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Theorem minveclem4a 25757

Description: Lemma for minvec 25763. 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 7454	. . . . 5 ⊢ (𝐽 fLim (𝑋filGen𝐹)) ∈ V
3	2	uniex 7759	. . . 4 ⊢ ∪ (𝐽 fLim (𝑋filGen𝐹)) ∈ V
4	3	snid 4619	. . 3 ⊢ ∪ (𝐽 fLim (𝑋filGen𝐹)) ∈ {∪ (𝐽 fLim (𝑋filGen𝐹))}
5		minvec.u	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑈 ∈ ℂPreHil)
6		cphngp 25500	. . . . . . . . . . . 12 ⊢ (𝑈 ∈ ℂPreHil → 𝑈 ∈ NrmGrp)
7		ngpxms 24926	. . . . . . . . . . . 12 ⊢ (𝑈 ∈ NrmGrp → 𝑈 ∈ ∞MetSp)
8	5, 6, 7	3syl 19	. . . . . . . . . . 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 24776	. . . . . . . . . . 11 ⊢ (𝑈 ∈ ∞MetSp → 𝐽 = (MetOpen‘𝐷))
13	8, 12	syl 18	. . . . . . . . . 10 ⊢ (𝜑 → 𝐽 = (MetOpen‘𝐷))
14	13	oveq1d 7436	. . . . . . . . 9 ⊢ (𝜑 → (𝐽 ↾_t 𝑌) = ((MetOpen‘𝐷) ↾_t 𝑌))
15	10, 11	xmsxmet 24781	. . . . . . . . . . 11 ⊢ (𝑈 ∈ ∞MetSp → 𝐷 ∈ (∞Met‘𝑋))
16	8, 15	syl 18	. . . . . . . . . 10 ⊢ (𝜑 → 𝐷 ∈ (∞Met‘𝑋))
17		minvec.y	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑌 ∈ (LSubSp‘𝑈))
18		eqid 2760	. . . . . . . . . . . 12 ⊢ (LSubSp‘𝑈) = (LSubSp‘𝑈)
19	10, 18	lssss 21217	. . . . . . . . . . 11 ⊢ (𝑌 ∈ (LSubSp‘𝑈) → 𝑌 ⊆ 𝑋)
20	17, 19	syl 18	. . . . . . . . . 10 ⊢ (𝜑 → 𝑌 ⊆ 𝑋)
21		eqid 2760	. . . . . . . . . . 11 ⊢ (𝐷 ↾ (𝑌 × 𝑌)) = (𝐷 ↾ (𝑌 × 𝑌))
22		eqid 2760	. . . . . . . . . . 11 ⊢ (MetOpen‘𝐷) = (MetOpen‘𝐷)
23		eqid 2760	. . . . . . . . . . 11 ⊢ (MetOpen‘(𝐷 ↾ (𝑌 × 𝑌))) = (MetOpen‘(𝐷 ↾ (𝑌 × 𝑌)))
24	21, 22, 23	metrest 24849	. . . . . . . . . 10 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑌 ⊆ 𝑋) → ((MetOpen‘𝐷) ↾_t 𝑌) = (MetOpen‘(𝐷 ↾ (𝑌 × 𝑌))))
25	16, 20, 24	syl2anc 596	. . . . . . . . 9 ⊢ (𝜑 → ((MetOpen‘𝐷) ↾_t 𝑌) = (MetOpen‘(𝐷 ↾ (𝑌 × 𝑌))))
26	14, 25	eqtr2d 2796	. . . . . . . 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 25755	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐹 ∈ (fBas‘𝑌))
35		fgcl 24203	. . . . . . . . . . 11 ⊢ (𝐹 ∈ (fBas‘𝑌) → (𝑌filGen𝐹) ∈ (Fil‘𝑌))
