Theorem lfladdcl 40128

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 486	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅))) → 𝑊 ∈ LMod)
3		simprl 783	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅))) → 𝑥 ∈ (Base‘𝑅))
4		simprr 785	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅))) → 𝑦 ∈ (Base‘𝑅))
5		lfladdcl.r	. . . . 5 ⊢ 𝑅 = (Scalar‘𝑊)
6		eqid 2761	. . . . 5 ⊢ (Base‘𝑅) = (Base‘𝑅)
7		lfladdcl.p	. . . . 5 ⊢ + = (+g‘𝑅)
8	5, 6, 7	lmodacl 21147	. . . 4 ⊢ ((𝑊 ∈ LMod ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) → (𝑥 + 𝑦) ∈ (Base‘𝑅))
9	2, 3, 4, 8	syl3anc 1398	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅))) → (𝑥 + 𝑦) ∈ (Base‘𝑅))
10		lfladdcl.g	. . . 4 ⊢ (𝜑 → 𝐺 ∈ 𝐹)
11		eqid 2761	. . . . 5 ⊢ (Base‘𝑊) = (Base‘𝑊)
12		lfladdcl.f	. . . . 5 ⊢ 𝐹 = (LFnl‘𝑊)
13	5, 6, 11, 12	lflf 40120	. . . 4 ⊢ ((𝑊 ∈ LMod ∧ 𝐺 ∈ 𝐹) → 𝐺:(Base‘𝑊)⟶(Base‘𝑅))
14	1, 10, 13	syl2anc 596	. . 3 ⊢ (𝜑 → 𝐺:(Base‘𝑊)⟶(Base‘𝑅))
15		lfladdcl.h	. . . 4 ⊢ (𝜑 → 𝐻 ∈ 𝐹)
16	5, 6, 11, 12	lflf 40120	. . . 4 ⊢ ((𝑊 ∈ LMod ∧ 𝐻 ∈ 𝐹) → 𝐻:(Base‘𝑊)⟶(Base‘𝑅))
17	1, 15, 16	syl2anc 596	. . 3 ⊢ (𝜑 → 𝐻:(Base‘𝑊)⟶(Base‘𝑅))
18		fvexd 6900	. . 3 ⊢ (𝜑 → (Base‘𝑊) ∈ V)
19		inidm 4172	. . 3 ⊢ ((Base‘𝑊) ∩ (Base‘𝑊)) = (Base‘𝑊)
20	9, 14, 17, 18, 18, 19	off 7711	. 2 ⊢ (𝜑 → (𝐺 ∘f + 𝐻):(Base‘𝑊)⟶(Base‘𝑅))
21	1	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑊) ∧ 𝑧 ∈ (Base‘𝑊))) → 𝑊 ∈ LMod)
22		simpr1 1213	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑊) ∧ 𝑧 ∈ (Base‘𝑊))) → 𝑥 ∈ (Base‘𝑅))
23		simpr2 1214	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑊) ∧ 𝑧 ∈ (Base‘𝑊))) → 𝑦 ∈ (Base‘𝑊))
24		eqid 2761	. . . . . . . 8 ⊢ ( ·𝑠 ‘𝑊) = ( ·𝑠 ‘𝑊)
25	11, 5, 24, 6	lmodvscl 21153	. . . . . . 7 ⊢ ((𝑊 ∈ LMod ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑊)) → (𝑥( ·𝑠 ‘𝑊)𝑦) ∈ (Base‘𝑊))
26	21, 22, 23, 25	syl3anc 1398	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑊) ∧ 𝑧 ∈ (Base‘𝑊))) → (𝑥( ·𝑠 ‘𝑊)𝑦) ∈ (Base‘𝑊))
27		simpr3 1215	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑊) ∧ 𝑧 ∈ (Base‘𝑊))) → 𝑧 ∈ (Base‘𝑊))
28		eqid 2761	. . . . . . 7 ⊢ (+g‘𝑊) = (+g‘𝑊)
29	11, 28	lmodvacl 21150	. . . . . 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 1398	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑊) ∧ 𝑧 ∈ (Base‘𝑊))) → ((𝑥( ·𝑠 ‘𝑊)𝑦)(+g‘𝑊)𝑧) ∈ (Base‘𝑊))
31	14	ffnd 6710	. . . . . 6 ⊢ (𝜑 → 𝐺 Fn (Base‘𝑊))
32	17	ffnd 6710	. . . . . 6 ⊢ (𝜑 → 𝐻 Fn (Base‘𝑊))
33		eqidd 2762	. . . . . 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 2762	. . . . . 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 7704	. . . . 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 603	. . . 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 2762	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ (Base‘𝑊)) → (𝐺‘𝑦) = (𝐺‘𝑦))
38		eqidd 2762	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ (Base‘𝑊)) → (𝐻‘𝑦) = (𝐻‘𝑦))
39	31, 32, 18, 18, 19, 37, 38	ofval 7704	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ (Base‘𝑊)) → ((𝐺 ∘f + 𝐻)‘𝑦) = ((𝐺‘𝑦) + (𝐻‘𝑦)))
40	23, 39	syldan 603	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑊) ∧ 𝑧 ∈ (Base‘𝑊))) → ((𝐺 ∘f + 𝐻)‘𝑦) = ((𝐺‘𝑦) + (𝐻‘𝑦)))
