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Theorem lmhmvsca 20800
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 2730 . 2 ( ·𝑠 β€˜π‘€) = ( ·𝑠 β€˜π‘€)
3 lmhmvsca.s . 2 Β· = ( ·𝑠 β€˜π‘)
4 eqid 2730 . 2 (Scalarβ€˜π‘€) = (Scalarβ€˜π‘€)
5 lmhmvsca.j . 2 𝐽 = (Scalarβ€˜π‘)
6 eqid 2730 . 2 (Baseβ€˜(Scalarβ€˜π‘€)) = (Baseβ€˜(Scalarβ€˜π‘€))
7 lmhmlmod1 20788 . . 3 (𝐹 ∈ (𝑀 LMHom 𝑁) β†’ 𝑀 ∈ LMod)
873ad2ant3 1133 . 2 ((𝐽 ∈ CRing ∧ 𝐴 ∈ 𝐾 ∧ 𝐹 ∈ (𝑀 LMHom 𝑁)) β†’ 𝑀 ∈ LMod)
9 lmhmlmod2 20787 . . 3 (𝐹 ∈ (𝑀 LMHom 𝑁) β†’ 𝑁 ∈ LMod)
1093ad2ant3 1133 . 2 ((𝐽 ∈ CRing ∧ 𝐴 ∈ 𝐾 ∧ 𝐹 ∈ (𝑀 LMHom 𝑁)) β†’ 𝑁 ∈ LMod)
114, 5lmhmsca 20785 . . 3 (𝐹 ∈ (𝑀 LMHom 𝑁) β†’ 𝐽 = (Scalarβ€˜π‘€))
12113ad2ant3 1133 . 2 ((𝐽 ∈ CRing ∧ 𝐴 ∈ 𝐾 ∧ 𝐹 ∈ (𝑀 LMHom 𝑁)) β†’ 𝐽 = (Scalarβ€˜π‘€))
131fvexi 6904 . . . . . 6 𝑉 ∈ V
1413a1i 11 . . . . 5 ((𝐽 ∈ CRing ∧ 𝐴 ∈ 𝐾 ∧ 𝐹 ∈ (𝑀 LMHom 𝑁)) β†’ 𝑉 ∈ V)
15 simpl2 1190 . . . . 5 (((𝐽 ∈ CRing ∧ 𝐴 ∈ 𝐾 ∧ 𝐹 ∈ (𝑀 LMHom 𝑁)) ∧ 𝑣 ∈ 𝑉) β†’ 𝐴 ∈ 𝐾)
16 eqid 2730 . . . . . . . 8 (Baseβ€˜π‘) = (Baseβ€˜π‘)
171, 16lmhmf 20789 . . . . . . 7 (𝐹 ∈ (𝑀 LMHom 𝑁) β†’ 𝐹:π‘‰βŸΆ(Baseβ€˜π‘))
18173ad2ant3 1133 . . . . . 6 ((𝐽 ∈ CRing ∧ 𝐴 ∈ 𝐾 ∧ 𝐹 ∈ (𝑀 LMHom 𝑁)) β†’ 𝐹:π‘‰βŸΆ(Baseβ€˜π‘))
1918ffvelcdmda 7085 . . . . 5 (((𝐽 ∈ CRing ∧ 𝐴 ∈ 𝐾 ∧ 𝐹 ∈ (𝑀 LMHom 𝑁)) ∧ 𝑣 ∈ 𝑉) β†’ (πΉβ€˜π‘£) ∈ (Baseβ€˜π‘))
20 fconstmpt 5737 . . . . . 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 6959 . . . . 5 ((𝐽 ∈ CRing ∧ 𝐴 ∈ 𝐾 ∧ 𝐹 ∈ (𝑀 LMHom 𝑁)) β†’ 𝐹 = (𝑣 ∈ 𝑉 ↦ (πΉβ€˜π‘£)))
2314, 15, 19, 21, 22offval2 7692 . . . 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 2731 . . . . 5 ((𝐽 ∈ CRing ∧ 𝐴 ∈ 𝐾 ∧ 𝐹 ∈ (𝑀 LMHom 𝑁)) β†’ (𝑒 ∈ (Baseβ€˜π‘) ↦ (𝐴 Β· 𝑒)) = (𝑒 ∈ (Baseβ€˜π‘) ↦ (𝐴 Β· 𝑒)))
25 oveq2 7419 . . . . 5 (𝑒 = (πΉβ€˜π‘£) β†’ (𝐴 Β· 𝑒) = (𝐴 Β· (πΉβ€˜π‘£)))
2619, 22, 24, 25fmptco 7128 . . . 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 2773 . . 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 1135 . . . . 5 ((𝐽 ∈ CRing ∧ 𝐴 ∈ 𝐾 ∧ 𝐹 ∈ (𝑀 LMHom 𝑁)) β†’ 𝐴 ∈ 𝐾)
