Theorem pj1lmhm 21375

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 2761	. . 3 ⊢ (+g‘𝑊) = (+g‘𝑊)
2		pj1lmhm.s	. . 3 ⊢ ⊕ = (LSSum‘𝑊)
3		pj1lmhm.z	. . 3 ⊢ 0 = (0g‘𝑊)
4		eqid 2761	. . 3 ⊢ (Cntz‘𝑊) = (Cntz‘𝑊)
5		pj1lmhm.1	. . . . 5 ⊢ (𝜑 → 𝑊 ∈ LMod)
6		pj1lmhm.l	. . . . . 6 ⊢ 𝐿 = (LSubSp‘𝑊)
7	6	lsssssubg 21233	. . . . 5 ⊢ (𝑊 ∈ LMod → 𝐿 ⊆ (SubGrp‘𝑊))
8	5, 7	syl 18	. . . 4 ⊢ (𝜑 → 𝐿 ⊆ (SubGrp‘𝑊))
9		pj1lmhm.2	. . . 4 ⊢ (𝜑 → 𝑇 ∈ 𝐿)
10	8, 9	sseldd 3932	. . 3 ⊢ (𝜑 → 𝑇 ∈ (SubGrp‘𝑊))
11		pj1lmhm.3	. . . 4 ⊢ (𝜑 → 𝑈 ∈ 𝐿)
12	8, 11	sseldd 3932	. . 3 ⊢ (𝜑 → 𝑈 ∈ (SubGrp‘𝑊))
13		pj1lmhm.4	. . 3 ⊢ (𝜑 → (𝑇 ∩ 𝑈) = { 0 })
14		lmodabl 21184	. . . . 5 ⊢ (𝑊 ∈ LMod → 𝑊 ∈ Abel)
15	5, 14	syl 18	. . . 4 ⊢ (𝜑 → 𝑊 ∈ Abel)
16	4, 15, 10, 12	ablcntzd 20071	. . 3 ⊢ (𝜑 → 𝑇 ⊆ ((Cntz‘𝑊)‘𝑈))
17		pj1lmhm.p	. . 3 ⊢ 𝑃 = (proj1‘𝑊)
18	1, 2, 3, 4, 10, 12, 13, 16, 17	pj1ghm 19917	. 2 ⊢ (𝜑 → (𝑇𝑃𝑈) ∈ ((𝑊 ↾s (𝑇 ⊕ 𝑈)) GrpHom 𝑊))
19		eqid 2761	. . 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 19913	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑇 ⊕ 𝑈)) → 𝑦 = (((𝑇𝑃𝑈)‘𝑦)(+g‘𝑊)((𝑈𝑃𝑇)‘𝑦)))
22	21	adantrl 729	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑦 ∈ (𝑇 ⊕ 𝑈))) → 𝑦 = (((𝑇𝑃𝑈)‘𝑦)(+g‘𝑊)((𝑈𝑃𝑇)‘𝑦)))
23	22	oveq2d 7436	. . . . . . 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 486	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑦 ∈ (𝑇 ⊕ 𝑈))) → 𝑊 ∈ LMod)
25		simprl 783	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑦 ∈ (𝑇 ⊕ 𝑈))) → 𝑥 ∈ (Base‘(Scalar‘𝑊)))
26	9	adantr 486	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑦 ∈ (𝑇 ⊕ 𝑈))) → 𝑇 ∈ 𝐿)
27		eqid 2761	. . . . . . . . . . 11 ⊢ (Base‘𝑊) = (Base‘𝑊)
28	27, 6	lssss 21211	. . . . . . . . . 10 ⊢ (𝑇 ∈ 𝐿 → 𝑇 ⊆ (Base‘𝑊))
29	26, 28	syl 18	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑦 ∈ (𝑇 ⊕ 𝑈))) → 𝑇 ⊆ (Base‘𝑊))
30	10	adantr 486	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑦 ∈ (𝑇 ⊕ 𝑈))) → 𝑇 ∈ (SubGrp‘𝑊))
31	12	adantr 486	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑦 ∈ (𝑇 ⊕ 𝑈))) → 𝑈 ∈ (SubGrp‘𝑊))
32	13	adantr 486	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑦 ∈ (𝑇 ⊕ 𝑈))) → (𝑇 ∩ 𝑈) = { 0 })
33	16	adantr 486	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑦 ∈ (𝑇 ⊕ 𝑈))) → 𝑇 ⊆ ((Cntz‘𝑊)‘𝑈))
34	1, 2, 3, 4, 30, 31, 32, 33, 17	pj1f 19911	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑦 ∈ (𝑇 ⊕ 𝑈))) → (𝑇𝑃𝑈):(𝑇 ⊕ 𝑈)⟶𝑇)
