Theorem heiborlem9 33618

Description: Lemma for heibor 33620. Discharge the hypotheses of heiborlem8 33617 by applying caubl 23106 to get a convergent point and adding the open cover assumption. (Contributed by Jeff Madsen, 20-Jan-2014.)

Hypotheses

Ref	Expression
heibor.1	⊢ 𝐽 = (MetOpen‘𝐷)
heibor.3	⊢ 𝐾 = {𝑢 ∣ ¬ ∃𝑣 ∈ (𝒫 𝑈 ∩ Fin)𝑢 ⊆ ∪ 𝑣}
heibor.4	⊢ 𝐺 = {⟨𝑦, 𝑛⟩ ∣ (𝑛 ∈ ℕ0 ∧ 𝑦 ∈ (𝐹‘𝑛) ∧ (𝑦𝐵𝑛) ∈ 𝐾)}
heibor.5	⊢ 𝐵 = (𝑧 ∈ 𝑋, 𝑚 ∈ ℕ0 ↦ (𝑧(ball‘𝐷)(1 / (2↑𝑚))))
heibor.6	⊢ (𝜑 → 𝐷 ∈ (CMet‘𝑋))
heibor.7	⊢ (𝜑 → 𝐹:ℕ0⟶(𝒫 𝑋 ∩ Fin))
heibor.8	⊢ (𝜑 → ∀𝑛 ∈ ℕ0 𝑋 = ∪ 𝑦 ∈ (𝐹‘𝑛)(𝑦𝐵𝑛))
heibor.9	⊢ (𝜑 → ∀𝑥 ∈ 𝐺 ((𝑇‘𝑥)𝐺((2nd ‘𝑥) + 1) ∧ ((𝐵‘𝑥) ∩ ((𝑇‘𝑥)𝐵((2nd ‘𝑥) + 1))) ∈ 𝐾))
heibor.10	⊢ (𝜑 → 𝐶𝐺0)
heibor.11	⊢ 𝑆 = seq0(𝑇, (𝑚 ∈ ℕ0 ↦ if(𝑚 = 0, 𝐶, (𝑚 − 1))))
heibor.12	⊢ 𝑀 = (𝑛 ∈ ℕ ↦ ⟨(𝑆‘𝑛), (3 / (2↑𝑛))⟩)
heibor.13	⊢ (𝜑 → 𝑈 ⊆ 𝐽)
heiborlem9.14	⊢ (𝜑 → ∪ 𝑈 = 𝑋)

Assertion

Ref	Expression
heiborlem9	⊢ (𝜑 → 𝜓)

Distinct variable groups:   𝑥,𝑛,𝑦,𝑢,𝐹   𝑥,𝐺   𝜑,𝑥   𝑚,𝑛,𝑢,𝑣,𝑥,𝑦,𝑧,𝐷   𝑚,𝑀,𝑢,𝑥,𝑦,𝑧   𝑇,𝑚,𝑛,𝑥,𝑦,𝑧   𝐵,𝑛,𝑢,𝑣,𝑦   𝑚,𝐽,𝑛,𝑢,𝑣,𝑥,𝑦,𝑧   𝑈,𝑛,𝑢,𝑣,𝑥,𝑦,𝑧   𝜓,𝑦,𝑧   𝑆,𝑚,𝑛,𝑢,𝑣,𝑥,𝑦,𝑧   𝑚,𝑋,𝑛,𝑢,𝑣,𝑥,𝑦,𝑧   𝐶,𝑚,𝑛,𝑢,𝑣,𝑦   𝑛,𝐾,𝑥,𝑦,𝑧   𝑥,𝐵

Allowed substitution hints:   𝜑(𝑦,𝑧,𝑣,𝑢,𝑚,𝑛)   𝜓(𝑥,𝑣,𝑢,𝑚,𝑛)   𝐵(𝑧,𝑚)   𝐶(𝑥,𝑧)   𝑇(𝑣,𝑢)   𝑈(𝑚)   𝐹(𝑧,𝑣,𝑚)   𝐺(𝑦,𝑧,𝑣,𝑢,𝑚,𝑛)   𝐾(𝑣,𝑢,𝑚)   𝑀(𝑣,𝑛)

