Theorem lsmcss 21853

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 3935	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ ( ⊥ ‘( ⊥ ‘𝑆)) → 𝑥 ∈ (𝑆 ⊕ ( ⊥ ‘𝑆))))
3		lsmcss.1	. . . . . . . 8 ⊢ (𝜑 → 𝑊 ∈ PreHil)
4		phllmod 21791	. . . . . . . 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 21831	. . . . . . . 8 ⊢ ( ⊥ ‘𝑆) ⊆ 𝑉
10	9	a1i 11	. . . . . . 7 ⊢ (𝜑 → ( ⊥ ‘𝑆) ⊆ 𝑉)
11		eqid 2762	. . . . . . . 8 ⊢ (+g‘𝑊) = (+g‘𝑊)
12		lsmcss.p	. . . . . . . 8 ⊢ ⊕ = (LSSum‘𝑊)
13	7, 11, 12	lsmelvalx 19716	. . . . . . 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 1397	. . . . . 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 738	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑧 ∈ ( ⊥ ‘𝑆))) ∧ (𝑦(+g‘𝑊)𝑧) ∈ ( ⊥ ‘( ⊥ ‘𝑆))) → 𝑊 ∈ PreHil)
17	6	ad2antrr 738	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑧 ∈ ( ⊥ ‘𝑆))) ∧ (𝑦(+g‘𝑊)𝑧) ∈ ( ⊥ ‘( ⊥ ‘𝑆))) → 𝑆 ⊆ 𝑉)
18		simplrl 788	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑧 ∈ ( ⊥ ‘𝑆))) ∧ (𝑦(+g‘𝑊)𝑧) ∈ ( ⊥ ‘( ⊥ ‘𝑆))) → 𝑦 ∈ 𝑆)
19	17, 18	sseldd 3937	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑧 ∈ ( ⊥ ‘𝑆))) ∧ (𝑦(+g‘𝑊)𝑧) ∈ ( ⊥ ‘( ⊥ ‘𝑆))) → 𝑦 ∈ 𝑉)
20		simplrr 789	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑧 ∈ ( ⊥ ‘𝑆))) ∧ (𝑦(+g‘𝑊)𝑧) ∈ ( ⊥ ‘( ⊥ ‘𝑆))) → 𝑧 ∈ ( ⊥ ‘𝑆))
21	9, 20	sselid 3934	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑧 ∈ ( ⊥ ‘𝑆))) ∧ (𝑦(+g‘𝑊)𝑧) ∈ ( ⊥ ‘( ⊥ ‘𝑆))) → 𝑧 ∈ 𝑉)
22		eqid 2762	. . . . . . . . . . . . . . . 16 ⊢ (Scalar‘𝑊) = (Scalar‘𝑊)
23		eqid 2762	. . . . . . . . . . . . . . . 16 ⊢ (·𝑖‘𝑊) = (·𝑖‘𝑊)
24		eqid 2762	. . . . . . . . . . . . . . . 16 ⊢ (+g‘(Scalar‘𝑊)) = (+g‘(Scalar‘𝑊))
25	22, 23, 7, 11, 24	ipdir 21800	. . . . . . . . . . . . . . 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 1398	. . . . . . . . . . . . . 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 2762	. . . . . . . . . . . . . . . . . 18 ⊢ (0g‘(Scalar‘𝑊)) = (0g‘(Scalar‘𝑊))
28	7, 23, 22, 27, 8	ocvi 21830	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑧 ∈ ( ⊥ ‘𝑆) ∧ 𝑦 ∈ 𝑆) → (𝑧(·𝑖‘𝑊)𝑦) = (0g‘(Scalar‘𝑊)))
29	20, 18, 28	syl2anc 595	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑧 ∈ ( ⊥ ‘𝑆))) ∧ (𝑦(+g‘𝑊)𝑧) ∈ ( ⊥ ‘( ⊥ ‘𝑆))) → (𝑧(·𝑖‘𝑊)𝑦) = (0g‘(Scalar‘𝑊)))
30	22, 23, 7, 27	iporthcom 21796	. . . . . . . . . . . . . . . . 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 1397	. . . . . . . . . . . . . . . 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 7427	. . . . . . . . . . . . . 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 21001	. . . . . . . . . . . . . . . 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 2762	. . . . . . . . . . . . . . . . 17 ⊢ (Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊))
