Theorem lfladdcl 37936

Description: Closure of addition of two functionals. (Contributed by NM, 19-Oct-2014.)

Hypotheses

Ref	Expression
lfladdcl.r	⊢ 𝑅 = (Scalar‘𝑊)
lfladdcl.p	⊢ + = (+g‘𝑅)
lfladdcl.f	⊢ 𝐹 = (LFnl‘𝑊)
lfladdcl.w	⊢ (𝜑 → 𝑊 ∈ LMod)
lfladdcl.g	⊢ (𝜑 → 𝐺 ∈ 𝐹)
lfladdcl.h	⊢ (𝜑 → 𝐻 ∈ 𝐹)

Assertion

Ref	Expression
lfladdcl	⊢ (𝜑 → (𝐺 ∘f + 𝐻) ∈ 𝐹)

Proof of Theorem lfladdcl

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

Step	Hyp	Ref	Expression
1		lfladdcl.w	. . . . 5 ⊢ (𝜑 → 𝑊 ∈ LMod)
2	1	adantr 481	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅))) → 𝑊 ∈ LMod)
3		simprl 769	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅))) → 𝑥 ∈ (Base‘𝑅))
4		simprr 771	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅))) → 𝑦 ∈ (Base‘𝑅))
5		lfladdcl.r	. . . . 5 ⊢ 𝑅 = (Scalar‘𝑊)
6		eqid 2732	. . . . 5 ⊢ (Base‘𝑅) = (Base‘𝑅)
7		lfladdcl.p	. . . . 5 ⊢ + = (+g‘𝑅)
8	5, 6, 7	lmodacl 20482	. . . 4 ⊢ ((𝑊 ∈ LMod ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) → (𝑥 + 𝑦) ∈ (Base‘𝑅))
9	2, 3, 4, 8	syl3anc 1371	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅))) → (𝑥 + 𝑦) ∈ (Base‘𝑅))
10		lfladdcl.g	. . . 4 ⊢ (𝜑 → 𝐺 ∈ 𝐹)
11		eqid 2732	. . . . 5 ⊢ (Base‘𝑊) = (Base‘𝑊)
12		lfladdcl.f	. . . . 5 ⊢ 𝐹 = (LFnl‘𝑊)
13	5, 6, 11, 12	lflf 37928	. . . 4 ⊢ ((𝑊 ∈ LMod ∧ 𝐺 ∈ 𝐹) → 𝐺:(Base‘𝑊)⟶(Base‘𝑅))
14	1, 10, 13	syl2anc 584	. . 3 ⊢ (𝜑 → 𝐺:(Base‘𝑊)⟶(Base‘𝑅))
15		lfladdcl.h	. . . 4 ⊢ (𝜑 → 𝐻 ∈ 𝐹)
16	5, 6, 11, 12	lflf 37928	. . . 4 ⊢ ((𝑊 ∈ LMod ∧ 𝐻 ∈ 𝐹) → 𝐻:(Base‘𝑊)⟶(Base‘𝑅))
17	1, 15, 16	syl2anc 584	. . 3 ⊢ (𝜑 → 𝐻:(Base‘𝑊)⟶(Base‘𝑅))
18		fvexd 6906	. . 3 ⊢ (𝜑 → (Base‘𝑊) ∈ V)
19		inidm 4218	. . 3 ⊢ ((Base‘𝑊) ∩ (Base‘𝑊)) = (Base‘𝑊)
20	9, 14, 17, 18, 18, 19	off 7687	. 2 ⊢ (𝜑 → (𝐺 ∘f + 𝐻):(Base‘𝑊)⟶(Base‘𝑅))
21	1	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑊) ∧ 𝑧 ∈ (Base‘𝑊))) → 𝑊 ∈ LMod)
