (((𝑓 finSupp 0 ∧ (𝑓( linC β€˜π‘€){𝑋}) = 𝑍) β†’ (π‘“β€˜π‘‹) = 0 ) β†’ ((𝑓( linC β€˜π‘€){𝑋}) = 𝑍 β†’ (π‘“β€˜π‘‹) = 0 )))
8079ralimdva 3161 . . . . . 6 ((𝑀 ∈ LMod ∧ 𝑋 ∈ 𝐡) β†’ (βˆ€π‘“ ∈ (𝑆 ↑m {𝑋})((𝑓 finSupp 0 ∧ (𝑓( linC β€˜π‘€){𝑋}) = 𝑍) β†’ (π‘“β€˜π‘‹) = 0 ) β†’ βˆ€π‘“ ∈ (𝑆 ↑m {𝑋})((𝑓( linC β€˜π‘€){𝑋}) = 𝑍 β†’ (π‘“β€˜π‘‹) = 0 )))
81 snlindsntor.z . . . . . . 7 𝑍 = (0gβ€˜π‘€)
8217, 9, 8, 73, 81, 51snlindsntorlem 46641 . . . . . 6 ((𝑀 ∈ LMod ∧ 𝑋 ∈ 𝐡) β†’ (βˆ€π‘“ ∈ (𝑆 ↑m {𝑋})((𝑓( linC β€˜π‘€){𝑋}) = 𝑍 β†’ (π‘“β€˜π‘‹) = 0 ) β†’ βˆ€π‘  ∈ 𝑆 ((𝑠 Β· 𝑋) = 𝑍 β†’ 𝑠 = 0 )))
8380, 82syld 47 . . . . 5 ((𝑀 ∈ LMod ∧ 𝑋 ∈ 𝐡) β†’ (βˆ€π‘“ ∈ (𝑆 ↑m {𝑋})((𝑓 finSupp 0 ∧ (𝑓( linC β€˜π‘€){𝑋}) = 𝑍) β†’ (π‘“β€˜π‘‹) = 0 ) β†’ βˆ€π‘  ∈ 𝑆 ((𝑠 Β· 𝑋) = 𝑍 β†’ 𝑠 = 0 )))
8468, 83impbid 211 . . . 4 ((𝑀 ∈ LMod ∧ 𝑋 ∈ 𝐡) β†’ (βˆ€π‘  ∈ 𝑆 ((𝑠 Β· 𝑋) = 𝑍 β†’ 𝑠 = 0 ) ↔ βˆ€π‘“ ∈ (𝑆 ↑m {𝑋})((𝑓 finSupp 0 ∧ (𝑓( linC β€˜π‘€){𝑋}) = 𝑍) β†’ (π‘“β€˜π‘‹) = 0 )))
85 fveqeq2 6855 . . . . . . . . 9 (𝑦 = 𝑋 β†’ ((π‘“β€˜π‘¦) = 0 ↔ (π‘“β€˜π‘‹) = 0 ))
8685ralsng 4638 . . . . . . . 8 (𝑋 ∈ 𝐡 β†’ (βˆ€π‘¦ ∈ {𝑋} (π‘“β€˜π‘¦) = 0 ↔ (π‘“β€˜π‘‹) = 0 ))
8786adantl 483 . . . . . . 7 ((𝑀 ∈ LMod ∧ 𝑋 ∈ 𝐡) β†’ (βˆ€π‘¦ ∈ {𝑋} (π‘“β€˜π‘¦) = 0 ↔ (π‘“β€˜π‘‹) = 0 ))
8887bicomd 222 . . . . . 6 ((𝑀 ∈ LMod ∧ 𝑋 ∈ 𝐡) β†’ ((π‘“β€˜π‘‹) = 0 ↔ βˆ€π‘¦ ∈ {𝑋} (π‘“β€˜π‘¦) = 0 ))
8988imbi2d 341 . . . . 5 ((𝑀 ∈ LMod ∧ 𝑋 ∈ 𝐡) β†’ (((𝑓 finSupp 0 ∧ (𝑓( linC β€˜π‘€){𝑋}) = 𝑍) β†’ (π‘“β€˜π‘‹) = 0 ) ↔ ((𝑓 finSupp 0 ∧ (𝑓( linC β€˜π‘€){𝑋}) = 𝑍) β†’ βˆ€π‘¦ ∈ {𝑋} (π‘“β€˜π‘¦) = 0 )))
9089ralbidv 3171 . . . 4 ((𝑀 ∈ LMod ∧ 𝑋 ∈ 𝐡) β†’ (βˆ€π‘“ ∈ (𝑆 ↑m {𝑋})((𝑓 finSupp 0 ∧ (𝑓( linC β€˜π‘€){𝑋}) = 𝑍) β†’ (π‘“β€˜π‘‹) = 0 ) ↔ βˆ€π‘“ ∈ (𝑆 ↑m {𝑋})((𝑓 finSupp 0 ∧ (𝑓( linC β€˜π‘€){𝑋}) = 𝑍) β†’ βˆ€π‘¦ ∈ {𝑋} (π‘“β€˜π‘¦) = 0 )))
91 snelpwi 5404 . . . . . 6 (𝑋 ∈ 𝐡 β†’ {𝑋} ∈ 𝒫 𝐡)
