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Theorem nlmvscnlem2 24202

Description: Lemma for nlmvscn 24204. Compare this proof with the similar elementary proof mulcn2 15540 for continuity of multiplication on ℂ. (Contributed by Mario Carneiro, 5-Oct-2015.)

**Hypotheses**
Ref	Expression
nlmvscn.f	⊢ 𝐹 = (Scalar‘𝑊)
nlmvscn.v	⊢ 𝑉 = (Base‘𝑊)
nlmvscn.k	⊢ 𝐾 = (Base‘𝐹)
nlmvscn.d	⊢ 𝐷 = (dist‘𝑊)
nlmvscn.e	⊢ 𝐸 = (dist‘𝐹)
nlmvscn.n	⊢ 𝑁 = (norm‘𝑊)
nlmvscn.a	⊢ 𝐴 = (norm‘𝐹)
nlmvscn.s	⊢ · = ( ·_𝑠 ‘𝑊)
nlmvscn.t	⊢ 𝑇 = ((𝑅 / 2) / ((𝐴‘𝐵) + 1))
nlmvscn.u	⊢ 𝑈 = ((𝑅 / 2) / ((𝑁‘𝑋) + 𝑇))
nlmvscn.w	⊢ (𝜑 → 𝑊 ∈ NrmMod)
nlmvscn.r	⊢ (𝜑 → 𝑅 ∈ ℝ⁺)
nlmvscn.b	⊢ (𝜑 → 𝐵 ∈ 𝐾)
nlmvscn.x	⊢ (𝜑 → 𝑋 ∈ 𝑉)
nlmvscn.c	⊢ (𝜑 → 𝐶 ∈ 𝐾)
nlmvscn.y	⊢ (𝜑 → 𝑌 ∈ 𝑉)
nlmvscn.1	⊢ (𝜑 → (𝐵𝐸𝐶) < 𝑈)
nlmvscn.2	⊢ (𝜑 → (𝑋𝐷𝑌) < 𝑇)

**Assertion**
Ref	Expression
nlmvscnlem2	⊢ (𝜑 → ((𝐵 · 𝑋)𝐷(𝐶 · 𝑌)) < 𝑅)

**Proof of Theorem nlmvscnlem2**
Step	Hyp	Ref	Expression
1		nlmvscn.w	. . . . 5 ⊢ (𝜑 → 𝑊 ∈ NrmMod)
2		nlmngp 24194	. . . . 5 ⊢ (𝑊 ∈ NrmMod → 𝑊 ∈ NrmGrp)
3	1, 2	syl 17	. . . 4 ⊢ (𝜑 → 𝑊 ∈ NrmGrp)
4		ngpms 24109	. . . 4 ⊢ (𝑊 ∈ NrmGrp → 𝑊 ∈ MetSp)
5	3, 4	syl 17	. . 3 ⊢ (𝜑 → 𝑊 ∈ MetSp)
6		nlmlmod 24195	. . . . 5 ⊢ (𝑊 ∈ NrmMod → 𝑊 ∈ LMod)
7	1, 6	syl 17	. . . 4 ⊢ (𝜑 → 𝑊 ∈ LMod)
8		nlmvscn.b	. . . 4 ⊢ (𝜑 → 𝐵 ∈ 𝐾)
9		nlmvscn.x	. . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝑉)
10		nlmvscn.v	. . . . 5 ⊢ 𝑉 = (Base‘𝑊)
11		nlmvscn.f	. . . . 5 ⊢ 𝐹 = (Scalar‘𝑊)
12		nlmvscn.s	. . . . 5 ⊢ · = ( ·_𝑠 ‘𝑊)
13		nlmvscn.k	. . . . 5 ⊢ 𝐾 = (Base‘𝐹)
14	10, 11, 12, 13	lmodvscl 20489	. . . 4 ⊢ ((𝑊 ∈ LMod ∧ 𝐵 ∈ 𝐾 ∧ 𝑋 ∈ 𝑉) → (𝐵 · 𝑋) ∈ 𝑉)
15	7, 8, 9, 14	syl3anc 1372	. . 3 ⊢ (𝜑 → (𝐵 · 𝑋) ∈ 𝑉)
16		nlmvscn.c	. . . 4 ⊢ (𝜑 → 𝐶 ∈ 𝐾)
17		nlmvscn.y	. . . 4 ⊢ (𝜑 → 𝑌 ∈ 𝑉)