22		simpr1 1194	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑊) ∧ 𝑧 ∈ (Base‘𝑊))) → 𝑥 ∈ (Base‘𝑅))
23		simpr2 1195	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑊) ∧ 𝑧 ∈ (Base‘𝑊))) → 𝑦 ∈ (Base‘𝑊))
24		eqid 2732	. . . . . . . 8 ⊢ ( ·𝑠 ‘𝑊) = ( ·𝑠 ‘𝑊)
25	11, 5, 24, 6	lmodvscl 20488	. . . . . . 7 ⊢ ((𝑊 ∈ LMod ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑊)) → (𝑥( ·𝑠 ‘𝑊)𝑦) ∈ (Base‘𝑊))
26	21, 22, 23, 25	syl3anc 1371	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑊) ∧ 𝑧 ∈ (Base‘𝑊))) → (𝑥( ·𝑠 ‘𝑊)𝑦) ∈ (Base‘𝑊))
27		simpr3 1196	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑊) ∧ 𝑧 ∈ (Base‘𝑊))) → 𝑧 ∈ (Base‘𝑊))
28		eqid 2732	. . . . . . 7 ⊢ (+g‘𝑊) = (+g‘𝑊)
29	11, 28	lmodvacl 20485	. . . . . 6 ⊢ ((𝑊 ∈ LMod ∧ (𝑥( ·𝑠 ‘𝑊)𝑦) ∈ (Base‘𝑊) ∧ 𝑧 ∈ (Base‘𝑊)) → ((𝑥( ·𝑠 ‘𝑊)𝑦)(+g‘𝑊)𝑧) ∈ (Base‘𝑊))
30	21, 26, 27, 29	syl3anc 1371	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑊) ∧ 𝑧 ∈ (Base‘𝑊))) → ((𝑥( ·𝑠 ‘𝑊)𝑦)(+g‘𝑊)𝑧) ∈ (Base‘𝑊))
31	14	ffnd 6718	. . . . . 6 ⊢ (𝜑 → 𝐺 Fn (Base‘𝑊))
32	17	ffnd 6718	. . . . . 6 ⊢ (𝜑 → 𝐻 Fn (Base‘𝑊))
33		eqidd 2733	. . . . . 6 ⊢ ((𝜑 ∧ ((𝑥( ·𝑠 ‘𝑊)𝑦)(+g‘𝑊)𝑧) ∈ (Base‘𝑊)) → (𝐺‘((𝑥( ·𝑠 ‘𝑊)𝑦)(+g‘𝑊)𝑧)) = (𝐺‘((𝑥( ·𝑠 ‘𝑊)𝑦)(+g‘𝑊)𝑧)))
34		eqidd 2733	. . . . . 6 ⊢ ((𝜑 ∧ ((𝑥( ·𝑠 ‘𝑊)𝑦)(+g‘𝑊)𝑧) ∈ (Base‘𝑊)) → (𝐻‘((𝑥( ·𝑠 ‘𝑊)𝑦)(+g‘𝑊)𝑧)) = (𝐻‘((𝑥( ·𝑠 ‘𝑊)𝑦)(+g‘𝑊)𝑧)))
35	31, 32, 18, 18, 19, 33, 34	ofval 7680	. . . . 5 ⊢ ((𝜑 ∧ ((𝑥( ·𝑠 ‘𝑊)𝑦)(+g‘𝑊)𝑧) ∈ (Base‘𝑊)) → ((𝐺 ∘f + 𝐻)‘((𝑥( ·𝑠 ‘𝑊)𝑦)(+g‘𝑊)𝑧)) = ((𝐺‘((𝑥( ·𝑠 ‘𝑊)𝑦)(+g‘𝑊)𝑧)) + (𝐻‘((𝑥( ·𝑠 ‘𝑊)𝑦)(+g‘𝑊)𝑧))))
36	30, 35	syldan 591	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑊) ∧ 𝑧 ∈ (Base‘𝑊))) → ((𝐺 ∘f + 𝐻)‘((𝑥( ·𝑠 ‘𝑊)𝑦)(+g‘𝑊)𝑧)) = ((𝐺‘((𝑥( ·𝑠 ‘𝑊)𝑦)(+g‘𝑊)𝑧)) + (𝐻‘((𝑥( ·𝑠 ‘𝑊)𝑦)(+g‘𝑊)𝑧))))
37		eqidd 2733	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ (Base‘𝑊)) → (𝐺‘𝑦) = (𝐺‘𝑦))