36	34, 35	syl 18	. . . . . . . . . 10 ⊢ (𝜑 → (𝑌filGen𝐹) ∈ (Fil‘𝑌))
37	10	fvexi 6900	. . . . . . . . . . 11 ⊢ 𝑋 ∈ V
38	37	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → 𝑋 ∈ V)
39		trfg 24216	. . . . . . . . . 10 ⊢ (((𝑌filGen𝐹) ∈ (Fil‘𝑌) ∧ 𝑌 ⊆ 𝑋 ∧ 𝑋 ∈ V) → ((𝑋filGen(𝑌filGen𝐹)) ↾_t 𝑌) = (𝑌filGen𝐹))
40	36, 20, 38, 39	syl3anc 1398	. . . . . . . . 9 ⊢ (𝜑 → ((𝑋filGen(𝑌filGen𝐹)) ↾_t 𝑌) = (𝑌filGen𝐹))
41		fgabs 24204	. . . . . . . . . . 11 ⊢ ((𝐹 ∈ (fBas‘𝑌) ∧ 𝑌 ⊆ 𝑋) → (𝑋filGen(𝑌filGen𝐹)) = (𝑋filGen𝐹))
42	34, 20, 41	syl2anc 596	. . . . . . . . . 10 ⊢ (𝜑 → (𝑋filGen(𝑌filGen𝐹)) = (𝑋filGen𝐹))
43	42	oveq1d 7436	. . . . . . . . 9 ⊢ (𝜑 → ((𝑋filGen(𝑌filGen𝐹)) ↾_t 𝑌) = ((𝑋filGen𝐹) ↾_t 𝑌))
44	40, 43	eqtr3d 2797	. . . . . . . 8 ⊢ (𝜑 → (𝑌filGen𝐹) = ((𝑋filGen𝐹) ↾_t 𝑌))
45	26, 44	oveq12d 7439	. . . . . . 7 ⊢ (𝜑 → ((MetOpen‘(𝐷 ↾ (𝑌 × 𝑌))) fLim (𝑌filGen𝐹)) = ((𝐽 ↾_t 𝑌) fLim ((𝑋filGen𝐹) ↾_t 𝑌)))
46		xmstps 24778	. . . . . . . . . 10 ⊢ (𝑈 ∈ ∞MetSp → 𝑈 ∈ TopSp)
47	8, 46	syl 18	. . . . . . . . 9 ⊢ (𝜑 → 𝑈 ∈ TopSp)
48	10, 9	istps 23258	. . . . . . . . 9 ⊢ (𝑈 ∈ TopSp ↔ 𝐽 ∈ (TopOn‘𝑋))
49	47, 48	sylib 221	. . . . . . . 8 ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))
50		fbsspw 24157	. . . . . . . . . . . 12 ⊢ (𝐹 ∈ (fBas‘𝑌) → 𝐹 ⊆ 𝒫 𝑌)
51	34, 50	syl 18	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐹 ⊆ 𝒫 𝑌)
52	20	sspwd 4569	. . . . . . . . . . 11 ⊢ (𝜑 → 𝒫 𝑌 ⊆ 𝒫 𝑋)
53	51, 52	sstrd 3918	. . . . . . . . . 10 ⊢ (𝜑 → 𝐹 ⊆ 𝒫 𝑋)
54		fbasweak 24190	. . . . . . . . . 10 ⊢ ((𝐹 ∈ (fBas‘𝑌) ∧ 𝐹 ⊆ 𝒫 𝑋 ∧ 𝑋 ∈ V) → 𝐹 ∈ (fBas‘𝑋))
55	34, 53, 38, 54	syl3anc 1398	. . . . . . . . 9 ⊢ (𝜑 → 𝐹 ∈ (fBas‘𝑋))
56		fgcl 24203	. . . . . . . . 9 ⊢ (𝐹 ∈ (fBas‘𝑋) → (𝑋filGen𝐹) ∈ (Fil‘𝑋))
57	55, 56	syl 18	. . . . . . . 8 ⊢ (𝜑 → (𝑋filGen𝐹) ∈ (Fil‘𝑋))
58		filfbas 24173	. . . . . . . . . . . . 13 ⊢ ((𝑌filGen𝐹) ∈ (Fil‘𝑌) → (𝑌filGen𝐹) ∈ (fBas‘𝑌))
59	34, 35, 58	3syl 19	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑌filGen𝐹) ∈ (fBas‘𝑌))
60		fbsspw 24157	. . . . . . . . . . . . . 14 ⊢ ((𝑌filGen𝐹) ∈ (fBas‘𝑌) → (𝑌filGen𝐹) ⊆ 𝒫 𝑌)
