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Theorem lmhmvsca 20521
Description: The pointwise scalar product of a linear function and a constant is linear, over a commutative ring. (Contributed by Mario Carneiro, 22-Sep-2015.)
Hypotheses
Ref Expression
lmhmvsca.v 𝑉 = (Baseβ€˜π‘€)
lmhmvsca.s Β· = ( ·𝑠 β€˜π‘)
lmhmvsca.j 𝐽 = (Scalarβ€˜π‘)
lmhmvsca.k 𝐾 = (Baseβ€˜π½)
Assertion
Ref Expression
lmhmvsca ((𝐽 ∈ CRing ∧ 𝐴 ∈ 𝐾 ∧ 𝐹 ∈ (𝑀 LMHom 𝑁)) β†’ ((𝑉 Γ— {𝐴}) ∘f Β· 𝐹) ∈ (𝑀 LMHom 𝑁))

Proof of Theorem lmhmvsca
Dummy variables 𝑣 𝑒 π‘₯ 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 lmhmvsca.v . 2 𝑉 = (Baseβ€˜π‘€)
2 eqid 2733 . 2 ( ·𝑠 β€˜π‘€) = ( ·𝑠 β€˜π‘€)
3 lmhmvsca.s . 2 Β· = ( ·𝑠 β€˜π‘)
4 eqid 2733 . 2 (Scalarβ€˜π‘€) = (Scalarβ€˜π‘€)
5 lmhmvsca.j . 2 𝐽 = (Scalarβ€˜π‘)
6 eqid 2733 . 2 (Baseβ€˜(Scalarβ€˜π‘€)) = (Baseβ€˜(Scalarβ€˜π‘€))
7 lmhmlmod1 20509 . . 3 (𝐹 ∈ (𝑀 LMHom 𝑁) β†’ 𝑀 ∈ LMod)
873ad2ant3 1136 . 2 ((𝐽 ∈ CRing ∧ 𝐴 ∈ 𝐾 ∧ 𝐹 ∈ (𝑀 LMHom 𝑁)) β†’ 𝑀 ∈ LMod)
9 lmhmlmod2 20508 . . 3 (𝐹 ∈ (𝑀 LMHom 𝑁) β†’ 𝑁 ∈ LMod)
1093ad2ant3 1136 . 2 ((𝐽 ∈ CRing ∧ 𝐴 ∈ 𝐾 ∧ 𝐹 ∈ (𝑀 LMHom 𝑁)) β†’ 𝑁 ∈ LMod)
114, 5lmhmsca 20506 . . 3 (𝐹 ∈ (𝑀 LMHom 𝑁) β†’ 𝐽 = (Scalarβ€˜π‘€))
12113ad2ant3 1136 . 2 ((𝐽 ∈ CRing ∧ 𝐴 ∈ 𝐾 ∧ 𝐹 ∈ (𝑀 LMHom 𝑁)) β†’ 𝐽 = (Scalarβ€˜π‘€))
131fvexi 6857 . . . . . 6 𝑉 ∈ V
1413a1i 11 . . . . 5 ((𝐽 ∈ CRing ∧ 𝐴 ∈ 𝐾 ∧ 𝐹 ∈ (𝑀 LMHom 𝑁)) β†’ 𝑉 ∈ V)
15 simpl2 1193 . . . . 5 (((𝐽 ∈ CRing ∧ 𝐴 ∈ 𝐾 ∧ 𝐹 ∈ (𝑀 LMHom 𝑁)) ∧ 𝑣 ∈ 𝑉) β†’ 𝐴 ∈ 𝐾)
16 eqid 2733 . . . . . . . 8 (Baseβ€˜π‘) = (Baseβ€˜π‘)
171, 16lmhmf 20510 . . . . . . 7 (𝐹 ∈ (𝑀 LMHom 𝑁) β†’ 𝐹:π‘‰βŸΆ(Baseβ€˜π‘))