35		simprr 785	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑦 ∈ (𝑇 ⊕ 𝑈))) → 𝑦 ∈ (𝑇 ⊕ 𝑈))
36	34, 35	ffvelcdmd 7085	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑦 ∈ (𝑇 ⊕ 𝑈))) → ((𝑇𝑃𝑈)‘𝑦) ∈ 𝑇)
37	29, 36	sseldd 3932	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑦 ∈ (𝑇 ⊕ 𝑈))) → ((𝑇𝑃𝑈)‘𝑦) ∈ (Base‘𝑊))
38	11	adantr 486	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑦 ∈ (𝑇 ⊕ 𝑈))) → 𝑈 ∈ 𝐿)
39	27, 6	lssss 21211	. . . . . . . . . 10 ⊢ (𝑈 ∈ 𝐿 → 𝑈 ⊆ (Base‘𝑊))
40	38, 39	syl 18	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑦 ∈ (𝑇 ⊕ 𝑈))) → 𝑈 ⊆ (Base‘𝑊))
41	1, 2, 3, 4, 30, 31, 32, 33, 17	pj2f 19912	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑦 ∈ (𝑇 ⊕ 𝑈))) → (𝑈𝑃𝑇):(𝑇 ⊕ 𝑈)⟶𝑈)
42	41, 35	ffvelcdmd 7085	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑦 ∈ (𝑇 ⊕ 𝑈))) → ((𝑈𝑃𝑇)‘𝑦) ∈ 𝑈)
43	40, 42	sseldd 3932	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑦 ∈ (𝑇 ⊕ 𝑈))) → ((𝑈𝑃𝑇)‘𝑦) ∈ (Base‘𝑊))
44		eqid 2761	. . . . . . . . 9 ⊢ ( ·𝑠 ‘𝑊) = ( ·𝑠 ‘𝑊)
45		eqid 2761	. . . . . . . . 9 ⊢ (Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊))
46	27, 1, 19, 44, 45	lmodvsdi 21160	. . . . . . . 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 1399	. . . . . . 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 2796	. . . . . 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 21358	. . . . . . . . . 10 ⊢ ((𝑊 ∈ LMod ∧ 𝑇 ∈ 𝐿 ∧ 𝑈 ∈ 𝐿) → (𝑇 ⊕ 𝑈) ∈ 𝐿)
50	5, 9, 11, 49	syl3anc 1398	. . . . . . . . 9 ⊢ (𝜑 → (𝑇 ⊕ 𝑈) ∈ 𝐿)
51	50	adantr 486	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑦 ∈ (𝑇 ⊕ 𝑈))) → (𝑇 ⊕ 𝑈) ∈ 𝐿)
52	19, 44, 45, 6	lssvscl 21230	. . . . . . . 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 852	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑦 ∈ (𝑇 ⊕ 𝑈))) → (𝑥( ·𝑠 ‘𝑊)𝑦) ∈ (𝑇 ⊕ 𝑈))
54	19, 44, 45, 6	lssvscl 21230	. . . . . . . 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 852	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑦 ∈ (𝑇 ⊕ 𝑈))) → (𝑥( ·𝑠 ‘𝑊)((𝑇𝑃𝑈)‘𝑦)) ∈ 𝑇)
56	19, 44, 45, 6	lssvscl 21230	. . . . . . . 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 852	. . . . . . 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 19914	. . . . . 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 235	. . . . 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 500	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑦 ∈ (𝑇 ⊕ 𝑈))) → ((𝑇𝑃𝑈)‘(𝑥( ·𝑠 ‘𝑊)𝑦)) = (𝑥( ·𝑠 ‘𝑊)((𝑇𝑃𝑈)‘𝑦)))
