ncvspi - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > ncvspi		Structured version Visualization version GIF version

Theorem ncvspi 25077

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 3961	. . . . . . 7 ⊢ (𝑊 ∈ (NrmVec ∩ ℂVec) ↔ (𝑊 ∈ NrmVec ∧ 𝑊 ∈ ℂVec))
2		nvcnlm 24606	. . . . . . . . 9 ⊢ (𝑊 ∈ NrmVec → 𝑊 ∈ NrmMod)
3		nlmngp 24587	. . . . . . . . 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 216	. . . . . 6 ⊢ (𝑊 ∈ (NrmVec ∩ ℂVec) → 𝑊 ∈ NrmGrp)
7	6	3ad2ant1 1131	. . . . 5 ⊢ ((𝑊 ∈ (NrmVec ∩ ℂVec) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ i ∈ 𝐾) → 𝑊 ∈ NrmGrp)
8		nvclmod 24608	. . . . . . . . . 10 ⊢ (𝑊 ∈ NrmVec → 𝑊 ∈ LMod)
9		lmodgrp 20743	. . . . . . . . . 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 216	. . . . . . 7 ⊢ (𝑊 ∈ (NrmVec ∩ ℂVec) → 𝑊 ∈ Grp)
13	12	3ad2ant1 1131	. . . . . 6 ⊢ ((𝑊 ∈ (NrmVec ∩ ℂVec) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ i ∈ 𝐾) → 𝑊 ∈ Grp)
14		simp2l 1197	. . . . . 6 ⊢ ((𝑊 ∈ (NrmVec ∩ ℂVec) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ i ∈ 𝐾) → 𝐴 ∈ 𝑉)
15		id 22	. . . . . . . . . 10 ⊢ (𝑊 ∈ ℂVec → 𝑊 ∈ ℂVec)
16	15	cvsclm 25046	. . . . . . . . 9 ⊢ (𝑊 ∈ ℂVec → 𝑊 ∈ ℂMod)
17	1, 16	simplbiim 504	. . . . . . . 8 ⊢ (𝑊 ∈ (NrmVec ∩ ℂVec) → 𝑊 ∈ ℂMod)
18	17	3ad2ant1 1131	. . . . . . 7 ⊢ ((𝑊 ∈ (NrmVec ∩ ℂVec) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ i ∈ 𝐾) → 𝑊 ∈ ℂMod)
19		simp3 1136	. . . . . . 7 ⊢ ((𝑊 ∈ (NrmVec ∩ ℂVec) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ i ∈ 𝐾) → i ∈ 𝐾)
20		simp2r 1198	. . . . . . 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 25008	. . . . . . 7 ⊢ ((𝑊 ∈ ℂMod ∧ i ∈ 𝐾 ∧ 𝐵 ∈ 𝑉) → (i · 𝐵) ∈ 𝑉)
26	18, 19, 20, 25	syl3anc 1369	. . . . . 6 ⊢ ((𝑊 ∈ (NrmVec ∩ ℂVec) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ i ∈ 𝐾) → (i · 𝐵) ∈ 𝑉)
27		ncvsdif.p	. . . . . . 7 ⊢ + = (+_g‘𝑊)
28	21, 27	grpcl 18891	. . . . . 6 ⊢ ((𝑊 ∈ Grp ∧ 𝐴 ∈ 𝑉 ∧ (i · 𝐵) ∈ 𝑉) → (𝐴 + (i · 𝐵)) ∈ 𝑉)
29	13, 14, 26, 28	syl3anc 1369	. . . . 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 24518	. . . . 5 ⊢ ((𝑊 ∈ NrmGrp ∧ (𝐴 + (i · 𝐵)) ∈ 𝑉) → (𝑁‘(𝐴 + (i · 𝐵))) ∈ ℝ)
32	7, 29, 31	syl2anc 583	. . . 4 ⊢ ((𝑊 ∈ (NrmVec ∩ ℂVec) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ i ∈ 𝐾) → (𝑁‘(𝐴 + (i · 𝐵))) ∈ ℝ)
33	32	recnd 11266	. . 3 ⊢ ((𝑊 ∈ (NrmVec ∩ ℂVec) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ i ∈ 𝐾) → (𝑁‘(𝐴 + (i · 𝐵))) ∈ ℂ)
34	33	mullidd 11256	. 2 ⊢ ((𝑊 ∈ (NrmVec ∩ ℂVec) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ i ∈ 𝐾) → (1 · (𝑁‘(𝐴 + (i · 𝐵)))) = (𝑁‘(𝐴 + (i · 𝐵))))
