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Theorem minveclem4a 25626

Description: Lemma for minvec 25632. 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 7456	. . . . 5 ⊢ (𝐽 fLim (𝑋filGen𝐹)) ∈ V
3	2	uniex 7752	. . . 4 ⊢ ∪ (𝐽 fLim (𝑋filGen𝐹)) ∈ V
4	3	snid 4633	. . 3 ⊢ ∪ (𝐽 fLim (𝑋filGen𝐹)) ∈ {∪ (𝐽 fLim (𝑋filGen𝐹))}
5		minvec.u	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑈 ∈ ℂPreHil)
6		cphngp 25369	. . . . . . . . . . . 12 ⊢ (𝑈 ∈ ℂPreHil → 𝑈 ∈ NrmGrp)
7		ngpxms 24795	. . . . . . . . . . . 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 24645	. . . . . . . . . . 11 ⊢ (𝑈 ∈ ∞MetSp → 𝐽 = (MetOpen‘𝐷))
13	8, 12	syl 18	. . . . . . . . . 10 ⊢ (𝜑 → 𝐽 = (MetOpen‘𝐷))
14	13	oveq1d 7438	. . . . . . . . 9 ⊢ (𝜑 → (𝐽 ↾_t 𝑌) = ((MetOpen‘𝐷) ↾_t 𝑌))
15	10, 11	xmsxmet 24650	. . . . . . . . . . 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 2766	. . . . . . . . . . . 12 ⊢ (LSubSp‘𝑈) = (LSubSp‘𝑈)
19	10, 18	lssss 21094	. . . . . . . . . . 11 ⊢ (𝑌 ∈ (LSubSp‘𝑈) → 𝑌 ⊆ 𝑋)
20	17, 19	syl 18	. . . . . . . . . 10 ⊢ (𝜑 → 𝑌 ⊆ 𝑋)
21		eqid 2766	. . . . . . . . . . 11 ⊢ (𝐷 ↾ (𝑌 × 𝑌)) = (𝐷 ↾ (𝑌 × 𝑌))
22		eqid 2766	. . . . . . . . . . 11 ⊢ (MetOpen‘𝐷) = (MetOpen‘𝐷)
23		eqid 2766	. . . . . . . . . . 11 ⊢ (MetOpen‘(𝐷 ↾ (𝑌 × 𝑌))) = (MetOpen‘(𝐷 ↾ (𝑌 × 𝑌)))
24	21, 22, 23	metrest 24718	. . . . . . . . . 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 2802	. . . . . . . 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 25624	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐹 ∈ (fBas‘𝑌))
35		fgcl 24072	. . . . . . . . . . 11 ⊢ (𝐹 ∈ (fBas‘𝑌) → (𝑌filGen𝐹) ∈ (Fil‘𝑌))
36	34, 35	syl 18	. . . . . . . . . 10 ⊢ (𝜑 → (𝑌filGen𝐹) ∈ (Fil‘𝑌))
37	10	fvexi 6902	. . . . . . . . . . 11 ⊢ 𝑋 ∈ V
38	37	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → 𝑋 ∈ V)
39		trfg 24085	. . . . . . . . . 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 24073	. . . . . . . . . . 11 ⊢ ((𝐹 ∈ (fBas‘𝑌) ∧ 𝑌 ⊆ 𝑋) → (𝑋filGen(𝑌filGen𝐹)) = (𝑋filGen𝐹))
42	34, 20, 41	syl2anc 596	. . . . . . . . . 10 ⊢ (𝜑 → (𝑋filGen(𝑌filGen𝐹)) = (𝑋filGen𝐹))
43	42	oveq1d 7438	. . . . . . . . 9 ⊢ (𝜑 → ((𝑋filGen(𝑌filGen𝐹)) ↾_t 𝑌) = ((𝑋filGen𝐹) ↾_t 𝑌))
44	40, 43	eqtr3d 2803	. . . . . . . 8 ⊢ (𝜑 → (𝑌filGen𝐹) = ((𝑋filGen𝐹) ↾_t 𝑌))
45	26, 44	oveq12d 7441	. . . . . . 7 ⊢ (𝜑 → ((MetOpen‘(𝐷 ↾ (𝑌 × 𝑌))) fLim (𝑌filGen𝐹)) = ((𝐽 ↾_t 𝑌) fLim ((𝑋filGen𝐹) ↾_t 𝑌)))
46		xmstps 24647	. . . . . . . . . 10 ⊢ (𝑈 ∈ ∞MetSp → 𝑈 ∈ TopSp)