38	22, 23, 7, 37	ipcl 21794	. . . . . . . . . . . . . . . 16 ⊢ ((𝑊 ∈ PreHil ∧ 𝑧 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉) → (𝑧(·𝑖‘𝑊)𝑧) ∈ (Base‘(Scalar‘𝑊)))
39	16, 21, 21, 38	syl3anc 1397	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑧 ∈ ( ⊥ ‘𝑆))) ∧ (𝑦(+g‘𝑊)𝑧) ∈ ( ⊥ ‘( ⊥ ‘𝑆))) → (𝑧(·𝑖‘𝑊)𝑧) ∈ (Base‘(Scalar‘𝑊)))
40	37, 24, 27	grplid 19040	. . . . . . . . . . . . . . 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 595	. . . . . . . . . . . . . 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 2801	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑧 ∈ ( ⊥ ‘𝑆))) ∧ (𝑦(+g‘𝑊)𝑧) ∈ ( ⊥ ‘( ⊥ ‘𝑆))) → ((𝑦(+g‘𝑊)𝑧)(·𝑖‘𝑊)𝑧) = (𝑧(·𝑖‘𝑊)𝑧))
43		simpr 489	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑧 ∈ ( ⊥ ‘𝑆))) ∧ (𝑦(+g‘𝑊)𝑧) ∈ ( ⊥ ‘( ⊥ ‘𝑆))) → (𝑦(+g‘𝑊)𝑧) ∈ ( ⊥ ‘( ⊥ ‘𝑆)))
44	7, 23, 22, 27, 8	ocvi 21830	. . . . . . . . . . . . . 14 ⊢ (((𝑦(+g‘𝑊)𝑧) ∈ ( ⊥ ‘( ⊥ ‘𝑆)) ∧ 𝑧 ∈ ( ⊥ ‘𝑆)) → ((𝑦(+g‘𝑊)𝑧)(·𝑖‘𝑊)𝑧) = (0g‘(Scalar‘𝑊)))
45	43, 20, 44	syl2anc 595	. . . . . . . . . . . . 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 2799	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑧 ∈ ( ⊥ ‘𝑆))) ∧ (𝑦(+g‘𝑊)𝑧) ∈ ( ⊥ ‘( ⊥ ‘𝑆))) → (𝑧(·𝑖‘𝑊)𝑧) = (0g‘(Scalar‘𝑊)))
47		eqid 2762	. . . . . . . . . . . . . 14 ⊢ (0g‘𝑊) = (0g‘𝑊)
48	22, 23, 7, 27, 47	ipeq0 21799	. . . . . . . . . . . . 13 ⊢ ((𝑊 ∈ PreHil ∧ 𝑧 ∈ 𝑉) → ((𝑧(·𝑖‘𝑊)𝑧) = (0g‘(Scalar‘𝑊)) ↔ 𝑧 = (0g‘𝑊)))
49	16, 21, 48	syl2anc 595	. . . . . . . . . . . 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 7428	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑧 ∈ ( ⊥ ‘𝑆))) ∧ (𝑦(+g‘𝑊)𝑧) ∈ ( ⊥ ‘( ⊥ ‘𝑆))) → (𝑦(+g‘𝑊)𝑧) = (𝑦(+g‘𝑊)(0g‘𝑊)))
52		lmodgrp 20999	. . . . . . . . . . . . 13 ⊢ (𝑊 ∈ LMod → 𝑊 ∈ Grp)
53	5, 52	syl 18	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑊 ∈ Grp)
54	53	ad2antrr 738	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑧 ∈ ( ⊥ ‘𝑆))) ∧ (𝑦(+g‘𝑊)𝑧) ∈ ( ⊥ ‘( ⊥ ‘𝑆))) → 𝑊 ∈ Grp)
55	7, 11, 47	grprid 19041	. . . . . . . . . . 11 ⊢ ((𝑊 ∈ Grp ∧ 𝑦 ∈ 𝑉) → (𝑦(+g‘𝑊)(0g‘𝑊)) = 𝑦)
56	54, 19, 55	syl2anc 595	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑧 ∈ ( ⊥ ‘𝑆))) ∧ (𝑦(+g‘𝑊)𝑧) ∈ ( ⊥ ‘( ⊥ ‘𝑆))) → (𝑦(+g‘𝑊)(0g‘𝑊)) = 𝑦)
57	51, 56	eqtrd 2797	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑧 ∈ ( ⊥ ‘𝑆))) ∧ (𝑦(+g‘𝑊)𝑧) ∈ ( ⊥ ‘( ⊥ ‘𝑆))) → (𝑦(+g‘𝑊)𝑧) = 𝑦)
58	57, 18	eqeltrd 2862	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑧 ∈ ( ⊥ ‘𝑆))) ∧ (𝑦(+g‘𝑊)𝑧) ∈ ( ⊥ ‘( ⊥ ‘𝑆))) → (𝑦(+g‘𝑊)𝑧) ∈ 𝑆)
59	58	ex 417	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑧 ∈ ( ⊥ ‘𝑆))) → ((𝑦(+g‘𝑊)𝑧) ∈ ( ⊥ ‘( ⊥ ‘𝑆)) → (𝑦(+g‘𝑊)𝑧) ∈ 𝑆))
60		eleq1 2850	. . . . . . . 8 ⊢ (𝑥 = (𝑦(+g‘𝑊)𝑧) → (𝑥 ∈ ( ⊥ ‘( ⊥ ‘𝑆)) ↔ (𝑦(+g‘𝑊)𝑧) ∈ ( ⊥ ‘( ⊥ ‘𝑆))))
61		eleq1 2850	. . . . . . . 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 3221	. . . . 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 3942	. 2 ⊢ (𝜑 → ( ⊥ ‘( ⊥ ‘𝑆)) ⊆ 𝑆)
