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Theorem minveclem4a 24499

Description: Lemma for minvec 24505. 𝐹 converges to a point 𝑃 in 𝑌. (Contributed by Mario Carneiro, 7-May-2014.) (Revised by Mario Carneiro, 15-Oct-2015.)

**Hypotheses**
Ref	Expression
minvec.x	⊢ 𝑋 = (Base‘𝑈)
minvec.m	⊢ − = (-_g‘𝑈)
minvec.n	⊢ 𝑁 = (norm‘𝑈)
minvec.u	⊢ (𝜑 → 𝑈 ∈ ℂPreHil)
minvec.y	⊢ (𝜑 → 𝑌 ∈ (LSubSp‘𝑈))
minvec.w	⊢ (𝜑 → (𝑈 ↾_s 𝑌) ∈ CMetSp)
minvec.a	⊢ (𝜑 → 𝐴 ∈ 𝑋)
minvec.j	⊢ 𝐽 = (TopOpen‘𝑈)
minvec.r	⊢ 𝑅 = ran (𝑦 ∈ 𝑌 ↦ (𝑁‘(𝐴 − 𝑦)))
minvec.s	⊢ 𝑆 = inf(𝑅, ℝ, < )
minvec.d	⊢ 𝐷 = ((dist‘𝑈) ↾ (𝑋 × 𝑋))
minvec.f	⊢ 𝐹 = ran (𝑟 ∈ ℝ⁺ ↦ {𝑦 ∈ 𝑌 ∣ ((𝐴𝐷𝑦)↑2) ≤ ((𝑆↑2) + 𝑟)})
minvec.p	⊢ 𝑃 = ∪ (𝐽 fLim (𝑋filGen𝐹))

**Assertion**
Ref	Expression
minveclem4a	⊢ (𝜑 → 𝑃 ∈ ((𝐽 fLim (𝑋filGen𝐹)) ∩ 𝑌))

Distinct variable groups: 𝑦, − 𝑦,𝑟,𝐴 𝐽,𝑟,𝑦 𝑦,𝑃 𝑦,𝐹 𝑦,𝑁 𝜑,𝑟,𝑦 𝑦,𝑅 𝑦,𝑈 𝑋,𝑟,𝑦 𝑌,𝑟,𝑦 𝐷,𝑟,𝑦 𝑆,𝑟,𝑦 Allowed substitution hints: 𝑃(𝑟) 𝑅(𝑟) 𝑈(𝑟) 𝐹(𝑟) − (𝑟) 𝑁(𝑟)

Proof of Theorem minveclem4a Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		minvec.p	. 2 ⊢ 𝑃 = ∪ (𝐽 fLim (𝑋filGen𝐹))
2		ovex 7288	. . . . 5 ⊢ (𝐽 fLim (𝑋filGen𝐹)) ∈ V
3	2	uniex 7572	. . . 4 ⊢ ∪ (𝐽 fLim (𝑋filGen𝐹)) ∈ V
4	3	snid 4594	. . 3 ⊢ ∪ (𝐽 fLim (𝑋filGen𝐹)) ∈ {∪ (𝐽 fLim (𝑋filGen𝐹))}
5		minvec.u	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑈 ∈ ℂPreHil)
6		cphngp 24242	. . . . . . . . . . . 12 ⊢ (𝑈 ∈ ℂPreHil → 𝑈 ∈ NrmGrp)
7		ngpxms 23663	. . . . . . . . . . . 12 ⊢ (𝑈 ∈ NrmGrp → 𝑈 ∈ ∞MetSp)
8	5, 6, 7	3syl 18	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑈 ∈ ∞MetSp)
9		minvec.j	. . . . . . . . . . . 12 ⊢ 𝐽 = (TopOpen‘𝑈)
10		minvec.x	. . . . . . . . . . . 12 ⊢ 𝑋 = (Base‘𝑈)
11		minvec.d	. . . . . . . . . . . 12 ⊢ 𝐷 = ((dist‘𝑈) ↾ (𝑋 × 𝑋))
12	9, 10, 11	xmstopn 23512	. . . . . . . . . . 11 ⊢ (𝑈 ∈ ∞MetSp → 𝐽 = (MetOpen‘𝐷))
13	8, 12	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 𝐽 = (MetOpen‘𝐷))
14	13	oveq1d 7270	. . . . . . . . 9 ⊢ (𝜑 → (𝐽 ↾_t 𝑌) = ((MetOpen‘𝐷) ↾_t 𝑌))
15	10, 11	xmsxmet 23517	. . . . . . . . . . 11 ⊢ (𝑈 ∈ ∞MetSp → 𝐷 ∈ (∞Met‘𝑋))