47	8, 46	syl 18	. . . . . . . . 9 ⊢ (𝜑 → 𝑈 ∈ TopSp)
48	10, 9	istps 23128	. . . . . . . . 9 ⊢ (𝑈 ∈ TopSp ↔ 𝐽 ∈ (TopOn‘𝑋))
49	47, 48	sylib 221	. . . . . . . 8 ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))
50		fbsspw 24026	. . . . . . . . . . . 12 ⊢ (𝐹 ∈ (fBas‘𝑌) → 𝐹 ⊆ 𝒫 𝑌)
51	34, 50	syl 18	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐹 ⊆ 𝒫 𝑌)
52	20	sspwd 4580	. . . . . . . . . . 11 ⊢ (𝜑 → 𝒫 𝑌 ⊆ 𝒫 𝑋)
53	51, 52	sstrd 3950	. . . . . . . . . 10 ⊢ (𝜑 → 𝐹 ⊆ 𝒫 𝑋)
54		fbasweak 24059	. . . . . . . . . 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 24072	. . . . . . . . 9 ⊢ (𝐹 ∈ (fBas‘𝑋) → (𝑋filGen𝐹) ∈ (Fil‘𝑋))
57	55, 56	syl 18	. . . . . . . 8 ⊢ (𝜑 → (𝑋filGen𝐹) ∈ (Fil‘𝑋))
58		filfbas 24042	. . . . . . . . . . . . 13 ⊢ ((𝑌filGen𝐹) ∈ (Fil‘𝑌) → (𝑌filGen𝐹) ∈ (fBas‘𝑌))
59	34, 35, 58	3syl 19	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑌filGen𝐹) ∈ (fBas‘𝑌))
60		fbsspw 24026	. . . . . . . . . . . . . 14 ⊢ ((𝑌filGen𝐹) ∈ (fBas‘𝑌) → (𝑌filGen𝐹) ⊆ 𝒫 𝑌)
61	59, 60	syl 18	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑌filGen𝐹) ⊆ 𝒫 𝑌)
62	61, 52	sstrd 3950	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑌filGen𝐹) ⊆ 𝒫 𝑋)
63		fbasweak 24059	. . . . . . . . . . . 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 24066	. . . . . . . . . . 11 ⊢ ((𝑌filGen𝐹) ∈ (fBas‘𝑋) → (𝑌filGen𝐹) ⊆ (𝑋filGen(𝑌filGen𝐹)))
66	64, 65	syl 18	. . . . . . . . . 10 ⊢ (𝜑 → (𝑌filGen𝐹) ⊆ (𝑋filGen(𝑌filGen𝐹)))
67	66, 42	sseqtrd 3976	. . . . . . . . 9 ⊢ (𝜑 → (𝑌filGen𝐹) ⊆ (𝑋filGen𝐹))
68		filtop 24049	. . . . . . . . . 10 ⊢ ((𝑌filGen𝐹) ∈ (Fil‘𝑌) → 𝑌 ∈ (𝑌filGen𝐹))
69	36, 68	syl 18	. . . . . . . . 9 ⊢ (𝜑 → 𝑌 ∈ (𝑌filGen𝐹))
70	67, 69	sseldd 3941	. . . . . . . 8 ⊢ (𝜑 → 𝑌 ∈ (𝑋filGen𝐹))
71		flimrest 24177	. . . . . . . 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 2801	. . . . . 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 25623	. . . . . . 7 ⊢ (𝜑 → (𝐷 ↾ (𝑌 × 𝑌)) ∈ (CMet‘𝑌))
75	10, 27, 28, 5, 17, 29, 30, 9, 31, 32, 11, 33	minveclem3 25625	. . . . . . 7 ⊢ (𝜑 → (𝑌filGen𝐹) ∈ (CauFil‘(𝐷 ↾ (𝑌 × 𝑌))))
76	23	cmetcvg 25481	. . . . . . 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 3029	. . . . 5 ⊢ (𝜑 → ((𝐽 fLim (𝑋filGen𝐹)) ∩ 𝑌) ≠ ∅)
79	78	neneqd 2966	. . . 4 ⊢ (𝜑 → ¬ ((𝐽 fLim (𝑋filGen𝐹)) ∩ 𝑌) = ∅)
80		inss1 4192	. . . . . . 7 ⊢ ((𝐽 fLim (𝑋filGen𝐹)) ∩ 𝑌) ⊆ (𝐽 fLim (𝑋filGen𝐹))
81	22	methaus 24714	. . . . . . . . . . . . 13 ⊢ (𝐷 ∈ (∞Met‘𝑋) → (MetOpen‘𝐷) ∈ Haus)
82	15, 81	syl 18	. . . . . . . . . . . 12 ⊢ (𝑈 ∈ ∞MetSp → (MetOpen‘𝐷) ∈ Haus)
