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Theorem lincscmcl 47103
Description: The multiplication of a linear combination with a scalar is a linear combination, see also the proof in [Lang] p. 129. (Contributed by AV, 11-Apr-2019.) (Proof shortened by AV, 28-Jul-2019.)
Hypotheses
Ref Expression
lincscmcl.s Β· = ( ·𝑠 β€˜π‘€)
lincscmcl.r 𝑅 = (Baseβ€˜(Scalarβ€˜π‘€))
Assertion
Ref Expression
lincscmcl (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Baseβ€˜π‘€)) ∧ 𝐢 ∈ 𝑅 ∧ 𝐷 ∈ (𝑀 LinCo 𝑉)) β†’ (𝐢 Β· 𝐷) ∈ (𝑀 LinCo 𝑉))

Proof of Theorem lincscmcl
Dummy variables 𝑠 𝑣 π‘₯ are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2732 . . . . 5 (Baseβ€˜π‘€) = (Baseβ€˜π‘€)
2 eqid 2732 . . . . 5 (Scalarβ€˜π‘€) = (Scalarβ€˜π‘€)
3 lincscmcl.r . . . . 5 𝑅 = (Baseβ€˜(Scalarβ€˜π‘€))
41, 2, 3lcoval 47083 . . . 4 ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Baseβ€˜π‘€)) β†’ (𝐷 ∈ (𝑀 LinCo 𝑉) ↔ (𝐷 ∈ (Baseβ€˜π‘€) ∧ βˆƒπ‘₯ ∈ (𝑅 ↑m 𝑉)(π‘₯ finSupp (0gβ€˜(Scalarβ€˜π‘€)) ∧ 𝐷 = (π‘₯( linC β€˜π‘€)𝑉)))))
54adantr 481 . . 3 (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Baseβ€˜π‘€)) ∧ 𝐢 ∈ 𝑅) β†’ (𝐷 ∈ (𝑀 LinCo 𝑉) ↔ (𝐷 ∈ (Baseβ€˜π‘€) ∧ βˆƒπ‘₯ ∈ (𝑅 ↑m 𝑉)(π‘₯ finSupp (0gβ€˜(Scalarβ€˜π‘€)) ∧ 𝐷 = (π‘₯( linC β€˜π‘€)𝑉)))))
6 simpl 483 . . . . . . 7 ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Baseβ€˜π‘€)) β†’ 𝑀 ∈ LMod)
76ad2antrr 724 . . . . . 6 ((((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Baseβ€˜π‘€)) ∧ 𝐢 ∈ 𝑅) ∧ (𝐷 ∈ (Baseβ€˜π‘€) ∧ βˆƒπ‘₯ ∈ (𝑅 ↑m 𝑉)(π‘₯ finSupp (0gβ€˜(Scalarβ€˜π‘€)) ∧ 𝐷 = (π‘₯( linC β€˜π‘€)𝑉)))) β†’ 𝑀 ∈ LMod)
8 simpr 485 . . . . . . 7 (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Baseβ€˜π‘€)) ∧ 𝐢 ∈ 𝑅) β†’ 𝐢 ∈ 𝑅)
98adantr 481 . . . . . 6 ((((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Baseβ€˜π‘€)) ∧ 𝐢 ∈ 𝑅) ∧ (𝐷 ∈ (Baseβ€˜π‘€) ∧ βˆƒπ‘₯ ∈ (𝑅 ↑m 𝑉)(π‘₯ finSupp (0gβ€˜(Scalarβ€˜π‘€)) ∧ 𝐷 = (π‘₯( linC β€˜π‘€)𝑉)))) β†’ 𝐢 ∈ 𝑅)
10 simprl 769 . . . . . 6 ((((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Baseβ€˜π‘€)) ∧ 𝐢 ∈ 𝑅) ∧ (𝐷 ∈ (Baseβ€˜π‘€) ∧ βˆƒπ‘₯ ∈ (𝑅 ↑m 𝑉)(π‘₯ finSupp (0gβ€˜(Scalarβ€˜π‘€)) ∧ 𝐷 = (π‘₯( linC β€˜π‘€)𝑉)))) β†’ 𝐷 ∈ (Baseβ€˜π‘€))
11 lincscmcl.s . . . . . . 7 Β· = ( ·𝑠 β€˜π‘€)
121, 2, 11, 3lmodvscl 20488 . . . . . 6 ((𝑀 ∈ LMod ∧ 𝐢 ∈ 𝑅 ∧ 𝐷 ∈ (Baseβ€˜π‘€)) β†’ (𝐢 Β· 𝐷) ∈ (Baseβ€˜π‘€))
