pjssmii - Hilbert Space Explorer

	Hilbert Space Explorer		< Previous Next > Nearby theorems
Mirrors > Home > HSE Home > Th. List > pjssmii		Structured version Visualization version GIF version

Theorem pjssmii 31760

Description: Projection meet property. Remark in [Kalmbach] p. 66. Also Theorem 4.5(i)->(iv) of [Beran] p. 112. (Contributed by NM, 31-Oct-1999.) (New usage is discouraged.)

**Hypotheses**
Ref	Expression
pjidm.1	⊢ 𝐻 ∈ C_ℋ
pjidm.2	⊢ 𝐴 ∈ ℋ
pjsslem.1	⊢ 𝐺 ∈ C_ℋ

**Assertion**
Ref	Expression
pjssmii	⊢ (𝐻 ⊆ 𝐺 → (((proj_ℎ‘𝐺)‘𝐴) −_ℎ ((proj_ℎ‘𝐻)‘𝐴)) = ((proj_ℎ‘(𝐺 ∩ (⊥‘𝐻)))‘𝐴))

**Proof of Theorem pjssmii**
Step	Hyp	Ref	Expression
1		pjsslem.1	. . . . . 6 ⊢ 𝐺 ∈ C_ℋ
2		pjidm.2	. . . . . 6 ⊢ 𝐴 ∈ ℋ
3	1, 2	pjclii 31500	. . . . 5 ⊢ ((proj_ℎ‘𝐺)‘𝐴) ∈ 𝐺
4		pjidm.1	. . . . . . 7 ⊢ 𝐻 ∈ C_ℋ
5	4, 2	pjclii 31500	. . . . . 6 ⊢ ((proj_ℎ‘𝐻)‘𝐴) ∈ 𝐻
6		ssel 3928	. . . . . 6 ⊢ (𝐻 ⊆ 𝐺 → (((proj_ℎ‘𝐻)‘𝐴) ∈ 𝐻 → ((proj_ℎ‘𝐻)‘𝐴) ∈ 𝐺))
7	5, 6	mpi 20	. . . . 5 ⊢ (𝐻 ⊆ 𝐺 → ((proj_ℎ‘𝐻)‘𝐴) ∈ 𝐺)
8	1	chshii 31306	. . . . . 6 ⊢ 𝐺 ∈ S_ℋ
9		shsubcl 31299	. . . . . 6 ⊢ ((𝐺 ∈ S_ℋ ∧ ((proj_ℎ‘𝐺)‘𝐴) ∈ 𝐺 ∧ ((proj_ℎ‘𝐻)‘𝐴) ∈ 𝐺) → (((proj_ℎ‘𝐺)‘𝐴) −_ℎ ((proj_ℎ‘𝐻)‘𝐴)) ∈ 𝐺)
10	8, 9	mp3an1 1451	. . . . 5 ⊢ ((((proj_ℎ‘𝐺)‘𝐴) ∈ 𝐺 ∧ ((proj_ℎ‘𝐻)‘𝐴) ∈ 𝐺) → (((proj_ℎ‘𝐺)‘𝐴) −_ℎ ((proj_ℎ‘𝐻)‘𝐴)) ∈ 𝐺)
11	3, 7, 10	sylancr 588	. . . 4 ⊢ (𝐻 ⊆ 𝐺 → (((proj_ℎ‘𝐺)‘𝐴) −_ℎ ((proj_ℎ‘𝐻)‘𝐴)) ∈ 𝐺)
12	4, 2, 1	pjsslem 31758	. . . . 5 ⊢ (((proj_ℎ‘(⊥‘𝐻))‘𝐴) −_ℎ ((proj_ℎ‘(⊥‘𝐺))‘𝐴)) = (((proj_ℎ‘𝐺)‘𝐴) −_ℎ ((proj_ℎ‘𝐻)‘𝐴))
13	4, 1	chsscon3i 31540	. . . . . 6 ⊢ (𝐻 ⊆ 𝐺 ↔ (⊥‘𝐺) ⊆ (⊥‘𝐻))
14	4	choccli 31386	. . . . . . . 8 ⊢ (⊥‘𝐻) ∈ C_ℋ
15	14, 2	pjclii 31500	. . . . . . 7 ⊢ ((proj_ℎ‘(⊥‘𝐻))‘𝐴) ∈ (⊥‘𝐻)
16	1	choccli 31386	. . . . . . . . 9 ⊢ (⊥‘𝐺) ∈ C_ℋ
17	16, 2	pjclii 31500	. . . . . . . 8 ⊢ ((proj_ℎ‘(⊥‘𝐺))‘𝐴) ∈ (⊥‘𝐺)