83	12, 82	eqeltrd 2866	. . . . . . . . . . 11 ⊢ (𝑈 ∈ ∞MetSp → 𝐽 ∈ Haus)
84		hausflimi 24174	. . . . . . . . . . 11 ⊢ (𝐽 ∈ Haus → ∃*𝑥 𝑥 ∈ (𝐽 fLim (𝑋filGen𝐹)))
85	8, 83, 84	3syl 19	. . . . . . . . . 10 ⊢ (𝜑 → ∃*𝑥 𝑥 ∈ (𝐽 fLim (𝑋filGen𝐹)))
86		ssn0 4365	. . . . . . . . . . . 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 4317	. . . . . . . . . . 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 9034	. . . . . . . . 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 9031	. . . . . . . 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 3982	. . . . . 6 ⊢ (𝜑 → ((𝐽 fLim (𝑋filGen𝐹)) ∩ 𝑌) ⊆ {∪ (𝐽 fLim (𝑋filGen𝐹))})
96		sssn 4797	. . . . . 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 2873	. 2 ⊢ (𝜑 → ∪ (𝐽 fLim (𝑋filGen𝐹)) ∈ ((𝐽 fLim (𝑋filGen𝐹)) ∩ 𝑌))
101	1, 100	eqeltrid 2870	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 1570 ∈ wcel 2146 ∃*wmo 2568 ∃!weu 2599 ≠ wne 2961 {crab 3419 Vcvv 3458 ∩ cin 3907 ⊆ wss 3908 ∅c0 4289 𝒫 cpw 4567 {csn 4594 ∪ cuni 4877 class class class wbr 5114 ↦ cmpt 5197 × cxp 5664 ran crn 5667 ↾ cres 5668 ‘cfv 6543 (class class class)co 7423 1_oc1o 8455 ≈ cen 8949 infcinf 9411 ℝcr 11117 + caddc 11121 < clt 11261 ≤ cle 11262 2c2 12313 ℝ⁺crp 13034 ↑cexp 14117 Basecbs 17294 ↾_s cress 17315 distcds 17344 ↾_t crest 17498 TopOpenctopn 17499 -_gcsg 19033 LSubSpclss 21089 ∞Metcxmet 21544 fBascfbas 21547 filGencfg 21548 MetOpencmopn 21549 TopOnctopon 23104 TopSpctps 23126 Hauscha 23502 Filcfil 24039 fLim cflim 24128 ∞MetSpcxms 24511 normcnm 24770 NrmGrpcngp 24771 ℂPreHilccph 25362 CauFilccfil 25448 CMetccmet 25450 CMetSpccms 25528

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2148 ax-9 2156 ax-10 2179 ax-11 2195 ax-12 2216 ax-ext 2738 ax-rep 5243 ax-sep 5262 ax-nul 5274 ax-pow 5341 ax-pr 5409 ax-un 7745 ax-cnex 11174 ax-resscn 11175 ax-1cn 11176 ax-icn 11177 ax-addcl 11178 ax-addrcl 11179 ax-mulcl 11180 ax-mulrcl 11181 ax-mulcom 11182 ax-addass 11183 ax-mulass 11184 ax-distr 11185 ax-i2m1 11186 ax-1ne0 11187 ax-1rid 11188 ax-rnegex 11189 ax-rrecex 11190 ax-cnre 11191 ax-pre-lttri 11192 ax-pre-lttrn 11193 ax-pre-ltadd 11194 ax-pre-mulgt0 11195 ax-pre-sup 11196 ax-addf 11197 ax-mulf 11198