137, 9, 10, 12syl3anc 1371 . . . . 5 ((((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Baseβ€˜π‘€)) ∧ 𝐢 ∈ 𝑅) ∧ (𝐷 ∈ (Baseβ€˜π‘€) ∧ βˆƒπ‘₯ ∈ (𝑅 ↑m 𝑉)(π‘₯ finSupp (0gβ€˜(Scalarβ€˜π‘€)) ∧ 𝐷 = (π‘₯( linC β€˜π‘€)𝑉)))) β†’ (𝐢 Β· 𝐷) ∈ (Baseβ€˜π‘€))
142lmodring 20478 . . . . . . . . . . . . . . . . 17 (𝑀 ∈ LMod β†’ (Scalarβ€˜π‘€) ∈ Ring)
1514ad2antrr 724 . . . . . . . . . . . . . . . 16 (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Baseβ€˜π‘€)) ∧ 𝐢 ∈ 𝑅) β†’ (Scalarβ€˜π‘€) ∈ Ring)
1615adantl 482 . . . . . . . . . . . . . . 15 ((((π‘₯ ∈ (𝑅 ↑m 𝑉) ∧ (π‘₯ finSupp (0gβ€˜(Scalarβ€˜π‘€)) ∧ 𝐷 = (π‘₯( linC β€˜π‘€)𝑉))) ∧ 𝐷 ∈ (Baseβ€˜π‘€)) ∧ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Baseβ€˜π‘€)) ∧ 𝐢 ∈ 𝑅)) β†’ (Scalarβ€˜π‘€) ∈ Ring)
1716adantr 481 . . . . . . . . . . . . . 14 (((((π‘₯ ∈ (𝑅 ↑m 𝑉) ∧ (π‘₯ finSupp (0gβ€˜(Scalarβ€˜π‘€)) ∧ 𝐷 = (π‘₯( linC β€˜π‘€)𝑉))) ∧ 𝐷 ∈ (Baseβ€˜π‘€)) ∧ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Baseβ€˜π‘€)) ∧ 𝐢 ∈ 𝑅)) ∧ 𝑣 ∈ 𝑉) β†’ (Scalarβ€˜π‘€) ∈ Ring)
188adantl 482 . . . . . . . . . . . . . . 15 ((((π‘₯ ∈ (𝑅 ↑m 𝑉) ∧ (π‘₯ finSupp (0gβ€˜(Scalarβ€˜π‘€)) ∧ 𝐷 = (π‘₯( linC β€˜π‘€)𝑉))) ∧ 𝐷 ∈ (Baseβ€˜π‘€)) ∧ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Baseβ€˜π‘€)) ∧ 𝐢 ∈ 𝑅)) β†’ 𝐢 ∈ 𝑅)
1918adantr 481 . . . . . . . . . . . . . 14 (((((π‘₯ ∈ (𝑅 ↑m 𝑉) ∧ (π‘₯ finSupp (0gβ€˜(Scalarβ€˜π‘€)) ∧ 𝐷 = (π‘₯( linC β€˜π‘€)𝑉))) ∧ 𝐷 ∈ (Baseβ€˜π‘€)) ∧ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Baseβ€˜π‘€)) ∧ 𝐢 ∈ 𝑅)) ∧ 𝑣 ∈ 𝑉) β†’ 𝐢 ∈ 𝑅)
20 elmapi 8842 . . . . . . . . . . . . . . . . . 18 (π‘₯ ∈ (𝑅 ↑m 𝑉) β†’ π‘₯:π‘‰βŸΆπ‘…)
21 ffvelcdm 7083 . . . . . . . . . . . . . . . . . . 19 ((π‘₯:π‘‰βŸΆπ‘… ∧ 𝑣 ∈ 𝑉) β†’ (π‘₯β€˜π‘£) ∈ 𝑅)
2221ex 413 . . . . . . . . . . . . . . . . . 18 (π‘₯:π‘‰βŸΆπ‘… β†’ (𝑣 ∈ 𝑉 β†’ (π‘₯β€˜π‘£) ∈ 𝑅))
2320, 22syl 17 . . . . . . . . . . . . . . . . 17 (π‘₯ ∈ (𝑅 ↑m 𝑉) β†’ (𝑣 ∈ 𝑉 β†’ (π‘₯β€˜π‘£) ∈ 𝑅))
2423adantr 481 . . . . . . . . . . . . . . . 16 ((π‘₯ ∈ (𝑅 ↑m 𝑉) ∧ (π‘₯ finSupp (0gβ€˜(Scalarβ€˜π‘€)) ∧ 𝐷 = (π‘₯( linC β€˜π‘€)𝑉))) β†’ (𝑣 ∈ 𝑉 β†’ (π‘₯β€˜π‘£) ∈ 𝑅))
2524ad2antrr 724 . . . . . . . . . . . . . . 15 ((((π‘₯ ∈ (𝑅 ↑m 𝑉) ∧ (π‘₯ finSupp (0gβ€˜(Scalarβ€˜π‘€)) ∧ 𝐷 = (π‘₯( linC β€˜π‘€)𝑉))) ∧ 𝐷 ∈ (Baseβ€˜π‘€)) ∧ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Baseβ€˜π‘€)) ∧ 𝐢 ∈ 𝑅)) β†’ (𝑣 ∈ 𝑉 β†’ (π‘₯β€˜π‘£) ∈ 𝑅))
