ncvspi - Metamath Proof Explorer

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Theorem ncvspi 22764

Description: The norm of a vector plus the imaginary scalar product of another. (Contributed by NM, 2-Feb-2007.) (Revised by AV, 8-Oct-2021.)

**Hypotheses**
Ref	Expression
ncvsprp.v	⊢ 𝑉 = (Base‘𝑊)
ncvsprp.n	⊢ 𝑁 = (norm‘𝑊)
ncvsprp.s	⊢ · = ( ·_𝑠 ‘𝑊)
ncvsdif.p	⊢ + = (+_g‘𝑊)
ncvspi.f	⊢ 𝐹 = (Scalar‘𝑊)
ncvspi.k	⊢ 𝐾 = (Base‘𝐹)

**Assertion**
Ref	Expression
ncvspi	⊢ ((𝑊 ∈ (NrmVec ∩ ℂVec) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ i ∈ 𝐾) → (𝑁‘(𝐴 + (i · 𝐵))) = (𝑁‘(𝐵 + (-i · 𝐴))))

**Proof of Theorem ncvspi**
Step	Hyp	Ref	Expression
1		elin 3758	. . . . . . 7 ⊢ (𝑊 ∈ (NrmVec ∩ ℂVec) ↔ (𝑊 ∈ NrmVec ∧ 𝑊 ∈ ℂVec))
2		nvcnlm 22310	. . . . . . . . 9 ⊢ (𝑊 ∈ NrmVec → 𝑊 ∈ NrmMod)
3		nlmngp 22291	. . . . . . . . 9 ⊢ (𝑊 ∈ NrmMod → 𝑊 ∈ NrmGrp)
4	2, 3	syl 17	. . . . . . . 8 ⊢ (𝑊 ∈ NrmVec → 𝑊 ∈ NrmGrp)
5	4	adantr 480	. . . . . . 7 ⊢ ((𝑊 ∈ NrmVec ∧ 𝑊 ∈ ℂVec) → 𝑊 ∈ NrmGrp)
6	1, 5	sylbi 206	. . . . . 6 ⊢ (𝑊 ∈ (NrmVec ∩ ℂVec) → 𝑊 ∈ NrmGrp)
7	6	3ad2ant1 1075	. . . . 5 ⊢ ((𝑊 ∈ (NrmVec ∩ ℂVec) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ i ∈ 𝐾) → 𝑊 ∈ NrmGrp)
8		nvclmod 22312	. . . . . . . . . 10 ⊢ (𝑊 ∈ NrmVec → 𝑊 ∈ LMod)
9		lmodgrp 18693	. . . . . . . . . 10 ⊢ (𝑊 ∈ LMod → 𝑊 ∈ Grp)
10	8, 9	syl 17	. . . . . . . . 9 ⊢ (𝑊 ∈ NrmVec → 𝑊 ∈ Grp)
11	10	adantr 480	. . . . . . . 8 ⊢ ((𝑊 ∈ NrmVec ∧ 𝑊 ∈ ℂVec) → 𝑊 ∈ Grp)
12	1, 11	sylbi 206	. . . . . . 7 ⊢ (𝑊 ∈ (NrmVec ∩ ℂVec) → 𝑊 ∈ Grp)
13	12	3ad2ant1 1075	. . . . . 6 ⊢ ((𝑊 ∈ (NrmVec ∩ ℂVec) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ i ∈ 𝐾) → 𝑊 ∈ Grp)
14		simp2l 1080	. . . . . 6 ⊢ ((𝑊 ∈ (NrmVec ∩ ℂVec) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ i ∈ 𝐾) → 𝐴 ∈ 𝑉)
15		id 22	. . . . . . . . . 10 ⊢ (𝑊 ∈ ℂVec → 𝑊 ∈ ℂVec)
16	15	cvsclm 22734	. . . . . . . . 9 ⊢ (𝑊 ∈ ℂVec → 𝑊 ∈ ℂMod)
17	1, 16	simplbiim 657	. . . . . . . 8 ⊢ (𝑊 ∈ (NrmVec ∩ ℂVec) → 𝑊 ∈ ℂMod)
18	17	3ad2ant1 1075	. . . . . . 7 ⊢ ((𝑊 ∈ (NrmVec ∩ ℂVec) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ i ∈ 𝐾) → 𝑊 ∈ ℂMod)
19		simp3 1056	. . . . . . 7 ⊢ ((𝑊 ∈ (NrmVec ∩ ℂVec) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ i ∈ 𝐾) → i ∈ 𝐾)
20		simp2r 1081	. . . . . . 7 ⊢ ((𝑊 ∈ (NrmVec ∩ ℂVec) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ i ∈ 𝐾) → 𝐵 ∈ 𝑉)
