Theorem lmhmvsca 21335

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 2763	. 2 ⊢ ( ·𝑠 ‘𝑀) = ( ·𝑠 ‘𝑀)
3		lmhmvsca.s	. 2 ⊢ · = ( ·𝑠 ‘𝑁)
4		eqid 2763	. 2 ⊢ (Scalar‘𝑀) = (Scalar‘𝑀)
5		lmhmvsca.j	. 2 ⊢ 𝐽 = (Scalar‘𝑁)
6		eqid 2763	. 2 ⊢ (Base‘(Scalar‘𝑀)) = (Base‘(Scalar‘𝑀))
7		lmhmlmod1 21323	. . 3 ⊢ (𝐹 ∈ (𝑀 LMHom 𝑁) → 𝑀 ∈ LMod)
8	7	3ad2ant3 1153	. 2 ⊢ ((𝐽 ∈ CRing ∧ 𝐴 ∈ 𝐾 ∧ 𝐹 ∈ (𝑀 LMHom 𝑁)) → 𝑀 ∈ LMod)
9		lmhmlmod2 21322	. . 3 ⊢ (𝐹 ∈ (𝑀 LMHom 𝑁) → 𝑁 ∈ LMod)
10	9	3ad2ant3 1153	. 2 ⊢ ((𝐽 ∈ CRing ∧ 𝐴 ∈ 𝐾 ∧ 𝐹 ∈ (𝑀 LMHom 𝑁)) → 𝑁 ∈ LMod)
11	4, 5	lmhmsca 21320	. . 3 ⊢ (𝐹 ∈ (𝑀 LMHom 𝑁) → 𝐽 = (Scalar‘𝑀))
12	11	3ad2ant3 1153	. 2 ⊢ ((𝐽 ∈ CRing ∧ 𝐴 ∈ 𝐾 ∧ 𝐹 ∈ (𝑀 LMHom 𝑁)) → 𝐽 = (Scalar‘𝑀))
13	1	fvexi 6910	. . . . . 6 ⊢ 𝑉 ∈ V
14	13	a1i 11	. . . . 5 ⊢ ((𝐽 ∈ CRing ∧ 𝐴 ∈ 𝐾 ∧ 𝐹 ∈ (𝑀 LMHom 𝑁)) → 𝑉 ∈ V)
15		simpl2 1211	. . . . 5 ⊢ (((𝐽 ∈ CRing ∧ 𝐴 ∈ 𝐾 ∧ 𝐹 ∈ (𝑀 LMHom 𝑁)) ∧ 𝑣 ∈ 𝑉) → 𝐴 ∈ 𝐾)
16		eqid 2763	. . . . . . . 8 ⊢ (Base‘𝑁) = (Base‘𝑁)
17	1, 16	lmhmf 21324	. . . . . . 7 ⊢ (𝐹 ∈ (𝑀 LMHom 𝑁) → 𝐹:𝑉⟶(Base‘𝑁))
18	17	3ad2ant3 1153	. . . . . 6 ⊢ ((𝐽 ∈ CRing ∧ 𝐴 ∈ 𝐾 ∧ 𝐹 ∈ (𝑀 LMHom 𝑁)) → 𝐹:𝑉⟶(Base‘𝑁))
19	18	ffvelcdmda 7095	. . . . 5 ⊢ (((𝐽 ∈ CRing ∧ 𝐴 ∈ 𝐾 ∧ 𝐹 ∈ (𝑀 LMHom 𝑁)) ∧ 𝑣 ∈ 𝑉) → (𝐹‘𝑣) ∈ (Base‘𝑁))
20		fconstmpt 5717	. . . . . 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 6964	. . . . 5 ⊢ ((𝐽 ∈ CRing ∧ 𝐴 ∈ 𝐾 ∧ 𝐹 ∈ (𝑀 LMHom 𝑁)) → 𝐹 = (𝑣 ∈ 𝑉 ↦ (𝐹‘𝑣)))
23	14, 15, 19, 21, 22	offval2 7723	. . . 4 ⊢ ((𝐽 ∈ CRing ∧ 𝐴 ∈ 𝐾 ∧ 𝐹 ∈ (𝑀 LMHom 𝑁)) → ((𝑉 × {𝐴}) ∘f · 𝐹) = (𝑣 ∈ 𝑉 ↦ (𝐴 · (𝐹‘𝑣))))
24		eqidd 2764	. . . . 5 ⊢ ((𝐽 ∈ CRing ∧ 𝐴 ∈ 𝐾 ∧ 𝐹 ∈ (𝑀 LMHom 𝑁)) → (𝑢 ∈ (Base‘𝑁) ↦ (𝐴 · 𝑢)) = (𝑢 ∈ (Base‘𝑁) ↦ (𝐴 · 𝑢)))
25		oveq2 7438	. . . . 5 ⊢ (𝑢 = (𝐹‘𝑣) → (𝐴 · 𝑢) = (𝐴 · (𝐹‘𝑣)))
26	19, 22, 24, 25	fmptco 7141	. . . 4 ⊢ ((𝐽 ∈ CRing ∧ 𝐴 ∈ 𝐾 ∧ 𝐹 ∈ (𝑀 LMHom 𝑁)) → ((𝑢 ∈ (Base‘𝑁) ↦ (𝐴 · 𝑢)) ∘ 𝐹) = (𝑣 ∈ 𝑉 ↦ (𝐴 · (𝐹‘𝑣))))
27	23, 26	eqtr4d 2801	. . 3 ⊢ ((𝐽 ∈ CRing ∧ 𝐴 ∈ 𝐾 ∧ 𝐹 ∈ (𝑀 LMHom 𝑁)) → ((𝑉 × {𝐴}) ∘f · 𝐹) = ((𝑢 ∈ (Base‘𝑁) ↦ (𝐴 · 𝑢)) ∘ 𝐹))
28		simp2 1155	. . . . 5 ⊢ ((𝐽 ∈ CRing ∧ 𝐴 ∈ 𝐾 ∧ 𝐹 ∈ (𝑀 LMHom 𝑁)) → 𝐴 ∈ 𝐾)
29		lmhmvsca.k	. . . . . 6 ⊢ 𝐾 = (Base‘𝐽)