18173ad2ant3 1136 . . . . . 6 ((𝐽 ∈ CRing ∧ 𝐴 ∈ 𝐾 ∧ 𝐹 ∈ (𝑀 LMHom 𝑁)) β†’ 𝐹:π‘‰βŸΆ(Baseβ€˜π‘))
1918ffvelcdmda 7036 . . . . 5 (((𝐽 ∈ CRing ∧ 𝐴 ∈ 𝐾 ∧ 𝐹 ∈ (𝑀 LMHom 𝑁)) ∧ 𝑣 ∈ 𝑉) β†’ (πΉβ€˜π‘£) ∈ (Baseβ€˜π‘))
20 fconstmpt 5695 . . . . . 6 (𝑉 Γ— {𝐴}) = (𝑣 ∈ 𝑉 ↦ 𝐴)
2120a1i 11 . . . . 5 ((𝐽 ∈ CRing ∧ 𝐴 ∈ 𝐾 ∧ 𝐹 ∈ (𝑀 LMHom 𝑁)) β†’ (𝑉 Γ— {𝐴}) = (𝑣 ∈ 𝑉 ↦ 𝐴))
2218feqmptd 6911 . . . . 5 ((𝐽 ∈ CRing ∧ 𝐴 ∈ 𝐾 ∧ 𝐹 ∈ (𝑀 LMHom 𝑁)) β†’ 𝐹 = (𝑣 ∈ 𝑉 ↦ (πΉβ€˜π‘£)))
2314, 15, 19, 21, 22offval2 7638 . . . 4 ((𝐽 ∈ CRing ∧ 𝐴 ∈ 𝐾 ∧ 𝐹 ∈ (𝑀 LMHom 𝑁)) β†’ ((𝑉 Γ— {𝐴}) ∘f Β· 𝐹) = (𝑣 ∈ 𝑉 ↦ (𝐴 Β· (πΉβ€˜π‘£))))
24 eqidd 2734 . . . . 5 ((𝐽 ∈ CRing ∧ 𝐴 ∈ 𝐾 ∧ 𝐹 ∈ (𝑀 LMHom 𝑁)) β†’ (𝑒 ∈ (Baseβ€˜π‘) ↦ (𝐴 Β· 𝑒)) = (𝑒 ∈ (Baseβ€˜π‘) ↦ (𝐴 Β· 𝑒)))
25 oveq2 7366 . . . . 5 (𝑒 = (πΉβ€˜π‘£) β†’ (𝐴 Β· 𝑒) = (𝐴 Β· (πΉβ€˜π‘£)))
2619, 22, 24, 25fmptco 7076 . . . 4 ((𝐽 ∈ CRing ∧ 𝐴 ∈ 𝐾 ∧ 𝐹 ∈ (𝑀 LMHom 𝑁)) β†’ ((𝑒 ∈ (Baseβ€˜π‘) ↦ (𝐴 Β· 𝑒)) ∘ 𝐹) = (𝑣 ∈ 𝑉 ↦ (𝐴 Β· (πΉβ€˜π‘£))))
2723, 26eqtr4d 2776 . . 3 ((𝐽 ∈ CRing ∧ 𝐴 ∈ 𝐾 ∧ 𝐹 ∈ (𝑀 LMHom 𝑁)) β†’ ((𝑉 Γ— {𝐴}) ∘f Β· 𝐹) = ((𝑒 ∈ (Baseβ€˜π‘) ↦ (𝐴 Β· 𝑒)) ∘ 𝐹))
28 simp2 1138 . . . . 5 ((𝐽 ∈ CRing ∧ 𝐴 ∈ 𝐾 ∧ 𝐹 ∈ (𝑀 LMHom 𝑁)) β†’ 𝐴 ∈ 𝐾)
29 lmhmvsca.k . . . . . 6 𝐾 = (Baseβ€˜π½)
3016, 5, 3, 29lmodvsghm 20398 . . . . 5 ((𝑁 ∈ LMod ∧ 𝐴 ∈ 𝐾) β†’ (𝑒 ∈ (Baseβ€˜π‘) ↦ (𝐴 Β· 𝑒)) ∈ (𝑁 GrpHom 𝑁))
3110, 28, 30syl2anc 585 . . . 4 ((𝐽 ∈ CRing ∧ 𝐴 ∈ 𝐾 ∧ 𝐹 ∈ (𝑀 LMHom 𝑁)) β†’ (𝑒 ∈ (Baseβ€˜π‘) ↦ (𝐴 Β· 𝑒)) ∈ (𝑁 GrpHom 𝑁))
32 lmghm 20507 . . . . 5 (𝐹 ∈ (𝑀 LMHom 𝑁) β†’ 𝐹 ∈ (𝑀 GrpHom 𝑁))
33323ad2ant3 1136 . . . 4 ((𝐽 ∈ CRing ∧ 𝐴 ∈ 𝐾 ∧ 𝐹 ∈ (𝑀 LMHom 𝑁)) β†’ 𝐹 ∈ (𝑀 GrpHom 𝑁))