21		ncvsprp.v	. . . . . . . 8 ⊢ 𝑉 = (Base‘𝑊)
22		ncvspi.f	. . . . . . . 8 ⊢ 𝐹 = (Scalar‘𝑊)
23		ncvsprp.s	. . . . . . . 8 ⊢ · = ( ·_𝑠 ‘𝑊)
24		ncvspi.k	. . . . . . . 8 ⊢ 𝐾 = (Base‘𝐹)
25	21, 22, 23, 24	clmvscl 22696	. . . . . . 7 ⊢ ((𝑊 ∈ ℂMod ∧ i ∈ 𝐾 ∧ 𝐵 ∈ 𝑉) → (i · 𝐵) ∈ 𝑉)
26	18, 19, 20, 25	syl3anc 1318	. . . . . 6 ⊢ ((𝑊 ∈ (NrmVec ∩ ℂVec) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ i ∈ 𝐾) → (i · 𝐵) ∈ 𝑉)
27		ncvsdif.p	. . . . . . 7 ⊢ + = (+_g‘𝑊)
28	21, 27	grpcl 17253	. . . . . 6 ⊢ ((𝑊 ∈ Grp ∧ 𝐴 ∈ 𝑉 ∧ (i · 𝐵) ∈ 𝑉) → (𝐴 + (i · 𝐵)) ∈ 𝑉)
29	13, 14, 26, 28	syl3anc 1318	. . . . 5 ⊢ ((𝑊 ∈ (NrmVec ∩ ℂVec) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ i ∈ 𝐾) → (𝐴 + (i · 𝐵)) ∈ 𝑉)
30		ncvsprp.n	. . . . . 6 ⊢ 𝑁 = (norm‘𝑊)
31	21, 30	nmcl 22230	. . . . 5 ⊢ ((𝑊 ∈ NrmGrp ∧ (𝐴 + (i · 𝐵)) ∈ 𝑉) → (𝑁‘(𝐴 + (i · 𝐵))) ∈ ℝ)
32	7, 29, 31	syl2anc 691	. . . 4 ⊢ ((𝑊 ∈ (NrmVec ∩ ℂVec) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ i ∈ 𝐾) → (𝑁‘(𝐴 + (i · 𝐵))) ∈ ℝ)
33	32	recnd 9947	. . 3 ⊢ ((𝑊 ∈ (NrmVec ∩ ℂVec) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ i ∈ 𝐾) → (𝑁‘(𝐴 + (i · 𝐵))) ∈ ℂ)
34	33	mulid2d 9937	. 2 ⊢ ((𝑊 ∈ (NrmVec ∩ ℂVec) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ i ∈ 𝐾) → (1 · (𝑁‘(𝐴 + (i · 𝐵)))) = (𝑁‘(𝐴 + (i · 𝐵))))
35		ax-icn 9874	. . . . . 6 ⊢ i ∈ ℂ
36	35	absnegi 13987	. . . . 5 ⊢ (abs‘-i) = (abs‘i)
37		absi 13874	. . . . 5 ⊢ (abs‘i) = 1
38	36, 37	eqtri 2632	. . . 4 ⊢ (abs‘-i) = 1
39	38	oveq1i 6559	. . 3 ⊢ ((abs‘-i) · (𝑁‘(𝐴 + (i · 𝐵)))) = (1 · (𝑁‘(𝐴 + (i · 𝐵))))
40		simp1 1054	. . . . 5 ⊢ ((𝑊 ∈ (NrmVec ∩ ℂVec) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ i ∈ 𝐾) → 𝑊 ∈ (NrmVec ∩ ℂVec))
41	22, 24	clmneg 22689	. . . . . . . . . . 11 ⊢ ((𝑊 ∈ ℂMod ∧ i ∈ 𝐾) → -i = ((inv_g‘𝐹)‘i))
42	16, 41	sylan 487	. . . . . . . . . 10 ⊢ ((𝑊 ∈ ℂVec ∧ i ∈ 𝐾) → -i = ((inv_g‘𝐹)‘i))
43	22	clmfgrp 22679	. . . . . . . . . . . 12 ⊢ (𝑊 ∈ ℂMod → 𝐹 ∈ Grp)
44	16, 43	syl 17	. . . . . . . . . . 11 ⊢ (𝑊 ∈ ℂVec → 𝐹 ∈ Grp)
45		eqid 2610	. . . . . . . . . . . 12 ⊢ (inv_g‘𝐹) = (inv_g‘𝐹)
46	24, 45	grpinvcl 17290	. . . . . . . . . . 11 ⊢ ((𝐹 ∈ Grp ∧ i ∈ 𝐾) → ((inv_g‘𝐹)‘i) ∈ 𝐾)
47	44, 46	sylan 487	. . . . . . . . . 10 ⊢ ((𝑊 ∈ ℂVec ∧ i ∈ 𝐾) → ((inv_g‘𝐹)‘i) ∈ 𝐾)
48	42, 47	eqeltrd 2688	. . . . . . . . 9 ⊢ ((𝑊 ∈ ℂVec ∧ i ∈ 𝐾) → -i ∈ 𝐾)
49	48	ex 449	. . . . . . . 8 ⊢ (𝑊 ∈ ℂVec → (i ∈ 𝐾 → -i ∈ 𝐾))