18		ssel 3928	. . . . . . . 8 ⊢ ((⊥‘𝐺) ⊆ (⊥‘𝐻) → (((proj_ℎ‘(⊥‘𝐺))‘𝐴) ∈ (⊥‘𝐺) → ((proj_ℎ‘(⊥‘𝐺))‘𝐴) ∈ (⊥‘𝐻)))
19	17, 18	mpi 20	. . . . . . 7 ⊢ ((⊥‘𝐺) ⊆ (⊥‘𝐻) → ((proj_ℎ‘(⊥‘𝐺))‘𝐴) ∈ (⊥‘𝐻))
20	14	chshii 31306	. . . . . . . 8 ⊢ (⊥‘𝐻) ∈ S_ℋ
21		shsubcl 31299	. . . . . . . 8 ⊢ (((⊥‘𝐻) ∈ S_ℋ ∧ ((proj_ℎ‘(⊥‘𝐻))‘𝐴) ∈ (⊥‘𝐻) ∧ ((proj_ℎ‘(⊥‘𝐺))‘𝐴) ∈ (⊥‘𝐻)) → (((proj_ℎ‘(⊥‘𝐻))‘𝐴) −_ℎ ((proj_ℎ‘(⊥‘𝐺))‘𝐴)) ∈ (⊥‘𝐻))
22	20, 21	mp3an1 1451	. . . . . . 7 ⊢ ((((proj_ℎ‘(⊥‘𝐻))‘𝐴) ∈ (⊥‘𝐻) ∧ ((proj_ℎ‘(⊥‘𝐺))‘𝐴) ∈ (⊥‘𝐻)) → (((proj_ℎ‘(⊥‘𝐻))‘𝐴) −_ℎ ((proj_ℎ‘(⊥‘𝐺))‘𝐴)) ∈ (⊥‘𝐻))
23	15, 19, 22	sylancr 588	. . . . . 6 ⊢ ((⊥‘𝐺) ⊆ (⊥‘𝐻) → (((proj_ℎ‘(⊥‘𝐻))‘𝐴) −_ℎ ((proj_ℎ‘(⊥‘𝐺))‘𝐴)) ∈ (⊥‘𝐻))
24	13, 23	sylbi 217	. . . . 5 ⊢ (𝐻 ⊆ 𝐺 → (((proj_ℎ‘(⊥‘𝐻))‘𝐴) −_ℎ ((proj_ℎ‘(⊥‘𝐺))‘𝐴)) ∈ (⊥‘𝐻))
25	12, 24	eqeltrrid 2842	. . . 4 ⊢ (𝐻 ⊆ 𝐺 → (((proj_ℎ‘𝐺)‘𝐴) −_ℎ ((proj_ℎ‘𝐻)‘𝐴)) ∈ (⊥‘𝐻))
26	11, 25	jca 511	. . 3 ⊢ (𝐻 ⊆ 𝐺 → ((((proj_ℎ‘𝐺)‘𝐴) −_ℎ ((proj_ℎ‘𝐻)‘𝐴)) ∈ 𝐺 ∧ (((proj_ℎ‘𝐺)‘𝐴) −_ℎ ((proj_ℎ‘𝐻)‘𝐴)) ∈ (⊥‘𝐻)))
27		elin 3918	. . . 4 ⊢ ((((proj_ℎ‘𝐺)‘𝐴) −_ℎ ((proj_ℎ‘𝐻)‘𝐴)) ∈ (𝐺 ∩ (⊥‘𝐻)) ↔ ((((proj_ℎ‘𝐺)‘𝐴) −_ℎ ((proj_ℎ‘𝐻)‘𝐴)) ∈ 𝐺 ∧ (((proj_ℎ‘𝐺)‘𝐴) −_ℎ ((proj_ℎ‘𝐻)‘𝐴)) ∈ (⊥‘𝐻)))
28	1, 14	chincli 31539	. . . . 5 ⊢ (𝐺 ∩ (⊥‘𝐻)) ∈ C_ℋ
29	1, 2	pjhclii 31501	. . . . . 6 ⊢ ((proj_ℎ‘𝐺)‘𝐴) ∈ ℋ
30	4, 2	pjhclii 31501	. . . . . 6 ⊢ ((proj_ℎ‘𝐻)‘𝐴) ∈ ℋ
31	29, 30	hvsubcli 31100	. . . . 5 ⊢ (((proj_ℎ‘𝐺)‘𝐴) −_ℎ ((proj_ℎ‘𝐻)‘𝐴)) ∈ ℋ
32	28, 31	pjchi 31511	. . . 4 ⊢ ((((proj_ℎ‘𝐺)‘𝐴) −_ℎ ((proj_ℎ‘𝐻)‘𝐴)) ∈ (𝐺 ∩ (⊥‘𝐻)) ↔ ((proj_ℎ‘(𝐺 ∩ (⊥‘𝐻)))‘(((proj_ℎ‘𝐺)‘𝐴) −_ℎ ((proj_ℎ‘𝐻)‘𝐴))) = (((proj_ℎ‘𝐺)‘𝐴) −_ℎ ((proj_ℎ‘𝐻)‘𝐴)))
33	27, 32	bitr3i 277	. . 3 ⊢ (((((proj_ℎ‘𝐺)‘𝐴) −_ℎ ((proj_ℎ‘𝐻)‘𝐴)) ∈ 𝐺 ∧ (((proj_ℎ‘𝐺)‘𝐴) −_ℎ ((proj_ℎ‘𝐻)‘𝐴)) ∈ (⊥‘𝐻)) ↔ ((proj_ℎ‘(𝐺 ∩ (⊥‘𝐻)))‘(((proj_ℎ‘𝐺)‘𝐴) −_ℎ ((proj_ℎ‘𝐻)‘𝐴))) = (((proj_ℎ‘𝐺)‘𝐴) −_ℎ ((proj_ℎ‘𝐻)‘𝐴)))