2625imp 407 . . . . . . . . . . . . . 14 (((((π‘₯ ∈ (𝑅 ↑m 𝑉) ∧ (π‘₯ finSupp (0gβ€˜(Scalarβ€˜π‘€)) ∧ 𝐷 = (π‘₯( linC β€˜π‘€)𝑉))) ∧ 𝐷 ∈ (Baseβ€˜π‘€)) ∧ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Baseβ€˜π‘€)) ∧ 𝐢 ∈ 𝑅)) ∧ 𝑣 ∈ 𝑉) β†’ (π‘₯β€˜π‘£) ∈ 𝑅)
27 eqid 2732 . . . . . . . . . . . . . . 15 (.rβ€˜(Scalarβ€˜π‘€)) = (.rβ€˜(Scalarβ€˜π‘€))
283, 27ringcl 20072 . . . . . . . . . . . . . 14 (((Scalarβ€˜π‘€) ∈ Ring ∧ 𝐢 ∈ 𝑅 ∧ (π‘₯β€˜π‘£) ∈ 𝑅) β†’ (𝐢(.rβ€˜(Scalarβ€˜π‘€))(π‘₯β€˜π‘£)) ∈ 𝑅)
2917, 19, 26, 28syl3anc 1371 . . . . . . . . . . . . 13 (((((π‘₯ ∈ (𝑅 ↑m 𝑉) ∧ (π‘₯ finSupp (0gβ€˜(Scalarβ€˜π‘€)) ∧ 𝐷 = (π‘₯( linC β€˜π‘€)𝑉))) ∧ 𝐷 ∈ (Baseβ€˜π‘€)) ∧ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Baseβ€˜π‘€)) ∧ 𝐢 ∈ 𝑅)) ∧ 𝑣 ∈ 𝑉) β†’ (𝐢(.rβ€˜(Scalarβ€˜π‘€))(π‘₯β€˜π‘£)) ∈ 𝑅)
3029fmpttd 7114 . . . . . . . . . . . 12 ((((π‘₯ ∈ (𝑅 ↑m 𝑉) ∧ (π‘₯ finSupp (0gβ€˜(Scalarβ€˜π‘€)) ∧ 𝐷 = (π‘₯( linC β€˜π‘€)𝑉))) ∧ 𝐷 ∈ (Baseβ€˜π‘€)) ∧ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Baseβ€˜π‘€)) ∧ 𝐢 ∈ 𝑅)) β†’ (𝑣 ∈ 𝑉 ↦ (𝐢(.rβ€˜(Scalarβ€˜π‘€))(π‘₯β€˜π‘£))):π‘‰βŸΆπ‘…)
313fvexi 6905 . . . . . . . . . . . . 13 𝑅 ∈ V
32 simpr 485 . . . . . . . . . . . . . . 15 ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Baseβ€˜π‘€)) β†’ 𝑉 ∈ 𝒫 (Baseβ€˜π‘€))
3332adantr 481 . . . . . . . . . . . . . 14 (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Baseβ€˜π‘€)) ∧ 𝐢 ∈ 𝑅) β†’ 𝑉 ∈ 𝒫 (Baseβ€˜π‘€))
3433adantl 482 . . . . . . . . . . . . 13 ((((π‘₯ ∈ (𝑅 ↑m 𝑉) ∧ (π‘₯ finSupp (0gβ€˜(Scalarβ€˜π‘€)) ∧ 𝐷 = (π‘₯( linC β€˜π‘€)𝑉))) ∧ 𝐷 ∈ (Baseβ€˜π‘€)) ∧ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Baseβ€˜π‘€)) ∧ 𝐢 ∈ 𝑅)) β†’ 𝑉 ∈ 𝒫 (Baseβ€˜π‘€))
35 elmapg 8832 . . . . . . . . . . . . 13 ((𝑅 ∈ V ∧ 𝑉 ∈ 𝒫 (Baseβ€˜π‘€)) β†’ ((𝑣 ∈ 𝑉 ↦ (𝐢(.rβ€˜(Scalarβ€˜π‘€))(π‘₯β€˜π‘£))) ∈ (𝑅 ↑m 𝑉) ↔ (𝑣 ∈ 𝑉 ↦ (𝐢(.rβ€˜(Scalarβ€˜π‘€))(π‘₯β€˜π‘£))):π‘‰βŸΆπ‘…))
3631, 34, 35sylancr 587 . . . . . . . . . . . 12 ((((π‘₯ ∈ (𝑅 ↑m 𝑉) ∧ (π‘₯ finSupp (0gβ€˜(Scalarβ€˜π‘€)) ∧ 𝐷 = (π‘₯( linC β€˜π‘€)𝑉))) ∧ 𝐷 ∈ (Baseβ€˜π‘€)) ∧ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Baseβ€˜π‘€)) ∧ 𝐢 ∈ 𝑅)) β†’ ((𝑣 ∈ 𝑉 ↦ (𝐢(.rβ€˜(Scalarβ€˜π‘€))(π‘₯β€˜π‘£))) ∈ (𝑅 ↑m 𝑉) ↔ (𝑣 ∈ 𝑉 ↦ (𝐢(.rβ€˜(Scalarβ€˜π‘€))(π‘₯β€˜π‘£))):π‘‰βŸΆπ‘…))