68		lsmcss.c	. . . 4 ⊢ 𝐶 = (ClSubSp‘𝑊)
69	7, 68, 8	iscss2 21847	. . 3 ⊢ ((𝑊 ∈ PreHil ∧ 𝑆 ⊆ 𝑉) → (𝑆 ∈ 𝐶 ↔ ( ⊥ ‘( ⊥ ‘𝑆)) ⊆ 𝑆))
70	3, 6, 69	syl2anc 595	. 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 400   = wceq 1569   ∈ wcel 2142  ∃wrex 3088   ⊆ wss 3904  ‘cfv 6536  (class class class)co 7412  Basecbs 17275  +_gcplusg 17316  Scalarcsca 17319  ·_𝑖cip 17321  0_gc0g 17498  Grpcgrp 19006  LSSumclsm 19710  LModclmod 20992  PreHilcphl 21785  ocvcocv 21821  ClSubSpccss 21822

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-rep 5237  ax-sep 5256  ax-nul 5268  ax-pow 5335  ax-pr 5403  ax-un 7734  ax-cnex 11162  ax-resscn 11163  ax-1cn 11164  ax-icn 11165  ax-addcl 11166  ax-addrcl 11167  ax-mulcl 11168  ax-mulrcl 11169  ax-mulcom 11170  ax-addass 11171  ax-mulass 11172  ax-distr 11173  ax-i2m1 11174  ax-1ne0 11175  ax-1rid 11176  ax-rnegex 11177  ax-rrecex 11178  ax-cnre 11179  ax-pre-lttri 11180  ax-pre-lttrn 11181  ax-pre-ltadd 11182  ax-pre-mulgt0 11183

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1103  df-3an 1104  df-tru 1572  df-fal 1582  df-ex 1809  df-nf 1813  df-sb 2096  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-nel 3064  df-ral 3079  df-rex 3089  df-rmo 3368  df-reu 3369  df-rab 3416  df-v 3456  df-sbc 3744  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-pss 3924  df-nul 4286  df-if 4487  df-pw 4563  df-sn 4589  df-pr 4591  df-op 4595  df-uni 4872  df-iun 4957  df-br 5109  df-opab 5173  df-mpt 5192  df-tr 5218  df-id 5555  df-eprel 5560  df-po 5568  df-so 5569  df-fr 5613  df-we 5615  df-xp 5666  df-rel 5667  df-cnv 5668  df-co 5669  df-dm 5670  df-rn 5671  df-res 5672  df-ima 5673  df-pred 6302  df-ord 6363  df-on 6364  df-lim 6365  df-suc 6366  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-riota 7369  df-ov 7415  df-oprab 7416  df-mpo 7417  df-om 7861  df-1st 7984  df-2nd 7985  df-tpos 8220  df-frecs 8276  df-wrecs 8307  df-recs 8356  df-rdg 8395  df-er 8692  df-map 8824  df-en 8942  df-dom 8943  df-sdom 8944  df-pnf 11251  df-mnf 11252  df-xr 11253  df-ltxr 11254  df-le 11255  df-sub 11449  df-neg 11450  df-nn 12240  df-2 12309  df-3 12310  df-4 12311  df-5 12312  df-6 12313  df-7 12314  df-8 12315  df-sets 17230  df-slot 17248  df-ndx 17260  df-base 17276  df-plusg 17329  df-mulr 17330  df-sca 17332  df-vsca 17333  df-ip 17334  df-0g 17500  df-mgm 18704  df-sgrp 18783  df-mnd 18799  df-mhm 18847  df-grp 19009  df-ghm 19290  df-lsm 19712  df-mgp 20223  df-ur 20270  df-ring 20323  df-oppr 20426  df-rhm 20561  df-staf 20953  df-srng 20954  df-lmod 20994  df-lmhm 21154  df-lvec 21235  df-sra 21305  df-rgmod 21306  df-phl 21787  df-ocv 21824  df-css 21825

This theorem is used by:  pjcss  21877