38		eqidd 2733	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ (Base‘𝑊)) → (𝐻‘𝑦) = (𝐻‘𝑦))
39	31, 32, 18, 18, 19, 37, 38	ofval 7680	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ (Base‘𝑊)) → ((𝐺 ∘f + 𝐻)‘𝑦) = ((𝐺‘𝑦) + (𝐻‘𝑦)))
40	23, 39	syldan 591	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑊) ∧ 𝑧 ∈ (Base‘𝑊))) → ((𝐺 ∘f + 𝐻)‘𝑦) = ((𝐺‘𝑦) + (𝐻‘𝑦)))
41	40	oveq2d 7424	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑊) ∧ 𝑧 ∈ (Base‘𝑊))) → (𝑥(.r‘𝑅)((𝐺 ∘f + 𝐻)‘𝑦)) = (𝑥(.r‘𝑅)((𝐺‘𝑦) + (𝐻‘𝑦))))
42		eqidd 2733	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧 ∈ (Base‘𝑊)) → (𝐺‘𝑧) = (𝐺‘𝑧))
43		eqidd 2733	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧 ∈ (Base‘𝑊)) → (𝐻‘𝑧) = (𝐻‘𝑧))
44	31, 32, 18, 18, 19, 42, 43	ofval 7680	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ (Base‘𝑊)) → ((𝐺 ∘f + 𝐻)‘𝑧) = ((𝐺‘𝑧) + (𝐻‘𝑧)))
45	27, 44	syldan 591	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑊) ∧ 𝑧 ∈ (Base‘𝑊))) → ((𝐺 ∘f + 𝐻)‘𝑧) = ((𝐺‘𝑧) + (𝐻‘𝑧)))
46	41, 45	oveq12d 7426	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑊) ∧ 𝑧 ∈ (Base‘𝑊))) → ((𝑥(.r‘𝑅)((𝐺 ∘f + 𝐻)‘𝑦)) + ((𝐺 ∘f + 𝐻)‘𝑧)) = ((𝑥(.r‘𝑅)((𝐺‘𝑦) + (𝐻‘𝑦))) + ((𝐺‘𝑧) + (𝐻‘𝑧))))
47	10	adantr 481	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑊) ∧ 𝑧 ∈ (Base‘𝑊))) → 𝐺 ∈ 𝐹)
48	5, 7, 11, 28, 12	lfladd 37931	. . . . . . . 8 ⊢ ((𝑊 ∈ LMod ∧ 𝐺 ∈ 𝐹 ∧ ((𝑥( ·𝑠 ‘𝑊)𝑦) ∈ (Base‘𝑊) ∧ 𝑧 ∈ (Base‘𝑊))) → (𝐺‘((𝑥( ·𝑠 ‘𝑊)𝑦)(+g‘𝑊)𝑧)) = ((𝐺‘(𝑥( ·𝑠 ‘𝑊)𝑦)) + (𝐺‘𝑧)))
49	21, 47, 26, 27, 48	syl112anc 1374	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑊) ∧ 𝑧 ∈ (Base‘𝑊))) → (𝐺‘((𝑥( ·𝑠 ‘𝑊)𝑦)(+g‘𝑊)𝑧)) = ((𝐺‘(𝑥( ·𝑠 ‘𝑊)𝑦)) + (𝐺‘𝑧)))
50	15	adantr 481	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑊) ∧ 𝑧 ∈ (Base‘𝑊))) → 𝐻 ∈ 𝐹)
