Theorem pj1lmhm 19872

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 2821	. . 3 ⊢ (+g‘𝑊) = (+g‘𝑊)
2		pj1lmhm.s	. . 3 ⊢ ⊕ = (LSSum‘𝑊)
3		pj1lmhm.z	. . 3 ⊢ 0 = (0g‘𝑊)
4		eqid 2821	. . 3 ⊢ (Cntz‘𝑊) = (Cntz‘𝑊)
5		pj1lmhm.1	. . . . 5 ⊢ (𝜑 → 𝑊 ∈ LMod)
6		pj1lmhm.l	. . . . . 6 ⊢ 𝐿 = (LSubSp‘𝑊)
7	6	lsssssubg 19730	. . . . 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 3968	. . 3 ⊢ (𝜑 → 𝑇 ∈ (SubGrp‘𝑊))
11		pj1lmhm.3	. . . 4 ⊢ (𝜑 → 𝑈 ∈ 𝐿)
12	8, 11	sseldd 3968	. . 3 ⊢ (𝜑 → 𝑈 ∈ (SubGrp‘𝑊))
13		pj1lmhm.4	. . 3 ⊢ (𝜑 → (𝑇 ∩ 𝑈) = { 0 })
14		lmodabl 19681	. . . . 5 ⊢ (𝑊 ∈ LMod → 𝑊 ∈ Abel)
15	5, 14	syl 17	. . . 4 ⊢ (𝜑 → 𝑊 ∈ Abel)
16	4, 15, 10, 12	ablcntzd 18977	. . 3 ⊢ (𝜑 → 𝑇 ⊆ ((Cntz‘𝑊)‘𝑈))
17		pj1lmhm.p	. . 3 ⊢ 𝑃 = (proj1‘𝑊)
18	1, 2, 3, 4, 10, 12, 13, 16, 17	pj1ghm 18829	. 2 ⊢ (𝜑 → (𝑇𝑃𝑈) ∈ ((𝑊 ↾s (𝑇 ⊕ 𝑈)) GrpHom 𝑊))
19		eqid 2821	. . 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 18825	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑇 ⊕ 𝑈)) → 𝑦 = (((𝑇𝑃𝑈)‘𝑦)(+g‘𝑊)((𝑈𝑃𝑇)‘𝑦)))
22	21	adantrl 714	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑦 ∈ (𝑇 ⊕ 𝑈))) → 𝑦 = (((𝑇𝑃𝑈)‘𝑦)(+g‘𝑊)((𝑈𝑃𝑇)‘𝑦)))
23	22	oveq2d 7172	. . . . . . 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 483	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑦 ∈ (𝑇 ⊕ 𝑈))) → 𝑊 ∈ LMod)
25		simprl 769	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑦 ∈ (𝑇 ⊕ 𝑈))) → 𝑥 ∈ (Base‘(Scalar‘𝑊)))
26	9	adantr 483	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑦 ∈ (𝑇 ⊕ 𝑈))) → 𝑇 ∈ 𝐿)
27		eqid 2821	. . . . . . . . . . 11 ⊢ (Base‘𝑊) = (Base‘𝑊)
28	27, 6	lssss 19708	. . . . . . . . . 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 483	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑦 ∈ (𝑇 ⊕ 𝑈))) → 𝑇 ∈ (SubGrp‘𝑊))
31	12	adantr 483	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑦 ∈ (𝑇 ⊕ 𝑈))) → 𝑈 ∈ (SubGrp‘𝑊))
32	13	adantr 483	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑦 ∈ (𝑇 ⊕ 𝑈))) → (𝑇 ∩ 𝑈) = { 0 })
33	16	adantr 483	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑦 ∈ (𝑇 ⊕ 𝑈))) → 𝑇 ⊆ ((Cntz‘𝑊)‘𝑈))
34	1, 2, 3, 4, 30, 31, 32, 33, 17	pj1f 18823	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑦 ∈ (𝑇 ⊕ 𝑈))) → (𝑇𝑃𝑈):(𝑇 ⊕ 𝑈)⟶𝑇)
35		simprr 771	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑦 ∈ (𝑇 ⊕ 𝑈))) → 𝑦 ∈ (𝑇 ⊕ 𝑈))
36	34, 35	ffvelrnd 6852	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑦 ∈ (𝑇 ⊕ 𝑈))) → ((𝑇𝑃𝑈)‘𝑦) ∈ 𝑇)
37	29, 36	sseldd 3968	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑦 ∈ (𝑇 ⊕ 𝑈))) → ((𝑇𝑃𝑈)‘𝑦) ∈ (Base‘𝑊))
38	11	adantr 483	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑦 ∈ (𝑇 ⊕ 𝑈))) → 𝑈 ∈ 𝐿)
39	27, 6	lssss 19708	. . . . . . . . . 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 18824	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑦 ∈ (𝑇 ⊕ 𝑈))) → (𝑈𝑃𝑇):(𝑇 ⊕ 𝑈)⟶𝑈)
