4cphipval2 - Metamath Proof Explorer

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Theorem 4cphipval2 25276

Description: Four times the inner product value cphipval2 25275. (Contributed by NM, 1-Feb-2008.) (Revised by AV, 18-Oct-2021.)

**Hypotheses**
Ref	Expression
cphipfval.x	⊢ 𝑋 = (Base‘𝑊)
cphipfval.p	⊢ + = (+_g‘𝑊)
cphipfval.s	⊢ · = ( ·_𝑠 ‘𝑊)
cphipfval.n	⊢ 𝑁 = (norm‘𝑊)
cphipfval.i	⊢ , = (·_𝑖‘𝑊)
cphipval2.m	⊢ − = (-_g‘𝑊)
cphipval2.f	⊢ 𝐹 = (Scalar‘𝑊)
cphipval2.k	⊢ 𝐾 = (Base‘𝐹)

**Assertion**
Ref	Expression
4cphipval2	⊢ (((𝑊 ∈ ℂPreHil ∧ i ∈ 𝐾) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (4 · (𝐴 , 𝐵)) = ((((𝑁‘(𝐴 + 𝐵))↑2) − ((𝑁‘(𝐴 − 𝐵))↑2)) + (i · (((𝑁‘(𝐴 + (i · 𝐵)))↑2) − ((𝑁‘(𝐴 − (i · 𝐵)))↑2)))))