18	10, 11, 12, 13	lmodvscl 20489	. . . 4 ⊢ ((𝑊 ∈ LMod ∧ 𝐶 ∈ 𝐾 ∧ 𝑌 ∈ 𝑉) → (𝐶 · 𝑌) ∈ 𝑉)
19	7, 16, 17, 18	syl3anc 1372	. . 3 ⊢ (𝜑 → (𝐶 · 𝑌) ∈ 𝑉)
20		nlmvscn.d	. . . 4 ⊢ 𝐷 = (dist‘𝑊)
21	10, 20	mscl 23967	. . 3 ⊢ ((𝑊 ∈ MetSp ∧ (𝐵 · 𝑋) ∈ 𝑉 ∧ (𝐶 · 𝑌) ∈ 𝑉) → ((𝐵 · 𝑋)𝐷(𝐶 · 𝑌)) ∈ ℝ)
22	5, 15, 19, 21	syl3anc 1372	. 2 ⊢ (𝜑 → ((𝐵 · 𝑋)𝐷(𝐶 · 𝑌)) ∈ ℝ)
23	10, 11, 12, 13	lmodvscl 20489	. . . . 5 ⊢ ((𝑊 ∈ LMod ∧ 𝐵 ∈ 𝐾 ∧ 𝑌 ∈ 𝑉) → (𝐵 · 𝑌) ∈ 𝑉)
24	7, 8, 17, 23	syl3anc 1372	. . . 4 ⊢ (𝜑 → (𝐵 · 𝑌) ∈ 𝑉)
25	10, 20	mscl 23967	. . . 4 ⊢ ((𝑊 ∈ MetSp ∧ (𝐵 · 𝑋) ∈ 𝑉 ∧ (𝐵 · 𝑌) ∈ 𝑉) → ((𝐵 · 𝑋)𝐷(𝐵 · 𝑌)) ∈ ℝ)
26	5, 15, 24, 25	syl3anc 1372	. . 3 ⊢ (𝜑 → ((𝐵 · 𝑋)𝐷(𝐵 · 𝑌)) ∈ ℝ)
27	10, 20	mscl 23967	. . . 4 ⊢ ((𝑊 ∈ MetSp ∧ (𝐵 · 𝑌) ∈ 𝑉 ∧ (𝐶 · 𝑌) ∈ 𝑉) → ((𝐵 · 𝑌)𝐷(𝐶 · 𝑌)) ∈ ℝ)
28	5, 24, 19, 27	syl3anc 1372	. . 3 ⊢ (𝜑 → ((𝐵 · 𝑌)𝐷(𝐶 · 𝑌)) ∈ ℝ)
29	26, 28	readdcld 11243	. 2 ⊢ (𝜑 → (((𝐵 · 𝑋)𝐷(𝐵 · 𝑌)) + ((𝐵 · 𝑌)𝐷(𝐶 · 𝑌))) ∈ ℝ)
30		nlmvscn.r	. . 3 ⊢ (𝜑 → 𝑅 ∈ ℝ⁺)
31	30	rpred 13016	. 2 ⊢ (𝜑 → 𝑅 ∈ ℝ)
32	10, 20	mstri 23975	. . 3 ⊢ ((𝑊 ∈ MetSp ∧ ((𝐵 · 𝑋) ∈ 𝑉 ∧ (𝐶 · 𝑌) ∈ 𝑉 ∧ (𝐵 · 𝑌) ∈ 𝑉)) → ((𝐵 · 𝑋)𝐷(𝐶 · 𝑌)) ≤ (((𝐵 · 𝑋)𝐷(𝐵 · 𝑌)) + ((𝐵 · 𝑌)𝐷(𝐶 · 𝑌))))
33	5, 15, 19, 24, 32	syl13anc 1373	. 2 ⊢ (𝜑 → ((𝐵 · 𝑋)𝐷(𝐶 · 𝑌)) ≤ (((𝐵 · 𝑋)𝐷(𝐵 · 𝑌)) + ((𝐵 · 𝑌)𝐷(𝐶 · 𝑌))))
34	11	nlmngp2 24197	. . . . . . . . 9 ⊢ (𝑊 ∈ NrmMod → 𝐹 ∈ NrmGrp)
35	1, 34	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝐹 ∈ NrmGrp)
36		nlmvscn.a	. . . . . . . . 9 ⊢ 𝐴 = (norm‘𝐹)
37	13, 36	nmcl 24125	. . . . . . . 8 ⊢ ((𝐹 ∈ NrmGrp ∧ 𝐵 ∈ 𝐾) → (𝐴‘𝐵) ∈ ℝ)
38	35, 8, 37	syl2anc 585	. . . . . . 7 ⊢ (𝜑 → (𝐴‘𝐵) ∈ ℝ)
39	13, 36	nmge0 24126	. . . . . . . 8 ⊢ ((𝐹 ∈ NrmGrp ∧ 𝐵 ∈ 𝐾) → 0 ≤ (𝐴‘𝐵))
40	35, 8, 39	syl2anc 585	. . . . . . 7 ⊢ (𝜑 → 0 ≤ (𝐴‘𝐵))
41	38, 40	ge0p1rpd 13046	. . . . . 6 ⊢ (𝜑 → ((𝐴‘𝐵) + 1) ∈ ℝ⁺)
42	41	rpred 13016	. . . . 5 ⊢ (𝜑 → ((𝐴‘𝐵) + 1) ∈ ℝ)