3730, 36mpbird 256 . . . . . . . . . . 11 ((((π‘₯ ∈ (𝑅 ↑m 𝑉) ∧ (π‘₯ finSupp (0gβ€˜(Scalarβ€˜π‘€)) ∧ 𝐷 = (π‘₯( linC β€˜π‘€)𝑉))) ∧ 𝐷 ∈ (Baseβ€˜π‘€)) ∧ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Baseβ€˜π‘€)) ∧ 𝐢 ∈ 𝑅)) β†’ (𝑣 ∈ 𝑉 ↦ (𝐢(.rβ€˜(Scalarβ€˜π‘€))(π‘₯β€˜π‘£))) ∈ (𝑅 ↑m 𝑉))
3815, 33, 83jca 1128 . . . . . . . . . . . . 13 (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Baseβ€˜π‘€)) ∧ 𝐢 ∈ 𝑅) β†’ ((Scalarβ€˜π‘€) ∈ Ring ∧ 𝑉 ∈ 𝒫 (Baseβ€˜π‘€) ∧ 𝐢 ∈ 𝑅))
3938adantl 482 . . . . . . . . . . . 12 ((((π‘₯ ∈ (𝑅 ↑m 𝑉) ∧ (π‘₯ finSupp (0gβ€˜(Scalarβ€˜π‘€)) ∧ 𝐷 = (π‘₯( linC β€˜π‘€)𝑉))) ∧ 𝐷 ∈ (Baseβ€˜π‘€)) ∧ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Baseβ€˜π‘€)) ∧ 𝐢 ∈ 𝑅)) β†’ ((Scalarβ€˜π‘€) ∈ Ring ∧ 𝑉 ∈ 𝒫 (Baseβ€˜π‘€) ∧ 𝐢 ∈ 𝑅))
40 simpl 483 . . . . . . . . . . . . 13 ((π‘₯ ∈ (𝑅 ↑m 𝑉) ∧ (π‘₯ finSupp (0gβ€˜(Scalarβ€˜π‘€)) ∧ 𝐷 = (π‘₯( linC β€˜π‘€)𝑉))) β†’ π‘₯ ∈ (𝑅 ↑m 𝑉))
4140ad2antrr 724 . . . . . . . . . . . 12 ((((π‘₯ ∈ (𝑅 ↑m 𝑉) ∧ (π‘₯ finSupp (0gβ€˜(Scalarβ€˜π‘€)) ∧ 𝐷 = (π‘₯( linC β€˜π‘€)𝑉))) ∧ 𝐷 ∈ (Baseβ€˜π‘€)) ∧ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Baseβ€˜π‘€)) ∧ 𝐢 ∈ 𝑅)) β†’ π‘₯ ∈ (𝑅 ↑m 𝑉))
42 simprl 769 . . . . . . . . . . . . 13 ((π‘₯ ∈ (𝑅 ↑m 𝑉) ∧ (π‘₯ finSupp (0gβ€˜(Scalarβ€˜π‘€)) ∧ 𝐷 = (π‘₯( linC β€˜π‘€)𝑉))) β†’ π‘₯ finSupp (0gβ€˜(Scalarβ€˜π‘€)))
4342ad2antrr 724 . . . . . . . . . . . 12 ((((π‘₯ ∈ (𝑅 ↑m 𝑉) ∧ (π‘₯ finSupp (0gβ€˜(Scalarβ€˜π‘€)) ∧ 𝐷 = (π‘₯( linC β€˜π‘€)𝑉))) ∧ 𝐷 ∈ (Baseβ€˜π‘€)) ∧ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Baseβ€˜π‘€)) ∧ 𝐢 ∈ 𝑅)) β†’ π‘₯ finSupp (0gβ€˜(Scalarβ€˜π‘€)))
443rmfsupp 47040 . . . . . . . . . . . 12 ((((Scalarβ€˜π‘€) ∈ Ring ∧ 𝑉 ∈ 𝒫 (Baseβ€˜π‘€) ∧ 𝐢 ∈ 𝑅) ∧ π‘₯ ∈ (𝑅 ↑m 𝑉) ∧ π‘₯ finSupp (0gβ€˜(Scalarβ€˜π‘€))) β†’ (𝑣 ∈ 𝑉 ↦ (𝐢(.rβ€˜(Scalarβ€˜π‘€))(π‘₯β€˜π‘£))) finSupp (0gβ€˜(Scalarβ€˜π‘€)))
4539, 41, 43, 44syl3anc 1371 . . . . . . . . . . 11 ((((π‘₯ ∈ (𝑅 ↑m 𝑉) ∧ (π‘₯ finSupp (0gβ€˜(Scalarβ€˜π‘€)) ∧ 𝐷 = (π‘₯( linC β€˜π‘€)𝑉))) ∧ 𝐷 ∈ (Baseβ€˜π‘€)) ∧ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Baseβ€˜π‘€)) ∧ 𝐢 ∈ 𝑅)) β†’ (𝑣 ∈ 𝑉 ↦ (𝐢(.rβ€˜(Scalarβ€˜π‘€))(π‘₯β€˜π‘£))) finSupp (0gβ€˜(Scalarβ€˜π‘€)))
46 oveq2 7416 . . . . . . . . . . . . . . 15 (𝐷 = (π‘₯( linC β€˜π‘€)𝑉) β†’ (𝐢 Β· 𝐷) = (𝐢 Β· (π‘₯( linC β€˜π‘€)𝑉)))