35		ax-icn 11191	. . . . . 6 ⊢ i ∈ ℂ
36	35	absnegi 15373	. . . . 5 ⊢ (abs‘-i) = (abs‘i)
37		absi 15259	. . . . 5 ⊢ (abs‘i) = 1
38	36, 37	eqtri 2756	. . . 4 ⊢ (abs‘-i) = 1
39	38	oveq1i 7424	. . 3 ⊢ ((abs‘-i) · (𝑁‘(𝐴 + (i · 𝐵)))) = (1 · (𝑁‘(𝐴 + (i · 𝐵))))
40		simp1 1134	. . . . 5 ⊢ ((𝑊 ∈ (NrmVec ∩ ℂVec) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ i ∈ 𝐾) → 𝑊 ∈ (NrmVec ∩ ℂVec))
41	22, 24	clmneg 25001	. . . . . . . . . . 11 ⊢ ((𝑊 ∈ ℂMod ∧ i ∈ 𝐾) → -i = ((inv_g‘𝐹)‘i))
42	16, 41	sylan 579	. . . . . . . . . 10 ⊢ ((𝑊 ∈ ℂVec ∧ i ∈ 𝐾) → -i = ((inv_g‘𝐹)‘i))
43	22	clmfgrp 24991	. . . . . . . . . . . 12 ⊢ (𝑊 ∈ ℂMod → 𝐹 ∈ Grp)
44	16, 43	syl 17	. . . . . . . . . . 11 ⊢ (𝑊 ∈ ℂVec → 𝐹 ∈ Grp)
45		eqid 2728	. . . . . . . . . . . 12 ⊢ (inv_g‘𝐹) = (inv_g‘𝐹)
46	24, 45	grpinvcl 18937	. . . . . . . . . . 11 ⊢ ((𝐹 ∈ Grp ∧ i ∈ 𝐾) → ((inv_g‘𝐹)‘i) ∈ 𝐾)
47	44, 46	sylan 579	. . . . . . . . . 10 ⊢ ((𝑊 ∈ ℂVec ∧ i ∈ 𝐾) → ((inv_g‘𝐹)‘i) ∈ 𝐾)
48	42, 47	eqeltrd 2829	. . . . . . . . 9 ⊢ ((𝑊 ∈ ℂVec ∧ i ∈ 𝐾) → -i ∈ 𝐾)
49	48	ex 412	. . . . . . . 8 ⊢ (𝑊 ∈ ℂVec → (i ∈ 𝐾 → -i ∈ 𝐾))
50	1, 49	simplbiim 504	. . . . . . 7 ⊢ (𝑊 ∈ (NrmVec ∩ ℂVec) → (i ∈ 𝐾 → -i ∈ 𝐾))
51	50	imp 406	. . . . . 6 ⊢ ((𝑊 ∈ (NrmVec ∩ ℂVec) ∧ i ∈ 𝐾) → -i ∈ 𝐾)
52	51	3adant2 1129	. . . . 5 ⊢ ((𝑊 ∈ (NrmVec ∩ ℂVec) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ i ∈ 𝐾) → -i ∈ 𝐾)
53	21, 30, 23, 22, 24	ncvsprp 25073	. . . . 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 1369	. . . 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 25012	. . . . . . 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 1370	. . . . . 6 ⊢ ((𝑊 ∈ (NrmVec ∩ ℂVec) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ i ∈ 𝐾) → (-i · (𝐴 + (i · 𝐵))) = ((-i · 𝐴) + (-i · (i · 𝐵))))
57	35, 35	mulneg1i 11684	. . . . . . . . . 10 ⊢ (-i · i) = -(i · i)
58		ixi 11867	. . . . . . . . . . . 12 ⊢ (i · i) = -1
59	58	negeqi 11477	. . . . . . . . . . 11 ⊢ -(i · i) = --1
60		negneg1e1 12354	. . . . . . . . . . 11 ⊢ --1 = 1
61	59, 60	eqtri 2756	. . . . . . . . . 10 ⊢ -(i · i) = 1
62	57, 61	eqtri 2756	. . . . . . . . 9 ⊢ (-i · i) = 1
63	62	oveq1i 7424	. . . . . . . 8 ⊢ ((-i · i) · 𝐵) = (1 · 𝐵)
64	21, 22, 23, 24	clmvsass 25009	. . . . . . . . 9 ⊢ ((𝑊 ∈ ℂMod ∧ (-i ∈ 𝐾 ∧ i ∈ 𝐾 ∧ 𝐵 ∈ 𝑉)) → ((-i · i) · 𝐵) = (-i · (i · 𝐵)))
65	18, 52, 19, 20, 64	syl13anc 1370	. . . . . . . 8 ⊢ ((𝑊 ∈ (NrmVec ∩ ℂVec) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ i ∈ 𝐾) → ((-i · i) · 𝐵) = (-i · (i · 𝐵)))