34 ghmco 19033 . . . 4 (((𝑒 ∈ (Baseβ€˜π‘) ↦ (𝐴 Β· 𝑒)) ∈ (𝑁 GrpHom 𝑁) ∧ 𝐹 ∈ (𝑀 GrpHom 𝑁)) β†’ ((𝑒 ∈ (Baseβ€˜π‘) ↦ (𝐴 Β· 𝑒)) ∘ 𝐹) ∈ (𝑀 GrpHom 𝑁))
3531, 33, 34syl2anc 585 . . 3 ((𝐽 ∈ CRing ∧ 𝐴 ∈ 𝐾 ∧ 𝐹 ∈ (𝑀 LMHom 𝑁)) β†’ ((𝑒 ∈ (Baseβ€˜π‘) ↦ (𝐴 Β· 𝑒)) ∘ 𝐹) ∈ (𝑀 GrpHom 𝑁))
3627, 35eqeltrd 2834 . 2 ((𝐽 ∈ CRing ∧ 𝐴 ∈ 𝐾 ∧ 𝐹 ∈ (𝑀 LMHom 𝑁)) β†’ ((𝑉 Γ— {𝐴}) ∘f Β· 𝐹) ∈ (𝑀 GrpHom 𝑁))
37 simpl1 1192 . . . . . 6 (((𝐽 ∈ CRing ∧ 𝐴 ∈ 𝐾 ∧ 𝐹 ∈ (𝑀 LMHom 𝑁)) ∧ (π‘₯ ∈ (Baseβ€˜(Scalarβ€˜π‘€)) ∧ 𝑦 ∈ 𝑉)) β†’ 𝐽 ∈ CRing)
38 simpl2 1193 . . . . . 6 (((𝐽 ∈ CRing ∧ 𝐴 ∈ 𝐾 ∧ 𝐹 ∈ (𝑀 LMHom 𝑁)) ∧ (π‘₯ ∈ (Baseβ€˜(Scalarβ€˜π‘€)) ∧ 𝑦 ∈ 𝑉)) β†’ 𝐴 ∈ 𝐾)
39 simprl 770 . . . . . . 7 (((𝐽 ∈ CRing ∧ 𝐴 ∈ 𝐾 ∧ 𝐹 ∈ (𝑀 LMHom 𝑁)) ∧ (π‘₯ ∈ (Baseβ€˜(Scalarβ€˜π‘€)) ∧ 𝑦 ∈ 𝑉)) β†’ π‘₯ ∈ (Baseβ€˜(Scalarβ€˜π‘€)))
4012fveq2d 6847 . . . . . . . . 9 ((𝐽 ∈ CRing ∧ 𝐴 ∈ 𝐾 ∧ 𝐹 ∈ (𝑀 LMHom 𝑁)) β†’ (Baseβ€˜π½) = (Baseβ€˜(Scalarβ€˜π‘€)))
4129, 40eqtrid 2785 . . . . . . . 8 ((𝐽 ∈ CRing ∧ 𝐴 ∈ 𝐾 ∧ 𝐹 ∈ (𝑀 LMHom 𝑁)) β†’ 𝐾 = (Baseβ€˜(Scalarβ€˜π‘€)))
4241adantr 482 . . . . . . 7 (((𝐽 ∈ CRing ∧ 𝐴 ∈ 𝐾 ∧ 𝐹 ∈ (𝑀 LMHom 𝑁)) ∧ (π‘₯ ∈ (Baseβ€˜(Scalarβ€˜π‘€)) ∧ 𝑦 ∈ 𝑉)) β†’ 𝐾 = (Baseβ€˜(Scalarβ€˜π‘€)))
4339, 42eleqtrrd 2837 . . . . . 6 (((𝐽 ∈ CRing ∧ 𝐴 ∈ 𝐾 ∧ 𝐹 ∈ (𝑀 LMHom 𝑁)) ∧ (π‘₯ ∈ (Baseβ€˜(Scalarβ€˜π‘€)) ∧ 𝑦 ∈ 𝑉)) β†’ π‘₯ ∈ 𝐾)
44 eqid 2733 . . . . . . 7 (.rβ€˜π½) = (.rβ€˜π½)
4529, 44crngcom 19987 . . . . . 6 ((𝐽 ∈ CRing ∧ 𝐴 ∈ 𝐾 ∧ π‘₯ ∈ 𝐾) β†’ (𝐴(.rβ€˜π½)π‘₯) = (π‘₯(.rβ€˜π½)𝐴))
4637, 38, 43, 45syl3anc 1372 . . . . 5 (((𝐽 ∈ CRing ∧ 𝐴 ∈ 𝐾 ∧ 𝐹 ∈ (𝑀 LMHom 𝑁)) ∧ (π‘₯ ∈ (Baseβ€˜(Scalarβ€˜π‘€)) ∧ 𝑦 ∈ 𝑉)) β†’ (𝐴(.rβ€˜π½)π‘₯) = (π‘₯(.rβ€˜π½)𝐴))