34	26, 33	sylib 218	. 2 ⊢ (𝐻 ⊆ 𝐺 → ((proj_ℎ‘(𝐺 ∩ (⊥‘𝐻)))‘(((proj_ℎ‘𝐺)‘𝐴) −_ℎ ((proj_ℎ‘𝐻)‘𝐴))) = (((proj_ℎ‘𝐺)‘𝐴) −_ℎ ((proj_ℎ‘𝐻)‘𝐴)))
35	28, 29, 30	pjsubii 31757	. . 3 ⊢ ((proj_ℎ‘(𝐺 ∩ (⊥‘𝐻)))‘(((proj_ℎ‘𝐺)‘𝐴) −_ℎ ((proj_ℎ‘𝐻)‘𝐴))) = (((proj_ℎ‘(𝐺 ∩ (⊥‘𝐻)))‘((proj_ℎ‘𝐺)‘𝐴)) −_ℎ ((proj_ℎ‘(𝐺 ∩ (⊥‘𝐻)))‘((proj_ℎ‘𝐻)‘𝐴)))
36	28, 29	pjhclii 31501	. . . . 5 ⊢ ((proj_ℎ‘(𝐺 ∩ (⊥‘𝐻)))‘((proj_ℎ‘𝐺)‘𝐴)) ∈ ℋ
37	28, 30	pjhclii 31501	. . . . 5 ⊢ ((proj_ℎ‘(𝐺 ∩ (⊥‘𝐻)))‘((proj_ℎ‘𝐻)‘𝐴)) ∈ ℋ
38	36, 37	hvsubvali 31099	. . . 4 ⊢ (((proj_ℎ‘(𝐺 ∩ (⊥‘𝐻)))‘((proj_ℎ‘𝐺)‘𝐴)) −_ℎ ((proj_ℎ‘(𝐺 ∩ (⊥‘𝐻)))‘((proj_ℎ‘𝐻)‘𝐴))) = (((proj_ℎ‘(𝐺 ∩ (⊥‘𝐻)))‘((proj_ℎ‘𝐺)‘𝐴)) +_ℎ (-1 ·_ℎ ((proj_ℎ‘(𝐺 ∩ (⊥‘𝐻)))‘((proj_ℎ‘𝐻)‘𝐴))))
39		inss1 4190	. . . . . . 7 ⊢ (𝐺 ∩ (⊥‘𝐻)) ⊆ 𝐺
40	28, 2, 1	pjss2i 31759	. . . . . . 7 ⊢ ((𝐺 ∩ (⊥‘𝐻)) ⊆ 𝐺 → ((proj_ℎ‘(𝐺 ∩ (⊥‘𝐻)))‘((proj_ℎ‘𝐺)‘𝐴)) = ((proj_ℎ‘(𝐺 ∩ (⊥‘𝐻)))‘𝐴))
41	39, 40	ax-mp 5	. . . . . 6 ⊢ ((proj_ℎ‘(𝐺 ∩ (⊥‘𝐻)))‘((proj_ℎ‘𝐺)‘𝐴)) = ((proj_ℎ‘(𝐺 ∩ (⊥‘𝐻)))‘𝐴)
42	4	chshii 31306	. . . . . . . . . . . 12 ⊢ 𝐻 ∈ S_ℋ
43		shococss 31373	. . . . . . . . . . . 12 ⊢ (𝐻 ∈ S_ℋ → 𝐻 ⊆ (⊥‘(⊥‘𝐻)))
44	42, 43	ax-mp 5	. . . . . . . . . . 11 ⊢ 𝐻 ⊆ (⊥‘(⊥‘𝐻))
45		inss2 4191	. . . . . . . . . . . 12 ⊢ (𝐺 ∩ (⊥‘𝐻)) ⊆ (⊥‘𝐻)
46	28, 14	chsscon3i 31540	. . . . . . . . . . . 12 ⊢ ((𝐺 ∩ (⊥‘𝐻)) ⊆ (⊥‘𝐻) ↔ (⊥‘(⊥‘𝐻)) ⊆ (⊥‘(𝐺 ∩ (⊥‘𝐻))))
47	45, 46	mpbi 230	. . . . . . . . . . 11 ⊢ (⊥‘(⊥‘𝐻)) ⊆ (⊥‘(𝐺 ∩ (⊥‘𝐻)))
48	44, 47	sstri 3944	. . . . . . . . . 10 ⊢ 𝐻 ⊆ (⊥‘(𝐺 ∩ (⊥‘𝐻)))
49	48, 5	sselii 3931	. . . . . . . . 9 ⊢ ((proj_ℎ‘𝐻)‘𝐴) ∈ (⊥‘(𝐺 ∩ (⊥‘𝐻)))
50	28, 30	pjoc2i 31517	. . . . . . . . 9 ⊢ (((proj_ℎ‘𝐻)‘𝐴) ∈ (⊥‘(𝐺 ∩ (⊥‘𝐻))) ↔ ((proj_ℎ‘(𝐺 ∩ (⊥‘𝐻)))‘((proj_ℎ‘𝐻)‘𝐴)) = 0_ℎ)
51	49, 50	mpbi 230	. . . . . . . 8 ⊢ ((proj_ℎ‘(𝐺 ∩ (⊥‘𝐻)))‘((proj_ℎ‘𝐻)‘𝐴)) = 0_ℎ
52	51	oveq2i 7371	. . . . . . 7 ⊢ (-1 ·_ℎ ((proj_ℎ‘(𝐺 ∩ (⊥‘𝐻)))‘((proj_ℎ‘𝐻)‘𝐴))) = (-1 ·_ℎ 0_ℎ)
