Theorem lsmcss 21096

Description: A subset of a pre-Hilbert space whose double orthocomplement has a projection decomposition is a closed subspace. This is the core of the proof that a topologically closed subspace is algebraically closed in a Hilbert space. (Contributed by Mario Carneiro, 13-Oct-2015.)

Hypotheses

Ref	Expression
lsmcss.c	⊢ 𝐶 = (ClSubSp‘𝑊)
lsmcss.j	⊢ 𝑉 = (Base‘𝑊)
lsmcss.o	⊢ ⊥ = (ocv‘𝑊)
lsmcss.p	⊢ ⊕ = (LSSum‘𝑊)
lsmcss.1	⊢ (𝜑 → 𝑊 ∈ PreHil)
lsmcss.2	⊢ (𝜑 → 𝑆 ⊆ 𝑉)
lsmcss.3	⊢ (𝜑 → ( ⊥ ‘( ⊥ ‘𝑆)) ⊆ (𝑆 ⊕ ( ⊥ ‘𝑆)))

Assertion

Ref	Expression
lsmcss	⊢ (𝜑 → 𝑆 ∈ 𝐶)

Proof of Theorem lsmcss

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

Step	Hyp	Ref	Expression
1		lsmcss.3	. . . . . . 7 ⊢ (𝜑 → ( ⊥ ‘( ⊥ ‘𝑆)) ⊆ (𝑆 ⊕ ( ⊥ ‘𝑆)))
2	1	sseld 3943	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ ( ⊥ ‘( ⊥ ‘𝑆)) → 𝑥 ∈ (𝑆 ⊕ ( ⊥ ‘𝑆))))
3		lsmcss.1	. . . . . . . 8 ⊢ (𝜑 → 𝑊 ∈ PreHil)
4		phllmod 21034	. . . . . . . 8 ⊢ (𝑊 ∈ PreHil → 𝑊 ∈ LMod)
5	3, 4	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝑊 ∈ LMod)
6		lsmcss.2	. . . . . . 7 ⊢ (𝜑 → 𝑆 ⊆ 𝑉)
7		lsmcss.j	. . . . . . . . 9 ⊢ 𝑉 = (Base‘𝑊)
8		lsmcss.o	. . . . . . . . 9 ⊢ ⊥ = (ocv‘𝑊)
9	7, 8	ocvss 21074	. . . . . . . 8 ⊢ ( ⊥ ‘𝑆) ⊆ 𝑉
10	9	a1i 11	. . . . . . 7 ⊢ (𝜑 → ( ⊥ ‘𝑆) ⊆ 𝑉)
11		eqid 2736	. . . . . . . 8 ⊢ (+g‘𝑊) = (+g‘𝑊)
12		lsmcss.p	. . . . . . . 8 ⊢ ⊕ = (LSSum‘𝑊)
13	7, 11, 12	lsmelvalx 19422	. . . . . . 7 ⊢ ((𝑊 ∈ LMod ∧ 𝑆 ⊆ 𝑉 ∧ ( ⊥ ‘𝑆) ⊆ 𝑉) → (𝑥 ∈ (𝑆 ⊕ ( ⊥ ‘𝑆)) ↔ ∃𝑦 ∈ 𝑆 ∃𝑧 ∈ ( ⊥ ‘𝑆)𝑥 = (𝑦(+g‘𝑊)𝑧)))
14	5, 6, 10, 13	syl3anc 1371	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ (𝑆 ⊕ ( ⊥ ‘𝑆)) ↔ ∃𝑦 ∈ 𝑆 ∃𝑧 ∈ ( ⊥ ‘𝑆)𝑥 = (𝑦(+g‘𝑊)𝑧)))
15	2, 14	sylibd 238	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ ( ⊥ ‘( ⊥ ‘𝑆)) → ∃𝑦 ∈ 𝑆 ∃𝑧 ∈ ( ⊥ ‘𝑆)𝑥 = (𝑦(+g‘𝑊)𝑧)))
16	3	ad2antrr 724	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑧 ∈ ( ⊥ ‘𝑆))) ∧ (𝑦(+g‘𝑊)𝑧) ∈ ( ⊥ ‘( ⊥ ‘𝑆))) → 𝑊 ∈ PreHil)
17	6	ad2antrr 724	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑧 ∈ ( ⊥ ‘𝑆))) ∧ (𝑦(+g‘𝑊)𝑧) ∈ ( ⊥ ‘( ⊥ ‘𝑆))) → 𝑆 ⊆ 𝑉)
18		simplrl 775	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑧 ∈ ( ⊥ ‘𝑆))) ∧ (𝑦(+g‘𝑊)𝑧) ∈ ( ⊥ ‘( ⊥ ‘𝑆))) → 𝑦 ∈ 𝑆)
