Theorem lmhmvsca 20952

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 2729	. 2 ⊢ ( ·𝑠 ‘𝑀) = ( ·𝑠 ‘𝑀)
3		lmhmvsca.s	. 2 ⊢ · = ( ·𝑠 ‘𝑁)
4		eqid 2729	. 2 ⊢ (Scalar‘𝑀) = (Scalar‘𝑀)
5		lmhmvsca.j	. 2 ⊢ 𝐽 = (Scalar‘𝑁)
6		eqid 2729	. 2 ⊢ (Base‘(Scalar‘𝑀)) = (Base‘(Scalar‘𝑀))
7		lmhmlmod1 20940	. . 3 ⊢ (𝐹 ∈ (𝑀 LMHom 𝑁) → 𝑀 ∈ LMod)
8	7	3ad2ant3 1135	. 2 ⊢ ((𝐽 ∈ CRing ∧ 𝐴 ∈ 𝐾 ∧ 𝐹 ∈ (𝑀 LMHom 𝑁)) → 𝑀 ∈ LMod)
9		lmhmlmod2 20939	. . 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 20937	. . 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 6872	. . . . . 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 2729	. . . . . . . 8 ⊢ (Base‘𝑁) = (Base‘𝑁)
17	1, 16	lmhmf 20941	. . . . . . 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 7056	. . . . 5 ⊢ (((𝐽 ∈ CRing ∧ 𝐴 ∈ 𝐾 ∧ 𝐹 ∈ (𝑀 LMHom 𝑁)) ∧ 𝑣 ∈ 𝑉) → (𝐹‘𝑣) ∈ (Base‘𝑁))
20		fconstmpt 5700	. . . . . 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 6929	. . . . 5 ⊢ ((𝐽 ∈ CRing ∧ 𝐴 ∈ 𝐾 ∧ 𝐹 ∈ (𝑀 LMHom 𝑁)) → 𝐹 = (𝑣 ∈ 𝑉 ↦ (𝐹‘𝑣)))
23	14, 15, 19, 21, 22	offval2 7673	. . . 4 ⊢ ((𝐽 ∈ CRing ∧ 𝐴 ∈ 𝐾 ∧ 𝐹 ∈ (𝑀 LMHom 𝑁)) → ((𝑉 × {𝐴}) ∘f · 𝐹) = (𝑣 ∈ 𝑉 ↦ (𝐴 · (𝐹‘𝑣))))
24		eqidd 2730	. . . . 5 ⊢ ((𝐽 ∈ CRing ∧ 𝐴 ∈ 𝐾 ∧ 𝐹 ∈ (𝑀 LMHom 𝑁)) → (𝑢 ∈ (Base‘𝑁) ↦ (𝐴 · 𝑢)) = (𝑢 ∈ (Base‘𝑁) ↦ (𝐴 · 𝑢)))
25		oveq2 7395	. . . . 5 ⊢ (𝑢 = (𝐹‘𝑣) → (𝐴 · 𝑢) = (𝐴 · (𝐹‘𝑣)))
26	19, 22, 24, 25	fmptco 7101	. . . 4 ⊢ ((𝐽 ∈ CRing ∧ 𝐴 ∈ 𝐾 ∧ 𝐹 ∈ (𝑀 LMHom 𝑁)) → ((𝑢 ∈ (Base‘𝑁) ↦ (𝐴 · 𝑢)) ∘ 𝐹) = (𝑣 ∈ 𝑉 ↦ (𝐴 · (𝐹‘𝑣))))
27	23, 26	eqtr4d 2767	. . 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 20829	. . . . 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 20938	. . . . 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 19168	. . . 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 2828	. 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 6862	. . . . . . . . 9 ⊢ ((𝐽 ∈ CRing ∧ 𝐴 ∈ 𝐾 ∧ 𝐹 ∈ (𝑀 LMHom 𝑁)) → (Base‘𝐽) = (Base‘(Scalar‘𝑀)))
41	29, 40	eqtrid 2776	. . . . . . . 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 2831	. . . . . 6 ⊢ (((𝐽 ∈ CRing ∧ 𝐴 ∈ 𝐾 ∧ 𝐹 ∈ (𝑀 LMHom 𝑁)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦 ∈ 𝑉)) → 𝑥 ∈ 𝐾)
44		eqid 2729	. . . . . . 7 ⊢ (.r‘𝐽) = (.r‘𝐽)
45	29, 44	crngcom 20160	. . . . . 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 7402	. . . 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 7057	. . . . 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 20793	. . . . 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 20793	. . . . 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 2772	. . 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 20784	. . . . . 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 6689	. . . . . 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 20942	. . . . . . . 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 7681	. . . 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 2730	. . . . . 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 7681	. . . . 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 7403	. . 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 2774	. 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 20946	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 2109  Vcvv 3447  {csn 4589   ↦ cmpt 5188   × cxp 5636   ∘ ccom 5642   Fn wfn 6506  ⟶wf 6507  ‘cfv 6511  (class class class)co 7387   ∘_f cof 7651  Basecbs 17179  ._rcmulr 17221  Scalarcsca 17223   ·_𝑠 cvsca 17224   GrpHom cghm 19144  CRingccrg 20143  LModclmod 20766   LMHom clmhm 20926

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 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5234  ax-sep 5251  ax-nul 5261  ax-pow 5320  ax-pr 5387  ax-un 7711  ax-cnex 11124  ax-resscn 11125  ax-1cn 11126  ax-icn 11127  ax-addcl 11128  ax-addrcl 11129  ax-mulcl 11130  ax-mulrcl 11131  ax-mulcom 11132  ax-addass 11133  ax-mulass 11134  ax-distr 11135  ax-i2m1 11136  ax-1ne0 11137  ax-1rid 11138  ax-rnegex 11139  ax-rrecex 11140  ax-cnre 11141  ax-pre-lttri 11142  ax-pre-lttrn 11143  ax-pre-ltadd 11144  ax-pre-mulgt0 11145

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 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-nel 3030  df-ral 3045  df-rex 3054  df-rmo 3354  df-reu 3355  df-rab 3406  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3934  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-iun 4957  df-br 5108  df-opab 5170  df-mpt 5189  df-tr 5215  df-id 5533  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5591  df-we 5593  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-pred 6274  df-ord 6335  df-on 6336  df-lim 6337  df-suc 6338  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-f1 6516  df-fo 6517  df-f1o 6518  df-fv 6519  df-riota 7344  df-ov 7390  df-oprab 7391  df-mpo 7392  df-of 7653  df-om 7843  df-1st 7968  df-2nd 7969  df-frecs 8260  df-wrecs 8291  df-recs 8340  df-rdg 8378  df-er 8671  df-map 8801  df-en 8919  df-dom 8920  df-sdom 8921  df-pnf 11210  df-mnf 11211  df-xr 11212  df-ltxr 11213  df-le 11214  df-sub 11407  df-neg 11408  df-nn 12187  df-2 12249  df-sets 17134  df-slot 17152  df-ndx 17164  df-base 17180  df-plusg 17233  df-0g 17404  df-mgm 18567  df-sgrp 18646  df-mnd 18662  df-mhm 18710  df-grp 18868  df-ghm 19145  df-cmn 19712  df-mgp 20050  df-cring 20145  df-lmod 20768  df-lmhm 20929

This theorem is referenced by:  mendlmod  43178