53		neg1cn 12134	. . . . . . . 8 ⊢ -1 ∈ ℂ
54		hvmul0 31103	. . . . . . . 8 ⊢ (-1 ∈ ℂ → (-1 ·_ℎ 0_ℎ) = 0_ℎ)
55	53, 54	ax-mp 5	. . . . . . 7 ⊢ (-1 ·_ℎ 0_ℎ) = 0_ℎ
56	52, 55	eqtri 2760	. . . . . 6 ⊢ (-1 ·_ℎ ((proj_ℎ‘(𝐺 ∩ (⊥‘𝐻)))‘((proj_ℎ‘𝐻)‘𝐴))) = 0_ℎ
57	41, 56	oveq12i 7372	. . . . 5 ⊢ (((proj_ℎ‘(𝐺 ∩ (⊥‘𝐻)))‘((proj_ℎ‘𝐺)‘𝐴)) +_ℎ (-1 ·_ℎ ((proj_ℎ‘(𝐺 ∩ (⊥‘𝐻)))‘((proj_ℎ‘𝐻)‘𝐴)))) = (((proj_ℎ‘(𝐺 ∩ (⊥‘𝐻)))‘𝐴) +_ℎ 0_ℎ)
58	28, 2	pjhclii 31501	. . . . . 6 ⊢ ((proj_ℎ‘(𝐺 ∩ (⊥‘𝐻)))‘𝐴) ∈ ℋ
59		ax-hvaddid 31083	. . . . . 6 ⊢ (((proj_ℎ‘(𝐺 ∩ (⊥‘𝐻)))‘𝐴) ∈ ℋ → (((proj_ℎ‘(𝐺 ∩ (⊥‘𝐻)))‘𝐴) +_ℎ 0_ℎ) = ((proj_ℎ‘(𝐺 ∩ (⊥‘𝐻)))‘𝐴))
60	58, 59	ax-mp 5	. . . . 5 ⊢ (((proj_ℎ‘(𝐺 ∩ (⊥‘𝐻)))‘𝐴) +_ℎ 0_ℎ) = ((proj_ℎ‘(𝐺 ∩ (⊥‘𝐻)))‘𝐴)
61	57, 60	eqtri 2760	. . . 4 ⊢ (((proj_ℎ‘(𝐺 ∩ (⊥‘𝐻)))‘((proj_ℎ‘𝐺)‘𝐴)) +_ℎ (-1 ·_ℎ ((proj_ℎ‘(𝐺 ∩ (⊥‘𝐻)))‘((proj_ℎ‘𝐻)‘𝐴)))) = ((proj_ℎ‘(𝐺 ∩ (⊥‘𝐻)))‘𝐴)
62	38, 61	eqtri 2760	. . 3 ⊢ (((proj_ℎ‘(𝐺 ∩ (⊥‘𝐻)))‘((proj_ℎ‘𝐺)‘𝐴)) −_ℎ ((proj_ℎ‘(𝐺 ∩ (⊥‘𝐻)))‘((proj_ℎ‘𝐻)‘𝐴))) = ((proj_ℎ‘(𝐺 ∩ (⊥‘𝐻)))‘𝐴)
63	35, 62	eqtri 2760	. 2 ⊢ ((proj_ℎ‘(𝐺 ∩ (⊥‘𝐻)))‘(((proj_ℎ‘𝐺)‘𝐴) −_ℎ ((proj_ℎ‘𝐻)‘𝐴))) = ((proj_ℎ‘(𝐺 ∩ (⊥‘𝐻)))‘𝐴)
64	34, 63	eqtr3di 2787	1 ⊢ (𝐻 ⊆ 𝐺 → (((proj_ℎ‘𝐺)‘𝐴) −_ℎ ((proj_ℎ‘𝐻)‘𝐴)) = ((proj_ℎ‘(𝐺 ∩ (⊥‘𝐻)))‘𝐴))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1542 ∈ wcel 2114 ∩ cin 3901 ⊆ wss 3902 ‘cfv 6493 (class class class)co 7360 ℂcc 11028 1c1 11031 -cneg 11369 ℋchba 30998 +_ℎ cva 30999 ·_ℎ csm 31000 0_ℎc0v 31003 −_ℎ cmv 31004 S_ℋ csh 31007 C_ℋ cch 31008 ⊥cort 31009 proj_ℎcpjh 31016

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 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-rep 5225 ax-sep 5242 ax-nul 5252 ax-pow 5311 ax-pr 5378 ax-un 7682 ax-inf2 9554 ax-cc 10349 ax-cnex 11086 ax-resscn 11087 ax-1cn 11088 ax-icn 11089 ax-addcl 11090 ax-addrcl 11091 ax-mulcl 11092 ax-mulrcl 11093 