19	17, 18	sseldd 3945	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑧 ∈ ( ⊥ ‘𝑆))) ∧ (𝑦(+g‘𝑊)𝑧) ∈ ( ⊥ ‘( ⊥ ‘𝑆))) → 𝑦 ∈ 𝑉)
20		simplrr 776	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑧 ∈ ( ⊥ ‘𝑆))) ∧ (𝑦(+g‘𝑊)𝑧) ∈ ( ⊥ ‘( ⊥ ‘𝑆))) → 𝑧 ∈ ( ⊥ ‘𝑆))
21	9, 20	sselid 3942	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑧 ∈ ( ⊥ ‘𝑆))) ∧ (𝑦(+g‘𝑊)𝑧) ∈ ( ⊥ ‘( ⊥ ‘𝑆))) → 𝑧 ∈ 𝑉)
22		eqid 2736	. . . . . . . . . . . . . . . 16 ⊢ (Scalar‘𝑊) = (Scalar‘𝑊)
23		eqid 2736	. . . . . . . . . . . . . . . 16 ⊢ (·𝑖‘𝑊) = (·𝑖‘𝑊)
24		eqid 2736	. . . . . . . . . . . . . . . 16 ⊢ (+g‘(Scalar‘𝑊)) = (+g‘(Scalar‘𝑊))
25	22, 23, 7, 11, 24	ipdir 21043	. . . . . . . . . . . . . . 15 ⊢ ((𝑊 ∈ PreHil ∧ (𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → ((𝑦(+g‘𝑊)𝑧)(·𝑖‘𝑊)𝑧) = ((𝑦(·𝑖‘𝑊)𝑧)(+g‘(Scalar‘𝑊))(𝑧(·𝑖‘𝑊)𝑧)))
26	16, 19, 21, 21, 25	syl13anc 1372	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑧 ∈ ( ⊥ ‘𝑆))) ∧ (𝑦(+g‘𝑊)𝑧) ∈ ( ⊥ ‘( ⊥ ‘𝑆))) → ((𝑦(+g‘𝑊)𝑧)(·𝑖‘𝑊)𝑧) = ((𝑦(·𝑖‘𝑊)𝑧)(+g‘(Scalar‘𝑊))(𝑧(·𝑖‘𝑊)𝑧)))
27		eqid 2736	. . . . . . . . . . . . . . . . . 18 ⊢ (0g‘(Scalar‘𝑊)) = (0g‘(Scalar‘𝑊))
28	7, 23, 22, 27, 8	ocvi 21073	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑧 ∈ ( ⊥ ‘𝑆) ∧ 𝑦 ∈ 𝑆) → (𝑧(·𝑖‘𝑊)𝑦) = (0g‘(Scalar‘𝑊)))
29	20, 18, 28	syl2anc 584	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑧 ∈ ( ⊥ ‘𝑆))) ∧ (𝑦(+g‘𝑊)𝑧) ∈ ( ⊥ ‘( ⊥ ‘𝑆))) → (𝑧(·𝑖‘𝑊)𝑦) = (0g‘(Scalar‘𝑊)))
30	22, 23, 7, 27	iporthcom 21039	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑊 ∈ PreHil ∧ 𝑧 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) → ((𝑧(·𝑖‘𝑊)𝑦) = (0g‘(Scalar‘𝑊)) ↔ (𝑦(·𝑖‘𝑊)𝑧) = (0g‘(Scalar‘𝑊))))
31	16, 21, 19, 30	syl3anc 1371	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑧 ∈ ( ⊥ ‘𝑆))) ∧ (𝑦(+g‘𝑊)𝑧) ∈ ( ⊥ ‘( ⊥ ‘𝑆))) → ((𝑧(·𝑖‘𝑊)𝑦) = (0g‘(Scalar‘𝑊)) ↔ (𝑦(·𝑖‘𝑊)𝑧) = (0g‘(Scalar‘𝑊))))
32	29, 31	mpbid 231	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑧 ∈ ( ⊥ ‘𝑆))) ∧ (𝑦(+g‘𝑊)𝑧) ∈ ( ⊥ ‘( ⊥ ‘𝑆))) → (𝑦(·𝑖‘𝑊)𝑧) = (0g‘(Scalar‘𝑊)))
33	32	oveq1d 7372	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑧 ∈ ( ⊥ ‘𝑆))) ∧ (𝑦(+g‘𝑊)𝑧) ∈ ( ⊥ ‘( ⊥ ‘𝑆))) → ((𝑦(·𝑖‘𝑊)𝑧)(+g‘(Scalar‘𝑊))(𝑧(·𝑖‘𝑊)𝑧)) = ((0g‘(Scalar‘𝑊))(+g‘(Scalar‘𝑊))(𝑧(·𝑖‘𝑊)𝑧)))
34	16, 4	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑧 ∈ ( ⊥ ‘𝑆))) ∧ (𝑦(+g‘𝑊)𝑧) ∈ ( ⊥ ‘( ⊥ ‘𝑆))) → 𝑊 ∈ LMod)