**Proof of Theorem 4cphipval2**
Step	Hyp	Ref	Expression
1		cphipfval.x	. . . 4 ⊢ 𝑋 = (Base‘𝑊)
2		cphipfval.p	. . . 4 ⊢ + = (+_g‘𝑊)
3		cphipfval.s	. . . 4 ⊢ · = ( ·_𝑠 ‘𝑊)
4		cphipfval.n	. . . 4 ⊢ 𝑁 = (norm‘𝑊)
5		cphipfval.i	. . . 4 ⊢ , = (·_𝑖‘𝑊)
6		cphipval2.m	. . . 4 ⊢ − = (-_g‘𝑊)
7		cphipval2.f	. . . 4 ⊢ 𝐹 = (Scalar‘𝑊)
8		cphipval2.k	. . . 4 ⊢ 𝐾 = (Base‘𝐹)
9	1, 2, 3, 4, 5, 6, 7, 8	cphipval2 25275	. . 3 ⊢ (((𝑊 ∈ ℂPreHil ∧ i ∈ 𝐾) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴 , 𝐵) = (((((𝑁‘(𝐴 + 𝐵))↑2) − ((𝑁‘(𝐴 − 𝐵))↑2)) + (i · (((𝑁‘(𝐴 + (i · 𝐵)))↑2) − ((𝑁‘(𝐴 − (i · 𝐵)))↑2)))) / 4))
10	9	oveq2d 7447	. 2 ⊢ (((𝑊 ∈ ℂPreHil ∧ i ∈ 𝐾) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (4 · (𝐴 , 𝐵)) = (4 · (((((𝑁‘(𝐴 + 𝐵))↑2) − ((𝑁‘(𝐴 − 𝐵))↑2)) + (i · (((𝑁‘(𝐴 + (i · 𝐵)))↑2) − ((𝑁‘(𝐴 − (i · 𝐵)))↑2)))) / 4)))
11	7, 8	cphsubrg 25214	. . . . . . . . . 10 ⊢ (𝑊 ∈ ℂPreHil → 𝐾 ∈ (SubRing‘ℂ_fld))
12		cnfldbas 21368	. . . . . . . . . . 11 ⊢ ℂ = (Base‘ℂ_fld)
13	12	subrgss 20572	. . . . . . . . . 10 ⊢ (𝐾 ∈ (SubRing‘ℂ_fld) → 𝐾 ⊆ ℂ)
14	11, 13	syl 17	. . . . . . . . 9 ⊢ (𝑊 ∈ ℂPreHil → 𝐾 ⊆ ℂ)
15	14	adantr 480	. . . . . . . 8 ⊢ ((𝑊 ∈ ℂPreHil ∧ i ∈ 𝐾) → 𝐾 ⊆ ℂ)
16	15	3ad2ant1 1134	. . . . . . 7 ⊢ (((𝑊 ∈ ℂPreHil ∧ i ∈ 𝐾) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → 𝐾 ⊆ ℂ)
17		simp1l 1198	. . . . . . . 8 ⊢ (((𝑊 ∈ ℂPreHil ∧ i ∈ 𝐾) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → 𝑊 ∈ ℂPreHil)
18		cphngp 25207	. . . . . . . . . . 11 ⊢ (𝑊 ∈ ℂPreHil → 𝑊 ∈ NrmGrp)
19		ngpgrp 24612	. . . . . . . . . . 11 ⊢ (𝑊 ∈ NrmGrp → 𝑊 ∈ Grp)
20	18, 19	syl 17	. . . . . . . . . 10 ⊢ (𝑊 ∈ ℂPreHil → 𝑊 ∈ Grp)
21	20	adantr 480	. . . . . . . . 9 ⊢ ((𝑊 ∈ ℂPreHil ∧ i ∈ 𝐾) → 𝑊 ∈ Grp)
22	1, 2	grpcl 18959	. . . . . . . . 9 ⊢ ((𝑊 ∈ Grp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴 + 𝐵) ∈ 𝑋)
23	21, 22	syl3an1 1164	. . . . . . . 8 ⊢ (((𝑊 ∈ ℂPreHil ∧ i ∈ 𝐾) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴 + 𝐵) ∈ 𝑋)
24	1, 5, 4, 7, 8	cphnmcl 25230	. . . . . . . 8 ⊢ ((𝑊 ∈ ℂPreHil ∧ (𝐴 + 𝐵) ∈ 𝑋) → (𝑁‘(𝐴 + 𝐵)) ∈ 𝐾)
25	17, 23, 24	syl2anc 584	. . . . . . 7 ⊢ (((𝑊 ∈ ℂPreHil ∧ i ∈ 𝐾) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝑁‘(𝐴 + 𝐵)) ∈ 𝐾)
26	16, 25	sseldd 3984	. . . . . 6 ⊢ (((𝑊 ∈ ℂPreHil ∧ i ∈ 𝐾) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝑁‘(𝐴 + 𝐵)) ∈ ℂ)
27	26	sqcld 14184	. . . . 5 ⊢ (((𝑊 ∈ ℂPreHil ∧ i ∈ 𝐾) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((𝑁‘(𝐴 + 𝐵))↑2) ∈ ℂ)
28	1, 6	grpsubcl 19038	. . . . . . . . 9 ⊢ ((𝑊 ∈ Grp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴 − 𝐵) ∈ 𝑋)
29	21, 28	syl3an1 1164	. . . . . . . 8 ⊢ (((𝑊 ∈ ℂPreHil ∧ i ∈ 𝐾) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴 − 𝐵) ∈ 𝑋)