Proof of Theorem heiborlem9

Dummy variables 𝑡 𝑘 𝑟 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		heibor.6	. . . . . . 7 ⊢ (𝜑 → 𝐷 ∈ (CMet‘𝑋))
2		cmetmet 23084	. . . . . . 7 ⊢ (𝐷 ∈ (CMet‘𝑋) → 𝐷 ∈ (Met‘𝑋))
3		metxmet 22139	. . . . . . 7 ⊢ (𝐷 ∈ (Met‘𝑋) → 𝐷 ∈ (∞Met‘𝑋))
4	1, 2, 3	3syl 18	. . . . . 6 ⊢ (𝜑 → 𝐷 ∈ (∞Met‘𝑋))
5		heibor.1	. . . . . . 7 ⊢ 𝐽 = (MetOpen‘𝐷)
6	5	mopntopon 22244	. . . . . 6 ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝐽 ∈ (TopOn‘𝑋))
7	4, 6	syl 17	. . . . 5 ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))
8		heibor.3	. . . . . . . . 9 ⊢ 𝐾 = {𝑢 ∣ ¬ ∃𝑣 ∈ (𝒫 𝑈 ∩ Fin)𝑢 ⊆ ∪ 𝑣}
9		heibor.4	. . . . . . . . 9 ⊢ 𝐺 = {⟨𝑦, 𝑛⟩ ∣ (𝑛 ∈ ℕ0 ∧ 𝑦 ∈ (𝐹‘𝑛) ∧ (𝑦𝐵𝑛) ∈ 𝐾)}
10		heibor.5	. . . . . . . . 9 ⊢ 𝐵 = (𝑧 ∈ 𝑋, 𝑚 ∈ ℕ0 ↦ (𝑧(ball‘𝐷)(1 / (2↑𝑚))))
11		heibor.7	. . . . . . . . 9 ⊢ (𝜑 → 𝐹:ℕ0⟶(𝒫 𝑋 ∩ Fin))
12		heibor.8	. . . . . . . . 9 ⊢ (𝜑 → ∀𝑛 ∈ ℕ0 𝑋 = ∪ 𝑦 ∈ (𝐹‘𝑛)(𝑦𝐵𝑛))
13		heibor.9	. . . . . . . . 9 ⊢ (𝜑 → ∀𝑥 ∈ 𝐺 ((𝑇‘𝑥)𝐺((2nd ‘𝑥) + 1) ∧ ((𝐵‘𝑥) ∩ ((𝑇‘𝑥)𝐵((2nd ‘𝑥) + 1))) ∈ 𝐾))
14		heibor.10	. . . . . . . . 9 ⊢ (𝜑 → 𝐶𝐺0)
15		heibor.11	. . . . . . . . 9 ⊢ 𝑆 = seq0(𝑇, (𝑚 ∈ ℕ0 ↦ if(𝑚 = 0, 𝐶, (𝑚 − 1))))
16		heibor.12	. . . . . . . . 9 ⊢ 𝑀 = (𝑛 ∈ ℕ ↦ ⟨(𝑆‘𝑛), (3 / (2↑𝑛))⟩)
17	5, 8, 9, 10, 1, 11, 12, 13, 14, 15, 16	heiborlem5 33614	. . . . . . . 8 ⊢ (𝜑 → 𝑀:ℕ⟶(𝑋 × ℝ+))
18	5, 8, 9, 10, 1, 11, 12, 13, 14, 15, 16	heiborlem6 33615	. . . . . . . 8 ⊢ (𝜑 → ∀𝑘 ∈ ℕ ((ball‘𝐷)‘(𝑀‘(𝑘 + 1))) ⊆ ((ball‘𝐷)‘(𝑀‘𝑘)))
19	5, 8, 9, 10, 1, 11, 12, 13, 14, 15, 16	heiborlem7 33616	. . . . . . . . 9 ⊢ ∀𝑟 ∈ ℝ+ ∃𝑘 ∈ ℕ (2nd ‘(𝑀‘𝑘)) < 𝑟
20	19	a1i 11	. . . . . . . 8 ⊢ (𝜑 → ∀𝑟 ∈ ℝ+ ∃𝑘 ∈ ℕ (2nd ‘(𝑀‘𝑘)) < 𝑟)
21	4, 17, 18, 20	caubl 23106	. . . . . . 7 ⊢ (𝜑 → (1st ∘ 𝑀) ∈ (Cau‘𝐷))
22	5	cmetcau 23087	. . . . . . 7 ⊢ ((𝐷 ∈ (CMet‘𝑋) ∧ (1st ∘ 𝑀) ∈ (Cau‘𝐷)) → (1st ∘ 𝑀) ∈ dom (⇝𝑡‘𝐽))
23	1, 21, 22	syl2anc 693	. . . . . 6 ⊢ (𝜑 → (1st ∘ 𝑀) ∈ dom (⇝𝑡‘𝐽))