51	5, 7, 11, 28, 12	lfladd 37931	. . . . . . . 8 ⊢ ((𝑊 ∈ LMod ∧ 𝐻 ∈ 𝐹 ∧ ((𝑥( ·𝑠 ‘𝑊)𝑦) ∈ (Base‘𝑊) ∧ 𝑧 ∈ (Base‘𝑊))) → (𝐻‘((𝑥( ·𝑠 ‘𝑊)𝑦)(+g‘𝑊)𝑧)) = ((𝐻‘(𝑥( ·𝑠 ‘𝑊)𝑦)) + (𝐻‘𝑧)))
52	21, 50, 26, 27, 51	syl112anc 1374	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑊) ∧ 𝑧 ∈ (Base‘𝑊))) → (𝐻‘((𝑥( ·𝑠 ‘𝑊)𝑦)(+g‘𝑊)𝑧)) = ((𝐻‘(𝑥( ·𝑠 ‘𝑊)𝑦)) + (𝐻‘𝑧)))
53	49, 52	oveq12d 7426	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑊) ∧ 𝑧 ∈ (Base‘𝑊))) → ((𝐺‘((𝑥( ·𝑠 ‘𝑊)𝑦)(+g‘𝑊)𝑧)) + (𝐻‘((𝑥( ·𝑠 ‘𝑊)𝑦)(+g‘𝑊)𝑧))) = (((𝐺‘(𝑥( ·𝑠 ‘𝑊)𝑦)) + (𝐺‘𝑧)) + ((𝐻‘(𝑥( ·𝑠 ‘𝑊)𝑦)) + (𝐻‘𝑧))))
54	5	lmodring 20478	. . . . . . . . 9 ⊢ (𝑊 ∈ LMod → 𝑅 ∈ Ring)
55	21, 54	syl 17	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑊) ∧ 𝑧 ∈ (Base‘𝑊))) → 𝑅 ∈ Ring)
56		ringcmn 20098	. . . . . . . 8 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ CMnd)
57	55, 56	syl 17	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑊) ∧ 𝑧 ∈ (Base‘𝑊))) → 𝑅 ∈ CMnd)
58	5, 6, 11, 12	lflcl 37929	. . . . . . . 8 ⊢ ((𝑊 ∈ LMod ∧ 𝐺 ∈ 𝐹 ∧ (𝑥( ·𝑠 ‘𝑊)𝑦) ∈ (Base‘𝑊)) → (𝐺‘(𝑥( ·𝑠 ‘𝑊)𝑦)) ∈ (Base‘𝑅))
59	21, 47, 26, 58	syl3anc 1371	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑊) ∧ 𝑧 ∈ (Base‘𝑊))) → (𝐺‘(𝑥( ·𝑠 ‘𝑊)𝑦)) ∈ (Base‘𝑅))
60	5, 6, 11, 12	lflcl 37929	. . . . . . . 8 ⊢ ((𝑊 ∈ LMod ∧ 𝐺 ∈ 𝐹 ∧ 𝑧 ∈ (Base‘𝑊)) → (𝐺‘𝑧) ∈ (Base‘𝑅))
61	21, 47, 27, 60	syl3anc 1371	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑊) ∧ 𝑧 ∈ (Base‘𝑊))) → (𝐺‘𝑧) ∈ (Base‘𝑅))
62	5, 6, 11, 12	lflcl 37929	. . . . . . . 8 ⊢ ((𝑊 ∈ LMod ∧ 𝐻 ∈ 𝐹 ∧ (𝑥( ·𝑠 ‘𝑊)𝑦) ∈ (Base‘𝑊)) → (𝐻‘(𝑥( ·𝑠 ‘𝑊)𝑦)) ∈ (Base‘𝑅))
63	21, 50, 26, 62	syl3anc 1371	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑊) ∧ 𝑧 ∈ (Base‘𝑊))) → (𝐻‘(𝑥( ·𝑠 ‘𝑊)𝑦)) ∈ (Base‘𝑅))
64	5, 6, 11, 12	lflcl 37929	. . . . . . . 8 ⊢ ((𝑊 ∈ LMod ∧ 𝐻 ∈ 𝐹 ∧ 𝑧 ∈ (Base‘𝑊)) → (𝐻‘𝑧) ∈ (Base‘𝑅))
65	21, 50, 27, 64	syl3anc 1371	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑊) ∧ 𝑧 ∈ (Base‘𝑊))) → (𝐻‘𝑧) ∈ (Base‘𝑅))