29 lmhmvsca.k . . . . . 6 𝐾 = (Baseβ€˜π½)
3016, 5, 3, 29lmodvsghm 20677 . . . . 5 ((𝑁 ∈ LMod ∧ 𝐴 ∈ 𝐾) β†’ (𝑒 ∈ (Baseβ€˜π‘) ↦ (𝐴 Β· 𝑒)) ∈ (𝑁 GrpHom 𝑁))
3110, 28, 30syl2anc 582 . . . 4 ((𝐽 ∈ CRing ∧ 𝐴 ∈ 𝐾 ∧ 𝐹 ∈ (𝑀 LMHom 𝑁)) β†’ (𝑒 ∈ (Baseβ€˜π‘) ↦ (𝐴 Β· 𝑒)) ∈ (𝑁 GrpHom 𝑁))
32 lmghm 20786 . . . . 5 (𝐹 ∈ (𝑀 LMHom 𝑁) β†’ 𝐹 ∈ (𝑀 GrpHom 𝑁))
33323ad2ant3 1133 . . . 4 ((𝐽 ∈ CRing ∧ 𝐴 ∈ 𝐾 ∧ 𝐹 ∈ (𝑀 LMHom 𝑁)) β†’ 𝐹 ∈ (𝑀 GrpHom 𝑁))
34 ghmco 19150 . . . 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 582 . . 3 ((𝐽 ∈ CRing ∧ 𝐴 ∈ 𝐾 ∧ 𝐹 ∈ (𝑀 LMHom 𝑁)) β†’ ((𝑒 ∈ (Baseβ€˜π‘) ↦ (𝐴 Β· 𝑒)) ∘ 𝐹) ∈ (𝑀 GrpHom 𝑁))
3627, 35eqeltrd 2831 . 2 ((𝐽 ∈ CRing ∧ 𝐴 ∈ 𝐾 ∧ 𝐹 ∈ (𝑀 LMHom 𝑁)) β†’ ((𝑉 Γ— {𝐴}) ∘f Β· 𝐹) ∈ (𝑀 GrpHom 𝑁))
37 simpl1 1189 . . . . . 6 (((𝐽 ∈ CRing ∧ 𝐴 ∈ 𝐾 ∧ 𝐹 ∈ (𝑀 LMHom 𝑁)) ∧ (π‘₯ ∈ (Baseβ€˜(Scalarβ€˜π‘€)) ∧ 𝑦 ∈ 𝑉)) β†’ 𝐽 ∈ CRing)
38 simpl2 1190 . . . . . 6 (((𝐽 ∈ CRing ∧ 𝐴 ∈ 𝐾 ∧ 𝐹 ∈ (𝑀 LMHom 𝑁)) ∧ (π‘₯ ∈ (Baseβ€˜(Scalarβ€˜π‘€)) ∧ 𝑦 ∈ 𝑉)) β†’ 𝐴 ∈ 𝐾)
39 simprl 767 . . . . . . 7 (((𝐽 ∈ CRing ∧ 𝐴 ∈ 𝐾 ∧ 𝐹 ∈ (𝑀 LMHom 𝑁)) ∧ (π‘₯ ∈ (Baseβ€˜(Scalarβ€˜π‘€)) ∧ 𝑦 ∈ 𝑉)) β†’ π‘₯ ∈ (Baseβ€˜(Scalarβ€˜π‘€)))
4012fveq2d 6894 . . . . . . . . 9 ((𝐽 ∈ CRing ∧ 𝐴 ∈ 𝐾 ∧ 𝐹 ∈ (𝑀 LMHom 𝑁)) β†’ (Baseβ€˜π½) = (Baseβ€˜(Scalarβ€˜π‘€)))
4129, 40eqtrid 2782 . . . . . . . 8 ((𝐽 ∈ CRing ∧ 𝐴 ∈ 𝐾 ∧ 𝐹 ∈ (𝑀 LMHom 𝑁)) β†’ 𝐾 = (Baseβ€˜(Scalarβ€˜π‘€)))
4241adantr 479 . . . . . . 7 (((𝐽 ∈ CRing ∧ 𝐴 ∈ 𝐾 ∧ 𝐹 ∈ (𝑀 LMHom 𝑁)) ∧ (π‘₯ ∈ (Baseβ€˜(Scalarβ€˜π‘€)) ∧ 𝑦 ∈ 𝑉)) β†’ 𝐾 = (Baseβ€˜(Scalarβ€˜π‘€)))
4339, 42eleqtrrd 2834 . . . . . 6 (((𝐽 ∈ CRing ∧ 𝐴 ∈ 𝐾 ∧ 𝐹 ∈ (𝑀 LMHom 𝑁)) ∧ (π‘₯ ∈ (Baseβ€˜(Scalarβ€˜π‘€)) ∧ 𝑦 ∈ 𝑉)) β†’ π‘₯ ∈ 𝐾)
44 eqid 2730 . . . . . . 7 (.rβ€˜π½) = (.rβ€˜π½)
4529, 44crngcom 20145 . . . . . 6 ((𝐽 ∈ CRing ∧ 𝐴 ∈ 𝐾 ∧ π‘₯ ∈ 𝐾) β†’ (𝐴(.rβ€˜π½)π‘₯) = (π‘₯(.rβ€˜π½)𝐴))