24	5	methaus 22325	. . . . . . . 8 ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝐽 ∈ Haus)
25	4, 24	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝐽 ∈ Haus)
26		lmfun 21185	. . . . . . 7 ⊢ (𝐽 ∈ Haus → Fun (⇝𝑡‘𝐽))
27		funfvbrb 6330	. . . . . . 7 ⊢ (Fun (⇝𝑡‘𝐽) → ((1st ∘ 𝑀) ∈ dom (⇝𝑡‘𝐽) ↔ (1st ∘ 𝑀)(⇝𝑡‘𝐽)((⇝𝑡‘𝐽)‘(1st ∘ 𝑀))))
28	25, 26, 27	3syl 18	. . . . . 6 ⊢ (𝜑 → ((1st ∘ 𝑀) ∈ dom (⇝𝑡‘𝐽) ↔ (1st ∘ 𝑀)(⇝𝑡‘𝐽)((⇝𝑡‘𝐽)‘(1st ∘ 𝑀))))
29	23, 28	mpbid 222	. . . . 5 ⊢ (𝜑 → (1st ∘ 𝑀)(⇝𝑡‘𝐽)((⇝𝑡‘𝐽)‘(1st ∘ 𝑀)))
30		lmcl 21101	. . . . 5 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ (1st ∘ 𝑀)(⇝𝑡‘𝐽)((⇝𝑡‘𝐽)‘(1st ∘ 𝑀))) → ((⇝𝑡‘𝐽)‘(1st ∘ 𝑀)) ∈ 𝑋)
31	7, 29, 30	syl2anc 693	. . . 4 ⊢ (𝜑 → ((⇝𝑡‘𝐽)‘(1st ∘ 𝑀)) ∈ 𝑋)
32		heiborlem9.14	. . . 4 ⊢ (𝜑 → ∪ 𝑈 = 𝑋)
33	31, 32	eleqtrrd 2704	. . 3 ⊢ (𝜑 → ((⇝𝑡‘𝐽)‘(1st ∘ 𝑀)) ∈ ∪ 𝑈)
34		eluni2 4440	. . 3 ⊢ (((⇝𝑡‘𝐽)‘(1st ∘ 𝑀)) ∈ ∪ 𝑈 ↔ ∃𝑡 ∈ 𝑈 ((⇝𝑡‘𝐽)‘(1st ∘ 𝑀)) ∈ 𝑡)
35	33, 34	sylib 208	. 2 ⊢ (𝜑 → ∃𝑡 ∈ 𝑈 ((⇝𝑡‘𝐽)‘(1st ∘ 𝑀)) ∈ 𝑡)
36	1	adantr 481	. . 3 ⊢ ((𝜑 ∧ (𝑡 ∈ 𝑈 ∧ ((⇝𝑡‘𝐽)‘(1st ∘ 𝑀)) ∈ 𝑡)) → 𝐷 ∈ (CMet‘𝑋))
37	11	adantr 481	. . 3 ⊢ ((𝜑 ∧ (𝑡 ∈ 𝑈 ∧ ((⇝𝑡‘𝐽)‘(1st ∘ 𝑀)) ∈ 𝑡)) → 𝐹:ℕ0⟶(𝒫 𝑋 ∩ Fin))
38	12	adantr 481	. . 3 ⊢ ((𝜑 ∧ (𝑡 ∈ 𝑈 ∧ ((⇝𝑡‘𝐽)‘(1st ∘ 𝑀)) ∈ 𝑡)) → ∀𝑛 ∈ ℕ0 𝑋 = ∪ 𝑦 ∈ (𝐹‘𝑛)(𝑦𝐵𝑛))
39	13	adantr 481	. . 3 ⊢ ((𝜑 ∧ (𝑡 ∈ 𝑈 ∧ ((⇝𝑡‘𝐽)‘(1st ∘ 𝑀)) ∈ 𝑡)) → ∀𝑥 ∈ 𝐺 ((𝑇‘𝑥)𝐺((2nd ‘𝑥) + 1) ∧ ((𝐵‘𝑥) ∩ ((𝑇‘𝑥)𝐵((2nd ‘𝑥) + 1))) ∈ 𝐾))
40	14	adantr 481	. . 3 ⊢ ((𝜑 ∧ (𝑡 ∈ 𝑈 ∧ ((⇝𝑡‘𝐽)‘(1st ∘ 𝑀)) ∈ 𝑡)) → 𝐶𝐺0)
41		heibor.13	. . . 4 ⊢ (𝜑 → 𝑈 ⊆ 𝐽)
42	41	adantr 481	. . 3 ⊢ ((𝜑 ∧ (𝑡 ∈ 𝑈 ∧ ((⇝𝑡‘𝐽)‘(1st ∘ 𝑀)) ∈ 𝑡)) → 𝑈 ⊆ 𝐽)
43		fvex 6201	. . 3 ⊢ ((⇝𝑡‘𝐽)‘(1st ∘ 𝑀)) ∈ V
44		simprr 796	. . 3 ⊢ ((𝜑 ∧ (𝑡 ∈ 𝑈 ∧ ((⇝𝑡‘𝐽)‘(1st ∘ 𝑀)) ∈ 𝑡)) → ((⇝𝑡‘𝐽)‘(1st ∘ 𝑀)) ∈ 𝑡)
45		simprl 794	. . 3 ⊢ ((𝜑 ∧ (𝑡 ∈ 𝑈 ∧ ((⇝𝑡‘𝐽)‘(1st ∘ 𝑀)) ∈ 𝑡)) → 𝑡 ∈ 𝑈)
46	29	adantr 481	. . 3 ⊢ ((𝜑 ∧ (𝑡 ∈ 𝑈 ∧ ((⇝𝑡‘𝐽)‘(1st ∘ 𝑀)) ∈ 𝑡)) → (1st ∘ 𝑀)(⇝𝑡‘𝐽)((⇝𝑡‘𝐽)‘(1st ∘ 𝑀)))