43	10, 20	mscl 23967	. . . . . 6 ⊢ ((𝑊 ∈ MetSp ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → (𝑋𝐷𝑌) ∈ ℝ)
44	5, 9, 17, 43	syl3anc 1372	. . . . 5 ⊢ (𝜑 → (𝑋𝐷𝑌) ∈ ℝ)
45	42, 44	remulcld 11244	. . . 4 ⊢ (𝜑 → (((𝐴‘𝐵) + 1) · (𝑋𝐷𝑌)) ∈ ℝ)
46	31	rehalfcld 12459	. . . 4 ⊢ (𝜑 → (𝑅 / 2) ∈ ℝ)
47	10, 12, 11, 13, 20, 36	nlmdsdi 24198	. . . . . 6 ⊢ ((𝑊 ∈ NrmMod ∧ (𝐵 ∈ 𝐾 ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉)) → ((𝐴‘𝐵) · (𝑋𝐷𝑌)) = ((𝐵 · 𝑋)𝐷(𝐵 · 𝑌)))
48	1, 8, 9, 17, 47	syl13anc 1373	. . . . 5 ⊢ (𝜑 → ((𝐴‘𝐵) · (𝑋𝐷𝑌)) = ((𝐵 · 𝑋)𝐷(𝐵 · 𝑌)))
49		msxms 23960	. . . . . . . 8 ⊢ (𝑊 ∈ MetSp → 𝑊 ∈ ∞MetSp)
50	5, 49	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝑊 ∈ ∞MetSp)
51	10, 20	xmsge0 23969	. . . . . . 7 ⊢ ((𝑊 ∈ ∞MetSp ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → 0 ≤ (𝑋𝐷𝑌))
52	50, 9, 17, 51	syl3anc 1372	. . . . . 6 ⊢ (𝜑 → 0 ≤ (𝑋𝐷𝑌))
53	38	lep1d 12145	. . . . . 6 ⊢ (𝜑 → (𝐴‘𝐵) ≤ ((𝐴‘𝐵) + 1))
54	38, 42, 44, 52, 53	lemul1ad 12153	. . . . 5 ⊢ (𝜑 → ((𝐴‘𝐵) · (𝑋𝐷𝑌)) ≤ (((𝐴‘𝐵) + 1) · (𝑋𝐷𝑌)))
55	48, 54	eqbrtrrd 5173	. . . 4 ⊢ (𝜑 → ((𝐵 · 𝑋)𝐷(𝐵 · 𝑌)) ≤ (((𝐴‘𝐵) + 1) · (𝑋𝐷𝑌)))
56		nlmvscn.2	. . . . . 6 ⊢ (𝜑 → (𝑋𝐷𝑌) < 𝑇)
57		nlmvscn.t	. . . . . 6 ⊢ 𝑇 = ((𝑅 / 2) / ((𝐴‘𝐵) + 1))
58	56, 57	breqtrdi 5190	. . . . 5 ⊢ (𝜑 → (𝑋𝐷𝑌) < ((𝑅 / 2) / ((𝐴‘𝐵) + 1)))
59	44, 46, 41	ltmuldiv2d 13064	. . . . 5 ⊢ (𝜑 → ((((𝐴‘𝐵) + 1) · (𝑋𝐷𝑌)) < (𝑅 / 2) ↔ (𝑋𝐷𝑌) < ((𝑅 / 2) / ((𝐴‘𝐵) + 1))))
60	58, 59	mpbird 257	. . . 4 ⊢ (𝜑 → (((𝐴‘𝐵) + 1) · (𝑋𝐷𝑌)) < (𝑅 / 2))
61	26, 45, 46, 55, 60	lelttrd 11372	. . 3 ⊢ (𝜑 → ((𝐵 · 𝑋)𝐷(𝐵 · 𝑌)) < (𝑅 / 2))
62		ngpms 24109	. . . . . . 7 ⊢ (𝐹 ∈ NrmGrp → 𝐹 ∈ MetSp)
63	35, 62	syl 17	. . . . . 6 ⊢ (𝜑 → 𝐹 ∈ MetSp)
64		nlmvscn.e	. . . . . . 7 ⊢ 𝐸 = (dist‘𝐹)
65	13, 64	mscl 23967	. . . . . 6 ⊢ ((𝐹 ∈ MetSp ∧ 𝐵 ∈ 𝐾 ∧ 𝐶 ∈ 𝐾) → (𝐵𝐸𝐶) ∈ ℝ)
