Theorem pj1lmhm 18921

Description: The left projection function is a linear operator. (Contributed by Mario Carneiro, 15-Oct-2015.) (Revised by Mario Carneiro, 21-Apr-2016.)

Hypotheses

Ref	Expression
pj1lmhm.l	⊢ 𝐿 = (LSubSp‘𝑊)
pj1lmhm.s	⊢ ⊕ = (LSSum‘𝑊)
pj1lmhm.z	⊢ 0 = (0g‘𝑊)
pj1lmhm.p	⊢ 𝑃 = (proj1‘𝑊)
pj1lmhm.1	⊢ (𝜑 → 𝑊 ∈ LMod)
pj1lmhm.2	⊢ (𝜑 → 𝑇 ∈ 𝐿)
pj1lmhm.3	⊢ (𝜑 → 𝑈 ∈ 𝐿)
pj1lmhm.4	⊢ (𝜑 → (𝑇 ∩ 𝑈) = { 0 })

Assertion

Ref	Expression
pj1lmhm	⊢ (𝜑 → (𝑇𝑃𝑈) ∈ ((𝑊 ↾s (𝑇 ⊕ 𝑈)) LMHom 𝑊))

Proof of Theorem pj1lmhm

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

Step	Hyp	Ref	Expression
1		eqid 2610	. . 3 ⊢ (+g‘𝑊) = (+g‘𝑊)
2		pj1lmhm.s	. . 3 ⊢ ⊕ = (LSSum‘𝑊)
3		pj1lmhm.z	. . 3 ⊢ 0 = (0g‘𝑊)
4		eqid 2610	. . 3 ⊢ (Cntz‘𝑊) = (Cntz‘𝑊)
5		pj1lmhm.1	. . . . 5 ⊢ (𝜑 → 𝑊 ∈ LMod)
6		pj1lmhm.l	. . . . . 6 ⊢ 𝐿 = (LSubSp‘𝑊)
7	6	lsssssubg 18779	. . . . 5 ⊢ (𝑊 ∈ LMod → 𝐿 ⊆ (SubGrp‘𝑊))
8	5, 7	syl 17	. . . 4 ⊢ (𝜑 → 𝐿 ⊆ (SubGrp‘𝑊))
9		pj1lmhm.2	. . . 4 ⊢ (𝜑 → 𝑇 ∈ 𝐿)
10	8, 9	sseldd 3569	. . 3 ⊢ (𝜑 → 𝑇 ∈ (SubGrp‘𝑊))
11		pj1lmhm.3	. . . 4 ⊢ (𝜑 → 𝑈 ∈ 𝐿)
12	8, 11	sseldd 3569	. . 3 ⊢ (𝜑 → 𝑈 ∈ (SubGrp‘𝑊))
13		pj1lmhm.4	. . 3 ⊢ (𝜑 → (𝑇 ∩ 𝑈) = { 0 })
14		lmodabl 18733	. . . . 5 ⊢ (𝑊 ∈ LMod → 𝑊 ∈ Abel)
15	5, 14	syl 17	. . . 4 ⊢ (𝜑 → 𝑊 ∈ Abel)
16	4, 15, 10, 12	ablcntzd 18083	. . 3 ⊢ (𝜑 → 𝑇 ⊆ ((Cntz‘𝑊)‘𝑈))
17		pj1lmhm.p	. . 3 ⊢ 𝑃 = (proj1‘𝑊)
18	1, 2, 3, 4, 10, 12, 13, 16, 17	pj1ghm 17939	. 2 ⊢ (𝜑 → (𝑇𝑃𝑈) ∈ ((𝑊 ↾s (𝑇 ⊕ 𝑈)) GrpHom 𝑊))
19		eqid 2610	. . 3 ⊢ (Scalar‘𝑊) = (Scalar‘𝑊)
20	19	a1i 11	. 2 ⊢ (𝜑 → (Scalar‘𝑊) = (Scalar‘𝑊))
21	1, 2, 3, 4, 10, 12, 13, 16, 17	pj1id 17935	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑇 ⊕ 𝑈)) → 𝑦 = (((𝑇𝑃𝑈)‘𝑦)(+g‘𝑊)((𝑈𝑃𝑇)‘𝑦)))
