Theorem lsmcss 21959

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 3929	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ ( ⊥ ‘( ⊥ ‘𝑆)) → 𝑥 ∈ (𝑆 ⊕ ( ⊥ ‘𝑆))))
3		lsmcss.1	. . . . . . . 8 ⊢ (𝜑 → 𝑊 ∈ PreHil)
4		phllmod 21897	. . . . . . . 8 ⊢ (𝑊 ∈ PreHil → 𝑊 ∈ LMod)
5	3, 4	syl 18	. . . . . . 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 21937	. . . . . . . 8 ⊢ ( ⊥ ‘𝑆) ⊆ 𝑉
10	9	a1i 11	. . . . . . 7 ⊢ (𝜑 → ( ⊥ ‘𝑆) ⊆ 𝑉)
11		eqid 2760	. . . . . . . 8 ⊢ (+g‘𝑊) = (+g‘𝑊)
12		lsmcss.p	. . . . . . . 8 ⊢ ⊕ = (LSSum‘𝑊)
13	7, 11, 12	lsmelvalx 19815	. . . . . . 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 1398	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ (𝑆 ⊕ ( ⊥ ‘𝑆)) ↔ ∃𝑦 ∈ 𝑆 ∃𝑧 ∈ ( ⊥ ‘𝑆)𝑥 = (𝑦(+g‘𝑊)𝑧)))
15	2, 14	sylibd 242	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ ( ⊥ ‘( ⊥ ‘𝑆)) → ∃𝑦 ∈ 𝑆 ∃𝑧 ∈ ( ⊥ ‘𝑆)𝑥 = (𝑦(+g‘𝑊)𝑧)))
16	3	ad2antrr 739	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑧 ∈ ( ⊥ ‘𝑆))) ∧ (𝑦(+g‘𝑊)𝑧) ∈ ( ⊥ ‘( ⊥ ‘𝑆))) → 𝑊 ∈ PreHil)
17	6	ad2antrr 739	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑧 ∈ ( ⊥ ‘𝑆))) ∧ (𝑦(+g‘𝑊)𝑧) ∈ ( ⊥ ‘( ⊥ ‘𝑆))) → 𝑆 ⊆ 𝑉)
18		simplrl 789	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑧 ∈ ( ⊥ ‘𝑆))) ∧ (𝑦(+g‘𝑊)𝑧) ∈ ( ⊥ ‘( ⊥ ‘𝑆))) → 𝑦 ∈ 𝑆)
19	17, 18	sseldd 3931	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑧 ∈ ( ⊥ ‘𝑆))) ∧ (𝑦(+g‘𝑊)𝑧) ∈ ( ⊥ ‘( ⊥ ‘𝑆))) → 𝑦 ∈ 𝑉)
20		simplrr 790	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑧 ∈ ( ⊥ ‘𝑆))) ∧ (𝑦(+g‘𝑊)𝑧) ∈ ( ⊥ ‘( ⊥ ‘𝑆))) → 𝑧 ∈ ( ⊥ ‘𝑆))
21	9, 20	sselid 3928	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑧 ∈ ( ⊥ ‘𝑆))) ∧ (𝑦(+g‘𝑊)𝑧) ∈ ( ⊥ ‘( ⊥ ‘𝑆))) → 𝑧 ∈ 𝑉)
22		eqid 2760	. . . . . . . . . . . . . . . 16 ⊢ (Scalar‘𝑊) = (Scalar‘𝑊)
23		eqid 2760	. . . . . . . . . . . . . . . 16 ⊢ (·𝑖‘𝑊) = (·𝑖‘𝑊)
24		eqid 2760	. . . . . . . . . . . . . . . 16 ⊢ (+g‘(Scalar‘𝑊)) = (+g‘(Scalar‘𝑊))
25	22, 23, 7, 11, 24	ipdir 21906	. . . . . . . . . . . . . . 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 1399	. . . . . . . . . . . . . 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 2760	. . . . . . . . . . . . . . . . . 18 ⊢ (0g‘(Scalar‘𝑊)) = (0g‘(Scalar‘𝑊))
28	7, 23, 22, 27, 8	ocvi 21936	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑧 ∈ ( ⊥ ‘𝑆) ∧ 𝑦 ∈ 𝑆) → (𝑧(·𝑖‘𝑊)𝑦) = (0g‘(Scalar‘𝑊)))