30	16, 5, 3, 29	lmodvsghm 21213	. . . . 5 ⊢ ((𝑁 ∈ LMod ∧ 𝐴 ∈ 𝐾) → (𝑢 ∈ (Base‘𝑁) ↦ (𝐴 · 𝑢)) ∈ (𝑁 GrpHom 𝑁))
31	10, 28, 30	syl2anc 596	. . . 4 ⊢ ((𝐽 ∈ CRing ∧ 𝐴 ∈ 𝐾 ∧ 𝐹 ∈ (𝑀 LMHom 𝑁)) → (𝑢 ∈ (Base‘𝑁) ↦ (𝐴 · 𝑢)) ∈ (𝑁 GrpHom 𝑁))
32		lmghm 21321	. . . . 5 ⊢ (𝐹 ∈ (𝑀 LMHom 𝑁) → 𝐹 ∈ (𝑀 GrpHom 𝑁))
33	32	3ad2ant3 1153	. . . 4 ⊢ ((𝐽 ∈ CRing ∧ 𝐴 ∈ 𝐾 ∧ 𝐹 ∈ (𝑀 LMHom 𝑁)) → 𝐹 ∈ (𝑀 GrpHom 𝑁))
34		ghmco 19465	. . . 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 596	. . 3 ⊢ ((𝐽 ∈ CRing ∧ 𝐴 ∈ 𝐾 ∧ 𝐹 ∈ (𝑀 LMHom 𝑁)) → ((𝑢 ∈ (Base‘𝑁) ↦ (𝐴 · 𝑢)) ∘ 𝐹) ∈ (𝑀 GrpHom 𝑁))
36	27, 35	eqeltrd 2864	. 2 ⊢ ((𝐽 ∈ CRing ∧ 𝐴 ∈ 𝐾 ∧ 𝐹 ∈ (𝑀 LMHom 𝑁)) → ((𝑉 × {𝐴}) ∘f · 𝐹) ∈ (𝑀 GrpHom 𝑁))
37		simpl1 1210	. . . . . 6 ⊢ (((𝐽 ∈ CRing ∧ 𝐴 ∈ 𝐾 ∧ 𝐹 ∈ (𝑀 LMHom 𝑁)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦 ∈ 𝑉)) → 𝐽 ∈ CRing)
38		simpl2 1211	. . . . . 6 ⊢ (((𝐽 ∈ CRing ∧ 𝐴 ∈ 𝐾 ∧ 𝐹 ∈ (𝑀 LMHom 𝑁)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦 ∈ 𝑉)) → 𝐴 ∈ 𝐾)
39		simprl 783	. . . . . . 7 ⊢ (((𝐽 ∈ CRing ∧ 𝐴 ∈ 𝐾 ∧ 𝐹 ∈ (𝑀 LMHom 𝑁)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦 ∈ 𝑉)) → 𝑥 ∈ (Base‘(Scalar‘𝑀)))
40	12	fveq2d 6900	. . . . . . . . 9 ⊢ ((𝐽 ∈ CRing ∧ 𝐴 ∈ 𝐾 ∧ 𝐹 ∈ (𝑀 LMHom 𝑁)) → (Base‘𝐽) = (Base‘(Scalar‘𝑀)))
41	29, 40	eqtrid 2810	. . . . . . . 8 ⊢ ((𝐽 ∈ CRing ∧ 𝐴 ∈ 𝐾 ∧ 𝐹 ∈ (𝑀 LMHom 𝑁)) → 𝐾 = (Base‘(Scalar‘𝑀)))
42	41	adantr 486	. . . . . . 7 ⊢ (((𝐽 ∈ CRing ∧ 𝐴 ∈ 𝐾 ∧ 𝐹 ∈ (𝑀 LMHom 𝑁)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦 ∈ 𝑉)) → 𝐾 = (Base‘(Scalar‘𝑀)))
43	39, 42	eleqtrrd 2867	. . . . . 6 ⊢ (((𝐽 ∈ CRing ∧ 𝐴 ∈ 𝐾 ∧ 𝐹 ∈ (𝑀 LMHom 𝑁)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦 ∈ 𝑉)) → 𝑥 ∈ 𝐾)
44		eqid 2763	. . . . . . 7 ⊢ (.r‘𝐽) = (.r‘𝐽)
45	29, 44	crngcom 20493	. . . . . 6 ⊢ ((𝐽 ∈ CRing ∧ 𝐴 ∈ 𝐾 ∧ 𝑥 ∈ 𝐾) → (𝐴(.r‘𝐽)𝑥) = (𝑥(.r‘𝐽)𝐴))
46	37, 38, 43, 45	syl3anc 1398	. . . . 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 7445	. . . 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 486	. . . . 5 ⊢ (((𝐽 ∈ CRing ∧ 𝐴 ∈ 𝐾 ∧ 𝐹 ∈ (𝑀 LMHom 𝑁)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦 ∈ 𝑉)) → 𝑁 ∈ LMod)