30	1, 5, 4, 7, 8	cphnmcl 25230	. . . . . . . 8 ⊢ ((𝑊 ∈ ℂPreHil ∧ (𝐴 − 𝐵) ∈ 𝑋) → (𝑁‘(𝐴 − 𝐵)) ∈ 𝐾)
31	17, 29, 30	syl2anc 584	. . . . . . 7 ⊢ (((𝑊 ∈ ℂPreHil ∧ i ∈ 𝐾) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝑁‘(𝐴 − 𝐵)) ∈ 𝐾)
32	16, 31	sseldd 3984	. . . . . 6 ⊢ (((𝑊 ∈ ℂPreHil ∧ i ∈ 𝐾) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝑁‘(𝐴 − 𝐵)) ∈ ℂ)
33	32	sqcld 14184	. . . . 5 ⊢ (((𝑊 ∈ ℂPreHil ∧ i ∈ 𝐾) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((𝑁‘(𝐴 − 𝐵))↑2) ∈ ℂ)
34	27, 33	subcld 11620	. . . 4 ⊢ (((𝑊 ∈ ℂPreHil ∧ i ∈ 𝐾) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (((𝑁‘(𝐴 + 𝐵))↑2) − ((𝑁‘(𝐴 − 𝐵))↑2)) ∈ ℂ)
35		ax-icn 11214	. . . . . 6 ⊢ i ∈ ℂ
36	35	a1i 11	. . . . 5 ⊢ (((𝑊 ∈ ℂPreHil ∧ i ∈ 𝐾) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → i ∈ ℂ)
37	17, 20	syl 17	. . . . . . . . . 10 ⊢ (((𝑊 ∈ ℂPreHil ∧ i ∈ 𝐾) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → 𝑊 ∈ Grp)
38		simp2 1138	. . . . . . . . . 10 ⊢ (((𝑊 ∈ ℂPreHil ∧ i ∈ 𝐾) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → 𝐴 ∈ 𝑋)
39		cphlmod 25208	. . . . . . . . . . . . 13 ⊢ (𝑊 ∈ ℂPreHil → 𝑊 ∈ LMod)
40	39	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝑊 ∈ ℂPreHil ∧ i ∈ 𝐾) → 𝑊 ∈ LMod)
41	40	3ad2ant1 1134	. . . . . . . . . . 11 ⊢ (((𝑊 ∈ ℂPreHil ∧ i ∈ 𝐾) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → 𝑊 ∈ LMod)
42		simp1r 1199	. . . . . . . . . . 11 ⊢ (((𝑊 ∈ ℂPreHil ∧ i ∈ 𝐾) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → i ∈ 𝐾)
43		simp3 1139	. . . . . . . . . . 11 ⊢ (((𝑊 ∈ ℂPreHil ∧ i ∈ 𝐾) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → 𝐵 ∈ 𝑋)
44	1, 7, 3, 8	lmodvscl 20876	. . . . . . . . . . 11 ⊢ ((𝑊 ∈ LMod ∧ i ∈ 𝐾 ∧ 𝐵 ∈ 𝑋) → (i · 𝐵) ∈ 𝑋)
45	41, 42, 43, 44	syl3anc 1373	. . . . . . . . . 10 ⊢ (((𝑊 ∈ ℂPreHil ∧ i ∈ 𝐾) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (i · 𝐵) ∈ 𝑋)
46	1, 2	grpcl 18959	. . . . . . . . . 10 ⊢ ((𝑊 ∈ Grp ∧ 𝐴 ∈ 𝑋 ∧ (i · 𝐵) ∈ 𝑋) → (𝐴 + (i · 𝐵)) ∈ 𝑋)
47	37, 38, 45, 46	syl3anc 1373	. . . . . . . . 9 ⊢ (((𝑊 ∈ ℂPreHil ∧ i ∈ 𝐾) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴 + (i · 𝐵)) ∈ 𝑋)
48	1, 5, 4, 7, 8	cphnmcl 25230	. . . . . . . . 9 ⊢ ((𝑊 ∈ ℂPreHil ∧ (𝐴 + (i · 𝐵)) ∈ 𝑋) → (𝑁‘(𝐴 + (i · 𝐵))) ∈ 𝐾)
49	17, 47, 48	syl2anc 584	. . . . . . . 8 ⊢ (((𝑊 ∈ ℂPreHil ∧ i ∈ 𝐾) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝑁‘(𝐴 + (i · 𝐵))) ∈ 𝐾)
50	16, 49	sseldd 3984	. . . . . . 7 ⊢ (((𝑊 ∈ ℂPreHil ∧ i ∈ 𝐾) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝑁‘(𝐴 + (i · 𝐵))) ∈ ℂ)
51	50	sqcld 14184	. . . . . 6 ⊢ (((𝑊 ∈ ℂPreHil ∧ i ∈ 𝐾) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((𝑁‘(𝐴 + (i · 𝐵)))↑2) ∈ ℂ)
