Theorem lsmcss 20381

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 3914	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ ( ⊥ ‘( ⊥ ‘𝑆)) → 𝑥 ∈ (𝑆 ⊕ ( ⊥ ‘𝑆))))
3		lsmcss.1	. . . . . . . 8 ⊢ (𝜑 → 𝑊 ∈ PreHil)
4		phllmod 20319	. . . . . . . 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 20359	. . . . . . . 8 ⊢ ( ⊥ ‘𝑆) ⊆ 𝑉
10	9	a1i 11	. . . . . . 7 ⊢ (𝜑 → ( ⊥ ‘𝑆) ⊆ 𝑉)
11		eqid 2798	. . . . . . . 8 ⊢ (+g‘𝑊) = (+g‘𝑊)
12		lsmcss.p	. . . . . . . 8 ⊢ ⊕ = (LSSum‘𝑊)
13	7, 11, 12	lsmelvalx 18757	. . . . . . 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 1368	. . . . . 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 725	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑧 ∈ ( ⊥ ‘𝑆))) ∧ (𝑦(+g‘𝑊)𝑧) ∈ ( ⊥ ‘( ⊥ ‘𝑆))) → 𝑊 ∈ PreHil)
17	6	ad2antrr 725	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑧 ∈ ( ⊥ ‘𝑆))) ∧ (𝑦(+g‘𝑊)𝑧) ∈ ( ⊥ ‘( ⊥ ‘𝑆))) → 𝑆 ⊆ 𝑉)
18		simplrl 776	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑧 ∈ ( ⊥ ‘𝑆))) ∧ (𝑦(+g‘𝑊)𝑧) ∈ ( ⊥ ‘( ⊥ ‘𝑆))) → 𝑦 ∈ 𝑆)
19	17, 18	sseldd 3916	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑧 ∈ ( ⊥ ‘𝑆))) ∧ (𝑦(+g‘𝑊)𝑧) ∈ ( ⊥ ‘( ⊥ ‘𝑆))) → 𝑦 ∈ 𝑉)
20		simplrr 777	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑧 ∈ ( ⊥ ‘𝑆))) ∧ (𝑦(+g‘𝑊)𝑧) ∈ ( ⊥ ‘( ⊥ ‘𝑆))) → 𝑧 ∈ ( ⊥ ‘𝑆))
21	9, 20	sseldi 3913	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑧 ∈ ( ⊥ ‘𝑆))) ∧ (𝑦(+g‘𝑊)𝑧) ∈ ( ⊥ ‘( ⊥ ‘𝑆))) → 𝑧 ∈ 𝑉)
22		eqid 2798	. . . . . . . . . . . . . . . 16 ⊢ (Scalar‘𝑊) = (Scalar‘𝑊)
23		eqid 2798	. . . . . . . . . . . . . . . 16 ⊢ (·𝑖‘𝑊) = (·𝑖‘𝑊)
24		eqid 2798	. . . . . . . . . . . . . . . 16 ⊢ (+g‘(Scalar‘𝑊)) = (+g‘(Scalar‘𝑊))
25	22, 23, 7, 11, 24	ipdir 20328	. . . . . . . . . . . . . . 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 1369	. . . . . . . . . . . . . 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 2798	. . . . . . . . . . . . . . . . . 18 ⊢ (0g‘(Scalar‘𝑊)) = (0g‘(Scalar‘𝑊))
28	7, 23, 22, 27, 8	ocvi 20358	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑧 ∈ ( ⊥ ‘𝑆) ∧ 𝑦 ∈ 𝑆) → (𝑧(·𝑖‘𝑊)𝑦) = (0g‘(Scalar‘𝑊)))
29	20, 18, 28	syl2anc 587	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑧 ∈ ( ⊥ ‘𝑆))) ∧ (𝑦(+g‘𝑊)𝑧) ∈ ( ⊥ ‘( ⊥ ‘𝑆))) → (𝑧(·𝑖‘𝑊)𝑦) = (0g‘(Scalar‘𝑊)))
30	22, 23, 7, 27	iporthcom 20324	. . . . . . . . . . . . . . . . 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 1368	. . . . . . . . . . . . . . . 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 7150	. . . . . . . . . . . . . 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 19636	. . . . . . . . . . . . . . . 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 2798	. . . . . . . . . . . . . . . . 17 ⊢ (Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊))
38	22, 23, 7, 37	ipcl 20322	. . . . . . . . . . . . . . . 16 ⊢ ((𝑊 ∈ PreHil ∧ 𝑧 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉) → (𝑧(·𝑖‘𝑊)𝑧) ∈ (Base‘(Scalar‘𝑊)))