4746adantl 482 . . . . . . . . . . . . . 14 ((π‘₯ finSupp (0gβ€˜(Scalarβ€˜π‘€)) ∧ 𝐷 = (π‘₯( linC β€˜π‘€)𝑉)) β†’ (𝐢 Β· 𝐷) = (𝐢 Β· (π‘₯( linC β€˜π‘€)𝑉)))
4847adantl 482 . . . . . . . . . . . . 13 ((π‘₯ ∈ (𝑅 ↑m 𝑉) ∧ (π‘₯ finSupp (0gβ€˜(Scalarβ€˜π‘€)) ∧ 𝐷 = (π‘₯( linC β€˜π‘€)𝑉))) β†’ (𝐢 Β· 𝐷) = (𝐢 Β· (π‘₯( linC β€˜π‘€)𝑉)))
4948ad2antrr 724 . . . . . . . . . . . 12 ((((π‘₯ ∈ (𝑅 ↑m 𝑉) ∧ (π‘₯ finSupp (0gβ€˜(Scalarβ€˜π‘€)) ∧ 𝐷 = (π‘₯( linC β€˜π‘€)𝑉))) ∧ 𝐷 ∈ (Baseβ€˜π‘€)) ∧ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Baseβ€˜π‘€)) ∧ 𝐢 ∈ 𝑅)) β†’ (𝐢 Β· 𝐷) = (𝐢 Β· (π‘₯( linC β€˜π‘€)𝑉)))
50 simprl 769 . . . . . . . . . . . . 13 ((((π‘₯ ∈ (𝑅 ↑m 𝑉) ∧ (π‘₯ finSupp (0gβ€˜(Scalarβ€˜π‘€)) ∧ 𝐷 = (π‘₯( linC β€˜π‘€)𝑉))) ∧ 𝐷 ∈ (Baseβ€˜π‘€)) ∧ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Baseβ€˜π‘€)) ∧ 𝐢 ∈ 𝑅)) β†’ (𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Baseβ€˜π‘€)))
5140adantr 481 . . . . . . . . . . . . . 14 (((π‘₯ ∈ (𝑅 ↑m 𝑉) ∧ (π‘₯ finSupp (0gβ€˜(Scalarβ€˜π‘€)) ∧ 𝐷 = (π‘₯( linC β€˜π‘€)𝑉))) ∧ 𝐷 ∈ (Baseβ€˜π‘€)) β†’ π‘₯ ∈ (𝑅 ↑m 𝑉))
5251, 8anim12i 613 . . . . . . . . . . . . 13 ((((π‘₯ ∈ (𝑅 ↑m 𝑉) ∧ (π‘₯ finSupp (0gβ€˜(Scalarβ€˜π‘€)) ∧ 𝐷 = (π‘₯( linC β€˜π‘€)𝑉))) ∧ 𝐷 ∈ (Baseβ€˜π‘€)) ∧ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Baseβ€˜π‘€)) ∧ 𝐢 ∈ 𝑅)) β†’ (π‘₯ ∈ (𝑅 ↑m 𝑉) ∧ 𝐢 ∈ 𝑅))
53 eqid 2732 . . . . . . . . . . . . . 14 (π‘₯( linC β€˜π‘€)𝑉) = (π‘₯( linC β€˜π‘€)𝑉)
54 eqid 2732 . . . . . . . . . . . . . 14 (𝑣 ∈ 𝑉 ↦ (𝐢(.rβ€˜(Scalarβ€˜π‘€))(π‘₯β€˜π‘£))) = (𝑣 ∈ 𝑉 ↦ (𝐢(.rβ€˜(Scalarβ€˜π‘€))(π‘₯β€˜π‘£)))
5511, 27, 53, 3, 54lincscm 47101 . . . . . . . . . . . . 13 (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Baseβ€˜π‘€)) ∧ (π‘₯ ∈ (𝑅 ↑m 𝑉) ∧ 𝐢 ∈ 𝑅) ∧ π‘₯ finSupp (0gβ€˜(Scalarβ€˜π‘€))) β†’ (𝐢 Β· (π‘₯( linC β€˜π‘€)𝑉)) = ((𝑣 ∈ 𝑉 ↦ (𝐢(.rβ€˜(Scalarβ€˜π‘€))(π‘₯β€˜π‘£)))( linC β€˜π‘€)𝑉))
5650, 52, 43, 55syl3anc 1371 . . . . . . . . . . . 12 ((((π‘₯ ∈ (𝑅 ↑m 𝑉) ∧ (π‘₯ finSupp (0gβ€˜(Scalarβ€˜π‘€)) ∧ 𝐷 = (π‘₯( linC β€˜π‘€)𝑉))) ∧ 𝐷 ∈ (Baseβ€˜π‘€)) ∧ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Baseβ€˜π‘€)) ∧ 𝐢 ∈ 𝑅)) β†’ (𝐢 Β· (π‘₯( linC β€˜π‘€)𝑉)) = ((𝑣 ∈ 𝑉 ↦ (𝐢(.rβ€˜(Scalarβ€˜π‘€))(π‘₯β€˜π‘£)))( linC β€˜π‘€)𝑉))