52	1, 6	grpsubcl 19038	. . . . . . . . . 10 ⊢ ((𝑊 ∈ Grp ∧ 𝐴 ∈ 𝑋 ∧ (i · 𝐵) ∈ 𝑋) → (𝐴 − (i · 𝐵)) ∈ 𝑋)
53	37, 38, 45, 52	syl3anc 1373	. . . . . . . . 9 ⊢ (((𝑊 ∈ ℂPreHil ∧ i ∈ 𝐾) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴 − (i · 𝐵)) ∈ 𝑋)
54	1, 5, 4, 7, 8	cphnmcl 25230	. . . . . . . . 9 ⊢ ((𝑊 ∈ ℂPreHil ∧ (𝐴 − (i · 𝐵)) ∈ 𝑋) → (𝑁‘(𝐴 − (i · 𝐵))) ∈ 𝐾)
55	17, 53, 54	syl2anc 584	. . . . . . . 8 ⊢ (((𝑊 ∈ ℂPreHil ∧ i ∈ 𝐾) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝑁‘(𝐴 − (i · 𝐵))) ∈ 𝐾)
56	16, 55	sseldd 3984	. . . . . . 7 ⊢ (((𝑊 ∈ ℂPreHil ∧ i ∈ 𝐾) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝑁‘(𝐴 − (i · 𝐵))) ∈ ℂ)
57	56	sqcld 14184	. . . . . 6 ⊢ (((𝑊 ∈ ℂPreHil ∧ i ∈ 𝐾) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((𝑁‘(𝐴 − (i · 𝐵)))↑2) ∈ ℂ)
58	51, 57	subcld 11620	. . . . 5 ⊢ (((𝑊 ∈ ℂPreHil ∧ i ∈ 𝐾) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (((𝑁‘(𝐴 + (i · 𝐵)))↑2) − ((𝑁‘(𝐴 − (i · 𝐵)))↑2)) ∈ ℂ)
59	36, 58	mulcld 11281	. . . 4 ⊢ (((𝑊 ∈ ℂPreHil ∧ i ∈ 𝐾) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (i · (((𝑁‘(𝐴 + (i · 𝐵)))↑2) − ((𝑁‘(𝐴 − (i · 𝐵)))↑2))) ∈ ℂ)
60	34, 59	addcld 11280	. . 3 ⊢ (((𝑊 ∈ ℂPreHil ∧ i ∈ 𝐾) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((((𝑁‘(𝐴 + 𝐵))↑2) − ((𝑁‘(𝐴 − 𝐵))↑2)) + (i · (((𝑁‘(𝐴 + (i · 𝐵)))↑2) − ((𝑁‘(𝐴 − (i · 𝐵)))↑2)))) ∈ ℂ)
61		4cn 12351	. . . 4 ⊢ 4 ∈ ℂ
62	61	a1i 11	. . 3 ⊢ (((𝑊 ∈ ℂPreHil ∧ i ∈ 𝐾) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → 4 ∈ ℂ)
63		4ne0 12374	. . . 4 ⊢ 4 ≠ 0
64	63	a1i 11	. . 3 ⊢ (((𝑊 ∈ ℂPreHil ∧ i ∈ 𝐾) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → 4 ≠ 0)
65	60, 62, 64	divcan2d 12045	. 2 ⊢ (((𝑊 ∈ ℂPreHil ∧ i ∈ 𝐾) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (4 · (((((𝑁‘(𝐴 + 𝐵))↑2) − ((𝑁‘(𝐴 − 𝐵))↑2)) + (i · (((𝑁‘(𝐴 + (i · 𝐵)))↑2) − ((𝑁‘(𝐴 − (i · 𝐵)))↑2)))) / 4)) = ((((𝑁‘(𝐴 + 𝐵))↑2) − ((𝑁‘(𝐴 − 𝐵))↑2)) + (i · (((𝑁‘(𝐴 + (i · 𝐵)))↑2) − ((𝑁‘(𝐴 − (i · 𝐵)))↑2)))))
66	10, 65	eqtrd 2777	1 ⊢ (((𝑊 ∈ ℂPreHil ∧ i ∈ 𝐾) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (4 · (𝐴 , 𝐵)) = ((((𝑁‘(𝐴 + 𝐵))↑2) − ((𝑁‘(𝐴 − 𝐵))↑2)) + (i · (((𝑁‘(𝐴 + (i · 𝐵)))↑2) − ((𝑁‘(𝐴 − (i · 𝐵)))↑2)))))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1087 = wceq 1540 ∈ wcel 2108 ≠ wne 2940 ⊆ wss 3951 ‘cfv 6561 (class class class)co 7431 ℂcc 11153 0cc0 11155 ici 11157 + caddc 11158 · cmul 11160 − cmin 11492 / cdiv 11920 2c2 12321 4c4 12323 ↑cexp 14102 Basecbs 17247 +_gcplusg 17297 Scalarcsca 17300 ·_𝑠 cvsca 17301 ·_𝑖cip 17302 Grpcgrp 18951 -_gcsg 18953 SubRingcsubrg 20569 LModclmod 20858 ℂ_fldccnfld 21364 normcnm 24589 NrmGrpcngp 24590 ℂPreHilccph 25200