22	21	adantrl 748	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑦 ∈ (𝑇 ⊕ 𝑈))) → 𝑦 = (((𝑇𝑃𝑈)‘𝑦)(+g‘𝑊)((𝑈𝑃𝑇)‘𝑦)))
23	22	oveq2d 6565	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑦 ∈ (𝑇 ⊕ 𝑈))) → (𝑥( ·𝑠 ‘𝑊)𝑦) = (𝑥( ·𝑠 ‘𝑊)(((𝑇𝑃𝑈)‘𝑦)(+g‘𝑊)((𝑈𝑃𝑇)‘𝑦))))
24	5	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑦 ∈ (𝑇 ⊕ 𝑈))) → 𝑊 ∈ LMod)
25		simprl 790	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑦 ∈ (𝑇 ⊕ 𝑈))) → 𝑥 ∈ (Base‘(Scalar‘𝑊)))
26	9	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑦 ∈ (𝑇 ⊕ 𝑈))) → 𝑇 ∈ 𝐿)
27		eqid 2610	. . . . . . . . . . 11 ⊢ (Base‘𝑊) = (Base‘𝑊)
28	27, 6	lssss 18758	. . . . . . . . . 10 ⊢ (𝑇 ∈ 𝐿 → 𝑇 ⊆ (Base‘𝑊))
29	26, 28	syl 17	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑦 ∈ (𝑇 ⊕ 𝑈))) → 𝑇 ⊆ (Base‘𝑊))
30	10	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑦 ∈ (𝑇 ⊕ 𝑈))) → 𝑇 ∈ (SubGrp‘𝑊))
31	12	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑦 ∈ (𝑇 ⊕ 𝑈))) → 𝑈 ∈ (SubGrp‘𝑊))
32	13	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑦 ∈ (𝑇 ⊕ 𝑈))) → (𝑇 ∩ 𝑈) = { 0 })
33	16	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑦 ∈ (𝑇 ⊕ 𝑈))) → 𝑇 ⊆ ((Cntz‘𝑊)‘𝑈))
34	1, 2, 3, 4, 30, 31, 32, 33, 17	pj1f 17933	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑦 ∈ (𝑇 ⊕ 𝑈))) → (𝑇𝑃𝑈):(𝑇 ⊕ 𝑈)⟶𝑇)
35		simprr 792	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑦 ∈ (𝑇 ⊕ 𝑈))) → 𝑦 ∈ (𝑇 ⊕ 𝑈))
36	34, 35	ffvelrnd 6268	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑦 ∈ (𝑇 ⊕ 𝑈))) → ((𝑇𝑃𝑈)‘𝑦) ∈ 𝑇)
37	29, 36	sseldd 3569	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑦 ∈ (𝑇 ⊕ 𝑈))) → ((𝑇𝑃𝑈)‘𝑦) ∈ (Base‘𝑊))
38	11	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑦 ∈ (𝑇 ⊕ 𝑈))) → 𝑈 ∈ 𝐿)