35	22	lmodfgrp 20331	. . . . . . . . . . . . . . . 16 ⊢ (𝑊 ∈ LMod → (Scalar‘𝑊) ∈ Grp)
36	34, 35	syl 17	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑧 ∈ ( ⊥ ‘𝑆))) ∧ (𝑦(+g‘𝑊)𝑧) ∈ ( ⊥ ‘( ⊥ ‘𝑆))) → (Scalar‘𝑊) ∈ Grp)
37		eqid 2736	. . . . . . . . . . . . . . . . 17 ⊢ (Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊))
38	22, 23, 7, 37	ipcl 21037	. . . . . . . . . . . . . . . 16 ⊢ ((𝑊 ∈ PreHil ∧ 𝑧 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉) → (𝑧(·𝑖‘𝑊)𝑧) ∈ (Base‘(Scalar‘𝑊)))
39	16, 21, 21, 38	syl3anc 1371	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑧 ∈ ( ⊥ ‘𝑆))) ∧ (𝑦(+g‘𝑊)𝑧) ∈ ( ⊥ ‘( ⊥ ‘𝑆))) → (𝑧(·𝑖‘𝑊)𝑧) ∈ (Base‘(Scalar‘𝑊)))
40	37, 24, 27	grplid 18780	. . . . . . . . . . . . . . 15 ⊢ (((Scalar‘𝑊) ∈ Grp ∧ (𝑧(·𝑖‘𝑊)𝑧) ∈ (Base‘(Scalar‘𝑊))) → ((0g‘(Scalar‘𝑊))(+g‘(Scalar‘𝑊))(𝑧(·𝑖‘𝑊)𝑧)) = (𝑧(·𝑖‘𝑊)𝑧))
41	36, 39, 40	syl2anc 584	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑧 ∈ ( ⊥ ‘𝑆))) ∧ (𝑦(+g‘𝑊)𝑧) ∈ ( ⊥ ‘( ⊥ ‘𝑆))) → ((0g‘(Scalar‘𝑊))(+g‘(Scalar‘𝑊))(𝑧(·𝑖‘𝑊)𝑧)) = (𝑧(·𝑖‘𝑊)𝑧))
42	26, 33, 41	3eqtrd 2780	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑧 ∈ ( ⊥ ‘𝑆))) ∧ (𝑦(+g‘𝑊)𝑧) ∈ ( ⊥ ‘( ⊥ ‘𝑆))) → ((𝑦(+g‘𝑊)𝑧)(·𝑖‘𝑊)𝑧) = (𝑧(·𝑖‘𝑊)𝑧))
43		simpr 485	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑧 ∈ ( ⊥ ‘𝑆))) ∧ (𝑦(+g‘𝑊)𝑧) ∈ ( ⊥ ‘( ⊥ ‘𝑆))) → (𝑦(+g‘𝑊)𝑧) ∈ ( ⊥ ‘( ⊥ ‘𝑆)))
44	7, 23, 22, 27, 8	ocvi 21073	. . . . . . . . . . . . . 14 ⊢ (((𝑦(+g‘𝑊)𝑧) ∈ ( ⊥ ‘( ⊥ ‘𝑆)) ∧ 𝑧 ∈ ( ⊥ ‘𝑆)) → ((𝑦(+g‘𝑊)𝑧)(·𝑖‘𝑊)𝑧) = (0g‘(Scalar‘𝑊)))
45	43, 20, 44	syl2anc 584	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑧 ∈ ( ⊥ ‘𝑆))) ∧ (𝑦(+g‘𝑊)𝑧) ∈ ( ⊥ ‘( ⊥ ‘𝑆))) → ((𝑦(+g‘𝑊)𝑧)(·𝑖‘𝑊)𝑧) = (0g‘(Scalar‘𝑊)))
46	42, 45	eqtr3d 2778	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑧 ∈ ( ⊥ ‘𝑆))) ∧ (𝑦(+g‘𝑊)𝑧) ∈ ( ⊥ ‘( ⊥ ‘𝑆))) → (𝑧(·𝑖‘𝑊)𝑧) = (0g‘(Scalar‘𝑊)))
47		eqid 2736	. . . . . . . . . . . . . 14 ⊢ (0g‘𝑊) = (0g‘𝑊)
48	22, 23, 7, 27, 47	ipeq0 21042	. . . . . . . . . . . . 13 ⊢ ((𝑊 ∈ PreHil ∧ 𝑧 ∈ 𝑉) → ((𝑧(·𝑖‘𝑊)𝑧) = (0g‘(Scalar‘𝑊)) ↔ 𝑧 = (0g‘𝑊)))
49	16, 21, 48	syl2anc 584	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑧 ∈ ( ⊥ ‘𝑆))) ∧ (𝑦(+g‘𝑊)𝑧) ∈ ( ⊥ ‘( ⊥ ‘𝑆))) → ((𝑧(·𝑖‘𝑊)𝑧) = (0g‘(Scalar‘𝑊)) ↔ 𝑧 = (0g‘𝑊)))