16	8, 15	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 𝐷 ∈ (∞Met‘𝑋))
17		minvec.y	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑌 ∈ (LSubSp‘𝑈))
18		eqid 2738	. . . . . . . . . . . 12 ⊢ (LSubSp‘𝑈) = (LSubSp‘𝑈)
19	10, 18	lssss 20113	. . . . . . . . . . 11 ⊢ (𝑌 ∈ (LSubSp‘𝑈) → 𝑌 ⊆ 𝑋)
20	17, 19	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 𝑌 ⊆ 𝑋)
21		eqid 2738	. . . . . . . . . . 11 ⊢ (𝐷 ↾ (𝑌 × 𝑌)) = (𝐷 ↾ (𝑌 × 𝑌))
22		eqid 2738	. . . . . . . . . . 11 ⊢ (MetOpen‘𝐷) = (MetOpen‘𝐷)
23		eqid 2738	. . . . . . . . . . 11 ⊢ (MetOpen‘(𝐷 ↾ (𝑌 × 𝑌))) = (MetOpen‘(𝐷 ↾ (𝑌 × 𝑌)))
24	21, 22, 23	metrest 23586	. . . . . . . . . 10 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑌 ⊆ 𝑋) → ((MetOpen‘𝐷) ↾_t 𝑌) = (MetOpen‘(𝐷 ↾ (𝑌 × 𝑌))))
25	16, 20, 24	syl2anc 583	. . . . . . . . 9 ⊢ (𝜑 → ((MetOpen‘𝐷) ↾_t 𝑌) = (MetOpen‘(𝐷 ↾ (𝑌 × 𝑌))))
26	14, 25	eqtr2d 2779	. . . . . . . 8 ⊢ (𝜑 → (MetOpen‘(𝐷 ↾ (𝑌 × 𝑌))) = (𝐽 ↾_t 𝑌))
27		minvec.m	. . . . . . . . . . . 12 ⊢ − = (-_g‘𝑈)
28		minvec.n	. . . . . . . . . . . 12 ⊢ 𝑁 = (norm‘𝑈)
29		minvec.w	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑈 ↾_s 𝑌) ∈ CMetSp)
30		minvec.a	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐴 ∈ 𝑋)
31		minvec.r	. . . . . . . . . . . 12 ⊢ 𝑅 = ran (𝑦 ∈ 𝑌 ↦ (𝑁‘(𝐴 − 𝑦)))
32		minvec.s	. . . . . . . . . . . 12 ⊢ 𝑆 = inf(𝑅, ℝ, < )
33		minvec.f	. . . . . . . . . . . 12 ⊢ 𝐹 = ran (𝑟 ∈ ℝ⁺ ↦ {𝑦 ∈ 𝑌 ∣ ((𝐴𝐷𝑦)↑2) ≤ ((𝑆↑2) + 𝑟)})
34	10, 27, 28, 5, 17, 29, 30, 9, 31, 32, 11, 33	minveclem3b 24497	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐹 ∈ (fBas‘𝑌))
35		fgcl 22937	. . . . . . . . . . 11 ⊢ (𝐹 ∈ (fBas‘𝑌) → (𝑌filGen𝐹) ∈ (Fil‘𝑌))
36	34, 35	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → (𝑌filGen𝐹) ∈ (Fil‘𝑌))
37	10	fvexi 6770	. . . . . . . . . . 11 ⊢ 𝑋 ∈ V
38	37	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → 𝑋 ∈ V)
39		trfg 22950	. . . . . . . . . 10 ⊢ (((𝑌filGen𝐹) ∈ (Fil‘𝑌) ∧ 𝑌 ⊆ 𝑋 ∧ 𝑋 ∈ V) → ((𝑋filGen(𝑌filGen𝐹)) ↾_t 𝑌) = (𝑌filGen𝐹))