29	20, 18, 28	syl2anc 596	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑧 ∈ ( ⊥ ‘𝑆))) ∧ (𝑦(+g‘𝑊)𝑧) ∈ ( ⊥ ‘( ⊥ ‘𝑆))) → (𝑧(·𝑖‘𝑊)𝑦) = (0g‘(Scalar‘𝑊)))
30	22, 23, 7, 27	iporthcom 21902	. . . . . . . . . . . . . . . . 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 1398	. . . . . . . . . . . . . . . 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 235	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑧 ∈ ( ⊥ ‘𝑆))) ∧ (𝑦(+g‘𝑊)𝑧) ∈ ( ⊥ ‘( ⊥ ‘𝑆))) → (𝑦(·𝑖‘𝑊)𝑧) = (0g‘(Scalar‘𝑊)))
33	32	oveq1d 7423	. . . . . . . . . . . . . 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 18	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑧 ∈ ( ⊥ ‘𝑆))) ∧ (𝑦(+g‘𝑊)𝑧) ∈ ( ⊥ ‘( ⊥ ‘𝑆))) → 𝑊 ∈ LMod)
35	22	lmodfgrp 21105	. . . . . . . . . . . . . . . 16 ⊢ (𝑊 ∈ LMod → (Scalar‘𝑊) ∈ Grp)
36	34, 35	syl 18	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑧 ∈ ( ⊥ ‘𝑆))) ∧ (𝑦(+g‘𝑊)𝑧) ∈ ( ⊥ ‘( ⊥ ‘𝑆))) → (Scalar‘𝑊) ∈ Grp)
37		eqid 2760	. . . . . . . . . . . . . . . . 17 ⊢ (Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊))
38	22, 23, 7, 37	ipcl 21900	. . . . . . . . . . . . . . . 16 ⊢ ((𝑊 ∈ PreHil ∧ 𝑧 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉) → (𝑧(·𝑖‘𝑊)𝑧) ∈ (Base‘(Scalar‘𝑊)))
39	16, 21, 21, 38	syl3anc 1398	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑧 ∈ ( ⊥ ‘𝑆))) ∧ (𝑦(+g‘𝑊)𝑧) ∈ ( ⊥ ‘( ⊥ ‘𝑆))) → (𝑧(·𝑖‘𝑊)𝑧) ∈ (Base‘(Scalar‘𝑊)))
40	37, 24, 27	grplid 19139	. . . . . . . . . . . . . . 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 596	. . . . . . . . . . . . . 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 2799	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑧 ∈ ( ⊥ ‘𝑆))) ∧ (𝑦(+g‘𝑊)𝑧) ∈ ( ⊥ ‘( ⊥ ‘𝑆))) → ((𝑦(+g‘𝑊)𝑧)(·𝑖‘𝑊)𝑧) = (𝑧(·𝑖‘𝑊)𝑧))
43		simpr 490	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑧 ∈ ( ⊥ ‘𝑆))) ∧ (𝑦(+g‘𝑊)𝑧) ∈ ( ⊥ ‘( ⊥ ‘𝑆))) → (𝑦(+g‘𝑊)𝑧) ∈ ( ⊥ ‘( ⊥ ‘𝑆)))
44	7, 23, 22, 27, 8	ocvi 21936	. . . . . . . . . . . . . 14 ⊢ (((𝑦(+g‘𝑊)𝑧) ∈ ( ⊥ ‘( ⊥ ‘𝑆)) ∧ 𝑧 ∈ ( ⊥ ‘𝑆)) → ((𝑦(+g‘𝑊)𝑧)(·𝑖‘𝑊)𝑧) = (0g‘(Scalar‘𝑊)))
45	43, 20, 44	syl2anc 596	. . . . . . . . . . . . 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 2797	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑧 ∈ ( ⊥ ‘𝑆))) ∧ (𝑦(+g‘𝑊)𝑧) ∈ ( ⊥ ‘( ⊥ ‘𝑆))) → (𝑧(·𝑖‘𝑊)𝑧) = (0g‘(Scalar‘𝑊)))
47		eqid 2760	. . . . . . . . . . . . . 14 ⊢ (0g‘𝑊) = (0g‘𝑊)