50	46, 49	mpbid 231	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑧 ∈ ( ⊥ ‘𝑆))) ∧ (𝑦(+g‘𝑊)𝑧) ∈ ( ⊥ ‘( ⊥ ‘𝑆))) → 𝑧 = (0g‘𝑊))
51	50	oveq2d 7373	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑧 ∈ ( ⊥ ‘𝑆))) ∧ (𝑦(+g‘𝑊)𝑧) ∈ ( ⊥ ‘( ⊥ ‘𝑆))) → (𝑦(+g‘𝑊)𝑧) = (𝑦(+g‘𝑊)(0g‘𝑊)))
52		lmodgrp 20329	. . . . . . . . . . . . 13 ⊢ (𝑊 ∈ LMod → 𝑊 ∈ Grp)
53	5, 52	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑊 ∈ Grp)
54	53	ad2antrr 724	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑧 ∈ ( ⊥ ‘𝑆))) ∧ (𝑦(+g‘𝑊)𝑧) ∈ ( ⊥ ‘( ⊥ ‘𝑆))) → 𝑊 ∈ Grp)
55	7, 11, 47	grprid 18781	. . . . . . . . . . 11 ⊢ ((𝑊 ∈ Grp ∧ 𝑦 ∈ 𝑉) → (𝑦(+g‘𝑊)(0g‘𝑊)) = 𝑦)
56	54, 19, 55	syl2anc 584	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑧 ∈ ( ⊥ ‘𝑆))) ∧ (𝑦(+g‘𝑊)𝑧) ∈ ( ⊥ ‘( ⊥ ‘𝑆))) → (𝑦(+g‘𝑊)(0g‘𝑊)) = 𝑦)
57	51, 56	eqtrd 2776	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑧 ∈ ( ⊥ ‘𝑆))) ∧ (𝑦(+g‘𝑊)𝑧) ∈ ( ⊥ ‘( ⊥ ‘𝑆))) → (𝑦(+g‘𝑊)𝑧) = 𝑦)
58	57, 18	eqeltrd 2838	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑧 ∈ ( ⊥ ‘𝑆))) ∧ (𝑦(+g‘𝑊)𝑧) ∈ ( ⊥ ‘( ⊥ ‘𝑆))) → (𝑦(+g‘𝑊)𝑧) ∈ 𝑆)
59	58	ex 413	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑧 ∈ ( ⊥ ‘𝑆))) → ((𝑦(+g‘𝑊)𝑧) ∈ ( ⊥ ‘( ⊥ ‘𝑆)) → (𝑦(+g‘𝑊)𝑧) ∈ 𝑆))
60		eleq1 2825	. . . . . . . 8 ⊢ (𝑥 = (𝑦(+g‘𝑊)𝑧) → (𝑥 ∈ ( ⊥ ‘( ⊥ ‘𝑆)) ↔ (𝑦(+g‘𝑊)𝑧) ∈ ( ⊥ ‘( ⊥ ‘𝑆))))
61		eleq1 2825	. . . . . . . 8 ⊢ (𝑥 = (𝑦(+g‘𝑊)𝑧) → (𝑥 ∈ 𝑆 ↔ (𝑦(+g‘𝑊)𝑧) ∈ 𝑆))
62	60, 61	imbi12d 344	. . . . . . 7 ⊢ (𝑥 = (𝑦(+g‘𝑊)𝑧) → ((𝑥 ∈ ( ⊥ ‘( ⊥ ‘𝑆)) → 𝑥 ∈ 𝑆) ↔ ((𝑦(+g‘𝑊)𝑧) ∈ ( ⊥ ‘( ⊥ ‘𝑆)) → (𝑦(+g‘𝑊)𝑧) ∈ 𝑆)))
63	59, 62	syl5ibrcom 246	. . . . . 6 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑧 ∈ ( ⊥ ‘𝑆))) → (𝑥 = (𝑦(+g‘𝑊)𝑧) → (𝑥 ∈ ( ⊥ ‘( ⊥ ‘𝑆)) → 𝑥 ∈ 𝑆)))
64	63	rexlimdvva 3205	. . . . 5 ⊢ (𝜑 → (∃𝑦 ∈ 𝑆 ∃𝑧 ∈ ( ⊥ ‘𝑆)𝑥 = (𝑦(+g‘𝑊)𝑧) → (𝑥 ∈ ( ⊥ ‘( ⊥ ‘𝑆)) → 𝑥 ∈ 𝑆)))
65	15, 64	syld 47	. . . 4 ⊢ (𝜑 → (𝑥 ∈ ( ⊥ ‘( ⊥ ‘𝑆)) → (𝑥 ∈ ( ⊥ ‘( ⊥ ‘𝑆)) → 𝑥 ∈ 𝑆)))
66	65	pm2.43d 53	. . 3 ⊢ (𝜑 → (𝑥 ∈ ( ⊥ ‘( ⊥ ‘𝑆)) → 𝑥 ∈ 𝑆))
67	66	ssrdv 3950	. 2 ⊢ (𝜑 → ( ⊥ ‘( ⊥ ‘𝑆)) ⊆ 𝑆)
68		lsmcss.c	. . . 4 ⊢ 𝐶 = (ClSubSp‘𝑊)
69	7, 68, 8	iscss2 21090	. . 3 ⊢ ((𝑊 ∈ PreHil ∧ 𝑆 ⊆ 𝑉) → (𝑆 ∈ 𝐶 ↔ ( ⊥ ‘( ⊥ ‘𝑆)) ⊆ 𝑆))
70	3, 6, 69	syl2anc 584	. 2 ⊢ (𝜑 → (𝑆 ∈ 𝐶 ↔ ( ⊥ ‘( ⊥ ‘𝑆)) ⊆ 𝑆))
71	67, 70	mpbird 256	1 ⊢ (𝜑 → 𝑆 ∈ 𝐶)