42	41, 35	ffvelrnd 6852	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑦 ∈ (𝑇 ⊕ 𝑈))) → ((𝑈𝑃𝑇)‘𝑦) ∈ 𝑈)
43	40, 42	sseldd 3968	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑦 ∈ (𝑇 ⊕ 𝑈))) → ((𝑈𝑃𝑇)‘𝑦) ∈ (Base‘𝑊))
44		eqid 2821	. . . . . . . . 9 ⊢ ( ·𝑠 ‘𝑊) = ( ·𝑠 ‘𝑊)
45		eqid 2821	. . . . . . . . 9 ⊢ (Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊))
46	27, 1, 19, 44, 45	lmodvsdi 19657	. . . . . . . 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 1368	. . . . . . 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 2856	. . . . . 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 19855	. . . . . . . . . 10 ⊢ ((𝑊 ∈ LMod ∧ 𝑇 ∈ 𝐿 ∧ 𝑈 ∈ 𝐿) → (𝑇 ⊕ 𝑈) ∈ 𝐿)
50	5, 9, 11, 49	syl3anc 1367	. . . . . . . . 9 ⊢ (𝜑 → (𝑇 ⊕ 𝑈) ∈ 𝐿)
51	50	adantr 483	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑦 ∈ (𝑇 ⊕ 𝑈))) → (𝑇 ⊕ 𝑈) ∈ 𝐿)
52	19, 44, 45, 6	lssvscl 19727	. . . . . . . 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 836	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑦 ∈ (𝑇 ⊕ 𝑈))) → (𝑥( ·𝑠 ‘𝑊)𝑦) ∈ (𝑇 ⊕ 𝑈))
54	19, 44, 45, 6	lssvscl 19727	. . . . . . . 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 836	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑦 ∈ (𝑇 ⊕ 𝑈))) → (𝑥( ·𝑠 ‘𝑊)((𝑇𝑃𝑈)‘𝑦)) ∈ 𝑇)
56	19, 44, 45, 6	lssvscl 19727	. . . . . . . 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 836	. . . . . . 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 18826	. . . . . 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 234	. . . . 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 497	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑦 ∈ (𝑇 ⊕ 𝑈))) → ((𝑇𝑃𝑈)‘(𝑥( ·𝑠 ‘𝑊)𝑦)) = (𝑥( ·𝑠 ‘𝑊)((𝑇𝑃𝑈)‘𝑦)))
61	60	ralrimivva 3191	. . 3 ⊢ (𝜑 → ∀𝑥 ∈ (Base‘(Scalar‘𝑊))∀𝑦 ∈ (𝑇 ⊕ 𝑈)((𝑇𝑃𝑈)‘(𝑥( ·𝑠 ‘𝑊)𝑦)) = (𝑥( ·𝑠 ‘𝑊)((𝑇𝑃𝑈)‘𝑦)))
62	8, 50	sseldd 3968	. . . . . 6 ⊢ (𝜑 → (𝑇 ⊕ 𝑈) ∈ (SubGrp‘𝑊))
63		eqid 2821	. . . . . . 7 ⊢ (𝑊 ↾s (𝑇 ⊕ 𝑈)) = (𝑊 ↾s (𝑇 ⊕ 𝑈))
64	63	subgbas 18283	. . . . . 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 3415	. . . 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 3197	. . 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 234	. 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 19732	. . . 4 ⊢ ((𝑊 ∈ LMod ∧ (𝑇 ⊕ 𝑈) ∈ 𝐿) → (𝑊 ↾s (𝑇 ⊕ 𝑈)) ∈ LMod)
70	5, 50, 69	syl2anc 586	. . 3 ⊢ (𝜑 → (𝑊 ↾s (𝑇 ⊕ 𝑈)) ∈ LMod)
71		ovex 7189	. . . . 5 ⊢ (𝑇 ⊕ 𝑈) ∈ V
72	63, 19	resssca 16650	. . . . 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 2821	. . . 4 ⊢ (Base‘(𝑊 ↾s (𝑇 ⊕ 𝑈))) = (Base‘(𝑊 ↾s (𝑇 ⊕ 𝑈)))