61	59, 60	syl 18	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑌filGen𝐹) ⊆ 𝒫 𝑌)
62	61, 52	sstrd 3918	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑌filGen𝐹) ⊆ 𝒫 𝑋)
63		fbasweak 24190	. . . . . . . . . . . 12 ⊢ (((𝑌filGen𝐹) ∈ (fBas‘𝑌) ∧ (𝑌filGen𝐹) ⊆ 𝒫 𝑋 ∧ 𝑋 ∈ V) → (𝑌filGen𝐹) ∈ (fBas‘𝑋))
64	59, 62, 38, 63	syl3anc 1398	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑌filGen𝐹) ∈ (fBas‘𝑋))
65		ssfg 24197	. . . . . . . . . . 11 ⊢ ((𝑌filGen𝐹) ∈ (fBas‘𝑋) → (𝑌filGen𝐹) ⊆ (𝑋filGen(𝑌filGen𝐹)))
66	64, 65	syl 18	. . . . . . . . . 10 ⊢ (𝜑 → (𝑌filGen𝐹) ⊆ (𝑋filGen(𝑌filGen𝐹)))
67	66, 42	sseqtrd 3944	. . . . . . . . 9 ⊢ (𝜑 → (𝑌filGen𝐹) ⊆ (𝑋filGen𝐹))
68		filtop 24180	. . . . . . . . . 10 ⊢ ((𝑌filGen𝐹) ∈ (Fil‘𝑌) → 𝑌 ∈ (𝑌filGen𝐹))
69	36, 68	syl 18	. . . . . . . . 9 ⊢ (𝜑 → 𝑌 ∈ (𝑌filGen𝐹))
70	67, 69	sseldd 3909	. . . . . . . 8 ⊢ (𝜑 → 𝑌 ∈ (𝑋filGen𝐹))
71		flimrest 24308	. . . . . . . 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 1398	. . . . . . 7 ⊢ (𝜑 → ((𝐽 ↾_t 𝑌) fLim ((𝑋filGen𝐹) ↾_t 𝑌)) = ((𝐽 fLim (𝑋filGen𝐹)) ∩ 𝑌))
73	45, 72	eqtrd 2795	. . . . . 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 25754	. . . . . . 7 ⊢ (𝜑 → (𝐷 ↾ (𝑌 × 𝑌)) ∈ (CMet‘𝑌))
75	10, 27, 28, 5, 17, 29, 30, 9, 31, 32, 11, 33	minveclem3 25756	. . . . . . 7 ⊢ (𝜑 → (𝑌filGen𝐹) ∈ (CauFil‘(𝐷 ↾ (𝑌 × 𝑌))))
76	23	cmetcvg 25612	. . . . . . 7 ⊢ (((𝐷 ↾ (𝑌 × 𝑌)) ∈ (CMet‘𝑌) ∧ (𝑌filGen𝐹) ∈ (CauFil‘(𝐷 ↾ (𝑌 × 𝑌)))) → ((MetOpen‘(𝐷 ↾ (𝑌 × 𝑌))) fLim (𝑌filGen𝐹)) ≠ ∅)
77	74, 75, 76	syl2anc 596	. . . . . 6 ⊢ (𝜑 → ((MetOpen‘(𝐷 ↾ (𝑌 × 𝑌))) fLim (𝑌filGen𝐹)) ≠ ∅)
78	73, 77	eqnetrrd 3023	. . . . 5 ⊢ (𝜑 → ((𝐽 fLim (𝑋filGen𝐹)) ∩ 𝑌) ≠ ∅)
79	78	neneqd 2960	. . . 4 ⊢ (𝜑 → ¬ ((𝐽 fLim (𝑋filGen𝐹)) ∩ 𝑌) = ∅)
80		inss1 4133	. . . . . . 7 ⊢ ((𝐽 fLim (𝑋filGen𝐹)) ∩ 𝑌) ⊆ (𝐽 fLim (𝑋filGen𝐹))
81	22	methaus 24845	. . . . . . . . . . . . 13 ⊢ (𝐷 ∈ (∞Met‘𝑋) → (MetOpen‘𝐷) ∈ Haus)
82	15, 81	syl 18	. . . . . . . . . . . 12 ⊢ (𝑈 ∈ ∞MetSp → (MetOpen‘𝐷) ∈ Haus)
83	12, 82	eqeltrd 2860	. . . . . . . . . . 11 ⊢ (𝑈 ∈ ∞MetSp → 𝐽 ∈ Haus)