40	36, 20, 38, 39	syl3anc 1369	. . . . . . . . 9 ⊢ (𝜑 → ((𝑋filGen(𝑌filGen𝐹)) ↾_t 𝑌) = (𝑌filGen𝐹))
41		fgabs 22938	. . . . . . . . . . 11 ⊢ ((𝐹 ∈ (fBas‘𝑌) ∧ 𝑌 ⊆ 𝑋) → (𝑋filGen(𝑌filGen𝐹)) = (𝑋filGen𝐹))
42	34, 20, 41	syl2anc 583	. . . . . . . . . 10 ⊢ (𝜑 → (𝑋filGen(𝑌filGen𝐹)) = (𝑋filGen𝐹))
43	42	oveq1d 7270	. . . . . . . . 9 ⊢ (𝜑 → ((𝑋filGen(𝑌filGen𝐹)) ↾_t 𝑌) = ((𝑋filGen𝐹) ↾_t 𝑌))
44	40, 43	eqtr3d 2780	. . . . . . . 8 ⊢ (𝜑 → (𝑌filGen𝐹) = ((𝑋filGen𝐹) ↾_t 𝑌))
45	26, 44	oveq12d 7273	. . . . . . 7 ⊢ (𝜑 → ((MetOpen‘(𝐷 ↾ (𝑌 × 𝑌))) fLim (𝑌filGen𝐹)) = ((𝐽 ↾_t 𝑌) fLim ((𝑋filGen𝐹) ↾_t 𝑌)))
46		xmstps 23514	. . . . . . . . . 10 ⊢ (𝑈 ∈ ∞MetSp → 𝑈 ∈ TopSp)
47	8, 46	syl 17	. . . . . . . . 9 ⊢ (𝜑 → 𝑈 ∈ TopSp)
48	10, 9	istps 21991	. . . . . . . . 9 ⊢ (𝑈 ∈ TopSp ↔ 𝐽 ∈ (TopOn‘𝑋))
49	47, 48	sylib 217	. . . . . . . 8 ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))
50		fbsspw 22891	. . . . . . . . . . . 12 ⊢ (𝐹 ∈ (fBas‘𝑌) → 𝐹 ⊆ 𝒫 𝑌)
51	34, 50	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐹 ⊆ 𝒫 𝑌)
52	20	sspwd 4545	. . . . . . . . . . 11 ⊢ (𝜑 → 𝒫 𝑌 ⊆ 𝒫 𝑋)
53	51, 52	sstrd 3927	. . . . . . . . . 10 ⊢ (𝜑 → 𝐹 ⊆ 𝒫 𝑋)
54		fbasweak 22924	. . . . . . . . . 10 ⊢ ((𝐹 ∈ (fBas‘𝑌) ∧ 𝐹 ⊆ 𝒫 𝑋 ∧ 𝑋 ∈ V) → 𝐹 ∈ (fBas‘𝑋))
55	34, 53, 38, 54	syl3anc 1369	. . . . . . . . 9 ⊢ (𝜑 → 𝐹 ∈ (fBas‘𝑋))
56		fgcl 22937	. . . . . . . . 9 ⊢ (𝐹 ∈ (fBas‘𝑋) → (𝑋filGen𝐹) ∈ (Fil‘𝑋))
57	55, 56	syl 17	. . . . . . . 8 ⊢ (𝜑 → (𝑋filGen𝐹) ∈ (Fil‘𝑋))
58		filfbas 22907	. . . . . . . . . . . . 13 ⊢ ((𝑌filGen𝐹) ∈ (Fil‘𝑌) → (𝑌filGen𝐹) ∈ (fBas‘𝑌))
59	34, 35, 58	3syl 18	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑌filGen𝐹) ∈ (fBas‘𝑌))
60		fbsspw 22891	. . . . . . . . . . . . . 14 ⊢ ((𝑌filGen𝐹) ∈ (fBas‘𝑌) → (𝑌filGen𝐹) ⊆ 𝒫 𝑌)
61	59, 60	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑌filGen𝐹) ⊆ 𝒫 𝑌)
62	61, 52	sstrd 3927	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑌filGen𝐹) ⊆ 𝒫 𝑋)
63		fbasweak 22924	. . . . . . . . . . . 12 ⊢ (((𝑌filGen𝐹) ∈ (fBas‘𝑌) ∧ (𝑌filGen𝐹) ⊆ 𝒫 𝑋 ∧ 𝑋 ∈ V) → (𝑌filGen𝐹) ∈ (fBas‘𝑋))
