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Theorem pjssmii 30043

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 29783	. . . . 5 ⊢ ((proj_ℎ‘𝐺)‘𝐴) ∈ 𝐺
4		pjidm.1	. . . . . . 7 ⊢ 𝐻 ∈ C_ℋ
5	4, 2	pjclii 29783	. . . . . 6 ⊢ ((proj_ℎ‘𝐻)‘𝐴) ∈ 𝐻
6		ssel 3914	. . . . . 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 29589	. . . . . 6 ⊢ 𝐺 ∈ S_ℋ
9		shsubcl 29582	. . . . . 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 1447	. . . . 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 587	. . . 4 ⊢ (𝐻 ⊆ 𝐺 → (((proj_ℎ‘𝐺)‘𝐴) −_ℎ ((proj_ℎ‘𝐻)‘𝐴)) ∈ 𝐺)
12	4, 2, 1	pjsslem 30041	. . . . 5 ⊢ (((proj_ℎ‘(⊥‘𝐻))‘𝐴) −_ℎ ((proj_ℎ‘(⊥‘𝐺))‘𝐴)) = (((proj_ℎ‘𝐺)‘𝐴) −_ℎ ((proj_ℎ‘𝐻)‘𝐴))
13	4, 1	chsscon3i 29823	. . . . . 6 ⊢ (𝐻 ⊆ 𝐺 ↔ (⊥‘𝐺) ⊆ (⊥‘𝐻))
14	4	choccli 29669	. . . . . . . 8 ⊢ (⊥‘𝐻) ∈ C_ℋ
15	14, 2	pjclii 29783	. . . . . . 7 ⊢ ((proj_ℎ‘(⊥‘𝐻))‘𝐴) ∈ (⊥‘𝐻)
16	1	choccli 29669	. . . . . . . . 9 ⊢ (⊥‘𝐺) ∈ C_ℋ
17	16, 2	pjclii 29783	. . . . . . . 8 ⊢ ((proj_ℎ‘(⊥‘𝐺))‘𝐴) ∈ (⊥‘𝐺)
18		ssel 3914	. . . . . . . 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 29589	. . . . . . . 8 ⊢ (⊥‘𝐻) ∈ S_ℋ
21		shsubcl 29582	. . . . . . . 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 1447	. . . . . . 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 587	. . . . . 6 ⊢ ((⊥‘𝐺) ⊆ (⊥‘𝐻) → (((proj_ℎ‘(⊥‘𝐻))‘𝐴) −_ℎ ((proj_ℎ‘(⊥‘𝐺))‘𝐴)) ∈ (⊥‘𝐻))
24	13, 23	sylbi 216	. . . . 5 ⊢ (𝐻 ⊆ 𝐺 → (((proj_ℎ‘(⊥‘𝐻))‘𝐴) −_ℎ ((proj_ℎ‘(⊥‘𝐺))‘𝐴)) ∈ (⊥‘𝐻))
25	12, 24	eqeltrrid 2844	. . . 4 ⊢ (𝐻 ⊆ 𝐺 → (((proj_ℎ‘𝐺)‘𝐴) −_ℎ ((proj_ℎ‘𝐻)‘𝐴)) ∈ (⊥‘𝐻))
26	11, 25	jca 512	. . 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 3903	. . . 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 29822	. . . . 5 ⊢ (𝐺 ∩ (⊥‘𝐻)) ∈ C_ℋ
29	1, 2	pjhclii 29784	. . . . . 6 ⊢ ((proj_ℎ‘𝐺)‘𝐴) ∈ ℋ
30	4, 2	pjhclii 29784	. . . . . 6 ⊢ ((proj_ℎ‘𝐻)‘𝐴) ∈ ℋ
31	29, 30	hvsubcli 29383	. . . . 5 ⊢ (((proj_ℎ‘𝐺)‘𝐴) −_ℎ ((proj_ℎ‘𝐻)‘𝐴)) ∈ ℋ
32	28, 31	pjchi 29794	. . . 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 276	. . 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 217	. 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 30040	. . 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 29784	. . . . 5 ⊢ ((proj_ℎ‘(𝐺 ∩ (⊥‘𝐻)))‘((proj_ℎ‘𝐺)‘𝐴)) ∈ ℋ
