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Theorem minveclem4a 25360

Description: Lemma for minvec 25366. 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 7387	. . . . 5 ⊢ (𝐽 fLim (𝑋filGen𝐹)) ∈ V
3	2	uniex 7682	. . . 4 ⊢ ∪ (𝐽 fLim (𝑋filGen𝐹)) ∈ V
4	3	snid 4616	. . 3 ⊢ ∪ (𝐽 fLim (𝑋filGen𝐹)) ∈ {∪ (𝐽 fLim (𝑋filGen𝐹))}
5		minvec.u	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑈 ∈ ℂPreHil)
6		cphngp 25103	. . . . . . . . . . . 12 ⊢ (𝑈 ∈ ℂPreHil → 𝑈 ∈ NrmGrp)
7		ngpxms 24519	. . . . . . . . . . . 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 24369	. . . . . . . . . . 11 ⊢ (𝑈 ∈ ∞MetSp → 𝐽 = (MetOpen‘𝐷))
13	8, 12	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 𝐽 = (MetOpen‘𝐷))
14	13	oveq1d 7369	. . . . . . . . 9 ⊢ (𝜑 → (𝐽 ↾_t 𝑌) = ((MetOpen‘𝐷) ↾_t 𝑌))
15	10, 11	xmsxmet 24374	. . . . . . . . . . 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 2733	. . . . . . . . . . . 12 ⊢ (LSubSp‘𝑈) = (LSubSp‘𝑈)
19	10, 18	lssss 20873	. . . . . . . . . . 11 ⊢ (𝑌 ∈ (LSubSp‘𝑈) → 𝑌 ⊆ 𝑋)
20	17, 19	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 𝑌 ⊆ 𝑋)
21		eqid 2733	. . . . . . . . . . 11 ⊢ (𝐷 ↾ (𝑌 × 𝑌)) = (𝐷 ↾ (𝑌 × 𝑌))
22		eqid 2733	. . . . . . . . . . 11 ⊢ (MetOpen‘𝐷) = (MetOpen‘𝐷)
23		eqid 2733	. . . . . . . . . . 11 ⊢ (MetOpen‘(𝐷 ↾ (𝑌 × 𝑌))) = (MetOpen‘(𝐷 ↾ (𝑌 × 𝑌)))
24	21, 22, 23	metrest 24442	. . . . . . . . . 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 2769	. . . . . . . 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 25358	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐹 ∈ (fBas‘𝑌))
35		fgcl 23796	. . . . . . . . . . 11 ⊢ (𝐹 ∈ (fBas‘𝑌) → (𝑌filGen𝐹) ∈ (Fil‘𝑌))
36	34, 35	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → (𝑌filGen𝐹) ∈ (Fil‘𝑌))
37	10	fvexi 6844	. . . . . . . . . . 11 ⊢ 𝑋 ∈ V
38	37	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → 𝑋 ∈ V)
39		trfg 23809	. . . . . . . . . 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 23797	. . . . . . . . . . 11 ⊢ ((𝐹 ∈ (fBas‘𝑌) ∧ 𝑌 ⊆ 𝑋) → (𝑋filGen(𝑌filGen𝐹)) = (𝑋filGen𝐹))
42	34, 20, 41	syl2anc 584	. . . . . . . . . 10 ⊢ (𝜑 → (𝑋filGen(𝑌filGen𝐹)) = (𝑋filGen𝐹))
43	42	oveq1d 7369	. . . . . . . . 9 ⊢ (𝜑 → ((𝑋filGen(𝑌filGen𝐹)) ↾_t 𝑌) = ((𝑋filGen𝐹) ↾_t 𝑌))
44	40, 43	eqtr3d 2770	. . . . . . . 8 ⊢ (𝜑 → (𝑌filGen𝐹) = ((𝑋filGen𝐹) ↾_t 𝑌))
45	26, 44	oveq12d 7372	. . . . . . 7 ⊢ (𝜑 → ((MetOpen‘(𝐷 ↾ (𝑌 × 𝑌))) fLim (𝑌filGen𝐹)) = ((𝐽 ↾_t 𝑌) fLim ((𝑋filGen𝐹) ↾_t 𝑌)))
46		xmstps 24371	. . . . . . . . . 10 ⊢ (𝑈 ∈ ∞MetSp → 𝑈 ∈ TopSp)
