Theorem cnextucn 24560

Description: Extension by continuity. Proposition 11 of [BourbakiTop1] p. II.20. Given a topology 𝐽 on 𝑋, a subset 𝐴 dense in 𝑋, this states a condition for 𝐹 from 𝐴 to a space 𝑌 Hausdorff and complete to be extensible by continuity. (Contributed by Thierry Arnoux, 4-Dec-2017.)

Hypotheses

Ref	Expression
cnextucn.x	⊢ 𝑋 = (Base‘𝑉)
cnextucn.y	⊢ 𝑌 = (Base‘𝑊)
cnextucn.j	⊢ 𝐽 = (TopOpen‘𝑉)
cnextucn.k	⊢ 𝐾 = (TopOpen‘𝑊)
cnextucn.u	⊢ 𝑈 = (UnifSt‘𝑊)
cnextucn.v	⊢ (𝜑 → 𝑉 ∈ TopSp)
cnextucn.t	⊢ (𝜑 → 𝑊 ∈ TopSp)
cnextucn.w	⊢ (𝜑 → 𝑊 ∈ CUnifSp)
cnextucn.h	⊢ (𝜑 → 𝐾 ∈ Haus)
cnextucn.a	⊢ (𝜑 → 𝐴 ⊆ 𝑋)
cnextucn.f	⊢ (𝜑 → 𝐹:𝐴⟶𝑌)
cnextucn.c	⊢ (𝜑 → ((cls‘𝐽)‘𝐴) = 𝑋)
cnextucn.l	⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ((𝑌 FilMap 𝐹)‘(((nei‘𝐽)‘{𝑥}) ↾t 𝐴)) ∈ (CauFilu‘𝑈))

Assertion

Ref	Expression
cnextucn	⊢ (𝜑 → ((𝐽CnExt𝐾)‘𝐹) ∈ (𝐽 Cn 𝐾))

Distinct variable groups:   𝑥,𝐴   𝑥,𝐹   𝑥,𝐽   𝑥,𝐾   𝜑,𝑥

Allowed substitution hints:   𝑈(𝑥)   𝑉(𝑥)   𝑊(𝑥)   𝑋(𝑥)   𝑌(𝑥)

