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Theorem pjssmii 30965

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 30705	. . . . 5 ⊢ ((proj_ℎ‘𝐺)‘𝐴) ∈ 𝐺
4		pjidm.1	. . . . . . 7 ⊢ 𝐻 ∈ C_ℋ
5	4, 2	pjclii 30705	. . . . . 6 ⊢ ((proj_ℎ‘𝐻)‘𝐴) ∈ 𝐻
6		ssel 3976	. . . . . 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 30511	. . . . . 6 ⊢ 𝐺 ∈ S_ℋ
9		shsubcl 30504	. . . . . 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 1449	. . . . 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 30963	. . . . 5 ⊢ (((proj_ℎ‘(⊥‘𝐻))‘𝐴) −_ℎ ((proj_ℎ‘(⊥‘𝐺))‘𝐴)) = (((proj_ℎ‘𝐺)‘𝐴) −_ℎ ((proj_ℎ‘𝐻)‘𝐴))
13	4, 1	chsscon3i 30745	. . . . . 6 ⊢ (𝐻 ⊆ 𝐺 ↔ (⊥‘𝐺) ⊆ (⊥‘𝐻))
14	4	choccli 30591	. . . . . . . 8 ⊢ (⊥‘𝐻) ∈ C_ℋ
15	14, 2	pjclii 30705	. . . . . . 7 ⊢ ((proj_ℎ‘(⊥‘𝐻))‘𝐴) ∈ (⊥‘𝐻)
16	1	choccli 30591	. . . . . . . . 9 ⊢ (⊥‘𝐺) ∈ C_ℋ
17	16, 2	pjclii 30705	. . . . . . . 8 ⊢ ((proj_ℎ‘(⊥‘𝐺))‘𝐴) ∈ (⊥‘𝐺)
18		ssel 3976	. . . . . . . 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 30511	. . . . . . . 8 ⊢ (⊥‘𝐻) ∈ S_ℋ
21		shsubcl 30504	. . . . . . . 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 1449	. . . . . . 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 216	. . . . 5 ⊢ (𝐻 ⊆ 𝐺 → (((proj_ℎ‘(⊥‘𝐻))‘𝐴) −_ℎ ((proj_ℎ‘(⊥‘𝐺))‘𝐴)) ∈ (⊥‘𝐻))
25	12, 24	eqeltrrid 2839	. . . 4 ⊢ (𝐻 ⊆ 𝐺 → (((proj_ℎ‘𝐺)‘𝐴) −_ℎ ((proj_ℎ‘𝐻)‘𝐴)) ∈ (⊥‘𝐻))
26	11, 25	jca 513	. . 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 3965	. . . 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 30744	. . . . 5 ⊢ (𝐺 ∩ (⊥‘𝐻)) ∈ C_ℋ
29	1, 2	pjhclii 30706	. . . . . 6 ⊢ ((proj_ℎ‘𝐺)‘𝐴) ∈ ℋ
30	4, 2	pjhclii 30706	. . . . . 6 ⊢ ((proj_ℎ‘𝐻)‘𝐴) ∈ ℋ
31	29, 30	hvsubcli 30305	. . . . 5 ⊢ (((proj_ℎ‘𝐺)‘𝐴) −_ℎ ((proj_ℎ‘𝐻)‘𝐴)) ∈ ℋ
32	28, 31	pjchi 30716	. . . 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 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 30962	. . 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 30706	. . . . 5 ⊢ ((proj_ℎ‘(𝐺 ∩ (⊥‘𝐻)))‘((proj_ℎ‘𝐺)‘𝐴)) ∈ ℋ