66	6, 7	cmn4 19668	. . . . . . 7 ⊢ ((𝑅 ∈ CMnd ∧ ((𝐺‘(𝑥( ·𝑠 ‘𝑊)𝑦)) ∈ (Base‘𝑅) ∧ (𝐺‘𝑧) ∈ (Base‘𝑅)) ∧ ((𝐻‘(𝑥( ·𝑠 ‘𝑊)𝑦)) ∈ (Base‘𝑅) ∧ (𝐻‘𝑧) ∈ (Base‘𝑅))) → (((𝐺‘(𝑥( ·𝑠 ‘𝑊)𝑦)) + (𝐺‘𝑧)) + ((𝐻‘(𝑥( ·𝑠 ‘𝑊)𝑦)) + (𝐻‘𝑧))) = (((𝐺‘(𝑥( ·𝑠 ‘𝑊)𝑦)) + (𝐻‘(𝑥( ·𝑠 ‘𝑊)𝑦))) + ((𝐺‘𝑧) + (𝐻‘𝑧))))
67	57, 59, 61, 63, 65, 66	syl122anc 1379	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑊) ∧ 𝑧 ∈ (Base‘𝑊))) → (((𝐺‘(𝑥( ·𝑠 ‘𝑊)𝑦)) + (𝐺‘𝑧)) + ((𝐻‘(𝑥( ·𝑠 ‘𝑊)𝑦)) + (𝐻‘𝑧))) = (((𝐺‘(𝑥( ·𝑠 ‘𝑊)𝑦)) + (𝐻‘(𝑥( ·𝑠 ‘𝑊)𝑦))) + ((𝐺‘𝑧) + (𝐻‘𝑧))))
68		eqid 2732	. . . . . . . . . . 11 ⊢ (.r‘𝑅) = (.r‘𝑅)
69	5, 6, 68, 11, 24, 12	lflmul 37933	. . . . . . . . . 10 ⊢ ((𝑊 ∈ LMod ∧ 𝐺 ∈ 𝐹 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑊))) → (𝐺‘(𝑥( ·𝑠 ‘𝑊)𝑦)) = (𝑥(.r‘𝑅)(𝐺‘𝑦)))
70	21, 47, 22, 23, 69	syl112anc 1374	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑊) ∧ 𝑧 ∈ (Base‘𝑊))) → (𝐺‘(𝑥( ·𝑠 ‘𝑊)𝑦)) = (𝑥(.r‘𝑅)(𝐺‘𝑦)))
71	5, 6, 68, 11, 24, 12	lflmul 37933	. . . . . . . . . 10 ⊢ ((𝑊 ∈ LMod ∧ 𝐻 ∈ 𝐹 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑊))) → (𝐻‘(𝑥( ·𝑠 ‘𝑊)𝑦)) = (𝑥(.r‘𝑅)(𝐻‘𝑦)))
72	21, 50, 22, 23, 71	syl112anc 1374	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑊) ∧ 𝑧 ∈ (Base‘𝑊))) → (𝐻‘(𝑥( ·𝑠 ‘𝑊)𝑦)) = (𝑥(.r‘𝑅)(𝐻‘𝑦)))
73	70, 72	oveq12d 7426	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑊) ∧ 𝑧 ∈ (Base‘𝑊))) → ((𝐺‘(𝑥( ·𝑠 ‘𝑊)𝑦)) + (𝐻‘(𝑥( ·𝑠 ‘𝑊)𝑦))) = ((𝑥(.r‘𝑅)(𝐺‘𝑦)) + (𝑥(.r‘𝑅)(𝐻‘𝑦))))
74	5, 6, 11, 12	lflcl 37929	. . . . . . . . . 10 ⊢ ((𝑊 ∈ LMod ∧ 𝐺 ∈ 𝐹 ∧ 𝑦 ∈ (Base‘𝑊)) → (𝐺‘𝑦) ∈ (Base‘𝑅))
75	21, 47, 23, 74	syl3anc 1371	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑊) ∧ 𝑧 ∈ (Base‘𝑊))) → (𝐺‘𝑦) ∈ (Base‘𝑅))
76	5, 6, 11, 12	lflcl 37929	. . . . . . . . . 10 ⊢ ((𝑊 ∈ LMod ∧ 𝐻 ∈ 𝐹 ∧ 𝑦 ∈ (Base‘𝑊)) → (𝐻‘𝑦) ∈ (Base‘𝑅))
77	21, 50, 23, 76	syl3anc 1371	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑊) ∧ 𝑧 ∈ (Base‘𝑊))) → (𝐻‘𝑦) ∈ (Base‘𝑅))