84		hausflimi 24305	. . . . . . . . . . 11 ⊢ (𝐽 ∈ Haus → ∃*𝑥 𝑥 ∈ (𝐽 fLim (𝑋filGen𝐹)))
85	8, 83, 84	3syl 19	. . . . . . . . . 10 ⊢ (𝜑 → ∃*𝑥 𝑥 ∈ (𝐽 fLim (𝑋filGen𝐹)))
86		ssn0 4354	. . . . . . . . . . . 12 ⊢ ((((𝐽 fLim (𝑋filGen𝐹)) ∩ 𝑌) ⊆ (𝐽 fLim (𝑋filGen𝐹)) ∧ ((𝐽 fLim (𝑋filGen𝐹)) ∩ 𝑌) ≠ ∅) → (𝐽 fLim (𝑋filGen𝐹)) ≠ ∅)
87	80, 78, 86	sylancr 599	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐽 fLim (𝑋filGen𝐹)) ≠ ∅)
88		n0moeu 4306	. . . . . . . . . . 11 ⊢ ((𝐽 fLim (𝑋filGen𝐹)) ≠ ∅ → (∃*𝑥 𝑥 ∈ (𝐽 fLim (𝑋filGen𝐹)) ↔ ∃!𝑥 𝑥 ∈ (𝐽 fLim (𝑋filGen𝐹))))
89	87, 88	syl 18	. . . . . . . . . 10 ⊢ (𝜑 → (∃*𝑥 𝑥 ∈ (𝐽 fLim (𝑋filGen𝐹)) ↔ ∃!𝑥 𝑥 ∈ (𝐽 fLim (𝑋filGen𝐹))))
90	85, 89	mpbid 235	. . . . . . . . 9 ⊢ (𝜑 → ∃!𝑥 𝑥 ∈ (𝐽 fLim (𝑋filGen𝐹)))
91		euen1b 9059	. . . . . . . . 9 ⊢ ((𝐽 fLim (𝑋filGen𝐹)) ≈ 1_o ↔ ∃!𝑥 𝑥 ∈ (𝐽 fLim (𝑋filGen𝐹)))
92	90, 91	sylibr 237	. . . . . . . 8 ⊢ (𝜑 → (𝐽 fLim (𝑋filGen𝐹)) ≈ 1_o)
93		en1b 9056	. . . . . . . 8 ⊢ ((𝐽 fLim (𝑋filGen𝐹)) ≈ 1_o ↔ (𝐽 fLim (𝑋filGen𝐹)) = {∪ (𝐽 fLim (𝑋filGen𝐹))})
94	92, 93	sylib 221	. . . . . . 7 ⊢ (𝜑 → (𝐽 fLim (𝑋filGen𝐹)) = {∪ (𝐽 fLim (𝑋filGen𝐹))})
95	80, 94	sseqtrid 3950	. . . . . 6 ⊢ (𝜑 → ((𝐽 fLim (𝑋filGen𝐹)) ∩ 𝑌) ⊆ {∪ (𝐽 fLim (𝑋filGen𝐹))})
96		sssn 4781	. . . . . 6 ⊢ (((𝐽 fLim (𝑋filGen𝐹)) ∩ 𝑌) ⊆ {∪ (𝐽 fLim (𝑋filGen𝐹))} ↔ (((𝐽 fLim (𝑋filGen𝐹)) ∩ 𝑌) = ∅ ∨ ((𝐽 fLim (𝑋filGen𝐹)) ∩ 𝑌) = {∪ (𝐽 fLim (𝑋filGen𝐹))}))
97	95, 96	sylib 221	. . . . 5 ⊢ (𝜑 → (((𝐽 fLim (𝑋filGen𝐹)) ∩ 𝑌) = ∅ ∨ ((𝐽 fLim (𝑋filGen𝐹)) ∩ 𝑌) = {∪ (𝐽 fLim (𝑋filGen𝐹))}))
98	97	ord 878	. . . 4 ⊢ (𝜑 → (¬ ((𝐽 fLim (𝑋filGen𝐹)) ∩ 𝑌) = ∅ → ((𝐽 fLim (𝑋filGen𝐹)) ∩ 𝑌) = {∪ (𝐽 fLim (𝑋filGen𝐹))}))
99	79, 98	mpd 16	. . 3 ⊢ (𝜑 → ((𝐽 fLim (𝑋filGen𝐹)) ∩ 𝑌) = {∪ (𝐽 fLim (𝑋filGen𝐹))})
100	4, 99	eleqtrrid 2867	. 2 ⊢ (𝜑 → ∪ (𝐽 fLim (𝑋filGen𝐹)) ∈ ((𝐽 fLim (𝑋filGen𝐹)) ∩ 𝑌))
101	1, 100	eqeltrid 2864	1 ⊢ (𝜑 → 𝑃 ∈ ((𝐽 fLim (𝑋filGen𝐹)) ∩ 𝑌))

Colors of variables: wff setvar class

