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Theorem lmhmvsca 20655
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 2732 . 2 ( ·𝑠 β€˜π‘€) = ( ·𝑠 β€˜π‘€)
3 lmhmvsca.s . 2 Β· = ( ·𝑠 β€˜π‘)
4 eqid 2732 . 2 (Scalarβ€˜π‘€) = (Scalarβ€˜π‘€)
5 lmhmvsca.j . 2 𝐽 = (Scalarβ€˜π‘)
6 eqid 2732 . 2 (Baseβ€˜(Scalarβ€˜π‘€)) = (Baseβ€˜(Scalarβ€˜π‘€))
7 lmhmlmod1 20643 . . 3 (𝐹 ∈ (𝑀 LMHom 𝑁) β†’ 𝑀 ∈ LMod)
873ad2ant3 1135 . 2 ((𝐽 ∈ CRing ∧ 𝐴 ∈ 𝐾 ∧ 𝐹 ∈ (𝑀 LMHom 𝑁)) β†’ 𝑀 ∈ LMod)
9 lmhmlmod2 20642 . . 3 (𝐹 ∈ (𝑀 LMHom 𝑁) β†’ 𝑁 ∈ LMod)
1093ad2ant3 1135 . 2 ((𝐽 ∈ CRing ∧ 𝐴 ∈ 𝐾 ∧ 𝐹 ∈ (𝑀 LMHom 𝑁)) β†’ 𝑁 ∈ LMod)
114, 5lmhmsca 20640 . . 3 (𝐹 ∈ (𝑀 LMHom 𝑁) β†’ 𝐽 = (Scalarβ€˜π‘€))
12113ad2ant3 1135 . 2 ((𝐽 ∈ CRing ∧ 𝐴 ∈ 𝐾 ∧ 𝐹 ∈ (𝑀 LMHom 𝑁)) β†’ 𝐽 = (Scalarβ€˜π‘€))
131fvexi 6905 . . . . . 6 𝑉 ∈ V
1413a1i 11 . . . . 5 ((𝐽 ∈ CRing ∧ 𝐴 ∈ 𝐾 ∧ 𝐹 ∈ (𝑀 LMHom 𝑁)) β†’ 𝑉 ∈ V)
15 simpl2 1192 . . . . 5 (((𝐽 ∈ CRing ∧ 𝐴 ∈ 𝐾 ∧ 𝐹 ∈ (𝑀 LMHom 𝑁)) ∧ 𝑣 ∈ 𝑉) β†’ 𝐴 ∈ 𝐾)
16 eqid 2732 . . . . . . . 8 (Baseβ€˜π‘) = (Baseβ€˜π‘)
171, 16lmhmf 20644 . . . . . . 7 (𝐹 ∈ (𝑀 LMHom 𝑁) β†’ 𝐹:π‘‰βŸΆ(Baseβ€˜π‘))
18173ad2ant3 1135 . . . . . 6 ((𝐽 ∈ CRing ∧ 𝐴 ∈ 𝐾 ∧ 𝐹 ∈ (𝑀 LMHom 𝑁)) β†’ 𝐹:π‘‰βŸΆ(Baseβ€˜π‘))
1918ffvelcdmda 7086 . . . . 5 (((𝐽 ∈ CRing ∧ 𝐴 ∈ 𝐾 ∧ 𝐹 ∈ (𝑀 LMHom 𝑁)) ∧ 𝑣 ∈ 𝑉) β†’ (πΉβ€˜π‘£) ∈ (Baseβ€˜π‘))
20 fconstmpt 5738 . . . . . 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 6960 . . . . 5 ((𝐽 ∈ CRing ∧ 𝐴 ∈ 𝐾 ∧ 𝐹 ∈ (𝑀 LMHom 𝑁)) β†’ 𝐹 = (𝑣 ∈ 𝑉 ↦ (πΉβ€˜π‘£)))
2314, 15, 19, 21, 22offval2 7689 . . . 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 2733 . . . . 5 ((𝐽 ∈ CRing ∧ 𝐴 ∈ 𝐾 ∧ 𝐹 ∈ (𝑀 LMHom 𝑁)) β†’ (𝑒 ∈ (Baseβ€˜π‘) ↦ (𝐴 Β· 𝑒)) = (𝑒 ∈ (Baseβ€˜π‘) ↦ (𝐴 Β· 𝑒)))
25 oveq2 7416 . . . . 5 (𝑒 = (πΉβ€˜π‘£) β†’ (𝐴 Β· 𝑒) = (𝐴 Β· (πΉβ€˜π‘£)))
2619, 22, 24, 25fmptco 7126 . . . 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 2775 . . 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 1137 . . . . 5 ((𝐽 ∈ CRing ∧ 𝐴 ∈ 𝐾 ∧ 𝐹 ∈ (𝑀 LMHom 𝑁)) β†’ 𝐴 ∈ 𝐾)
29 lmhmvsca.k . . . . . 6 𝐾 = (Baseβ€˜π½)
3016, 5, 3, 29lmodvsghm 20532 . . . . 5 ((𝑁 ∈ LMod ∧ 𝐴 ∈ 𝐾) β†’ (𝑒 ∈ (Baseβ€˜π‘) ↦ (𝐴 Β· 𝑒)) ∈ (𝑁 GrpHom 𝑁))
3110, 28, 30syl2anc 584 . . . 4 ((𝐽 ∈ CRing ∧ 𝐴 ∈ 𝐾 ∧ 𝐹 ∈ (𝑀 LMHom 𝑁)) β†’ (𝑒 ∈ (Baseβ€˜π‘) ↦ (𝐴 Β· 𝑒)) ∈ (𝑁 GrpHom 𝑁))
32 lmghm 20641 . . . . 5 (𝐹 ∈ (𝑀 LMHom 𝑁) β†’ 𝐹 ∈ (𝑀 GrpHom 𝑁))
33323ad2ant3 1135 . . . 4 ((𝐽 ∈ CRing ∧ 𝐴 ∈ 𝐾 ∧ 𝐹 ∈ (𝑀 LMHom 𝑁)) β†’ 𝐹 ∈ (𝑀 GrpHom 𝑁))