49	18	adantr 486	. . . . . 6 ⊢ (((𝐽 ∈ CRing ∧ 𝐴 ∈ 𝐾 ∧ 𝐹 ∈ (𝑀 LMHom 𝑁)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦 ∈ 𝑉)) → 𝐹:𝑉⟶(Base‘𝑁))
50		simprr 785	. . . . . 6 ⊢ (((𝐽 ∈ CRing ∧ 𝐴 ∈ 𝐾 ∧ 𝐹 ∈ (𝑀 LMHom 𝑁)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦 ∈ 𝑉)) → 𝑦 ∈ 𝑉)
51	49, 50	ffvelcdmd 7096	. . . . 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 21177	. . . . 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 1399	. . . 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 21177	. . . . 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 1399	. . . 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 2806	. . 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 21168	. . . . . 6 ⊢ ((𝑀 ∈ LMod ∧ 𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦 ∈ 𝑉) → (𝑥( ·𝑠 ‘𝑀)𝑦) ∈ 𝑉)
58	57	3expb 1138	. . . . 5 ⊢ ((𝑀 ∈ LMod ∧ (𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦 ∈ 𝑉)) → (𝑥( ·𝑠 ‘𝑀)𝑦) ∈ 𝑉)
59	8, 58	sylan 592	. . . 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 6721	. . . . . 6 ⊢ ((𝐽 ∈ CRing ∧ 𝐴 ∈ 𝐾 ∧ 𝐹 ∈ (𝑀 LMHom 𝑁)) → 𝐹 Fn 𝑉)
62	61	adantr 486	. . . . 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 21325	. . . . . . . 8 ⊢ ((𝐹 ∈ (𝑀 LMHom 𝑁) ∧ 𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦 ∈ 𝑉) → (𝐹‘(𝑥( ·𝑠 ‘𝑀)𝑦)) = (𝑥 · (𝐹‘𝑦)))
64	63	3expb 1138	. . . . . . 7 ⊢ ((𝐹 ∈ (𝑀 LMHom 𝑁) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦 ∈ 𝑉)) → (𝐹‘(𝑥( ·𝑠 ‘𝑀)𝑦)) = (𝑥 · (𝐹‘𝑦)))
65	64	3ad2antl3 1206	. . . . . 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 486	. . . . 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 7731	. . . 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 700	. . 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 2764	. . . . . 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 7731	. . . . 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 700	. . . 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 7446	. . 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 2808	. 