39	16, 21, 21, 38	syl3anc 1368	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑧 ∈ ( ⊥ ‘𝑆))) ∧ (𝑦(+g‘𝑊)𝑧) ∈ ( ⊥ ‘( ⊥ ‘𝑆))) → (𝑧(·𝑖‘𝑊)𝑧) ∈ (Base‘(Scalar‘𝑊)))
40	37, 24, 27	grplid 18125	. . . . . . . . . . . . . . 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 587	. . . . . . . . . . . . . 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 2837	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑧 ∈ ( ⊥ ‘𝑆))) ∧ (𝑦(+g‘𝑊)𝑧) ∈ ( ⊥ ‘( ⊥ ‘𝑆))) → ((𝑦(+g‘𝑊)𝑧)(·𝑖‘𝑊)𝑧) = (𝑧(·𝑖‘𝑊)𝑧))
43		simpr 488	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑧 ∈ ( ⊥ ‘𝑆))) ∧ (𝑦(+g‘𝑊)𝑧) ∈ ( ⊥ ‘( ⊥ ‘𝑆))) → (𝑦(+g‘𝑊)𝑧) ∈ ( ⊥ ‘( ⊥ ‘𝑆)))
44	7, 23, 22, 27, 8	ocvi 20358	. . . . . . . . . . . . . 14 ⊢ (((𝑦(+g‘𝑊)𝑧) ∈ ( ⊥ ‘( ⊥ ‘𝑆)) ∧ 𝑧 ∈ ( ⊥ ‘𝑆)) → ((𝑦(+g‘𝑊)𝑧)(·𝑖‘𝑊)𝑧) = (0g‘(Scalar‘𝑊)))
45	43, 20, 44	syl2anc 587	. . . . . . . . . . . . 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 2835	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑧 ∈ ( ⊥ ‘𝑆))) ∧ (𝑦(+g‘𝑊)𝑧) ∈ ( ⊥ ‘( ⊥ ‘𝑆))) → (𝑧(·𝑖‘𝑊)𝑧) = (0g‘(Scalar‘𝑊)))
47		eqid 2798	. . . . . . . . . . . . . 14 ⊢ (0g‘𝑊) = (0g‘𝑊)
48	22, 23, 7, 27, 47	ipeq0 20327	. . . . . . . . . . . . 13 ⊢ ((𝑊 ∈ PreHil ∧ 𝑧 ∈ 𝑉) → ((𝑧(·𝑖‘𝑊)𝑧) = (0g‘(Scalar‘𝑊)) ↔ 𝑧 = (0g‘𝑊)))
49	16, 21, 48	syl2anc 587	. . . . . . . . . . . 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 7151	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑧 ∈ ( ⊥ ‘𝑆))) ∧ (𝑦(+g‘𝑊)𝑧) ∈ ( ⊥ ‘( ⊥ ‘𝑆))) → (𝑦(+g‘𝑊)𝑧) = (𝑦(+g‘𝑊)(0g‘𝑊)))
52		lmodgrp 19634	. . . . . . . . . . . . 13 ⊢ (𝑊 ∈ LMod → 𝑊 ∈ Grp)
53	5, 52	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑊 ∈ Grp)
54	53	ad2antrr 725	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑧 ∈ ( ⊥ ‘𝑆))) ∧ (𝑦(+g‘𝑊)𝑧) ∈ ( ⊥ ‘( ⊥ ‘𝑆))) → 𝑊 ∈ Grp)
55	7, 11, 47	grprid 18126	. . . . . . . . . . 11 ⊢ ((𝑊 ∈ Grp ∧ 𝑦 ∈ 𝑉) → (𝑦(+g‘𝑊)(0g‘𝑊)) = 𝑦)
56	54, 19, 55	syl2anc 587	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑧 ∈ ( ⊥ ‘𝑆))) ∧ (𝑦(+g‘𝑊)𝑧) ∈ ( ⊥ ‘( ⊥ ‘𝑆))) → (𝑦(+g‘𝑊)(0g‘𝑊)) = 𝑦)
57	51, 56	eqtrd 2833	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑧 ∈ ( ⊥ ‘𝑆))) ∧ (𝑦(+g‘𝑊)𝑧) ∈ ( ⊥ ‘( ⊥ ‘𝑆))) → (𝑦(+g‘𝑊)𝑧) = 𝑦)
58	57, 18	eqeltrd 2890	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑧 ∈ ( ⊥ ‘𝑆))) ∧ (𝑦(+g‘𝑊)𝑧) ∈ ( ⊥ ‘( ⊥ ‘𝑆))) → (𝑦(+g‘𝑊)𝑧) ∈ 𝑆)
59	58	ex 416	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑧 ∈ ( ⊥ ‘𝑆))) → ((𝑦(+g‘𝑊)𝑧) ∈ ( ⊥ ‘( ⊥ ‘𝑆)) → (𝑦(+g‘𝑊)𝑧) ∈ 𝑆))
60		eleq1 2877	. . . . . . . 8 ⊢ (𝑥 = (𝑦(+g‘𝑊)𝑧) → (𝑥 ∈ ( ⊥ ‘( ⊥ ‘𝑆)) ↔ (𝑦(+g‘𝑊)𝑧) ∈ ( ⊥ ‘( ⊥ ‘𝑆))))
61		eleq1 2877	. . . . . . . 8 ⊢ (𝑥 = (𝑦(+g‘𝑊)𝑧) → (𝑥 ∈ 𝑆 ↔ (𝑦(+g‘𝑊)𝑧) ∈ 𝑆))
62	60, 61	imbi12d 348	. . . . . . 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 3253	. . . . 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 3921	. 2 ⊢ (𝜑 → ( ⊥ ‘( ⊥ ‘𝑆)) ⊆ 𝑆)
68		lsmcss.c	. . . 4 ⊢ 𝐶 = (ClSubSp‘𝑊)
69	7, 68, 8	iscss2 20375	. . 3 ⊢ ((𝑊 ∈ PreHil ∧ 𝑆 ⊆ 𝑉) → (𝑆 ∈ 𝐶 ↔ ( ⊥ ‘( ⊥ ‘𝑆)) ⊆ 𝑆))
70	3, 6, 69	syl2anc 587	. 2 ⊢ (𝜑 → (𝑆 ∈ 𝐶 ↔ ( ⊥ ‘( ⊥ ‘𝑆)) ⊆ 𝑆))
71	67, 70	mpbird 260	1 ⊢ (𝜑 → 𝑆 ∈ 𝐶)