ax-mulcom 11094 ax-addass 11095 ax-mulass 11096 ax-distr 11097 ax-i2m1 11098 ax-1ne0 11099 ax-1rid 11100 ax-rnegex 11101 ax-rrecex 11102 ax-cnre 11103 ax-pre-lttri 11104 ax-pre-lttrn 11105 ax-pre-ltadd 11106 ax-pre-mulgt0 11107 ax-pre-sup 11108 ax-addf 11109 ax-mulf 11110 ax-hilex 31078 ax-hfvadd 31079 ax-hvcom 31080 ax-hvass 31081 ax-hv0cl 31082 ax-hvaddid 31083 ax-hfvmul 31084 ax-hvmulid 31085 ax-hvmulass 31086 ax-hvdistr1 31087 ax-hvdistr2 31088 ax-hvmul0 31089 ax-hfi 31158 ax-his1 31161 ax-his2 31162 ax-his3 31163 ax-his4 31164 ax-hcompl 31281

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3062 df-rmo 3351 df-reu 3352 df-rab 3401 df-v 3443 df-sbc 3742 df-csb 3851 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-pss 3922 df-nul 4287 df-if 4481 df-pw 4557 df-sn 4582 df-pr 4584 df-tp 4586 df-op 4588 df-uni 4865 df-int 4904 df-iun 4949 df-iin 4950 df-br 5100 df-opab 5162 df-mpt 5181 df-tr 5207 df-id 5520 df-eprel 5525 df-po 5533 df-so 5534 df-fr 5578 df-se 5579 df-we 5580 df-xp 5631 df-rel 5632 df-cnv 5633 df-co 5634 df-dm 5635 df-rn 5636 df-res 5637 df-ima 5638 df-pred 6260 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6495 df-fn 6496 df-f 6497 df-f1 6498 df-fo 6499 df-f1o 6500 df-fv 6501 df-isom 6502 df-riota 7317 df-ov 7363 df-oprab 7364 df-mpo 7365 df-of 7624 df-om 7811 df-1st 7935 df-2nd 7936 df-supp 8105 df-frecs 8225 df-wrecs 8256 df-recs 8305 df-rdg 8343 df-1o 8399 df-2o 8400 df-oadd 8403 df-omul 8404 df-er 8637 df-map 8769 df-pm 8770 df-ixp 8840 df-en 8888 df-dom 8889 df-sdom 8890 df-fin 8891 df-fsupp 9269 df-fi 9318 df-sup 9349 df-inf 9350 df-oi 9419 df-card 9855 df-acn 9858 df-pnf 11172 df-mnf 11173 df-xr 11174 df-ltxr 11175 df-le 11176 df-sub 11370 df-neg 11371 df-div 11799 df-nn 12150 df-2 12212 df-3 12213 df-4 12214 df-5 12215 