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Theorem lmhmvsca 20656
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 20644 . . 3 (𝐹 ∈ (𝑀 LMHom 𝑁) β†’ 𝑀 ∈ LMod)
873ad2ant3 1136 . 2 ((𝐽 ∈ CRing ∧ 𝐴 ∈ 𝐾 ∧ 𝐹 ∈ (𝑀 LMHom 𝑁)) β†’ 𝑀 ∈ LMod)
9 lmhmlmod2 20643 . . 3 (𝐹 ∈ (𝑀 LMHom 𝑁) β†’ 𝑁 ∈ LMod)
1093ad2ant3 1136 . 2 ((𝐽 ∈ CRing ∧ 𝐴 ∈ 𝐾 ∧ 𝐹 ∈ (𝑀 LMHom 𝑁)) β†’ 𝑁 ∈ LMod)
114, 5lmhmsca 20641 . . 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 6906 . . . . . 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 20645 . . . . . . 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 7087 . . . . 5 (((𝐽 ∈ CRing ∧ 𝐴 ∈ 𝐾 ∧ 𝐹 ∈ (𝑀 LMHom 𝑁)) ∧ 𝑣 ∈ 𝑉) β†’ (πΉβ€˜π‘£) ∈ (Baseβ€˜π‘))
20 fconstmpt 5739 . . . . . 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 6961 . . . . 5 ((𝐽 ∈ CRing ∧ 𝐴 ∈ 𝐾 ∧ 𝐹 ∈ (𝑀 LMHom 𝑁)) β†’ 𝐹 = (𝑣 ∈ 𝑉 ↦ (πΉβ€˜π‘£)))
2314, 15, 19, 21, 22offval2 7690 . . . 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 7417 . . . . 5 (𝑒 = (πΉβ€˜π‘£) β†’ (𝐴 Β· 𝑒) = (𝐴 Β· (πΉβ€˜π‘£)))
2619, 22, 24, 25fmptco 7127 . . . 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 20533 . . . . 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 20642 . . . . 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 19112 . . . 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 6896 . . . . . . . . 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 20074 . . . . . 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 7424 . . . 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 7088 . . . . 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 20497 . . . . 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 20497 . . . . 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 20489 . . . . . 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 6719 . . . . . 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 20646 . . . . . . . 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 7696 . . . 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 7696 . . . . 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 7425 . . 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 20650 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 3475  {csn 4629   ↦ cmpt 5232   Γ— cxp 5675   ∘ ccom 5681   Fn wfn 6539  βŸΆwf 6540  β€˜cfv 6544  (class class class)co 7409   ∘f cof 7668  Basecbs 17144  .rcmulr 17198  Scalarcsca 17200   ·𝑠 cvsca 17201   GrpHom cghm 19089  CRingccrg 20057  LModclmod 20471   LMHom clmhm 20630
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 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725  ax-cnex 11166  ax-resscn 11167  ax-1cn 11168  ax-icn 11169  ax-addcl 11170  ax-addrcl 11171  ax-mulcl 11172  ax-mulrcl 11173  ax-mulcom 11174  ax-addass 11175  ax-mulass 11176  ax-distr 11177  ax-i2m1 11178  ax-1ne0 11179  ax-1rid 11180  ax-rnegex 11181  ax-rrecex 11182  ax-cnre 11183  ax-pre-lttri 11184  ax-pre-lttrn 11185  ax-pre-ltadd 11186  ax-pre-mulgt0 11187
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 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-rmo 3377  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-pred 6301  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-riota 7365  df-ov 7412  df-oprab 7413  df-mpo 7414  df-of 7670  df-om 7856  df-2nd 7976  df-frecs 8266  df-wrecs 8297  df-recs 8371  df-rdg 8410  df-er 8703  df-map 8822  df-en 8940  df-dom 8941  df-sdom 8942  df-pnf 11250  df-mnf 11251  df-xr 11252  df-ltxr 11253  df-le 11254  df-sub 11446  df-neg 11447  df-nn 12213  df-2 12275  df-sets 17097  df-slot 17115  df-ndx 17127  df-base 17145  df-plusg 17210  df-0g 17387  df-mgm 18561  df-sgrp 18610  df-mnd 18626  df-mhm 18671  df-grp 18822  df-ghm 19090  df-cmn 19650  df-mgp 19988  df-cring 20059  df-lmod 20473  df-lmhm 20633
This theorem is referenced by:  mendlmod  41935
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