Theorem lsmcss 21622

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 3931	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ ( ⊥ ‘( ⊥ ‘𝑆)) → 𝑥 ∈ (𝑆 ⊕ ( ⊥ ‘𝑆))))
3		lsmcss.1	. . . . . . . 8 ⊢ (𝜑 → 𝑊 ∈ PreHil)
4		phllmod 21560	. . . . . . . 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 21600	. . . . . . . 8 ⊢ ( ⊥ ‘𝑆) ⊆ 𝑉
10	9	a1i 11	. . . . . . 7 ⊢ (𝜑 → ( ⊥ ‘𝑆) ⊆ 𝑉)
11		eqid 2730	. . . . . . . 8 ⊢ (+g‘𝑊) = (+g‘𝑊)
12		lsmcss.p	. . . . . . . 8 ⊢ ⊕ = (LSSum‘𝑊)
13	7, 11, 12	lsmelvalx 19545	. . . . . . 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 1373	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ (𝑆 ⊕ ( ⊥ ‘𝑆)) ↔ ∃𝑦 ∈ 𝑆 ∃𝑧 ∈ ( ⊥ ‘𝑆)𝑥 = (𝑦(+g‘𝑊)𝑧)))
15	2, 14	sylibd 239	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ ( ⊥ ‘( ⊥ ‘𝑆)) → ∃𝑦 ∈ 𝑆 ∃𝑧 ∈ ( ⊥ ‘𝑆)𝑥 = (𝑦(+g‘𝑊)𝑧)))
16	3	ad2antrr 726	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑧 ∈ ( ⊥ ‘𝑆))) ∧ (𝑦(+g‘𝑊)𝑧) ∈ ( ⊥ ‘( ⊥ ‘𝑆))) → 𝑊 ∈ PreHil)
17	6	ad2antrr 726	. . . . . . . . . . . . . . . 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 3933	. . . . . . . . . . . . . . 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	sselid 3930	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑧 ∈ ( ⊥ ‘𝑆))) ∧ (𝑦(+g‘𝑊)𝑧) ∈ ( ⊥ ‘( ⊥ ‘𝑆))) → 𝑧 ∈ 𝑉)
22		eqid 2730	. . . . . . . . . . . . . . . 16 ⊢ (Scalar‘𝑊) = (Scalar‘𝑊)
23		eqid 2730	. . . . . . . . . . . . . . . 16 ⊢ (·𝑖‘𝑊) = (·𝑖‘𝑊)
24		eqid 2730	. . . . . . . . . . . . . . . 16 ⊢ (+g‘(Scalar‘𝑊)) = (+g‘(Scalar‘𝑊))
25	22, 23, 7, 11, 24	ipdir 21569	. . . . . . . . . . . . . . 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 1374	. . . . . . . . . . . . . 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 2730	. . . . . . . . . . . . . . . . . 18 ⊢ (0g‘(Scalar‘𝑊)) = (0g‘(Scalar‘𝑊))
28	7, 23, 22, 27, 8	ocvi 21599	. . . . . . . . . . . . . . . . 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 21565	. . . . . . . . . . . . . . . . 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 1373	. . . . . . . . . . . . . . . 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 232	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑧 ∈ ( ⊥ ‘𝑆))) ∧ (𝑦(+g‘𝑊)𝑧) ∈ ( ⊥ ‘( ⊥ ‘𝑆))) → (𝑦(·𝑖‘𝑊)𝑧) = (0g‘(Scalar‘𝑊)))
33	32	oveq1d 7356	. . . . . . . . . . . . . 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 20795	. . . . . . . . . . . . . . . 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 2730	. . . . . . . . . . . . . . . . 17 ⊢ (Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊))
38	22, 23, 7, 37	ipcl 21563	. . . . . . . . . . . . . . . 16 ⊢ ((𝑊 ∈ PreHil ∧ 𝑧 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉) → (𝑧(·𝑖‘𝑊)𝑧) ∈ (Base‘(Scalar‘𝑊)))
