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Theorem minveclem4a 25330

Description: Lemma for minvec 25336. 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 7420	. . . . 5 ⊢ (𝐽 fLim (𝑋filGen𝐹)) ∈ V
3	2	uniex 7717	. . . 4 ⊢ ∪ (𝐽 fLim (𝑋filGen𝐹)) ∈ V
4	3	snid 4626	. . 3 ⊢ ∪ (𝐽 fLim (𝑋filGen𝐹)) ∈ {∪ (𝐽 fLim (𝑋filGen𝐹))}
5		minvec.u	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑈 ∈ ℂPreHil)
6		cphngp 25073	. . . . . . . . . . . 12 ⊢ (𝑈 ∈ ℂPreHil → 𝑈 ∈ NrmGrp)
7		ngpxms 24489	. . . . . . . . . . . 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 24339	. . . . . . . . . . 11 ⊢ (𝑈 ∈ ∞MetSp → 𝐽 = (MetOpen‘𝐷))
13	8, 12	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 𝐽 = (MetOpen‘𝐷))
14	13	oveq1d 7402	. . . . . . . . 9 ⊢ (𝜑 → (𝐽 ↾_t 𝑌) = ((MetOpen‘𝐷) ↾_t 𝑌))
15	10, 11	xmsxmet 24344	. . . . . . . . . . 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 2729	. . . . . . . . . . . 12 ⊢ (LSubSp‘𝑈) = (LSubSp‘𝑈)
19	10, 18	lssss 20842	. . . . . . . . . . 11 ⊢ (𝑌 ∈ (LSubSp‘𝑈) → 𝑌 ⊆ 𝑋)
20	17, 19	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 𝑌 ⊆ 𝑋)
21		eqid 2729	. . . . . . . . . . 11 ⊢ (𝐷 ↾ (𝑌 × 𝑌)) = (𝐷 ↾ (𝑌 × 𝑌))
22		eqid 2729	. . . . . . . . . . 11 ⊢ (MetOpen‘𝐷) = (MetOpen‘𝐷)
23		eqid 2729	. . . . . . . . . . 11 ⊢ (MetOpen‘(𝐷 ↾ (𝑌 × 𝑌))) = (MetOpen‘(𝐷 ↾ (𝑌 × 𝑌)))
24	21, 22, 23	metrest 24412	. . . . . . . . . 10 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑌 ⊆ 𝑋) → ((MetOpen‘𝐷) ↾_t 𝑌) = (MetOpen‘(𝐷 ↾ (𝑌 × 𝑌))))
25	16, 20, 24	syl2anc 584	. . . . . . . . 9 ⊢ (𝜑 → ((MetOpen‘𝐷) ↾_t 𝑌) = (MetOpen‘(𝐷 ↾ (𝑌 × 𝑌))))
26	14, 25	eqtr2d 2765	. . . . . . . 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 25328	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐹 ∈ (fBas‘𝑌))
35		fgcl 23765	. . . . . . . . . . 11 ⊢ (𝐹 ∈ (fBas‘𝑌) → (𝑌filGen𝐹) ∈ (Fil‘𝑌))
36	34, 35	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → (𝑌filGen𝐹) ∈ (Fil‘𝑌))
37	10	fvexi 6872	. . . . . . . . . . 11 ⊢ 𝑋 ∈ V
38	37	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → 𝑋 ∈ V)
39		trfg 23778	. . . . . . . . . 10 ⊢ (((𝑌filGen𝐹) ∈ (Fil‘𝑌) ∧ 𝑌 ⊆ 𝑋 ∧ 𝑋 ∈ V) → ((𝑋filGen(𝑌filGen𝐹)) ↾_t 𝑌) = (𝑌filGen𝐹))
40	36, 20, 38, 39	syl3anc 1373	. . . . . . . . 9 ⊢ (𝜑 → ((𝑋filGen(𝑌filGen𝐹)) ↾_t 𝑌) = (𝑌filGen𝐹))