50	1, 49	simplbiim 657	. . . . . . 7 ⊢ (𝑊 ∈ (NrmVec ∩ ℂVec) → (i ∈ 𝐾 → -i ∈ 𝐾))
51	50	imp 444	. . . . . 6 ⊢ ((𝑊 ∈ (NrmVec ∩ ℂVec) ∧ i ∈ 𝐾) → -i ∈ 𝐾)
52	51	3adant2 1073	. . . . 5 ⊢ ((𝑊 ∈ (NrmVec ∩ ℂVec) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ i ∈ 𝐾) → -i ∈ 𝐾)
53	21, 30, 23, 22, 24	ncvsprp 22760	. . . . 5 ⊢ ((𝑊 ∈ (NrmVec ∩ ℂVec) ∧ -i ∈ 𝐾 ∧ (𝐴 + (i · 𝐵)) ∈ 𝑉) → (𝑁‘(-i · (𝐴 + (i · 𝐵)))) = ((abs‘-i) · (𝑁‘(𝐴 + (i · 𝐵)))))
54	40, 52, 29, 53	syl3anc 1318	. . . 4 ⊢ ((𝑊 ∈ (NrmVec ∩ ℂVec) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ i ∈ 𝐾) → (𝑁‘(-i · (𝐴 + (i · 𝐵)))) = ((abs‘-i) · (𝑁‘(𝐴 + (i · 𝐵)))))
55	21, 22, 23, 24, 27	clmvsdi 22700	. . . . . . 7 ⊢ ((𝑊 ∈ ℂMod ∧ (-i ∈ 𝐾 ∧ 𝐴 ∈ 𝑉 ∧ (i · 𝐵) ∈ 𝑉)) → (-i · (𝐴 + (i · 𝐵))) = ((-i · 𝐴) + (-i · (i · 𝐵))))
56	18, 52, 14, 26, 55	syl13anc 1320	. . . . . 6 ⊢ ((𝑊 ∈ (NrmVec ∩ ℂVec) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ i ∈ 𝐾) → (-i · (𝐴 + (i · 𝐵))) = ((-i · 𝐴) + (-i · (i · 𝐵))))
57	35, 35	mulneg1i 10355	. . . . . . . . . 10 ⊢ (-i · i) = -(i · i)
58		ixi 10535	. . . . . . . . . . . 12 ⊢ (i · i) = -1
59	58	negeqi 10153	. . . . . . . . . . 11 ⊢ -(i · i) = --1
60		negneg1e1 11005	. . . . . . . . . . 11 ⊢ --1 = 1
61	59, 60	eqtri 2632	. . . . . . . . . 10 ⊢ -(i · i) = 1
62	57, 61	eqtri 2632	. . . . . . . . 9 ⊢ (-i · i) = 1
63	62	oveq1i 6559	. . . . . . . 8 ⊢ ((-i · i) · 𝐵) = (1 · 𝐵)
64	21, 22, 23, 24	clmvsass 22697	. . . . . . . . 9 ⊢ ((𝑊 ∈ ℂMod ∧ (-i ∈ 𝐾 ∧ i ∈ 𝐾 ∧ 𝐵 ∈ 𝑉)) → ((-i · i) · 𝐵) = (-i · (i · 𝐵)))
65	18, 52, 19, 20, 64	syl13anc 1320	. . . . . . . 8 ⊢ ((𝑊 ∈ (NrmVec ∩ ℂVec) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ i ∈ 𝐾) → ((-i · i) · 𝐵) = (-i · (i · 𝐵)))
66		simpr 476	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → 𝐵 ∈ 𝑉)
67	17, 66	anim12i 588	. . . . . . . . . 10 ⊢ ((𝑊 ∈ (NrmVec ∩ ℂVec) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉)) → (𝑊 ∈ ℂMod ∧ 𝐵 ∈ 𝑉))
68	67	3adant3 1074	. . . . . . . . 9 ⊢ ((𝑊 ∈ (NrmVec ∩ ℂVec) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ i ∈ 𝐾) → (𝑊 ∈ ℂMod ∧ 𝐵 ∈ 𝑉))
69	21, 23	clmvs1 22701	. . . . . . . . 9 ⊢ ((𝑊 ∈ ℂMod ∧ 𝐵 ∈ 𝑉) → (1 · 𝐵) = 𝐵)
70	68, 69	syl 17	. . . . . . . 8 ⊢ ((𝑊 ∈ (NrmVec ∩ ℂVec) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ i ∈ 𝐾) → (1 · 𝐵) = 𝐵)
71	63, 65, 70	3eqtr3a 2668	. . . . . . 7 ⊢ ((𝑊 ∈ (NrmVec ∩ ℂVec) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ i ∈ 𝐾) → (-i · (i · 𝐵)) = 𝐵)