df-6 12216 df-7 12217 df-8 12218 df-9 12219 df-n0 12406 df-z 12493 df-dec 12612 df-uz 12756 df-q 12866 df-rp 12910 df-xneg 13030 df-xadd 13031 df-xmul 13032 df-ioo 13269 df-ico 13271 df-icc 13272 df-fz 13428 df-fzo 13575 df-fl 13716 df-seq 13929 df-exp 13989 df-hash 14258 df-cj 15026 df-re 15027 df-im 15028 df-sqrt 15162 df-abs 15163 df-clim 15415 df-rlim 15416 df-sum 15614 df-struct 17078 df-sets 17095 df-slot 17113 df-ndx 17125 df-base 17141 df-ress 17162 df-plusg 17194 df-mulr 17195 df-starv 17196 df-sca 17197 df-vsca 17198 df-ip 17199 df-tset 17200 df-ple 17201 df-ds 17203 df-unif 17204 df-hom 17205 df-cco 17206 df-rest 17346 df-topn 17347 df-0g 17365 df-gsum 17366 df-topgen 17367 df-pt 17368 df-prds 17371 df-xrs 17427 df-qtop 17432 df-imas 17433 df-xps 17435 df-mre 17509 df-mrc 17510 df-acs 17512 df-mgm 18569 df-sgrp 18648 df-mnd 18664 df-submnd 18713 df-mulg 19002 df-cntz 19250 df-cmn 19715 df-psmet 21305 df-xmet 21306 df-met 21307 df-bl 21308 df-mopn 21309 df-fbas 21310 df-fg 21311 df-cnfld 21314 df-top 22842 df-topon 22859 df-topsp 22881 df-bases 22894 df-cld 22967 df-ntr 22968 df-cls 22969 df-nei 23046 df-cn 23175 df-cnp 23176 df-lm 23177 df-haus 23263 df-tx 23510 df-hmeo 23703 df-fil 23794 df-fm 23886 df-flim 23887 df-flf 23888 df-xms 24268 df-ms 24269 df-tms 24270 df-cfil 25215 df-cau 25216 df-cmet 25217 df-grpo 30572 df-gid 30573 df-ginv 30574 df-gdiv 30575 df-ablo 30624 df-vc 30638 df-nv 30671 df-va 30674 df-ba 30675 df-sm 30676 df-0v 30677 df-vs 30678 df-nmcv 30679 df-ims 30680 df-dip 30780 df-ssp 30801 df-ph 30892 df-cbn 30942 df-hnorm 31047 df-hba 31048 df-hvsub 31050 df-hlim 31051 df-hcau 31052 df-sh 31286 df-ch 31300 df-oc 31331 df-ch0 31332 df-shs 31387 df-pjh 31474

This theorem is referenced by: pjcji 31763 pjssmi 32244

W3C validator