Theorem lmhmvsca 20949

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 20937	. . 3 ⊢ (𝐹 ∈ (𝑀 LMHom 𝑁) → 𝑀 ∈ LMod)
8	7	3ad2ant3 1135	. 2 ⊢ ((𝐽 ∈ CRing ∧ 𝐴 ∈ 𝐾 ∧ 𝐹 ∈ (𝑀 LMHom 𝑁)) → 𝑀 ∈ LMod)
9		lmhmlmod2 20936	. . 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 20934	. . 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 6836	. . . . . 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 20938	. . . . . . 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 7018	. . . . 5 ⊢ (((𝐽 ∈ CRing ∧ 𝐴 ∈ 𝐾 ∧ 𝐹 ∈ (𝑀 LMHom 𝑁)) ∧ 𝑣 ∈ 𝑉) → (𝐹‘𝑣) ∈ (Base‘𝑁))
20		fconstmpt 5681	. . . . . 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 6891	. . . . 5 ⊢ ((𝐽 ∈ CRing ∧ 𝐴 ∈ 𝐾 ∧ 𝐹 ∈ (𝑀 LMHom 𝑁)) → 𝐹 = (𝑣 ∈ 𝑉 ↦ (𝐹‘𝑣)))
23	14, 15, 19, 21, 22	offval2 7633	. . . 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 7357	. . . . 5 ⊢ (𝑢 = (𝐹‘𝑣) → (𝐴 · 𝑢) = (𝐴 · (𝐹‘𝑣)))
26	19, 22, 24, 25	fmptco 7063	. . . 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 20826	. . . . 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 20935	. . . . 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 19115	. . . 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 6826	. . . . . . . . 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 20136	. . . . . 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 7364	. . . 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 7019	. . . . 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 20790	. . . . 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 20790	. . . . 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 20781	. . . . . 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 6653	. . . . . 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 20939	. . . . . . . 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 7641	. . . 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 7641	. . . . 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 7365	. . 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 20943	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 3436  {csn 4577   ↦ cmpt 5173   × cxp 5617   ∘ ccom 5623   Fn wfn 6477  ⟶wf 6478  ‘cfv 6482  (class class class)co 7349   ∘_f cof 7611  Basecbs 17120  ._rcmulr 17162  Scalarcsca 17164   ·_𝑠 cvsca 17165   GrpHom cghm 19091  CRingccrg 20119  LModclmod 20763   LMHom clmhm 20923

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 5218  ax-sep 5235  ax-nul 5245  ax-pow 5304  ax-pr 5371  ax-un 7671  ax-cnex 11065  ax-resscn 11066  ax-1cn 11067  ax-icn 11068  ax-addcl 11069  ax-addrcl 11070  ax-mulcl 11071  ax-mulrcl 11072  ax-mulcom 11073  ax-addass 11074  ax-mulass 11075  ax-distr 11076  ax-i2m1 11077  ax-1ne0 11078  ax-1rid 11079  ax-rnegex 11080  ax-rrecex 11081  ax-cnre 11082  ax-pre-lttri 11083  ax-pre-lttrn 11084  ax-pre-ltadd 11085  ax-pre-mulgt0 11086

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 3343  df-reu 3344  df-rab 3395  df-v 3438  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4285  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4859  df-iun 4943  df-br 5093  df-opab 5155  df-mpt 5174  df-tr 5200  df-id 5514  df-eprel 5519  df-po 5527  df-so 5528  df-fr 5572  df-we 5574  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-pred 6249  df-ord 6310  df-on 6311  df-lim 6312  df-suc 6313  df-iota 6438  df-fun 6484  df-fn 6485  df-f 6486  df-f1 6487  df-fo 6488  df-f1o 6489  df-fv 6490  df-riota 7306  df-ov 7352  df-oprab 7353  df-mpo 7354  df-of 7613  df-om 7800  df-1st 7924  df-2nd 7925  df-frecs 8214  df-wrecs 8245  df-recs 8294  df-rdg 8332  df-er 8625  df-map 8755  df-en 8873  df-dom 8874  df-sdom 8875  df-pnf 11151  df-mnf 11152  df-xr 11153  df-ltxr 11154  df-le 11155  df-sub 11349  df-neg 11350  df-nn 12129  df-2 12191  df-sets 17075  df-slot 17093  df-ndx 17105  df-base 17121  df-plusg 17174  df-0g 17345  df-mgm 18514  df-sgrp 18593  df-mnd 18609  df-mhm 18657  df-grp 18815  df-ghm 19092  df-cmn 19661  df-mgp 20026  df-cring 20121  df-lmod 20765  df-lmhm 20926

This theorem is referenced by:  mendlmod  43166