64	59, 62, 38, 63	syl3anc 1369	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑌filGen𝐹) ∈ (fBas‘𝑋))
65		ssfg 22931	. . . . . . . . . . 11 ⊢ ((𝑌filGen𝐹) ∈ (fBas‘𝑋) → (𝑌filGen𝐹) ⊆ (𝑋filGen(𝑌filGen𝐹)))
66	64, 65	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → (𝑌filGen𝐹) ⊆ (𝑋filGen(𝑌filGen𝐹)))
67	66, 42	sseqtrd 3957	. . . . . . . . 9 ⊢ (𝜑 → (𝑌filGen𝐹) ⊆ (𝑋filGen𝐹))
68		filtop 22914	. . . . . . . . . 10 ⊢ ((𝑌filGen𝐹) ∈ (Fil‘𝑌) → 𝑌 ∈ (𝑌filGen𝐹))
69	36, 68	syl 17	. . . . . . . . 9 ⊢ (𝜑 → 𝑌 ∈ (𝑌filGen𝐹))
70	67, 69	sseldd 3918	. . . . . . . 8 ⊢ (𝜑 → 𝑌 ∈ (𝑋filGen𝐹))
71		flimrest 23042	. . . . . . . 8 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑋filGen𝐹) ∈ (Fil‘𝑋) ∧ 𝑌 ∈ (𝑋filGen𝐹)) → ((𝐽 ↾_t 𝑌) fLim ((𝑋filGen𝐹) ↾_t 𝑌)) = ((𝐽 fLim (𝑋filGen𝐹)) ∩ 𝑌))
72	49, 57, 70, 71	syl3anc 1369	. . . . . . 7 ⊢ (𝜑 → ((𝐽 ↾_t 𝑌) fLim ((𝑋filGen𝐹) ↾_t 𝑌)) = ((𝐽 fLim (𝑋filGen𝐹)) ∩ 𝑌))
73	45, 72	eqtrd 2778	. . . . . 6 ⊢ (𝜑 → ((MetOpen‘(𝐷 ↾ (𝑌 × 𝑌))) fLim (𝑌filGen𝐹)) = ((𝐽 fLim (𝑋filGen𝐹)) ∩ 𝑌))
74	10, 27, 28, 5, 17, 29, 30, 9, 31, 32, 11	minveclem3a 24496	. . . . . . 7 ⊢ (𝜑 → (𝐷 ↾ (𝑌 × 𝑌)) ∈ (CMet‘𝑌))
75	10, 27, 28, 5, 17, 29, 30, 9, 31, 32, 11, 33	minveclem3 24498	. . . . . . 7 ⊢ (𝜑 → (𝑌filGen𝐹) ∈ (CauFil‘(𝐷 ↾ (𝑌 × 𝑌))))
76	23	cmetcvg 24354	. . . . . . 7 ⊢ (((𝐷 ↾ (𝑌 × 𝑌)) ∈ (CMet‘𝑌) ∧ (𝑌filGen𝐹) ∈ (CauFil‘(𝐷 ↾ (𝑌 × 𝑌)))) → ((MetOpen‘(𝐷 ↾ (𝑌 × 𝑌))) fLim (𝑌filGen𝐹)) ≠ ∅)
77	74, 75, 76	syl2anc 583	. . . . . 6 ⊢ (𝜑 → ((MetOpen‘(𝐷 ↾ (𝑌 × 𝑌))) fLim (𝑌filGen𝐹)) ≠ ∅)
78	73, 77	eqnetrrd 3011	. . . . 5 ⊢ (𝜑 → ((𝐽 fLim (𝑋filGen𝐹)) ∩ 𝑌) ≠ ∅)
79	78	neneqd 2947	. . . 4 ⊢ (𝜑 → ¬ ((𝐽 fLim (𝑋filGen𝐹)) ∩ 𝑌) = ∅)
80		inss1 4159	. . . . . . 7 ⊢ ((𝐽 fLim (𝑋filGen𝐹)) ∩ 𝑌) ⊆ (𝐽 fLim (𝑋filGen𝐹))
81	22	methaus 23582	. . . . . . . . . . . . 13 ⊢ (𝐷 ∈ (∞Met‘𝑋) → (MetOpen‘𝐷) ∈ Haus)
82	15, 81	syl 17	. . . . . . . . . . . 12 ⊢ (𝑈 ∈ ∞MetSp → (MetOpen‘𝐷) ∈ Haus)
83	12, 82	eqeltrd 2839	. . . . . . . . . . 11 ⊢ (𝑈 ∈ ∞MetSp → 𝐽 ∈ Haus)