4746oveq1d 7373 . . . 4 (((𝐽 ∈ CRing ∧ 𝐴 ∈ 𝐾 ∧ 𝐹 ∈ (𝑀 LMHom 𝑁)) ∧ (π‘₯ ∈ (Baseβ€˜(Scalarβ€˜π‘€)) ∧ 𝑦 ∈ 𝑉)) β†’ ((𝐴(.rβ€˜π½)π‘₯) Β· (πΉβ€˜π‘¦)) = ((π‘₯(.rβ€˜π½)𝐴) Β· (πΉβ€˜π‘¦)))
4810adantr 482 . . . . 5 (((𝐽 ∈ CRing ∧ 𝐴 ∈ 𝐾 ∧ 𝐹 ∈ (𝑀 LMHom 𝑁)) ∧ (π‘₯ ∈ (Baseβ€˜(Scalarβ€˜π‘€)) ∧ 𝑦 ∈ 𝑉)) β†’ 𝑁 ∈ LMod)
4918adantr 482 . . . . . 6 (((𝐽 ∈ CRing ∧ 𝐴 ∈ 𝐾 ∧ 𝐹 ∈ (𝑀 LMHom 𝑁)) ∧ (π‘₯ ∈ (Baseβ€˜(Scalarβ€˜π‘€)) ∧ 𝑦 ∈ 𝑉)) β†’ 𝐹:π‘‰βŸΆ(Baseβ€˜π‘))
50 simprr 772 . . . . . 6 (((𝐽 ∈ CRing ∧ 𝐴 ∈ 𝐾 ∧ 𝐹 ∈ (𝑀 LMHom 𝑁)) ∧ (π‘₯ ∈ (Baseβ€˜(Scalarβ€˜π‘€)) ∧ 𝑦 ∈ 𝑉)) β†’ 𝑦 ∈ 𝑉)
5149, 50ffvelcdmd 7037 . . . . 5 (((𝐽 ∈ CRing ∧ 𝐴 ∈ 𝐾 ∧ 𝐹 ∈ (𝑀 LMHom 𝑁)) ∧ (π‘₯ ∈ (Baseβ€˜(Scalarβ€˜π‘€)) ∧ 𝑦 ∈ 𝑉)) β†’ (πΉβ€˜π‘¦) ∈ (Baseβ€˜π‘))
5216, 5, 3, 29, 44lmodvsass 20362 . . . . 5 ((𝑁 ∈ LMod ∧ (𝐴 ∈ 𝐾 ∧ π‘₯ ∈ 𝐾 ∧ (πΉβ€˜π‘¦) ∈ (Baseβ€˜π‘))) β†’ ((𝐴(.rβ€˜π½)π‘₯) Β· (πΉβ€˜π‘¦)) = (𝐴 Β· (π‘₯ Β· (πΉβ€˜π‘¦))))
5348, 38, 43, 51, 52syl13anc 1373 . . . 4 (((𝐽 ∈ CRing ∧ 𝐴 ∈ 𝐾 ∧ 𝐹 ∈ (𝑀 LMHom 𝑁)) ∧ (π‘₯ ∈ (Baseβ€˜(Scalarβ€˜π‘€)) ∧ 𝑦 ∈ 𝑉)) β†’ ((𝐴(.rβ€˜π½)π‘₯) Β· (πΉβ€˜π‘¦)) = (𝐴 Β· (π‘₯ Β· (πΉβ€˜π‘¦))))
5416, 5, 3, 29, 44lmodvsass 20362 . . . . 5 ((𝑁 ∈ LMod ∧ (π‘₯ ∈ 𝐾 ∧ 𝐴 ∈ 𝐾 ∧ (πΉβ€˜π‘¦) ∈ (Baseβ€˜π‘))) β†’ ((π‘₯(.rβ€˜π½)𝐴) Β· (πΉβ€˜π‘¦)) = (π‘₯ Β· (𝐴 Β· (πΉβ€˜π‘¦))))
5548, 43, 38, 51, 54syl13anc 1373 . . . 4 (((𝐽 ∈ CRing ∧ 𝐴 ∈ 𝐾 ∧ 𝐹 ∈ (𝑀 LMHom 𝑁)) ∧ (π‘₯ ∈ (Baseβ€˜(Scalarβ€˜π‘€)) ∧ 𝑦 ∈ 𝑉)) β†’ ((π‘₯(.rβ€˜π½)𝐴) Β· (πΉβ€˜π‘¦)) = (π‘₯ Β· (𝐴 Β· (πΉβ€˜π‘¦))))
5647, 53, 553eqtr3d 2781 . . 3 (((𝐽 ∈ CRing ∧ 𝐴 ∈ 𝐾 ∧ 𝐹 ∈ (𝑀 LMHom 𝑁)) ∧ (π‘₯ ∈ (Baseβ€˜(Scalarβ€˜π‘€)) ∧ 𝑦 ∈ 𝑉)) β†’ (𝐴 Β· (π‘₯ Β· (πΉβ€˜π‘¦))) = (π‘₯ Β· (𝐴 Β· (πΉβ€˜π‘¦))))