66	63, 8, 16, 65	syl3anc 1372	. . . . 5 ⊢ (𝜑 → (𝐵𝐸𝐶) ∈ ℝ)
67		nlmvscn.n	. . . . . . . 8 ⊢ 𝑁 = (norm‘𝑊)
68	10, 67	nmcl 24125	. . . . . . 7 ⊢ ((𝑊 ∈ NrmGrp ∧ 𝑋 ∈ 𝑉) → (𝑁‘𝑋) ∈ ℝ)
69	3, 9, 68	syl2anc 585	. . . . . 6 ⊢ (𝜑 → (𝑁‘𝑋) ∈ ℝ)
70	30	rphalfcld 13028	. . . . . . . . 9 ⊢ (𝜑 → (𝑅 / 2) ∈ ℝ⁺)
71	70, 41	rpdivcld 13033	. . . . . . . 8 ⊢ (𝜑 → ((𝑅 / 2) / ((𝐴‘𝐵) + 1)) ∈ ℝ⁺)
72	57, 71	eqeltrid 2838	. . . . . . 7 ⊢ (𝜑 → 𝑇 ∈ ℝ⁺)
73	72	rpred 13016	. . . . . 6 ⊢ (𝜑 → 𝑇 ∈ ℝ)
74	69, 73	readdcld 11243	. . . . 5 ⊢ (𝜑 → ((𝑁‘𝑋) + 𝑇) ∈ ℝ)
75	66, 74	remulcld 11244	. . . 4 ⊢ (𝜑 → ((𝐵𝐸𝐶) · ((𝑁‘𝑋) + 𝑇)) ∈ ℝ)
76	10, 12, 11, 13, 20, 67, 64	nlmdsdir 24199	. . . . . 6 ⊢ ((𝑊 ∈ NrmMod ∧ (𝐵 ∈ 𝐾 ∧ 𝐶 ∈ 𝐾 ∧ 𝑌 ∈ 𝑉)) → ((𝐵𝐸𝐶) · (𝑁‘𝑌)) = ((𝐵 · 𝑌)𝐷(𝐶 · 𝑌)))
77	1, 8, 16, 17, 76	syl13anc 1373	. . . . 5 ⊢ (𝜑 → ((𝐵𝐸𝐶) · (𝑁‘𝑌)) = ((𝐵 · 𝑌)𝐷(𝐶 · 𝑌)))
78	10, 67	nmcl 24125	. . . . . . 7 ⊢ ((𝑊 ∈ NrmGrp ∧ 𝑌 ∈ 𝑉) → (𝑁‘𝑌) ∈ ℝ)
79	3, 17, 78	syl2anc 585	. . . . . 6 ⊢ (𝜑 → (𝑁‘𝑌) ∈ ℝ)
80		msxms 23960	. . . . . . . 8 ⊢ (𝐹 ∈ MetSp → 𝐹 ∈ ∞MetSp)
81	63, 80	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝐹 ∈ ∞MetSp)
82	13, 64	xmsge0 23969	. . . . . . 7 ⊢ ((𝐹 ∈ ∞MetSp ∧ 𝐵 ∈ 𝐾 ∧ 𝐶 ∈ 𝐾) → 0 ≤ (𝐵𝐸𝐶))
83	81, 8, 16, 82	syl3anc 1372	. . . . . 6 ⊢ (𝜑 → 0 ≤ (𝐵𝐸𝐶))
84	79, 69	resubcld 11642	. . . . . . . 8 ⊢ (𝜑 → ((𝑁‘𝑌) − (𝑁‘𝑋)) ∈ ℝ)
85		eqid 2733	. . . . . . . . . . 11 ⊢ (-_g‘𝑊) = (-_g‘𝑊)
86	10, 67, 85	nm2dif 24134	. . . . . . . . . 10 ⊢ ((𝑊 ∈ NrmGrp ∧ 𝑌 ∈ 𝑉 ∧ 𝑋 ∈ 𝑉) → ((𝑁‘𝑌) − (𝑁‘𝑋)) ≤ (𝑁‘(𝑌(-_g‘𝑊)𝑋)))
87	3, 17, 9, 86	syl3anc 1372	. . . . . . . . 9 ⊢ (𝜑 → ((𝑁‘𝑌) − (𝑁‘𝑋)) ≤ (𝑁‘(𝑌(-_g‘𝑊)𝑋)))
88	67, 10, 85, 20	ngpdsr 24114	. . . . . . . . . 10 ⊢ ((𝑊 ∈ NrmGrp ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → (𝑋𝐷𝑌) = (𝑁‘(𝑌(-_g‘𝑊)𝑋)))