39	27, 6	lssss 18758	. . . . . . . . . 10 ⊢ (𝑈 ∈ 𝐿 → 𝑈 ⊆ (Base‘𝑊))
40	38, 39	syl 17	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑦 ∈ (𝑇 ⊕ 𝑈))) → 𝑈 ⊆ (Base‘𝑊))
41	1, 2, 3, 4, 30, 31, 32, 33, 17	pj2f 17934	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑦 ∈ (𝑇 ⊕ 𝑈))) → (𝑈𝑃𝑇):(𝑇 ⊕ 𝑈)⟶𝑈)
42	41, 35	ffvelrnd 6268	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑦 ∈ (𝑇 ⊕ 𝑈))) → ((𝑈𝑃𝑇)‘𝑦) ∈ 𝑈)
43	40, 42	sseldd 3569	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑦 ∈ (𝑇 ⊕ 𝑈))) → ((𝑈𝑃𝑇)‘𝑦) ∈ (Base‘𝑊))
44		eqid 2610	. . . . . . . . 9 ⊢ ( ·𝑠 ‘𝑊) = ( ·𝑠 ‘𝑊)
45		eqid 2610	. . . . . . . . 9 ⊢ (Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊))
46	27, 1, 19, 44, 45	lmodvsdi 18709	. . . . . . . 8 ⊢ ((𝑊 ∈ LMod ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ ((𝑇𝑃𝑈)‘𝑦) ∈ (Base‘𝑊) ∧ ((𝑈𝑃𝑇)‘𝑦) ∈ (Base‘𝑊))) → (𝑥( ·𝑠 ‘𝑊)(((𝑇𝑃𝑈)‘𝑦)(+g‘𝑊)((𝑈𝑃𝑇)‘𝑦))) = ((𝑥( ·𝑠 ‘𝑊)((𝑇𝑃𝑈)‘𝑦))(+g‘𝑊)(𝑥( ·𝑠 ‘𝑊)((𝑈𝑃𝑇)‘𝑦))))
47	24, 25, 37, 43, 46	syl13anc 1320	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑦 ∈ (𝑇 ⊕ 𝑈))) → (𝑥( ·𝑠 ‘𝑊)(((𝑇𝑃𝑈)‘𝑦)(+g‘𝑊)((𝑈𝑃𝑇)‘𝑦))) = ((𝑥( ·𝑠 ‘𝑊)((𝑇𝑃𝑈)‘𝑦))(+g‘𝑊)(𝑥( ·𝑠 ‘𝑊)((𝑈𝑃𝑇)‘𝑦))))
48	23, 47	eqtrd 2644	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑦 ∈ (𝑇 ⊕ 𝑈))) → (𝑥( ·𝑠 ‘𝑊)𝑦) = ((𝑥( ·𝑠 ‘𝑊)((𝑇𝑃𝑈)‘𝑦))(+g‘𝑊)(𝑥( ·𝑠 ‘𝑊)((𝑈𝑃𝑇)‘𝑦))))
49	6, 2	lsmcl 18904	. . . . . . . . . 10 ⊢ ((𝑊 ∈ LMod ∧ 𝑇 ∈ 𝐿 ∧ 𝑈 ∈ 𝐿) → (𝑇 ⊕ 𝑈) ∈ 𝐿)
50	5, 9, 11, 49	syl3anc 1318	. . . . . . . . 9 ⊢ (𝜑 → (𝑇 ⊕ 𝑈) ∈ 𝐿)
51	50	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑦 ∈ (𝑇 ⊕ 𝑈))) → (𝑇 ⊕ 𝑈) ∈ 𝐿)
52	19, 44, 45, 6	lssvscl 18776	. . . . . . . 8 ⊢ (((𝑊 ∈ LMod ∧ (𝑇 ⊕ 𝑈) ∈ 𝐿) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑦 ∈ (𝑇 ⊕ 𝑈))) → (𝑥( ·𝑠 ‘𝑊)𝑦) ∈ (𝑇 ⊕ 𝑈))