48	22, 23, 7, 27, 47	ipeq0 21905	. . . . . . . . . . . . 13 ⊢ ((𝑊 ∈ PreHil ∧ 𝑧 ∈ 𝑉) → ((𝑧(·𝑖‘𝑊)𝑧) = (0g‘(Scalar‘𝑊)) ↔ 𝑧 = (0g‘𝑊)))
49	16, 21, 48	syl2anc 596	. . . . . . . . . . . 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 235	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑧 ∈ ( ⊥ ‘𝑆))) ∧ (𝑦(+g‘𝑊)𝑧) ∈ ( ⊥ ‘( ⊥ ‘𝑆))) → 𝑧 = (0g‘𝑊))
51	50	oveq2d 7424	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑧 ∈ ( ⊥ ‘𝑆))) ∧ (𝑦(+g‘𝑊)𝑧) ∈ ( ⊥ ‘( ⊥ ‘𝑆))) → (𝑦(+g‘𝑊)𝑧) = (𝑦(+g‘𝑊)(0g‘𝑊)))
52		lmodgrp 21103	. . . . . . . . . . . . 13 ⊢ (𝑊 ∈ LMod → 𝑊 ∈ Grp)
53	5, 52	syl 18	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑊 ∈ Grp)
54	53	ad2antrr 739	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑧 ∈ ( ⊥ ‘𝑆))) ∧ (𝑦(+g‘𝑊)𝑧) ∈ ( ⊥ ‘( ⊥ ‘𝑆))) → 𝑊 ∈ Grp)
55	7, 11, 47	grprid 19140	. . . . . . . . . . 11 ⊢ ((𝑊 ∈ Grp ∧ 𝑦 ∈ 𝑉) → (𝑦(+g‘𝑊)(0g‘𝑊)) = 𝑦)
56	54, 19, 55	syl2anc 596	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑧 ∈ ( ⊥ ‘𝑆))) ∧ (𝑦(+g‘𝑊)𝑧) ∈ ( ⊥ ‘( ⊥ ‘𝑆))) → (𝑦(+g‘𝑊)(0g‘𝑊)) = 𝑦)
57	51, 56	eqtrd 2795	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑧 ∈ ( ⊥ ‘𝑆))) ∧ (𝑦(+g‘𝑊)𝑧) ∈ ( ⊥ ‘( ⊥ ‘𝑆))) → (𝑦(+g‘𝑊)𝑧) = 𝑦)
58	57, 18	eqeltrd 2860	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑧 ∈ ( ⊥ ‘𝑆))) ∧ (𝑦(+g‘𝑊)𝑧) ∈ ( ⊥ ‘( ⊥ ‘𝑆))) → (𝑦(+g‘𝑊)𝑧) ∈ 𝑆)
59	58	ex 418	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑧 ∈ ( ⊥ ‘𝑆))) → ((𝑦(+g‘𝑊)𝑧) ∈ ( ⊥ ‘( ⊥ ‘𝑆)) → (𝑦(+g‘𝑊)𝑧) ∈ 𝑆))
60		eleq1 2848	. . . . . . . 8 ⊢ (𝑥 = (𝑦(+g‘𝑊)𝑧) → (𝑥 ∈ ( ⊥ ‘( ⊥ ‘𝑆)) ↔ (𝑦(+g‘𝑊)𝑧) ∈ ( ⊥ ‘( ⊥ ‘𝑆))))
61		eleq1 2848	. . . . . . . 8 ⊢ (𝑥 = (𝑦(+g‘𝑊)𝑧) → (𝑥 ∈ 𝑆 ↔ (𝑦(+g‘𝑊)𝑧) ∈ 𝑆))
62	60, 61	imbi12d 347	. . . . . . 7 ⊢ (𝑥 = (𝑦(+g‘𝑊)𝑧) → ((𝑥 ∈ ( ⊥ ‘( ⊥ ‘𝑆)) → 𝑥 ∈ 𝑆) ↔ ((𝑦(+g‘𝑊)𝑧) ∈ ( ⊥ ‘( ⊥ ‘𝑆)) → (𝑦(+g‘𝑊)𝑧) ∈ 𝑆)))
63	59, 62	syl5ibrcom 250	. . . . . 6 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑧 ∈ ( ⊥ ‘𝑆))) → (𝑥 = (𝑦(+g‘𝑊)𝑧) → (𝑥 ∈ ( ⊥ ‘( ⊥ ‘𝑆)) → 𝑥 ∈ 𝑆)))
64	63	rexlimdvva 3219	. . . . 5 ⊢ (𝜑 → (∃𝑦 ∈ 𝑆 ∃𝑧 ∈ ( ⊥ ‘𝑆)𝑥 = (𝑦(+g‘𝑊)𝑧) → (𝑥 ∈ ( ⊥ ‘( ⊥ ‘𝑆)) → 𝑥 ∈ 𝑆)))
65	15, 64	syld 48	. . . 4 ⊢ (𝜑 → (𝑥 ∈ ( ⊥ ‘( ⊥ ‘𝑆)) → (𝑥 ∈ ( ⊥ ‘( ⊥ ‘𝑆)) → 𝑥 ∈ 𝑆)))
66	65	pm2.43d 54	. . 3 ⊢ (𝜑 → (𝑥 ∈ ( ⊥ ‘( ⊥ ‘𝑆)) → 𝑥 ∈ 𝑆))
67	66	ssrdv 3936	. 2 ⊢ (𝜑 → ( ⊥ ‘( ⊥ ‘𝑆)) ⊆ 𝑆)
68		lsmcss.c	. . . 4 ⊢ 𝐶 = (ClSubSp‘𝑊)
69	7, 68, 8	iscss2 21953	. . 3 ⊢ ((𝑊 ∈ PreHil ∧ 𝑆 ⊆ 𝑉) → (𝑆 ∈ 𝐶 ↔ ( ⊥ ‘( ⊥ ‘𝑆)) ⊆ 𝑆))
70	3, 6, 69	syl2anc 596	. 2 ⊢ (𝜑 → (𝑆 ∈ 𝐶 ↔ ( ⊥ ‘( ⊥ ‘𝑆)) ⊆ 𝑆))