75	63, 44	ressvsca 16651	. . . . 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 19800	. . 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 586	. 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 1338	1 ⊢ (𝜑 → (𝑇𝑃𝑈) ∈ ((𝑊 ↾s (𝑇 ⊕ 𝑈)) LMHom 𝑊))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   ∧ w3a 1083   = wceq 1537   ∈ wcel 2114  ∀wral 3138  Vcvv 3494   ∩ cin 3935   ⊆ wss 3936  {csn 4567  ‘cfv 6355  (class class class)co 7156  Basecbs 16483   ↾_s cress 16484  +_gcplusg 16565  Scalarcsca 16568   ·_𝑠 cvsca 16569  0_gc0g 16713  SubGrpcsubg 18273   GrpHom cghm 18355  Cntzccntz 18445  LSSumclsm 18759  proj₁cpj1 18760  Abelcabl 18907  LModclmod 19634  LSubSpclss 19703   LMHom clmhm 19791

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793  ax-rep 5190  ax-sep 5203  ax-nul 5210  ax-pow 5266  ax-pr 5330  ax-un 7461  ax-cnex 10593  ax-resscn 10594  ax-1cn 10595  ax-icn 10596  ax-addcl 10597  ax-addrcl 10598  ax-mulcl 10599  ax-mulrcl 10600  ax-mulcom 10601  ax-addass 10602  ax-mulass 10603  ax-distr 10604  ax-i2m1 10605  ax-1ne0 10606  ax-1rid 10607  ax-rnegex 10608  ax-rrecex 10609  ax-cnre 10610  ax-pre-lttri 10611  ax-pre-lttrn 10612  ax-pre-ltadd 10613  ax-pre-mulgt0 10614

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-nel 3124  df-ral 3143  df-rex 3144  df-reu 3145  df-rmo 3146  df-rab 3147  df-v 3496  df-sbc 3773  df-csb 3884  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-pss 3954  df-nul 4292  df-if 4468  df-pw 4541  df-sn 4568  df-pr 4570  df-tp 4572  df-op 4574  df-uni 4839  df-iun 4921  df-br 5067  df-opab 5129  df-mpt 5147  df-tr 5173  df-id 5460  df-eprel 5465  df-po 5474  df-so 5475  df-fr 5514  df-we 5516  df-xp 5561  df-rel 5562  df-cnv 5563  df-co 5564  df-dm 5565  df-rn 5566  df-res 5567  df-ima 5568  df-pred 6148  df-ord 6194  df-on 6195  df-lim 6196  df-suc 6197  df-iota 6314  df-fun 6357  df-fn 6358  df-f 6359  df-f1 6360  df-fo 6361  df-f1o 6362  df-fv 6363  df-riota 7114  df-ov 7159  df-oprab 7160  df-mpo 7161  df-om 7581  df-1st 7689  df-2nd 7690  df-wrecs 7947  df-recs 8008  df-rdg 8046  df-er 8289  df-en 8510  df-dom 8511  df-sdom 8512  df-pnf 10677  df-mnf 10678  df-xr 10679  df-ltxr 10680  df-le 10681  df-sub 10872  df-neg 10873  df-nn 11639  df-2 11701  df-3 11702  df-4 11703  df-5 11704  df-6 11705  df-ndx 16486  df-slot 16487  df-base 16489  df-sets 16490  df-ress 16491  df-plusg 16578  df-sca 16581  df-vsca 16582  df-0g 16715  df-mgm 17852  df-sgrp 17901  df-mnd 17912  df-submnd 17957  df-grp 18106  df-minusg 18107  df-sbg 18108  df-subg 18276  df-ghm 18356  df-cntz 18447  df-lsm 18761  df-pj1 18762  df-cmn 18908  df-abl 18909  df-mgp 19240  df-ur 19252  df-ring 19299  df-lmod 19636  df-lss 19704  df-lmhm 19794

This theorem is referenced by:  pj1lmhm2  19873  pjff  20856