41	40	oveq2d 7436	. . . . . 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 2762	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧 ∈ (Base‘𝑊)) → (𝐺‘𝑧) = (𝐺‘𝑧))
43		eqidd 2762	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧 ∈ (Base‘𝑊)) → (𝐻‘𝑧) = (𝐻‘𝑧))
44	31, 32, 18, 18, 19, 42, 43	ofval 7704	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ (Base‘𝑊)) → ((𝐺 ∘f + 𝐻)‘𝑧) = ((𝐺‘𝑧) + (𝐻‘𝑧)))
45	27, 44	syldan 603	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑊) ∧ 𝑧 ∈ (Base‘𝑊))) → ((𝐺 ∘f + 𝐻)‘𝑧) = ((𝐺‘𝑧) + (𝐻‘𝑧)))
46	41, 45	oveq12d 7438	. . . . 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 486	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑊) ∧ 𝑧 ∈ (Base‘𝑊))) → 𝐺 ∈ 𝐹)
48	5, 7, 11, 28, 12	lfladd 40123	. . . . . . . 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 1401	. . . . . . 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 486	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑊) ∧ 𝑧 ∈ (Base‘𝑊))) → 𝐻 ∈ 𝐹)
51	5, 7, 11, 28, 12	lfladd 40123	. . . . . . . 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 1401	. . . . . . 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 7438	. . . . . 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 21143	. . . . . . . . 9 ⊢ (𝑊 ∈ LMod → 𝑅 ∈ Ring)
55	21, 54	syl 18	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑊) ∧ 𝑧 ∈ (Base‘𝑊))) → 𝑅 ∈ Ring)
56		ringcmn 20511	. . . . . . . 8 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ CMnd)
57	55, 56	syl 18	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑊) ∧ 𝑧 ∈ (Base‘𝑊))) → 𝑅 ∈ CMnd)
58	5, 6, 11, 12	lflcl 40121	. . . . . . . 8 ⊢ ((𝑊 ∈ LMod ∧ 𝐺 ∈ 𝐹 ∧ (𝑥( ·𝑠 ‘𝑊)𝑦) ∈ (Base‘𝑊)) → (𝐺‘(𝑥( ·𝑠 ‘𝑊)𝑦)) ∈ (Base‘𝑅))
59	21, 47, 26, 58	syl3anc 1398	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑊) ∧ 𝑧 ∈ (Base‘𝑊))) → (𝐺‘(𝑥( ·𝑠 ‘𝑊)𝑦)) ∈ (Base‘𝑅))
60	5, 6, 11, 12	lflcl 40121	. . . . . . . 8 ⊢ ((𝑊 ∈ LMod ∧ 𝐺 ∈ 𝐹 ∧ 𝑧 ∈ (Base‘𝑊)) → (𝐺‘𝑧) ∈ (Base‘𝑅))
61	21, 47, 27, 60	syl3anc 1398	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑊) ∧ 𝑧 ∈ (Base‘𝑊))) → (𝐺‘𝑧) ∈ (Base‘𝑅))
62	5, 6, 11, 12	lflcl 40121	. . . . . . . 8 ⊢ ((𝑊 ∈ LMod ∧ 𝐻 ∈ 𝐹 ∧ (𝑥( ·𝑠 ‘𝑊)𝑦) ∈ (Base‘𝑊)) → (𝐻‘(𝑥( ·𝑠 ‘𝑊)𝑦)) ∈ (Base‘𝑅))
63	21, 50, 26, 62	syl3anc 1398	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑊) ∧ 𝑧 ∈ (Base‘𝑊))) → (𝐻‘(𝑥( ·𝑠 ‘𝑊)𝑦)) ∈ (Base‘𝑅))
64	5, 6, 11, 12	lflcl 40121	. . . . . . . 8 ⊢ ((𝑊 ∈ LMod ∧ 𝐻 ∈ 𝐹 ∧ 𝑧 ∈ (Base‘𝑊)) → (𝐻‘𝑧) ∈ (Base‘𝑅))
65	21, 50, 27, 64	syl3anc 1398	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑊) ∧ 𝑧 ∈ (Base‘𝑊))) → (𝐻‘𝑧) ∈ (Base‘𝑅))
66	6, 7	cmn4 20015	. . . . . . 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 1406	. . . . . 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 2761	. . . . . . . . . . 11 ⊢ (.r‘𝑅) = (.r‘𝑅)
69	5, 6, 68, 11, 24, 12	lflmul 40125	. . . . . . . . . 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 1401	. . . . . . . . 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 40125	. . . . . . . . . 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 1401	. . . . . . . . 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 7438	. . . . . . . 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 40121	. . . . . . . . . 10 ⊢ ((𝑊 ∈ LMod ∧ 𝐺 ∈ 𝐹 ∧ 𝑦 ∈ (Base‘𝑊)) → (𝐺‘𝑦) ∈ (Base‘𝑅))