9291adantl 483 . . . . 5 ((𝑀 ∈ LMod ∧ 𝑋 ∈ 𝐡) β†’ {𝑋} ∈ 𝒫 𝐡)
9392biantrurd 534 . . . 4 ((𝑀 ∈ LMod ∧ 𝑋 ∈ 𝐡) β†’ (βˆ€π‘“ ∈ (𝑆 ↑m {𝑋})((𝑓 finSupp 0 ∧ (𝑓( linC β€˜π‘€){𝑋}) = 𝑍) β†’ βˆ€π‘¦ ∈ {𝑋} (π‘“β€˜π‘¦) = 0 ) ↔ ({𝑋} ∈ 𝒫 𝐡 ∧ βˆ€π‘“ ∈ (𝑆 ↑m {𝑋})((𝑓 finSupp 0 ∧ (𝑓( linC β€˜π‘€){𝑋}) = 𝑍) β†’ βˆ€π‘¦ ∈ {𝑋} (π‘“β€˜π‘¦) = 0 ))))
9484, 90, 933bitrd 305 . . 3 ((𝑀 ∈ LMod ∧ 𝑋 ∈ 𝐡) β†’ (βˆ€π‘  ∈ 𝑆 ((𝑠 Β· 𝑋) = 𝑍 β†’ 𝑠 = 0 ) ↔ ({𝑋} ∈ 𝒫 𝐡 ∧ βˆ€π‘“ ∈ (𝑆 ↑m {𝑋})((𝑓 finSupp 0 ∧ (𝑓( linC β€˜π‘€){𝑋}) = 𝑍) β†’ βˆ€π‘¦ ∈ {𝑋} (π‘“β€˜π‘¦) = 0 ))))
954, 94bitrid 283 . 2 ((𝑀 ∈ LMod ∧ 𝑋 ∈ 𝐡) β†’ (βˆ€π‘  ∈ (𝑆 βˆ– { 0 })(𝑠 Β· 𝑋) β‰  𝑍 ↔ ({𝑋} ∈ 𝒫 𝐡 ∧ βˆ€π‘“ ∈ (𝑆 ↑m {𝑋})((𝑓 finSupp 0 ∧ (𝑓( linC β€˜π‘€){𝑋}) = 𝑍) β†’ βˆ€π‘¦ ∈ {𝑋} (π‘“β€˜π‘¦) = 0 ))))
96 snex 5392 . . 3 {𝑋} ∈ V
9717, 81, 9, 8, 73islininds 46617 . . 3 (({𝑋} ∈ V ∧ 𝑀 ∈ LMod) β†’ ({𝑋} linIndS 𝑀 ↔ ({𝑋} ∈ 𝒫 𝐡 ∧ βˆ€π‘“ ∈ (𝑆 ↑m {𝑋})((𝑓 finSupp 0 ∧ (𝑓( linC β€˜π‘€){𝑋}) = 𝑍) β†’ βˆ€π‘¦ ∈ {𝑋} (π‘“β€˜π‘¦) = 0 ))))
9896, 5, 97sylancr 588 . 2 ((𝑀 ∈ LMod ∧ 𝑋 ∈ 𝐡) β†’ ({𝑋} linIndS 𝑀 ↔ ({𝑋} ∈ 𝒫 𝐡 ∧ βˆ€π‘“ ∈ (𝑆 ↑m {𝑋})((𝑓 finSupp 0 ∧ (𝑓( linC β€˜π‘€){𝑋}) = 𝑍) β†’ βˆ€π‘¦ ∈ {𝑋} (π‘“β€˜π‘¦) = 0 ))))
9995, 98bitr4d 282 1 ((𝑀 ∈ LMod ∧ 𝑋 ∈ 𝐡) β†’ (βˆ€π‘  ∈ (𝑆 βˆ– { 0 })(𝑠 Β· 𝑋) β‰  𝑍 ↔ {𝑋} linIndS 𝑀))
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ↔ wb 205   ∧ wa 397   = wceq 1542   ∈ wcel 2107   β‰  wne 2940  βˆ€wral 3061  Vcvv 3447   βˆ– cdif 3911  π’« cpw 4564  {csn 4590   class class class wbr 5109   ↦ cmpt 5192  βŸΆwf 6496  β€˜cfv 6500  (class class class)co 7361   ↑m cmap 8771  Fincfn 8889   finSupp cfsupp 9311  Basecbs 17091  Scalarcsca 17144   ·𝑠 cvsca 17145  0gc0g 17329   Ξ£g cgsu 17330  Mndcmnd 18564  LModclmod 20365   linC clinc 46575   linIndS clininds 46611