5749, 56eqtrd 2772 . . . . . . . . . . 11 ((((π‘₯ ∈ (𝑅 ↑m 𝑉) ∧ (π‘₯ finSupp (0gβ€˜(Scalarβ€˜π‘€)) ∧ 𝐷 = (π‘₯( linC β€˜π‘€)𝑉))) ∧ 𝐷 ∈ (Baseβ€˜π‘€)) ∧ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Baseβ€˜π‘€)) ∧ 𝐢 ∈ 𝑅)) β†’ (𝐢 Β· 𝐷) = ((𝑣 ∈ 𝑉 ↦ (𝐢(.rβ€˜(Scalarβ€˜π‘€))(π‘₯β€˜π‘£)))( linC β€˜π‘€)𝑉))
58 breq1 5151 . . . . . . . . . . . . 13 (𝑠 = (𝑣 ∈ 𝑉 ↦ (𝐢(.rβ€˜(Scalarβ€˜π‘€))(π‘₯β€˜π‘£))) β†’ (𝑠 finSupp (0gβ€˜(Scalarβ€˜π‘€)) ↔ (𝑣 ∈ 𝑉 ↦ (𝐢(.rβ€˜(Scalarβ€˜π‘€))(π‘₯β€˜π‘£))) finSupp (0gβ€˜(Scalarβ€˜π‘€))))
59 oveq1 7415 . . . . . . . . . . . . . 14 (𝑠 = (𝑣 ∈ 𝑉 ↦ (𝐢(.rβ€˜(Scalarβ€˜π‘€))(π‘₯β€˜π‘£))) β†’ (𝑠( linC β€˜π‘€)𝑉) = ((𝑣 ∈ 𝑉 ↦ (𝐢(.rβ€˜(Scalarβ€˜π‘€))(π‘₯β€˜π‘£)))( linC β€˜π‘€)𝑉))
6059eqeq2d 2743 . . . . . . . . . . . . 13 (𝑠 = (𝑣 ∈ 𝑉 ↦ (𝐢(.rβ€˜(Scalarβ€˜π‘€))(π‘₯β€˜π‘£))) β†’ ((𝐢 Β· 𝐷) = (𝑠( linC β€˜π‘€)𝑉) ↔ (𝐢 Β· 𝐷) = ((𝑣 ∈ 𝑉 ↦ (𝐢(.rβ€˜(Scalarβ€˜π‘€))(π‘₯β€˜π‘£)))( linC β€˜π‘€)𝑉)))
6158, 60anbi12d 631 . . . . . . . . . . . 12 (𝑠 = (𝑣 ∈ 𝑉 ↦ (𝐢(.rβ€˜(Scalarβ€˜π‘€))(π‘₯β€˜π‘£))) β†’ ((𝑠 finSupp (0gβ€˜(Scalarβ€˜π‘€)) ∧ (𝐢 Β· 𝐷) = (𝑠( linC β€˜π‘€)𝑉)) ↔ ((𝑣 ∈ 𝑉 ↦ (𝐢(.rβ€˜(Scalarβ€˜π‘€))(π‘₯β€˜π‘£))) finSupp (0gβ€˜(Scalarβ€˜π‘€)) ∧ (𝐢 Β· 𝐷) = ((𝑣 ∈ 𝑉 ↦ (𝐢(.rβ€˜(Scalarβ€˜π‘€))(π‘₯β€˜π‘£)))( linC β€˜π‘€)𝑉))))
6261rspcev 3612 . . . . . . . . . . 11 (((𝑣 ∈ 𝑉 ↦ (𝐢(.rβ€˜(Scalarβ€˜π‘€))(π‘₯β€˜π‘£))) ∈ (𝑅 ↑m 𝑉) ∧ ((𝑣 ∈ 𝑉 ↦ (𝐢(.rβ€˜(Scalarβ€˜π‘€))(π‘₯β€˜π‘£))) finSupp (0gβ€˜(Scalarβ€˜π‘€)) ∧ (𝐢 Β· 𝐷) = ((𝑣 ∈ 𝑉 ↦ (𝐢(.rβ€˜(Scalarβ€˜π‘€))(π‘₯β€˜π‘£)))( linC β€˜π‘€)𝑉))) β†’ βˆƒπ‘  ∈ (𝑅 ↑m 𝑉)(𝑠 finSupp (0gβ€˜(Scalarβ€˜π‘€)) ∧ (𝐢 Β· 𝐷) = (𝑠( linC β€˜π‘€)𝑉)))
6337, 45, 57, 62syl12anc 835 . . . . . . . . . 10 ((((π‘₯ ∈ (𝑅 ↑m 𝑉) ∧ (π‘₯ finSupp (0gβ€˜(Scalarβ€˜π‘€)) ∧ 𝐷 = (π‘₯( linC β€˜π‘€)𝑉))) ∧ 𝐷 ∈ (Baseβ€˜π‘€)) ∧ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Baseβ€˜π‘€)) ∧ 𝐢 ∈ 𝑅)) β†’ βˆƒπ‘  ∈ (𝑅 ↑m 𝑉)(𝑠 finSupp (0gβ€˜(Scalarβ€˜π‘€)) ∧ (𝐢 Β· 𝐷) = (𝑠( linC β€˜π‘€)𝑉)))
6463ex 413 . . . . . . . . 9 (((π‘₯ ∈ (𝑅 ↑m 𝑉) ∧ (π‘₯ finSupp (0gβ€˜(Scalarβ€˜π‘€)) ∧ 𝐷 = (π‘₯( linC β€˜π‘€)𝑉))) ∧ 𝐷 ∈ (Baseβ€˜π‘€)) β†’ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Baseβ€˜π‘€)) ∧ 𝐢 ∈ 𝑅) β†’ βˆƒπ‘  ∈ (𝑅 ↑m 𝑉)(𝑠 finSupp (0gβ€˜(Scalarβ€˜π‘€)) ∧ (𝐢 Β· 𝐷) = (𝑠( linC β€˜π‘€)𝑉))))