66		simpr 484	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → 𝐵 ∈ 𝑉)
67	17, 66	anim12i 612	. . . . . . . . . 10 ⊢ ((𝑊 ∈ (NrmVec ∩ ℂVec) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉)) → (𝑊 ∈ ℂMod ∧ 𝐵 ∈ 𝑉))
68	67	3adant3 1130	. . . . . . . . 9 ⊢ ((𝑊 ∈ (NrmVec ∩ ℂVec) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ i ∈ 𝐾) → (𝑊 ∈ ℂMod ∧ 𝐵 ∈ 𝑉))
69	21, 23	clmvs1 25013	. . . . . . . . 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 2792	. . . . . . 7 ⊢ ((𝑊 ∈ (NrmVec ∩ ℂVec) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ i ∈ 𝐾) → (-i · (i · 𝐵)) = 𝐵)
72	71	oveq2d 7430	. . . . . 6 ⊢ ((𝑊 ∈ (NrmVec ∩ ℂVec) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ i ∈ 𝐾) → ((-i · 𝐴) + (-i · (i · 𝐵))) = ((-i · 𝐴) + 𝐵))
73		clmabl 24989	. . . . . . . . . 10 ⊢ (𝑊 ∈ ℂMod → 𝑊 ∈ Abel)
74	16, 73	syl 17	. . . . . . . . 9 ⊢ (𝑊 ∈ ℂVec → 𝑊 ∈ Abel)
75	1, 74	simplbiim 504	. . . . . . . 8 ⊢ (𝑊 ∈ (NrmVec ∩ ℂVec) → 𝑊 ∈ Abel)
76	75	3ad2ant1 1131	. . . . . . 7 ⊢ ((𝑊 ∈ (NrmVec ∩ ℂVec) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ i ∈ 𝐾) → 𝑊 ∈ Abel)
77	21, 22, 23, 24	clmvscl 25008	. . . . . . . 8 ⊢ ((𝑊 ∈ ℂMod ∧ -i ∈ 𝐾 ∧ 𝐴 ∈ 𝑉) → (-i · 𝐴) ∈ 𝑉)
78	18, 52, 14, 77	syl3anc 1369	. . . . . . 7 ⊢ ((𝑊 ∈ (NrmVec ∩ ℂVec) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ i ∈ 𝐾) → (-i · 𝐴) ∈ 𝑉)
79	21, 27	ablcom 19747	. . . . . . 7 ⊢ ((𝑊 ∈ Abel ∧ (-i · 𝐴) ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → ((-i · 𝐴) + 𝐵) = (𝐵 + (-i · 𝐴)))
80	76, 78, 20, 79	syl3anc 1369	. . . . . 6 ⊢ ((𝑊 ∈ (NrmVec ∩ ℂVec) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ i ∈ 𝐾) → ((-i · 𝐴) + 𝐵) = (𝐵 + (-i · 𝐴)))
81	56, 72, 80	3eqtrd 2772	. . . . 5 ⊢ ((𝑊 ∈ (NrmVec ∩ ℂVec) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ i ∈ 𝐾) → (-i · (𝐴 + (i · 𝐵))) = (𝐵 + (-i · 𝐴)))
82	81	fveq2d 6895	. . . 4 ⊢ ((𝑊 ∈ (NrmVec ∩ ℂVec) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ i ∈ 𝐾) → (𝑁‘(-i · (𝐴 + (i · 𝐵)))) = (𝑁‘(𝐵 + (-i · 𝐴))))
83	54, 82	eqtr3d 2770	. . 3 ⊢ ((𝑊 ∈ (NrmVec ∩ ℂVec) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ i ∈ 𝐾) → ((abs‘-i) · (𝑁‘(𝐴 + (i · 𝐵)))) = (𝑁‘(𝐵 + (-i · 𝐴))))
84	39, 83	eqtr3id 2782	. 2 ⊢ ((𝑊 ∈ (NrmVec ∩ ℂVec) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ i ∈ 𝐾) → (1 · (𝑁‘(𝐴 + (i · 𝐵)))) = (𝑁‘(𝐵 + (-i · 𝐴))))