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 ax-pre-sup 11233 ax-addf 11234 ax-mulf 11235

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-tp 4631 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8014 df-2nd 8015 df-tpos 8251 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-er 8745 df-map 8868 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-sup 9482 df-inf 9483 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-div 11921 df-nn 12267 df-2 12329 df-3 12330 df-4 12331 df-5 12332 df-6 12333 df-7 12334 df-8 12335 df-9 12336 df-n0 12527 df-z 12614 df-dec 12734 df-uz 12879 df-q 12991 df-rp 13035 df-xneg 13154 df-xadd 13155 df-xmul 13156 df-ico 13393 df-fz 13548 df-seq 14043 df-exp 14103 df-cj 15138 df-re 15139 df-im 15140 df-sqrt 15274 df-abs 15275 df-struct 17184 df-sets 17201 df-slot 17219 df-ndx 17231 df-base 17248 df-ress 17275 df-plusg 17310 df-mulr 17311 df-starv 17312 df-sca 17313 df-vsca 17314 df-ip 17315 df-tset 17316 df-ple 17317 df-ds 17319 df-unif 17320 df-0g 17486 df-topgen 17488 df-mgm 18653 df-sgrp 18732 df-mnd 18748 df-mhm 18796 df-grp 18954 df-minusg 18955 df-sbg 18956 df-subg 19141 df-ghm 19231 df-cmn 19800 df-abl 19801 df-mgp 20138 df-rng 20150 df-ur 20179 df-ring 20232 df-cring 20233 df-oppr 20334 df-dvdsr 20357 df-unit 20358 df-rhm 20472 df-subrg 20570 df-drng 20731 df-staf 20840 df-srng 20841 df-lmod 20860 df-lmhm 21021 df-lvec 21102 df-sra 21172 df-rgmod 21173 df-psmet 21356 df-xmet 21357 df-met 21358 df-bl 21359 df-mopn 21360 df-cnfld 21365 df-phl 21644 df-top 22900 df-topon 22917 df-topsp 22939 df-bases 22953 df-xms 24330 df-ms 24331 df-nm 24595 df-ngp 24596 df-nlm 24599 df-clm 25096 df-cph 25202

This theorem is referenced by: (None)

W3C validator