Theorem lmhmvsca 21003

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 = (Base‘𝑀)
lmhmvsca.s	⊢ · = ( ·𝑠 ‘𝑁)
lmhmvsca.j	⊢ 𝐽 = (Scalar‘𝑁)
lmhmvsca.k	⊢ 𝐾 = (Base‘𝐽)

Assertion

Ref	Expression
lmhmvsca	⊢ ((𝐽 ∈ CRing ∧ 𝐴 ∈ 𝐾 ∧ 𝐹 ∈ (𝑀 LMHom 𝑁)) → ((𝑉 × {𝐴}) ∘f · 𝐹) ∈ (𝑀 LMHom 𝑁))

Proof of Theorem lmhmvsca

Dummy variables 𝑣 𝑢 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		lmhmvsca.v	. 2 ⊢ 𝑉 = (Base‘𝑀)
2		eqid 2735	. 2 ⊢ ( ·𝑠 ‘𝑀) = ( ·𝑠 ‘𝑀)
3		lmhmvsca.s	. 2 ⊢ · = ( ·𝑠 ‘𝑁)
4		eqid 2735	. 2 ⊢ (Scalar‘𝑀) = (Scalar‘𝑀)
5		lmhmvsca.j	. 2 ⊢ 𝐽 = (Scalar‘𝑁)
6		eqid 2735	. 2 ⊢ (Base‘(Scalar‘𝑀)) = (Base‘(Scalar‘𝑀))
7		lmhmlmod1 20991	. . 3 ⊢ (𝐹 ∈ (𝑀 LMHom 𝑁) → 𝑀 ∈ LMod)
8	7	3ad2ant3 1135	. 2 ⊢ ((𝐽 ∈ CRing ∧ 𝐴 ∈ 𝐾 ∧ 𝐹 ∈ (𝑀 LMHom 𝑁)) → 𝑀 ∈ LMod)
9		lmhmlmod2 20990	. . 3 ⊢ (𝐹 ∈ (𝑀 LMHom 𝑁) → 𝑁 ∈ LMod)
10	9	3ad2ant3 1135	. 2 ⊢ ((𝐽 ∈ CRing ∧ 𝐴 ∈ 𝐾 ∧ 𝐹 ∈ (𝑀 LMHom 𝑁)) → 𝑁 ∈ LMod)
11	4, 5	lmhmsca 20988	. . 3 ⊢ (𝐹 ∈ (𝑀 LMHom 𝑁) → 𝐽 = (Scalar‘𝑀))
12	11	3ad2ant3 1135	. 2 ⊢ ((𝐽 ∈ CRing ∧ 𝐴 ∈ 𝐾 ∧ 𝐹 ∈ (𝑀 LMHom 𝑁)) → 𝐽 = (Scalar‘𝑀))
13	1	fvexi 6890	. . . . . 6 ⊢ 𝑉 ∈ V
14	13	a1i 11	. . . . 5 ⊢ ((𝐽 ∈ CRing ∧ 𝐴 ∈ 𝐾 ∧ 𝐹 ∈ (𝑀 LMHom 𝑁)) → 𝑉 ∈ V)
15		simpl2 1193	. . . . 5 ⊢ (((𝐽 ∈ CRing ∧ 𝐴 ∈ 𝐾 ∧ 𝐹 ∈ (𝑀 LMHom 𝑁)) ∧ 𝑣 ∈ 𝑉) → 𝐴 ∈ 𝐾)
16		eqid 2735	. . . . . . . 8 ⊢ (Base‘𝑁) = (Base‘𝑁)
17	1, 16	lmhmf 20992	. . . . . . 7 ⊢ (𝐹 ∈ (𝑀 LMHom 𝑁) → 𝐹:𝑉⟶(Base‘𝑁))
18	17	3ad2ant3 1135	. . . . . 6 ⊢ ((𝐽 ∈ CRing ∧ 𝐴 ∈ 𝐾 ∧ 𝐹 ∈ (𝑀 LMHom 𝑁)) → 𝐹:𝑉⟶(Base‘𝑁))
19	18	ffvelcdmda 7074	. . . . 5 ⊢ (((𝐽 ∈ CRing ∧ 𝐴 ∈ 𝐾 ∧ 𝐹 ∈ (𝑀 LMHom 𝑁)) ∧ 𝑣 ∈ 𝑉) → (𝐹‘𝑣) ∈ (Base‘𝑁))
20		fconstmpt 5716	. . . . . 6 ⊢ (𝑉 × {𝐴}) = (𝑣 ∈ 𝑉 ↦ 𝐴)
21	20	a1i 11	. . . . 5 ⊢ ((𝐽 ∈ CRing ∧ 𝐴 ∈ 𝐾 ∧ 𝐹 ∈ (𝑀 LMHom 𝑁)) → (𝑉 × {𝐴}) = (𝑣 ∈ 𝑉 ↦ 𝐴))
22	18	feqmptd 6947	. . . . 5 ⊢ ((𝐽 ∈ CRing ∧ 𝐴 ∈ 𝐾 ∧ 𝐹 ∈ (𝑀 LMHom 𝑁)) → 𝐹 = (𝑣 ∈ 𝑉 ↦ (𝐹‘𝑣)))
23	14, 15, 19, 21, 22	offval2 7691	. . . 4 ⊢ ((𝐽 ∈ CRing ∧ 𝐴 ∈ 𝐾 ∧ 𝐹 ∈ (𝑀 LMHom 𝑁)) → ((𝑉 × {𝐴}) ∘f · 𝐹) = (𝑣 ∈ 𝑉 ↦ (𝐴 · (𝐹‘𝑣))))