78	6, 7, 68	ringdi 20080	. . . . . . . . 9 ⊢ ((𝑅 ∈ Ring ∧ (𝑥 ∈ (Base‘𝑅) ∧ (𝐺‘𝑦) ∈ (Base‘𝑅) ∧ (𝐻‘𝑦) ∈ (Base‘𝑅))) → (𝑥(.r‘𝑅)((𝐺‘𝑦) + (𝐻‘𝑦))) = ((𝑥(.r‘𝑅)(𝐺‘𝑦)) + (𝑥(.r‘𝑅)(𝐻‘𝑦))))
79	55, 22, 75, 77, 78	syl13anc 1372	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑊) ∧ 𝑧 ∈ (Base‘𝑊))) → (𝑥(.r‘𝑅)((𝐺‘𝑦) + (𝐻‘𝑦))) = ((𝑥(.r‘𝑅)(𝐺‘𝑦)) + (𝑥(.r‘𝑅)(𝐻‘𝑦))))
80	73, 79	eqtr4d 2775	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑊) ∧ 𝑧 ∈ (Base‘𝑊))) → ((𝐺‘(𝑥( ·𝑠 ‘𝑊)𝑦)) + (𝐻‘(𝑥( ·𝑠 ‘𝑊)𝑦))) = (𝑥(.r‘𝑅)((𝐺‘𝑦) + (𝐻‘𝑦))))
81	80	oveq1d 7423	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑊) ∧ 𝑧 ∈ (Base‘𝑊))) → (((𝐺‘(𝑥( ·𝑠 ‘𝑊)𝑦)) + (𝐻‘(𝑥( ·𝑠 ‘𝑊)𝑦))) + ((𝐺‘𝑧) + (𝐻‘𝑧))) = ((𝑥(.r‘𝑅)((𝐺‘𝑦) + (𝐻‘𝑦))) + ((𝐺‘𝑧) + (𝐻‘𝑧))))
82	53, 67, 81	3eqtrd 2776	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑊) ∧ 𝑧 ∈ (Base‘𝑊))) → ((𝐺‘((𝑥( ·𝑠 ‘𝑊)𝑦)(+g‘𝑊)𝑧)) + (𝐻‘((𝑥( ·𝑠 ‘𝑊)𝑦)(+g‘𝑊)𝑧))) = ((𝑥(.r‘𝑅)((𝐺‘𝑦) + (𝐻‘𝑦))) + ((𝐺‘𝑧) + (𝐻‘𝑧))))
83	46, 82	eqtr4d 2775	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑊) ∧ 𝑧 ∈ (Base‘𝑊))) → ((𝑥(.r‘𝑅)((𝐺 ∘f + 𝐻)‘𝑦)) + ((𝐺 ∘f + 𝐻)‘𝑧)) = ((𝐺‘((𝑥( ·𝑠 ‘𝑊)𝑦)(+g‘𝑊)𝑧)) + (𝐻‘((𝑥( ·𝑠 ‘𝑊)𝑦)(+g‘𝑊)𝑧))))
84	36, 83	eqtr4d 2775	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑊) ∧ 𝑧 ∈ (Base‘𝑊))) → ((𝐺 ∘f + 𝐻)‘((𝑥( ·𝑠 ‘𝑊)𝑦)(+g‘𝑊)𝑧)) = ((𝑥(.r‘𝑅)((𝐺 ∘f + 𝐻)‘𝑦)) + ((𝐺 ∘f + 𝐻)‘𝑧)))
85	84	ralrimivvva 3203	. 2 ⊢ (𝜑 → ∀𝑥 ∈ (Base‘𝑅)∀𝑦 ∈ (Base‘𝑊)∀𝑧 ∈ (Base‘𝑊)((𝐺 ∘f + 𝐻)‘((𝑥( ·𝑠 ‘𝑊)𝑦)(+g‘𝑊)𝑧)) = ((𝑥(.r‘𝑅)((𝐺 ∘f + 𝐻)‘𝑦)) + ((𝐺 ∘f + 𝐻)‘𝑧)))
86	11, 28, 5, 24, 6, 7, 68, 12	islfl 37925	. . 3 ⊢ (𝑊 ∈ LMod → ((𝐺 ∘f + 𝐻) ∈ 𝐹 ↔ ((𝐺 ∘f + 𝐻):(Base‘𝑊)⟶(Base‘𝑅) ∧ ∀𝑥 ∈ (Base‘𝑅)∀𝑦 ∈ (Base‘𝑊)∀𝑧 ∈ (Base‘𝑊)((𝐺 ∘f + 𝐻)‘((𝑥( ·𝑠 ‘𝑊)𝑦)(+g‘𝑊)𝑧)) = ((𝑥(.r‘𝑅)((𝐺 ∘f + 𝐻)‘𝑦)) + ((𝐺 ∘f + 𝐻)‘𝑧)))))