47	8, 46	syl 17	. . . . . . . . 9 ⊢ (𝜑 → 𝑈 ∈ TopSp)
48	10, 9	istps 22852	. . . . . . . . 9 ⊢ (𝑈 ∈ TopSp ↔ 𝐽 ∈ (TopOn‘𝑋))
49	47, 48	sylib 218	. . . . . . . 8 ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))
50		fbsspw 23750	. . . . . . . . . . . 12 ⊢ (𝐹 ∈ (fBas‘𝑌) → 𝐹 ⊆ 𝒫 𝑌)
51	34, 50	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐹 ⊆ 𝒫 𝑌)
52	20	sspwd 4564	. . . . . . . . . . 11 ⊢ (𝜑 → 𝒫 𝑌 ⊆ 𝒫 𝑋)
53	51, 52	sstrd 3941	. . . . . . . . . 10 ⊢ (𝜑 → 𝐹 ⊆ 𝒫 𝑋)
54		fbasweak 23783	. . . . . . . . . 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 23796	. . . . . . . . 9 ⊢ (𝐹 ∈ (fBas‘𝑋) → (𝑋filGen𝐹) ∈ (Fil‘𝑋))
57	55, 56	syl 17	. . . . . . . 8 ⊢ (𝜑 → (𝑋filGen𝐹) ∈ (Fil‘𝑋))
58		filfbas 23766	. . . . . . . . . . . . 13 ⊢ ((𝑌filGen𝐹) ∈ (Fil‘𝑌) → (𝑌filGen𝐹) ∈ (fBas‘𝑌))
59	34, 35, 58	3syl 18	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑌filGen𝐹) ∈ (fBas‘𝑌))
60		fbsspw 23750	. . . . . . . . . . . . . 14 ⊢ ((𝑌filGen𝐹) ∈ (fBas‘𝑌) → (𝑌filGen𝐹) ⊆ 𝒫 𝑌)
61	59, 60	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑌filGen𝐹) ⊆ 𝒫 𝑌)
62	61, 52	sstrd 3941	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑌filGen𝐹) ⊆ 𝒫 𝑋)
63		fbasweak 23783	. . . . . . . . . . . 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 23790	. . . . . . . . . . 11 ⊢ ((𝑌filGen𝐹) ∈ (fBas‘𝑋) → (𝑌filGen𝐹) ⊆ (𝑋filGen(𝑌filGen𝐹)))
66	64, 65	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → (𝑌filGen𝐹) ⊆ (𝑋filGen(𝑌filGen𝐹)))
67	66, 42	sseqtrd 3967	. . . . . . . . 9 ⊢ (𝜑 → (𝑌filGen𝐹) ⊆ (𝑋filGen𝐹))
68		filtop 23773	. . . . . . . . . 10 ⊢ ((𝑌filGen𝐹) ∈ (Fil‘𝑌) → 𝑌 ∈ (𝑌filGen𝐹))
69	36, 68	syl 17	. . . . . . . . 9 ⊢ (𝜑 → 𝑌 ∈ (𝑌filGen𝐹))
70	67, 69	sseldd 3931	. . . . . . . 8 ⊢ (𝜑 → 𝑌 ∈ (𝑋filGen𝐹))
71		flimrest 23901	. . . . . . . 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 2768	. . . . . 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 25357	. . . . . . 7 ⊢ (𝜑 → (𝐷 ↾ (𝑌 × 𝑌)) ∈ (CMet‘𝑌))
75	10, 27, 28, 5, 17, 29, 30, 9, 31, 32, 11, 33	minveclem3 25359	. . . . . . 7 ⊢ (𝜑 → (𝑌filGen𝐹) ∈ (CauFil‘(𝐷 ↾ (𝑌 × 𝑌))))
76	23	cmetcvg 25215	. . . . . . 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 2997	. . . . 5 ⊢ (𝜑 → ((𝐽 fLim (𝑋filGen𝐹)) ∩ 𝑌) ≠ ∅)
79	78	neneqd 2934	. . . 4 ⊢ (𝜑 → ¬ ((𝐽 fLim (𝑋filGen𝐹)) ∩ 𝑌) = ∅)
80		inss1 4186	. . . . . . 7 ⊢ ((𝐽 fLim (𝑋filGen𝐹)) ∩ 𝑌) ⊆ (𝐽 fLim (𝑋filGen𝐹))
81	22	methaus 24438	. . . . . . . . . . . . 13 ⊢ (𝐷 ∈ (∞Met‘𝑋) → (MetOpen‘𝐷) ∈ Haus)