39	16, 21, 21, 38	syl3anc 1373	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑧 ∈ ( ⊥ ‘𝑆))) ∧ (𝑦(+g‘𝑊)𝑧) ∈ ( ⊥ ‘( ⊥ ‘𝑆))) → (𝑧(·𝑖‘𝑊)𝑧) ∈ (Base‘(Scalar‘𝑊)))
40	37, 24, 27	grplid 18872	. . . . . . . . . . . . . . 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 2769	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑧 ∈ ( ⊥ ‘𝑆))) ∧ (𝑦(+g‘𝑊)𝑧) ∈ ( ⊥ ‘( ⊥ ‘𝑆))) → ((𝑦(+g‘𝑊)𝑧)(·𝑖‘𝑊)𝑧) = (𝑧(·𝑖‘𝑊)𝑧))
43		simpr 484	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑧 ∈ ( ⊥ ‘𝑆))) ∧ (𝑦(+g‘𝑊)𝑧) ∈ ( ⊥ ‘( ⊥ ‘𝑆))) → (𝑦(+g‘𝑊)𝑧) ∈ ( ⊥ ‘( ⊥ ‘𝑆)))
44	7, 23, 22, 27, 8	ocvi 21599	. . . . . . . . . . . . . 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 2767	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑧 ∈ ( ⊥ ‘𝑆))) ∧ (𝑦(+g‘𝑊)𝑧) ∈ ( ⊥ ‘( ⊥ ‘𝑆))) → (𝑧(·𝑖‘𝑊)𝑧) = (0g‘(Scalar‘𝑊)))
47		eqid 2730	. . . . . . . . . . . . . 14 ⊢ (0g‘𝑊) = (0g‘𝑊)
48	22, 23, 7, 27, 47	ipeq0 21568	. . . . . . . . . . . . 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 232	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑧 ∈ ( ⊥ ‘𝑆))) ∧ (𝑦(+g‘𝑊)𝑧) ∈ ( ⊥ ‘( ⊥ ‘𝑆))) → 𝑧 = (0g‘𝑊))
51	50	oveq2d 7357	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑧 ∈ ( ⊥ ‘𝑆))) ∧ (𝑦(+g‘𝑊)𝑧) ∈ ( ⊥ ‘( ⊥ ‘𝑆))) → (𝑦(+g‘𝑊)𝑧) = (𝑦(+g‘𝑊)(0g‘𝑊)))
52		lmodgrp 20793	. . . . . . . . . . . . 13 ⊢ (𝑊 ∈ LMod → 𝑊 ∈ Grp)
53	5, 52	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑊 ∈ Grp)
54	53	ad2antrr 726	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑧 ∈ ( ⊥ ‘𝑆))) ∧ (𝑦(+g‘𝑊)𝑧) ∈ ( ⊥ ‘( ⊥ ‘𝑆))) → 𝑊 ∈ Grp)
55	7, 11, 47	grprid 18873	. . . . . . . . . . 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 2765	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑧 ∈ ( ⊥ ‘𝑆))) ∧ (𝑦(+g‘𝑊)𝑧) ∈ ( ⊥ ‘( ⊥ ‘𝑆))) → (𝑦(+g‘𝑊)𝑧) = 𝑦)
58	57, 18	eqeltrd 2829	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑧 ∈ ( ⊥ ‘𝑆))) ∧ (𝑦(+g‘𝑊)𝑧) ∈ ( ⊥ ‘( ⊥ ‘𝑆))) → (𝑦(+g‘𝑊)𝑧) ∈ 𝑆)
59	58	ex 412	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑧 ∈ ( ⊥ ‘𝑆))) → ((𝑦(+g‘𝑊)𝑧) ∈ ( ⊥ ‘( ⊥ ‘𝑆)) → (𝑦(+g‘𝑊)𝑧) ∈ 𝑆))
60		eleq1 2817	. . . . . . . 8 ⊢ (𝑥 = (𝑦(+g‘𝑊)𝑧) → (𝑥 ∈ ( ⊥ ‘( ⊥ ‘𝑆)) ↔ (𝑦(+g‘𝑊)𝑧) ∈ ( ⊥ ‘( ⊥ ‘𝑆))))
61		eleq1 2817	. . . . . . . 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 247	. . . . . 6 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑧 ∈ ( ⊥ ‘𝑆))) → (𝑥 = (𝑦(+g‘𝑊)𝑧) → (𝑥 ∈ ( ⊥ ‘( ⊥ ‘𝑆)) → 𝑥 ∈ 𝑆)))
64	63	rexlimdvva 3187	. . . . 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 3938	. 2 ⊢ (𝜑 → ( ⊥ ‘( ⊥ ‘𝑆)) ⊆ 𝑆)
68		lsmcss.c	. . . 4 ⊢ 𝐶 = (ClSubSp‘𝑊)