41		fgabs 23766	. . . . . . . . . . 11 ⊢ ((𝐹 ∈ (fBas‘𝑌) ∧ 𝑌 ⊆ 𝑋) → (𝑋filGen(𝑌filGen𝐹)) = (𝑋filGen𝐹))
42	34, 20, 41	syl2anc 584	. . . . . . . . . 10 ⊢ (𝜑 → (𝑋filGen(𝑌filGen𝐹)) = (𝑋filGen𝐹))
43	42	oveq1d 7402	. . . . . . . . 9 ⊢ (𝜑 → ((𝑋filGen(𝑌filGen𝐹)) ↾_t 𝑌) = ((𝑋filGen𝐹) ↾_t 𝑌))
44	40, 43	eqtr3d 2766	. . . . . . . 8 ⊢ (𝜑 → (𝑌filGen𝐹) = ((𝑋filGen𝐹) ↾_t 𝑌))
45	26, 44	oveq12d 7405	. . . . . . 7 ⊢ (𝜑 → ((MetOpen‘(𝐷 ↾ (𝑌 × 𝑌))) fLim (𝑌filGen𝐹)) = ((𝐽 ↾_t 𝑌) fLim ((𝑋filGen𝐹) ↾_t 𝑌)))
46		xmstps 24341	. . . . . . . . . 10 ⊢ (𝑈 ∈ ∞MetSp → 𝑈 ∈ TopSp)
47	8, 46	syl 17	. . . . . . . . 9 ⊢ (𝜑 → 𝑈 ∈ TopSp)
48	10, 9	istps 22821	. . . . . . . . 9 ⊢ (𝑈 ∈ TopSp ↔ 𝐽 ∈ (TopOn‘𝑋))
49	47, 48	sylib 218	. . . . . . . 8 ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))
50		fbsspw 23719	. . . . . . . . . . . 12 ⊢ (𝐹 ∈ (fBas‘𝑌) → 𝐹 ⊆ 𝒫 𝑌)
51	34, 50	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐹 ⊆ 𝒫 𝑌)
52	20	sspwd 4576	. . . . . . . . . . 11 ⊢ (𝜑 → 𝒫 𝑌 ⊆ 𝒫 𝑋)
53	51, 52	sstrd 3957	. . . . . . . . . 10 ⊢ (𝜑 → 𝐹 ⊆ 𝒫 𝑋)
54		fbasweak 23752	. . . . . . . . . 10 ⊢ ((𝐹 ∈ (fBas‘𝑌) ∧ 𝐹 ⊆ 𝒫 𝑋 ∧ 𝑋 ∈ V) → 𝐹 ∈ (fBas‘𝑋))
55	34, 53, 38, 54	syl3anc 1373	. . . . . . . . 9 ⊢ (𝜑 → 𝐹 ∈ (fBas‘𝑋))
56		fgcl 23765	. . . . . . . . 9 ⊢ (𝐹 ∈ (fBas‘𝑋) → (𝑋filGen𝐹) ∈ (Fil‘𝑋))
57	55, 56	syl 17	. . . . . . . 8 ⊢ (𝜑 → (𝑋filGen𝐹) ∈ (Fil‘𝑋))
58		filfbas 23735	. . . . . . . . . . . . 13 ⊢ ((𝑌filGen𝐹) ∈ (Fil‘𝑌) → (𝑌filGen𝐹) ∈ (fBas‘𝑌))
59	34, 35, 58	3syl 18	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑌filGen𝐹) ∈ (fBas‘𝑌))
60		fbsspw 23719	. . . . . . . . . . . . . 14 ⊢ ((𝑌filGen𝐹) ∈ (fBas‘𝑌) → (𝑌filGen𝐹) ⊆ 𝒫 𝑌)
61	59, 60	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑌filGen𝐹) ⊆ 𝒫 𝑌)
62	61, 52	sstrd 3957	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑌filGen𝐹) ⊆ 𝒫 𝑋)
63		fbasweak 23752	. . . . . . . . . . . 12 ⊢ (((𝑌filGen𝐹) ∈ (fBas‘𝑌) ∧ (𝑌filGen𝐹) ⊆ 𝒫 𝑋 ∧ 𝑋 ∈ V) → (𝑌filGen𝐹) ∈ (fBas‘𝑋))
64	59, 62, 38, 63	syl3anc 1373	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑌filGen𝐹) ∈ (fBas‘𝑋))
65		ssfg 23759	. . . . . . . . . . 11 ⊢ ((𝑌filGen𝐹) ∈ (fBas‘𝑋) → (𝑌filGen𝐹) ⊆ (𝑋filGen(𝑌filGen𝐹)))
66	64, 65	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → (𝑌filGen𝐹) ⊆ (𝑋filGen(𝑌filGen𝐹)))
67	66, 42	sseqtrd 3983	. . . . . . . . 9 ⊢ (𝜑 → (𝑌filGen𝐹) ⊆ (𝑋filGen𝐹))