24		eqidd 2736	. . . . 5 ⊢ ((𝐽 ∈ CRing ∧ 𝐴 ∈ 𝐾 ∧ 𝐹 ∈ (𝑀 LMHom 𝑁)) → (𝑢 ∈ (Base‘𝑁) ↦ (𝐴 · 𝑢)) = (𝑢 ∈ (Base‘𝑁) ↦ (𝐴 · 𝑢)))
25		oveq2 7413	. . . . 5 ⊢ (𝑢 = (𝐹‘𝑣) → (𝐴 · 𝑢) = (𝐴 · (𝐹‘𝑣)))
26	19, 22, 24, 25	fmptco 7119	. . . 4 ⊢ ((𝐽 ∈ CRing ∧ 𝐴 ∈ 𝐾 ∧ 𝐹 ∈ (𝑀 LMHom 𝑁)) → ((𝑢 ∈ (Base‘𝑁) ↦ (𝐴 · 𝑢)) ∘ 𝐹) = (𝑣 ∈ 𝑉 ↦ (𝐴 · (𝐹‘𝑣))))
27	23, 26	eqtr4d 2773	. . 3 ⊢ ((𝐽 ∈ CRing ∧ 𝐴 ∈ 𝐾 ∧ 𝐹 ∈ (𝑀 LMHom 𝑁)) → ((𝑉 × {𝐴}) ∘f · 𝐹) = ((𝑢 ∈ (Base‘𝑁) ↦ (𝐴 · 𝑢)) ∘ 𝐹))
28		simp2 1137	. . . . 5 ⊢ ((𝐽 ∈ CRing ∧ 𝐴 ∈ 𝐾 ∧ 𝐹 ∈ (𝑀 LMHom 𝑁)) → 𝐴 ∈ 𝐾)
29		lmhmvsca.k	. . . . . 6 ⊢ 𝐾 = (Base‘𝐽)
30	16, 5, 3, 29	lmodvsghm 20880	. . . . 5 ⊢ ((𝑁 ∈ LMod ∧ 𝐴 ∈ 𝐾) → (𝑢 ∈ (Base‘𝑁) ↦ (𝐴 · 𝑢)) ∈ (𝑁 GrpHom 𝑁))
31	10, 28, 30	syl2anc 584	. . . 4 ⊢ ((𝐽 ∈ CRing ∧ 𝐴 ∈ 𝐾 ∧ 𝐹 ∈ (𝑀 LMHom 𝑁)) → (𝑢 ∈ (Base‘𝑁) ↦ (𝐴 · 𝑢)) ∈ (𝑁 GrpHom 𝑁))
32		lmghm 20989	. . . . 5 ⊢ (𝐹 ∈ (𝑀 LMHom 𝑁) → 𝐹 ∈ (𝑀 GrpHom 𝑁))
33	32	3ad2ant3 1135	. . . 4 ⊢ ((𝐽 ∈ CRing ∧ 𝐴 ∈ 𝐾 ∧ 𝐹 ∈ (𝑀 LMHom 𝑁)) → 𝐹 ∈ (𝑀 GrpHom 𝑁))
34		ghmco 19219	. . . 4 ⊢ (((𝑢 ∈ (Base‘𝑁) ↦ (𝐴 · 𝑢)) ∈ (𝑁 GrpHom 𝑁) ∧ 𝐹 ∈ (𝑀 GrpHom 𝑁)) → ((𝑢 ∈ (Base‘𝑁) ↦ (𝐴 · 𝑢)) ∘ 𝐹) ∈ (𝑀 GrpHom 𝑁))
35	31, 33, 34	syl2anc 584	. . 3 ⊢ ((𝐽 ∈ CRing ∧ 𝐴 ∈ 𝐾 ∧ 𝐹 ∈ (𝑀 LMHom 𝑁)) → ((𝑢 ∈ (Base‘𝑁) ↦ (𝐴 · 𝑢)) ∘ 𝐹) ∈ (𝑀 GrpHom 𝑁))
36	27, 35	eqeltrd 2834	. 2 ⊢ ((𝐽 ∈ CRing ∧ 𝐴 ∈ 𝐾 ∧ 𝐹 ∈ (𝑀 LMHom 𝑁)) → ((𝑉 × {𝐴}) ∘f · 𝐹) ∈ (𝑀 GrpHom 𝑁))
37		simpl1 1192	. . . . . 6 ⊢ (((𝐽 ∈ CRing ∧ 𝐴 ∈ 𝐾 ∧ 𝐹 ∈ (𝑀 LMHom 𝑁)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦 ∈ 𝑉)) → 𝐽 ∈ CRing)
38		simpl2 1193	. . . . . 6 ⊢ (((𝐽 ∈ CRing ∧ 𝐴 ∈ 𝐾 ∧ 𝐹 ∈ (𝑀 LMHom 𝑁)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦 ∈ 𝑉)) → 𝐴 ∈ 𝐾)
39		simprl 770	. . . . . . 7 ⊢ (((𝐽 ∈ CRing ∧ 𝐴 ∈ 𝐾 ∧ 𝐹 ∈ (𝑀 LMHom 𝑁)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦 ∈ 𝑉)) → 𝑥 ∈ (Base‘(Scalar‘𝑀)))
40	12	fveq2d 6880	. . . . . . . . 9 ⊢ ((𝐽 ∈ CRing ∧ 𝐴 ∈ 𝐾 ∧ 𝐹 ∈ (𝑀 LMHom 𝑁)) → (Base‘𝐽) = (Base‘(Scalar‘𝑀)))