571, 4, 2, 6lmodvscl 20354 . . . . . 6 ((𝑀 ∈ LMod ∧ π‘₯ ∈ (Baseβ€˜(Scalarβ€˜π‘€)) ∧ 𝑦 ∈ 𝑉) β†’ (π‘₯( ·𝑠 β€˜π‘€)𝑦) ∈ 𝑉)
58573expb 1121 . . . . 5 ((𝑀 ∈ LMod ∧ (π‘₯ ∈ (Baseβ€˜(Scalarβ€˜π‘€)) ∧ 𝑦 ∈ 𝑉)) β†’ (π‘₯( ·𝑠 β€˜π‘€)𝑦) ∈ 𝑉)
598, 58sylan 581 . . . 4 (((𝐽 ∈ CRing ∧ 𝐴 ∈ 𝐾 ∧ 𝐹 ∈ (𝑀 LMHom 𝑁)) ∧ (π‘₯ ∈ (Baseβ€˜(Scalarβ€˜π‘€)) ∧ 𝑦 ∈ 𝑉)) β†’ (π‘₯( ·𝑠 β€˜π‘€)𝑦) ∈ 𝑉)
6013a1i 11 . . . . 5 (((𝐽 ∈ CRing ∧ 𝐴 ∈ 𝐾 ∧ 𝐹 ∈ (𝑀 LMHom 𝑁)) ∧ (π‘₯ ∈ (Baseβ€˜(Scalarβ€˜π‘€)) ∧ 𝑦 ∈ 𝑉)) β†’ 𝑉 ∈ V)
6118ffnd 6670 . . . . . 6 ((𝐽 ∈ CRing ∧ 𝐴 ∈ 𝐾 ∧ 𝐹 ∈ (𝑀 LMHom 𝑁)) β†’ 𝐹 Fn 𝑉)
6261adantr 482 . . . . 5 (((𝐽 ∈ CRing ∧ 𝐴 ∈ 𝐾 ∧ 𝐹 ∈ (𝑀 LMHom 𝑁)) ∧ (π‘₯ ∈ (Baseβ€˜(Scalarβ€˜π‘€)) ∧ 𝑦 ∈ 𝑉)) β†’ 𝐹 Fn 𝑉)
634, 6, 1, 2, 3lmhmlin 20511 . . . . . . . 8 ((𝐹 ∈ (𝑀 LMHom 𝑁) ∧ π‘₯ ∈ (Baseβ€˜(Scalarβ€˜π‘€)) ∧ 𝑦 ∈ 𝑉) β†’ (πΉβ€˜(π‘₯( ·𝑠 β€˜π‘€)𝑦)) = (π‘₯ Β· (πΉβ€˜π‘¦)))
64633expb 1121 . . . . . . 7 ((𝐹 ∈ (𝑀 LMHom 𝑁) ∧ (π‘₯ ∈ (Baseβ€˜(Scalarβ€˜π‘€)) ∧ 𝑦 ∈ 𝑉)) β†’ (πΉβ€˜(π‘₯( ·𝑠 β€˜π‘€)𝑦)) = (π‘₯ Β· (πΉβ€˜π‘¦)))
65643ad2antl3 1188 . . . . . 6 (((𝐽 ∈ CRing ∧ 𝐴 ∈ 𝐾 ∧ 𝐹 ∈ (𝑀 LMHom 𝑁)) ∧ (π‘₯ ∈ (Baseβ€˜(Scalarβ€˜π‘€)) ∧ 𝑦 ∈ 𝑉)) β†’ (πΉβ€˜(π‘₯( ·𝑠 β€˜π‘€)𝑦)) = (π‘₯ Β· (πΉβ€˜π‘¦)))
6665adantr 482 . . . . 5 ((((𝐽 ∈ CRing ∧ 𝐴 ∈ 𝐾 ∧ 𝐹 ∈ (𝑀 LMHom 𝑁)) ∧ (π‘₯ ∈ (Baseβ€˜(Scalarβ€˜π‘€)) ∧ 𝑦 ∈ 𝑉)) ∧ (π‘₯( ·𝑠 β€˜π‘€)𝑦) ∈ 𝑉) β†’ (πΉβ€˜(π‘₯( ·𝑠 β€˜π‘€)𝑦)) = (π‘₯ Β· (πΉβ€˜π‘¦)))
6760, 38, 62, 66ofc1 7644 . . . 4 ((((𝐽 ∈ CRing ∧ 𝐴 ∈ 𝐾 ∧ 𝐹 ∈ (𝑀 LMHom 𝑁)) ∧ (π‘₯ ∈ (Baseβ€˜(Scalarβ€˜π‘€)) ∧ 𝑦 ∈ 𝑉)) ∧ (π‘₯( ·𝑠 β€˜π‘€)𝑦) ∈ 𝑉) β†’ (((𝑉 Γ— {𝐴}) ∘f Β· 𝐹)β€˜(π‘₯( ·𝑠 β€˜π‘€)𝑦)) = (𝐴 Β· (π‘₯ Β· (πΉβ€˜π‘¦))))