37	28, 30	pjhclii 30706	. . . . 5 ⊢ ((proj_ℎ‘(𝐺 ∩ (⊥‘𝐻)))‘((proj_ℎ‘𝐻)‘𝐴)) ∈ ℋ
38	36, 37	hvsubvali 30304	. . . 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 4229	. . . . . . 7 ⊢ (𝐺 ∩ (⊥‘𝐻)) ⊆ 𝐺
40	28, 2, 1	pjss2i 30964	. . . . . . 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 30511	. . . . . . . . . . . 12 ⊢ 𝐻 ∈ S_ℋ
43		shococss 30578	. . . . . . . . . . . 12 ⊢ (𝐻 ∈ S_ℋ → 𝐻 ⊆ (⊥‘(⊥‘𝐻)))
44	42, 43	ax-mp 5	. . . . . . . . . . 11 ⊢ 𝐻 ⊆ (⊥‘(⊥‘𝐻))
45		inss2 4230	. . . . . . . . . . . 12 ⊢ (𝐺 ∩ (⊥‘𝐻)) ⊆ (⊥‘𝐻)
46	28, 14	chsscon3i 30745	. . . . . . . . . . . 12 ⊢ ((𝐺 ∩ (⊥‘𝐻)) ⊆ (⊥‘𝐻) ↔ (⊥‘(⊥‘𝐻)) ⊆ (⊥‘(𝐺 ∩ (⊥‘𝐻))))
47	45, 46	mpbi 229	. . . . . . . . . . 11 ⊢ (⊥‘(⊥‘𝐻)) ⊆ (⊥‘(𝐺 ∩ (⊥‘𝐻)))
48	44, 47	sstri 3992	. . . . . . . . . 10 ⊢ 𝐻 ⊆ (⊥‘(𝐺 ∩ (⊥‘𝐻)))
49	48, 5	sselii 3980	. . . . . . . . 9 ⊢ ((proj_ℎ‘𝐻)‘𝐴) ∈ (⊥‘(𝐺 ∩ (⊥‘𝐻)))
50	28, 30	pjoc2i 30722	. . . . . . . . 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 7420	. . . . . . 7 ⊢ (-1 ·_ℎ ((proj_ℎ‘(𝐺 ∩ (⊥‘𝐻)))‘((proj_ℎ‘𝐻)‘𝐴))) = (-1 ·_ℎ 0_ℎ)
53		neg1cn 12326	. . . . . . . 8 ⊢ -1 ∈ ℂ
54		hvmul0 30308	. . . . . . . 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 2761	. . . . . 6 ⊢ (-1 ·_ℎ ((proj_ℎ‘(𝐺 ∩ (⊥‘𝐻)))‘((proj_ℎ‘𝐻)‘𝐴))) = 0_ℎ
57	41, 56	oveq12i 7421	. . . . 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 30706	. . . . . 6 ⊢ ((proj_ℎ‘(𝐺 ∩ (⊥‘𝐻)))‘𝐴) ∈ ℋ
59		ax-hvaddid 30288	. . . . . 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 2761	. . . 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 2761	. . 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 2761	. 2 ⊢ ((proj_ℎ‘(𝐺 ∩ (⊥‘𝐻)))‘(((proj_ℎ‘𝐺)‘𝐴) −_ℎ ((proj_ℎ‘𝐻)‘𝐴))) = ((proj_ℎ‘(𝐺 ∩ (⊥‘𝐻)))‘𝐴)
64	34, 63	eqtr3di 2788	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 397 = wceq 1542 ∈ wcel 2107 ∩ cin 3948 ⊆ wss 3949 ‘cfv 6544 (class class class)co 7409 ℂcc 11108 1c1 11111 -cneg 11445 ℋchba 30203 +_ℎ cva 30204 ·_ℎ csm 30205 0_ℎc0v 30208 −_ℎ cmv 30209 S_ℋ csh 30212 C_ℋ cch 30213 ⊥cort 30214 proj_ℎcpjh 30221