89	3, 9, 17, 88	syl3anc 1372	. . . . . . . . 9 ⊢ (𝜑 → (𝑋𝐷𝑌) = (𝑁‘(𝑌(-_g‘𝑊)𝑋)))
90	87, 89	breqtrrd 5177	. . . . . . . 8 ⊢ (𝜑 → ((𝑁‘𝑌) − (𝑁‘𝑋)) ≤ (𝑋𝐷𝑌))
91	44, 73, 56	ltled 11362	. . . . . . . 8 ⊢ (𝜑 → (𝑋𝐷𝑌) ≤ 𝑇)
92	84, 44, 73, 90, 91	letrd 11371	. . . . . . 7 ⊢ (𝜑 → ((𝑁‘𝑌) − (𝑁‘𝑋)) ≤ 𝑇)
93	79, 69, 73	lesubadd2d 11813	. . . . . . 7 ⊢ (𝜑 → (((𝑁‘𝑌) − (𝑁‘𝑋)) ≤ 𝑇 ↔ (𝑁‘𝑌) ≤ ((𝑁‘𝑋) + 𝑇)))
94	92, 93	mpbid 231	. . . . . 6 ⊢ (𝜑 → (𝑁‘𝑌) ≤ ((𝑁‘𝑋) + 𝑇))
95	79, 74, 66, 83, 94	lemul2ad 12154	. . . . 5 ⊢ (𝜑 → ((𝐵𝐸𝐶) · (𝑁‘𝑌)) ≤ ((𝐵𝐸𝐶) · ((𝑁‘𝑋) + 𝑇)))
96	77, 95	eqbrtrrd 5173	. . . 4 ⊢ (𝜑 → ((𝐵 · 𝑌)𝐷(𝐶 · 𝑌)) ≤ ((𝐵𝐸𝐶) · ((𝑁‘𝑋) + 𝑇)))
97		nlmvscn.1	. . . . . 6 ⊢ (𝜑 → (𝐵𝐸𝐶) < 𝑈)
98		nlmvscn.u	. . . . . 6 ⊢ 𝑈 = ((𝑅 / 2) / ((𝑁‘𝑋) + 𝑇))
99	97, 98	breqtrdi 5190	. . . . 5 ⊢ (𝜑 → (𝐵𝐸𝐶) < ((𝑅 / 2) / ((𝑁‘𝑋) + 𝑇)))
100		0red 11217	. . . . . . 7 ⊢ (𝜑 → 0 ∈ ℝ)
101	10, 67	nmge0 24126	. . . . . . . 8 ⊢ ((𝑊 ∈ NrmGrp ∧ 𝑋 ∈ 𝑉) → 0 ≤ (𝑁‘𝑋))
102	3, 9, 101	syl2anc 585	. . . . . . 7 ⊢ (𝜑 → 0 ≤ (𝑁‘𝑋))
103	69, 72	ltaddrpd 13049	. . . . . . 7 ⊢ (𝜑 → (𝑁‘𝑋) < ((𝑁‘𝑋) + 𝑇))
104	100, 69, 74, 102, 103	lelttrd 11372	. . . . . 6 ⊢ (𝜑 → 0 < ((𝑁‘𝑋) + 𝑇))
105		ltmuldiv 12087	. . . . . 6 ⊢ (((𝐵𝐸𝐶) ∈ ℝ ∧ (𝑅 / 2) ∈ ℝ ∧ (((𝑁‘𝑋) + 𝑇) ∈ ℝ ∧ 0 < ((𝑁‘𝑋) + 𝑇))) → (((𝐵𝐸𝐶) · ((𝑁‘𝑋) + 𝑇)) < (𝑅 / 2) ↔ (𝐵𝐸𝐶) < ((𝑅 / 2) / ((𝑁‘𝑋) + 𝑇))))
106	66, 46, 74, 104, 105	syl112anc 1375	. . . . 5 ⊢ (𝜑 → (((𝐵𝐸𝐶) · ((𝑁‘𝑋) + 𝑇)) < (𝑅 / 2) ↔ (𝐵𝐸𝐶) < ((𝑅 / 2) / ((𝑁‘𝑋) + 𝑇))))
107	99, 106	mpbird 257	. . . 4 ⊢ (𝜑 → ((𝐵𝐸𝐶) · ((𝑁‘𝑋) + 𝑇)) < (𝑅 / 2))
108	28, 75, 46, 96, 107	lelttrd 11372	. . 3 ⊢ (𝜑 → ((𝐵 · 𝑌)𝐷(𝐶 · 𝑌)) < (𝑅 / 2))
109	26, 28, 31, 61, 108	lt2halvesd 12460	. 2 ⊢ (𝜑 → (((𝐵 · 𝑋)𝐷(𝐵 · 𝑌)) + ((𝐵 · 𝑌)𝐷(𝐶 · 𝑌))) < 𝑅)
