Theorem pj1lmhm 21095

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 2736	. . 3 ⊢ (+g‘𝑊) = (+g‘𝑊)
2		pj1lmhm.s	. . 3 ⊢ ⊕ = (LSSum‘𝑊)
3		pj1lmhm.z	. . 3 ⊢ 0 = (0g‘𝑊)
4		eqid 2736	. . 3 ⊢ (Cntz‘𝑊) = (Cntz‘𝑊)
5		pj1lmhm.1	. . . . 5 ⊢ (𝜑 → 𝑊 ∈ LMod)
6		pj1lmhm.l	. . . . . 6 ⊢ 𝐿 = (LSubSp‘𝑊)
7	6	lsssssubg 20953	. . . . 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 3922	. . 3 ⊢ (𝜑 → 𝑇 ∈ (SubGrp‘𝑊))
11		pj1lmhm.3	. . . 4 ⊢ (𝜑 → 𝑈 ∈ 𝐿)
12	8, 11	sseldd 3922	. . 3 ⊢ (𝜑 → 𝑈 ∈ (SubGrp‘𝑊))
13		pj1lmhm.4	. . 3 ⊢ (𝜑 → (𝑇 ∩ 𝑈) = { 0 })
14		lmodabl 20904	. . . . 5 ⊢ (𝑊 ∈ LMod → 𝑊 ∈ Abel)
15	5, 14	syl 17	. . . 4 ⊢ (𝜑 → 𝑊 ∈ Abel)
16	4, 15, 10, 12	ablcntzd 19832	. . 3 ⊢ (𝜑 → 𝑇 ⊆ ((Cntz‘𝑊)‘𝑈))
17		pj1lmhm.p	. . 3 ⊢ 𝑃 = (proj1‘𝑊)
18	1, 2, 3, 4, 10, 12, 13, 16, 17	pj1ghm 19678	. 2 ⊢ (𝜑 → (𝑇𝑃𝑈) ∈ ((𝑊 ↾s (𝑇 ⊕ 𝑈)) GrpHom 𝑊))
19		eqid 2736	. . 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 19674	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑇 ⊕ 𝑈)) → 𝑦 = (((𝑇𝑃𝑈)‘𝑦)(+g‘𝑊)((𝑈𝑃𝑇)‘𝑦)))
22	21	adantrl 717	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑦 ∈ (𝑇 ⊕ 𝑈))) → 𝑦 = (((𝑇𝑃𝑈)‘𝑦)(+g‘𝑊)((𝑈𝑃𝑇)‘𝑦)))
23	22	oveq2d 7383	. . . . . . 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 771	. . . . . . . 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 2736	. . . . . . . . . . 11 ⊢ (Base‘𝑊) = (Base‘𝑊)
28	27, 6	lssss 20931	. . . . . . . . . 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 19672	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑦 ∈ (𝑇 ⊕ 𝑈))) → (𝑇𝑃𝑈):(𝑇 ⊕ 𝑈)⟶𝑇)
35		simprr 773	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑦 ∈ (𝑇 ⊕ 𝑈))) → 𝑦 ∈ (𝑇 ⊕ 𝑈))
36	34, 35	ffvelcdmd 7037	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑦 ∈ (𝑇 ⊕ 𝑈))) → ((𝑇𝑃𝑈)‘𝑦) ∈ 𝑇)
37	29, 36	sseldd 3922	. . . . . . . 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 20931	. . . . . . . . . 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 19673	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑦 ∈ (𝑇 ⊕ 𝑈))) → (𝑈𝑃𝑇):(𝑇 ⊕ 𝑈)⟶𝑈)
42	41, 35	ffvelcdmd 7037	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑦 ∈ (𝑇 ⊕ 𝑈))) → ((𝑈𝑃𝑇)‘𝑦) ∈ 𝑈)
43	40, 42	sseldd 3922	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑦 ∈ (𝑇 ⊕ 𝑈))) → ((𝑈𝑃𝑇)‘𝑦) ∈ (Base‘𝑊))
44		eqid 2736	. . . . . . . . 9 ⊢ ( ·𝑠 ‘𝑊) = ( ·𝑠 ‘𝑊)
45		eqid 2736	. . . . . . . . 9 ⊢ (Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊))
46	27, 1, 19, 44, 45	lmodvsdi 20880	. . . . . . . 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 1375	. . . . . . 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 2771	. . . . . 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 21078	. . . . . . . . . 10 ⊢ ((𝑊 ∈ LMod ∧ 𝑇 ∈ 𝐿 ∧ 𝑈 ∈ 𝐿) → (𝑇 ⊕ 𝑈) ∈ 𝐿)
50	5, 9, 11, 49	syl3anc 1374	. . . . . . . . 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 20950	. . . . . . . 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 839	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑦 ∈ (𝑇 ⊕ 𝑈))) → (𝑥( ·𝑠 ‘𝑊)𝑦) ∈ (𝑇 ⊕ 𝑈))
54	19, 44, 45, 6	lssvscl 20950	. . . . . . . 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 839	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑦 ∈ (𝑇 ⊕ 𝑈))) → (𝑥( ·𝑠 ‘𝑊)((𝑇𝑃𝑈)‘𝑦)) ∈ 𝑇)
56	19, 44, 45, 6	lssvscl 20950	. . . . . . . 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 839	. . . . . . 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 19675	. . . . . 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 232	. . . . 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 494	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑦 ∈ (𝑇 ⊕ 𝑈))) → ((𝑇𝑃𝑈)‘(𝑥( ·𝑠 ‘𝑊)𝑦)) = (𝑥( ·𝑠 ‘𝑊)((𝑇𝑃𝑈)‘𝑦)))
61	60	ralrimivva 3180	. . 3 ⊢ (𝜑 → ∀𝑥 ∈ (Base‘(Scalar‘𝑊))∀𝑦 ∈ (𝑇 ⊕ 𝑈)((𝑇𝑃𝑈)‘(𝑥( ·𝑠 ‘𝑊)𝑦)) = (𝑥( ·𝑠 ‘𝑊)((𝑇𝑃𝑈)‘𝑦)))
62	8, 50	sseldd 3922	. . . . . 6 ⊢ (𝜑 → (𝑇 ⊕ 𝑈) ∈ (SubGrp‘𝑊))
63		eqid 2736	. . . . . . 7 ⊢ (𝑊 ↾s (𝑇 ⊕ 𝑈)) = (𝑊 ↾s (𝑇 ⊕ 𝑈))
64	63	subgbas 19106	. . . . . 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 3295	. . . 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 3160	. . 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 232	. 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 20955	. . . 4 ⊢ ((𝑊 ∈ LMod ∧ (𝑇 ⊕ 𝑈) ∈ 𝐿) → (𝑊 ↾s (𝑇 ⊕ 𝑈)) ∈ LMod)
70	5, 50, 69	syl2anc 585	. . 3 ⊢ (𝜑 → (𝑊 ↾s (𝑇 ⊕ 𝑈)) ∈ LMod)
71		ovex 7400	. . . . 5 ⊢ (𝑇 ⊕ 𝑈) ∈ V
72	63, 19	resssca 17306	. . . . 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 2736	. . . 4 ⊢ (Base‘(𝑊 ↾s (𝑇 ⊕ 𝑈))) = (Base‘(𝑊 ↾s (𝑇 ⊕ 𝑈)))
75	63, 44	ressvsca 17307	. . . . 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 21023	. . 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 585	. 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 1344	1 ⊢ (𝜑 → (𝑇𝑃𝑈) ∈ ((𝑊 ↾s (𝑇 ⊕ 𝑈)) LMHom 𝑊))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114  ∀wral 3051  Vcvv 3429   ∩ cin 3888   ⊆ wss 3889  {csn 4567  ‘cfv 6498  (class class class)co 7367  Basecbs 17179   ↾_s cress 17200  +_gcplusg 17220  Scalarcsca 17223   ·_𝑠 cvsca 17224  0_gc0g 17402  SubGrpcsubg 19096   GrpHom cghm 19187  Cntzccntz 19290  LSSumclsm 19609  proj₁cpj1 19610  Abelcabl 19756  LModclmod 20855  LSubSpclss 20926   LMHom clmhm 21014