41	29, 40	eqtrid 2782	. . . . . . . 8 ⊢ ((𝐽 ∈ CRing ∧ 𝐴 ∈ 𝐾 ∧ 𝐹 ∈ (𝑀 LMHom 𝑁)) → 𝐾 = (Base‘(Scalar‘𝑀)))
42	41	adantr 480	. . . . . . 7 ⊢ (((𝐽 ∈ CRing ∧ 𝐴 ∈ 𝐾 ∧ 𝐹 ∈ (𝑀 LMHom 𝑁)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦 ∈ 𝑉)) → 𝐾 = (Base‘(Scalar‘𝑀)))
43	39, 42	eleqtrrd 2837	. . . . . 6 ⊢ (((𝐽 ∈ CRing ∧ 𝐴 ∈ 𝐾 ∧ 𝐹 ∈ (𝑀 LMHom 𝑁)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦 ∈ 𝑉)) → 𝑥 ∈ 𝐾)
44		eqid 2735	. . . . . . 7 ⊢ (.r‘𝐽) = (.r‘𝐽)
45	29, 44	crngcom 20211	. . . . . 6 ⊢ ((𝐽 ∈ CRing ∧ 𝐴 ∈ 𝐾 ∧ 𝑥 ∈ 𝐾) → (𝐴(.r‘𝐽)𝑥) = (𝑥(.r‘𝐽)𝐴))
46	37, 38, 43, 45	syl3anc 1373	. . . . 5 ⊢ (((𝐽 ∈ CRing ∧ 𝐴 ∈ 𝐾 ∧ 𝐹 ∈ (𝑀 LMHom 𝑁)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦 ∈ 𝑉)) → (𝐴(.r‘𝐽)𝑥) = (𝑥(.r‘𝐽)𝐴))
47	46	oveq1d 7420	. . . 4 ⊢ (((𝐽 ∈ CRing ∧ 𝐴 ∈ 𝐾 ∧ 𝐹 ∈ (𝑀 LMHom 𝑁)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦 ∈ 𝑉)) → ((𝐴(.r‘𝐽)𝑥) · (𝐹‘𝑦)) = ((𝑥(.r‘𝐽)𝐴) · (𝐹‘𝑦)))
48	10	adantr 480	. . . . 5 ⊢ (((𝐽 ∈ CRing ∧ 𝐴 ∈ 𝐾 ∧ 𝐹 ∈ (𝑀 LMHom 𝑁)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦 ∈ 𝑉)) → 𝑁 ∈ LMod)
49	18	adantr 480	. . . . . 6 ⊢ (((𝐽 ∈ CRing ∧ 𝐴 ∈ 𝐾 ∧ 𝐹 ∈ (𝑀 LMHom 𝑁)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦 ∈ 𝑉)) → 𝐹:𝑉⟶(Base‘𝑁))
50		simprr 772	. . . . . 6 ⊢ (((𝐽 ∈ CRing ∧ 𝐴 ∈ 𝐾 ∧ 𝐹 ∈ (𝑀 LMHom 𝑁)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦 ∈ 𝑉)) → 𝑦 ∈ 𝑉)
51	49, 50	ffvelcdmd 7075	. . . . 5 ⊢ (((𝐽 ∈ CRing ∧ 𝐴 ∈ 𝐾 ∧ 𝐹 ∈ (𝑀 LMHom 𝑁)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦 ∈ 𝑉)) → (𝐹‘𝑦) ∈ (Base‘𝑁))
52	16, 5, 3, 29, 44	lmodvsass 20844	. . . . 5 ⊢ ((𝑁 ∈ LMod ∧ (𝐴 ∈ 𝐾 ∧ 𝑥 ∈ 𝐾 ∧ (𝐹‘𝑦) ∈ (Base‘𝑁))) → ((𝐴(.r‘𝐽)𝑥) · (𝐹‘𝑦)) = (𝐴 · (𝑥 · (𝐹‘𝑦))))
53	48, 38, 43, 51, 52	syl13anc 1374	. . . 4 ⊢ (((𝐽 ∈ CRing ∧ 𝐴 ∈ 𝐾 ∧ 𝐹 ∈ (𝑀 LMHom 𝑁)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦 ∈ 𝑉)) → ((𝐴(.r‘𝐽)𝑥) · (𝐹‘𝑦)) = (𝐴 · (𝑥 · (𝐹‘𝑦))))