85	34, 84	eqtr3d 2770	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 395 ∧ w3a 1085 = wceq 1534 ∈ wcel 2099 ∩ cin 3944 ‘cfv 6542 (class class class)co 7414 ℝcr 11131 1c1 11133 ici 11134 · cmul 11137 -cneg 11469 abscabs 15207 Basecbs 17173 +_gcplusg 17226 Scalarcsca 17229 ·_𝑠 cvsca 17230 Grpcgrp 18883 inv_gcminusg 18884 Abelcabl 19729 LModclmod 20736 normcnm 24478 NrmGrpcngp 24479 NrmModcnlm 24482 NrmVeccnvc 24483 ℂModcclm 24982 ℂVecccvs 25043

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7734 ax-cnex 11188 ax-resscn 11189 ax-1cn 11190 ax-icn 11191 ax-addcl 11192 ax-addrcl 11193 ax-mulcl 11194 ax-mulrcl 11195 ax-mulcom 11196 ax-addass 11197 ax-mulass 11198 ax-distr 11199 ax-i2m1 11200 ax-1ne0 11201 ax-1rid 11202 ax-rnegex 11203 ax-rrecex 11204 ax-cnre 11205 ax-pre-lttri 11206 ax-pre-lttrn 11207 ax-pre-ltadd 11208 ax-pre-mulgt0 11209 ax-pre-sup 11210 ax-addf 11211 ax-mulf 11212

This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2937 df-nel 3043 df-ral 3058 df-rex 3067 df-rmo 3372 df-reu 3373 df-rab 3429 df-v 3472 df-sbc 3776 df-csb 3891 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-pss 3964 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-tp 4629 df-op 4631 df-uni 4904 df-iun 4993 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-om 7865 df-1st 7987 df-2nd 7988 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-1o 8480 df-er 8718 df-map 8840 df-en 8958 df-dom 8959 df-sdom 8960 df-fin 8961 df-sup 9459 df-inf 9460 df-pnf 11274 df-mnf 11275 df-xr 11276 df-ltxr 11277 df-le 11278 df-sub 11470 df-neg 11471 df-div 11896 df-nn 12237 df-2 12299 df-3 12300 df-4 12301 df-5 12302 df-6 12303 df-7 12304 df-8 12305 df-9 12306 df-n0 12497 df-z 12583 df-dec 12702 df-uz 12847 df-q 12957 df-rp 13001 df-xneg 13118 df-xadd 13119 df-xmul 13120 df-fz 13511 df-seq 13993 df-exp 14053 df-cj 15072 df-re 15073 df-im 15074 df-sqrt 15208 df-abs 15209 df-struct 17109 df-sets 17126 df-slot 17144 df-ndx 17156 df-base 17174 df-ress 17203 df-plusg 17239 df-mulr 17240 df-starv 17241 df-tset 17245 df-ple 17246 df-ds 17248 df-unif 17249 df-0g 17416 df-topgen 17418 df-mgm 18593 df-sgrp 18672 df-mnd 18688 df-grp 18886 df-minusg 18887 df-subg 19071 df-cmn 19730 df-abl 19731 df-mgp 20068 df-ur 20115 df-ring 20168 df-cring 20169 df-subrg 20501 df-lmod 20738 df-psmet 21264 df-xmet 21265 df-met 21266 df-bl 21267 df-mopn 21268 df-cnfld 21273 df-top 22789 df-topon 22806 df-topsp 22828 df-bases 22842 df-xms 24219 df-ms 24220 df-nm 24484 df-ngp 24485 df-nlm 24488 df-nvc 24489 df-clm 24983 df-cvs 25044

This theorem is referenced by: (None)

W3C validator