87	1, 86	syl 17	. 2 ⊢ (𝜑 → ((𝐺 ∘f + 𝐻) ∈ 𝐹 ↔ ((𝐺 ∘f + 𝐻):(Base‘𝑊)⟶(Base‘𝑅) ∧ ∀𝑥 ∈ (Base‘𝑅)∀𝑦 ∈ (Base‘𝑊)∀𝑧 ∈ (Base‘𝑊)((𝐺 ∘f + 𝐻)‘((𝑥( ·𝑠 ‘𝑊)𝑦)(+g‘𝑊)𝑧)) = ((𝑥(.r‘𝑅)((𝐺 ∘f + 𝐻)‘𝑦)) + ((𝐺 ∘f + 𝐻)‘𝑧)))))
88	20, 85, 87	mpbir2and 711	1 ⊢ (𝜑 → (𝐺 ∘f + 𝐻) ∈ 𝐹)

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106  ∀wral 3061  Vcvv 3474  ⟶wf 6539  ‘cfv 6543  (class class class)co 7408   ∘_f cof 7667  Basecbs 17143  +_gcplusg 17196  ._rcmulr 17197  Scalarcsca 17199   ·_𝑠 cvsca 17200  CMndccmn 19647  Ringcrg 20055  LModclmod 20470  LFnlclfn 37922

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 2703  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7724  ax-cnex 11165  ax-resscn 11166  ax-1cn 11167  ax-icn 11168  ax-addcl 11169  ax-addrcl 11170  ax-mulcl 11171  ax-mulrcl 11172  ax-mulcom 11173  ax-addass 11174  ax-mulass 11175  ax-distr 11176  ax-i2m1 11177  ax-1ne0 11178  ax-1rid 11179  ax-rnegex 11180  ax-rrecex 11181  ax-cnre 11182  ax-pre-lttri 11183  ax-pre-lttrn 11184  ax-pre-ltadd 11185  ax-pre-mulgt0 11186

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 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3376  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7364  df-ov 7411  df-oprab 7412  df-mpo 7413  df-of 7669  df-om 7855  df-1st 7974  df-2nd 7975  df-frecs 8265  df-wrecs 8296  df-recs 8370  df-rdg 8409  df-er 8702  df-map 8821  df-en 8939  df-dom 8940  df-sdom 8941  df-pnf 11249  df-mnf 11250  df-xr 11251  df-ltxr 11252  df-le 11253  df-sub 11445  df-neg 11446  df-nn 12212  df-2 12274  df-sets 17096  df-slot 17114  df-ndx 17126  df-base 17144  df-plusg 17209  df-0g 17386  df-mgm 18560  df-sgrp 18609  df-mnd 18625  df-grp 18821  df-minusg 18822  df-sbg 18823  df-cmn 19649  df-abl 19650  df-mgp 19987  df-ur 20004  df-ring 20057  df-lmod 20472  df-lfl 37923

This theorem is referenced by:  ldualvaddcl  37995