4637, 38, 43, 45syl3anc 1369 . . . . 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 7426 . . . 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 479 . . . . 5 (((𝐽 ∈ CRing ∧ 𝐴 ∈ 𝐾 ∧ 𝐹 ∈ (𝑀 LMHom 𝑁)) ∧ (π‘₯ ∈ (Baseβ€˜(Scalarβ€˜π‘€)) ∧ 𝑦 ∈ 𝑉)) β†’ 𝑁 ∈ LMod)
4918adantr 479 . . . . . 6 (((𝐽 ∈ CRing ∧ 𝐴 ∈ 𝐾 ∧ 𝐹 ∈ (𝑀 LMHom 𝑁)) ∧ (π‘₯ ∈ (Baseβ€˜(Scalarβ€˜π‘€)) ∧ 𝑦 ∈ 𝑉)) β†’ 𝐹:π‘‰βŸΆ(Baseβ€˜π‘))
50 simprr 769 . . . . . 6 (((𝐽 ∈ CRing ∧ 𝐴 ∈ 𝐾 ∧ 𝐹 ∈ (𝑀 LMHom 𝑁)) ∧ (π‘₯ ∈ (Baseβ€˜(Scalarβ€˜π‘€)) ∧ 𝑦 ∈ 𝑉)) β†’ 𝑦 ∈ 𝑉)
5149, 50ffvelcdmd 7086 . . . . 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 20641 . . . . 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 1370 . . . 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 20641 . . . . 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 1370 . . . 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 2778 . . 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 20632 . . . . . 6 ((𝑀 ∈ LMod ∧ π‘₯ ∈ (Baseβ€˜(Scalarβ€˜π‘€)) ∧ 𝑦 ∈ 𝑉) β†’ (π‘₯( ·𝑠 β€˜π‘€)𝑦) ∈ 𝑉)
58573expb 1118 . . . . 5 ((𝑀 ∈ LMod ∧ (π‘₯ ∈ (Baseβ€˜(Scalarβ€˜π‘€)) ∧ 𝑦 ∈ 𝑉)) β†’ (π‘₯( ·𝑠 β€˜π‘€)𝑦) ∈ 𝑉)
598, 58sylan 578 . . . 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 6717 . . . . . 6 ((𝐽 ∈ CRing ∧ 𝐴 ∈ 𝐾 ∧ 𝐹 ∈ (𝑀 LMHom 𝑁)) β†’ 𝐹 Fn 𝑉)
6261adantr 479 . . . . 5 (((𝐽 ∈ CRing ∧ 𝐴 ∈ 𝐾 ∧ 𝐹 ∈ (𝑀 LMHom 𝑁)) ∧ (π‘₯ ∈ (Baseβ€˜(Scalarβ€˜π‘€)) ∧ 𝑦 ∈ 𝑉)) β†’ 𝐹 Fn 𝑉)
634, 6, 1, 2, 3lmhmlin 20790 . . . . . . . 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 1118 . . . . . . 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 1185 . . . . . 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 479 . . . . 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 7698 . . . 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 683 . . 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 2731 . . . . . 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 7698 . . . . 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 683 . . . 