68		filtop 23742	. . . . . . . . . 10 ⊢ ((𝑌filGen𝐹) ∈ (Fil‘𝑌) → 𝑌 ∈ (𝑌filGen𝐹))
69	36, 68	syl 17	. . . . . . . . 9 ⊢ (𝜑 → 𝑌 ∈ (𝑌filGen𝐹))
70	67, 69	sseldd 3947	. . . . . . . 8 ⊢ (𝜑 → 𝑌 ∈ (𝑋filGen𝐹))
71		flimrest 23870	. . . . . . . 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 1373	. . . . . . 7 ⊢ (𝜑 → ((𝐽 ↾_t 𝑌) fLim ((𝑋filGen𝐹) ↾_t 𝑌)) = ((𝐽 fLim (𝑋filGen𝐹)) ∩ 𝑌))
73	45, 72	eqtrd 2764	. . . . . 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 25327	. . . . . . 7 ⊢ (𝜑 → (𝐷 ↾ (𝑌 × 𝑌)) ∈ (CMet‘𝑌))
75	10, 27, 28, 5, 17, 29, 30, 9, 31, 32, 11, 33	minveclem3 25329	. . . . . . 7 ⊢ (𝜑 → (𝑌filGen𝐹) ∈ (CauFil‘(𝐷 ↾ (𝑌 × 𝑌))))
76	23	cmetcvg 25185	. . . . . . 7 ⊢ (((𝐷 ↾ (𝑌 × 𝑌)) ∈ (CMet‘𝑌) ∧ (𝑌filGen𝐹) ∈ (CauFil‘(𝐷 ↾ (𝑌 × 𝑌)))) → ((MetOpen‘(𝐷 ↾ (𝑌 × 𝑌))) fLim (𝑌filGen𝐹)) ≠ ∅)
77	74, 75, 76	syl2anc 584	. . . . . 6 ⊢ (𝜑 → ((MetOpen‘(𝐷 ↾ (𝑌 × 𝑌))) fLim (𝑌filGen𝐹)) ≠ ∅)
78	73, 77	eqnetrrd 2993	. . . . 5 ⊢ (𝜑 → ((𝐽 fLim (𝑋filGen𝐹)) ∩ 𝑌) ≠ ∅)
79	78	neneqd 2930	. . . 4 ⊢ (𝜑 → ¬ ((𝐽 fLim (𝑋filGen𝐹)) ∩ 𝑌) = ∅)
80		inss1 4200	. . . . . . 7 ⊢ ((𝐽 fLim (𝑋filGen𝐹)) ∩ 𝑌) ⊆ (𝐽 fLim (𝑋filGen𝐹))
81	22	methaus 24408	. . . . . . . . . . . . 13 ⊢ (𝐷 ∈ (∞Met‘𝑋) → (MetOpen‘𝐷) ∈ Haus)
82	15, 81	syl 17	. . . . . . . . . . . 12 ⊢ (𝑈 ∈ ∞MetSp → (MetOpen‘𝐷) ∈ Haus)
83	12, 82	eqeltrd 2828	. . . . . . . . . . 11 ⊢ (𝑈 ∈ ∞MetSp → 𝐽 ∈ Haus)
84		hausflimi 23867	. . . . . . . . . . 11 ⊢ (𝐽 ∈ Haus → ∃*𝑥 𝑥 ∈ (𝐽 fLim (𝑋filGen𝐹)))
85	8, 83, 84	3syl 18	. . . . . . . . . 10 ⊢ (𝜑 → ∃*𝑥 𝑥 ∈ (𝐽 fLim (𝑋filGen𝐹)))
86		ssn0 4367	. . . . . . . . . . . 12 ⊢ ((((𝐽 fLim (𝑋filGen𝐹)) ∩ 𝑌) ⊆ (𝐽 fLim (𝑋filGen𝐹)) ∧ ((𝐽 fLim (𝑋filGen𝐹)) ∩ 𝑌) ≠ ∅) → (𝐽 fLim (𝑋filGen𝐹)) ≠ ∅)
87	80, 78, 86	sylancr 587	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐽 fLim (𝑋filGen𝐹)) ≠ ∅)
88		n0moeu 4322	. . . . . . . . . . 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 232	. . . . . . . . 9 ⊢ (𝜑 → ∃!𝑥 𝑥 ∈ (𝐽 fLim (𝑋filGen𝐹)))
91		euen1b 8999	. . . . . . . . 9 ⊢ ((𝐽 fLim (𝑋filGen𝐹)) ≈ 1_o ↔ ∃!𝑥 𝑥 ∈ (𝐽 fLim (𝑋filGen𝐹)))
92	90, 91	sylibr 234	. . . . . . . 8 ⊢ (𝜑 → (𝐽 fLim (𝑋filGen𝐹)) ≈ 1_o)
93		en1b 8996	. . . . . . . 8 ⊢ ((𝐽 fLim (𝑋filGen𝐹)) ≈ 1_o ↔ (𝐽 fLim (𝑋filGen𝐹)) = {∪ (𝐽 fLim (𝑋filGen𝐹))})
94	92, 93	sylib 218	. . . . . . 7 ⊢ (𝜑 → (𝐽 fLim (𝑋filGen𝐹)) = {∪ (𝐽 fLim (𝑋filGen𝐹))})