54	16, 5, 3, 29, 44	lmodvsass 20844	. . . . 5 ⊢ ((𝑁 ∈ LMod ∧ (𝑥 ∈ 𝐾 ∧ 𝐴 ∈ 𝐾 ∧ (𝐹‘𝑦) ∈ (Base‘𝑁))) → ((𝑥(.r‘𝐽)𝐴) · (𝐹‘𝑦)) = (𝑥 · (𝐴 · (𝐹‘𝑦))))
55	48, 43, 38, 51, 54	syl13anc 1374	. . . 4 ⊢ (((𝐽 ∈ CRing ∧ 𝐴 ∈ 𝐾 ∧ 𝐹 ∈ (𝑀 LMHom 𝑁)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦 ∈ 𝑉)) → ((𝑥(.r‘𝐽)𝐴) · (𝐹‘𝑦)) = (𝑥 · (𝐴 · (𝐹‘𝑦))))
56	47, 53, 55	3eqtr3d 2778	. . 3 ⊢ (((𝐽 ∈ CRing ∧ 𝐴 ∈ 𝐾 ∧ 𝐹 ∈ (𝑀 LMHom 𝑁)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦 ∈ 𝑉)) → (𝐴 · (𝑥 · (𝐹‘𝑦))) = (𝑥 · (𝐴 · (𝐹‘𝑦))))
57	1, 4, 2, 6	lmodvscl 20835	. . . . . 6 ⊢ ((𝑀 ∈ LMod ∧ 𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦 ∈ 𝑉) → (𝑥( ·𝑠 ‘𝑀)𝑦) ∈ 𝑉)
58	57	3expb 1120	. . . . 5 ⊢ ((𝑀 ∈ LMod ∧ (𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦 ∈ 𝑉)) → (𝑥( ·𝑠 ‘𝑀)𝑦) ∈ 𝑉)
59	8, 58	sylan 580	. . . 4 ⊢ (((𝐽 ∈ CRing ∧ 𝐴 ∈ 𝐾 ∧ 𝐹 ∈ (𝑀 LMHom 𝑁)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦 ∈ 𝑉)) → (𝑥( ·𝑠 ‘𝑀)𝑦) ∈ 𝑉)
60	13	a1i 11	. . . . 5 ⊢ (((𝐽 ∈ CRing ∧ 𝐴 ∈ 𝐾 ∧ 𝐹 ∈ (𝑀 LMHom 𝑁)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦 ∈ 𝑉)) → 𝑉 ∈ V)
61	18	ffnd 6707	. . . . . 6 ⊢ ((𝐽 ∈ CRing ∧ 𝐴 ∈ 𝐾 ∧ 𝐹 ∈ (𝑀 LMHom 𝑁)) → 𝐹 Fn 𝑉)
62	61	adantr 480	. . . . 5 ⊢ (((𝐽 ∈ CRing ∧ 𝐴 ∈ 𝐾 ∧ 𝐹 ∈ (𝑀 LMHom 𝑁)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦 ∈ 𝑉)) → 𝐹 Fn 𝑉)
63	4, 6, 1, 2, 3	lmhmlin 20993	. . . . . . . 8 ⊢ ((𝐹 ∈ (𝑀 LMHom 𝑁) ∧ 𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦 ∈ 𝑉) → (𝐹‘(𝑥( ·𝑠 ‘𝑀)𝑦)) = (𝑥 · (𝐹‘𝑦)))
64	63	3expb 1120	. . . . . . 7 ⊢ ((𝐹 ∈ (𝑀 LMHom 𝑁) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦 ∈ 𝑉)) → (𝐹‘(𝑥( ·𝑠 ‘𝑀)𝑦)) = (𝑥 · (𝐹‘𝑦)))
65	64	3ad2antl3 1188	. . . . . 6 ⊢ (((𝐽 ∈ CRing ∧ 𝐴 ∈ 𝐾 ∧ 𝐹 ∈ (𝑀 LMHom 𝑁)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦 ∈ 𝑉)) → (𝐹‘(𝑥( ·𝑠 ‘𝑀)𝑦)) = (𝑥 · (𝐹‘𝑦)))
66	65	adantr 480	. . . . 5 ⊢ ((((𝐽 ∈ CRing ∧ 𝐴 ∈ 𝐾 ∧ 𝐹 ∈ (𝑀 LMHom 𝑁)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦 ∈ 𝑉)) ∧ (𝑥( ·𝑠 ‘𝑀)𝑦) ∈ 𝑉) → (𝐹‘(𝑥( ·𝑠 ‘𝑀)𝑦)) = (𝑥 · (𝐹‘𝑦)))