72	71	oveq2d 6565	. . . . . 6 ⊢ ((𝑊 ∈ (NrmVec ∩ ℂVec) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ i ∈ 𝐾) → ((-i · 𝐴) + (-i · (i · 𝐵))) = ((-i · 𝐴) + 𝐵))
73		clmabl 22677	. . . . . . . . . 10 ⊢ (𝑊 ∈ ℂMod → 𝑊 ∈ Abel)
74	16, 73	syl 17	. . . . . . . . 9 ⊢ (𝑊 ∈ ℂVec → 𝑊 ∈ Abel)
75	1, 74	simplbiim 657	. . . . . . . 8 ⊢ (𝑊 ∈ (NrmVec ∩ ℂVec) → 𝑊 ∈ Abel)
76	75	3ad2ant1 1075	. . . . . . 7 ⊢ ((𝑊 ∈ (NrmVec ∩ ℂVec) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ i ∈ 𝐾) → 𝑊 ∈ Abel)
77	21, 22, 23, 24	clmvscl 22696	. . . . . . . 8 ⊢ ((𝑊 ∈ ℂMod ∧ -i ∈ 𝐾 ∧ 𝐴 ∈ 𝑉) → (-i · 𝐴) ∈ 𝑉)
78	18, 52, 14, 77	syl3anc 1318	. . . . . . 7 ⊢ ((𝑊 ∈ (NrmVec ∩ ℂVec) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ i ∈ 𝐾) → (-i · 𝐴) ∈ 𝑉)
79	21, 27	ablcom 18033	. . . . . . 7 ⊢ ((𝑊 ∈ Abel ∧ (-i · 𝐴) ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → ((-i · 𝐴) + 𝐵) = (𝐵 + (-i · 𝐴)))
80	76, 78, 20, 79	syl3anc 1318	. . . . . 6 ⊢ ((𝑊 ∈ (NrmVec ∩ ℂVec) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ i ∈ 𝐾) → ((-i · 𝐴) + 𝐵) = (𝐵 + (-i · 𝐴)))
81	56, 72, 80	3eqtrd 2648	. . . . 5 ⊢ ((𝑊 ∈ (NrmVec ∩ ℂVec) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ i ∈ 𝐾) → (-i · (𝐴 + (i · 𝐵))) = (𝐵 + (-i · 𝐴)))
82	81	fveq2d 6107	. . . 4 ⊢ ((𝑊 ∈ (NrmVec ∩ ℂVec) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ i ∈ 𝐾) → (𝑁‘(-i · (𝐴 + (i · 𝐵)))) = (𝑁‘(𝐵 + (-i · 𝐴))))
83	54, 82	eqtr3d 2646	. . 3 ⊢ ((𝑊 ∈ (NrmVec ∩ ℂVec) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ i ∈ 𝐾) → ((abs‘-i) · (𝑁‘(𝐴 + (i · 𝐵)))) = (𝑁‘(𝐵 + (-i · 𝐴))))
84	39, 83	syl5eqr 2658	. 2 ⊢ ((𝑊 ∈ (NrmVec ∩ ℂVec) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ i ∈ 𝐾) → (1 · (𝑁‘(𝐴 + (i · 𝐵)))) = (𝑁‘(𝐵 + (-i · 𝐴))))
85	34, 84	eqtr3d 2646	1 ⊢ ((𝑊 ∈ (NrmVec ∩ ℂVec) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ i ∈ 𝐾) → (𝑁‘(𝐴 + (i · 𝐵))) = (𝑁‘(𝐵 + (-i · 𝐴))))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 383 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 ∩ cin 3539 ‘cfv 5804 (class class class)co 6549 ℝcr 9814 1c1 9816 ici 9817 · cmul 9820 -cneg 10146 abscabs 13822 Basecbs 15695 +_gcplusg 15768 Scalarcsca 15771 ·_𝑠 cvsca 15772 Grpcgrp 17245 inv_gcminusg 17246 Abelcabl 18017 LModclmod 18686 normcnm 22191 NrmGrpcngp 22192 NrmModcnlm 22195 NrmVeccnvc 22196 ℂModcclm 22670 ℂVecccvs 22731