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2570 df-eu 2600 df-clab 2745 df-cleq 2758 df-clel 2841 df-nfc 2915 df-ne 2962 df-nel 3068 df-ral 3083 df-rex 3093 df-rmo 3372 df-reu 3373 df-rab 3420 df-v 3460 df-sbc 3748 df-csb 3857 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-pss 3928 df-nul 4290 df-if 4493 df-pw 4569 df-sn 4595 df-pr 4597 df-tp 4599 df-op 4601 df-uni 4878 df-int 4918 df-iun 4963 df-br 5115 df-opab 5179 df-mpt 5198 df-tr 5224 df-id 5561 df-eprel 5566 df-po 5574 df-so 5575 df-fr 5619 df-we 5621 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-res 5678 df-ima 5679 df-pred 6309 df-ord 6370 df-on 6371 df-lim 6372 df-suc 6373 df-iota 6499 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7380 df-ov 7426 df-oprab 7427 df-mpo 7428 df-om 7872 df-1st 7995 df-2nd 7996 df-tpos 8231 df-frecs 8287 df-wrecs 8318 df-recs 8367 df-rdg 8406 df-1o 8462 df-er 8703 df-map 8835 df-en 8953 df-dom 8954 df-sdom 8955 df-fin 8956 df-fi 9381 df-sup 9412 df-inf 9413 df-pnf 11263 df-mnf 11264 df-xr 11265 df-ltxr 11266 df-le 11267 df-sub 11461 df-neg 11462 df-div 11890 df-nn 12252 df-2 12321 df-3 12322 df-4 12323 df-5 12324 df-6 12325 df-7 12326 df-8 12327 df-9 12328 df-n0 12523 df-z 12610 df-dec 12730 df-uz 12881 df-q 12991 df-rp 13035 df-xneg 13155 df-xadd 13156 df-xmul 13157 df-ico 13396 df-icc 13397 df-fz 13554 df-seq 14058 df-exp 14118 df-cj 15176 df-re 15177 df-im 15178 df-sqrt 15312 df-abs 15313 df-struct 17232 df-sets 17249 df-slot 17267 df-ndx 17279 df-base 17295 df-ress 17316 df-plusg 17348 df-mulr 17349 df-starv 17350 df-sca 17351 df-vsca 17352 df-ip 17353 df-tset 17354 df-ple 17355 df-ds 17357 df-unif 17358 df-rest 17500 df-0g 17519 df-topgen 17521 df-mgm 18723 df-sgrp 18806 df-mnd 18822 df-mhm 18872 df-grp 19034 df-minusg 19035 df-sbg 19036 df-mulg 19165 df-subg 19220 df-ghm 19315 df-cmn 19883 df-abl 19884 df-mgp 20248 df-rng 20262 df-ur 20295 df-ring 20348 df-cring 20349 df-oppr 20452 df-dvdsr 20472 df-unit 20473 df-invr 20503 df-dvr 20516 df-rhm 20587 df-subrg 20706 df-drng 20866 df-staf 20979 df-srng 20980 df-lmod 21020 df-lss 21090 df-lmhm 21180 df-lvec 21261 df-sra 21331 df-rgmod 21332 df-psmet 21551 df-xmet 21552 df-met 21553 df-bl 21554 df-mopn 21555 df-fbas 21556 df-fg 21557 df-cnfld 21560 df-phl 21813 df-top 23088 df-topon 23105 df-topsp 23127 df-bases 23140 df-ntr 23214 df-nei 23292 df-haus 23509 df-fil 24040 df-flim 24133 df-xms 24514 df-ms 24515 df-nm 24776 df-ngp 24777 df-nlm 24780 df-clm 25259 df-cph 25364 df-cfil 25451 df-cmet 25453 df-cms 25531

This theorem is used by: minveclem4b 25627 minveclem4 25628

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