Theorem lfladdcl 39060

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 480	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅))) → 𝑊 ∈ LMod)
3		simprl 770	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅))) → 𝑥 ∈ (Base‘𝑅))
4		simprr 772	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅))) → 𝑦 ∈ (Base‘𝑅))
5		lfladdcl.r	. . . . 5 ⊢ 𝑅 = (Scalar‘𝑊)
6		eqid 2729	. . . . 5 ⊢ (Base‘𝑅) = (Base‘𝑅)
7		lfladdcl.p	. . . . 5 ⊢ + = (+g‘𝑅)
8	5, 6, 7	lmodacl 20775	. . . 4 ⊢ ((𝑊 ∈ LMod ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) → (𝑥 + 𝑦) ∈ (Base‘𝑅))
9	2, 3, 4, 8	syl3anc 1373	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅))) → (𝑥 + 𝑦) ∈ (Base‘𝑅))
10		lfladdcl.g	. . . 4 ⊢ (𝜑 → 𝐺 ∈ 𝐹)
11		eqid 2729	. . . . 5 ⊢ (Base‘𝑊) = (Base‘𝑊)
12		lfladdcl.f	. . . . 5 ⊢ 𝐹 = (LFnl‘𝑊)
13	5, 6, 11, 12	lflf 39052	. . . 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 39052	. . . 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 6837	. . 3 ⊢ (𝜑 → (Base‘𝑊) ∈ V)
19		inidm 4178	. . 3 ⊢ ((Base‘𝑊) ∩ (Base‘𝑊)) = (Base‘𝑊)
20	9, 14, 17, 18, 18, 19	off 7631	. 2 ⊢ (𝜑 → (𝐺 ∘f + 𝐻):(Base‘𝑊)⟶(Base‘𝑅))
21	1	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑊) ∧ 𝑧 ∈ (Base‘𝑊))) → 𝑊 ∈ LMod)
22		simpr1 1195	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑊) ∧ 𝑧 ∈ (Base‘𝑊))) → 𝑥 ∈ (Base‘𝑅))
23		simpr2 1196	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑊) ∧ 𝑧 ∈ (Base‘𝑊))) → 𝑦 ∈ (Base‘𝑊))
24		eqid 2729	. . . . . . . 8 ⊢ ( ·𝑠 ‘𝑊) = ( ·𝑠 ‘𝑊)
25	11, 5, 24, 6	lmodvscl 20781	. . . . . . 7 ⊢ ((𝑊 ∈ LMod ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑊)) → (𝑥( ·𝑠 ‘𝑊)𝑦) ∈ (Base‘𝑊))
26	21, 22, 23, 25	syl3anc 1373	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑊) ∧ 𝑧 ∈ (Base‘𝑊))) → (𝑥( ·𝑠 ‘𝑊)𝑦) ∈ (Base‘𝑊))
27		simpr3 1197	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑊) ∧ 𝑧 ∈ (Base‘𝑊))) → 𝑧 ∈ (Base‘𝑊))
28		eqid 2729	. . . . . . 7 ⊢ (+g‘𝑊) = (+g‘𝑊)
29	11, 28	lmodvacl 20778	. . . . . 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 1373	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑊) ∧ 𝑧 ∈ (Base‘𝑊))) → ((𝑥( ·𝑠 ‘𝑊)𝑦)(+g‘𝑊)𝑧) ∈ (Base‘𝑊))
31	14	ffnd 6653	. . . . . 6 ⊢ (𝜑 → 𝐺 Fn (Base‘𝑊))
32	17	ffnd 6653	. . . . . 6 ⊢ (𝜑 → 𝐻 Fn (Base‘𝑊))
33		eqidd 2730	. . . . . 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 2730	. . . . . 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 7624	. . . . 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 2730	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ (Base‘𝑊)) → (𝐺‘𝑦) = (𝐺‘𝑦))
38		eqidd 2730	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ (Base‘𝑊)) → (𝐻‘𝑦) = (𝐻‘𝑦))
39	31, 32, 18, 18, 19, 37, 38	ofval 7624	. . . . . . . 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 7365	. . . . . 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 2730	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧 ∈ (Base‘𝑊)) → (𝐺‘𝑧) = (𝐺‘𝑧))
43		eqidd 2730	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧 ∈ (Base‘𝑊)) → (𝐻‘𝑧) = (𝐻‘𝑧))
44	31, 32, 18, 18, 19, 42, 43	ofval 7624	. . . . . . 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 7367	. . . . 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 480	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑊) ∧ 𝑧 ∈ (Base‘𝑊))) → 𝐺 ∈ 𝐹)
48	5, 7, 11, 28, 12	lfladd 39055	. . . . . . . 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 1376	. . . . . . 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 480	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑊) ∧ 𝑧 ∈ (Base‘𝑊))) → 𝐻 ∈ 𝐹)