84		hausflimi 23039	. . . . . . . . . . 11 ⊢ (𝐽 ∈ Haus → ∃*𝑥 𝑥 ∈ (𝐽 fLim (𝑋filGen𝐹)))
85	8, 83, 84	3syl 18	. . . . . . . . . 10 ⊢ (𝜑 → ∃*𝑥 𝑥 ∈ (𝐽 fLim (𝑋filGen𝐹)))
86		ssn0 4331	. . . . . . . . . . . 12 ⊢ ((((𝐽 fLim (𝑋filGen𝐹)) ∩ 𝑌) ⊆ (𝐽 fLim (𝑋filGen𝐹)) ∧ ((𝐽 fLim (𝑋filGen𝐹)) ∩ 𝑌) ≠ ∅) → (𝐽 fLim (𝑋filGen𝐹)) ≠ ∅)
87	80, 78, 86	sylancr 586	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐽 fLim (𝑋filGen𝐹)) ≠ ∅)
88		n0moeu 4287	. . . . . . . . . . 11 ⊢ ((𝐽 fLim (𝑋filGen𝐹)) ≠ ∅ → (∃*𝑥 𝑥 ∈ (𝐽 fLim (𝑋filGen𝐹)) ↔ ∃!𝑥 𝑥 ∈ (𝐽 fLim (𝑋filGen𝐹))))
89	87, 88	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → (∃*𝑥 𝑥 ∈ (𝐽 fLim (𝑋filGen𝐹)) ↔ ∃!𝑥 𝑥 ∈ (𝐽 fLim (𝑋filGen𝐹))))
90	85, 89	mpbid 231	. . . . . . . . 9 ⊢ (𝜑 → ∃!𝑥 𝑥 ∈ (𝐽 fLim (𝑋filGen𝐹)))
91		euen1b 8771	. . . . . . . . 9 ⊢ ((𝐽 fLim (𝑋filGen𝐹)) ≈ 1_o ↔ ∃!𝑥 𝑥 ∈ (𝐽 fLim (𝑋filGen𝐹)))
92	90, 91	sylibr 233	. . . . . . . 8 ⊢ (𝜑 → (𝐽 fLim (𝑋filGen𝐹)) ≈ 1_o)
93		en1b 8767	. . . . . . . 8 ⊢ ((𝐽 fLim (𝑋filGen𝐹)) ≈ 1_o ↔ (𝐽 fLim (𝑋filGen𝐹)) = {∪ (𝐽 fLim (𝑋filGen𝐹))})
94	92, 93	sylib 217	. . . . . . 7 ⊢ (𝜑 → (𝐽 fLim (𝑋filGen𝐹)) = {∪ (𝐽 fLim (𝑋filGen𝐹))})
95	80, 94	sseqtrid 3969	. . . . . 6 ⊢ (𝜑 → ((𝐽 fLim (𝑋filGen𝐹)) ∩ 𝑌) ⊆ {∪ (𝐽 fLim (𝑋filGen𝐹))})
96		sssn 4756	. . . . . 6 ⊢ (((𝐽 fLim (𝑋filGen𝐹)) ∩ 𝑌) ⊆ {∪ (𝐽 fLim (𝑋filGen𝐹))} ↔ (((𝐽 fLim (𝑋filGen𝐹)) ∩ 𝑌) = ∅ ∨ ((𝐽 fLim (𝑋filGen𝐹)) ∩ 𝑌) = {∪ (𝐽 fLim (𝑋filGen𝐹))}))
97	95, 96	sylib 217	. . . . 5 ⊢ (𝜑 → (((𝐽 fLim (𝑋filGen𝐹)) ∩ 𝑌) = ∅ ∨ ((𝐽 fLim (𝑋filGen𝐹)) ∩ 𝑌) = {∪ (𝐽 fLim (𝑋filGen𝐹))}))
98	97	ord 860	. . . 4 ⊢ (𝜑 → (¬ ((𝐽 fLim (𝑋filGen𝐹)) ∩ 𝑌) = ∅ → ((𝐽 fLim (𝑋filGen𝐹)) ∩ 𝑌) = {∪ (𝐽 fLim (𝑋filGen𝐹))}))
99	79, 98	mpd 15	. . 3 ⊢ (𝜑 → ((𝐽 fLim (𝑋filGen𝐹)) ∩ 𝑌) = {∪ (𝐽 fLim (𝑋filGen𝐹))})
100	4, 99	eleqtrrid 2846	. 2 ⊢ (𝜑 → ∪ (𝐽 fLim (𝑋filGen𝐹)) ∈ ((𝐽 fLim (𝑋filGen𝐹)) ∩ 𝑌))