82	15, 81	syl 17	. . . . . . . . . . . 12 ⊢ (𝑈 ∈ ∞MetSp → (MetOpen‘𝐷) ∈ Haus)
83	12, 82	eqeltrd 2833	. . . . . . . . . . 11 ⊢ (𝑈 ∈ ∞MetSp → 𝐽 ∈ Haus)
84		hausflimi 23898	. . . . . . . . . . 11 ⊢ (𝐽 ∈ Haus → ∃*𝑥 𝑥 ∈ (𝐽 fLim (𝑋filGen𝐹)))
85	8, 83, 84	3syl 18	. . . . . . . . . 10 ⊢ (𝜑 → ∃*𝑥 𝑥 ∈ (𝐽 fLim (𝑋filGen𝐹)))
86		ssn0 4353	. . . . . . . . . . . 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 4308	. . . . . . . . . . 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 8959	. . . . . . . . 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 8956	. . . . . . . 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 3973	. . . . . 6 ⊢ (𝜑 → ((𝐽 fLim (𝑋filGen𝐹)) ∩ 𝑌) ⊆ {∪ (𝐽 fLim (𝑋filGen𝐹))})
96		sssn 4779	. . . . . 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 2840	. 2 ⊢ (𝜑 → ∪ (𝐽 fLim (𝑋filGen𝐹)) ∈ ((𝐽 fLim (𝑋filGen𝐹)) ∩ 𝑌))
101	1, 100	eqeltrid 2837	1 ⊢ (𝜑 → 𝑃 ∈ ((𝐽 fLim (𝑋filGen𝐹)) ∩ 𝑌))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∨ wo 847 = wceq 1541 ∈ wcel 2113 ∃*wmo 2535 ∃!weu 2565 ≠ wne 2929 {crab 3396 Vcvv 3437 ∩ cin 3897 ⊆ wss 3898 ∅c0 4282 𝒫 cpw 4551 {csn 4577 ∪ cuni 4860 class class class wbr 5095 ↦ cmpt 5176 × cxp 5619 ran crn 5622 ↾ cres 5623 ‘cfv 6488 (class class class)co 7354 1_oc1o 8386 ≈ cen 8874 infcinf 9334 ℝcr 11014 + caddc 11018 < clt 11155 ≤ cle 11156 2c2 12189 ℝ⁺crp 12894 ↑cexp 13972 Basecbs 17124 ↾_s cress 17145 distcds 17174 ↾_t crest 17328 TopOpenctopn 17329 -_gcsg 18852 LSubSpclss 20868 ∞Metcxmet 21280 fBascfbas 21283 filGencfg 21284 MetOpencmopn 21285 TopOnctopon 22828 TopSpctps 22850 Hauscha 23226 Filcfil 23763 fLim cflim 23852 ∞MetSpcxms 24235 normcnm 24494 NrmGrpcngp 24495 ℂPreHilccph 25096 CauFilccfil 25182 CMetccmet 25184 CMetSpccms 25262

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2182 ax-ext 2705 ax-rep 5221 ax-sep 5238 ax-nul 5248 ax-pow 5307 ax-pr 5374 ax-un 7676 ax-cnex 11071 ax-resscn 11072 ax-1cn 11073 ax-icn 11074 ax-addcl 11075 ax-addrcl 11076 ax-mulcl 11077 ax-mulrcl 11078 ax-mulcom 11079 ax-addass 11080 ax-mulass 11081 ax-distr 11082 ax-i2m1 11083 ax-1ne0 11084 ax-1rid 11085 ax-rnegex 11086 ax-rrecex 11087 ax-cnre 11088 ax-pre-lttri 11089 ax-pre-lttrn 11090 ax-pre-ltadd 11091 ax-pre-mulgt0 11092 ax-pre-sup 11093 ax-addf 11094 ax-mulf 11095