6564ex 413 . . . . . . . 8 ((π‘₯ ∈ (𝑅 ↑m 𝑉) ∧ (π‘₯ finSupp (0gβ€˜(Scalarβ€˜π‘€)) ∧ 𝐷 = (π‘₯( linC β€˜π‘€)𝑉))) β†’ (𝐷 ∈ (Baseβ€˜π‘€) β†’ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Baseβ€˜π‘€)) ∧ 𝐢 ∈ 𝑅) β†’ βˆƒπ‘  ∈ (𝑅 ↑m 𝑉)(𝑠 finSupp (0gβ€˜(Scalarβ€˜π‘€)) ∧ (𝐢 Β· 𝐷) = (𝑠( linC β€˜π‘€)𝑉)))))
6665rexlimiva 3147 . . . . . . 7 (βˆƒπ‘₯ ∈ (𝑅 ↑m 𝑉)(π‘₯ finSupp (0gβ€˜(Scalarβ€˜π‘€)) ∧ 𝐷 = (π‘₯( linC β€˜π‘€)𝑉)) β†’ (𝐷 ∈ (Baseβ€˜π‘€) β†’ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Baseβ€˜π‘€)) ∧ 𝐢 ∈ 𝑅) β†’ βˆƒπ‘  ∈ (𝑅 ↑m 𝑉)(𝑠 finSupp (0gβ€˜(Scalarβ€˜π‘€)) ∧ (𝐢 Β· 𝐷) = (𝑠( linC β€˜π‘€)𝑉)))))
6766impcom 408 . . . . . 6 ((𝐷 ∈ (Baseβ€˜π‘€) ∧ βˆƒπ‘₯ ∈ (𝑅 ↑m 𝑉)(π‘₯ finSupp (0gβ€˜(Scalarβ€˜π‘€)) ∧ 𝐷 = (π‘₯( linC β€˜π‘€)𝑉))) β†’ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Baseβ€˜π‘€)) ∧ 𝐢 ∈ 𝑅) β†’ βˆƒπ‘  ∈ (𝑅 ↑m 𝑉)(𝑠 finSupp (0gβ€˜(Scalarβ€˜π‘€)) ∧ (𝐢 Β· 𝐷) = (𝑠( linC β€˜π‘€)𝑉))))
6867impcom 408 . . . . 5 ((((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Baseβ€˜π‘€)) ∧ 𝐢 ∈ 𝑅) ∧ (𝐷 ∈ (Baseβ€˜π‘€) ∧ βˆƒπ‘₯ ∈ (𝑅 ↑m 𝑉)(π‘₯ finSupp (0gβ€˜(Scalarβ€˜π‘€)) ∧ 𝐷 = (π‘₯( linC β€˜π‘€)𝑉)))) β†’ βˆƒπ‘  ∈ (𝑅 ↑m 𝑉)(𝑠 finSupp (0gβ€˜(Scalarβ€˜π‘€)) ∧ (𝐢 Β· 𝐷) = (𝑠( linC β€˜π‘€)𝑉)))
691, 2, 3lcoval 47083 . . . . . 6 ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Baseβ€˜π‘€)) β†’ ((𝐢 Β· 𝐷) ∈ (𝑀 LinCo 𝑉) ↔ ((𝐢 Β· 𝐷) ∈ (Baseβ€˜π‘€) ∧ βˆƒπ‘  ∈ (𝑅 ↑m 𝑉)(𝑠 finSupp (0gβ€˜(Scalarβ€˜π‘€)) ∧ (𝐢 Β· 𝐷) = (𝑠( linC β€˜π‘€)𝑉)))))
7069ad2antrr 724 . . . . 5 ((((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Baseβ€˜π‘€)) ∧ 𝐢 ∈ 𝑅) ∧ (𝐷 ∈ (Baseβ€˜π‘€) ∧ βˆƒπ‘₯ ∈ (𝑅 ↑m 𝑉)(π‘₯ finSupp (0gβ€˜(Scalarβ€˜π‘€)) ∧ 𝐷 = (π‘₯( linC β€˜π‘€)𝑉)))) β†’ ((𝐢 Β· 𝐷) ∈ (𝑀 LinCo 𝑉) ↔ ((𝐢 Β· 𝐷) ∈ (Baseβ€˜π‘€) ∧ βˆƒπ‘  ∈ (𝑅 ↑m 𝑉)(𝑠 finSupp (0gβ€˜(Scalarβ€˜π‘€)) ∧ (𝐢 Β· 𝐷) = (𝑠( linC β€˜π‘€)𝑉)))))
7113, 68, 70mpbir2and 711 . . . 4 ((((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Baseβ€˜π‘€)) ∧ 𝐢 ∈ 𝑅) ∧ (𝐷 ∈ (Baseβ€˜π‘€) ∧ βˆƒπ‘₯ ∈ (𝑅 ↑m 𝑉)(π‘₯ finSupp (0gβ€˜(Scalarβ€˜π‘€)) ∧ 𝐷 = (π‘₯( linC β€˜π‘€)𝑉)))) β†’ (𝐢 Β· 𝐷) ∈ (𝑀 LinCo 𝑉))
7271ex 413 . . 3 (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Baseβ€˜π‘€)) ∧ 𝐢 ∈ 𝑅) β†’ ((𝐷 ∈ (Baseβ€˜π‘€) ∧ βˆƒπ‘₯ ∈ (𝑅 ↑m 𝑉)(π‘₯ finSupp (0gβ€˜(Scalarβ€˜π‘€)) ∧ 𝐷 = (π‘₯( linC β€˜π‘€)𝑉))) β†’ (𝐢 Β· 𝐷) ∈ (𝑀 LinCo 𝑉)))