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1538   ∈ wcel 2111  ∃wrex 3107   ⊆ wss 3881  ‘cfv 6324  (class class class)co 7135  Basecbs 16475  +_gcplusg 16557  Scalarcsca 16560  ·_𝑖cip 16562  0_gc0g 16705  Grpcgrp 18095  LSSumclsm 18751  LModclmod 19627  PreHilcphl 20313  ocvcocv 20349  ClSubSpccss 20350

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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441  ax-cnex 10582  ax-resscn 10583  ax-1cn 10584  ax-icn 10585  ax-addcl 10586  ax-addrcl 10587  ax-mulcl 10588  ax-mulrcl 10589  ax-mulcom 10590  ax-addass 10591  ax-mulass 10592  ax-distr 10593  ax-i2m1 10594  ax-1ne0 10595  ax-1rid 10596  ax-rnegex 10597  ax-rrecex 10598  ax-cnre 10599  ax-pre-lttri 10600  ax-pre-lttrn 10601  ax-pre-ltadd 10602  ax-pre-mulgt0 10603

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-nel 3092  df-ral 3111  df-rex 3112  df-reu 3113  df-rmo 3114  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-pss 3900  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4801  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-tr 5137  df-id 5425  df-eprel 5430  df-po 5438  df-so 5439  df-fr 5478  df-we 5480  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-pred 6116  df-ord 6162  df-on 6163  df-lim 6164  df-suc 6165  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-riota 7093  df-ov 7138  df-oprab 7139  df-mpo 7140  df-om 7561  df-1st 7671  df-2nd 7672  df-tpos 7875  df-wrecs 7930  df-recs 7991  df-rdg 8029  df-er 8272  df-map 8391  df-en 8493  df-dom 8494  df-sdom 8495  df-pnf 10666  df-mnf 10667  df-xr 10668  df-ltxr 10669  df-le 10670  df-sub 10861  df-neg 10862  df-nn 11626  df-2 11688  df-3 11689  df-4 11690  df-5 11691  df-6 11692  df-7 11693  df-8 11694  df-ndx 16478  df-slot 16479  df-base 16481  df-sets 16482  df-plusg 16570  df-mulr 16571  df-sca 16573  df-vsca 16574  df-ip 16575  df-0g 16707  df-mgm 17844  df-sgrp 17893  df-mnd 17904  df-mhm 17948  df-grp 18098  df-ghm 18348  df-lsm 18753  df-mgp 19233  df-ur 19245  df-ring 19292  df-oppr 19369  df-rnghom 19463  df-staf 19609  df-srng 19610  df-lmod 19629  df-lmhm 19787  df-lvec 19868  df-sra 19937  df-rgmod 19938  df-phl 20315  df-ocv 20352  df-css 20353

This theorem is referenced by:  pjcss  20405