51	5, 7, 11, 28, 12	lfladd 39055	. . . . . . . 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 1376	. . . . . . 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 7367	. . . . . 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 20771	. . . . . . . . 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 20167	. . . . . . . 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 39053	. . . . . . . 8 ⊢ ((𝑊 ∈ LMod ∧ 𝐺 ∈ 𝐹 ∧ (𝑥( ·𝑠 ‘𝑊)𝑦) ∈ (Base‘𝑊)) → (𝐺‘(𝑥( ·𝑠 ‘𝑊)𝑦)) ∈ (Base‘𝑅))
59	21, 47, 26, 58	syl3anc 1373	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑊) ∧ 𝑧 ∈ (Base‘𝑊))) → (𝐺‘(𝑥( ·𝑠 ‘𝑊)𝑦)) ∈ (Base‘𝑅))
60	5, 6, 11, 12	lflcl 39053	. . . . . . . 8 ⊢ ((𝑊 ∈ LMod ∧ 𝐺 ∈ 𝐹 ∧ 𝑧 ∈ (Base‘𝑊)) → (𝐺‘𝑧) ∈ (Base‘𝑅))
61	21, 47, 27, 60	syl3anc 1373	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑊) ∧ 𝑧 ∈ (Base‘𝑊))) → (𝐺‘𝑧) ∈ (Base‘𝑅))
62	5, 6, 11, 12	lflcl 39053	. . . . . . . 8 ⊢ ((𝑊 ∈ LMod ∧ 𝐻 ∈ 𝐹 ∧ (𝑥( ·𝑠 ‘𝑊)𝑦) ∈ (Base‘𝑊)) → (𝐻‘(𝑥( ·𝑠 ‘𝑊)𝑦)) ∈ (Base‘𝑅))
63	21, 50, 26, 62	syl3anc 1373	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑊) ∧ 𝑧 ∈ (Base‘𝑊))) → (𝐻‘(𝑥( ·𝑠 ‘𝑊)𝑦)) ∈ (Base‘𝑅))
64	5, 6, 11, 12	lflcl 39053	. . . . . . . 8 ⊢ ((𝑊 ∈ LMod ∧ 𝐻 ∈ 𝐹 ∧ 𝑧 ∈ (Base‘𝑊)) → (𝐻‘𝑧) ∈ (Base‘𝑅))
65	21, 50, 27, 64	syl3anc 1373	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑊) ∧ 𝑧 ∈ (Base‘𝑊))) → (𝐻‘𝑧) ∈ (Base‘𝑅))
66	6, 7	cmn4 19680	. . . . . . 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 1381	. . . . . 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 2729	. . . . . . . . . . 11 ⊢ (.r‘𝑅) = (.r‘𝑅)
69	5, 6, 68, 11, 24, 12	lflmul 39057	. . . . . . . . . 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 1376	. . . . . . . . 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 39057	. . . . . . . . . 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 1376	. . . . . . . . 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 7367	. . . . . . . 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 39053	. . . . . . . . . 10 ⊢ ((𝑊 ∈ LMod ∧ 𝐺 ∈ 𝐹 ∧ 𝑦 ∈ (Base‘𝑊)) → (𝐺‘𝑦) ∈ (Base‘𝑅))
75	21, 47, 23, 74	syl3anc 1373	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑊) ∧ 𝑧 ∈ (Base‘𝑊))) → (𝐺‘𝑦) ∈ (Base‘𝑅))
76	5, 6, 11, 12	lflcl 39053	. . . . . . . . . 10 ⊢ ((𝑊 ∈ LMod ∧ 𝐻 ∈ 𝐹 ∧ 𝑦 ∈ (Base‘𝑊)) → (𝐻‘𝑦) ∈ (Base‘𝑅))
77	21, 50, 23, 76	syl3anc 1373	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑊) ∧ 𝑧 ∈ (Base‘𝑊))) → (𝐻‘𝑦) ∈ (Base‘𝑅))
78	6, 7, 68	ringdi 20146	. . . . . . . . 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 1374	. . . . . . . 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 2767	. . . . . . 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 7364	. . . . . 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 2768	. . . . 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 2767	. . . 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 2767	. . 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 3175	. 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 39049	. . 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 713	1 ⊢ (𝜑 → (𝐺 ∘f + 𝐻) ∈ 𝐹)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109  ∀wral 3044  Vcvv 3436  ⟶wf 6478  ‘cfv 6482  (class class class)co 7349   ∘_f cof 7611  Basecbs 17120  +_gcplusg 17161  ._rcmulr 17162  Scalarcsca 17164   ·_𝑠 cvsca 17165  CMndccmn 19659  Ringcrg 20118  LModclmod 20763  LFnlclfn 39046

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-grp 18815  df-minusg 18816  df-sbg 18817  df-cmn 19661  df-abl 19662  df-mgp 20026  df-ur 20067  df-ring 20120  df-lmod 20765  df-lfl 39047

This theorem is referenced by:  ldualvaddcl  39119