101	1, 100	eqeltrid 2843	1 ⊢ (𝜑 → 𝑃 ∈ ((𝐽 fLim (𝑋filGen𝐹)) ∩ 𝑌))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∨ wo 843 = wceq 1539 ∈ wcel 2108 ∃*wmo 2538 ∃!weu 2568 ≠ wne 2942 {crab 3067 Vcvv 3422 ∩ cin 3882 ⊆ wss 3883 ∅c0 4253 𝒫 cpw 4530 {csn 4558 ∪ cuni 4836 class class class wbr 5070 ↦ cmpt 5153 × cxp 5578 ran crn 5581 ↾ cres 5582 ‘cfv 6418 (class class class)co 7255 1_oc1o 8260 ≈ cen 8688 infcinf 9130 ℝcr 10801 + caddc 10805 < clt 10940 ≤ cle 10941 2c2 11958 ℝ⁺crp 12659 ↑cexp 13710 Basecbs 16840 ↾_s cress 16867 distcds 16897 ↾_t crest 17048 TopOpenctopn 17049 -_gcsg 18494 LSubSpclss 20108 ∞Metcxmet 20495 fBascfbas 20498 filGencfg 20499 MetOpencmopn 20500 TopOnctopon 21967 TopSpctps 21989 Hauscha 22367 Filcfil 22904 fLim cflim 22993 ∞MetSpcxms 23378 normcnm 23638 NrmGrpcngp 23639 ℂPreHilccph 24235 CauFilccfil 24321 CMetccmet 24323 CMetSpccms 24401

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1799 ax-4 1813 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2156 ax-12 2173 ax-ext 2709 ax-rep 5205 ax-sep 5218 ax-nul 5225 ax-pow 5283 ax-pr 5347 ax-un 7566 ax-cnex 10858 ax-resscn 10859 ax-1cn 10860 ax-icn 10861 ax-addcl 10862 ax-addrcl 10863 ax-mulcl 10864 ax-mulrcl 10865 ax-mulcom 10866 ax-addass 10867 ax-mulass 10868 ax-distr 10869 ax-i2m1 10870 ax-1ne0 10871 ax-1rid 10872 ax-rnegex 10873 ax-rrecex 10874 ax-cnre 10875 ax-pre-lttri 10876 ax-pre-lttrn 10877 ax-pre-ltadd 10878 ax-pre-mulgt0 10879 ax-pre-sup 10880 ax-addf 10881 ax-mulf 10882

This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1784 df-nf 1788 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2888 df-ne 2943 df-nel 3049 df-ral 3068 df-rex 3069 df-reu 3070 df-rmo 3071 df-rab 3072 df-v 3424 df-sbc 3712 df-csb 3829 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3902 df-nul 4254 df-if 4457 df-pw 4532 df-sn 4559 df-pr 4561 df-tp 4563 df-op 4565 df-uni 4837 df-int 4877 df-iun 4923 df-br 5071 df-opab 5133 df-mpt 5154 df-tr 5188 df-id 5480 df-eprel 5486 df-po 5494 df-so 5495 df-fr 5535 df-we 5537 df-xp 5586 df-rel 5587 df-cnv 5588 df-co 5589 df-dm 5590 df-rn 5591 df-res 5592 df-ima 5593 df-pred 6191 df-ord 6254 df-on 6255 df-lim 6256 df-suc 6257 df-iota 6376 df-fun 6420 