34 ghmco 19111 . . . 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 584 . . 3 ((𝐽 ∈ CRing ∧ 𝐴 ∈ 𝐾 ∧ 𝐹 ∈ (𝑀 LMHom 𝑁)) β†’ ((𝑒 ∈ (Baseβ€˜π‘) ↦ (𝐴 Β· 𝑒)) ∘ 𝐹) ∈ (𝑀 GrpHom 𝑁))
3627, 35eqeltrd 2833 . 2 ((𝐽 ∈ CRing ∧ 𝐴 ∈ 𝐾 ∧ 𝐹 ∈ (𝑀 LMHom 𝑁)) β†’ ((𝑉 Γ— {𝐴}) ∘f Β· 𝐹) ∈ (𝑀 GrpHom 𝑁))
37 simpl1 1191 . . . . . 6 (((𝐽 ∈ CRing ∧ 𝐴 ∈ 𝐾 ∧ 𝐹 ∈ (𝑀 LMHom 𝑁)) ∧ (π‘₯ ∈ (Baseβ€˜(Scalarβ€˜π‘€)) ∧ 𝑦 ∈ 𝑉)) β†’ 𝐽 ∈ CRing)
38 simpl2 1192 . . . . . 6 (((𝐽 ∈ CRing ∧ 𝐴 ∈ 𝐾 ∧ 𝐹 ∈ (𝑀 LMHom 𝑁)) ∧ (π‘₯ ∈ (Baseβ€˜(Scalarβ€˜π‘€)) ∧ 𝑦 ∈ 𝑉)) β†’ 𝐴 ∈ 𝐾)
39 simprl 769 . . . . . . 7 (((𝐽 ∈ CRing ∧ 𝐴 ∈ 𝐾 ∧ 𝐹 ∈ (𝑀 LMHom 𝑁)) ∧ (π‘₯ ∈ (Baseβ€˜(Scalarβ€˜π‘€)) ∧ 𝑦 ∈ 𝑉)) β†’ π‘₯ ∈ (Baseβ€˜(Scalarβ€˜π‘€)))
4012fveq2d 6895 . . . . . . . . 9 ((𝐽 ∈ CRing ∧ 𝐴 ∈ 𝐾 ∧ 𝐹 ∈ (𝑀 LMHom 𝑁)) β†’ (Baseβ€˜π½) = (Baseβ€˜(Scalarβ€˜π‘€)))
4129, 40eqtrid 2784 . . . . . . . 8 ((𝐽 ∈ CRing ∧ 𝐴 ∈ 𝐾 ∧ 𝐹 ∈ (𝑀 LMHom 𝑁)) β†’ 𝐾 = (Baseβ€˜(Scalarβ€˜π‘€)))
4241adantr 481 . . . . . . 7 (((𝐽 ∈ CRing ∧ 𝐴 ∈ 𝐾 ∧ 𝐹 ∈ (𝑀 LMHom 𝑁)) ∧ (π‘₯ ∈ (Baseβ€˜(Scalarβ€˜π‘€)) ∧ 𝑦 ∈ 𝑉)) β†’ 𝐾 = (Baseβ€˜(Scalarβ€˜π‘€)))
4339, 42eleqtrrd 2836 . . . . . 6 (((𝐽 ∈ CRing ∧ 𝐴 ∈ 𝐾 ∧ 𝐹 ∈ (𝑀 LMHom 𝑁)) ∧ (π‘₯ ∈ (Baseβ€˜(Scalarβ€˜π‘€)) ∧ 𝑦 ∈ 𝑉)) β†’ π‘₯ ∈ 𝐾)
44 eqid 2732 . . . . . . 7 (.rβ€˜π½) = (.rβ€˜π½)
4529, 44crngcom 20073 . . . . . 6 ((𝐽 ∈ CRing ∧ 𝐴 ∈ 𝐾 ∧ π‘₯ ∈ 𝐾) β†’ (𝐴(.rβ€˜π½)π‘₯) = (π‘₯(.rβ€˜π½)𝐴))
4637, 38, 43, 45syl3anc 1371 . . . . 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 7423 . . . 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 481 . . . . 5 (((𝐽 ∈ CRing ∧ 𝐴 ∈ 𝐾 ∧ 𝐹 ∈ (𝑀 LMHom 𝑁)) ∧ (π‘₯ ∈ (Baseβ€˜(Scalarβ€˜π‘€)) ∧ 𝑦 ∈ 𝑉)) β†’ 𝑁 ∈ LMod)
4918adantr 481 . . . . . 6 (((𝐽 ∈ CRing ∧ 𝐴 ∈ 𝐾 ∧ 𝐹 ∈ (𝑀 LMHom 𝑁)) ∧ (π‘₯ ∈ (Baseβ€˜(Scalarβ€˜π‘€)) ∧ 𝑦 ∈ 𝑉)) β†’ 𝐹:π‘‰βŸΆ(Baseβ€˜π‘))
50 simprr 771 . . . . . 6 (((𝐽 ∈ CRing ∧ 𝐴 ∈ 𝐾 ∧ 𝐹 ∈ (𝑀 LMHom 𝑁)) ∧ (π‘₯ ∈ (Baseβ€˜(Scalarβ€˜π‘€)) ∧ 𝑦 ∈ 𝑉)) β†’ 𝑦 ∈ 𝑉)
5149, 50ffvelcdmd 7087 . . . . 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 20496 . . . . 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 1372 . . . 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 20496 . . . . 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 1372 . . . 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 2780 . . 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 20488 . . . . . 6 ((𝑀 ∈ LMod ∧ π‘₯ ∈ (Baseβ€˜(Scalarβ€˜π‘€)) ∧ 𝑦 ∈ 𝑉) β†’ (π‘₯( ·𝑠 β€˜π‘€)𝑦) ∈ 𝑉)