37	28, 30	pjhclii 29784	. . . . 5 ⊢ ((proj_ℎ‘(𝐺 ∩ (⊥‘𝐻)))‘((proj_ℎ‘𝐻)‘𝐴)) ∈ ℋ
38	36, 37	hvsubvali 29382	. . . 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 4162	. . . . . . 7 ⊢ (𝐺 ∩ (⊥‘𝐻)) ⊆ 𝐺
40	28, 2, 1	pjss2i 30042	. . . . . . 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 29589	. . . . . . . . . . . 12 ⊢ 𝐻 ∈ S_ℋ
43		shococss 29656	. . . . . . . . . . . 12 ⊢ (𝐻 ∈ S_ℋ → 𝐻 ⊆ (⊥‘(⊥‘𝐻)))
44	42, 43	ax-mp 5	. . . . . . . . . . 11 ⊢ 𝐻 ⊆ (⊥‘(⊥‘𝐻))
45		inss2 4163	. . . . . . . . . . . 12 ⊢ (𝐺 ∩ (⊥‘𝐻)) ⊆ (⊥‘𝐻)
46	28, 14	chsscon3i 29823	. . . . . . . . . . . 12 ⊢ ((𝐺 ∩ (⊥‘𝐻)) ⊆ (⊥‘𝐻) ↔ (⊥‘(⊥‘𝐻)) ⊆ (⊥‘(𝐺 ∩ (⊥‘𝐻))))
47	45, 46	mpbi 229	. . . . . . . . . . 11 ⊢ (⊥‘(⊥‘𝐻)) ⊆ (⊥‘(𝐺 ∩ (⊥‘𝐻)))
48	44, 47	sstri 3930	. . . . . . . . . 10 ⊢ 𝐻 ⊆ (⊥‘(𝐺 ∩ (⊥‘𝐻)))
49	48, 5	sselii 3918	. . . . . . . . 9 ⊢ ((proj_ℎ‘𝐻)‘𝐴) ∈ (⊥‘(𝐺 ∩ (⊥‘𝐻)))
50	28, 30	pjoc2i 29800	. . . . . . . . 9 ⊢ (((proj_ℎ‘𝐻)‘𝐴) ∈ (⊥‘(𝐺 ∩ (⊥‘𝐻))) ↔ ((proj_ℎ‘(𝐺 ∩ (⊥‘𝐻)))‘((proj_ℎ‘𝐻)‘𝐴)) = 0_ℎ)
51	49, 50	mpbi 229	. . . . . . . 8 ⊢ ((proj_ℎ‘(𝐺 ∩ (⊥‘𝐻)))‘((proj_ℎ‘𝐻)‘𝐴)) = 0_ℎ
52	51	oveq2i 7286	. . . . . . 7 ⊢ (-1 ·_ℎ ((proj_ℎ‘(𝐺 ∩ (⊥‘𝐻)))‘((proj_ℎ‘𝐻)‘𝐴))) = (-1 ·_ℎ 0_ℎ)
53		neg1cn 12087	. . . . . . . 8 ⊢ -1 ∈ ℂ
54		hvmul0 29386	. . . . . . . 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 2766	. . . . . 6 ⊢ (-1 ·_ℎ ((proj_ℎ‘(𝐺 ∩ (⊥‘𝐻)))‘((proj_ℎ‘𝐻)‘𝐴))) = 0_ℎ
57	41, 56	oveq12i 7287	. . . . 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 29784	. . . . . 6 ⊢ ((proj_ℎ‘(𝐺 ∩ (⊥‘𝐻)))‘𝐴) ∈ ℋ
59		ax-hvaddid 29366	. . . . . 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 2766	. . . 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 2766	. . 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 2766	. 2 ⊢ ((proj_ℎ‘(𝐺 ∩ (⊥‘𝐻)))‘(((proj_ℎ‘𝐺)‘𝐴) −_ℎ ((proj_ℎ‘𝐻)‘𝐴))) = ((proj_ℎ‘(𝐺 ∩ (⊥‘𝐻)))‘𝐴)
64	34, 63	eqtr3di 2793	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 396 = wceq 1539 ∈ wcel 2106 ∩ cin 3886 ⊆ wss 3887 ‘cfv 6433 (class class class)co 7275 ℂcc 10869 1c1 10872 -cneg 11206 ℋchba 29281 +_ℎ cva 29282 ·_ℎ csm 29283 0_ℎc0v 29286 −_ℎ cmv 29287 S_ℋ csh 29290 C_ℋ cch 29291 ⊥cort 29292 proj_ℎcpjh 29299