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2725 df-clel 2808 df-nfc 2882 df-ne 2930 df-nel 3034 df-ral 3049 df-rex 3058 df-rmo 3347 df-reu 3348 df-rab 3397 df-v 3439 df-sbc 3738 df-csb 3847 df-dif 3901 df-un 3903 df-in 3905 df-ss 3915 df-pss 3918 df-nul 4283 df-if 4477 df-pw 4553 df-sn 4578 df-pr 4580 df-tp 4582 df-op 4584 df-uni 4861 df-int 4900 df-iun 4945 df-br 5096 df-opab 5158 df-mpt 5177 df-tr 5203 df-id 5516 df-eprel 5521 df-po 5529 df-so 5530 df-fr 5574 df-we 5576 df-xp 5627 df-rel 5628 df-cnv 5629 df-co 5630 df-dm 5631 df-rn 5632 df-res 5633 df-ima 5634 df-pred 6255 df-ord 6316 df-on 6317 df-lim 6318 df-suc 6319 df-iota 6444 df-fun 6490 df-fn 6491 df-f 6492 df-f1 6493 df-fo 6494 df-f1o 6495 df-fv 6496 df-riota 7311 df-ov 7357 df-oprab 7358 df-mpo 7359 df-om 7805 df-1st 7929 df-2nd 7930 df-tpos 8164 df-frecs 8219 df-wrecs 8250 df-recs 8299 df-rdg 8337 df-1o 8393 df-er 8630 df-map 8760 df-en 8878 df-dom 8879 df-sdom 8880 df-fin 8881 df-fi 9304 df-sup 9335 df-inf 9336 df-pnf 11157 df-mnf 11158 df-xr 11159 df-ltxr 11160 df-le 11161 df-sub 11355 df-neg 11356 df-div 11784 df-nn 12135 df-2 12197 df-3 12198 df-4 12199 df-5 12200 df-6 12201 df-7 12202 df-8 12203 df-9 12204 df-n0 12391 df-z 12478 df-dec 12597 df-uz 12741 df-q 12851 df-rp 12895 df-xneg 13015 df-xadd 13016 df-xmul 13017 df-ico 13255 df-icc 13256 df-fz 13412 df-seq 13913 df-exp 13973 df-cj 15010 df-re 15011 df-im 15012 df-sqrt 15146 df-abs 15147 df-struct 17062 df-sets 17079 df-slot 17097 df-ndx 17109 df-base 17125 df-ress 17146 df-plusg 17178 df-mulr 17179 df-starv 17180 df-sca 17181 df-vsca 17182 df-ip 17183 df-tset 17184 df-ple 17185 df-ds 17187 df-unif 17188 df-rest 17330 df-0g 17349 df-topgen 17351 df-mgm 18552 df-sgrp 18631 df-mnd 18647 df-mhm 18695 df-grp 18853 df-minusg 18854 df-sbg 18855 df-mulg 18985 df-subg 19040 df-ghm 19129 df-cmn 19698 df-abl 19699 df-mgp 20063 df-rng 20075 df-ur 20104 df-ring 20157 df-cring 20158 df-oppr 20259 df-dvdsr 20279 df-unit 20280 df-invr 20310 df-dvr 20323 df-rhm 20394 df-subrg 20489 df-drng 20650 df-staf 20758 df-srng 20759 df-lmod 20799 df-lss 20869 df-lmhm 20960 df-lvec 21041 df-sra 21111 df-rgmod 21112 df-psmet 21287 df-xmet 21288 df-met 21289 df-bl 21290 df-mopn 21291 df-fbas 21292 df-fg 21293 df-cnfld 21296 df-phl 21567 df-top 22812 df-topon 22829 df-topsp 22851 df-bases 22864 df-ntr 22938 df-nei 23016 df-haus 23233 df-fil 23764 df-flim 23857 df-xms 24238 df-ms 24239 df-nm 24500 df-ngp 24501 df-nlm 24504 df-clm 24993 df-cph 25098 df-cfil 25185 df-cmet 25187 df-cms 25265

This theorem is referenced by: minveclem4b 25361 minveclem4 25362

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