67	60, 38, 62, 66	ofc1 7699	. . . 4 ⊢ ((((𝐽 ∈ CRing ∧ 𝐴 ∈ 𝐾 ∧ 𝐹 ∈ (𝑀 LMHom 𝑁)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦 ∈ 𝑉)) ∧ (𝑥( ·𝑠 ‘𝑀)𝑦) ∈ 𝑉) → (((𝑉 × {𝐴}) ∘f · 𝐹)‘(𝑥( ·𝑠 ‘𝑀)𝑦)) = (𝐴 · (𝑥 · (𝐹‘𝑦))))
68	59, 67	mpdan 687	. . 3 ⊢ (((𝐽 ∈ CRing ∧ 𝐴 ∈ 𝐾 ∧ 𝐹 ∈ (𝑀 LMHom 𝑁)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦 ∈ 𝑉)) → (((𝑉 × {𝐴}) ∘f · 𝐹)‘(𝑥( ·𝑠 ‘𝑀)𝑦)) = (𝐴 · (𝑥 · (𝐹‘𝑦))))
69		eqidd 2736	. . . . . 6 ⊢ ((((𝐽 ∈ CRing ∧ 𝐴 ∈ 𝐾 ∧ 𝐹 ∈ (𝑀 LMHom 𝑁)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦 ∈ 𝑉)) ∧ 𝑦 ∈ 𝑉) → (𝐹‘𝑦) = (𝐹‘𝑦))
70	60, 38, 62, 69	ofc1 7699	. . . . 5 ⊢ ((((𝐽 ∈ CRing ∧ 𝐴 ∈ 𝐾 ∧ 𝐹 ∈ (𝑀 LMHom 𝑁)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦 ∈ 𝑉)) ∧ 𝑦 ∈ 𝑉) → (((𝑉 × {𝐴}) ∘f · 𝐹)‘𝑦) = (𝐴 · (𝐹‘𝑦)))
71	50, 70	mpdan 687	. . . 4 ⊢ (((𝐽 ∈ CRing ∧ 𝐴 ∈ 𝐾 ∧ 𝐹 ∈ (𝑀 LMHom 𝑁)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦 ∈ 𝑉)) → (((𝑉 × {𝐴}) ∘f · 𝐹)‘𝑦) = (𝐴 · (𝐹‘𝑦)))
72	71	oveq2d 7421	. . 3 ⊢ (((𝐽 ∈ CRing ∧ 𝐴 ∈ 𝐾 ∧ 𝐹 ∈ (𝑀 LMHom 𝑁)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦 ∈ 𝑉)) → (𝑥 · (((𝑉 × {𝐴}) ∘f · 𝐹)‘𝑦)) = (𝑥 · (𝐴 · (𝐹‘𝑦))))
73	56, 68, 72	3eqtr4d 2780	. 2 ⊢ (((𝐽 ∈ CRing ∧ 𝐴 ∈ 𝐾 ∧ 𝐹 ∈ (𝑀 LMHom 𝑁)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦 ∈ 𝑉)) → (((𝑉 × {𝐴}) ∘f · 𝐹)‘(𝑥( ·𝑠 ‘𝑀)𝑦)) = (𝑥 · (((𝑉 × {𝐴}) ∘f · 𝐹)‘𝑦)))
74	1, 2, 3, 4, 5, 6, 8, 10, 12, 36, 73	islmhmd 20997	1 ⊢ ((𝐽 ∈ CRing ∧ 𝐴 ∈ 𝐾 ∧ 𝐹 ∈ (𝑀 LMHom 𝑁)) → ((𝑉 × {𝐴}) ∘f · 𝐹) ∈ (𝑀 LMHom 𝑁))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2108  Vcvv 3459  {csn 4601   ↦ cmpt 5201   × cxp 5652   ∘ ccom 5658   Fn wfn 6526  ⟶wf 6527  ‘cfv 6531  (class class class)co 7405   ∘_f cof 7669  Basecbs 17228  ._rcmulr 17272  Scalarcsca 17274   ·_𝑠 cvsca 17275   GrpHom cghm 19195  CRingccrg 20194  LModclmod 20817   LMHom clmhm 20977