47	5, 8, 9, 10, 36, 37, 38, 39, 40, 15, 16, 42, 43, 44, 45, 46	heiborlem8 33617	. 2 ⊢ ((𝜑 ∧ (𝑡 ∈ 𝑈 ∧ ((⇝𝑡‘𝐽)‘(1st ∘ 𝑀)) ∈ 𝑡)) → 𝜓)
48	35, 47	rexlimddv 3035	1 ⊢ (𝜑 → 𝜓)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∧ wa 384   ∧ w3a 1037   = wceq 1483   ∈ wcel 1990  {cab 2608  ∀wral 2912  ∃wrex 2913   ∩ cin 3573   ⊆ wss 3574  ifcif 4086  𝒫 cpw 4158  ⟨cop 4183  ∪ cuni 4436  ∪ ciun 4520   class class class wbr 4653  {copab 4712   ↦ cmpt 4729  dom cdm 5114   ∘ ccom 5118  Fun wfun 5882  ⟶wf 5884  ‘cfv 5888  (class class class)co 6650   ↦ cmpt2 6652  1^st c1st 7166  2^nd c2nd 7167  Fincfn 7955  0cc0 9936  1c1 9937   + caddc 9939   < clt 10074   − cmin 10266   / cdiv 10684  ℕcn 11020  2c2 11070  3c3 11071  ℕ₀cn0 11292  ℝ⁺crp 11832  seqcseq 12801  ↑cexp 12860  ∞Metcxmt 19731  Metcme 19732  ballcbl 19733  MetOpencmopn 19736  TopOnctopon 20715  ⇝_𝑡clm 21030  Hauscha 21112  Caucca 23051  CMetcms 23052

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-iin 4523  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-er 7742  df-map 7859  df-pm 7860  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-sup 8348  df-inf 8349  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-n0 11293  df-z 11378  df-uz 11688  df-q 11789  df-rp 11833  df-xneg 11946  df-xadd 11947  df-xmul 11948  df-ico 12181  df-icc 12182  df-fl 12593  df-seq 12802  df-exp 12861  df-rest 16083  df-topgen 16104  df-psmet 19738  df-xmet 19739  df-met 19740  df-bl 19741  df-mopn 19742  df-fbas 19743  df-fg 19744  df-top 20699  df-topon 20716  df-bases 20750  df-cld 20823  df-ntr 20824  df-cls 20825  df-nei 20902  df-lm 21033  df-haus 21119  df-fil 21650  df-fm 21742  df-flim 21743  df-flf 21744  df-cfil 23053  df-cau 23054  df-cmet 23055

This theorem is referenced by:  heiborlem10  33619