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 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725 ax-inf2 9636 ax-cc 10430 ax-cnex 11166 ax-resscn 11167 ax-1cn 11168 ax-icn 11169 ax-addcl 11170 ax-addrcl 11171 ax-mulcl 11172 ax-mulrcl 11173 ax-mulcom 11174 ax-addass 11175 ax-mulass 11176 ax-distr 11177 ax-i2m1 11178 ax-1ne0 11179 ax-1rid 11180 ax-rnegex 11181 ax-rrecex 11182 ax-cnre 11183 ax-pre-lttri 11184 ax-pre-lttrn 11185 ax-pre-ltadd 11186 ax-pre-mulgt0 11187 ax-pre-sup 11188 ax-addf 11189 ax-mulf 11190 ax-hilex 30283 ax-hfvadd 30284 ax-hvcom 30285 ax-hvass 30286 ax-hv0cl 30287 ax-hvaddid 30288 ax-hfvmul 30289 ax-hvmulid 30290 ax-hvmulass 30291 ax-hvdistr1 30292 ax-hvdistr2 30293 ax-hvmul0 30294 ax-hfi 30363 ax-his1 30366 ax-his2 30367 ax-his3 30368 ax-his4 30369 ax-hcompl 30486

This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-tp 4634 df-op 4636 df-uni 4910 df-int 4952 df-iun 5000 df-iin 5001 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5575 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-se 5633 df-we 5634 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-isom 6553 df-riota 7365 df-ov 7412 df-oprab 7413 df-mpo 7414 df-of 7670 df-om 7856 df-1st 7975 df-2nd 7976 df-supp 8147 df-frecs 8266 df-wrecs 8297 df-recs 8371 df-rdg 8410 df-1o 8466 df-2o 8467 df-oadd 8470 df-omul 8471 df-er 8703 df-map 8822 df-pm 8823 df-ixp 8892 df-en 8940 df-dom 8941 df-sdom 8942 df-fin 8943 df-fsupp 9362 df-fi 9406 df-sup 9437 df-inf 9438 df-oi 9505 df-card 9934 df-acn 9937 df-pnf 11250 df-mnf 11251 df-xr 11252 df-ltxr 11253 df-le 11254 df-sub 11446 df-neg 11447 df-div 11872 df-nn 12213 df-2 12275 df-3 12276 df-4 12277 df-5 12278 df-6 12279 df-7 12280 df-8 12281 df-9 12282 df-n0 12473 df-z 12559 df-dec 12678 df-uz 12823 df-q 12933 df-rp 12975 df-xneg 13092 df-xadd 13093 df-xmul 13094 df-ioo 13328 df-ico 13330 df-icc 13331 df-fz 13485 df-fzo 13628 df-fl 13757 df-seq 13967 df-exp 14028 df-hash 14291 df-cj 15046 df-re 15047 df-im 15048 df-sqrt 15182 df-abs 15183 df-clim 15432 df-rlim 15433 df-sum 15633 df-struct 17080 df-sets 17097 df-slot 17115 df-ndx 17127 df-base 17145 df-ress 17174 df-plusg 17210 df-mulr 17211 df-starv 17212 df-sca 17213 df-vsca 17214 df-ip 17215 df-tset 17216 df-ple 17217 df-ds 17219 df-unif 17220 df-hom 17221 df-cco 17222 df-rest 17368 df-topn 17369 df-0g 17387 df-gsum 17388 df-topgen 17389 df-pt 17390 df-prds 17393 df-xrs 17448 df-qtop 17453 df-imas 17454 df-xps 17456 df-mre 17530 df-mrc 17531 df-acs 17533 df-mgm 18561 df-sgrp 18610 df-mnd 18626 df-submnd 18672 df-mulg 18951 df-cntz 19181 df-cmn 19650 df-psmet 20936 df-xmet 20937 df-met 20938 df-bl 20939 df-mopn 20940 df-fbas 20941 df-fg 20942 df-cnfld 20945 df-top 22396 df-topon 22413 df-topsp 22435 df-bases 22449 df-cld 22523 df-ntr 22524 df-cls 22525 df-nei 22602 df-cn 22731 df-cnp 22732 df-lm 22733 df-haus 22819 df-tx 23066 df-hmeo 23259 df-fil 23350 df-fm 23442 df-flim 23443 df-flf 23444 df-xms 23826 df-ms 23827 df-tms 23828 df-cfil 24772 df-cau 24773 df-cmet 24774 df-grpo 29777 df-gid 29778 df-ginv 29779 df-gdiv 29780 df-ablo 29829 df-vc 29843 df-nv 29876 df-va 29879 df-ba 29880 df-sm 29881 df-0v 29882 df-vs 29883 df-nmcv 29884 df-ims 29885 df-dip 29985 df-ssp 30006 df-ph 30097 df-cbn 30147 df-hnorm 30252 df-hba 30253 df-hvsub 30255 df-hlim 30256 df-hcau 30257 df-sh 30491 df-ch 30505 df-oc 30536 df-ch0 30537 df-shs 30592 df-pjh 30679

This theorem is referenced by: pjcji 30968 pjssmi 31449

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