Theorem lmhmvsca 20506

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 2736	. 2 ⊢ ( ·𝑠 ‘𝑀) = ( ·𝑠 ‘𝑀)
3		lmhmvsca.s	. 2 ⊢ · = ( ·𝑠 ‘𝑁)
4		eqid 2736	. 2 ⊢ (Scalar‘𝑀) = (Scalar‘𝑀)
5		lmhmvsca.j	. 2 ⊢ 𝐽 = (Scalar‘𝑁)
6		eqid 2736	. 2 ⊢ (Base‘(Scalar‘𝑀)) = (Base‘(Scalar‘𝑀))
7		lmhmlmod1 20494	. . 3 ⊢ (𝐹 ∈ (𝑀 LMHom 𝑁) → 𝑀 ∈ LMod)
8	7	3ad2ant3 1135	. 2 ⊢ ((𝐽 ∈ CRing ∧ 𝐴 ∈ 𝐾 ∧ 𝐹 ∈ (𝑀 LMHom 𝑁)) → 𝑀 ∈ LMod)
9		lmhmlmod2 20493	. . 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 20491	. . 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 6856	. . . . . 6 ⊢ 𝑉 ∈ V
14	13	a1i 11	. . . . 5 ⊢ ((𝐽 ∈ CRing ∧ 𝐴 ∈ 𝐾 ∧ 𝐹 ∈ (𝑀 LMHom 𝑁)) → 𝑉 ∈ V)
15		simpl2 1192	. . . . 5 ⊢ (((𝐽 ∈ CRing ∧ 𝐴 ∈ 𝐾 ∧ 𝐹 ∈ (𝑀 LMHom 𝑁)) ∧ 𝑣 ∈ 𝑉) → 𝐴 ∈ 𝐾)
16		eqid 2736	. . . . . . . 8 ⊢ (Base‘𝑁) = (Base‘𝑁)
17	1, 16	lmhmf 20495	. . . . . . 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 7035	. . . . 5 ⊢ (((𝐽 ∈ CRing ∧ 𝐴 ∈ 𝐾 ∧ 𝐹 ∈ (𝑀 LMHom 𝑁)) ∧ 𝑣 ∈ 𝑉) → (𝐹‘𝑣) ∈ (Base‘𝑁))
20		fconstmpt 5694	. . . . . 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 6910	. . . . 5 ⊢ ((𝐽 ∈ CRing ∧ 𝐴 ∈ 𝐾 ∧ 𝐹 ∈ (𝑀 LMHom 𝑁)) → 𝐹 = (𝑣 ∈ 𝑉 ↦ (𝐹‘𝑣)))
23	14, 15, 19, 21, 22	offval2 7637	. . . 4 ⊢ ((𝐽 ∈ CRing ∧ 𝐴 ∈ 𝐾 ∧ 𝐹 ∈ (𝑀 LMHom 𝑁)) → ((𝑉 × {𝐴}) ∘f · 𝐹) = (𝑣 ∈ 𝑉 ↦ (𝐴 · (𝐹‘𝑣))))
24		eqidd 2737	. . . . 5 ⊢ ((𝐽 ∈ CRing ∧ 𝐴 ∈ 𝐾 ∧ 𝐹 ∈ (𝑀 LMHom 𝑁)) → (𝑢 ∈ (Base‘𝑁) ↦ (𝐴 · 𝑢)) = (𝑢 ∈ (Base‘𝑁) ↦ (𝐴 · 𝑢)))
25		oveq2 7365	. . . . 5 ⊢ (𝑢 = (𝐹‘𝑣) → (𝐴 · 𝑢) = (𝐴 · (𝐹‘𝑣)))
26	19, 22, 24, 25	fmptco 7075	. . . 4 ⊢ ((𝐽 ∈ CRing ∧ 𝐴 ∈ 𝐾 ∧ 𝐹 ∈ (𝑀 LMHom 𝑁)) → ((𝑢 ∈ (Base‘𝑁) ↦ (𝐴 · 𝑢)) ∘ 𝐹) = (𝑣 ∈ 𝑉 ↦ (𝐴 · (𝐹‘𝑣))))
27	23, 26	eqtr4d 2779	. . 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 20383	. . . . 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 20492	. . . . 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 19028	. . . 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 2838	. 2 ⊢ ((𝐽 ∈ CRing ∧ 𝐴 ∈ 𝐾 ∧ 𝐹 ∈ (𝑀 LMHom 𝑁)) → ((𝑉 × {𝐴}) ∘f · 𝐹) ∈ (𝑀 GrpHom 𝑁))
37		simpl1 1191	. . . . . 6 ⊢ (((𝐽 ∈ CRing ∧ 𝐴 ∈ 𝐾 ∧ 𝐹 ∈ (𝑀 LMHom 𝑁)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦 ∈ 𝑉)) → 𝐽 ∈ CRing)
38		simpl2 1192	. . . . . 6 ⊢ (((𝐽 ∈ CRing ∧ 𝐴 ∈ 𝐾 ∧ 𝐹 ∈ (𝑀 LMHom 𝑁)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦 ∈ 𝑉)) → 𝐴 ∈ 𝐾)
39		simprl 769	. . . . . . 7 ⊢ (((𝐽 ∈ CRing ∧ 𝐴 ∈ 𝐾 ∧ 𝐹 ∈ (𝑀 LMHom 𝑁)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦 ∈ 𝑉)) → 𝑥 ∈ (Base‘(Scalar‘𝑀)))
40	12	fveq2d 6846	. . . . . . . . 9 ⊢ ((𝐽 ∈ CRing ∧ 𝐴 ∈ 𝐾 ∧ 𝐹 ∈ (𝑀 LMHom 𝑁)) → (Base‘𝐽) = (Base‘(Scalar‘𝑀)))
41	29, 40	eqtrid 2788	. . . . . . . 8 ⊢ ((𝐽 ∈ CRing ∧ 𝐴 ∈ 𝐾 ∧ 𝐹 ∈ (𝑀 LMHom 𝑁)) → 𝐾 = (Base‘(Scalar‘𝑀)))
42	41	adantr 481	. . . . . . 7 ⊢ (((𝐽 ∈ CRing ∧ 𝐴 ∈ 𝐾 ∧ 𝐹 ∈ (𝑀 LMHom 𝑁)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦 ∈ 𝑉)) → 𝐾 = (Base‘(Scalar‘𝑀)))
43	39, 42	eleqtrrd 2841	. . . . . 6 ⊢ (((𝐽 ∈ CRing ∧ 𝐴 ∈ 𝐾 ∧ 𝐹 ∈ (𝑀 LMHom 𝑁)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦 ∈ 𝑉)) → 𝑥 ∈ 𝐾)
44		eqid 2736	. . . . . . 7 ⊢ (.r‘𝐽) = (.r‘𝐽)
45	29, 44	crngcom 19982	. . . . . 6 ⊢ ((𝐽 ∈ CRing ∧ 𝐴 ∈ 𝐾 ∧ 𝑥 ∈ 𝐾) → (𝐴(.r‘𝐽)𝑥) = (𝑥(.r‘𝐽)𝐴))