6859, 67mpdan 686 . . 3 (((𝐽 ∈ CRing ∧ 𝐴 ∈ 𝐾 ∧ 𝐹 ∈ (𝑀 LMHom 𝑁)) ∧ (π‘₯ ∈ (Baseβ€˜(Scalarβ€˜π‘€)) ∧ 𝑦 ∈ 𝑉)) β†’ (((𝑉 Γ— {𝐴}) ∘f Β· 𝐹)β€˜(π‘₯( ·𝑠 β€˜π‘€)𝑦)) = (𝐴 Β· (π‘₯ Β· (πΉβ€˜π‘¦))))
69 eqidd 2734 . . . . . 6 ((((𝐽 ∈ CRing ∧ 𝐴 ∈ 𝐾 ∧ 𝐹 ∈ (𝑀 LMHom 𝑁)) ∧ (π‘₯ ∈ (Baseβ€˜(Scalarβ€˜π‘€)) ∧ 𝑦 ∈ 𝑉)) ∧ 𝑦 ∈ 𝑉) β†’ (πΉβ€˜π‘¦) = (πΉβ€˜π‘¦))
7060, 38, 62, 69ofc1 7644 . . . . 5 ((((𝐽 ∈ CRing ∧ 𝐴 ∈ 𝐾 ∧ 𝐹 ∈ (𝑀 LMHom 𝑁)) ∧ (π‘₯ ∈ (Baseβ€˜(Scalarβ€˜π‘€)) ∧ 𝑦 ∈ 𝑉)) ∧ 𝑦 ∈ 𝑉) β†’ (((𝑉 Γ— {𝐴}) ∘f Β· 𝐹)β€˜π‘¦) = (𝐴 Β· (πΉβ€˜π‘¦)))
7150, 70mpdan 686 . . . 4 (((𝐽 ∈ CRing ∧ 𝐴 ∈ 𝐾 ∧ 𝐹 ∈ (𝑀 LMHom 𝑁)) ∧ (π‘₯ ∈ (Baseβ€˜(Scalarβ€˜π‘€)) ∧ 𝑦 ∈ 𝑉)) β†’ (((𝑉 Γ— {𝐴}) ∘f Β· 𝐹)β€˜π‘¦) = (𝐴 Β· (πΉβ€˜π‘¦)))
7271oveq2d 7374 . . 3 (((𝐽 ∈ CRing ∧ 𝐴 ∈ 𝐾 ∧ 𝐹 ∈ (𝑀 LMHom 𝑁)) ∧ (π‘₯ ∈ (Baseβ€˜(Scalarβ€˜π‘€)) ∧ 𝑦 ∈ 𝑉)) β†’ (π‘₯ Β· (((𝑉 Γ— {𝐴}) ∘f Β· 𝐹)β€˜π‘¦)) = (π‘₯ Β· (𝐴 Β· (πΉβ€˜π‘¦))))
7356, 68, 723eqtr4d 2783 . 2 (((𝐽 ∈ CRing ∧ 𝐴 ∈ 𝐾 ∧ 𝐹 ∈ (𝑀 LMHom 𝑁)) ∧ (π‘₯ ∈ (Baseβ€˜(Scalarβ€˜π‘€)) ∧ 𝑦 ∈ 𝑉)) β†’ (((𝑉 Γ— {𝐴}) ∘f Β· 𝐹)β€˜(π‘₯( ·𝑠 β€˜π‘€)𝑦)) = (π‘₯ Β· (((𝑉 Γ— {𝐴}) ∘f Β· 𝐹)β€˜π‘¦)))
741, 2, 3, 4, 5, 6, 8, 10, 12, 36, 73islmhmd 20515 1 ((𝐽 ∈ CRing ∧ 𝐴 ∈ 𝐾 ∧ 𝐹 ∈ (𝑀 LMHom 𝑁)) β†’ ((𝑉 Γ— {𝐴}) ∘f Β· 𝐹) ∈ (𝑀 LMHom 𝑁))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107  Vcvv 3444  {csn 4587   ↦ cmpt 5189   Γ— cxp 5632   ∘ ccom 5638   Fn wfn 6492  βŸΆwf 6493  β€˜cfv 6497  (class class class)co 7358   ∘f cof 7616  Basecbs 17088  .rcmulr 17139  Scalarcsca 17141   ·𝑠 cvsca 17142   GrpHom cghm 19010  CRingccrg 19970  LModclmod 20336   LMHom clmhm 20495