69	7, 68, 8	iscss2 21616	. . 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 257	1 ⊢ (𝜑 → 𝑆 ∈ 𝐶)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1541   ∈ wcel 2110  ∃wrex 3054   ⊆ wss 3900  ‘cfv 6477  (class class class)co 7341  Basecbs 17112  +_gcplusg 17153  Scalarcsca 17156  ·_𝑖cip 17158  0_gc0g 17335  Grpcgrp 18838  LSSumclsm 19539  LModclmod 20786  PreHilcphl 21554  ocvcocv 21590  ClSubSpccss 21591

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 1968  ax-7 2009  ax-8 2112  ax-9 2120  ax-10 2143  ax-11 2159  ax-12 2179  ax-ext 2702  ax-rep 5215  ax-sep 5232  ax-nul 5242  ax-pow 5301  ax-pr 5368  ax-un 7663  ax-cnex 11054  ax-resscn 11055  ax-1cn 11056  ax-icn 11057  ax-addcl 11058  ax-addrcl 11059  ax-mulcl 11060  ax-mulrcl 11061  ax-mulcom 11062  ax-addass 11063  ax-mulass 11064  ax-distr 11065  ax-i2m1 11066  ax-1ne0 11067  ax-1rid 11068  ax-rnegex 11069  ax-rrecex 11070  ax-cnre 11071  ax-pre-lttri 11072  ax-pre-lttrn 11073  ax-pre-ltadd 11074  ax-pre-mulgt0 11075

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-nel 3031  df-ral 3046  df-rex 3055  df-rmo 3344  df-reu 3345  df-rab 3394  df-v 3436  df-sbc 3740  df-csb 3849  df-dif 3903  df-un 3905  df-in 3907  df-ss 3917  df-pss 3920  df-nul 4282  df-if 4474  df-pw 4550  df-sn 4575  df-pr 4577  df-op 4581  df-uni 4858  df-iun 4941  df-br 5090  df-opab 5152  df-mpt 5171  df-tr 5197  df-id 5509  df-eprel 5514  df-po 5522  df-so 5523  df-fr 5567  df-we 5569  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-res 5626  df-ima 5627  df-pred 6244  df-ord 6305  df-on 6306  df-lim 6307  df-suc 6308  df-iota 6433  df-fun 6479  df-fn 6480  df-f 6481  df-f1 6482  df-fo 6483  df-f1o 6484  df-fv 6485  df-riota 7298  df-ov 7344  df-oprab 7345  df-mpo 7346  df-om 7792  df-1st 7916  df-2nd 7917  df-tpos 8151  df-frecs 8206  df-wrecs 8237  df-recs 8286  df-rdg 8324  df-er 8617  df-map 8747  df-en 8865  df-dom 8866  df-sdom 8867  df-pnf 11140  df-mnf 11141  df-xr 11142  df-ltxr 11143  df-le 11144  df-sub 11338  df-neg 11339  df-nn 12118  df-2 12180  df-3 12181  df-4 12182  df-5 12183  df-6 12184  df-7 12185  df-8 12186  df-sets 17067  df-slot 17085  df-ndx 17097  df-base 17113  df-plusg 17166  df-mulr 17167  df-sca 17169  df-vsca 17170  df-ip 17171  df-0g 17337  df-mgm 18540  df-sgrp 18619  df-mnd 18635  df-mhm 18683  df-grp 18841  df-ghm 19118  df-lsm 19541  df-mgp 20052  df-ur 20093  df-ring 20146  df-oppr 20248  df-rhm 20383  df-staf 20747  df-srng 20748  df-lmod 20788  df-lmhm 20949  df-lvec 21030  df-sra 21100  df-rgmod 21101  df-phl 21556  df-ocv 21593  df-css 21594

This theorem is referenced by:  pjcss  21646