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 7427 . . 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 2780 . 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 20794 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 394   ∧ w3a 1085   = wceq 1539   ∈ wcel 2104  Vcvv 3472  {csn 4627   ↦ cmpt 5230   Γ— cxp 5673   ∘ ccom 5679   Fn wfn 6537  βŸΆwf 6538  β€˜cfv 6542  (class class class)co 7411   ∘f cof 7670  Basecbs 17148  .rcmulr 17202  Scalarcsca 17204   ·𝑠 cvsca 17205   GrpHom cghm 19127  CRingccrg 20128  LModclmod 20614   LMHom clmhm 20774
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7727  ax-cnex 11168  ax-resscn 11169  ax-1cn 11170  ax-icn 11171  ax-addcl 11172  ax-addrcl 11173  ax-mulcl 11174  ax-mulrcl 11175  ax-mulcom 11176  ax-addass 11177  ax-mulass 11178  ax-distr 11179  ax-i2m1 11180  ax-1ne0 11181  ax-1rid 11182  ax-rnegex 11183  ax-rrecex 11184  ax-cnre 11185  ax-pre-lttri 11186  ax-pre-lttrn 11187  ax-pre-ltadd 11188  ax-pre-mulgt0 11189
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-rmo 3374  df-reu 3375  df-rab 3431  df-v 3474  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-pred 6299  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-riota 7367  df-ov 7414  df-oprab 7415  df-mpo 7416  df-of 7672  df-om 7858  df-2nd 7978  df-frecs 8268  df-wrecs 8299  df-recs 8373  df-rdg 8412  df-er 8705  df-map 8824  df-en 8942  df-dom 8943  df-sdom 8944  df-pnf 11254  df-mnf 11255  df-xr 11256  df-ltxr 11257  df-le 11258  df-sub 11450  df-neg 11451  df-nn 12217  df-2 12279  df-sets 17101  df-slot 17119  df-ndx 17131  df-base 17149  df-plusg 17214  df-0g 17391  df-mgm 18565  df-sgrp 18644  df-mnd 18660  df-mhm 18705  df-grp 18858  df-ghm 19128  df-cmn 19691  df-mgp 20029  df-cring 20130  df-lmod 20616  df-lmhm 20777
This theorem is referenced by:  mendlmod  42237
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