735, 72sylbid 239 . 2 (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Baseβ€˜π‘€)) ∧ 𝐢 ∈ 𝑅) β†’ (𝐷 ∈ (𝑀 LinCo 𝑉) β†’ (𝐢 Β· 𝐷) ∈ (𝑀 LinCo 𝑉)))
74733impia 1117 1 (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Baseβ€˜π‘€)) ∧ 𝐢 ∈ 𝑅 ∧ 𝐷 ∈ (𝑀 LinCo 𝑉)) β†’ (𝐢 Β· 𝐷) ∈ (𝑀 LinCo 𝑉))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106  βˆƒwrex 3070  Vcvv 3474  π’« cpw 4602   class class class wbr 5148   ↦ cmpt 5231  βŸΆwf 6539  β€˜cfv 6543  (class class class)co 7408   ↑m cmap 8819   finSupp cfsupp 9360  Basecbs 17143  .rcmulr 17197  Scalarcsca 17199   ·𝑠 cvsca 17200  0gc0g 17384  Ringcrg 20055  LModclmod 20470   linC clinc 47075   LinCo clinco 47076
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7724  ax-cnex 11165  ax-resscn 11166  ax-1cn 11167  ax-icn 11168  ax-addcl 11169  ax-addrcl 11170  ax-mulcl 11171  ax-mulrcl 11172  ax-mulcom 11173  ax-addass 11174  ax-mulass 11175  ax-distr 11176  ax-i2m1 11177  ax-1ne0 11178  ax-1rid 11179  ax-rnegex 11180  ax-rrecex 11181  ax-cnre 11182  ax-pre-lttri 11183  ax-pre-lttrn 11184  ax-pre-ltadd 11185  ax-pre-mulgt0 11186
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3376  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-int 4951  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-se 5632  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-isom 6552  df-riota 7364  df-ov 7411  df-oprab 7412  df-mpo 7413  df-om 7855  df-1st 7974  df-2nd 7975  df-supp 8146  df-frecs 8265  df-wrecs 8296  df-recs 8370  df-rdg 8409  df-1o 8465  df-er 8702  df-map 8821  df-en 8939  df-dom 8940  df-sdom 8941  df-fin 8942  df-fsupp 9361  df-oi 9504  df-card 9933  df-pnf 11249  df-mnf 11250  df-xr 11251  df-ltxr 11252  df-le 11253  df-sub 11445  df-neg 11446  df-nn 12212  df-2 12274  df-n0 12472  df-z 12558  df-uz 12822  df-fz 13484  df-fzo 13627  df-seq 13966  df-hash 14290  df-sets 17096  df-slot 17114  df-ndx 17126  df-base 17144  df-plusg 17209  df-0g 17386  df-gsum 17387  df-mgm 18560  df-sgrp 18609  df-mnd 18625  df-mhm 18670  df-grp 18821  df-minusg 18822  df-ghm 19089  df-cntz 19180  df-cmn 19649  df-abl 19650  df-mgp 19987  df-ur 20004  df-ring 20057  df-lmod 20472  df-linc 47077  df-lco 47078
This theorem is referenced by:  lincsumscmcl  47104
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