df-fn 6421 df-f 6422 df-f1 6423 df-fo 6424 df-f1o 6425 df-fv 6426 df-riota 7212 df-ov 7258 df-oprab 7259 df-mpo 7260 df-om 7688 df-1st 7804 df-2nd 7805 df-tpos 8013 df-frecs 8068 df-wrecs 8099 df-recs 8173 df-rdg 8212 df-1o 8267 df-er 8456 df-map 8575 df-en 8692 df-dom 8693 df-sdom 8694 df-fin 8695 df-fi 9100 df-sup 9131 df-inf 9132 df-pnf 10942 df-mnf 10943 df-xr 10944 df-ltxr 10945 df-le 10946 df-sub 11137 df-neg 11138 df-div 11563 df-nn 11904 df-2 11966 df-3 11967 df-4 11968 df-5 11969 df-6 11970 df-7 11971 df-8 11972 df-9 11973 df-n0 12164 df-z 12250 df-dec 12367 df-uz 12512 df-q 12618 df-rp 12660 df-xneg 12777 df-xadd 12778 df-xmul 12779 df-ico 13014 df-icc 13015 df-fz 13169 df-seq 13650 df-exp 13711 df-cj 14738 df-re 14739 df-im 14740 df-sqrt 14874 df-abs 14875 df-struct 16776 df-sets 16793 df-slot 16811 df-ndx 16823 df-base 16841 df-ress 16868 df-plusg 16901 df-mulr 16902 df-starv 16903 df-sca 16904 df-vsca 16905 df-ip 16906 df-tset 16907 df-ple 16908 df-ds 16910 df-unif 16911 df-rest 17050 df-0g 17069 df-topgen 17071 df-mgm 18241 df-sgrp 18290 df-mnd 18301 df-mhm 18345 df-grp 18495 df-minusg 18496 df-sbg 18497 df-mulg 18616 df-subg 18667 df-ghm 18747 df-cmn 19303 df-abl 19304 df-mgp 19636 df-ur 19653 df-ring 19700 df-cring 19701 df-oppr 19777 df-dvdsr 19798 df-unit 19799 df-invr 19829 df-dvr 19840 df-rnghom 19874 df-drng 19908 df-subrg 19937 df-staf 20020 df-srng 20021 df-lmod 20040 df-lss 20109 df-lmhm 20199 df-lvec 20280 df-sra 20349 df-rgmod 20350 df-psmet 20502 df-xmet 20503 df-met 20504 df-bl 20505 df-mopn 20506 df-fbas 20507 df-fg 20508 df-cnfld 20511 df-phl 20743 df-top 21951 df-topon 21968 df-topsp 21990 df-bases 22004 df-ntr 22079 df-nei 22157 df-haus 22374 df-fil 22905 df-flim 22998 df-xms 23381 df-ms 23382 df-nm 23644 df-ngp 23645 df-nlm 23648 df-clm 24132 df-cph 24237 df-cfil 24324 df-cmet 24326 df-cms 24404

This theorem is referenced by: minveclem4b 24500 minveclem4 24501

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