Proof of Theorem cnextucn

Step	Hyp	Ref	Expression
1		eqid 2760	. 2 ⊢ ∪ 𝐽 = ∪ 𝐽
2		eqid 2760	. 2 ⊢ ∪ 𝐾 = ∪ 𝐾
3		cnextucn.v	. . 3 ⊢ (𝜑 → 𝑉 ∈ TopSp)
4		cnextucn.j	. . . 4 ⊢ 𝐽 = (TopOpen‘𝑉)
5	4	tpstop 23194	. . 3 ⊢ (𝑉 ∈ TopSp → 𝐽 ∈ Top)
6	3, 5	syl 18	. 2 ⊢ (𝜑 → 𝐽 ∈ Top)
7		cnextucn.h	. 2 ⊢ (𝜑 → 𝐾 ∈ Haus)
8		cnextucn.f	. . 3 ⊢ (𝜑 → 𝐹:𝐴⟶𝑌)
9		cnextucn.t	. . . . 5 ⊢ (𝜑 → 𝑊 ∈ TopSp)
10		cnextucn.y	. . . . . 6 ⊢ 𝑌 = (Base‘𝑊)
11		cnextucn.k	. . . . . 6 ⊢ 𝐾 = (TopOpen‘𝑊)
12	10, 11	tpsuni 23193	. . . . 5 ⊢ (𝑊 ∈ TopSp → 𝑌 = ∪ 𝐾)
13	9, 12	syl 18	. . . 4 ⊢ (𝜑 → 𝑌 = ∪ 𝐾)
14	13	feq3d 6690	. . 3 ⊢ (𝜑 → (𝐹:𝐴⟶𝑌 ↔ 𝐹:𝐴⟶∪ 𝐾))
15	8, 14	mpbid 235	. 2 ⊢ (𝜑 → 𝐹:𝐴⟶∪ 𝐾)
16		cnextucn.a	. . 3 ⊢ (𝜑 → 𝐴 ⊆ 𝑋)
17		cnextucn.x	. . . . 5 ⊢ 𝑋 = (Base‘𝑉)
18	17, 4	tpsuni 23193	. . . 4 ⊢ (𝑉 ∈ TopSp → 𝑋 = ∪ 𝐽)
19	3, 18	syl 18	. . 3 ⊢ (𝜑 → 𝑋 = ∪ 𝐽)
20	16, 19	sseqtrd 3967	. 2 ⊢ (𝜑 → 𝐴 ⊆ ∪ 𝐽)
21		cnextucn.c	. . 3 ⊢ (𝜑 → ((cls‘𝐽)‘𝐴) = 𝑋)
22	21, 19	eqtrd 2795	. 2 ⊢ (𝜑 → ((cls‘𝐽)‘𝐴) = ∪ 𝐽)
23	10, 11	istps 23191	. . . . . 6 ⊢ (𝑊 ∈ TopSp ↔ 𝐾 ∈ (TopOn‘𝑌))
24	9, 23	sylib 221	. . . . 5 ⊢ (𝜑 → 𝐾 ∈ (TopOn‘𝑌))
25	24	adantr 486	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ∪ 𝐽) → 𝐾 ∈ (TopOn‘𝑌))
26	19	eleq2d 2846	. . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↔ 𝑥 ∈ ∪ 𝐽))
27	26	biimpar 483	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ∪ 𝐽) → 𝑥 ∈ 𝑋)
28	21	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ∪ 𝐽) → ((cls‘𝐽)‘𝐴) = 𝑋)
29	27, 28	eleqtrrd 2863	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ∪ 𝐽) → 𝑥 ∈ ((cls‘𝐽)‘𝐴))
30		toptopon2 23175	. . . . . . . . 9 ⊢ (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘∪ 𝐽))
31	6, 30	sylib 221	. . . . . . . 8 ⊢ (𝜑 → 𝐽 ∈ (TopOn‘∪ 𝐽))
32		fveq2 6881	. . . . . . . . . 10 ⊢ (𝑋 = ∪ 𝐽 → (TopOn‘𝑋) = (TopOn‘∪ 𝐽))
33	32	eleq2d 2846	. . . . . . . . 9 ⊢ (𝑋 = ∪ 𝐽 → (𝐽 ∈ (TopOn‘𝑋) ↔ 𝐽 ∈ (TopOn‘∪ 𝐽)))
34	19, 33	syl 18	. . . . . . . 8 ⊢ (𝜑 → (𝐽 ∈ (TopOn‘𝑋) ↔ 𝐽 ∈ (TopOn‘∪ 𝐽)))
35	31, 34	mpbird 260	. . . . . . 7 ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))
36	35	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ∪ 𝐽) → 𝐽 ∈ (TopOn‘𝑋))
37	16	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ∪ 𝐽) → 𝐴 ⊆ 𝑋)
38		trnei 24150	. . . . . 6 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋 ∧ 𝑥 ∈ 𝑋) → (𝑥 ∈ ((cls‘𝐽)‘𝐴) ↔ (((nei‘𝐽)‘{𝑥}) ↾t 𝐴) ∈ (Fil‘𝐴)))
39	36, 37, 27, 38	syl3anc 1398	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ∪ 𝐽) → (𝑥 ∈ ((cls‘𝐽)‘𝐴) ↔ (((nei‘𝐽)‘{𝑥}) ↾t 𝐴) ∈ (Fil‘𝐴)))
40	29, 39	mpbid 235	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ∪ 𝐽) → (((nei‘𝐽)‘{𝑥}) ↾t 𝐴) ∈ (Fil‘𝐴))
41	8	adantr 486	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ∪ 𝐽) → 𝐹:𝐴⟶𝑌)
42		flfval 24248	. . . 4 ⊢ ((𝐾 ∈ (TopOn‘𝑌) ∧ (((nei‘𝐽)‘{𝑥}) ↾t 𝐴) ∈ (Fil‘𝐴) ∧ 𝐹:𝐴⟶𝑌) → ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾t 𝐴))‘𝐹) = (𝐾 fLim ((𝑌 FilMap 𝐹)‘(((nei‘𝐽)‘{𝑥}) ↾t 𝐴))))
43	25, 40, 41, 42	syl3anc 1398	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ ∪ 𝐽) → ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾t 𝐴))‘𝐹) = (𝐾 fLim ((𝑌 FilMap 𝐹)‘(((nei‘𝐽)‘{𝑥}) ↾t 𝐴))))
44		cnextucn.w	. . . . 5 ⊢ (𝜑 → 𝑊 ∈ CUnifSp)
45	44	adantr 486	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ∪ 𝐽) → 𝑊 ∈ CUnifSp)
46		cnextucn.l	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ((𝑌 FilMap 𝐹)‘(((nei‘𝐽)‘{𝑥}) ↾t 𝐴)) ∈ (CauFilu‘𝑈))
47	27, 46	syldan 603	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ∪ 𝐽) → ((𝑌 FilMap 𝐹)‘(((nei‘𝐽)‘{𝑥}) ↾t 𝐴)) ∈ (CauFilu‘𝑈))
48		cnextucn.u	. . . . . 6 ⊢ 𝑈 = (UnifSt‘𝑊)
49	48	fveq2i 6884	. . . . 5 ⊢ (CauFilu‘𝑈) = (CauFilu‘(UnifSt‘𝑊))
50	47, 49	eleqtrdi 2870	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ∪ 𝐽) → ((𝑌 FilMap 𝐹)‘(((nei‘𝐽)‘{𝑥}) ↾t 𝐴)) ∈ (CauFilu‘(UnifSt‘𝑊)))
51	10	fvexi 6895	. . . . 5 ⊢ 𝑌 ∈ V
52		filfbas 24106	. . . . . 6 ⊢ ((((nei‘𝐽)‘{𝑥}) ↾t 𝐴) ∈ (Fil‘𝐴) → (((nei‘𝐽)‘{𝑥}) ↾t 𝐴) ∈ (fBas‘𝐴))
53	40, 52	syl 18	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ∪ 𝐽) → (((nei‘𝐽)‘{𝑥}) ↾t 𝐴) ∈ (fBas‘𝐴))
54		fmfil 24202	. . . . 5 ⊢ ((𝑌 ∈ V ∧ (((nei‘𝐽)‘{𝑥}) ↾t 𝐴) ∈ (fBas‘𝐴) ∧ 𝐹:𝐴⟶𝑌) → ((𝑌 FilMap 𝐹)‘(((nei‘𝐽)‘{𝑥}) ↾t 𝐴)) ∈ (Fil‘𝑌))
55	51, 53, 41, 54	mp3an2i 1495	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ∪ 𝐽) → ((𝑌 FilMap 𝐹)‘(((nei‘𝐽)‘{𝑥}) ↾t 𝐴)) ∈ (Fil‘𝑌))
56	10, 11	cuspcvg 24558	. . . 4 ⊢ ((𝑊 ∈ CUnifSp ∧ ((𝑌 FilMap 𝐹)‘(((nei‘𝐽)‘{𝑥}) ↾t 𝐴)) ∈ (CauFilu‘(UnifSt‘𝑊)) ∧ ((𝑌 FilMap 𝐹)‘(((nei‘𝐽)‘{𝑥}) ↾t 𝐴)) ∈ (Fil‘𝑌)) → (𝐾 fLim ((𝑌 FilMap 𝐹)‘(((nei‘𝐽)‘{𝑥}) ↾t 𝐴))) ≠ ∅)
57	45, 50, 55, 56	syl3anc 1398	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ ∪ 𝐽) → (𝐾 fLim ((𝑌 FilMap 𝐹)‘(((nei‘𝐽)‘{𝑥}) ↾t 𝐴))) ≠ ∅)
58	43, 57	eqnetrd 3022	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ ∪ 𝐽) → ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾t 𝐴))‘𝐹) ≠ ∅)
59		cuspusp 24557	. . . 4 ⊢ (𝑊 ∈ CUnifSp → 𝑊 ∈ UnifSp)
60	44, 59	syl 18	. . 3 ⊢ (𝜑 → 𝑊 ∈ UnifSp)
61	11	uspreg 24531	. . 3 ⊢ ((𝑊 ∈ UnifSp ∧ 𝐾 ∈ Haus) → 𝐾 ∈ Reg)
62	60, 7, 61	syl2anc 596	. 2 ⊢ (𝜑 → 𝐾 ∈ Reg)
63	1, 2, 6, 7, 15, 20, 22, 58, 62	cnextcn 24325	1 ⊢ (𝜑 → ((𝐽CnExt𝐾)‘𝐹) ∈ (𝐽 Cn 𝐾))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2145   ≠ wne 2955  Vcvv 3450   ⊆ wss 3899  ∅c0 4279  {csn 4584  ∪ cuni 4867  ⟶wf 6531  ‘cfv 6535  (class class class)co 7416  Basecbs 17326   ↾_t crest 17530  TopOpenctopn 17531  fBascfbas 21605  Topctop 23150  TopOnctopon 23167  TopSpctps 23189  clsccl 23275  neicnei 23354   Cn ccn 23481  Hauscha 23565  Regcreg 23566  Filcfil 24103   FilMap cfm 24191   fLim cflim 24192   fLimf cflf 24193  CnExtccnext 24317  UnifStcuss 24511  UnifSpcusp 24512  CauFil_uccfilu 24543  CUnifSpccusp 24554