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 21329	1 ⊢ ((𝐽 ∈ CRing ∧ 𝐴 ∈ 𝐾 ∧ 𝐹 ∈ (𝑀 LMHom 𝑁)) → ((𝑉 × {𝐴}) ∘f · 𝐹) ∈ (𝑀 LMHom 𝑁))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∧ w3a 1103   = wceq 1569   ∈ wcel 2146  Vcvv 3454  {csn 4588   ↦ cmpt 5189   × cxp 5653   ∘ ccom 5659   Fn wfn 6542  ⟶wf 6543  ‘cfv 6547  (class class class)co 7430   ∘_f cof 7701  Basecbs 17402  ._rcmulr 17444  Scalarcsca 17446   ·_𝑠 cvsca 17447   GrpHom cghm 19442  CRingccrg 20475  LModclmod 21150   LMHom clmhm 21309

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1944  ax-6 2001  ax-7 2042  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2196  ax-12 2215  ax-ext 2735  ax-rep 5235  ax-sep 5252  ax-nul 5263  ax-pow 5331  ax-pr 5395  ax-un 7761  ax-cnex 11269  ax-resscn 11270  ax-1cn 11271  ax-icn 11272  ax-addcl 11273  ax-addrcl 11274  ax-mulcl 11275  ax-mulrcl 11276  ax-mulcom 11277  ax-addass 11278  ax-mulass 11279  ax-distr 11280  ax-i2m1 11281  ax-1ne0 11282  ax-1rid 11283  ax-rnegex 11284  ax-rrecex 11285  ax-cnre 11286  ax-pre-lttri 11287  ax-pre-lttrn 11288  ax-pre-ltadd 11289  ax-pre-mulgt0 11290

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2101  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-nel 3066  df-ral 3081  df-rex 3091  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-iun 4956  df-br 5107  df-opab 5171  df-mpt 5190  df-tr 5216  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-pred 6311  df-ord 6372  df-on 6373  df-lim 6374  df-suc 6375  df-iota 6501  df-fun 6549  df-fn 6550  df-f 6551  df-f1 6552  df-fo 6553  df-f1o 6554  df-fv 6555  df-riota 7387  df-ov 7433  df-oprab 7434  df-mpo 7435  df-of 7703  df-om 7888  df-1st 8011  df-2nd 8012  df-frecs 8313  df-wrecs 8344  df-recs 8393  df-rdg 8432  df-er 8731  df-map 8863  df-en 8988  df-dom 8989  df-sdom 8990  df-pnf 11358  df-mnf 11359  df-xr 11360  df-ltxr 11361  df-le 11362  df-sub 11556  df-neg 11557  df-nn 12349  df-2 12418  df-sets 17357  df-slot 17375  df-ndx 17387  df-base 17403  df-plusg 17456  df-0g 17627  df-mgm 18831  df-sgrp 18923  df-mnd 18939  df-mhm 18993  df-grp 19162  df-ghm 19443  df-cmn 20011  df-mgp 20376  df-cring 20477  df-lmod 21152  df-lmhm 21312

This theorem is used by:  mendlmod  44334