53	24, 51, 25, 35, 52	syl22anc 1319	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑦 ∈ (𝑇 ⊕ 𝑈))) → (𝑥( ·𝑠 ‘𝑊)𝑦) ∈ (𝑇 ⊕ 𝑈))
54	19, 44, 45, 6	lssvscl 18776	. . . . . . . 8 ⊢ (((𝑊 ∈ LMod ∧ 𝑇 ∈ 𝐿) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ ((𝑇𝑃𝑈)‘𝑦) ∈ 𝑇)) → (𝑥( ·𝑠 ‘𝑊)((𝑇𝑃𝑈)‘𝑦)) ∈ 𝑇)
55	24, 26, 25, 36, 54	syl22anc 1319	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑦 ∈ (𝑇 ⊕ 𝑈))) → (𝑥( ·𝑠 ‘𝑊)((𝑇𝑃𝑈)‘𝑦)) ∈ 𝑇)
56	19, 44, 45, 6	lssvscl 18776	. . . . . . . 8 ⊢ (((𝑊 ∈ LMod ∧ 𝑈 ∈ 𝐿) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ ((𝑈𝑃𝑇)‘𝑦) ∈ 𝑈)) → (𝑥( ·𝑠 ‘𝑊)((𝑈𝑃𝑇)‘𝑦)) ∈ 𝑈)
57	24, 38, 25, 42, 56	syl22anc 1319	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑦 ∈ (𝑇 ⊕ 𝑈))) → (𝑥( ·𝑠 ‘𝑊)((𝑈𝑃𝑇)‘𝑦)) ∈ 𝑈)
58	1, 2, 3, 4, 30, 31, 32, 33, 17, 53, 55, 57	pj1eq 17936	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑦 ∈ (𝑇 ⊕ 𝑈))) → ((𝑥( ·𝑠 ‘𝑊)𝑦) = ((𝑥( ·𝑠 ‘𝑊)((𝑇𝑃𝑈)‘𝑦))(+g‘𝑊)(𝑥( ·𝑠 ‘𝑊)((𝑈𝑃𝑇)‘𝑦))) ↔ (((𝑇𝑃𝑈)‘(𝑥( ·𝑠 ‘𝑊)𝑦)) = (𝑥( ·𝑠 ‘𝑊)((𝑇𝑃𝑈)‘𝑦)) ∧ ((𝑈𝑃𝑇)‘(𝑥( ·𝑠 ‘𝑊)𝑦)) = (𝑥( ·𝑠 ‘𝑊)((𝑈𝑃𝑇)‘𝑦)))))
59	48, 58	mpbid 221	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑦 ∈ (𝑇 ⊕ 𝑈))) → (((𝑇𝑃𝑈)‘(𝑥( ·𝑠 ‘𝑊)𝑦)) = (𝑥( ·𝑠 ‘𝑊)((𝑇𝑃𝑈)‘𝑦)) ∧ ((𝑈𝑃𝑇)‘(𝑥( ·𝑠 ‘𝑊)𝑦)) = (𝑥( ·𝑠 ‘𝑊)((𝑈𝑃𝑇)‘𝑦))))
60	59	simpld 474	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑦 ∈ (𝑇 ⊕ 𝑈))) → ((𝑇𝑃𝑈)‘(𝑥( ·𝑠 ‘𝑊)𝑦)) = (𝑥( ·𝑠 ‘𝑊)((𝑇𝑃𝑈)‘𝑦)))
61	60	ralrimivva 2954	. . 3 ⊢ (𝜑 → ∀𝑥 ∈ (Base‘(Scalar‘𝑊))∀𝑦 ∈ (𝑇 ⊕ 𝑈)((𝑇𝑃𝑈)‘(𝑥( ·𝑠 ‘𝑊)𝑦)) = (𝑥( ·𝑠 ‘𝑊)((𝑇𝑃𝑈)‘𝑦)))
62	8, 50	sseldd 3569	. . . . . 6 ⊢ (𝜑 → (𝑇 ⊕ 𝑈) ∈ (SubGrp‘𝑊))
63		eqid 2610	. . . . . . 7 ⊢ (𝑊 ↾s (𝑇 ⊕ 𝑈)) = (𝑊 ↾s (𝑇 ⊕ 𝑈))
64	63	subgbas 17421	. . . . . 6 ⊢ ((𝑇 ⊕ 𝑈) ∈ (SubGrp‘𝑊) → (𝑇 ⊕ 𝑈) = (Base‘(𝑊 ↾s (𝑇 ⊕ 𝑈))))