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 5243  ax-sep 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385  ax-un 7673  ax-cnex 11112  ax-resscn 11113  ax-1cn 11114  ax-icn 11115  ax-addcl 11116  ax-addrcl 11117  ax-mulcl 11118  ax-mulrcl 11119  ax-mulcom 11120  ax-addass 11121  ax-mulass 11122  ax-distr 11123  ax-i2m1 11124  ax-1ne0 11125  ax-1rid 11126  ax-rnegex 11127  ax-rrecex 11128  ax-cnre 11129  ax-pre-lttri 11130  ax-pre-lttrn 11131  ax-pre-ltadd 11132  ax-pre-mulgt0 11133
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 3446  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3930  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-iun 4957  df-br 5107  df-opab 5169  df-mpt 5190  df-tr 5224  df-id 5532  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5589  df-we 5591  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-pred 6254  df-ord 6321  df-on 6322  df-lim 6323  df-suc 6324  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-f1 6502  df-fo 6503  df-f1o 6504  df-fv 6505  df-riota 7314  df-ov 7361  df-oprab 7362  df-mpo 7363  df-of 7618  df-om 7804  df-2nd 7923  df-frecs 8213  df-wrecs 8244  df-recs 8318  df-rdg 8357  df-er 8651  df-map 8770  df-en 8887  df-dom 8888  df-sdom 8889  df-pnf 11196  df-mnf 11197  df-xr 11198  df-ltxr 11199  df-le 11200  df-sub 11392  df-neg 11393  df-nn 12159  df-2 12221  df-sets 17041  df-slot 17059  df-ndx 17071  df-base 17089  df-plusg 17151  df-0g 17328  df-mgm 18502  df-sgrp 18551  df-mnd 18562  df-mhm 18606  df-grp 18756  df-ghm 19011  df-cmn 19569  df-mgp 19902  df-cring 19972  df-lmod 20338  df-lmhm 20498
This theorem is referenced by:  mendlmod  41563
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