46	37, 38, 43, 45	syl3anc 1371	. . . . 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 7372	. . . 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 481	. . . . 5 ⊢ (((𝐽 ∈ CRing ∧ 𝐴 ∈ 𝐾 ∧ 𝐹 ∈ (𝑀 LMHom 𝑁)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦 ∈ 𝑉)) → 𝑁 ∈ LMod)
49	18	adantr 481	. . . . . 6 ⊢ (((𝐽 ∈ CRing ∧ 𝐴 ∈ 𝐾 ∧ 𝐹 ∈ (𝑀 LMHom 𝑁)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦 ∈ 𝑉)) → 𝐹:𝑉⟶(Base‘𝑁))
50		simprr 771	. . . . . 6 ⊢ (((𝐽 ∈ CRing ∧ 𝐴 ∈ 𝐾 ∧ 𝐹 ∈ (𝑀 LMHom 𝑁)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦 ∈ 𝑉)) → 𝑦 ∈ 𝑉)
51	49, 50	ffvelcdmd 7036	. . . . 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 20347	. . . . 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 1372	. . . 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 20347	. . . . 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 1372	. . . 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 2784	. . 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 20339	. . . . . 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 6669	. . . . . 6 ⊢ ((𝐽 ∈ CRing ∧ 𝐴 ∈ 𝐾 ∧ 𝐹 ∈ (𝑀 LMHom 𝑁)) → 𝐹 Fn 𝑉)
62	61	adantr 481	. . . . 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 20496	. . . . . . . 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 1187	. . . . . 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 481	. . . . 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 7643	. . . 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 685	. . 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 2737	. . . . . 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 7643	. . . . 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 685	. . . 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 7373	. . 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 2786	. 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 20500	1 ⊢ ((𝐽 ∈ CRing ∧ 𝐴 ∈ 𝐾 ∧ 𝐹 ∈ (𝑀 LMHom 𝑁)) → ((𝑉 × {𝐴}) ∘f · 𝐹) ∈ (𝑀 LMHom 𝑁))

Syntax hints:   → wi 4   ∧ wa 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106  Vcvv 3445  {csn 4586   ↦ cmpt 5188   × cxp 5631   ∘ ccom 5637   Fn wfn 6491  ⟶wf 6492  ‘cfv 6496  (class class class)co 7357   ∘_f cof 7615  Basecbs 17083  ._rcmulr 17134  Scalarcsca 17136   ·_𝑠 cvsca 17137   GrpHom cghm 19005  CRingccrg 19965  LModclmod 20322   LMHom clmhm 20480

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 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128

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 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-of 7617  df-om 7803  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-er 8648  df-map 8767  df-en 8884  df-dom 8885  df-sdom 8886  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-nn 12154  df-2 12216  df-sets 17036  df-slot 17054  df-ndx 17066  df-base 17084  df-plusg 17146  df-0g 17323  df-mgm 18497  df-sgrp 18546  df-mnd 18557  df-mhm 18601  df-grp 18751  df-ghm 19006  df-cmn 19564  df-mgp 19897  df-cring 19967  df-lmod 20324  df-lmhm 20483

This theorem is referenced by:  mendlmod  41506