95	80, 94	sseqtrid 3989	. . . . . 6 ⊢ (𝜑 → ((𝐽 fLim (𝑋filGen𝐹)) ∩ 𝑌) ⊆ {∪ (𝐽 fLim (𝑋filGen𝐹))})
96		sssn 4790	. . . . . 6 ⊢ (((𝐽 fLim (𝑋filGen𝐹)) ∩ 𝑌) ⊆ {∪ (𝐽 fLim (𝑋filGen𝐹))} ↔ (((𝐽 fLim (𝑋filGen𝐹)) ∩ 𝑌) = ∅ ∨ ((𝐽 fLim (𝑋filGen𝐹)) ∩ 𝑌) = {∪ (𝐽 fLim (𝑋filGen𝐹))}))
97	95, 96	sylib 218	. . . . 5 ⊢ (𝜑 → (((𝐽 fLim (𝑋filGen𝐹)) ∩ 𝑌) = ∅ ∨ ((𝐽 fLim (𝑋filGen𝐹)) ∩ 𝑌) = {∪ (𝐽 fLim (𝑋filGen𝐹))}))
98	97	ord 864	. . . 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 2835	. 2 ⊢ (𝜑 → ∪ (𝐽 fLim (𝑋filGen𝐹)) ∈ ((𝐽 fLim (𝑋filGen𝐹)) ∩ 𝑌))
101	1, 100	eqeltrid 2832	1 ⊢ (𝜑 → 𝑃 ∈ ((𝐽 fLim (𝑋filGen𝐹)) ∩ 𝑌))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∨ wo 847 = wceq 1540 ∈ wcel 2109 ∃*wmo 2531 ∃!weu 2561 ≠ wne 2925 {crab 3405 Vcvv 3447 ∩ cin 3913 ⊆ wss 3914 ∅c0 4296 𝒫 cpw 4563 {csn 4589 ∪ cuni 4871 class class class wbr 5107 ↦ cmpt 5188 × cxp 5636 ran crn 5639 ↾ cres 5640 ‘cfv 6511 (class class class)co 7387 1_oc1o 8427 ≈ cen 8915 infcinf 9392 ℝcr 11067 + caddc 11071 < clt 11208 ≤ cle 11209 2c2 12241 ℝ⁺crp 12951 ↑cexp 14026 Basecbs 17179 ↾_s cress 17200 distcds 17229 ↾_t crest 17383 TopOpenctopn 17384 -_gcsg 18867 LSubSpclss 20837 ∞Metcxmet 21249 fBascfbas 21252 filGencfg 21253 MetOpencmopn 21254 TopOnctopon 22797 TopSpctps 22819 Hauscha 23195 Filcfil 23732 fLim cflim 23821 ∞MetSpcxms 24205 normcnm 24464 NrmGrpcngp 24465 ℂPreHilccph 25066 CauFilccfil 25152 CMetccmet 25154 CMetSpccms 25232

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 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-rep 5234 ax-sep 5251 ax-nul 5261 ax-pow 5320 ax-pr 5387 ax-un 7711 ax-cnex 11124 ax-resscn 11125 ax-1cn 11126 ax-icn 11127 ax-addcl 11128 ax-addrcl 11129 ax-mulcl 11130 ax-mulrcl 11131 ax-mulcom 11132 ax-addass 11133 ax-mulass 11134 ax-distr 11135 ax-i2m1 11136 ax-1ne0 11137 ax-1rid 11138 ax-rnegex 11139 ax-rrecex 11140 ax-cnre 11141 ax-pre-lttri 11142 ax-pre-lttrn 11143 ax-pre-ltadd 11144 ax-pre-mulgt0 11145 ax-pre-sup 11146 ax-addf 11147 ax-mulf 11148