61	60	ralrimivva 3206	. . 3 ⊢ (𝜑 → ∀𝑥 ∈ (Base‘(Scalar‘𝑊))∀𝑦 ∈ (𝑇 ⊕ 𝑈)((𝑇𝑃𝑈)‘(𝑥( ·𝑠 ‘𝑊)𝑦)) = (𝑥( ·𝑠 ‘𝑊)((𝑇𝑃𝑈)‘𝑦)))
62	8, 50	sseldd 3932	. . . . . 6 ⊢ (𝜑 → (𝑇 ⊕ 𝑈) ∈ (SubGrp‘𝑊))
63		eqid 2761	. . . . . . 7 ⊢ (𝑊 ↾s (𝑇 ⊕ 𝑈)) = (𝑊 ↾s (𝑇 ⊕ 𝑈))
64	63	subgbas 19340	. . . . . 6 ⊢ ((𝑇 ⊕ 𝑈) ∈ (SubGrp‘𝑊) → (𝑇 ⊕ 𝑈) = (Base‘(𝑊 ↾s (𝑇 ⊕ 𝑈))))
65	62, 64	syl 18	. . . . 5 ⊢ (𝜑 → (𝑇 ⊕ 𝑈) = (Base‘(𝑊 ↾s (𝑇 ⊕ 𝑈))))
66	65	raleqdv 3320	. . . 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 3186	. . 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 235	. 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 21235	. . . 4 ⊢ ((𝑊 ∈ LMod ∧ (𝑇 ⊕ 𝑈) ∈ 𝐿) → (𝑊 ↾s (𝑇 ⊕ 𝑈)) ∈ LMod)
70	5, 50, 69	syl2anc 596	. . 3 ⊢ (𝜑 → (𝑊 ↾s (𝑇 ⊕ 𝑈)) ∈ LMod)
71		ovex 7453	. . . . 5 ⊢ (𝑇 ⊕ 𝑈) ∈ V
72	63, 19	resssca 17514	. . . . 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 2761	. . . 4 ⊢ (Base‘(𝑊 ↾s (𝑇 ⊕ 𝑈))) = (Base‘(𝑊 ↾s (𝑇 ⊕ 𝑈)))
75	63, 44	ressvsca 17515	. . . . 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 21303	. . 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 596	. 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 1361	1 ⊢ (𝜑 → (𝑇𝑃𝑈) ∈ ((𝑊 ↾s (𝑇 ⊕ 𝑈)) LMHom 𝑊))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145  ∀wral 3077  Vcvv 3451   ∩ cin 3898   ⊆ wss 3899  {csn 4584  ‘cfv 6538  (class class class)co 7420  Basecbs 17387   ↾_s cress 17408  +_gcplusg 17428  Scalarcsca 17431   ·_𝑠 cvsca 17432  0_gc0g 17610  SubGrpcsubg 19330   GrpHom cghm 19427  Cntzccntz 19529  LSSumclsm 19848  proj₁cpj1 19849  Abelcabl 19995  LModclmod 21135  LSubSpclss 21206   LMHom clmhm 21294

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-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-3 12406  df-4 12407  df-5 12408  df-6 12409  df-sets 17342  df-slot 17360  df-ndx 17372  df-base 17388  df-ress 17409  df-plusg 17441  df-sca 17444  df-vsca 17445  df-0g 17612  df-mgm 18816  df-sgrp 18908  df-mnd 18924  df-submnd 18979  df-grp 19147  df-minusg 19148  df-sbg 19149  df-subg 19333  df-ghm 19428  df-cntz 19531  df-lsm 19850  df-pj1 19851  df-cmn 19996  df-abl 19997  df-mgp 20361  df-ur 20408  df-ring 20461  df-lmod 21137  df-lss 21207  df-lmhm 21297

This theorem is used by:  pj1lmhm2  21376  pjff  22018