75	21, 47, 23, 74	syl3anc 1398	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑊) ∧ 𝑧 ∈ (Base‘𝑊))) → (𝐺‘𝑦) ∈ (Base‘𝑅))
76	5, 6, 11, 12	lflcl 40121	. . . . . . . . . 10 ⊢ ((𝑊 ∈ LMod ∧ 𝐻 ∈ 𝐹 ∧ 𝑦 ∈ (Base‘𝑊)) → (𝐻‘𝑦) ∈ (Base‘𝑅))
77	21, 50, 23, 76	syl3anc 1398	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑊) ∧ 𝑧 ∈ (Base‘𝑊))) → (𝐻‘𝑦) ∈ (Base‘𝑅))
78	6, 7, 68	ringdi 20489	. . . . . . . . 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 1399	. . . . . . . 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 2799	. . . . . . 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 7435	. . . . . 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 2800	. . . . 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 2799	. . . 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 2799	. . 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 3209	. 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 40117	. . 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 18	. 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 726	1 ⊢ (𝜑 → (𝐺 ∘f + 𝐻) ∈ 𝐹)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145  ∀wral 3077  Vcvv 3451  ⟶wf 6534  ‘cfv 6538  (class class class)co 7420   ∘_f cof 7691  Basecbs 17387  +_gcplusg 17428  ._rcmulr 17429  Scalarcsca 17431   ·_𝑠 cvsca 17432  CMndccmn 19994  Ringcrg 20459  LModclmod 21135  LFnlclfn 40114

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-rep 5232  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7751  ax-cnex 11256  ax-resscn 11257  ax-1cn 11258  ax-icn 11259  ax-addcl 11260  ax-addrcl 11261  ax-mulcl 11262  ax-mulrcl 11263  ax-mulcom 11264  ax-addass 11265  ax-mulass 11266  ax-distr 11267  ax-i2m1 11268  ax-1ne0 11269  ax-1rid 11270  ax-rnegex 11271  ax-rrecex 11272  ax-cnre 11273  ax-pre-lttri 11274  ax-pre-lttrn 11275  ax-pre-ltadd 11276  ax-pre-mulgt0 11277

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-nel 3063  df-ral 3078  df-rex 3088  df-rmo 3366  df-reu 3367  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5546  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-we 5606  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-pred 6304  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-riota 7377  df-ov 7423  df-oprab 7424  df-mpo 7425  df-of 7693  df-om 7878  df-1st 8001  df-2nd 8002  df-frecs 8299  df-wrecs 8330  df-recs 8379  df-rdg 8418  df-er 8717  df-map 8849  df-en 8974  df-dom 8975  df-sdom 8976  df-pnf 11345  df-mnf 11346  df-xr 11347  df-ltxr 11348  df-le 11349  df-sub 11543  df-neg 11544  df-nn 12336  df-2 12405  df-sets 17342  df-slot 17360  df-ndx 17372  df-base 17388  df-plusg 17441  df-0g 17612  df-mgm 18816  df-sgrp 18908  df-mnd 18924  df-grp 19147  df-minusg 19148  df-sbg 19149  df-cmn 19996  df-abl 19997  df-mgp 20361  df-ur 20408  df-ring 20461  df-lmod 21137  df-lfl 40115

This theorem is used by:  ldualvaddcl  40187