110	22, 29, 31, 33, 109	lelttrd 11372	1 ⊢ (𝜑 → ((𝐵 · 𝑋)𝐷(𝐶 · 𝑌)) < 𝑅)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 = wceq 1542 ∈ wcel 2107 class class class wbr 5149 ‘cfv 6544 (class class class)co 7409 ℝcr 11109 0cc0 11110 1c1 11111 + caddc 11113 · cmul 11115 < clt 11248 ≤ cle 11249 − cmin 11444 / cdiv 11871 2c2 12267 ℝ⁺crp 12974 Basecbs 17144 Scalarcsca 17200 ·_𝑠 cvsca 17201 distcds 17206 -_gcsg 18821 LModclmod 20471 ∞MetSpcxms 23823 MetSpcms 23824 normcnm 24085 NrmGrpcngp 24086 NrmModcnlm 24089

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-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725 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

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-iun 5000 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-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-riota 7365 df-ov 7412 df-oprab 7413 df-mpo 7414 df-om 7856 df-1st 7975 df-2nd 7976 df-frecs 8266 df-wrecs 8297 df-recs 8371 df-rdg 8410 df-1o 8466 df-er 8703 df-map 8822 df-en 8940 df-dom 8941 df-sdom 8942 df-fin 8943 df-sup 9437 df-inf 9438 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-fz 13485 df-seq 13967 df-exp 14028 df-cj 15046 df-re 15047 df-im 15048 df-sqrt 15182 df-abs 15183 df-struct 17080 df-sets 17097 df-slot 17115 df-ndx 17127 df-base 17145 df-plusg 17210 df-mulr 17211 df-tset 17216 df-ple 17217 df-ds 17219 df-0g 17387 df-topgen 17389 df-xrs 17448 df-mgm 18561 df-sgrp 18610 df-mnd 18626 df-grp 18822 df-minusg 18823 df-sbg 18824 df-mgp 19988 df-ur 20005 df-ring 20058 df-lmod 20473 df-psmet 20936 df-xmet 20937 df-met 20938 df-bl 20939 df-mopn 20940 df-top 22396 df-topon 22413 df-topsp 22435 df-bases 22449 df-xms 23826 df-ms 23827 df-nm 24091 df-ngp 24092 df-nrg 24094 df-nlm 24095

This theorem is referenced by: nlmvscnlem1 24203

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