71	67, 70	mpbird 260	1 ⊢ (𝜑 → 𝑆 ∈ 𝐶)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2145  ∃wrex 3086   ⊆ wss 3898  ‘cfv 6527  (class class class)co 7408  Basecbs 17348  +_gcplusg 17389  Scalarcsca 17392  ·_𝑖cip 17394  0_gc0g 17571  Grpcgrp 19105  LSSumclsm 19809  LModclmod 21096  PreHilcphl 21891  ocvcocv 21927  ClSubSpccss 21928

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 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7734  ax-cnex 11227  ax-resscn 11228  ax-1cn 11229  ax-icn 11230  ax-addcl 11231  ax-addrcl 11232  ax-mulcl 11233  ax-mulrcl 11234  ax-mulcom 11235  ax-addass 11236  ax-mulass 11237  ax-distr 11238  ax-i2m1 11239  ax-1ne0 11240  ax-1rid 11241  ax-rnegex 11242  ax-rrecex 11243  ax-cnre 11244  ax-pre-lttri 11245  ax-pre-lttrn 11246  ax-pre-ltadd 11247  ax-pre-mulgt0 11248

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 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-nel 3062  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-pss 3918  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5542  df-eprel 5547  df-po 5555  df-so 5556  df-fr 5600  df-we 5602  df-xp 5653  df-rel 5654  df-cnv 5655  df-co 5656  df-dm 5657  df-rn 5658  df-res 5659  df-ima 5660  df-pred 6293  df-ord 6354  df-on 6355  df-lim 6356  df-suc 6357  df-iota 6483  df-fun 6529  df-fn 6530  df-f 6531  df-f1 6532  df-fo 6533  df-f1o 6534  df-fv 6535  df-riota 7365  df-ov 7411  df-oprab 7412  df-mpo 7413  df-om 7861  df-1st 7984  df-2nd 7985  df-tpos 8221  df-frecs 8277  df-wrecs 8308  df-recs 8357  df-rdg 8396  df-er 8695  df-map 8827  df-en 8952  df-dom 8953  df-sdom 8954  df-pnf 11316  df-mnf 11317  df-xr 11318  df-ltxr 11319  df-le 11320  df-sub 11514  df-neg 11515  df-nn 12305  df-2 12374  df-3 12375  df-4 12376  df-5 12377  df-6 12378  df-7 12379  df-8 12380  df-sets 17303  df-slot 17321  df-ndx 17333  df-base 17349  df-plusg 17402  df-mulr 17403  df-sca 17405  df-vsca 17406  df-ip 17407  df-0g 17573  df-mgm 18777  df-sgrp 18869  df-mnd 18885  df-mhm 18939  df-grp 19108  df-ghm 19389  df-lsm 19811  df-mgp 20322  df-ur 20369  df-ring 20422  df-oppr 20528  df-rhm 20663  df-staf 21057  df-srng 21058  df-lmod 21098  df-lmhm 21258  df-lvec 21339  df-sra 21409  df-rgmod 21410  df-phl 21893  df-ocv 21930  df-css 21931

This theorem is used by:  pjcss  21983