This proof depends on syntax axioms: ¬ wn 3 → wi 4 ↔ wb 209 ∨ wo 861 = wceq 1569 ∈ wcel 2144 ∃*wmo 2562 ∃!weu 2593 ≠ wne 2955 {crab 3412 Vcvv 3450 ⊆ wss 3895 ∩ cin 4089 ∅c0 4278 𝒫 cpw 4556 {csn 4584 ∪ cuni 4866 class class class wbr 5102 ↦ cmpt 5185 × cxp 5648 ran crn 5651 ↾ cres 5652 ‘cfv 6539 (class class class)co 7421 1_oc1o 8473 ≈ cen 8974 infcinf 9437 ℝcr 11203 + caddc 11207 < clt 11347 ≤ cle 11348 2c2 12401 ℝ⁺crp 13124 ↑cexp 14208 Basecbs 17393 ↾_s cress 17414 distcds 17443 ↾_t crest 17597 TopOpenctopn 17598 -_gcsg 19152 LSubSpclss 21212 ∞Metcxmet 21669 fBascfbas 21672 filGencfg 21673 MetOpencmopn 21674 TopOnctopon 23234 TopSpctps 23256 Hauscha 23632 Filcfil 24170 fLim cflim 24259 ∞MetSpcxms 24642 normcnm 24901 NrmGrpcngp 24902 ℂPreHilccph 25493 CauFilccfil 25579 CMetccmet 25581 CMetSpccms 25659

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1827 ax-4 1841 ax-5 1942 ax-6 1999 ax-7 2040 ax-8 2146 ax-9 2154 ax-10 2177 ax-11 2193 ax-12 2212 ax-ext 2732 ax-rep 5231 ax-sep 5248 ax-nul 5259 ax-pow 5326 ax-pr 5390 ax-un 7752 ax-cnex 11260 ax-resscn 11261 ax-1cn 11262 ax-icn 11263 ax-addcl 11264 ax-addrcl 11265 ax-mulcl 11266 ax-mulrcl 11267 ax-mulcom 11268 ax-addass 11269 ax-mulass 11270 ax-distr 11271 ax-i2m1 11272 ax-1ne0 11273 ax-1rid 11274 ax-rnegex 11275 ax-rrecex 11276 ax-cnre 11277 ax-pre-lttri 11278 ax-pre-lttrn 11279 ax-pre-ltadd 11280 ax-pre-mulgt0 11281 ax-pre-sup 11282 ax-addf 11283 ax-mulf 11284

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1572 df-fal 1582 df-ex 1812 df-nf 1816 df-sb 2099 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-nel 3062 df-ral 3077 df-rex 3087 df-rmo 3365 df-reu 3366 df-rab 3413 df-v 3452 df-sbc 3739 df-csb 3847 df-ss 3896 df-pss 4014 df-dif 4047 df-in 4091 df-un 4151 df-nul 4279 df-if 4482 df-pw 4558 df-sn 4585 df-pr 4587 df-tp 4589 df-op 4835 df-uni 4867 df-int 4907 df-iun 4952 df-br 5103 df-opab 5167 df-mpt 5186 df-tr 5212 df-id 5545 df-eprel 5550 df-po 5558 df-so 5559 df-fr 5603 df-we 5605 df-xp 5656 df-rel 5657 df-cnv 5658 df-co 5659 df-dm 5660 df-rn 5661 df-res 5662 df-ima 5663 df-pred 6304 df-ord 6365 df-on 6366 