65	62, 64	syl 17	. . . . 5 ⊢ (𝜑 → (𝑇 ⊕ 𝑈) = (Base‘(𝑊 ↾s (𝑇 ⊕ 𝑈))))
66	65	raleqdv 3121	. . . 4 ⊢ (𝜑 → (∀𝑦 ∈ (𝑇 ⊕ 𝑈)((𝑇𝑃𝑈)‘(𝑥( ·𝑠 ‘𝑊)𝑦)) = (𝑥( ·𝑠 ‘𝑊)((𝑇𝑃𝑈)‘𝑦)) ↔ ∀𝑦 ∈ (Base‘(𝑊 ↾s (𝑇 ⊕ 𝑈)))((𝑇𝑃𝑈)‘(𝑥( ·𝑠 ‘𝑊)𝑦)) = (𝑥( ·𝑠 ‘𝑊)((𝑇𝑃𝑈)‘𝑦))))
67	66	ralbidv 2969	. . 3 ⊢ (𝜑 → (∀𝑥 ∈ (Base‘(Scalar‘𝑊))∀𝑦 ∈ (𝑇 ⊕ 𝑈)((𝑇𝑃𝑈)‘(𝑥( ·𝑠 ‘𝑊)𝑦)) = (𝑥( ·𝑠 ‘𝑊)((𝑇𝑃𝑈)‘𝑦)) ↔ ∀𝑥 ∈ (Base‘(Scalar‘𝑊))∀𝑦 ∈ (Base‘(𝑊 ↾s (𝑇 ⊕ 𝑈)))((𝑇𝑃𝑈)‘(𝑥( ·𝑠 ‘𝑊)𝑦)) = (𝑥( ·𝑠 ‘𝑊)((𝑇𝑃𝑈)‘𝑦))))
68	61, 67	mpbid 221	. 2 ⊢ (𝜑 → ∀𝑥 ∈ (Base‘(Scalar‘𝑊))∀𝑦 ∈ (Base‘(𝑊 ↾s (𝑇 ⊕ 𝑈)))((𝑇𝑃𝑈)‘(𝑥( ·𝑠 ‘𝑊)𝑦)) = (𝑥( ·𝑠 ‘𝑊)((𝑇𝑃𝑈)‘𝑦)))
69	63, 6	lsslmod 18781	. . . 4 ⊢ ((𝑊 ∈ LMod ∧ (𝑇 ⊕ 𝑈) ∈ 𝐿) → (𝑊 ↾s (𝑇 ⊕ 𝑈)) ∈ LMod)
70	5, 50, 69	syl2anc 691	. . 3 ⊢ (𝜑 → (𝑊 ↾s (𝑇 ⊕ 𝑈)) ∈ LMod)
71		ovex 6577	. . . . 5 ⊢ (𝑇 ⊕ 𝑈) ∈ V
72	63, 19	resssca 15854	. . . . 5 ⊢ ((𝑇 ⊕ 𝑈) ∈ V → (Scalar‘𝑊) = (Scalar‘(𝑊 ↾s (𝑇 ⊕ 𝑈))))
73	71, 72	ax-mp 5	. . . 4 ⊢ (Scalar‘𝑊) = (Scalar‘(𝑊 ↾s (𝑇 ⊕ 𝑈)))
74		eqid 2610	. . . 4 ⊢ (Base‘(𝑊 ↾s (𝑇 ⊕ 𝑈))) = (Base‘(𝑊 ↾s (𝑇 ⊕ 𝑈)))
75	63, 44	ressvsca 15855	. . . . 5 ⊢ ((𝑇 ⊕ 𝑈) ∈ V → ( ·𝑠 ‘𝑊) = ( ·𝑠 ‘(𝑊 ↾s (𝑇 ⊕ 𝑈))))
76	71, 75	ax-mp 5	. . . 4 ⊢ ( ·𝑠 ‘𝑊) = ( ·𝑠 ‘(𝑊 ↾s (𝑇 ⊕ 𝑈)))
77	73, 19, 45, 74, 76, 44	islmhm3 18849	. . 3 ⊢ (((𝑊 ↾s (𝑇 ⊕ 𝑈)) ∈ LMod ∧ 𝑊 ∈ LMod) → ((𝑇𝑃𝑈) ∈ ((𝑊 ↾s (𝑇 ⊕ 𝑈)) LMHom 𝑊) ↔ ((𝑇𝑃𝑈) ∈ ((𝑊 ↾s (𝑇 ⊕ 𝑈)) GrpHom 𝑊) ∧ (Scalar‘𝑊) = (Scalar‘𝑊) ∧ ∀𝑥 ∈ (Base‘(Scalar‘𝑊))∀𝑦 ∈ (Base‘(𝑊 ↾s (𝑇 ⊕ 𝑈)))((𝑇𝑃𝑈)‘(𝑥( ·𝑠 ‘𝑊)𝑦)) = (𝑥( ·𝑠 ‘𝑊)((𝑇𝑃𝑈)‘𝑦)))))
78	70, 5, 77	syl2anc 691	. 2 ⊢ (𝜑 → ((𝑇𝑃𝑈) ∈ ((𝑊 ↾s (𝑇 ⊕ 𝑈)) LMHom 𝑊) ↔ ((𝑇𝑃𝑈) ∈ ((𝑊 ↾s (𝑇 ⊕ 𝑈)) GrpHom 𝑊) ∧ (Scalar‘𝑊) = (Scalar‘𝑊) ∧ ∀𝑥 ∈ (Base‘(Scalar‘𝑊))∀𝑦 ∈ (Base‘(𝑊 ↾s (𝑇 ⊕ 𝑈)))((𝑇𝑃𝑈)‘(𝑥( ·𝑠 ‘𝑊)𝑦)) = (𝑥( ·𝑠 ‘𝑊)((𝑇𝑃𝑈)‘𝑦)))))
79	18, 20, 68, 78	mpbir3and 1238	1 ⊢ (𝜑 → (𝑇𝑃𝑈) ∈ ((𝑊 ↾s (𝑇 ⊕ 𝑈)) LMHom 𝑊))

Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  ∀wral 2896  Vcvv 3173   ∩ cin 3539   ⊆ wss 3540  {csn 4125  ‘cfv 5804  (class class class)co 6549  Basecbs 15695   ↾_s cress 15696  +_gcplusg 15768  Scalarcsca 15771   ·_𝑠 cvsca 15772  0_gc0g 15923  SubGrpcsubg 17411   GrpHom cghm 17480  Cntzccntz 17571  LSSumclsm 17872  proj₁cpj1 17873  Abelcabl 18017  LModclmod 18686  LSubSpclss 18753   LMHom clmhm 18840

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-er 7629  df-en 7842  df-dom 7843  df-sdom 7844  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-2 10956  df-3 10957  df-4 10958  df-5 10959  df-6 10960  df-ndx 15698  df-slot 15699  df-base 15700  df-sets 15701  df-ress 15702  df-plusg 15781  df-sca 15784  df-vsca 15785  df-0g 15925  df-mgm 17065  df-sgrp 17107  df-mnd 17118  df-submnd 17159  df-grp 17248  df-minusg 17249  df-sbg 17250  df-subg 17414  df-ghm 17481  df-cntz 17573  df-lsm 17874  df-pj1 17875  df-cmn 18018  df-abl 18019  df-mgp 18313  df-ur 18325  df-ring 18372  df-lmod 18688  df-lss 18754  df-lmhm 18843

This theorem is referenced by:  pj1lmhm2  18922  pjff  19875