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-rmo 3354 df-reu 3355 df-rab 3406 df-v 3449 df-sbc 3754 df-csb 3863 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-pss 3934 df-nul 4297 df-if 4489 df-pw 4565 df-sn 4590 df-pr 4592 df-tp 4594 df-op 4596 df-uni 4872 df-int 4911 df-iun 4957 df-br 5108 df-opab 5170 df-mpt 5189 df-tr 5215 df-id 5533 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5591 df-we 5593 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-pred 6274 df-ord 6335 df-on 6336 df-lim 6337 df-suc 6338 df-iota 6464 df-fun 6513 df-fn 6514 df-f 6515 df-f1 6516 df-fo 6517 df-f1o 6518 df-fv 6519 df-riota 7344 df-ov 7390 df-oprab 7391 df-mpo 7392 df-om 7843 df-1st 7968 df-2nd 7969 df-tpos 8205 df-frecs 8260 df-wrecs 8291 df-recs 8340 df-rdg 8378 df-1o 8434 df-er 8671 df-map 8801 df-en 8919 df-dom 8920 df-sdom 8921 df-fin 8922 df-fi 9362 df-sup 9393 df-inf 9394 df-pnf 11210 df-mnf 11211 df-xr 11212 df-ltxr 11213 df-le 11214 df-sub 11407 df-neg 11408 df-div 11836 df-nn 12187 df-2 12249 df-3 12250 df-4 12251 df-5 12252 df-6 12253 df-7 12254 df-8 12255 df-9 12256 df-n0 12443 df-z 12530 df-dec 12650 df-uz 12794 df-q 12908 df-rp 12952 df-xneg 13072 df-xadd 13073 df-xmul 13074 df-ico 13312 df-icc 13313 df-fz 13469 df-seq 13967 df-exp 14027 df-cj 15065 df-re 15066 df-im 15067 df-sqrt 15201 df-abs 15202 df-struct 17117 df-sets 17134 df-slot 17152 df-ndx 17164 df-base 17180 df-ress 17201 df-plusg 17233 df-mulr 17234 df-starv 17235 df-sca 17236 df-vsca 17237 df-ip 17238 df-tset 17239 df-ple 17240 df-ds 17242 df-unif 17243 df-rest 17385 df-0g 17404 df-topgen 17406 df-mgm 18567 df-sgrp 18646 df-mnd 18662 df-mhm 18710 df-grp 18868 df-minusg 18869 df-sbg 18870 df-mulg 19000 df-subg 19055 df-ghm 19145 df-cmn 19712 df-abl 19713 df-mgp 20050 df-rng 20062 df-ur 20091 df-ring 20144 df-cring 20145 df-oppr 20246 df-dvdsr 20266 df-unit 20267 df-invr 20297 df-dvr 20310 df-rhm 20381 df-subrg 20479 df-drng 20640 df-staf 20748 df-srng 20749 df-lmod 20768 df-lss 20838 df-lmhm 20929 df-lvec 21010 df-sra 21080 df-rgmod 21081 df-psmet 21256 df-xmet 21257 df-met 21258 df-bl 21259 df-mopn 21260 df-fbas 21261 df-fg 21262 df-cnfld 21265 df-phl 21535 df-top 22781 df-topon 22798 df-topsp 22820 df-bases 22833 df-ntr 22907 df-nei 22985 df-haus 23202 df-fil 23733 df-flim 23826 df-xms 24208 df-ms 24209 df-nm 24470 df-ngp 24471 df-nlm 24474 df-clm 24963 df-cph 25068 df-cfil 25155 df-cmet 25157 df-cms 25235

This theorem is referenced by: minveclem4b 25331 minveclem4 25332

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