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-rep 5209 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 ax-inf2 9399 ax-cc 10191 ax-cnex 10927 ax-resscn 10928 ax-1cn 10929 ax-icn 10930 ax-addcl 10931 ax-addrcl 10932 ax-mulcl 10933 ax-mulrcl 10934 ax-mulcom 10935 ax-addass 10936 ax-mulass 10937 ax-distr 10938 ax-i2m1 10939 ax-1ne0 10940 ax-1rid 10941 ax-rnegex 10942 ax-rrecex 10943 ax-cnre 10944 ax-pre-lttri 10945 ax-pre-lttrn 10946 ax-pre-ltadd 10947 ax-pre-mulgt0 10948 ax-pre-sup 10949 ax-addf 10950 ax-mulf 10951 ax-hilex 29361 ax-hfvadd 29362 ax-hvcom 29363 ax-hvass 29364 ax-hv0cl 29365 ax-hvaddid 29366 ax-hfvmul 29367 ax-hvmulid 29368 ax-hvmulass 29369 ax-hvdistr1 29370 ax-hvdistr2 29371 ax-hvmul0 29372 ax-hfi 29441 ax-his1 29444 ax-his2 29445 ax-his3 29446 ax-his4 29447 ax-hcompl 29564

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-rmo 3071 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-pss 3906 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-tp 4566 df-op 4568 df-uni 4840 df-int 4880 df-iun 4926 df-iin 4927 df-br 5075 df-opab 5137 df-mpt 5158 df-tr 5192 df-id 5489 df-eprel 5495 df-po 5503 df-so 5504 df-fr 5544 df-se 5545 df-we 5546 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-pred 6202 df-ord 6269 df-on 6270 df-lim 6271 df-suc 6272 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-isom 6442 df-riota 7232 df-ov 7278 df-oprab 7279 df-mpo 7280 df-of 7533 df-om 7713 df-1st 7831 df-2nd 7832 df-supp 7978 df-frecs 8097 df-wrecs 8128 df-recs 8202 df-rdg 8241 df-1o 8297 df-2o 8298 df-oadd 8301 df-omul 8302 df-er 8498 df-map 8617 df-pm 8618 df-ixp 8686 df-en 8734 df-dom 8735 df-sdom 8736 df-fin 8737 df-fsupp 9129 df-fi 9170 df-sup 9201 df-inf 9202 df-oi 9269 df-card 9697 df-acn 9700 df-pnf 11011 df-mnf 11012 df-xr 11013 df-ltxr 11014 df-le 11015 df-sub 11207 df-neg 11208 df-div 11633 df-nn 11974 df-2 12036 df-3 12037 df-4 12038 df-5 12039 df-6 12040 df-7 12041 df-8 12042 df-9 12043 df-n0 12234 df-z 12320 df-dec 12438 df-uz 12583 df-q 12689 df-rp 12731 df-xneg 12848 df-xadd 12849 df-xmul 12850 df-ioo 13083 df-ico 13085 df-icc 13086 df-fz 13240 df-fzo 13383 df-fl 13512 df-seq 13722 df-exp 13783 df-hash 14045 df-cj 14810 df-re 14811 df-im 14812 df-sqrt 14946 df-abs 14947 df-clim 15197 df-rlim 15198 df-sum 15398 df-struct 16848 df-sets 16865 df-slot 16883 df-ndx 16895 df-base 16913 df-ress 16942 df-plusg 16975 df-mulr 16976 df-starv 16977 df-sca 16978 df-vsca 16979 df-ip 16980 df-tset 16981 df-ple 16982 df-ds 16984 df-unif 16985 df-hom 16986 df-cco 16987 df-rest 17133 df-topn 17134 df-0g 17152 df-gsum 17153 df-topgen 17154 df-pt 17155 df-prds 17158 df-xrs 17213 df-qtop 17218 df-imas 17219 df-xps 17221 df-mre 17295 df-mrc 17296 df-acs 17298 df-mgm 18326 df-sgrp 18375 df-mnd 18386 df-submnd 18431 df-mulg 18701 df-cntz 18923 df-cmn 19388 df-psmet 20589 df-xmet 20590 df-met 20591 df-bl 20592 df-mopn 20593 df-fbas 20594 df-fg 20595 df-cnfld 20598 df-top 22043 df-topon 22060 df-topsp 22082 df-bases 22096 df-cld 22170 df-ntr 22171 df-cls 22172 df-nei 22249 df-cn 22378 df-cnp 22379 df-lm 22380 df-haus 22466 df-tx 22713 df-hmeo 22906 df-fil 22997 df-fm 23089 df-flim 23090 df-flf 23091 df-xms 23473 df-ms 23474 df-tms 23475 df-cfil 24419 df-cau 24420 df-cmet 24421 df-grpo 28855 df-gid 28856 df-ginv 28857 df-gdiv 28858 df-ablo 28907 df-vc 28921 df-nv 28954 df-va 28957 df-ba 28958 df-sm 28959 df-0v 28960 df-vs 28961 df-nmcv 28962 df-ims 28963 df-dip 29063 df-ssp 29084 df-ph 29175 df-cbn 29225 df-hnorm 29330 df-hba 29331 df-hvsub 29333 df-hlim 29334 df-hcau 29335 df-sh 29569 df-ch 29583 df-oc 29614 df-ch0 29615 df-shs 29670 df-pjh 29757

This theorem is referenced by: pjcji 30046 pjssmi 30527

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