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 ax-pre-sup 9893 ax-addf 9894 ax-mulf 9895

This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-oadd 7451 df-er 7629 df-map 7746 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-sup 8231 df-inf 8232 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-div 10564 df-nn 10898 df-2 10956 df-3 10957 df-4 10958 df-5 10959 df-6 10960 df-7 10961 df-8 10962 df-9 10963 df-n0 11170 df-z 11255 df-dec 11370 df-uz 11564 df-q 11665 df-rp 11709 df-xneg 11822 df-xadd 11823 df-xmul 11824 df-fz 12198 df-seq 12664 df-exp 12723 df-cj 13687 df-re 13688 df-im 13689 df-sqrt 13823 df-abs 13824 df-struct 15697 df-ndx 15698 df-slot 15699 df-base 15700 df-sets 15701 df-ress 15702 df-plusg 15781 df-mulr 15782 df-starv 15783 df-tset 15787 df-ple 15788 df-ds 15791 df-unif 15792 df-0g 15925 df-topgen 15927 df-mgm 17065 df-sgrp 17107 df-mnd 17118 df-grp 17248 df-minusg 17249 df-subg 17414 df-cmn 18018 df-abl 18019 df-mgp 18313 df-ur 18325 df-ring 18372 df-cring 18373 df-subrg 18601 df-lmod 18688 df-psmet 19559 df-xmet 19560 df-met 19561 df-bl 19562 df-mopn 19563 df-cnfld 19568 df-top 20521 df-bases 20522 df-topon 20523 df-topsp 20524 df-xms 21935 df-ms 21936 df-nm 22197 df-ngp 22198 df-nlm 22201 df-nvc 22202 df-clm 22671 df-cvs 22732

This theorem is referenced by: (None)

W3C validator