df-lim 6367 df-suc 6368 df-iota 6494 df-fun 6541 df-fn 6542 df-f 6543 df-f1 6544 df-fo 6545 df-f1o 6546 df-fv 6547 df-riota 7378 df-ov 7424 df-oprab 7425 df-mpo 7426 df-om 7879 df-1st 8002 df-2nd 8003 df-tpos 8247 df-frecs 8303 df-wrecs 8334 df-recs 8383 df-rdg 8422 df-1o 8480 df-er 8721 df-map 8853 df-en 8978 df-dom 8979 df-sdom 8980 df-fin 8981 df-fi 9407 df-sup 9438 df-inf 9439 df-pnf 11349 df-mnf 11350 df-xr 11351 df-ltxr 11352 df-le 11353 df-sub 11547 df-neg 11548 df-div 11978 df-nn 12340 df-2 12409 df-3 12410 df-4 12411 df-5 12412 df-6 12413 df-7 12414 df-8 12415 df-9 12416 df-n0 12611 df-z 12698 df-dec 12819 df-uz 12970 df-q 13080 df-rp 13125 df-xneg 13245 df-xadd 13246 df-xmul 13247 df-ico 13486 df-icc 13487 df-fz 13644 df-seq 14149 df-exp 14209 df-cj 15272 df-re 15273 df-im 15274 df-sqrt 15408 df-abs 15409 df-struct 17331 df-sets 17348 df-slot 17366 df-ndx 17378 df-base 17394 df-ress 17415 df-plusg 17447 df-mulr 17448 df-starv 17449 df-sca 17450 df-vsca 17451 df-ip 17452 df-tset 17453 df-ple 17454 df-ds 17456 df-unif 17457 df-rest 17599 df-0g 17618 df-topgen 17620 df-mgm 18822 df-sgrp 18914 df-mnd 18930 df-mhm 18984 df-grp 19153 df-minusg 19154 df-sbg 19155 df-mulg 19284 df-subg 19339 df-ghm 19434 df-cmn 20002 df-abl 20003 df-mgp 20367 df-rng 20381 df-ur 20414 df-ring 20467 df-cring 20468 df-oppr 20573 df-dvdsr 20593 df-unit 20594 df-invr 20624 df-dvr 20637 df-rhm 20708 df-subrg 20828 df-drng 20988 df-staf 21102 df-srng 21103 df-lmod 21143 df-lss 21213 df-lmhm 21303 df-lvec 21384 df-sra 21454 df-rgmod 21455 df-psmet 21676 df-xmet 21677 df-met 21678 df-bl 21679 df-mopn 21680 df-fbas 21681 df-fg 21682 df-cnfld 21685 df-phl 21938 df-top 23218 df-topon 23235 df-topsp 23257 df-bases 23270 df-ntr 23344 df-nei 23422 df-haus 23639 df-fil 24171 df-flim 24264 df-xms 24645 df-ms 24646 df-nm 24907 df-ngp 24908 df-nlm 24911 df-clm 25390 df-cph 25495 df-cfil 25582 df-cmet 25584 df-cms 25662

This theorem is used by: minveclem4b 25758 minveclem4 25759

W3C validator