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 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-rep 5232  ax-sep 5251  ax-nul 5263  ax-pow 5330  ax-pr 5398  ax-un 7742

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 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-reu 3366  df-rab 3413  df-v 3452  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-int 4908  df-iun 4953  df-iin 4954  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-ord 6362  df-on 6363  df-lim 6364  df-suc 6365  df-iota 6491  df-fun 6537  df-fn 6538  df-f 6539  df-f1 6540  df-fo 6541  df-f1o 6542  df-fv 6543  df-ov 7419  df-oprab 7420  df-mpo 7421  df-om 7869  df-1st 7992  df-2nd 7993  df-1o 8462  df-2o 8463  df-map 8835  df-pm 8836  df-en 8960  df-fin 8963  df-fi 9388  df-rest 17532  df-topgen 17553  df-fbas 21614  df-fg 21615  df-top 23151  df-topon 23168  df-topsp 23190  df-bases 23203  df-cld 23276  df-ntr 23277  df-cls 23278  df-nei 23355  df-cn 23484  df-cnp 23485  df-haus 23572  df-reg 23573  df-tx 23820  df-fil 24104  df-fm 24196  df-flim 24197  df-flf 24198  df-cnext 24318  df-ust 24459  df-utop 24489  df-usp 24515  df-cusp 24555

This theorem is used by:  ucnextcn  24561