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5246  ax-sep 5260  ax-nul 5267  ax-pow 5324  ax-pr 5388  ax-un 7676  ax-cnex 11115  ax-resscn 11116  ax-1cn 11117  ax-icn 11118  ax-addcl 11119  ax-addrcl 11120  ax-mulcl 11121  ax-mulrcl 11122  ax-mulcom 11123  ax-addass 11124  ax-mulass 11125  ax-distr 11126  ax-i2m1 11127  ax-1ne0 11128  ax-1rid 11129  ax-rnegex 11130  ax-rrecex 11131  ax-cnre 11132  ax-pre-lttri 11133  ax-pre-lttrn 11134  ax-pre-ltadd 11135  ax-pre-mulgt0 11136
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3352  df-reu 3353  df-rab 3407  df-v 3449  df-sbc 3744  df-csb 3860  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3933  df-nul 4287  df-if 4491  df-pw 4566  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-int 4912  df-iun 4960  df-br 5110  df-opab 5172  df-mpt 5193  df-tr 5227  df-id 5535  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5592  df-se 5593  df-we 5594  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-rn 5648  df-res 5649  df-ima 5650  df-pred 6257  df-ord 6324  df-on 6325  df-lim 6326  df-suc 6327  df-iota 6452  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-isom 6509  df-riota 7317  df-ov 7364  df-oprab 7365  df-mpo 7366  df-om 7807  df-1st 7925  df-2nd 7926  df-supp 8097  df-frecs 8216  df-wrecs 8247  df-recs 8321  df-rdg 8360  df-1o 8416  df-er 8654  df-map 8773  df-en 8890  df-dom 8891  df-sdom 8892  df-fin 8893  df-fsupp 9312  df-oi 9454  df-card 9883  df-pnf 11199  df-mnf 11200  df-xr 11201  df-ltxr 11202  df-le 11203  df-sub 11395  df-neg 11396  df-nn 12162  df-n0 12422  df-z 12508  df-uz 12772  df-fz 13434  df-fzo 13577  df-seq 13916  df-hash 14240  df-0g 17331  df-gsum 17332  df-mgm 18505  df-sgrp 18554  df-mnd 18565  df-grp 18759  df-mulg 18881  df-cntz 19105  df-lmod 20367  df-linc 46577  df-lininds 46613
This theorem is referenced by:  lindssnlvec  46657
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