58573expb 1120 . . . . 5 ((𝑀 ∈ LMod ∧ (π‘₯ ∈ (Baseβ€˜(Scalarβ€˜π‘€)) ∧ 𝑦 ∈ 𝑉)) β†’ (π‘₯( ·𝑠 β€˜π‘€)𝑦) ∈ 𝑉)
598, 58sylan 580 . . . 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 6718 . . . . . 6 ((𝐽 ∈ CRing ∧ 𝐴 ∈ 𝐾 ∧ 𝐹 ∈ (𝑀 LMHom 𝑁)) β†’ 𝐹 Fn 𝑉)
6261adantr 481 . . . . 5 (((𝐽 ∈ CRing ∧ 𝐴 ∈ 𝐾 ∧ 𝐹 ∈ (𝑀 LMHom 𝑁)) ∧ (π‘₯ ∈ (Baseβ€˜(Scalarβ€˜π‘€)) ∧ 𝑦 ∈ 𝑉)) β†’ 𝐹 Fn 𝑉)
634, 6, 1, 2, 3lmhmlin 20645 . . . . . . . 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 1120 . . . . . . 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 1187 . . . . . 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 481 . . . . 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 7695 . . . 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 685 . . 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 2733 . . . . . 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 7695 . . . . 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 685 . . . 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 7424 . . 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 2782 . 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 20649 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 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106  Vcvv 3474  {csn 4628   ↦ cmpt 5231   Γ— cxp 5674   ∘ ccom 5680   Fn wfn 6538  βŸΆwf 6539  β€˜cfv 6543  (class class class)co 7408   ∘f cof 7667  Basecbs 17143  .rcmulr 17197  Scalarcsca 17199   ·𝑠 cvsca 17200   GrpHom cghm 19088  CRingccrg 20056  LModclmod 20470   LMHom clmhm 20629
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-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-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-riota 7364  df-ov 7411  df-oprab 7412  df-mpo 7413  df-of 7669  df-om 7855  df-2nd 7975  df-frecs 8265  df-wrecs 8296  df-recs 8370  df-rdg 8409  df-er 8702  df-map 8821  df-en 8939  df-dom 8940  df-sdom 8941  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-sets 17096  df-slot 17114  df-ndx 17126  df-base 17144  df-plusg 17209  df-0g 17386  df-mgm 18560  df-sgrp 18609  df-mnd 18625  df-mhm 18670  df-grp 18821  df-ghm 19089  df-cmn 19649  df-mgp 19987  df-cring 20058  df-lmod 20472  df-lmhm 20632
This theorem is referenced by:  mendlmod  41925
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