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 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707  ax-rep 5249  ax-sep 5266  ax-nul 5276  ax-pow 5335  ax-pr 5402  ax-un 7729  ax-cnex 11185  ax-resscn 11186  ax-1cn 11187  ax-icn 11188  ax-addcl 11189  ax-addrcl 11190  ax-mulcl 11191  ax-mulrcl 11192  ax-mulcom 11193  ax-addass 11194  ax-mulass 11195  ax-distr 11196  ax-i2m1 11197  ax-1ne0 11198  ax-1rid 11199  ax-rnegex 11200  ax-rrecex 11201  ax-cnre 11202  ax-pre-lttri 11203  ax-pre-lttrn 11204  ax-pre-ltadd 11205  ax-pre-mulgt0 11206

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ne 2933  df-nel 3037  df-ral 3052  df-rex 3061  df-rmo 3359  df-reu 3360  df-rab 3416  df-v 3461  df-sbc 3766  df-csb 3875  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-pss 3946  df-nul 4309  df-if 4501  df-pw 4577  df-sn 4602  df-pr 4604  df-op 4608  df-uni 4884  df-iun 4969  df-br 5120  df-opab 5182  df-mpt 5202  df-tr 5230  df-id 5548  df-eprel 5553  df-po 5561  df-so 5562  df-fr 5606  df-we 5608  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667  df-pred 6290  df-ord 6355  df-on 6356  df-lim 6357  df-suc 6358  df-iota 6484  df-fun 6533  df-fn 6534  df-f 6535  df-f1 6536  df-fo 6537  df-f1o 6538  df-fv 6539  df-riota 7362  df-ov 7408  df-oprab 7409  df-mpo 7410  df-of 7671  df-om 7862  df-1st 7988  df-2nd 7989  df-frecs 8280  df-wrecs 8311  df-recs 8385  df-rdg 8424  df-er 8719  df-map 8842  df-en 8960  df-dom 8961  df-sdom 8962  df-pnf 11271  df-mnf 11272  df-xr 11273  df-ltxr 11274  df-le 11275  df-sub 11468  df-neg 11469  df-nn 12241  df-2 12303  df-sets 17183  df-slot 17201  df-ndx 17213  df-base 17229  df-plusg 17284  df-0g 17455  df-mgm 18618  df-sgrp 18697  df-mnd 18713  df-mhm 18761  df-grp 18919  df-ghm 19196  df-cmn 19763  df-mgp 20101  df-cring 20196  df-lmod 20819  df-lmhm 20980

This theorem is referenced by:  mendlmod  43213