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5307  ax-pr 5375  ax-un 7689  ax-cnex 11094  ax-resscn 11095  ax-1cn 11096  ax-icn 11097  ax-addcl 11098  ax-addrcl 11099  ax-mulcl 11100  ax-mulrcl 11101  ax-mulcom 11102  ax-addass 11103  ax-mulass 11104  ax-distr 11105  ax-i2m1 11106  ax-1ne0 11107  ax-1rid 11108  ax-rnegex 11109  ax-rrecex 11110  ax-cnre 11111  ax-pre-lttri 11112  ax-pre-lttrn 11113  ax-pre-ltadd 11114  ax-pre-mulgt0 11115

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-nel 3037  df-ral 3052  df-rex 3062  df-rmo 3342  df-reu 3343  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-pss 3909  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-iun 4935  df-br 5086  df-opab 5148  df-mpt 5167  df-tr 5193  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6265  df-ord 6326  df-on 6327  df-lim 6328  df-suc 6329  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-f1 6503  df-fo 6504  df-f1o 6505  df-fv 6506  df-riota 7324  df-ov 7370  df-oprab 7371  df-mpo 7372  df-om 7818  df-1st 7942  df-2nd 7943  df-frecs 8231  df-wrecs 8262  df-recs 8311  df-rdg 8349  df-er 8643  df-map 8775  df-en 8894  df-dom 8895  df-sdom 8896  df-pnf 11181  df-mnf 11182  df-xr 11183  df-ltxr 11184  df-le 11185  df-sub 11379  df-neg 11380  df-nn 12175  df-2 12244  df-3 12245  df-4 12246  df-5 12247  df-6 12248  df-sets 17134  df-slot 17152  df-ndx 17164  df-base 17180  df-ress 17201  df-plusg 17233  df-sca 17236  df-vsca 17237  df-0g 17404  df-mgm 18608  df-sgrp 18687  df-mnd 18703  df-submnd 18752  df-grp 18912  df-minusg 18913  df-sbg 18914  df-subg 19099  df-ghm 19188  df-cntz 19292  df-lsm 19611  df-pj1 19612  df-cmn 19757  df-abl 19758  df-mgp 20122  df-ur 20163  df-ring 20216  df-lmod 20857  df-lss 20927  df-lmhm 21017

This theorem is referenced by:  pj1lmhm2  21096  pjff  21692