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   = wceq 1541   ∈ wcel 2106  ∃wrex 3073   ⊆ wss 3910  ‘cfv 6496  (class class class)co 7357  Basecbs 17083  +_gcplusg 17133  Scalarcsca 17136  ·_𝑖cip 17138  0_gc0g 17321  Grpcgrp 18748  LSSumclsm 19416  LModclmod 20322  PreHilcphl 21028  ocvcocv 21064  ClSubSpccss 21065

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7803  df-1st 7921  df-2nd 7922  df-tpos 8157  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-er 8648  df-map 8767  df-en 8884  df-dom 8885  df-sdom 8886  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-nn 12154  df-2 12216  df-3 12217  df-4 12218  df-5 12219  df-6 12220  df-7 12221  df-8 12222  df-sets 17036  df-slot 17054  df-ndx 17066  df-base 17084  df-plusg 17146  df-mulr 17147  df-sca 17149  df-vsca 17150  df-ip 17151  df-0g 17323  df-mgm 18497  df-sgrp 18546  df-mnd 18557  df-mhm 18601  df-grp 18751  df-ghm 19006  df-lsm 19418  df-mgp 19897  df-ur 19914  df-ring 19966  df-oppr 20049  df-rnghom 20146  df-staf 20304  df-srng 20305  df-lmod 20324  df-lmhm 20483  df-lvec 20564  df-sra 20633  df-rgmod 20634  df-phl 21030  df-ocv 21067  df-css 21068

This theorem is referenced by:  pjcss  21122